TIME-DEPENDANT BEHAVIOUR OF REINFORCED HIGH- Arangjelovski.pdf · the influence of the variable loads to time-dependant behaviour of the reinforced concrete structures. 24 reinforced

University "St.Cyril and Methodius Faculty of Civil Engineering-Skopje

TIME-DEPENDANT BEHAVIOUR OF REINFORCED HIGH-STRENGTH CONCRETE ELEMENTS UNDER ACTION OF

VARIABLE LOADS

Doctoral thesis

TONI ARANGJELOVSKI

September, 2011

Content v

Content:

Summary .............................................................................................................................. ix

List of figures ........................................................................................................................ xi

List of tables ....................................................................................................................... xvii

1. Introduction

1.1. Background and Significance ......................................................................................... 1

1.2. Objective and Scope ...................................................................................................... 4

1.3. Outline of Thesis ............................................................................................................ 9

2. Deformation properties of the concrete

2.1. General ......................................................................................................................... 11

2.2. Short-term behavior of concrete ................................................................................... 13

2.3. Long-term behavior of concrete ................................................................................... 15

2.4. Time-dependant deformations of concrete

2.4.1. Shrinkage of concrete ............................................................................................... 17

2.4.2. Autogenous shrinkage ............................................................................................... 17

2.4.3. Drying shrinkage ....................................................................................................... 19

2.4.4. Creep ......................................................................................................................... 21

3. Review of some experimental researches concerning creep of concrete

3.1. Introduction ................................................................................................................... 27

3.2. Influence of the long-term and repeated loads on the strains

3.2.1. Reinforcing strains and concrete strains ................................................................... 27

3.2.2. Average strain in the reinforcement, tension stiffening ............................................. 30

3.3. Crack width ................................................................................................................... 30

3.3.1. Long-term crack width ............................................................................................... 31

3.4. Deflection

3.4.1. Long-term deflections ................................................................................................ 31

3.5. Conclusion .................................................................................................................... 35

4. Stress-strain relations for concrete and reinforcement

4.1 Stress-strain relation for concrete at the short-term compressive load ......................... 37

4.2 Stress-strains relation for structural analysis according to European standard EN1992-1-1 EUROCODE 2 .......................................................................................... 38

4.3. Stress-strain relation at short-term tension load ........................................................... 41

4.4 Time-dependant stress-strains relations of concrete

4.4.1. Compliance function .................................................................................................. 42

Content vi

4.4.2 Principle of superposition ........................................................................................... 45

4.4.3 Actual models for designing the shrinkage and creeping of the concrete .................. 48

4.4.4 Model B3 .................................................................................................................... 48

4.4.4.1 Calculation of creep and time dependent strain components ................................. 49

4.4.4.2 Average shrinkage and creep of cross sections at drying ....................................... 50

4.4.4.3 Prediction of model parameters .............................................................................. 51

4.5 Stress-strain relation for reinforcement ......................................................................... 53

4.5.1 Relationships between stresses and strains in the reinforcement according to EN 1992-1-1 EUROCODE-2 ...................................................................................... 53

5 Methods for structural analysis of the creep

5.1 Effective modulus method (EMM) ................................................................................. 55

5.2 Age adjusted effective modulus method (AAEMM) ....................................................... 55

5.2.1. Practical calculation procedure ................................................................................. 58

5.2.2. Effects of variable loads ............................................................................................ 61

5.2.2.1. Iterative method ...................................................................................................... 61

5.2.2.2. Superposition of fictitious load effects .................................................................... 62

6. Serviceability limit state design

6.1. Limitation of stresses .................................................................................................... 65

6.1.1 Limitation of compressive stresses ............................................................................ 65

6.1.2. Limitation of tensile stresses ..................................................................................... 66

6.1.3. Procedure for calculation of the stresses .................................................................. 66

6.2. Control of cracking

6.2.1. Limitation of the crack width ...................................................................................... 67

6.2.2. Cracking caused by loading ...................................................................................... 67

6.2.3. Calculation of the crack width .................................................................................... 68

6.2.4. Minimum area of reinforcement ................................................................................. 70

6.3. Control of deflections .................................................................................................... 71

6.3.1. Limitation of the deflections ....................................................................................... 71

6.3.2. Deflection under the effect of short-term loading ...................................................... 72

6.3.3. Deflections under the effect of long-term loading ...................................................... 74

7. Experimental research of the variable loads effect on time-dependant behavior of concrete elements

7.1. Purpose of the experimental testing .............................................................................. 77

7.2. Description of the experimental program ..................................................................... 77

7.3. Materials and manufacturing method of reinforced concrete beams and control specimens ................................................................................................... 83

Content vii

7.4. Measurement technique ............................................................................................... 85

7.5. Loading method and testing of the elements ............................................................... 86

7.6. Experimentally determined mechanical and deformation properties of the concrete and the reinforcement .......................................................................... 91

7.7. Analysis of the results from experimental research .................................................... 105

7.7.1. Time-dependant deflections .................................................................................... 105

Deflections of the long-term effect of the load at series “B” beams .......................... 105

Deflections of long-term effect of the load at series “C” beams ................................ 106

Deflections of long-term effect of the load at series “D” beams ................................ 107

Deflections of the long-term effect of the load at series “E” beams .......................... 111

Comparison of the deflections of the long-term load effect for series of beams “B” and “C” ................................................................................ 115

Comparison of the deflections of the long-term effect of the permanent load and the effect of the variable load for the series of beams “C”, “D” and “E” ............. 117

7.7.2 Time-dependant cracks ............................................................................................ 121

7.7.3 Time-dependant strains in the concrete ................................................................... 128

Strains of the long-term effect of the load for series “B” beams ................................ 128

Strains of the long-term effect of the load at series “C” beams ................................ 132

Strains of the long-term effect of the load at series “D” beams ................................ 136

Strains of the long-term effect of the load at series “E” beams ................................ 142

Comparison of the measured strains at testing of long-term effect of the load for all series of beams ............................................................................. 148

8. Analytical solution of variable load effect to time-dependant behavior of concrete

8.1 Procedure with quasi-permanent load .......................................................................... 151

8.2 Analytical analysis ........................................................................................................ 153

9. Conclusions .................................................................................................................. 175

10. Literature ..................................................................................................................... 177

Content viii

Summary ix

Summary The creep and shrinkage of the concrete are long-term processes that have great influence to the behavior of the reinforced concrete structures and they cause permanent change of the state of stresses and deformations during time.

The usual influence of the creep of concrete is related to the action of the permanent loads on one structure, but influence of the variable loads effect to the development of the deformations is underestimated. However, in certain structures where we often have repeated variable loads, in the analysis are necessary to take into consideration the influence of the variable loads.

The design of the state of stresses and strains in the concrete, the crack width and deformations under the effect of different types of loads is very difficult due to the influence of interactive factors such as: tensile strength, shrinkage, creep, relaxation, decrease of the tensile stiffness at bending, changes in the value of the modulus of elasticity etc. These systems of factors are additionally complicated with introduction of the history of loading. All of this facts, point out that we are talking about non-linear behaviour of the concrete. Simple superposition cannot be applied, which is usually used in the algebraic methods for the design (AAEMM). The use of non-linear methods is unpractical at design of usual problems in the engineering practice, and therefore is better to accept simple assumptions in order to obtain linear dependencies whose accuracy can be checked with experimental tests.

Today for control of deformations from shrinkage and creep of concrete models are used that beside the effect of the permanent loads, also the effect of the variable loads is taken into consideration. Frequently in the analysis the variable loads are considered as quasi-permanent loads, i.e. it is assumed that one part of the variable load acts as permanent load defined by the coefficient of participation "2". This model for serviceability limit state design is presented in the European regulations EUROCODE-2 (EN 1992-1-1:2004).

In the framework of this research experimental program was carried out in order to determine the influence of the variable loads to time-dependant behaviour of the reinforced concrete structures. 24 reinforced concrete beams with dimensions 15/28/300cm have been made. In the tests different types of concrete and histories of loading were used. 6 series of beams were proposed. Beside the study of real loading history by variable loads, influence of the use of high-strength concrete class C70/85 was studied too. For comparison in the analysis concrete with normal strength, class CC-30 i.e. strength class of concrete C30/37 has been used. Each series of beams is composed of 2 beams of ordinary concrete and 2 beams of high-strength concrete. For studying of the influence of variable load two series of tests of beam have been defined on which permanent load “G” and repeated variable load “Q” is acting in cycles of loading/unloading for 24 hours, and for 48 hours appropriately. The beams were tested subjected to short-term load until failure at concrete age of t=40 days, and after the loading period, in which certain loading history was applied, beams were tested until failure at age of 400 days.

Analytical analysis was performed using B3 model and Age adjusted elasticity modulus method to verify experimental results and to propose values for the quasi-permanent coefficient 2. In this way, variable load effect will be included in the serviceability limit state analysis of time-dependant behaviour of reinforced concrete elements as quasi-permanent load instead of use of real loading history. Keywords: high-strength concrete, shrinkage, creep, dead load, variable load, quasi-

permanent load

Summary x

List of figures xi

List of figures:

Figure 1.1 Schematic Diagram of the effects from the dead and the life loads to the

reinforced concrete elements

Figure 1.2 Loading history on crane girder: a) Intensity of load F, b) Position of the crane x, c) Lateral forces H and d) Brake force B

Figure 1.3 Traffic loads on highway A64-France expressed through the maximum daily gross weight of vehicles

Figure 1.4 Different types of cyclic loads

Figure 2.1 Deformations in concrete: a) Shrinkage strain, b) Mechanical strain caused by the stress (elastic strain plus creep strain) and c) recovery strain after unloading

Figure 2.2 Shrinkage, creep at loading and creep at unloading

Figure 2.3 Stress – strains relation for the concrete

Figure 2.4 Concrete creep under long-term load

Figure 2.5 Shrinkage of normal and high strength concrete

Figure 2.6 Autogenous shrinkage of various ordinary concrete (W/C>0.45), Le Roy 1994

Figure 2.7 Autogenous shrinkage of various high-performance concrete (without silica fume), Le Roy 1994

Figure 2.8 Drying shrinkage during time

Figure 2.9 Diagram drying shrinkage during time and humidity

Figure 2.10 Components of total deformation

Figure 2.11 Creep-shrinkage coupling

Figure 2.12 Deformation of concrete creep: stabilization and non-stabilization process

Figure 2.13 Deformation of concrete creep during time under different stress history

Figure 3.1 Increase of the maximum cracks and middle span deflections under repeated loads – Lovegrove and Din (1982)

Figure 3.2 Modifications of strains under long-term and repeated loads

Figure 3.3 Increase in crack width under long term or repeated load; Rehm and Eligehausen (1977)

Figure 3.4 Increase in average crack width Lutz, Sharma and Gergely (1967)

Figure 3.5 Measured middle span deflections-time in singly and doubly reinforced beams; Lutz, Sharma and Gergely (1967)

Figure 3.6 Relation between long-term deflections (at or ∆at) and initial deflections (a0) evaluated from test results by Numbergerova and Hajek 1994

Figures 4.1 Typical stress-strain diagrams at uniaxial compressive test for different types of concrete compared to BS8110

Figure 4.2 Typical stresses-strains diagrams at uniaxial compressive testing according to various researchers

Figure 4.3 Stress-strain relation for structural analysis

List of figures xii

(Use of 0.4fcm for the definition of Ecm is approximate)

Figure 4.4 Parabola-rectangle diagram for concrete under compression

Figure 4.5 Procedure to derive an σ-w relation obtained from a deformation controlled uniaxial tension test

Figure 4.6 Creep isochrones and compliance curves for various ages t’ at loading

Figure 4.7 Decomposition of history of stresses on steps of stresses (left) and impulses of stresses (right)

Figure 4.8 Defining of time intervals for step-by-step analysis

Figure 4.9 Stress-strains diagrams for typical reinforcing steel: a) hot rolled steel; b) cold rolled steel

Figure 4.10 Idealized and designed stress-strain relation for reinforcement (At tension and at compression)

Figure 5.1 Strain histories for which AAEM is exact

Figure 5.2 Development of the strains and stresses in the cross-section for time t0-t1

Figure 5.3 Effects of variable actions using the method of the superposition of fictitious load effects

Figure 6.1 (a) Load-deformation diagram of a element subjected to permanent increasing load; (b) crack width-deformation in a load controlled test

Figure 6.2 Idealized load-deformation diagram of a reinforced concrete element

Figure 6.3 Model of behavior of reinforced concrete beam in phase 2

Figure 6.4 Alternative visualization of the model behavior of reinforcement in phase 2

Figure 7.1 Reinforced concrete beams for testing

Figure 7.2 Laboratory

Figure 7.3 Mean humidity and temperature in the laboratory

Figure 7.4 Placing of the strain gauges

Figure 7.5 Formwork and concreting of the reinforced concrete beams

Figure 7.6 Reinforced concrete beams curing

Figure 7.7 Gravity lever

Figure 7.8 Position of the measuring points on the reinforced concrete beam

Figure 7.9 Picture of the position of the measuring points of the reinforced concrete beam

Figure 7.10 Testing of the compressive strength

Figure 7.11 Testing of the flexure tensile strength

Figure 7.12 Testing of splitting tensile strength

Figure 7.13 Testing of the modulus of elasticity

Figure.14a) Testing of the autogenous shrinkage and drying shriknkage

Figure 14b) Testing of the creep of concrete

Figure 7.15 Preparation of the moulds for testing of the autogenous shrinkage

Figure 7.16 Diagram σc-εc for concrete class C30/37 at t=40 days and according EC-2

List of figures xiii

Figure 7.17 Diagram σc-εc for concrete CC-30, class C30/37 at t=400 days and according EC-2

Figure 7.18 Diagram σc-εc for concrete class C60/75 at t=40 days and according to EC-2

Figure 7.19 Diagram σc-εc for concrete class C6-/75 at t=400 days and according to EC-2

Figure 7.20 Diagram σc-εc for concrete CC-30, class C30/37 at t=40 and t=400 days and for concrete CC-70, class C60/75 at t=40 and t=400 days

Figure 7.21 Diagram εas-t of the autogenous shrinkage of the concrete class C30/37

Figure 7.22 Diagram εds-t of the drying shrinkage of concrete class C30/37

Figure 7.23 Diagram εcc-t from the creep of the concrete class C30/27

Figure 7.24 Diagram εas-t, εds-t and εcc-t of the autogenous shrinkage, drying shrinkage and creep of the concrete class C30/37

Figure 7.25 Diagram εas-t of the autogenous shrinkage of concrete class C60/75

Figure 7.26 Diagram εds-t from the drying shrinkage for concrete class C60/75

Figure 7.27 Diagram εcc-t from the creep of the concrete class C60/75

Figure 7.28 Diagram εas-t, εds-t and εcc-t of the autogenous shrinkage, drying shrinkage and creep of the concrete class C60/75

Figure 7.29 Diagram εas-t from the autogenous shrinkage of concrete C30/37 and C60/75

Figure 7.30 Diagram εds-t from the drying shrinkage of concrete C30/37 and C60/75

Figure 7.31 Diagram εcc-t from creep of the concrete C30/37 and C60/75

Figure 7.32 Diagram force-strain for reinforcement 10 and 12

Figure 7.33 Development of the deflections during the time for series “B” beams

Figure 7.34 Development of the deflections during the time for series “C” beams

Figure 7.35 Development of the deflections during the time for series of beams “D1”


Figure 7.37 Development of the deflections for beam “D1” in 20 days after the loading

Figure 7.38 Development of the deflections for beam “D3” in 20 days after the loading

Figure 7.39 Development of the deflections during the time for series of beams “D” presented with mean value of the beams D1 and D2 and for beams D3 and D4

Figure 7.40 Development of the deflections during the time for series of beams “E1”


Figure 7.42 Development of the deflections for beam “E1” in 20 days after the loading


Figure 7.44 Development of the deflections during the time for series of beams “E” presented with mean vale of the beams E1 and E2 and for the beams E3 and E4

Figure 7.45 Development of the deflections during the time for series of beams “B” and “C” made of ordinary concrete

List of figures xiv

Figure 7.46 Development of the deflections during the time for series of beams “B” and “C” made of high-strength concrete

Figure 7.47 Development of the deflection during the time for series of beams “C”, “D” and “E” made of ordinary concrete

Figure 7.48 Development of the deflections during the time for series of beams “C”, “D” and “E” made of high-strength concrete

Figure 7.49 Diagram of development of the crack width w – time t for beams C1 and C2

Figure 7.50 Diagram of development of the crack width w – time t for beam D1




Figure 7.54 Diagram of development of the crack width w – time t for beam E1




Figure 7.58 Measured strains during the time of the effect of the dead load FG for beam B1 at concrete age t=40-400 days




Figure 7.62 Measured strains during the time from the effect of the dead load FG+Q/2 for beam C1 at concrete age t=40-400 days




Figure 7.66 Measured strains during the time from the effect of the dead load FG and variable load FQ for beam D1 at concrete age t=40-400 days




Figure 7.70 Measured strains during the time from the effect of the dead load FG and variable load FQ for beam E1 at concrete age t=40-400 days


List of figures xv



Figure 8.1 Considered variation possibilities

Figure 8.2 Diagram as-t autogenous shrinkage, ds-t drying shrinkage and cc-t creep of concrete class C30/37

Figure 8.3 Diagram dc and cc-t creep of concrete class C30/37: experimental results, B3-model and B3-model improved

Figure 8.4 Diagram cc-t creep of concrete class C30/37: experimental results, B3-model and B3-model improved

Figure 8.5 Diagram dc and cc-t creep of concrete class C30/37: experimental results, B3-model improved and B3-model improved including real humidity

Figure 8.6 Diagram cc-t creep of concrete class C30/37: experimental results, B3-model improved and B3-model improved including real humidity

Figure 8.7 Diagram ds-t drying shrinkage of concrete class C30/37: experimental results, B3-model and B3-model improved

Figure 8.8 Diagram ds-t drying shrinkage of concrete class C30/37: experimental results, B3-model and B3-model improved including real humidity

Figure 8.9 Diagram Humidity vs. drying shrinkage of concrete class C30/37: experimental results

Figure 8.10 Diagram as-t autogenous shrinkage of concrete class C30/37: experimental results, B3-model and B3-model improved









Figure 8.19 Diagram as-t autogenous shrinkage of concrete class C60/75: experimental results, B3-model and B3-model improved including real humidity

Figure 8.20 Diagram development of the deflection during time for series "B" beams concrete class C30/37

List of figures xvi

Figure 8.21 Diagram normalized deflection a/amax vs. time for series "B" beams concrete class C30/37

Figure 8.22 Diagram development of the deflection during time for series "C" beams concrete class C30/37

Figure 8.23 Diagram normalized deflection a/amax vs. time for series "C" beams concrete class C30/37

Figure 8.24 Diagram normalized deflection a/amax vs. time for series "B" and "C" beams concrete class C30/37

Figure 8.25 Diagram development of the deflection during time for series "D" beams concrete class C30/37

Figure 8.26 Diagram normalized deflection a/amax vs. time for series "D" beams concrete class C30/37

Figure 8.27 Diagram development of the deflection during time for series "E" beams concrete class C30/37

Figure 8.28 Diagram normalized deflection a/amax vs. time for series "E" beams concrete class C30/37

Figure 8.29 Diagram normalized deflection a/amax vs. time for series "D" and "E" beams concrete class C30/37

Figure 8.30 Diagram development of the deflection during time for series "B" beams concrete class C60/75


Figure 8.32 Diagram development of the deflection during time for series "C" beams concrete class C60/75


Figure 8.34 Diagram development of the deflection during time for series "D" beams concrete class C60/75 under permanent load and quasi permanent load 2=0.5

Figure 8.35 Diagram development of the deflection during time for series "D" beams concrete class C60/75 under permanent load and quasi permanent load 2=1.0

Figure 8.36 Diagram normalized deflection a/amax vs. time for series "B" beams concrete classC30/37 and C60/75

Figure 8.37 Diagram normalized deflection a/amax vs. time for series "C" beams concrete class C30/37 and C60/75

List of tables xvii

List of tables:

Table 2.1 Materials, composition and strain of the autogenous shrinkage of the high strength, the self-compacting concrete and massive concrete

Table 2.2 Experimental characteristics of kinetics of the concrete creep

Table 4.1 Strength and deformation characteristics for concrete

Table 7.1 Experimental program

Table 7.2 History of loading for the elements of series A

Table 7.3 History of loading for the elements of series F

Table 7.4 History of loading for the elements of series B

Table 7.5 History of loading for the elements of series C

Table 7.6 History of loading for the elements of series D

Table 7.7 History of loading for the elements of series E

Table 7.8 Scheme of the control specimens taken for every type of concrete

Table 7.9 Concrete mix design

Table 7.10 Results from testing properties of concrete, concrete class C30/37

Table 7.11 Results from testing properties of concrete, concrete class C60/75

Table 7.12 Comparison of results for the ordinary concrete class C30/37


Table 7.14 Mechanical characteristics of the reinforcement

Table 7.15 Measured values of the deflection at t0=40 days and during the time t=400 days




Table 7.19 Comparison of the measured deflections for the beams of series “C”, “D” and “E” made of ordinary concrete

Table 7.20 Comparison of the measured deflections for beams of series “C”, “D” and “E” made of high-strength concrete

Table 7.21 Development of the crack width during the time

Table 7.22 Measured strains in the section of the series “B” beams at monitoring of the strains during the time under the effect of long-term dead load FG=(2x4)kN

Table 7.23 Measured strains in the section of the series “B” beams at testing of short-term loads to failure

Table 7.24 Measured strains in the section of the series “C” beams at monitoring of the strains during the time under the effect of the long-term dead load FG+Q/2=(2x7.8)kN

List of tables xviii

Table 7.25 Measured strains in the section of the series “C” beams at testing of short-term loads to failure

Table 7.26 Measured strains in the section of the series “D” beams at monitoring of the strains during the time under the effect of the dead long-term load FG±FQ/2=(2x4)kN±(2x7.6)kN

Table 7.27 Measured strains in the section of the series “D” beams at testing of short-term loads to failure

Table 7.28 Measured strains in the section of the series “E” beams at monitoring of the strains during the time under the effect of the dead long-term load FG±FQ/2=(2x4)kN±(2x7.6)kN

Table 7.29 Measured strains in the section of the series “E” beams at testing of short-term loads to failure

Table 7.30 Maximum measured strains in the cross-section of the beams made of ordinary concrete class C30/37 at testing to failure

Table 7.31 Maximum measured strains in the cross-section of the beams made of high strength concrete class C60/75 at testing to failure

Table 7.32 Change of the strains in the cross-section at beams made of ordinary concrete CC-30, class C30/37 during the time


Table 8.1

Analytically determined coefficients2 for reinforced concrete beams made of ordinary concrete for the series “D” and “E”

.

1.Introduction 1

1. Introduction 1.1. Background and Significance

Today reinforced concrete is the basic construction material, used for building of different types of concrete constructions. It is artificially composite material which is consists of hardened concrete and reinforcement embedded into the concrete. These two materials have different mechanical properties that compared to one another are opposite from each other. Concrete is a material ideal for withstanding compressive forces and reinforcing steel has equal carrying capacity in tension and in compression, which enables to compensate low tensile strength of concrete. The composite interaction between these two materials through their bond, the adhesion, is a reason for the effective transfer of load between the reinforcement and the concrete, and a precondition for the existence of the reinforced concrete.

Internal tensile forces, in reinforced concrete structural element cross-section, are effectively carried out by the reinforcement, when the tensile strength of the concrete is exceeded i.e. when cracks appears in tensile area of the concrete. The reinforcement could not prevent the crack appearance, but it helps to distribute the cracks alongside the elements because of actual bond behavior between reinforcing steel and concrete. Under normal in-service conditions crack appearance is inevitable in many reinforced concrete constructions. In most structures cracks appears due to application of variable loads and due to restrained deformations caused by shrinkage of concrete. Occurring of cracks is cause for the nonlinear behavior of the concrete. Beside the exceeded concrete tensile strength, the process of the cracks appearance and their growth is additionally complicated because of the shrinkage, decreasing of the bond between the concrete and the reinforcement and because of creep, which appears under influence of sustained load.

Cracks reduce stiffness of the reinforcing concrete elements in the tensile area and larger deflections occur. Permanent loads and cyclic repeated loads due to effect of creep additionally increase deflections.

Action of permanent and variable loads causes significant increase in concrete and reinforcing steel strains, increase in cracks width, increase in deflections, reduction of the tension stiffening and increase in bond-slip between the reinforcement and the concrete. These effects have lead to permanent changes of stresses in the reinforcement and in the concrete; increase of the curvature in the cross section and failure at cyclic load–fatigue.

All mentioned changes: increased strains, greater crack width and increased deflections as a result of action of different types of loads, endanger the ultimate limit state and serviceability limit state of the reinforced concrete structures (figure 1.1) [1].

Behavior of the reinforced concrete structural elements, subjected to service load, depends from character and duration of the load. It’s known that under short-term load and at relatively low level of stresses up to 0.4fcm the bearing capacity of the concrete and the steel is in proportion with their modulus of elasticity. But at long-term load reinforced concrete behavior two materials exists: concrete that acts like highly elastic-plastic material that has property of aging and steel that acts completely elastic. This difference in the acting of these two materials causes permanent distribution of the stresses in the cross-sections, redistribution of the static forces at statically indeterminate structure; increase in deflections and in opening of crack width, reduction of the critical forces at the thin elements etc [2].

For structure to remain in the serviceability limit state, crack width must be small enough to prevent required reliability, small enough to avoid waterproofing problems and small enough to prevent ingress of water which leads to corrosion of the reinforcement. Crack width and deflections should be in allowable limits to be acceptable from an aesthetic point of view.

1.Introduction 2

Figure 1.1 Schematic Diagram of the effects from the dead and the life loads to the reinforced concrete elements [1]

Shrinkage and creep of concrete have a very important role, sometimes crucial, for the design of serviceability limit state in many concrete structures, time-dependant behavior after the construction. Incorrect or inaccurate prediction of shrinkage and creep may have undesirable consequences for all involved parties including designers, constructors, owners, users and insurance companies [3]. In some extreme cases such as thin shells, underestimation of creep effects can cause catastrophic failure. In segmental construction bridges inaccurate creep analysis may cause excessive deflections, difficulties with closure or permanent deflections [3]. In prestressed elements the prestress force may be lower and cracks may appear in critical sections.

1.Introduction 3

In these and many others cases, using non-appropriate models or methods, or not devoting enough resources to an accurate study of creep and shrinkage effects may be cause of degradation, expensive and long repair process.

These important long-term effects are not simple phenomenon to understand, and even less simple to model and predict [3]. The obvious factors which increase shrinkage at given time after concreting are high water-cement ratio in the mixture, less quantity of aggregate, was the concrete cured and time duration of the curing, low relative humidity of the environment, small element size, high temperatures and low percentage of reinforcing. The factors which affects creep are: the quantity of the cement, mechanical characteristics of the aggregate (modulus of elasticity, porosity), relative humidity and temperature of the environment, dimensions and the form of the cross-sections, value of stresses, type of load and age of concrete at first loading.

But concrete mix proportioning, during the design of structure, is not known in advance and we don’t have information about the type and the chemical composition of the cement, granulometry of the aggregate, addition of mineral admixtures and chemical admixtures. However concrete in real structure is subjected to other effects like oscillation of environmental conditions and the aggressiveness of the environment. To make things more complicated, the laboratory experiments show that even under best controlled fabrication and constant hygrometer conditions, creep and shrinkage effects still have a degree of uncertainty which is much more higher than other properties, like for example the uniaxiall compressive strength of concrete [3].

Just like other engineering problems, one first step for understanding the phenomenon and the modeling of the creep and shrinkage is to identify the dominant physical phenomena that are involved, and then separate the material properties from the structural effects [3]. Most models for predicting behavior of concrete are empirical, and in order to obtain a model for practical use it’s required to compare designed values with experimental results. Thus we will get a model that can change because of the numerous results from the experimental tests and can be applicable to different types of concrete like: high strength concretes, high performance concrete, self-compacting concrete and fiber reinforced concrete. Therefore today models, which enable prediction of shrinkage and creep of different types of concrete, are taking the advantage.

Today for prediction of creep and shrinkage strains, in ordinary and high strength concrete, models where total shrinkage is a sum of autogenous shrinkage and drying shrinkage were used. For the creep of the concrete creep linearity model was used as in case of European structural codes EUROCODE-2 (EN1992-1-1:2004).

At the last 50 years different methods for analysis of creep were used that can be generally divided in two groups [4]:

Direct methods

Iterative or step by step methods.

The direct methods allow calculation of creep effects in one step of loading. Direct methods are: Method of effective modulus, Age-adjusting effective modulus etc.

In iterative methods period of time is divided into certain number of steps and special calculations are made for every step. Methods for the iterative solution of creep are: Method of superposition, rate-of-creep method, rate-of-flow method etc.

The calculation of stresses, strains, crack width and deflections under different types of loads is very difficult because of interaction effects like: tensile strength of concrete, shrinkage, creep, relaxation, tension softening and changes in the value of the modulus of elasticity. This system of effects is additionally complicated with introduction of load history. All this indicates that for nonlinear concrete behavior can’t be used simple superposition of the effects. On the other side, use of nonlinear methods is unpractical for solution of usual

1.Introduction 4

problems in engineering practice. It is better to accept simple presumptions in order to obtain linear relations whose accuracy can be proved by experimental test. The history of load in the design is a sum of states that consists actions of permanent and variable loads which increases initial variation of the stresses. In order to simplify the problem, the total value of the stresses is a sum of permanent load state and one state in which we have variable load.

Design codes are delicate compromise between the state of existing knowledge for the problem, which is continuously changing due to new experimental results and providing practically applicable methods that can be used by practicing engineer [3].

Structural engineer have obligation to consider, design and prove all limit states that are important for the structure. That should be done before the construction of the structure during design, then during the construction and in definite intervals during the expected service lifetime. We should have in mind that the ultimate limit state and serviceability limit state are equally important for reliability of structure and that they have different requirements concerning the behavior of structure. The requirements shouldn’t affect on the fact that in practice frequently only ultimate limit state is important for the design. It’s all on structural engineer experience; knowledge and responsibility how far to go deep in analysis of serviceability limit state in certain phases.

The development of rational and accurate models that will define time-dependant behavior of reinforced concrete structural elements needs a lot of experimental tests that will confirm the results of proposed procedures for design, and thus to obtain their verification.

For that purpose we can use two alternatives: laboratory experiment that gives real results, but also limited regarding the dimensions, loads and support conditions; or computer simulation with application of the finite elements method, that gives a virtual creation of the reinforced concrete structure and practically there aren’t any limitations. The first alternative testing time-dependant behavior of concrete is very expensive because it takes not just resources at the time of testing, but it also takes a large space in the laboratory. The second alternative is effective and cheap in comparison to the conventional experimental access. But identified critical parameters that affects on the accuracy of numerical model can be proved only from experiments: testing properties of concrete and real structural elements.

In one word, the design of reinforced concrete structure serviceability limit state always needs new theoretical and experimental tests concerning shrinkage and creep of various types of concrete subjected to different load history.

1.2. Objective and Scope

The shrinkage of concrete is a volume change due to loss of water by evaporation and intercourse with the humidity of the environment, or hydration of cement without presence of the load. The creep of concrete appears under presence of long-term external load, i.e. when the concrete is subjected to permanent stresses during time. These volume changes are long-term deformations and depend of the time, that's why are also called time-dependent. These processes are sensitive regarding curing of concrete, variations of humidity and temperature of the ambient, concrete mix design, properties of concrete: compressive strength, tensile strength and the modulus of elasticity. But one of the most important parameters that affects on the development of concrete creep deformation are history of the stresses (history of the application of the loads), intensity, type of loads and age of concrete at first loading[1].

The real value of shrinkage and creep strains has a big importance for the durability and service life of reinforced concrete structures, long-term stability and safety against failure. Neglecting of these phenomena that leads to excessive deflection and crack width is one of the reasons for numerous problems in civil engineering. The errors in calculation of shrinkage and creep strains are generally much bigger than the errors done during simplification of the analysis methods. For the sensitive structures, due to creep, there is no

1.Introduction 5

sense to use finite element method or another sophisticated model analysis if there is no real model to calculate input values of shrinkage and creep .

We can notice that reinforced concrete bridges are structures that are most sensitive on creep of concrete. Reason for that are character and intensity of the variable traffic loads. Besides bridges, this kind of load can appear at other types of reinforced concrete structures such as multi-store parking garages, multi-store warehouses and crane beams.

In the analysis of structures, loads are most important variables that affect on creep of concrete. The loads most frequently are defined like phenomenon that causes constant change of the state of stresses and state of strains [5]. Basic properties of the loads are: their appearance, how often they appear, intensity, time duration, geometry of action, (position and direction) and variation in intensity. The history of action of loads can be expressed with single or more component data [5].

In many cases the load intensity is presented only with one scalar, which is representative value. Example for that are the cases with permanent loads caused by self-weight of the structure. It can be determinate precisely and it is expected to stay unchanged in service life period of structure. The effects from creep of concrete generally are related to the long-term property of permanent load.

Against them are variable loads which intensity is changing in time. They have phenomenon of appearance in different time intervals that cannot be predicted and that are acting like random variables during the service life of structure.

Simple example to present complexity of the history of variable load is crane loads shown in figure 1.2. Although acts life load with intensity “F”, because of its movement alongside the girder, we have change of the position of the load “X” and appearance of horizontal lateral forces with changeable direction “±H”, and at stop we have appearance of brake force “±B”.

Figure 1.2 Loading history on crane girder: a) Intensity of load F, b) Position of the crane x, c) Lateral forces H and d) Brake force B [5]

1.Introduction 6

For the traffic loads besides the intensity it is useful to know the variations of the intensity during the time, average number and duration of the cycles of loading and unloading during one year. On figure 1.3 one example is shown of daily variations in load intensity on the highway A64 in France measured during 38 days (Arroyo, Hannachi, Siegert and Jacob, 2008). This type of load belongs to the group of cyclic loads.

Figure 1.3 Traffic loads on highway A64-France expressed through the maximum daily gross

weight of vehicles [6]

Experimental studies on time-dependant behavior of reinforced concrete structural elements under variable repeated load available in the literature shows increase in stresses, increase in crack width and increase in deflections. The elements were subjected to numerous cycles of loading - unloading during testing in order to obtain fatigue strength of concrete. In these tests there is no possibility for real development of deformation due to creep of concrete.

Time-dependant behavior of reinforced concrete elements, especially analyzing creep of concrete due to variable repeated load is a subject of very few researchers. In the literature effects of the variable load is replaced with action of repeated or cyclic loads. Typical examples of cyclic repeated loads are shown on diagram 1.4 [7].

Creep of concrete influence on the behavior of the reinforced concrete elements is proved by design of serviceability limit state: limitation of stresses, control of cracking and control of deflections. For this purpose according to Eurocode-2 (EN 1992-1) several combination of the loads are taken into consideration:

Characteristic combination of loads that is used for proving of the irreversible serviceability limit state, i.e. checking states such as micro-cracking or possible local non-catastrophic failure leading to large cracks in section [8].

Frequent combination of loads that is used for proving of the reversible serviceability limit state, i.e. checking some aspects of cracking and deflections [8].

Quasi-permanent combination provide an estimate of sustained loads on structure and is appropriate for the verification of creep [8].

1.Introduction 7

Figure 1.4 Different types of cyclic loads [7]

Variable loads had property of short-term loads in serviceability limit state design, neglecting creep of concrete consideration under variable load. In Eurocode-2 (EN1992-1) we have possibility to take into consideration creep of concrete effects of sustained permanent loads and influences from the variable load taken as a part of the intensity that will act like permanent load, i.e. his replacement with quasi-permanent load independently from loading history. The quasi-permanent load is determinate by the coefficient ψ2.

For various type of concrete structures, such as warehouses and multi-storey garages where variable load like traffic loads exists, the coefficient ψ2=0.8 for the surface of the warehouse, for parking garages where vehicles with weight less than 30kN ψ2=0.6, i.e. ψ2=0.3 for vehicles with weight greater than 30kN (but with maximum weight up to 160kN) [9].

For highway and railway bridges, in Eurocode-2 (EN1992-1), the variable loads as traffic loads are treated as short-time load ψ2=0 at highway bridges, but long-term sustained load for railway bridges ψ2=1 [10]. For certain bridge structures, like city bridges under severe traffic, there is possibility to propose the coefficient ψ2. According to the reports of DIN standards [11] related to the Eurocodes for city bridges proposed value ψ2=0.2. In National document for city bridges in Finland coefficient ψ2=0.3 [12].

There is a lack of data from experiments with real loading history of variable load and analysis of its influence on the structures sensitive to creep of concrete effects, especially for structures using low or high strength concrete. Practically there is no experimental research

1.Introduction 8

of concrete structural elements subjected to imposed loads or traffic loads loaded in cycles of loading-unloading. Because of more frequent use of high-strength concrete, especially for concrete structures where we have action of traffic loads, this parameter should also be included in investigation

Faculty of Civil Engineering in Skopje have great experience in experimental and theoretical research of time-dependant behaviour of reinforced and prestressed concrete. Based on professor's Sande Atanasovski original idea, to study the effects of creep on structural elements subjected to real loading history several research programs were realized to propose quasi-permanent coefficient. Following this idea, in the PhD thesis of Professor Goran Markovski an experimental program was carried out to analyze variable loads effects on time-dependant behaviour of prestressed concrete elements. In his experimental research he was used real loading history that acts on city bridges, on elements made of prestressed concrete. Loading history was consisting of action of long-term permanent dead load and repeated variable load that changes in cycles: loaded for 12 hours and unloaded for 12 hours. For this loading history following values of the quasi-permanent coefficient were proposed: ψ2=0.45 (where initial and time-dependant deflections were obtained by quasi-permanent combination of loads) and ψ2=0.60 (where only time-dependant deflection was obtained by quasi-permanent combination of loads) [13].

In continuation and in this thesis experimental programme was proposed to analyze effects of different types of loading histories that were acting as repeated load in cycles of loading and unloading on time-dependant behaviour of high-strength concrete structural elements. For the tests following loading histories were defined:

Short-term load to failure at concrete age of 28 days and 400 days. Long-term permanent load with intensity “G” which do not cause cracking in the

reinforced concrete beam. This load on the beam was applied for period of one year. Long-term loads with intensity equal to the sum of permanent loads “G” and variable

load “Q/2” i.e. 50% participation of the variable load that acts like quasi-permanent load (ψ2=0.50. This sustained load “G+Q/2” was applied for period of one year.

Combination of action of long-term permanent load with intensity “G” and repeated variable load “Q” which was applied in cycles of loading/unloading for 24 hours. This load“G±Q/2” was acting on the beam for period of one year.

Combination of action of long-term permanent load with intensity “G” and repeated variable load “Q” which was applied in cycles of loading/unloading for 48 hours. This load“G±Q/2” was acting on the beam for period of one year.

In this research besides loading histories one parameter more was included in the analysis related to compressive strength of concrete use of high-strength concrete class C60/75. For comparison, in the researches also concrete with usual compressive strength concrete class C30/37 was used.

In the experimental test increase in deflections, increase in crack width and strains in concrete and reinforcement were measured in the period of one year. After the period of one year every reinforced concrete beam was tested on short-term loading until failure.

From testing of these parameters sufficient data were obtained to propose quasi-permanent coefficient “ψ2” in order to define quasi-permanent combination of loads needed in analysis for verification of creep in serviceability limit state design.

1.Introduction 9

1.3. Outline of Thesis

In chapter 2 time-dependants deformation characteristics of the concrete were presented, subjected to short-term loads and sustained loads.

In chapter 3 preview of the theoretical and experimental research available in the literature were given concerning behaviour of reinforced concrete under permanent and variable loads. Analysis of experimental researches was focused on following parameters: strain in concrete and reinforcement, cracking and deflections.

In chapter 4 properties of concrete, according to EUROCODE-2, were given with analysis of compressive strength, tensile strength, stress-strain diagram under short-term load, modulus of elasticity and material models for concrete shrinkage and creep. Special attention was devoted to analysis of creep and shrinkage using model B3 (Bazant and Baweja) later used for comparison of numerical and experimental data.

In chapter 5 structural analysis according to Age adjusted effective modulus method was given.

In chapter 6 basis of serviceability limit state design according EUROCODE-2 are given.

In chapter 7 description of experimental program, results from the tests and analysis of the results from the testing of the reinforced concrete elements are given.

In chapter 8 numerical analysis using the model B3 was performed first to verify experimental results for creep and shrinkage of concrete updating compliance function J(t, t') on the basis of proposed parameters p1, p2 and p6. These results then were used in structural analysis using AAEM method and method of superposition to determine initial and time-dependant deflections under influence of variable loads. From these analysis, quasi-permanent coefficient were obtained for ordinary and high-strength concrete subjected to different loading history of repeated variable loads for beams series D and E.

Chapter 9 summarizes the conclusions obtained from analysis of numerical and experimental data and proposals for further research in this field.

1.Introduction 10

2 Deformation properties of the concrete 11

2. Deformation properties of the concrete 2.1. General

The deformations of concrete, changes in the volume are response of the material to the external loads that are acting in time and to external environment. The changes of the volume are expressed with strains that are defined like change of the length of one element in ratio to the unit length, and it is non-dimensional number.

When concrete element is subjected to load, at age t1, it deforms much more than the strain that appears when we don’t have load, when only shrinkage of concrete occurs. Shrinkage of concrete εcs(t) is shown on figure 2.1.(b) [14] (Jirasek, Bazant, 2001). The increase of deformation is a result of the mechanical strain caused from the stresses εσ(t) which is consists of [14]:

Elastic strain εe=σ/E(t1) in case of uniaxial stress state – “σ” which depends on “E(t1)"-modulus of elasticity at age t1, also named initial strain or short-term strain and[14]

Creep strain εcc(t) (or viscous elastic strain εcv(t)). The term viscoelastic strain is for the fact that the creep depends linearly from the stresses at the serviceability level. Modeling of this phenomenon is based on the linear viscoelasticity. That’s why sometimes in the literature creep strain is defined as εcv (the index “v” is in fact means linear viscoelasticity) [14].

Figure 2.1 Deformations in concrete: a) Shrinkage strain, b) Mechanical strain caused by the stress (elastic strain plus creep strain) and c) recovery strain after unloading [14]

At constant stress, the creep strain increases gradually and as it is known so far, without any bigger changes in time (figure 2.2 (c)). If the concrete is unloaded at age t2 (figure 2.2 (c)) initial elastic (reversible) strain appears which depends from the modulus of elasticity at age E(t2), followed with time depending reversible monotone creep [14].

On the basis of the previous perceptions, in general case the total strain εc(t) in time t, at uniaxially loaded concrete element in time t0 (t>t0) with constant stress σc(t0) can be presented as sum of the separated strains (CEB-FIP Model code 1990,1998) [15]: εc(t) = εci(t0) + εcc(t) + εcs(t) + εct(t) [15] ..................................................................... (2.1)

Or at constant temperature, when we don’t have temperature change:

εc(t) = εci(t0) + εcc(t) + εcs(t) [15] ............................................................................. (2.2)


Where:

εc(t) – is the total strain in the concrete in time (t)

εci(t0) – is initial strain in the concrete at loading in time (t0)

εcc(t) – is strain from the concrete creep in time (t)

εcs(t) – is strain from the concrete shrinkage in time (t)

εct(t) – is strain from the temperature change in time (t)

Figure 2.2 Shrinkage, creep at loading and creep at unloading [14] The equation (2.1) can also be written in the following form: εc(t) = εcσ(t) + εcn(t) [15] ......................................................................................... (2.3)

Where:

εcσ(t) - strains which depend on the stresses of the concrete

εcn(t) - strains which do not depend on the stresses of the concrete (strains that depend on the cement hydration and the conditions of the external environment)

εcσ(t) = εci(t) + εcc(t) [15] ...................................................................................... (2.4) εcn(t) = εcs(t) + εct(t) [15] ...................................................................................... (2.5)


2.2. Short-term behavior of concrete

When concrete is subjected to short-term load exhibits a continuous increase of deformation during the time [1]. The strains increase very rapidly within the first seconds or minutes after loading and then strain growth become slower [1].

The value of the strains and the velocity of its increase depend on various factors [1]:

Amplitude of loading Age of the concrete at the time of loading Is the concrete loaded for the first time or not The concrete mix design

Usually in the process of structural design detailed information for the concrete composition and its quality are not known, and thus the empirical constitutive laws are based on the compressive strength of the concrete and the elasticity module of the concrete. But constitutive law describing the behavior of concrete just after the loading is rather complex. Therefore the definition of what shot-term loading is seems to be of primary interest [1].

Typical diagram of stress-strain relation is shown in the figure 2.3. In practice it is assumed that, the stresses in the concrete are proportional with the strains for level of serviceability loads.

Figure 2.3 Stress – strains relation for the concrete [16] The strain that appears at applying of the stresses (or just after few seconds) is known as instantaneous (initial) strain and it is expressed with following equation:

0

00 tE

tt

c

cci

[16] .............................................................................................. (2.6)

Where: σc(t0) – stress in the concrete in time t0

Ec(t0) – modulus of elasticity of the concrete at age t0, time when the load is applied The constitutive law for the behavior of concrete at short-term load is defined by modulus of elasticity, at constant Poisson’s ratio.


If simple case is assumed, i.e. a concrete element subjected to constant stress, the rate of stress increase during loading process, the age of concrete and duration of loading are important factors determining strain development in time.

The constitutive law should be represented in general by a function, which takes into consideration the increase of the development of strains, in the literature known as compliance function.

This function in the older models was based on measurement in rather long time after loading, and therefore they didn't represent strain development immediately after the loading and during time [1]. The form of the function at this model is based on the equation [1]:

00c

0 t,t1tE

1t,tJ [1] ............................................................................. (2.7)

Where:

Εc(t0) – Modulus of elasticity

φ(t,t0) –Creep coefficient This form of the compliance function (equation 2.7.) has some disadvantage, if the strain immediately after the loading is to be calculated. The smallest strain is achieved if φ= φ(t0,t0)=0, which leads to [1]:

0c

0c0min,ci tE

tt

[1] ........................................................................................... (2.8)

But for the strain εci,min(t0)-there is no precise definition. The modulus of elasticity is usually taken as designed value, value determined by experiment, which leads to [1]:

)t(E 0c [1] ................................................................................................... (2.9)

Where: ∆σ and ∆ε are stress and strain increments, after several cycles of loading and unloading. This definition of modulus of elasticity is suitable for analysis of the deformations induced by loads that can occur many times. For analysis of the deformations caused by the permanent load which is applied only once, at the beginning of the service life, the designed value of the modulus of elasticity can be reduced so that plastic initial strains should be taken in consideration [1].

If short-term strains are calculated, the shortest time of load duration is given by value when the elastic strain is given with εci.min(t0). If shorter load duration is assumed the function given in equation (2.7.) doesn’t lead to real results [1].

The contemporary models can describe increase the strains immediately after the loading with better accuracy, like the models BP (Bazant and Panula, 1978) and B3 (Bazant, RILEM 1995). Mostly the researchers, who carried out experimental testings, because they can measure the strain and strain development during the time precisely, use these models. When the compliance function is based on short-term measurements, so in order to improve the accuracy of the analysis, the contemporary models are more convenient than those based directly on the modulus of elasticity (equation 2.7.)[1].These models use similar form of equation (2.7.) [1]:

00c

0 t,tCtE

1t,tJ [1] ............................................................................... (2.10)


Where:

Ec (t0) – asymptote initial modulus of elasticity

C (t.t0) – increment of the strain caused by unit stresses This form precisely defines a constitutive law for concrete. The duration of the loading can be changed from several seconds to several years, but allows to have one law for the behavior of the concrete at short-term and at long-term loads [1].

But anyway, from a practical point of view it is necessary to define one constant value the elastic strain, i.e. and the analysis in great majority of cases should be based on the Hook’s law. To avoided unclear definition of the modulus of elasticity, it is necessary to defined load duration, for which the value of the modulus of elasticity gives correct strains [1].

The value of the elasticity module at age of 28 days could be defined as follows:

000c t,ttJ

1tE

[1] .................................................................................... (2.11)

Where:

t0 = 28 days and

∆t - is duration of the load 2.3. Long-term behavior of concrete

Under constant stresses, the strains are increasing during the time because of concrete creep and total strain is sum of the initial strain and creep strain in time t (see figure2.4) [16].

0

0c

0cc t,t1

tE

tt

[16] .............................................................................. (2.12)

Where is:

φ (t,t0) – creep coefficient

The coefficient φ presents the influence of the creep in ratio to the initial strain, and its value increases if the load is applied at early age of the concrete t0 and increases when we have constant-permanent stresses in long-term period (t-t0) [16].

Figure 2.4 Concrete creep under long-term load [16]


At the loading of concrete different types of deformations in the concrete were observed. At long-term loads not just elastic but also relatively low plastic deformations were occurred and time dependant deformations that can be reversible or irreversible [1].

On the level of service load creep appears because of the creep of the cement paste, because the aggregate has no influence on the deformations. If greater loads are applied the cracks in the transit zone between the aggregate and the cement paste leads to increase of the time depending deformations [1].

The parameters, which have influence on the time-dependant deformations of concrete, can be classified into two groups. The first group includes so-called technological parameters. The most important are: water-cement ratio, stiffness of the aggregate, volume and quantity of aggregate and type of cement. On the other side are the parameters, which depend of the external factors like the age of the concrete at loading, relative humidity of the environment, geometry and dimension of the constructive elements. At constant environment conditions, the creep deformations increase if [1]:

Water-cement increases

Stiffness or amount of aggregate decreases

Hardening time of concrete is decreased

Relative humidity decreases

The thickness of the structural elements is smaller

Higher humidity content of the concrete at loading

The temperature increases

Age of the concrete at loading decreases

Increase of load occurs

At normal conditions the age of concrete have the greatest influence on the creep at loading.

The well-known linear dependence for calculation of the total deformation, neglecting concrete shrinkage is given by equation [1]:

00000 1 t,ttt,ttt,t elccel [1] ................................................ (2.13) For the time-dependant behavior of concrete important property is concrete aging, which is different from the aging of the other materials. In concrete, aging causes increase of compressive strength and modulus of elasticity at age t. This causes reduction of the creep for concrete loaded at higher age. An important cause for the aging is the chemical processes of hydration [14]. Product of the hydration is mostly alite hydrate gel, which gradually fills the pores of the hardened cement paste [14]. Because of that mean size of the capillary pores is reduced, and gradually increases the stiffness and strength of the microstructure [14]. But the process of aging continues to last many years, until the concrete is not dry under 80% relative humidity in the capillary water.

Because the aging has a very important role, the time is measured from the beginning of the setting of the concrete, i.e. typically few hours after the mixing [14].

This means that for the calculation of creep the variable t, always corresponds to the concrete aging. In certain period, at the beginning, the concrete is in formwork and cannot dry or if it is artificially protected from drying to achieve proper cure. For calculation of the shrinkage is important to know the approximate age t0, the age at beginning of drying which corresponds to the time of removal of formwork or at the end of curing.


2.4. Time-dependant deformations of concrete 2.4.1. Shrinkage of concrete

The concrete shrinkage is a result of evaporation of the water, which is in the capillary pores in the concrete connected with the surface, following the instability of the relative humidity between external environment and humidity in the pores.

Today the shrinkage is usually categorized as follows [17]:

Plastic shrinkage

Early age shrinkage – shrinkage of the concrete in the first 24 hours

Drying shrinkage of the concrete The shrinkage of the concrete is very simple phenomenon in its manifestation. Those are obvious changes of the concrete volume, but a very complex phenomenon when causes of its existence should be understood [18].

When we speak generally for the shrinkage, it is better to indicate that shrinkage of the concrete is a combination of several elementary types of shrinkage (Altcin, Neville and Acker, 1997) [18]:

Plastic shrinkage, which appears on the surface of the fresh concrete exposed to drying (Wittmann, 1976)

Autogenous shrinkage which appears due to hydration of the cement (sometimes it is called self-drying), but it should be separated from the chemical shrinkage (Tazawa, Myazawa and Kasai, 1995)

Shrinkage at drying, which appears because of the movement of the water through the hardened concrete, i.e. evaporation of the internal water in the external environment

Thermal shrinkage

Carbonic shrinkage To understood the origin of shrinkage and the main reasons for appearance of every elementary shrinkage, it is necessary to understood hydration and their physical, thermodynamically and mechanical consequences [18]. 2.4.2. Autogenous shrinkage

The autogenous shrinkage of the concrete is generally known very long, but until now on it wasn’t devoted enough attention, like it was devoted to drying shrinkage to secure concrete structures from cracking.

But today, according to experimental results, concrete which has low water-cement factor, addition of mineral admixtures exhibit larger value of autogenous shrinkage which leads to cracking under certain conditions. Autogenous shrinkage is very important for high-strength concrete, self-compacting concrete and massive concrete. For above mentioned concrete value of autogenous shrinkage is given in table 2.1. The ratio between the autogenous shrinkage and the drying shrinkage in the total shrinkage of the concrete is shown in figure 2.5.

The autogenous shrinkage is macroscopic reduction of the volume of the cement materials, at the start of hydration process, from the start of setting time of the cement. The autogenous shrinkage doesn’t include the changes of the volume, which appear due to loss, or penetration of substances, temperature changes, or from applying any external forces [19].


Table 2.1 Materials, composition and strain of the autogenous shrinkage of the high strength, the self-compacting concrete and massive concrete [19]

High-strength concrete

Content of binders: 450-600kg/m3 Water-cement factor: 0.25-0.40 Mineral additives: silicon dust, and fly ash Chemical additives: super-plasticizer

Strain from the autogenous shrinkage: εas=200-400x10-6

εas=0.20-0.40‰

Self-compacting concrete

Content of binders: 350-500kg/m3 Water-cement ratio: 0.25-0.40 Mineral additives: slag, fly ash and limestone powder Chemical additives: superplasticizer, additive for control of the viscosity

Strain from autogenous shrinkage: εas=100-400x10-6

εas=0.10-0.40‰

Massive concrete Content of binders: 250-350kg/m3

Water-cement factor: 0.45-0.60 Cement: cement with low heat of hydration, cement with additives Mineral additives: slag and fly ash

Strain from autogenous shrinkage:

εas=10x10-6 εas=0.10‰

Figure 2.5 Shrinkage of normal and high strength concrete [20] Autogenous shrinkage is less than 10-4 in concretes with W/C ratio greater than 0.45 for ordinary concrete (figure 2.6), but in increases very rapidly when this ratio falls below 0.40 for high-strength concrete (figure 2.7).

Figure 2.6 Autogenous shrinkage of various ordinary concrete (W/C>0.45), Le Roy 1994 [1]


Figure 2.7 Autogenous shrinkage of various high-performance concrete (without silica fume), Le Roy 1994 [1]

The autogenous shrinkage can be expressed like coefficient of the autogenous shrinkage i.e. like a percent of reduction of the volume or like one-dimensional change of the length-strain from autogenous shrinkage (εas) [19].

The cracks that appear due to the autogenous shrinkage are difficult to control because of the density of the microstructure of the concrete with low water-cement factor and use of mineral additives (fly ash, slag and silica fume). Proper curing of concrete cannot effectively reduce the appearance of these cracks [18].

The autogenous shrinkage of concrete depends on:

Mineral composition of the cement

Degree of hydration

Mix design

Water-cement factor

Curing conditions

2.4.3. Drying shrinkage

After 24 hours, the autogenous shrinkage doesn’t stop, development of the carbonic shrinkage starts as a reaction between the carbon dioxide from the atmosphere and the products of hydration, but all types of shrinkage are less important than the appearance of drying shrinkage which starts after curing of concrete is finished.

The drying shrinkage of concrete is just a part of the shrinkage of the cement paste. In the process of drying shrinkage the aggregate has an important role to separate the cement paste and to strengthen it.

In practice, to produce workable concrete, it is necessary approximately two times more water than the theoretically needed so that the cement hydration could continue. After concrete has been cured, drying shrinkage begins because of excessive water that has not reacted with the cement. It begins to migrate from the interior of concrete mass to the surface. As moisture evaporates, the concrete volume shrinks. From the cement chemistry it is known that the drying shrinkage is caused by the contraction of the hydrated hardened calcium silicate gel during the withdrawal of the water from the concrete [7].


When the concrete is exposed to drying, at state of equilibrium with 50% relative humidity on the external environment, drying shrinkage is within 0.04-0.08% (400 to 800 micro strains). The change of the volume due to shrinkage of the concrete is not equal with the loss of water volume. The initial loss of the water causes small or doesn’t cause shrinkage of the concrete because water first evaporates from larger pores [21]. Further evaporation of the absorbed water is reversible against the absorption of the additional moisture (Neville 1996) [21]. One source of the irreversible drying shrinkage is the loss of water from the small capillary pores of the hydrated cement paste (Mehta and Monteiro, 1993) [21]. The evaporation of the capillary water causes disjoining pressure due to the tension in the narrow space between two solid surfaces. The removal of the capillary water reduces the disjoining pressure and leads to shrinkage of the hydrated cement paste exposed to drying conditions [21]. In ratio to the capillary water, the meniscus of the water in the small pores (5-50nm) cause forces of hydro-static tensioning and the removal of this water causes compressive stresses on the walls of the capillary pores, which contributes to the total shrinkage of the system [21].

Typical development of drying shrinkage in time is shown on figure 2.8.

Figure 2.8 Drying shrinkage during time [22] The size of the drying shrinkage is influenced by many factors, including [7]:

Stiffness and quantity of the aggregate

Water-cement ratio

Total quantity of paste

Type and quantity of the chemical admixtures

Curing regime and the age of the concrete at which it is exposed on air

Type and quantity of the mineral additives

Theoretical length of the concrete element which is defined as the ratio of section area of the element to its semi-perimeter in contact with the atmosphere

Diameter, amount and distribution of reinforcing steel

Relative humidity and its change rate

Carbonation For known quantities of aggregate and cement paste, the drying shrinkage can increase two times if the quantity of aggregate is reduced from 80% to 50% [21].

Feldman (1969) has noticed that the concrete exhibits less drying shrinkage if quartz, limestone, granite or feldspar aggregate were used or larger drying shrinkage if we use aggregate as: sand, basalt and shale [17].


Mokhtarzadeh and French’s (2000) findings suggest that concrete compressive strength and composition of cementitious materials had no significant effect on drying shrinkage of high strength concretes [17].

Mineral additives can decrease, increased or have no influence on the drying shrinkage of the concrete. Pozzolans usually increase the drying shrinkage, due to forming of system, which contains smaller pores. The fly ash and the silica fume are causing the same effect. Chemical additives have tendency to increase the drying shrinkage [21].

The atmospheric diffusion of absorbed water present in the cement paste and water retained in the pores, due to the capillary forces of tensioning lasts longer and can be accelerated at high external temperatures, high porosity and low relative humidity of the environment. For different types of concrete which are being monitored for 20 years, it is determined that the drying shrinkage is about 20-25% for two weeks, 50-60% for three months and 75-90% for one year from total value of drying shrinkage. For the normal concrete the drying shrinkage of the concrete is within 400 to 700 micro strains at normal conditions. For reinforced concrete elements, the drying shrinkage is within 200 to 300 micro strains. It is noticed two times bigger drying shrinkage if the concrete at relative humidity of the environment of 45% compared with the drying shrinkage at relative humidity of the environment of 80% [21]. The shrinkage decreases if increases relative humidity. If the concrete is exposed to 100% relative humidity or if it is submerged in water, comes to increase of the volume because it doesn’t exist movement of the water to the outside (figure 2.9) [22].

Figure 2.9 Diagram drying shrinkage during time and humidity [22] The size and the form of the concrete elements have important effect to the size and the total drying shrinkage. The total shrinkage of the massive concrete sections is smaller in ratio to the smaller dimensions of the concrete elements, even if at massive sections lasts much longer in ratio to the shrinkage of the smaller sections (Nilsson 1995) [21]. 2.4.4. Creep

When concrete is subjected to permanent load, first initial strains appear which gradually increases caused by the creep of concrete. The creep is continuous deformation, which appears in the concrete under the influence of permanent load, and its consequences are visible after few years. The deformation can increase several times due to the creep of the concrete in ratio to the initial deformation at loading.

The origin of creep in the cement materials is still not known enough. Defining of the mechanisms, which take place in cement materials, are on much lower level, than the macro level, which is usually used at testing of the concrete creep on laboratory specimen. These micro mechanisms that causes time depending deformations are developed from specific behavior of concrete at short-term creep, for the first time mentioned by Ruetz, 1996 [23].


The lack of knowledge for the method of development of these micro-mechanisms or so called “real mechanisms” by Wittmann 1982 is maybe the reason for existence of great number of models for prediction of the creep in the concrete [23].

Studies of Ulm, Le Maou and Boulay 1999, are based on macroscopic kinetics of the basic creep. The characteristic of the kinetics of creep, which is shown in table 2.2, suggests that two mechanisms of creep appear in different time connected to different pores spaces in cement materials [23]. Table 2.2 Experimental characteristics of kinetics of the concrete creep [23]

Short-term creep Long-term creep

Reversible at reversible creep

Reduction of the size of the creep function J and the degree (dJ/dt) during the hydration

Asymptotic phenomenon

Size of the w/c factor

Irreversible

Kinetics at aging don’t depends on:

-viscosity history (t-t0)

-age at loading t0

-concrete composition

It is not asymptotic

Mechanism which occurs in the capillary space (1m)

Mechanism with occurs in the micro-pores of C-S-H (10-20 Å)

The mechanism of short-term creep occurs in the capillary space. This assumption is in accordance with the fact that the normal concrete has greater creep than high strength concretes. The basic difference between these two types of concrete is the capillary porosity (Broghel, 1994). Similar kinetics of the short-term creep and the reversible creep shows that the short-time creep is reversible. The fact that their size and kinetic energy depend on the reaction of hydration, mechanism of diffusion exists in the capillary pores caused by macroscopic loading (Wittmann, 1982). The water included in the micro-diffusion doesn’t have to be only capillary water, but also absorbed water on the capillary walls [23].

The kinetics of the long-term creep it seems that doesn’t depend on the concrete type. This shows that the mechanism of diffusion occurs in the normal space of the cement materials formed by C-S-H layers. This mechanism has mechanical origin. Bazant (1997) suggests the sliding mechanism, that mechanism of long-term creep is a result of the sliding of the C-S-H layers [23].

Time dependant strain in the concrete, besides the elastic strain, so called non-elastic strain, consists of: shrinkage strain εcs which is independent from the stresses σ and the additional mechanical strain εcc caused by the stresses, called concrete creep [14]. Existence of the shrinkage was known since the invention of the modern concrete, but existence of the creep was discovered by Hatt in 1907 year. For the level of the serviceability stresses, of concrete structures, generally is assumed that the concrete depends linearly on the stresses and the modeling of this phenomenon is based on the linear high elasticity [14].

The creep that appears at constant humidity (when there is no exchange of the humidity with the external environment) is called basic creep [14]. Basic creep is the total deformation of a loaded, sealed specimen, without elastic deformation and autogenous shrinkage. The simultaneous change of the humidity causes additional creep, called drying creep or Pickett effect (Pickett 1956). This creep has complex physical origin. Drying increases the local high stresses in the microstructure of the calcium silicate hydrated gel (Bazant et all.1997). Great part of the drying creep under the influence of compression is evident creep (Wittmann 1974, Bazant and Wu 1974), which in reality is stress caused by shrinkage, and it is treated as creep. Usually the testing of the concrete is conducted on specimens, which are exposed to external environment i.e. so called drying creep [14].


In one word, the total creep is in fact sum of basic creep and drying creep (Vincent et al. 2004) [17].

The interaction between the creep and the shrinkage has been matter of research especially for ordinary concrete under drying and wetting conditions (L’Hermite 1960 and Gamblr et al, 1978). The existence of proportionality between the drying creep and drying shrinkage gave basis to the concept of stress-induced shrinkage (Bazant, 1985, 1987. 1994 and Granger 1996) and its refinement in the micro-prestress solidification theory (Bazant, 1997). Under drying conditions, the proportionality means that both phenomena are controlled on the macro-scale of laboratory test specimen by the same rate-determining process: exchange with the exterior. With the recent event of high strength concretes, both experimental (SIcard et al, 1996) and theoretical (Hua, 1995) researches turned to the creep and shrinkage interactions under sealed conditions, i.e. to basic creep and autogenous shrinkage couplings (figures 2.10 and 2.11) [23].

Figure 2.10 Components of total deformation

Figure 2.11 Creep-shrinkage coupling [13] When the concrete is exposed to constant stress, infinitely long or long enough two processes can appear (figure 2.12) [22]:

Stabilization of the creep process–when the deformation is up to final deformation, where no failure appears and

Non-stabilization of the creep process-when the deformation is infinitely big which leads to failure.


The first case occurs at low values of stresses and the concrete can be considered that has this kind of behavior within the service stresses. The second case occurs at high level of stresses, but it is very difficult to determinate the line that separates this two cases. The high levels of stresses cause significantly non-linear increase of the deformations of the concrete [22].

Figure 2.12 Deformation of concrete creep: stabilization and non-stabilization process [22]

The concrete creep is closely related to the shrinkage and the two phenomena are related to the hydration of the cement paste. Concrete that has smaller deformation from the shrinkage has low potential for development of the creep [17].

Main parameter, which has influence to the creep, is the intensity of the load in function of time (figure 2.13), concrete composition, humidity and size of elements (Zia et al, 1997) [17].

Figure 2.13 Deformation of concrete creep during time under different stress history [13] The linear concrete creep (viscous creep) depends on the following factors [22]:

Temperature and humidity of the environment

Type and quantity of cement


Quantity of water (greater water-cement ratio greater the creep which is especially expressed at water-cement ratio > 0.6)

Granulometry of the aggregate

Dimensions of the elements

Concrete curing Testing of long-term deformations of beams of high strength concrete Paulson et al, 1991, experimentally proved that the creep coefficient of high strength concretes subjected to long-term compressive stresses is significantly smaller in comparison to creep coefficient of ordinary concrete [17].

Collins (1989) tested five mixtures of concrete, which at age of 28 days have compressive strength between 60MPa to 64MPa. Experiments have showed that the creep is smaller if we have smaller quantity of cement paste and greater maximum size of coarse aggregate [17].

Carette et al (1993) testing high-performance concrete adding fly ash noticed that they also have smaller creep compared to ordinary concrete [17].

In the concrete only the cement paste has creep, while the aggregate doesn’t have creep. Therefore the creep is considered as non-linear function in ratio to the volume of the cement paste in the concrete. Because of that creep depends on the concrete composition, content of the volume of non-hydrated cement paste, aggregate volume and certain physical characteristics of the aggregate: grain-size composition, maximum size of the coarse aggregate, shape and texture of the aggregate. Also very important factor is the modulus of elasticity of aggregate, which directly affects on the size of concrete creep, while the other characteristics affect indirectly on the concrete creep.

The type of cement also affects on the concrete creep because it affects on the concrete strength at the moment of loading. On the basis of many researches on concretes with different compressive strength, was proved that the size of the concrete creep is reverse proportional with compressive strength of the concrete at the moment of applying the load (Mehta and Monteiro, 2006) [7]. The fines of cement affect on the degree of hydration and the development of the strength at early age concrete and also affects on the creep. Extremely fines cements with specific surface to 740kg/m2 lead to appearance of early creep, but lower creep after one of two years after the loading (Bennett and Loat, 1970) [7].

The chemical admixtures superplasticizer and retarding admixture of setting time, lead to system of more fine pores in the products of hydration and increasing the creep in many cases, but not in all (Hope et el, 1967; Jessop et al 1967) [7].

The size and the shape of concrete elements have influence on the magnitude of creep because the degree of loss of water is obviously controlled by the length path of flow of water, which are related to the creep.


3 Review of some experimental researches concerning creep of concrete 27

3. Review of some experimental researches concerning creep of concrete 3.1. Introduction

Long-term and repeated loads may cause significant increase in crack width and deflections which affect on increase in concrete deformations, decrease in tension stiffening and increase in slip alongside the reinforcing bar, which results in drop of neutral axis, and as consequence increase in stresses appears in the reinforcement in the cracks and increase in curvature of the section [1].

The increase in the crack width and deflections can cause inadequate behavior of the elements in service or failure for fatigue, when they are loaded with dynamic load because of the fatigue of the material [1].

Several parameters have great influence on the increase of crack width and deflections of reinforced concrete elements under sustained and repeated loads [1]:

Level of loading, loading history

Reinforcement: reinforcing ratio, size of reinforcing bars, distribution of bars in the section and rib pattern

Concrete: strength, type of concrete (normal, lightweight, high-strength), shrinkage of concrete

Cross-section shape of the elements

State of the cracking by applying of long-term (sustained) and repeated loads Typical examples of test are the testing of Lovegrove and Din (1982) for development of the maximum cracks width and deformations in the middle of the spacing on beam which is exposed to repeated loads shown on the figure 3.1 [1]. 3.2. Influence of the long-term and repeated loads on the strains

The crack width and middle span deflections increase with the increase of number of load cycles, or in time under long-term loads [Soretz (1985), Stevens (1972), Rehm and Eligenhausen (1973), Balaguru and Shah (1982), Balazs and Eligehausen (1992) and others], although rate of development is decreasing in time in ratio to initial values.

In case of repeated loading, the highest values of crack width and deflections are observed under maximum value of number of cycles by repeated load (figure 3.1) [1].

Unloading decreases the crack width and deflections without complete closing of cracks, because bond stresses between reinforcement and concrete do not withdraw by unloading, because self-equilibrium of bond stresses remains between the cracks [1].

Test results by long-term and repeated loads are discussed on usual basis with measurements of crack width and deflections [1]. 3.2.1. Reinforcing strains and concrete strains

Shrinkage and creep or cyclic creep of concrete can cause a strain increase in compression zone of elements due to loss of tensile stress of concrete between the cracks [1].

This causes redistribution of stresses, drop in the neutral axis which cause increase in reinforcing steel in the cracks (figure 3.2). Lutz, Sharma and Gergely (1967) have observed 232% higher stresses at singly reinforced concrete beams, and at doubly reinforced concrete beams for 109% higher maximum compressive strains [1].

The increase of the tension reinforcing stress under long-term loads is large as 10% than the initial stresses for beams with usual dimensions measured after long time [1].


The strains in the cracks (obtained as ratio of the total crack width to the total length of element) increases more rapidly than the average strain at the level of the tension reinforcement which indicates loss of concrete tension between the cracks. The reinforcing strains in the cracks of the element subjected to bending exceed the theoretical value obtained by elastic analysis of the cracked cross-section. Rehm and Eligehausen (1977) have observed 10% to 15% increase above the theoretical value, which can be much bigger at sections with higher reinforcing ratio (1973) [1].

Figure 3.1 Increase of the maximum cracks and middle span deflections under repeated loads – Lovegrove and Din (1982) [1]

a) Sample

b) Diagram of force-maximum crack width and number of cycles of loading

c) Diagram of force-deflection in middle span and number of cycles of loading


Figure 3.2 Modifications of strains under long-term and repeated loads [1]

a) Diagram load-compressive strain (for the beam on figure 3.1.a)Lovegrove and Din (1982)

b) Diagram of compression strain-time, Lutz, Sharma and Gergely (1967)

c) Modification of strain distribution under long-term loading; Stevens (1972)

d) Diagram of calculated and measured strains in the reinforcement at cracks (average of three results of repeated loads); Rehm and Elgehausen (1973)

e) Increase of strain in the reinforcement at crack under long-term loads; Illstone, Stevens (1972)


3.2.2. Average strain in the reinforcement, tension stiffening

One part of the reinforcement stresses in cracks is transformed to the tensioned concrete by bond, causing reduction of reinforcement stresses between the cracks. Because of this contribution of tensioned concrete, the phenomena are generally referred as tension stiffening [1].

With increasing number of loading cycles or increase of time under long-term loading the tension stiffening decreases and leads to an increase in the average reinforcement strain and increase of the crack width [1].

The smaller the relative rib area of the reinforcing bar, the faster increases the average strains in the reinforcement. The increase of average strain is explained by Lutz and Sharma (1967) as a result of the appearance of micro cracks around the reinforcing bar [1].

Rostasy et al. (1976) have shown that the tension stiffening mostly depend on concrete tensile strength, the quality of bond stresses, reinforcement ratio and load history [1]. 3.3. Crack width

The increase of crack width observed in elements under long-term loads or constant amplitude load is rather large. This can be explained with the differences in the reinforcement ratio, rib pattern of reinforcing bars, concrete shrinkage or the state of the cracking by applying the long-term or repeated loads [1].

Many of the test results show that the increase of the characteristic crack width (wk) agrees with the increase of the average crack width rather well (Rehm and Eligehausen 1973). The characteristic crack width is defined by value of 95% fractal of crack width. The maximum crack width (wmax) is often used in the literature instead of the characteristic crack width, but they have similar significance. As long as probability level of wmax was not always given in the references, both notations wk and wmax were kept [1].

Figure 3.3 gives a comparison of increase in crack width caused by long-term load or repeated load. A 35% increase was observed in 21 month or in 5·105 load cycles (giving much shorter testing time), respectively under 270N/mm2 sustained load or maximum cyclic load [1].

Figure 3.3 Increase in crack width under long term or repeated load; Rehm and Eligehausen (1977) [1]


3.3.1. Long-term crack width

The increase of flexural crack width under long-term loading was experimentally studied by Lutz, Sharma and Gergely (1967) on single and doubly reinforced concrete beams. The increase of the average crack width at the end of 5 month period reached about 30% (figure 3.4). The average increase in crack width for single reinforced beams was 5% greater (in terms of initial values) than for doubly reinforced beams with equal tensile and compressive reinforcing [1].

Stevens, Bryden-Smith and Hunt (1972) have observed that the ultimate crack width of reinforced concrete beams under permanent load was about twice than the initial, and also greather for lower values of concrete cover [1].

Figure 3.4 Increase in average crack width Lutz, Sharma and Gergely (1967) [1]

a) Specimen

b) Percentage increase of average crack width in singly reinforced sections

c) Percentage increase of average crack width in doubly reinforced sections Jaccoud and Favre (1982) carried out long-term tests on reinforced concrete one way (Series C) and in two ways (Series B) [1].

The measured mean crack widths after one year were for 1.41 to 3.0 times bigger than the mean crack width after 5 minutes depending on the load level. This increase is relatively higher for slabs that were already in crack formation phase. At two-way slabs, the mean crack width was doubly increased after one year [1]. 3.4. Deflection

3.4.1. Long-term deflections

Lutz, Sharma and Gergely (1967) have observed that after 5 months from the tests by long-term load, the middle span deflections are 2.15 times the corresponding instantaneous value measured just after the loading in singly reinforced concrete beam and 1.62 in doubly reinforced concrete beams (figure 3.5). The larger increase of deflection in singly reinforced


concrete beams as compared to those for doubly reinforced concrete beams are partly due to unrestricted shrinkage that appears after loading in the compression zone of singly reinforced concrete beams and partly because of the influence of the compression reinforcement reducing creep deformations [1].

Today two different approaches exist for calculation of the long-term deformations [1]:

Direct calculation of the final deflection considering creep and shrinkage of concrete and deterioration of tension stiffening [1].

Determination of the increment of deflection under long-term loads using a multiplier (>1) applied to the calculated initial deformation [1].

The advantage of the second method (besides its simplicity) that in some cases the increment of the deflection is to be directly controlled. Several suggestions are available according to Branson 1977, ACI318-2008, Unger 1967, CEB-FIP MC 1990 to determine this multiplier [1].

Figure 3.5 Measured middle span deflections-time in singly and doubly reinforced beams; Lutz, Sharma and Gergely (1967) [1]

Sharaf (1992) studied the results of 46 own and others reinforced concrete beams under long-term load where the creep and shrinkage processes were parallel measured on cylinders. In a parameter analysis of 39 beams sections as variables were: shape and dimensions of the section, reinforcing ratio of tensile and compression reinforcement, concrete class, relative humidity and ratio of service to ultimate load. He concluded that [1]:

It is necessary to use modulus of elasticity reduced for 15% for the calculation of initial deflection, due to the rapid initial creep at the beginning of loading [1].

In order to be able to consider the type of the load, age at loading and duration of load it is necessary to include the creep coefficient as multiplier for calculation of the long-term deflections [1].

The compressive strength of concrete and form of the cross-section can be neglected in the multiplier, but steel ratios of tensile and compressed reinforcement should be taken into consideration [1].

In case of loads that are variable in time (especially those with opposite sign, for example: prestressing force and imposed load), the question is whether product model (CEB-FIP Model Code 1990) or summation model according to MC1978) is better approach [1].

The long-term measurements in period of 1 year were carried out by Jaccoud and Favre (1982) on one way reinforced concrete slabs (Series C) on six load levels: M/Mu=0.2; 0.3; 0.4; 0.5 and 0.6 (the lowest load level was bellow the cracking load). The long-term loads are applied at age of the concrete of 28 days. Measured deflections after one year are 1.77 to


4.18 times than the initial values of the deflections measured after 5 minutes. The relative increase of the deflections is higher for lower load level and lower for higher load level [1].

In the series B Jaccoud and Favre (1982) tested two way simply supported reinforced concrete slabs, loaded by long-term load on two load levels (M/Mu=0.345 and 0.6) at age of 14 days after the casting. The load level M/Mu=0.345 have been chosen just above the cracking load. The relative increase of deflections after one year related to the deflections measured after 1 day after applying the load were in the limits from 1.70 to 3.17 times depending on the load level [1].

From the tests they concluded that the relative increase of deflections under long-term loads due to creep and shrinkage of concrete depends on the load level and the reinforcement ratio. The relative increase of deflections is more obvious for elements in uncracked phase or in crack formations phase than for elements in stabilized cracking phase. The direct approach to calculate deflections based on the moment-curvature relationship with or without normal force was first developed and given in CEB Manual Cracking and Deflection (CEB, 1985). This basic model takes into consideration the effects of reinforcement, the global effect of cracking and the time-dependant effects of concrete. This model in a specific way was validated by test from Jaccoud and Favre (1982) and now is generally accepted in CEB-FIP Model code 1990 (CEB, 1993). More simplified methods like the bilinear method, global coefficient method (Favre and Charif 1994, Ghali and Favre 1994) were developed from this model [1].

One of the simplest approaches for calculation of the long-term deflections assumes that time dependent deflection (at) is a linear function of the initial deflection by applying of the long-term load (a0) (Numbergova and Hajek 1994). This assumption is shown on figure 3.6 where the relation between at and a0 is plotted evaluating test results from Jaccoud and Favre (1982) The point of intersection of the linear regression lines with the ordinate axis indicate deflection due to shrinkage. This assumption of Numbergova and Hajek (1994) is developed on basis of the observations of Corley and Sozen (1966), Neville and DIlger (1983), Clarke et al. (1988) and Hajek (1994) [1].

According to Corley and Sozen (1966) the long-term deflections of reinforced concrete beam can be divided into three components [1]:

1. Deflections resulting from instantaneous strains,

2. Deflection resulting from creep strains and

3. Deflections resulting from shrinkage strains

This approach can be expressed as follows:

sh0rtt aa1a [1] ...................................................................................... (3.1)

Where is: at-long-term deflection in a distance x from the beam support

a0-initial deflection in the time of load application

ash-deflection due to shrinkage

βrt-is multiplier of the initial deflection giving the deflection increment due to creep. If the coefficient βrt doesn’t depend on the load level in certain moment this equation will represent straight line and would mean linear dependence between at and a0


Figure 3.6 Relation between long-term deflections (at or ∆at) and initial deflections (a0) evaluated from test results by Numbergerova and Hajek 1994 [1]

a) Jaccoud and Favre (1982) Series C, one-way slabs after 28 and 365 days

b) Washa (1947)-1800 days

c) Hajnal-Kony (1963)-1770 days

d) Corely and Sozen (1966)-700 days

e) Figarivskij (1962)-240 days To support the hypothesis for the linear relationship between the long-term and initial deflections by Numbergova and Hajek (1994) and Hajek (1994) 114 reinforced concrete slabs were tested at the Institute of construction and architecture in Bratislava, which served for elaboration of design method given in the regulations of the Czechoslovakia CSN1988 [1].


On the basis of results they came to the conclusion that for the coefficient βrt (for long-term loaded elements) applied to the initial deflection can be obtained from following equation [1]:

0trt tt07.0exp1 [1] .................................................................... (3.2)

Where is:

t-t0 time duration of long-term load in days

φ-creep coefficient of concrete depending on the relative humidity

αt-coefficient which takes into consideration the age of concrete at the time of load application t0

0t t015.0exp08.015.0 [1] ...................................................................... (3.3)

On basis of the results from tests of 114 own and 100 other results on slabs and beams Nurmbergova and Hajek (1994) concluded that a linear relationship can be assumed between the time depending deflection at and the initial deflection a0 in range of service loads for cracked and for uncracked elements independently on the load level [1]. 3.5. Conclusion

Crack width and deflections predicted on the basis of short term tests do not provide a satisfactory guide to crack widths and deflections in service. The long-term and repeated loads cause significant increase of the deflections, increase of the crack width and increase of the strains in concrete and reinforcement. From the tests with long-term and repeated loads can be observed [1]:

Reinforcement stresses in the crack of an element subjected to bending, may increase above the theoretical value on the basis of cracked section analysis [1].

Crack width increases with loading time or with increasing number of load cycles and approximately is two times bigger than the initial value of crack width. Crack width at unloading also has increasing tendency. The range of increase of crack width depends due to differences in the reinforcement ratio, distance between the bars, shrinkage of concrete and state of cracking at applying the long-term or repeated loads. Repeated loads causes faster increase in crack width compared to long-term loads on same level and time interval [1].

Singly reinforced beams show a higher increase in deflections than doubly reinforced sections, which indicates that reinforcement ratio of compression reinforcement should be taken into consideration [1].

Simplest approach for long-term deflections is the hypothesis for linear relation between the long-term deflection at and the initial deflection a0 within the range of service loads for sections with or without the appearance of cracks, independently from the level of the load [1].

Also for the long-term cyclic dependent deflection an linear relation can be established in ratio to the initial deflection a0 [1].


4 Stress-strain relations for concrete and reinforcement 37

4. Stress-strain relations for concrete and reinforcement 4.1 Stress-strain relation for concrete at the short-term compressive load

The behavior of concrete is defined with stress-strain diagrams, and till today many relations were proposed (figures 4.1 and 4.2) (Ros, 1950; Hognestad, 1955; Smith and Young, 1956; Kriz and Lee, 1960, Desayi and Krishnan, 1982; Carriera and Kuang-Han, 1964; SInsha, 1964; Popovics, 1973; Wang, Shah and Naaman, 1978; Vangyset, 1996; Wee, Chin and Mansur, 1996; Tasnimi, 2004). The existence of long list of formulas corresponds to the fact that defining simple equation that will correspond to experimental results is not simple. Many parameters affects on the stress-strain diagram which are related to properties of concrete than to testing method, load or displacement control (Popovics, 1998) [24].

Besides load level shape of the stress-strain diagram depends on the size and form of testing samples, size and position of measurement points and type of load (Clark, 1967; Halasz, 1967; Rusch and Turk, 1959; Rasch, 1962). It also depends on the concrete mix design, age at testing (Helmuth, 1966) and to type and quantity of aggregate [24].

Rational analysis and design of the reinforced concrete structures are based on prediction of concrete behavior through mathematical function of stress-strain relation for short-term loading and relations that define long-term loading.

Figures 4.1 Typical stress-strain diagrams at uniaxial compressive test for different types of concrete compared to BS8110 [4]

The diagram stresses-strains at short-term loads of one axial compressive load show that the concrete behaves approximately linear to the level of (30-40%) of the ultimate load. Over this point comes to distortion of the diagram due to the two-phase system in the concrete (hardened cement paste and aggregate which have individually have approximately linear relation of stress-strain diagrams Johnson,1928; Larue,1958; Johnston,1970; Hobbs,1969). If these diagrams are combined they give one non-linear diagram. Occurrence of cracks between cement matrix and aggregate is reason for discontinuity which leads to failure of the specimen. Additional contribution to failure comes from intensive appearance of micro cracks in mortar and appearance of certain creep (Ross et al, 1968; L’Hermite, 1962) [24].


Figure 4.2 Typical stresses-strains diagrams at uniaxial compressive testing according to various researchers [4]

The non-linear behavior of concrete in post-elastic area, especially after point reaching maximum stress makes this relation very sensitive. This is actually the cause for the existence of many models. The concrete at effect of short-term loads can be considered to behave linear elastic to certain level of the stresses in ratio to the ultimate stresses. Elastic behavior of the concrete is defined with two basic parameters: modulus of elasticity and Poisson’s coefficient.

In continuation the mathematic functions of the dependency between the stresses and the strains which are mostly used in the structural design according to Eurocode-2 ((EN1992-1) will be given. 4.2 Stress-strains relation for structural analysis according to European standard

EN1992-1-1 EUROCODE 2

The relationship between the stresses σc and strains εc for short-term loads on pressure is shown on figure 4.3 and is given with the equation (4.1) [25]:

Figure 4.3 Stress-strain relation for structural analysis (Use of 0.4fcm for the definition of Ecm is approximate)[25]


2k1

k

f

2

cm

c

[25] ........................................................................................... (4.1)

Where:

η = εc/εct

εct is strain at maximum stresses according to the table 4.1

k = 1.05 Ecm x | εct|/fcm (fcm according to the table 4.1) The expression (4.1) is valid in case of 0 < |εc| < | εcu1| where εcu1 is nominal ultimate strain [25].

For dimensioning of the cross-sections can be used the following relationship of stresses and strains shown on figure 4.4 and given in the equations (4.2) and (4.3) [25]:

n

2c

ccdc 11f

for 2cc0 [25] .......................................................... (4.2)

cdc f for 2cuc2c [25] .............................................................................. (4.3) Where is:

N is exponent according to the table 4.1

εc2 is strain at maximum strength according to the table 4.1

εcu2 is ultimate strain according to the table 4.1

Figure 4.4 Parabola-rectangle diagram for concrete under compression [25] Other simplified stress-strain relations can be used beside parabola and rectangle relation like the bilinear diagram [25].


Table 4.1 Strength and deformation characteristics for concrete [25]

Strength classes for concrete

fck (MPa)

12 16 20 25 30 35 40 45 50 55 60 70 80 90

fck,cube (MPa)

15 20 25 30 37 45 50 55 60 67 75 85 95 105

fcm (MPa)

20 24 28 33 38 43 48 53 58 63 68 78 88 98

fctm (MPa)

1.6 1.9 2.2 2.6 2.9 3.2 3.5 3.8 4.1 4.2 4.4 4.6 4.8 5.0

fctk,0.05 (MPa)

1.1 1.3 1.5 1.8 2.0 2.2 2.5 2.7 2.9 3.0 3.1 3.2 3.4 3.5

fctk,0.95 (MPa)

2.0 2.5 2.9 3.3 3.8 4.2 4.6 4.9 5.3 5.5 5.7 6.0 6.3 6.6

Ecm (GPa)

27 29 30 31 33 34 35 36 37 38 39 41 42 44

c1 (‰)

1.8 1.9 2.0 2.1 2.2 2.25 2.3 2.4 2.45 2.5 2.6 2.7 2.8 2.8

cu1 (‰)

3.5 3.2 3.0 2.8 2.8 2.8

c2 (‰)

2 2.2 2.3 2.4 2.5 2.6

cu2 (‰)

3.5 3.1 2.9 2.7 2.6 2.6

n 2 1.75 1.6 1.45 1.4 1.4

c3 (‰)

1.75 1.8 1.9 2.0 2.2 2.3

cu3 (‰)

3.5 3.1 2.9 2.7 2.6 2.6


4.3. Stress-strain relation at short-term tension load

Tension stresses in the structures are caused by restrained shrinkage, drying shrinkage and by temperature changes. The cases of appearance of one axial tension are very rare in reinforced concrete structures except at action of seismic forces.

Concrete has small tension strength due to the limited interface between aggregate and cement paste, the existence of micro cracks and voids which are formed at hardened concrete. Usually the tension strength of concrete is between 8-15% of the compressive strength [26].

Therefore at design of reinforced concrete structures tension of concrete is neglected and the design is based on the theory of elasticity and plasticity. But the design can’t be rationally carried out using conventional methods for elements where cracks are developed. Therefore it is necessary to take into consideration properties that define the zone where crack is formed. The practical meaning of concrete tensile strength is that tensile strength has a fundamental role in the fracture mechanics of hardened concrete [26].

For testing of tensile strength several methods have been recommended that depend on type of stresses, distribution of stresses and calculation of stresses. Direct way of testing of tensile strength has certain difficulties and imperfections, so more often indirect methods as the tensile flexure strength and tensile splitting strength were used.

Stress-strain diagram obtained during uniaxial tension test has two specific points. After reaching maximum tension strength point we have gradual decrease due to the loss of the tensile strength which leads to decrease tension stiffness and longer part of the curve with crack width increase effect. This means that the concrete is not a completely rigid material by applying tension stresses, i.e. but possibility for constant transfer of stresses between the aggregate interlock zone appears at crack opening (figure 4.5).

During the tension test, fracture process zone does not establish itself instantaneously right across the specimen as the initial micro cracking activity is dispersed; the deformation becomes localized at fracture process zone after maximum load has been reached. Because the crack opening can’t be measured directly it must be derived from the total deformation consist of elastic deformation δe and any prepeak inelastic deformation δine according to figure 4.5 [26].

Figure 4.5 Procedure to derive an σ-w relation obtained from a deformation controlled uniaxial tension test [26]

The tension softening diagram expresses the relation between stresses and crack opening. The area enclosed by this tension-softening curve with horizontal axis is exactly the fracture energy GF defined by equation as follow [26]:

cw

0nF dwwG [26] ............................................................................................. (4.4)


4.4 Time-dependant stress-strains relations of concrete

Creep and shrinkage of concrete are complex phenomenon and a constitutive equation which will be applicable to define time-dependant stress-strain relation is difficult to formulate. Creep of concrete has nonlinear behavior even at short-term loading. Numerical creep analysis may be possible if integral-type from the preceding section are converted to a differential-type form consisting of a system of first order ordinary differential equations in time.

Most frequently the time-dependant stress-strain relation is expressed with compliance functions or with creep coefficient using the principle of linearity between stresses and strains at service load of structure. The calculation of creep caused by variable stress is facilitated by the principle of superposition. This principle is equivalent to the hypothesis of linearity of the constitutive equations that histories are the sum of responses to each of them taken separately.

For the analytical formulation of stress-strain relation the models of sum and the models of product were used. 4.4.1. Compliance function

If we consider only creep at constant stress using the principle of linearity between the stresses and strains at service stress range up to 40% of compressive strength, thus [27]:

'0 t,tJtt [27] ..................................................................................... (4.5) Where:

σ - is uniaxial stress

ε - axial strain

t - time which presents the age of the concrete

J(t, t’) is the compliance function; this function represents the strain (elastic plus creep) at time t caused by a unit constant uniaxial stress that has been acting since time t’. In the linear range, the creep at uniaxial stress is completely determined by function J(t, t’). Typical shape of this function is given on figure 4.6. The compliance function is often expressed as a sum of the elastic compliance and creep compliance [27]:

'

''

''

tE

t,t1t,tC

tE

1t,tJ

[27] .................................................................. (4.6)

An important property of the compliance function of concrete is that is a function of two variables, the current age t and the age at loading t’. That is main characteristic of the concrete that the compliance function cannot be considered as a function of one variable i.e. the time difference t-t’ as it is in classical viscoelasticity. The aging of concrete is a considerably obstacle to analytical solution of structural problem and necessitates that most real problems have to be solved by numerical methods [27].

As a consequence of shrinkage and creep, the stress in all structures varies with time even if the load is constant.


Figure 4.6 Creep isochrones and compliance curves for various ages t’ at loading [27] The calculation of the creep caused by variable stresses is greatly facilitated by the principle of superposition. This principle is usually assumed to apply to concrete within the service stress range. In case of variable stresses following equation can be obtained [27]:

tdt

,tJt,tJtt,ts

t

t000

0

[27] ............................................. (4.7)

εs(t) – strain which is not a result of stresses (shrinkage and/or thermal strain)

J(t,t’) – creep function or compliance function – total strain of single stress In the equation (4.7), the member σ(to) J(t, to) gives the assumed linearity of the creep strains at constant stresses in time as follows [27]:

dJ

)(

)(t,t

to

[27] ........................................................................................... (4.8)

This represents the principle of superposition in case of variable stresses.

The compliance function contains elastic (instantaneous) specific strain 1/E(to) and specific creep strain usually presented with the creep coefficient φ(t, to). If Eb(t) = Ebo than:

)]t,t(1[E

1t,tJ o

b0 [27] ................................................................................. (4.9)

And the equation (4.7) is as follows:


(t)d)(

s

t

to

o

})],t(1[)]t,t(1[{E

1t,t

bob

b0 [27] ...................... (4.10)

It is more convenient to define the compliance function by a formula. Between the simplest formulas for the compliance function, at constant humidity and temperature, are power curves of load duration t0-t and by inverse power curves at age t0 at loading [27].

Among simple formulas is the double power law (Bazant 1975, Bazant and Osman 1976, Bazant and Panula 1978) [27]:

n'm'

0

1

0

' tttEE

1t,tJ

[27] ................................................................. (4.11)

Where approximately n=1/8, m=0.05; ø1=3-6; Eo=1.5xEb28 [27].

The power law of load duration first suggested by Straub, 1930 and Shank 1935 follows theoretically certain hypothesis about micro structural creep mechanism called the rate process theory by Wittmann, 1971 and 1974, as well as a statistical model of creep mechanism Cinlar et. Al 1977 or some micro mechanics models (Bazant, 1979) using conventional modulus of elasticity 1/E instead of 1/Eo. This definition of the elastic term restricts the range of applicability [27].

For load duration that is long, the Logarithmic law predicts more real concrete creep effects. The equation of the logarithmic law is proposed by Hanson in 1953 and is given as follows [27]:

'0

n'm'1

10

' tftlntt1Et,tJ [27] ................................................... (4.12) Also many other formulas for the compliance function were proposed as hyperbolic function of Ross (1937) and Lorman (1940) given with the following equation [27]:

'' ttt,tba/tt,tC

[27] ......................................................................... (4.13)

This equation is convenient for fitting of test data but does not apply to long creep durations. In this type of equations also belongs the Dischinger formula (1937) [27]:

'' ttaexp1t,tC [27] ............................................................................. (4.14) Careful studying shows that the double power law exhibits certain deviations from experimental results which seem to be systematic errors so improvement can be obtained by triple power law Bazant and Chern, 1985. This formula specifies the unit creep rate i.e. the time derivative of the compliance function [27]:

n'n1'

m'

0

1''

t/ttt

t

Et,tJ

t

t,tJ

[27] ......................................................... (4.15)

In this equation we have one coefficient more ψ1, than the double power law [27].

With integration of this creep function we obtained the triple power law [27]:

)]n;'t,t(B)'tt)[('t(EE

1)'t,t(J nm

o

1

o

[27] ......................................... (4.16)


'tt;d])'t

't(1[n)n;'t,t(B 1nn

'tt

0

[27] ............................................ (4.17)

Function B(t, t’; n) represents the deviation from the double power law, which is binomial integral which can’t be expressed in a closed form for realistic values of the exponent n. The integral may be easily evaluated in terms of convergent power series or by step by step integration in logarithm time scale. The numerical evaluation of the binominal integral is easy, its use may be avoided by another formula which represents a smooth transition from the double power law to the logarithmic law log-double double power law) [27]:

])'tt)('t(1ln[EE

1)'t,t(J nm

1o

o

o

[27] .............................................. (4.18)

Where: ψo=ø1/ψ1

All formulas for predicting of the creep function are presented with age at loading and the time duration of the load in form of product, i.e. the creep for a given period of loading can be predicted as product of creep coefficient of creep which depends on the concrete age at loading and function which describes the development of the creep in time.

As alternative to this method, creep can be described also with the formulation of sum where the creep strain is a sum of delayed elastic and viscose strains. This method was developed due to the research by McHenry (1943), Masloy (1941), Arutyumian (1952), Bresler and Selna (1964), Selna (1967, 1969) and Makaddam and Bresler (1972, 1974) which use sum of exponentials of t-t’ with coefficient which depend on t’. This sum can be easily adapted to any result of the testing if sum of at least four exponential terms exist. Different expressions are used for special assignments to enabled simplification of the method for structural analysis of the creep. They include the expressions of Whitney (1932), Glanville (1930), Dischinger (1937, 1939), England and Illston (1965) and Nielsen (1970, 1977) which lead to development of the methods like: Classical theory of aging (Rate of creep) or improved method of Dischinger and Method of creep velocity (Rate of flow method) but also and other expressions by Chiorino and Levi, 1967, Arutyunian 1952, Levi and Pizzeti 1951.

It’s obvious that neither of these two solution approaches gives correct answer for every possible case which can appear in the real structure during the service life. Main reason for the errors in the models is the non-linear phenomenons of the concrete creep which can be precisely modeled only with apply of the non-linear solution approach. Additional errors of the concrete creep models are that concrete creep is closely related to the shrinkage and the elastic strain. Therefore scientific discussions continue on the dilemma which linear model is the best linear model for predicting creep of concrete. 4.4.2 Principle of superposition

In the framework of the service stress range which are 50% of compressive strength, the creep law generally presents linear relation between the stress history σ(t) and strains. The linear hypothesis agrees very well with test results in the case of basic creep (creep at constant humidity conditions), but has a significant error in case of drying, which is neglected.

The linearity, first, means that at constant uniaxial stress σ applied at age of concrete t’, the corresponding strain ε(t) at any time t≥t’ may be written as [14]:

't,tJt [14] ............................................................................................... (4.19)


Where J(t, t’) is the uniaxial compliance function, creep which characterizes property of material. The typical curves of compliance J versus current age t at various constant values of the age at load application t’ are shown in figure 4.6.

The equation (4.19) reflects the proportionality of the answer of the material [14].

The calculation of the concrete creep at variable stresses is related to use of the principle of superposition. The principle of superposition is equivalent to the linearity hypothesis so the total stress (or strain) is a sum of the separate stress histories applied at various times. That means that if the stress history σa(t) corresponds to the strain history εa(t) and stress history of stress σb(t) corresponds to the history of strain εb(t) then [14]:

ttt ba i.e. ttt ba [14] ...................................................... (4.20) This property of the material represents the principle of superposition in one time interval, which was proposed in general for non-aging phenomena by Boltzmann (1974), and at the aging phenomena by Volterra (1909) [14].

Every stress history can be decomposed into infinite small steps, dσ(t’), applied at various times t’ lesser than the current time t. According to equation (4.19) each step history causes infinite small strain histories dε(t)=J(t,t’)dσ(t’) (figure 4.7 (a)). Summing all these infinite small contributions and adding the initial strain ε0(t) it is obtained [14]:

t

ttdttJt0 0'', [14] .......................................................................... (4.21)

The initial strain ε0 generally includes not only shrinkage strain εsh, but also the thermal strain and cracking strain. Only the shrinkage strain will be considered here, in which case ε0(t)= εsh(t) [14].

The equation (4.21) is not only a consequence of the principle of superposition but it is also equivalent of alternative statement.

The integral in the equation (4.21) is not standard (Riemann) integral but a generalized type of integral called Stieltjes integral, in which the continuous history of stresses were admitted. If there is jump from σ- to σ+ in time t0 then the contribution of the integral (4.21) of the jump is (σ+-σ-)J(t,t’), which follows from (4.21) by integration of the increments dσ(tj) from σ- to σ+ at constant time tj [14].

For the frequent assumed cases that the concrete remains stress-free until at some time t1, the finite stress σ1 is applied suddenly (by a jump) and subsequently the stress σ(t) varies continuously, (4.21) can be written in terms of Riemann’s integral as follows[14]:

t

ttdt

dt

tdttJttJt

1011 '

'

'',, [14] ................................................... (4.22)

Where: t1

+ denotes the beginning of integration just after the initial jump.

When the strain history ε(t) is prescribed, then (4.21 or (4.22) represents integral equation for the stress history σ(t). This is well known type of Volterra integral equation, a special case of integral equations of second order. The compliance function J(t,t’) is called the kernel of the integral equation [14].

According to this principle, the strains caused by the stress history σ(t) can be calculate by its decomposition on small increments dσ(t’) applied at time t’ and superposition (4.22) of separate strains [14]:


t

0

(t)εt')J(t,t')dσ(ε(t)0

[14] ............................................................................... (4.23)

Figure 4.7 Decomposition of history of stresses on steps of stresses (left) and impulses of stresses (right) [27]

The total load is divided on certain time intervals steps whose length increases in time. The notations which are used at this step-by-step analysis are shown on picture 4.8.

For accurate results of the analysis at continuously variable load (stresses) time intervals ∆tj, should be selected to be with approximately equal length on the logarithmic scale of time. In the literature several different formulations of step-by-step method can be found. Here will mention two [13]:

Ghali et al – the increment of stresses Dσj which is applied in the middle of the time interval (time tj). The elastic component in that time step is calculated with E(time tj) and the creep begins in time tj [13].

Bazant and Najjar – designed trapezoidal rule. If relatively big time steps are used than Simpson’s rule should be applied for greater accuracy. According to this total strains at end of the jth interval are equal to the sum of the strains from the increase of the stresses ∆σj, obtained at the time of all previous steps [13].

The principle of superposition gives accurate predictions under the following conditions [13]:

1. Stresses are within the service stress range i.e. less than 0.4 of the strength

2. Unloading, i.e. decrease of the strains

3. There is no significant change in moisture content distribution during creep

4. There is no large sudden stress increase long after the initial loading

Figure 4.8 Defining of time intervals for step-by-step analysis [13]


4.4.3 Actual models for designing the shrinkage and creeping of the concrete

In practice the analysis of the concrete creep is carried out with use of creep coefficient φ(t,t0) which represents the ratio between the creep strain and the instantaneous elastic strain [1]:

)t(

)t,t()t,t(

oce

occo

[1].......................................................................................... (4.24)

)t(

)t,t()t,t(

28ce

occo

[1].......................................................................................... (4.25)

According to Bazant (Creep and Shrinkage of Concrete – Mathematical Modeling, 1986) this formulation of the creep (through the creep coefficient) is inappropriate because often leads to a combination of incompatible values of the modulus of elasticity Ec(t0) and the creep coefficient φ(t,t0). Therefore it is recommended in the calculations to be used experimental established or predicted value of creep function J(t,t0) through which the total strain is taken into consideration [27]. 4.4.4 Model B3

This model was developed by Z.P. Bazant ans S. Baweja (1995), which is based on [28]:

a) Existence of many experimental data of testing shrinkage and creep of concrete

b) Compilation of a computerized data bank

c) Development of computerized statistical methods for curve fitting and optimization

d) Improved understanding of the physical processes involved in shrinkage and creep of concrete, aging, diffusion process, thermally activated processes, micro cracking and their mathematical modeling

The model is in accordance with the general recommendations formulated by RILEM TC-107. Coefficients of variation of the errors of the predictions of shrinkage and creep are much smaller than CEB-FIP Model Code 1990 and American Institute for concrete ACI209 model.

The model B3 is restricted to the service range up to about 0.45fc where fc is compressive strength of cylinder at age of 28 days. Prediction of material properties strength and mix-design were restricted to use of Portland cement concrete with following parameters [28]:

85.0c/w35.0 [28] ...................................................................................... (4.26)

5.13c/a5.2 [28] ........................................................................................ (4.27)

3m/kg720c160 [28] .................................................................................. (4.28)

MPa70fMPa17 c [28] .................................................................................. (4.29) Where is:

w/c – water-cement ratio

a/c – aggregate-cement ratio by weight

c – cement content in kg/m3

fc – cylinder compressive strength at age of 28 days


Formulations in the model are also valid for concretes cured at least one day [28].

In the model, calculation of strains at constant stress applied at age of the concrete t’ can be calculated according to the following equation [28]:

tTt't,tJt sh [28] ..................................................................... (4.30) Where is:

J(t’t) – compliance function equal to the strain (elastic and creep strain) at time t caused by a unit uniaxial constant stress applied at age t’

σ –uniaxial stress

ε – strain

εsh – shrinkage strain

∆T(t) – temperature change from reference temperature at age t

– thermal expansion coefficient The compliance function further is decomposed on [28] :

0'

d'

01' t,t,tCt,tCqt,tJ [28] ................................................................. (4.31)

Where is:

q1 – instantaneous initial strain due to unit stress

C0(t,t’) – Compliance function for basic creep (creep at constant moisture content and no moisture movement through the material)

Cd(t,t’,t0) – additional compliance function for creep due to simultaneous drying

The creep coefficient (t,t’) which represents most convenient method for to introduce creep into structural analysis should be calculated from compliance [28]:

1t,tJtEt,t ''' [28] .................................................................................. (4.32) E(t’) – is static modulus of elasticity loading age t’ 4.4.4.1 Calculation of creep and time dependent strain components

The basic creep compliance function is defined by its time rate than its value [28]:

t

q

tttt

qtqnt,tC 4

n1''

3m

2'0

at to 1.0n,5.0m [28] ................................ (4.33)

Where:

t/t,tCt,tC '0

'

[28] .................................................................................... (4.34) t and t’ – time in days, m and n are empirical parameters whose value can be taken the same

for all normal concretes

q2, q3 and q4 – are empirical constitutive parameters Total basic creep compliance is obtained by integrating of equation (4.33) as follows [28]:


'4

n'3

'2

'0

t

tlnqtt1lnqt,tQqt,tC [28] ....................................... (4.35)

Where Q(t,t’) can be calculated from an approximate explicit formula in equation (4.36). The function Q(t,t’) can be easily obtained by numerical integration. The use of approximate formula (4.36) (developed by Bazant and Prasannan, 1989) have smaller error of 1% for m=0.5 and n=0.1 [28].

'' tr/1

tr

'

'f'

f'

t,tZ

tQ1tQt,tQ

[28] .......................................................... (4.36)

Where is:

8t7.1tr12.0'' [28] ..................................................................................... (4.37)

n'm'' tt1lntt,tZ [28] ....................................................................... (4.38)

19/4'9/2''f t21.1t086.0tQ

[28] ........................................................... (4.39)

4.4.4.2 Average shrinkage and creep of cross sections at drying

A) Shrinkage

tSkt,t hsh0sh [28] .................................................................................. (4.40) Where time dependence is given by following equation [28]:

sh

0tttanhtS

[28] ....................................................................................... (4.41)

Humidity dependence is given as follows [28]:

1h0.98 for ioninterpolat Linear

water) in (swelling 1h for 2.0

98.0h for h1

k

3

h [28] .................... (4.42)

Size dependence is given by following expressions [28]:

2stsh Dkk [28] ............................................................................................. (4.43) Where is:

v/s – volume to surface ratio of the concrete element

D=2v/s – effective cross-section thickness which coincides with the actual thickness in the case of slab

kt – is factor defined with the equation (4.54)

ks – is cross-shape factor: .


cube a for 55.1

sphere a for 30.1

prism square infinite an for 25.1

cylinder infinite an for 15.1

slab infinite an for 00.1

ks [28] ........................................................ (4.44)

For more simple analysis when high accuracy is not needed it can be assumed ks1 [28].

Time-dependence of total shrinkage [28]:

sh0

ssh tE

607E

2/1

t85.04

t28EtE

[28] ...................................... (4.45)

Where:

εs∞ – is a constant given by equation (4.53). This means that εs∞ =εsh∞ for t0=7 days and sh=600 days B) Additional creep due to drying (drying creep)

2/1'050

'd tH8exptH8expqt,t,tc [28] ............................................ (4.46)

0''

0 t,tmaxt If t0’ is the time at which drying and loading for the first time acting simultaneously [28]:

tS)h11tH [28] .................................................................................... (4.47)

4.4.4.3 Prediction of model parameters

The model parameters are estimation from concrete strength and composition [28]. A) For basic creep

286

1 E/106.0q [28] ....................................................................................... (4.48)

c28 f4734E [28] ............................................................................................. (4.49)

9.0c

5.02 fc4.185q [28] ....................................................................................... (4.50)

24

3 qc/w29.0q [28] ...................................................................................... (4.51)

7.04 c/a3.20q [28] ....................................................................................... (4.52)

B) For shrinkage

)10(270fw109.1 628.0c

1.2221s

[28] ............................................. (4.53)

4/1c

08.00t ft5.8k (days/cm2) [28] ........................................................................ (4.54)

00.11 for cement type I [28] .......................................................................... (4.55)

85.01 for cement type II [28] ........................................................................ (4.56)


10.11 for cement type III [28] ........................................................................ (4.57)

75.02 for steam curing [28] ........................................................................ (4.58)

20.12 for sealed or normal curing in air with initial protection against drying [28] ........................................................... (4.59)

00.12 for curing in water or at 100% relative humidity [28] ......................... (4.60) C) Drying creep

6.0sh

1c

55 f1057.7q

[28] ......................................................................... (4.61)

Model B3 allows updating of creep prediction on the basis of short-term measurements of creep defining two update parameter p1 and p2 as follows [28]:

'211

' t,tFppqt,tJ [28] .......................................................................... (4.62) 0

'd

'0

' t,t,tCt,tCt,tF [28] ....................................................................... (4.63)

According to well-known normal equations of at least square linear regression [28]:

2i

2i

iiii2

FFn

JFJFnp [28] ....................................................................... (4.64)

1

21 q

FpJp

[28] .............................................................................................. (4.65)

Where:

J -mean of all measured Ji values

F -mean of all corresponding Fi values. Updating of shrinkage predictions can be also defined on the basis of short-term measurements of shrinkage by parameter p6 as follows [28]:

i2shi

shii'shi

6p

[28] .......................................................................................... (4.66)

shi -shrinkage values calculated from model B3 'shi -measured short-time values of shrinkage

The model B3 can be used and for special types of concrete such as high strength concretes and fiber reinforced concretes if the parameters were calibrate by testing on short-term loads. Due to the autogenous shrinkage of the high strength concretes it is necessary total shrinkage to be obtained by following equation [28]:

0sha0totalsh t,ttt,t [28] .......................................................................... (4.67)

Where is:

εa –autogenous shrinkage strain

εsh – drying shrinkage strain


The value of the autogenous shrinkage strain can be calculated according to the following equation [28]:

tSh99.0t aaaa [28] ......................................................................... (4.68)

a

sa

tttanhtS

[28] ...................................................................................... (4.69)

Where is:

ts – time of final set of cement

εsh – drying shrinkage

a – half time of autogenous shrinkage

ha∞ –final self-desiccation humidity (it may be assumed to be about 80%) 4.5 Stress-strain relation for reinforcement

In structural analysis of reinforced concrete sections, besides stress-strains relation for concrete it is necessary stress-strains relation for reinforcement to be defined.

The behavior of the reinforcement is defined by following properties:

yield strength (fyk or f0.2k)

maximum actual yield strength (fy,max)

tensile strength (ft)

ductility (uk and ft/fyk)

fatigue strength

bond characteristics 4.5.1 Relationships between stresses and strains in the reinforcement according to EN 1992-1-1 EUROCODE-2

The application rules for design and detailing in this Eurocode are valid for a specified yield strength range fyk=400-600MPa. On figure 4.9 typical stresses-strains diagrams were shown for hot rolled steel and cold worked steel [25].

For usual designing of reinforced concrete sectios, are used the presumptions for behavior of the reinforcement accordingly to the picture 4.10 [25]:

a) An inclined top branch with a strain limit of εud and a maximum stress of kfyk/γs at strain εuk, where k=(ft/fy)k [25]

b) Horizontal top branch without the need to check strain limit [25]

Recommended value for εud=0.9uk [25].

Value of the characteristic strains under the effect of maximum force εuk(%) for reinforcing steel class A, B and C is ≥2.5; ≥5; ≥7.5% respectively [25].


Figure 4.9 Stress-strains diagrams for typical reinforcing steel: a) hot rolled steel; b) cold rolled steel [25]

Figure 4.10 Idealized and designed stress-strain relation for reinforcement (At tension and at compression) [25]

5 Models for structural analysis of the creep 55

5. Methods for structural analysis of the creep

For the analysis of the structural creep algebraic methods and simplified compliance functions were used. Algebraic methods are: Effective modulus method and Age adjusted effective modulus method. Methods with simplified creep functions are: Rate of creep method, Rate of flow method, the improved method of Dischinger, Arutyunyan etc. Below principal characteristics of the Age adjusted effective modulus method will be represented. 5.1 Effective modulus method (EMM)

The oldest simplified method is the effective modulus method (McMillan, 1916) in which the solution for the creep at time t is obtained by elastic structural analysis based on the so called effective modulus [14]:

't,t1

'tE

't,tJ

1Eef

[14] ............................................................................. (5.1)

If the stress in time is constant in the considered time interval the method gives accurate results. On the other side if the stress varies during the time, causes variations also in the calculation. Less variation in the strains produces less error in the design. As a rule if the variations in the strains don’t exceed the value of 15-20%, in ratio to the initial stresses, the error that is made at prediction of the strains can be neglected [14].

The error can be very large especially for long-time response of structures loaded at early age of concrete (Bazant and Naijar, 1973) [14]. 5.2 Age adjusted effective modulus method (AAEMM)

The Age adjusted effective modulus method is formulated for a one step loading history, i.e. for load that is applied in time t, which is constant until current time t or varies monotonically at a gradually decreasing rate. The response to multi step load histories can be obtained by superposing the solutions for several one-step histories. While the effective modulus method takes one step from the unstressed state of the structure at time t1

- before the first loading to the current state at time t, AAEMM takes one step from the initial stressed state at time t1

+ after application of the load to the current state at time t. Hence, the initial state just after the loading, which doesn’t have any importance in the effective modulus method, must be calculated separately, based on the standard elastic analysis of the structure on the initial modulus of elasticity E(t1) [14].

In the AAEM method history of stress and strain between the initial and the current state is approximated by a linear combination of creep at constant stress and relaxation at constant strain. The strain α, which is applied in time t1 and kept constant, causes stress history (t)=R(t,t1), where R is a relaxation function. Also, the stress β, applied in time t1 and kept constant causes strain history (t)=J(t,t1), where J is the compliance function. In accordance to the principles of superposition the stress and strain histories are [14]:

1t,tJt [14] ........................................................................................ (5.2) 1t,tRt za(t≥t1) [14] ........................................................................... (5.3)

Corresponding to each other (i.e. satisfy the viscoelastic constitutive equations). The coefficients α and β can be expressed in terms of the stress value (t1

+)=1, just after the application of the load, and (t)=1+, at time t. Substituting these values in (5.3) and calling that R(t1,t1)=1/J(t1,t1)=E(t1)=elasticity modulus at age t1, we obtain two linear equations for α and β [14]:


11tE [14] ............................................................................................ (5.4)

11t,tR [14] ............................................................................... (5.5) From which is possible to determine the coefficients:

11 tEt,tR

[14] ....................................................................................... (5.6)

111

1

t,tRtE

tE

[14] ............................................................................... (5.7)

With the use of this result the initial (elastic) strain can be obtained from equation (5.2) in :

1

1

11111 tEtE

t,tJt [14] ............................................ (5.8)

And the strain increment [14]:

111

11

111111 tE

1t,tJ

t,tRtE

1t,tJtEt,tJt,tJtt (5.9)

The expression E(t1)J(t,t1)-1 is a creep coefficient (t,t1). If we introduced the so called age adjusted effective modulus (Bazant, 1972) [14]:

1

11

11

111

''

t,t

t,tRtE

1t,tJtE

t,tRtEt,tE

[14] ................................................ (5.10)

And replace 1/E(t1) by 1 in accordance to equation (5.8) and equation (5.9) assumes a convenient form [14]:

111

t,tt,t''E

[14] ........................................................................... (5.11)

This is the basic equation of the AAEM method in which the increment of strain increase over the interval (t1,t) is equal to the increment of stress divided by the effective modulus plus the initial (elastic) strain multiplied by the creep coefficient [14].

To keep the previous derivation simple, we have neglected the shrinkage effect. Anyway the shrinkage effect can be automatically incorporated by replacing of (t) with the mechanical strain: 0(t)=(t)-sh(t) which means that the increment of shrinkage strain must be added to the right side of the equation: (5.11). The final form of the basic equation of AAEM (in case of uniaxial stress) is [14]:

tt,t t

t,t''E

tt sh11

1

1

11 tE

tt

[14] ........................... (5.12)

Where:

1shshsh11 ttt ,ttt ,ttt ...(5.13)


For simple use, the age adjusted effective modulus, whose primary definition is the equation (5.10) can be expressed in the following form [14]:

11

11 t,t t,t1

tEt,t''E

[14] ..................................................................... (5.14)

Which represents a adjustment of the effective modulus. This equation has advantage because the so called aging coefficient [14]:

111

11 t,t

1

t,tRtE

tEt,t

[14] .............................................................. (5.15)

varies relatively little, usually from 0.5 to 1.0 with 0.8 as most typical value [14]. The value =1 characterizes the limiting state of a non-aging material. Certainly if there is no aging, such as in the case of short-term creep of if concrete is loaded at old age, the optimal value of is close to 1 (around 0.992) and E" is approximately equal to Eef. This fact (which can’t be true if the exponent n in the creep formula of the model B3 ((Bazant, Baweja, 1996 and 2000) is much bigger than 0.1, for example 0.5) shows that is calculated principally for the effect of aging [14].

Tables of table, computed for certain compliance functions, have been included in ACI Committee 209 design recommendations. In order to avoid computer solution of the relaxation function R(t,t') for a given compliance function J(t,t') one may use the following semi-empirical approximate formula which correct asymptotical properties (Bazant and Kim, 1979) [14]:

2

tt ,1

t,tJ

t,tJ

1t,tJ

115.0

t,tJ

992.0t,tR 1

1

1

11

[14] ........... (5.16)

Figure 5.1 Strain histories for which AAEM is exact [14] For ordinary concretes, the error of this formula (recommended for calculation in CEB-FIP Model Code 1990, equations 5.8-7 and 5.8-3) is generally under 1%; and t and t1 must be given in days [15].

The basic equation (5.12) of AAEM is accurate for all the strain histories representing a linear transformation of the creep coefficient curve, i.e.


11sh tt t,t batt ...................................................... (5.17)

Where a and b are arbitrary constants. This includes wide range of strain histories shown on figure 5.1, for which AAEM is exact. The actual strains histories in the structures under permanent load are frequently well approximated with the equation (5.17). This equation gives satisfying accuracy of AAEM in wide range of the problems. 5.2.1. Practical calculation procedure

Basic assumptions included in AAEMM are [1]:

If the tensile stress in the concrete doesn’t exceed the strength fct, the section is uncracked

If the section is cracked, the concrete in tension is neglected

The hypothesis of superposition and linearity of the creep are applicable

In service condition, steel has a linearly elastic behaviour characterized by elasticity modulus Es

Plane sections before loading remain plane after loading

The stress diagram in the uncracked concrete is linear If we take into consideration section shown on figure 5.2. The x and y axes have their origin at the top level of the axis of symmetry. Asi denotes the area of the ordinary reinforcement and of the prestressing steel that are set at levels ysi. It is necessary to assume that the load effects remain constant over time t0-t1 [1].

If the stress-strain diagrams at the time t0 (i.e. the initial elastic solution due to bending moment M and normal force N) in the concrete accordingly to the figure 3.1 can be assumed that [1]:

ytr1tt 00c0 [1] .............................................................................. (5.18)

0n0c0 ty

y1tt [1] ........................................................................... (5.19)

Where c(t0) and c(t0) are strain and stress at the top level of concrete section respectively, and yn(t0) is the depth of the neutral axis. T final time t1 the significance of the symbols is the same and we obtain [1]: for strain ytr1tt 11c1 [1]...........................................................(5.20)

for the stress in the concrete

1n1c1 ty

y1tt [1]...........................................(5.21)

while in the reinforcement 01rel0s1ss01s ttttEtt [1].......(5.22)

Where rel(t1,t0) is relaxation in the strength of the previous stress [1].


Figure 5.2 Development of the strains and stresses in the cross-section for time t0-t1[1] Concrete strain affected by creep with creep coefficient (t1,t0), and by shrinkage cs(t1,t0) is assessed by means of AAEM method. Thus [1]:

01cs28,c

0101

0c01

28,c

01001 t,t

E

t,tt,t

tE

1tt

E

t,tttt

... (5.23)

For concrete fibres that are always under compression over the time interval being considered, can be applied the AAEM. But this can't be applied for concrete subjected to the gradually increasing compressive stresses produced by the displacement of the neutral axis from yn(t0) to yn(t1). The problem becomes much more complex and calls for an iterative procedure and has little importance on bending behaviour. If the load effects are constant for the equilibrium equation can be written [1]:

0dAtt 01A [1] .............................................................................. (5.24)

0dAytt 01A [1] .......................................................................... (5.25)

With the substitution of the equations (5.19), (5.21), (5.22) into (5.24) and (525) respectively, the following equation are obtained [1]:

01 tA 00n

0c11n

tA 1c tAdty

y1ttAd

ty

y1t

0t,tAytr1tr1ttAE 01relsii010c1csis [1] ...... (5.26)

0

0ntA 0c1

1ntA 1c tAyd

ty

y1ttAyd

ty

y1t

01

0yt,tAyytr1tr1ttAE i01relsiii010c1csis .. (5.27)

Where 0tA and 1tA are the reacting concrete areas at time t0 and t1 respectively [1].


The strains can be expressed by substituting equations (5.18), (5.19), (5.20) and (5.21) into the equation (5.23) [1]:

28,c

01

0n0c00c11c1 E

t,t

ty

y1tytr1tytr1tt

01cs28,c

0101

0c0n0c

1n1c t,t

E

t,tt,t

tE

1

ty

y1t

ty

y1t

[1] ................................................................................................................................... (5.28)

If we denote the static moment with S and the inertia moment with J , of reacting concrete

area A with respect to the referent axis x by substituting equation (5.28) into the equations (5.26) and (5.27) we obtain a two equations system with two unknowns c(t1) and yn(t1). The numerical solution is significantly simplified by assuming the following modular [1]:

1t,tE

t,t

tE

1Et 01

28,c

01

0cs0e

[1] ............................................... (5.29)

At the initial time t0:

si0e00 AttAtA [1] .......................................................................... (5.30)

sisi0e00 yAttStS [1] ...................................................................... (5.31)

2sisi0e00 yAttJtJ [1] ....................................................................... (5.32)

And at the final time t1:

0128,c

01

1cs1e t,t

E

t,t

tE

1Et

[1] ........................................................ (5.33)

In a similar way at time t0 we can obtain the geometrical characteristics A(t1), S(t1) and J(t1). Finally by setting the constants [1]:

01relsisi01css

0n

000c0 t,tAAt,tE

ty

tStAtN [1] ......... (5.34)

si01relsisisi01css

0n

000c0 yt,tAyAt,tE

ty

tJtStM

[1] (5.35)

The system can be written in the following form [1]:

0

1n

111c N

ty

tStAt

[1] ........................................................................... (5.36)


0

1n

111c M

ty

tJtSt

[1] ........................................................................... (5.37)

From which the parameters of stresses c(t1) and yn(t1) in time t1are derived. The problem is solved in closed analytical form when the section is uncracked, as the geometrical characteristics A(t1), S(t1) and J(t1) are calculated directly. On the other side if in time t1 the section is cracked, the same procedure is valid with reference to the reacting section. Then it is necessary to proceed tentatively by establishing the position of neutral axis, so as to define the reacting section and hence determine A(t1), S(t1) and J(t1). The following step is to determine whether the following condition [1]:

1010

10101n tSNtAM

tJNtSMty

[1] ...................................................................... (5.38)

Inferred from the system of equations (5.36) and (5.37) is satisfied. If it’s not, make new attempts until convergence of the solution is obtained. Finally determine stress c(t1) from the equation (5.36) or the equation (5.37). The strain parameters may be determined from equation (5.28) keeping in mind the principle of identity of polynomials [1]:

01cs28,c

0101

0c0c1c

28,c

010c0c1c t,t

E

t,tt,t

tE

1tt

E

t,tttt

[1]

....................................................................................................................................... (5.39)

28,c

0101

0c0n

0c

1n

1c

28,c

01

1n

0c01 E

t,tt,t

tE

1

ty

t

ty

t

E

t,t

ty

ttr1tr1

. (5.40)

Finally equation (5.22) enables us to calculate the stresses in the reinforcements [1]. This procedure makes it possible to determine the stresses and strains in cracked or uncracked sections subjected to constant load effects over the time t0-t1 [1]. 5.2.2. Effects of variable loads

The variable loads cause increase to the variations in stresses from M and N, modify the stresses and strains due to permanent loads. In uncracked elements the effects can be superimposed as the reacting section remains unchanged. When we have cracked section, this can’t be done due to the displacement of the neutral axis. The problem can be solved in two different ways leading to the same result: one of them is based on the formulation given in the AAEM method and the other one on superposition of the fictitious load effects. In both cases, we start with the stress determined at the end of the first stage at time t1 [1]. 5.2.2.1. Iterative method

With the formulation of the AAEM method the problem can be address to the instantaneous stress variations. In this case, the delayed strains are zero and due to that we set (t2,t1)=0, cs(t2,t1)=0 and rel(t2,t1)=0, and that should meet the following equations [1]:


A 01 NdAtt [1] ............................................................................. (5.41)

A 01 MdAytt [1] ........................................................................ (5.42)

The modular coefficient is 1c

s1e2e tE

Ett and constants N0 and M0, as they are

defined in the equations (3.14) and (3.15), become [1]:

Nty

tStAtN

0n

000c0

Mty

tJtStM

0n

000c0

[1].......(5.43)

The solution to this problem is with iterative procedure based on the equations (3.16) and (3.17) which give yn(t2) and c(t2) from which it is possible to obtained all the other values in with the formulas given in the AAEM method [1]. 5.2.2.2. Superposition of fictitious load effects

Starting from the stress present in the concrete c(t1) and the reinforcement s(t1), the load effect variations M and N give rise to stress variations which are in the elastic field and are represented by the real elasticity modulus of the material at the time being considered and by the modular coefficient e(t1)=Es/Ec(t1) [1].

Figure 5.3 shows the diagram of the stress c(t1) in the concrete and the stress s(t1) in the steel, through the coefficient e(t1) [1].

Due to the influences of the shrinkage and creep, the diagram of the stresses in uncracked concrete continues to be linear, while the diagram relating to steel stresses is no longer linear. The latter diagram must be divided into an initial linear part consisting of the stresses

1c t and 1s t (figure 5.3.b) and the other part including only the stress in the

reinforcement 1s1ss tt (see figure 5.3.c). The stress diagram “b” has as its

resultant N* and its resulting moment M* in respect to the external load-effects reference axis. N* and M* can be instantaneous load effects and summed up with the M and N variations [1].

If we consider the stress diagram arising by the load effects N*N and M*M (linear elastic calculation) (figure 5.3.d) to which we must add the diagram “c” of residual stresses

s in the reinforcement in order to obtained the final stress diagram (figure 5.3e). It can be

noted that the stress variations are not linear (figure 5.3f) due to the variation in the reacting section [1].

The final strains in the section are determined by summing the strains related to long-term actions and to the elastic strains due to instantaneous variation in load-effects. Actually for the extreme fibre of concrete in compression we get [1]:

1c1c2c1c2c tE/tttt [1] ...................................................... (5.44)

While in the reinforcement the strain is elastic one. [1]


Figure 5.3 Effects of variable actions using the method of the superposition of fictitious load effects [1]


.

6 Serviceability limit state design 65

6. Serviceability limit state design

At reinforced concrete structures, conformity criteria to the functional requirements are related to the behavior of the structure in service conditions, i.e. serviceability limit states design. The most significant criteria are: limitation of the maximum stresses in the material at different loading conditions, control of the cracking and control of the deflections. 6.1. Limitation of stresses

Most existing regulations address to the complex problem of the transfer of stresses from reinforcement in tension in the cracked section to the concrete in tension between cracks by introducing the notion of mean curvature (CEB-FIP Model Code 1990, Eurocode-2) or with mean moment of inertia (ACI Committee 435). Problem of bond between reinforcement and the concrete are not analyzed in details. These simplifications allow assessing section conditions solely at the cracked state, generally referred as state 2 or in the uncracked section, referred as state 1. Afterwards we continue with the evaluation of the intermediate state, by assessing the mean curvature or by the mean moment of inertia through simplified formula which either taking into consideration the bond quality in an extremely simplified manner (CEB-FIP Model Code 1990 and Eurocode-2) or neglecting it entirely (ACI) [1].

Serviceability limit state design requires definition of following parameters [8]:

Loading history and method of analysis to established design load effect

The designed material properties should be assumed in the design

Criteria defining the limit of satisfactory performance

Appropriate methods by which structural behavior may be predicted Calculation of these parameters exhibit certain difficulties due to the variety of interacting factors involved such as tensile strength, shrinkage, creep, relaxation, tension stiffening, properties of bond and variations in the elasticity modulus in time. The complexity of the solution and the method depends on the loading history.

Because of that the relationships between actions, load effects, stresses and deformation are non linear and the application and the simple superposition of the effects is no longer valid. In order to take into account all load effects, non-linear finite element method should be used, but this approach often proves unnecessary in actual engineering practice, and that’s why by introducing simplified assumptions whose accuracy in common design situations can be proved by experimental tests.

EUROCODE-2, for serviceability limit states design defines three combinations of actions [8]:

characteristic combination i,ki,o1,kj,k QQG

frequent combination i,ki,21,k1,1j,k QQG

quasi permanent combination i,ki,2j,k QG

The analysis should be carried out as elastic (without redistribution). This analysis is normally based on the stiffness of the uncracked section, but if cracking is to occur than a more realistic analysis should be used. 6.1.1 Limitation of compressive stresses

The limitation of stresses is caused by two reasons:

To avoid the formations of micro-cracks in the concrete, which might reduce the durability of the structures [8]

To avoid appearance of non-linear creep (excessive creep).


Micro-cracking will start to develop in the concrete when the compressive stresses exceed 70% of the compressive strength. The risk of appearance of micro-cracking is a function of the aggressiveness of the environment. If aggressiveness of the environment exists, in that case, the limitation of the stresses is up to 60% from the compressive strength (as well as increase in the cover to any reinforcement) [8].

The design methods used for verification of serviceability assume linear creep i.e. creep proportional to stress. This assumption is valid if the stresses are limited to 50% from the compressive strength. Therefore it is logical to impose a limit on compressive stresses under the quasi-permanent combination of loads, where a higher level of creep cause non-linear creep (higher deflections under creep which can seriously affect concrete behaviour) [8]. 6.1.2. Limitation of tensile stresses

Limitations may be defined for the tensile stresses in the concrete or in the reinforcement. In the case of reinforcement, this is necessary to ensure that inelastic deflections of the reinforcement are avoided under service load. This kind of deflection affects to the calculation of cracking or deflections which assume that the reinforcement behaves ideally elastic and can could result in excessively large crack width. This could be critical in areas subjected to frequent variations in loading. Because of that the limitation of the tensile stresses in the reinforcement should prevent inelastic deformation do not occur. The reinforcing bar should have 50% of its yield strength, and over 50% only for very insignificant period [8]. 6.1.3. Procedure for calculation of the stresses

Mostly the control of the stresses in reinforced concrete structures may be avoided if the following conditions have been met that [8]:

The designing for the ultimate limit state has been carried out in accordance with requirements for durability

Minimum reinforcement provisions were satisfied

Correct detailing

Not more than 30% of redistribution of the stresses has been allowed The calculation of the stresses should be carried out on the basis of the following assumptions [8]:

Plane sections remain plane

Reinforcement and concrete in tension are assumed to behave elastic

Concrete is assumed to be elastic up to its tensile strength fctm. If the stresses has exceeded fctm, in section is assumed to be cracked and concrete in tension is ignored.

Creep may generally be taken into account by assuming that the ratio of the elasticity modulus for steel to that for concrete is 15. Lower values which are based on the actual elasticity modulus of concrete may be used where less than 50% of the stresses arise from quasi-permanent loads. A more accurate determination of the effects of creep may be used if desired.

Usually the calculation of the stresses is done using standard elastic formula, for beam subjected only to flexure stresses may be calculated from:

Stress = My/I [8] .................................................................................................... (6.1)


Where:

M – bending moment, y - distance from the neutral axis to the point considered and l – moment of inertia of the section (which should be calculated on the basis of either for cracked or uncracked section as appropriate). 6.2. Control of cracking

6.2.1. Limitation of the crack width

There are many reasons to limit the width of crack to a relative small value. Usually The most common reasons are [8]:

To avoid possible corrosion damages to the reinforcement due to deleterious substances penetrating through the cracks to the reinforcement

To avoid, or limit leakage through cracks, which is usually critical for design consideration in water-retaining structures

To avoid an unsightly appearance of the structure The cracks appear due to : plastic shrinkage, reinforcement corrosion, alkali-silica reaction – expansive reactions, restrained deformations from shrinkage and temperature movements and loading Of the above items only restrained deformations and loading should be treated by the designer as two serious causes of cracking. 6.2.2. Cracking caused by loading

If a continuously increasing tension is applied to a tension element, the first crack will be formed when the tensile strength of the weakest section in the element is exceeded. The formation of this crack leads to a local redistribution of the stresses within the section. At the crack all tensile forces will be transferred to the reinforcement and the stresses in the concrete at crack must be equal to zero. With increasing distance from the crack force is transferred by bond from the reinforcement to the concrete until at some distance S0 from the crack, the stress distribution in the section is unchanged from what it was before the crack formed [8].

This local redistribution of the stresses in the region of the crack is accompanied by an extension of the element. This extension plus minor shortening of the concrete which has been relieved of the tensile stresses it was supporting is accommodated in the crack. The crack opens up to a finite width immediately on its formation. The formation of the crack and the resulting extension of the element also reduce the stiffness of the element. If further load is applied second crack will form at the next weakest section, although it will not form within distance S0 of the first crack since the stresses in this region are already reduced by the formation of the first crack. Further increase of loading leads to formation of further cracks until eventually no area of the member surface remains that is not within S0 of a previously formed crack. The formation of every crack will lead to a reduction in the element stiffness. When all cracks have been formed further loading will result in an increases of the existing crack width, but no new crack will appear. The stresses in the concrete will be relived by limited bond slip near the crack faces and by formation of internal micro cracks. This process leads to further reduction of the stiffness but clearly the stiffness cannot reduce below the stiffness of the reinforcement [8].

Figure 6.1 illustrates the behaviour described above. The stepped part of the load-deformation diagram, actually defines the appearance of cracks.


Figure 6.1 (a) Load-deformation diagram of a element subjected to permanent increasing load; (b) crack width-deformation in a load controlled test [8]

6.2.3. Calculation of the crack width

The development of formula for prediction of the crack width follows the form description of the cracking phenomena given above, depends on the distance between the cracks and strain at crack:

mrmSw [8] ....................................................................................................... (6.2) Where:

w-crack width, Srm average crack spacing and m average strain at crack.

This is statement of compatibility. Because no crack forms within S0 of an existing crack, this defines the minimum spacing between two cracks [8]. The maximum spacing is 2S0 since if a spacing existed wider than this a further crack could form. Therefore the spacing between the cracks is between S0 and 2S0 or often spacing is assumed of 1.5S0 [8].

It is in the calculation of Srm, that the most significant differences arise between the formula in national codes. The distance S0, i.e. Srm, depends on the rate at which stress can be transferred from the reinforcement, which is carrying all the force at a crack to the concrete. This transfer depend by bond stresses on the bar surface. If it is assumed that the bond


stress is constant along the length S0 and stresses will just reach the tensile strength of the concrete at a distance S0 from a crack, then [8]:

ctc0 fAS [8] .................................................................................................. (6.3)

Where: is bond stress, Ac is area of the concrete section and fct is tensile strength of the concrete.

If:

c2 A4/ [8] ................................................................................................. (6.4)

And substituting for Ac gives:

4/fS ct0 [8] ................................................................................................ (6.5)

And from this:

/k25.0Srm [8] ............................................................................................ (6.6) Where: k is constant that depends on the bond characteristics of the reinforcement. This is the oldest forms of relation for the predictions of crack spacing. With introduction of the cover to the reinforcement, above mentioned equation obtains the following form [8]:

/k25.0kcS 1rm [8] ................................................................................... (6.7)

Where: c is the cover to the reinforcement. This formula has been derived for elements subjected to pure tension. In order to be able to apply it to bending, it is necessary to introduce a further coefficient k2 and to define an effective reinforcement ratio r. That leads to the following formula [8]:

r21rm /kk25.0kcS [8] ................................................................................. (6.8) Where:

k1- coefficient taking account the bond properties of the reinforcement

k2- coefficient depending on the form of the stress distribution The average strain m is obtained by the equation () where the calculation of deflections is considered.

The equation for crack width gives an estimate of the mean crack width, which is not required in the designing i.e. but a width with a considerably lower likelihood of occurrence. This crack width is so called characteristic crack width (width with a 5% probability). Characteristic crack width according to EUROCODE 2 can be obtained by multiplying the mean width by 1.7.


6.2.4. Minimum area of reinforcement

An essential condition for the suitability of the formula for crack width calculation is that the reinforcement stays into the elastic area of behavior. If the reinforcement yields, the deformation becomes concentrated at the crack, where yielding is occurring, which is not the condition considered in the derivation of the formula. Most critical state is the state immediately after the form of the first crack. Should formation of the first crack lead to yield, than only a single crack will form and all the deformation will be concentrated at this crack. The principle involved here can be demonstrated by considering a element subjected to pure tension [8].

The force necessary to cause crack to element is given by [8]:

ctcr fAN [8] ........................................................................................................ (6.9) Where is:

Nc –cracking load;

Ac –area of concrete

fct -tensile strength of the concrete The strength of the steel is Asfy. Therefore, in order steel not to yield on first cracking and to spread cracking to develop [8]:

Asfy>Acfct [8] ......................................................................................................... (6.10) Or

As>Acfct/fy [8] ........................................................................................................ (6.11) This condition provides the minimum reinforcement area required for controlled cracking. Where cracking is caused by loading, this limitation has no importance, since no cracks will form under service conditions.

But if tensile stresses may be generated by restrained shrinkage or imposed deformations it is essential to ensure that at least the minimum reinforcement area is provided. This result is true for flexure and for pure tension.

Because this principle is applied for pure tension, the actual equations will differ for different types of elements. To avoid complexity in the rules, a factor kc is introduced into the above relation to adjust for different forms of stress redistribution.

A further coefficient k is included to allow for the influence of the internal self-equilibrium stresses caused by restrained deformation. These effects do not occur uniformly throughout the section, i.e. but will occur more rapidly near the member surface. As a result, the deformation of the surface of concrete will be restrained by the interior concrete and higher tension will developed near surface. The higher tensile stresses at the surface will cause occurring of cracking at a lower load than would be predicted on the basis of a linear distribution of stresses across section. Because the cracking load is smaller than the predicted, smaller amount of reinforcement is required to ensure that controlled cracking occurs. The function of the coefficient k is to reduce the minimum reinforcement area in cases where such non-linear stress distribution occurred [8].

At the end it is necessary to consider what value should be chosen for the tensile strength of the concrete fct. From the equations above minimum area of reinforcement is proportional to the tensile strength of the concrete. It seems logical to take as the tensile strength of concrete an estimate of its likely maximum value [8].


All these considerations lead to following formula:

sceff,ctcs /AkfkA [8] ....................................................................................... (6.12)

Where is: kc -coefficient which takes into account the form of loading

k -coefficient which takes into consideration the possible presence of non-linear stress distribution

fct,eff –is tensile strength of the concrete effective at the time when firs crack form

Ac –area of concrete in tension of concrete immediately before the formation of the first crack

s -is stress in the reinforcement which can be generally taken at the yield strength The value of the coefficient kc is equal to 1.0 for pure tension, 0.4 for pure flexure and between 0 and 0.40 for prestressed elements [8].

Also the value of kc=0 can be taken into the following cases [8]:

a) Under a rare combinations of loads, the whole section remains in compression or

b) Under the action of the estimated values of prestress, the depth of the tension zone calculated on the basis of a cracked section under the loading leading to the formation of the first crack does not exceed lesser of h/2 and 500mm

Similar to the determination of the minimum area of reinforcement at pure tension, can be carried out for bending, or bending combined with compression or tension. 6.3. Control of deflections Usually the control of deflections is determined by using simple rules such as limits to span/effective depth ratios span/height, which is adequate approach for all normal situations.

For serviceability limit states design there are four necessary elements [8]:

a) Criteria defining the limit to satisfactory behavior

b) Appropriate design loads

c) Appropriate design material properties

d) Means of predicting behavior 6.3.1. Limitation of the deflections

The selection of limit criteria to deflections which will ensure that the structure will be able to fulfill its required function is a complex process, and is not possible for a code to specify simple limits that will meet all requirements and still be economical. Therefore EUROCODE-2 makes it clear that is the responsibility of the designer to agree suitable values with the client taking into consideration the purpose of the structure. Limits are suggested in EC-2 but these are for general guidance's only, it remains the responsibility of the designer to check whether these are appropriate for the particular case considered or some other limits should be used [8].


6.3.2. Deflection under the effect of short-term loading

The behavior of reinforced concrete structures subjected to short-term loads is characterized by load-deformation diagram shown in figure 6.2. It is convenient to consider the curve to be made up of three phases.

Figure 6.2. Idealized load-deformation diagram of a reinforced concrete element [8] Phase 1 (uncracked): In this phase the tensile strength of the concrete has not been exceeded, the section behaves elastically and its behavior can be predicted on the basis of an uncracked section analysis but with allowance made for the reinforcement [8].

Phase 2 (cracked): In this phase the concrete has cracked in tension. The concrete in compression and the reinforcement may be considered to remain elastic. The behavior of the tension zone is very complex. At a crack the concrete in tension carries no stress. But between cracks, bond transfers stress from the reinforcement to the concrete so that with increasing distance from a crack, the tension carried by the concrete increases. The behavior shown on figure 6.2 for this phase reflects the average state where the tension zone carries some tension. This is satisfactory explanation for deflection calculation, since the calculation of the deflection requires the integration of the behavior over the length of the beam [8].

Phase 3 (inelastic): In this phase the steel has yielded or the concrete is stressed to a level where the assumptions of elasticity cease to be reasonable or both. This phase is reached only at loads well above those likely to occur in normal service, and so is not of interest for serviceability calculations [8].

The behavior of the concrete in phase 2 is that causes difficulties in the analysis and all current design approaches are empirical and approximate.

Certain limits to behavior can be defined as follows [8]:

At the instant of cracking, when the tensile strength of the concrete is just attained, the response of the element must lie on phase 1 line

Since cracking effectively reduced the stiffness of the element the behavior after cracking must lead to curvatures greater than phase 1 curvature

The maximum possible curvature corresponds to the condition where the concrete in tension carries absolutely no stress. This is the response that would be calculated on the basis of a cracked transformed section.


In practice the experiments have shown that at the cracking moment, the behavior lies on the phase 1. As the load is increased above the cracking load, the behavior of the element has tendency toward the fully cracked response. This can be expected since increase in cracking and bond slip in the region of cracks leads to an increasing loss of effectiveness of the concrete in tension [8].

The basic concept of the method is illustrated on figure 6.3 considering a length of a beam bounded by two cracks. It is assumed that some length close to the cracks is fully cracked while the remain part is uncracked [8].

Figure 6.3 Model of behavior of reinforced concrete beam in phase 2 [8] If the distance between cracks is S, the length of the section with cracks is s, and the length of the section without cracks is (1-)s.

At pure flexure the rotation over the length s is given by following equation:

12 r/1s1r/1s [8] ........................................................................... (6.13)

The average curvature is given by:

12m r/11r/1s/r/1 [8] .............................................................. (6.14) The average reinforcement strain may be calculated using:

2s1ssm 1 [8] ..................................................................................... (6.15) In above equations the indexes 1 and 2 indicate behavior calculated assuming the section to be uncracked and fully cracked respectively. An alternative way of visualizing the idealization is given in figure 6.4 for the strain in the reinforcement [8].

Figure 6.4 Alternative visualization of the model behavior of reinforcement in phase 2 [8]


EUROCODE-2 adopts the following equation [8]:

2ssr1 /1 [8] ....................................................................................... (6.16) Where:

1-coefficient characterizing of the bond properties of the reinforcement (1=0.5 for plain bars and 1=1 for ribbed bars)

sr- reinforcement stress calculated on the basis of a cracked section under the loading that just causes the tensile strength of the concrete to be attained at the section considered

s-reinforcement stress under the loading considered calculated on the basis of a cracked section. 6.3.3. Deflections under the effect of long-term loading

At calculation of long-term deflections it is necessary to take into consideration: creep, shrinkage and in cracked section reduction of the tensile stresses in the concrete in the tension zone due to spreading with time of cracking and local bond failure [8].

The creep is usually calculated with the effective modulus method which is not very accurate but is simple to use. The effective modulus of elasticity of the concrete, which takes into consideration the creep, can be calculated by the following equation:

1/EE cmeff,c [8] ........................................................................................ (6.17)

Where:

Ec,eff -is effective elasticity modulus of the concrete,

Ecm -short-term modulus of elasticity of concrete

-creep coefficient. The shrinkage of the concrete causes compressive stress in the reinforcement csEs and equivalent to force equal to:

sscscs AEN [8] ................................................................................................ (6.18) Where:

cs -is shrinkage strain

Es -modulus of elasticity of the reinforcement

As -area of the reinforcement Under the released force effects in reinforcement the beam will deform that leads to a curvature given by:

1cse1csscs1ccscs I/SIE/eAEIE/eNr/1 [8] ........................................ (6.19)

Where:

e -modular ratio e=Es/Ec

e -eccentricity of the reinforcement

I1 -second moment of area of the uncracked section


S -first moment of area of the reinforcement static moment of inertia of the reinforcement about the centroid of the concrete section (=Ase for singly reinforced sections)

In these equations Ec is the effective elasticity modulus should clearly be an effective value allowing for the effects of creep, since shrinkage is a long-term effect. This method can be applied also for cracked sections with substitution I1 to I2.

The effects of long-term or repeated loading are to reduce the effectiveness of the transfer of stress from the reinforcement to the concrete between the cracks. This effect can be seen to be modeled by increasing the distribution coefficient , with introduction of another coefficient 2 in the equation:

2ssr21 /1 [8] ................................................................................... (6.20)

2 -is coefficient which takes into account of duration of loading or of repeated loading. This coefficient takes value of 1 for monotonic short-term loading and 0.5 for long-term or repeated loads.


7. Experimental research of the variable loads effect on time-dependant behavior of concrete 77

7. Experimental research of the variable loads effect on time-dependant behaviour of concrete elements

7.1. Purpose of the experimental testing

The experimental testing have been carried out on the Faculty of Civil Engineering in Skopje, in the period between January 2007 and April 2008. The purpose of the experimental research is to determine the influence of the long-term permanent and variable loads to the time-dependent behaviour of the reinforced concrete elements made of ordinary and high-strength concrete. On the basis of the obtained results, for the solution of these kinds of problems in the practice, values of the quasi-permanent coefficient will be proposed to replace the action of variable loads as quasi-permanent loads according to Eurocode-2. 7.2. Description of the experimental program

In the experimental program 24 reinforced concrete beams were made with rectangular cross-section and dimensions width/height 15/28cm and total length l=300cm shown on figure 7.1. The chosen dimensions of the beams enable use of real concrete and reinforcement, as they are used in practice.

Of the total number, 12 beams are made of ordinary concrete, concrete class 30, i.e. C30/37, and 12 beams are made of high-strength concrete, concrete class 70, i.e. C60/75. Reinforced concrete beams are reinforced with ribbed bars, strength class 400/500, in the tensile zone with 212, and in the compression zone with 28. For shear reinforcement stirrups made of smooth reinforcement 240/360 6mm were used.

The dimensions of the reinforced concrete beams and the position of the reinforcement are shown on figure 7.1.

Figure 7.1 Reinforced concrete beams for testing Reinforced concrete beams are divided in six series A, B, C, D, E and F. Each series consists of two groups A: A1 and A2; B: B1 and B2; C: C1 and C2: D: D1 and D2; E: E1 and E2 and F: F1 and F2. The groups in each series are separating the beams according to the


concrete type (index 1 denotes the elements made of ordinary concrete, concrete class C30/37, and index 2 is for elements made of high-strength concrete, class C60/75).

Besides the strength characteristics of the concrete, variable parameter is also the type of the load:

short-term load

permanent load with intensity “G”

permanent load with intensity “G+Q/2”

long-term permanent load “G” and short-term repeated variable load with intensity “Q” The experimental program is shown in table 7.1.

Table 7.1 Experimental program

Ser

ies

Gro

up

Num

ber

of

elem

ents

Typ

e

of c

oncr

ete

Type of load Cycles of loading

Tim

e of

test

ing

of th

e el

emen

t

Tim

e of

fo

llow

ing

of th

e el

emen

t

A A1 2 C30/37

Short-term load / t=40

A2 2 C60/75 t=40

B B1 2 C30/37

Permanent load "G" / t=400 t=360

B2 2 C60/75 t=400 t=360

C C1 2 C30/37 Permanent load

"G+Q/2" /

t=400 t=360

C2 2 C60/75 t=400 t=360

D D1 2 C30/37

Permanent load "G" and Variable load "Q"

loading for t1=24 hours and unloading t1=24 hours

t=400 t=360

D2 2 C60/75 t=400 t=360

E

E1 2 C30/37 Permanent load "G" and Variable load "Q"

loading for t2=48 hours and unloading t2=48 hours

t=400 t=360

E2 2 C60/75 t=400 t=360

F

F1 2 C30/37 Short-term load

/ t=400 t=360

F2 2 C60/75 t=400 t=360

The loading is applied to the reinforced concrete beams with the action of two concentrated forces set on distance of 100cm from the supports and on mutual distance of 80cm, so the span of the beam is 280cm, simply supported beam. The time-dependent loads on the beams were applied by gravity lever which enable maximum increase of the applied load for 28 times.

For the needs of the experimental tests a correction of the gravity levers was made providing the necessary increase of 16.8 times of the initial load to obtain necessary test load.

For the determined beams dimensions, the quality of the used materials and the reinforcement, theoretical values of the bending moments were calculated as follows:


bending moment of self-weight of the beam Mg=1kNm

bending moment of action of the long-term permanent load MG=4kNm

bending crack moment:

for concrete class C30/37 Mcr=5.7kNm

for concrete class C60/75 Mcr=8.6kNm

bending moment of action of the repeated variable load MQ=7.6kNm

bending moment at service MG+Q=11.6kNm

ultimate bending moment Mu=21.7kNm

ratio between the moment of the permanent and the variable load MG/ MQ=0.53

The tests were carried out in the laboratories at the Faculty of Civil Engineering (figure 7.2) in environment of relative constant values of humidity and temperature (figure 7.3).

Figure 7.2 Laboratory

0102030405060708090

100

0 50 100 150 200 250 300 350 400 450t [days]

RH

[%

] /

T [o

C]

Temperature Tm=17oC

Relative humidity RHm=63%

Figure 7.3 Mean humidity and temperature in the laboratory

The elements of the series “A” were tested to short-term loads until failure at concrete age t=40 days (Table 7.2). With these testing relevant relations will be determined for the behaviour of the elements at this age and to make comparison between the behaviour of the elements made of ordinary concrete and the elements made of high-strength concrete.

Besides the elements of series “A” also the elements of series “F” were tested to short-term loads until failure at concrete age of t=400 days in order the effect of the concrete age on the behaviour of the elements made of ordinary and high-strength concrete to be determined (Table 7.3).


Table 7.2 Loading history for the elements of series A

Series Beams Number of elements

History of loading

A

A1 and A2-C30/37

2

A3 and A4-C60/75

2

Table 7.3 Loading history for the elements of series F


History of loading

F

F1 and F2-C30/37

2

F3 and F4-C60/75

2

On the elements of series “B” at concrete age t=40 days long-term permanent load was applied by intensity “G” chosen so that under its influence in the section of the elements doesn’t have appearance of cracks (7.4). During the period t=40÷400 the time-dependant behaviour of these elements was monitored measuring the deformations increase and the strain changes. After this period the elements are tested to short-term loads until failure at concrete age t=400 days.

Table 7.4 Loading history for the elements of series B


History of loading

B

B1 and B2-C30/37

2

B3 and B4-C60/75

2


In the series “C” on the beams, long-term permanent load was applied with intensity “G+Q/2” at the age of concrete t=40 days chosen so that under its influence in the section of the elements cracks appears as it is shown in table 7.5. Under the action of this load, crack appears only in the beams made of concrete class C30/37. During the period t=40÷400 days time-dependant behaviour of beams were monitored measuring the deformations increase, increase of crack width and strain changes. After this period the elements are tested to short-term loads until failure at concrete age t=400 days.

Table 7.5 Loading history for the elements of series C


History of loading

C

C1 and C2-C30/37

2

C3 and C4-C60/75

2

On the beams from series “D” long-term permanent load with intensity “G” and repeated variable load with intensity “Q” were applied at age of concrete t=40 days.. The repeated variable load “Q” is applied in cycles: loaded for ∆t1=24 hours and unloaded for ∆t1=24 (Table 7.6). During the period t=40÷400 days, the time-dependant behaviour of the beams were monitored measuring the deformations increase, increase of crack width and strain changes. After this period the elements are tested to short-term loads until failure at concrete age t=400 days.

Table 7.6 Loading history for the elements of series D


History of loading

D

D1 and D2-C30/37

2

D3 and D4-C60/75

2

On the beams from series “D” long-term permanent load with intensity “G” and repeated variable load with intensity “Q” were applied at age of concrete t=40 days. The repeated variable load “Q” is applied in cycles: loaded for ∆t1=48 hours and unloaded for ∆t1=48 (Table 7.7). During the period t=40÷400 days the time-dependant behaviour of the beams were monitored measuring the deformations increase, increase of crack width and strain changes. Then the elements are tested to short-term loads until failure at concrete age t=400 days.


Table 7.7 Loading history for the elements of series E


History of loading

E

E1 and E2-C30/37

2

E3 and E4-C60/75

2

With the establishment of this regime of the elements loading, the effect of the repeated variable loads to the behaviour of the reinforced concrete elements of ordinary and high-strength concrete during the time is expected to be determined and the behaviour of the elements made of ordinary concrete to be compared with elements made of high-strength concrete.

For the concrete classes C30/37 and C60/75 mix design of the concrete was carried out and initial testing were made to determine concrete quality through basic characteristics: defining of the workability of the fresh concrete, density of the fresh concrete and compressive strength of the hardened concrete. Beside that, for the needs of the autogenous shrinkage test determination of the initial setting time of the concrete was done. The previous testing of the concrete quality is done during December 2006.

During the casting of the reinforced concrete beams control specimens for testing of the mechanical and time dependant characteristics of the concrete were taken, in order to obtain the real quality of the concrete. Following properties of the hardened concrete were tested:

1) Mechanical characteristics:

Compressive strength

Flexural tensile strength

Splitting tensile strength

Modulus of elasticity

2) Deformation characteristics:

Autogenous shrinkage

Drying shrinkage

Creep

The type of the control specimens, number and the properties which were tested at certain age are shown in table 7.8.

Besides the testing of the concrete properties, also testing of the basic mechanical properties of the reinforcement ribbed bars strength class 400/500-2 were done, to determine yield strength f0.2k, tensile strength ft and modulus of elasticity for the reinforcement bars 8 and 12.


Table 7.8 Control specimens for testing properties of concrete

Characteristics Type of control specimen Number

of samples

Concrete age at testing [days]

Compressive strength Cubes a=15cm 3

3

40

400

Flexural tensile strength Prisms 10/10/40cm 3

3

40

400

Splitting tensile strength Cubes a=15cm 3

3

40

400

Modulus of elasticity Cylinders D/H=15/30cm 6

6

40

400

Autogenous shrinkage Prisms 10/10/50cm 2 400

Drying shrinkage Prisms 12/12/36cm 3 400

Creep Prisms 12/12/36cm 3 400

7.3. Materials and manufacturing method of reinforced concrete beams and control

specimens

For manufacturing of reinforced concrete beams, mix proportioning for concrete class C30/37 and concrete class C60/75 was designed. The mix design of the concrete is shown in table 7.9.

Table 7.9 Concrete mix design

Materials Type CC-30

class 30/37

CC-70

class C60/75

kg/m3 kg/m3

Cement (C) PC45s (CEM I 42.5R) 360 414

Mineral additives (SF)

silica fume 8% of the content of (C+SP)

/ 36

Water Of a well 210 158

Water-cement ratio W/C

0.58 0.35

Chemical additives superplasticizer 1.1% of the content of (C+SP)

/ 5

Aggregate 1850 1870

I (0-4)mm river sand 40% 37%

II (4-8)mm crushed limestone 22% 25%

III (8-16)mm crushed limestone 38% 38%

Ordinary Portland Cement PC45c (CEM I 42.5R) was used which is product of A.D. “USJE” -Skopje, group TITAN.

The silica fume and the superplasticizer C-21 are products of “ADING” a.d. Skopje.


The river aggregate I-fr. is from the borrow pit of the river Lepenec of the Factory for prefabricated concrete elements “KARPOS” A.D. Skopje, and the coarse aggregate II-fr. and III-fr. is crushed aggregate from limestone origin of the quarry of Banjani, Skopje, supplied by Civil construction company “BETON” A.D. Skopje.

For the tension and compression reinforcement, ribbed bars strength class 400/500-2 were used, and for the stirrups smooth reinforcement strength class 240/360 was used.

The reinforcement of the beams is done in the Factory for prefabricated concrete elements “KARPOS” A.D. Skopje. After the reinforcing on the tensile reinforcement, strain gauges were set and protected in the middle of the span of the reinforced concrete beams (figure 7.4).

Figure 7.4 Placing of the strain gauges The casting of the beams is also done in the factory “KARPOS” A.D. Skopje, in wooden mould which enables casting of 12 beams that should be made of one concrete class. The preparation of the concrete is done in the central factory for concrete, located in the range of the factory. The concrete placing is done on vibration platform, on which the wooden mould is set (figure 7.5). The filling of the mould is with crane and bucket in two layers, and then the compaction of each concrete layer is done with vibrations caused by the vibration platform with determined time of vibration.

Figure 7.5 Formwork and concreting of the reinforced concrete beams During the concreting of the beams all necessary control specimens were taken for testing of the mechanical properties and time dependant properties of shrinkage and creep at concrete age of 40 and 400 days.

In this way the control specimens for testing of the concrete characteristics are made of the same quality of concrete which is placed in the beams.


The reinforced concrete beams are cured 7 days in the wooden mould which was set in separate room-chamber, protected with expandable polystyrene d=5cm and covered with PVC-foil (figure 7.6). In this way approximately constant hygrometric conditions were provided with temperature between 12-16°C and relative humidity of 70-80%.

Figure 7.6 Reinforced concrete beams curing The control specimens for testing of the concrete quality are kept in same conditions in the first 24 hours, and then are transported in the Laboratory of the Faculty of Civil Engineering, Skopje, where are subjected to hygrometric conditions which will be the same as for the testing of beams.

The reinforced concrete beams after 7 days are transported to the Laboratory of the Faculty of Civil Engineering, Skopje. In the next 7 days the necessary notation and placement of contact gauge for measurement of the strains were done.

With the measuring of the beams strains, the monitoring of the strains of the concrete shrinkage is started from the 14th day after the concreting.

The first series of reinforced concrete beams were of concrete class C30/37. The second series of beams were of high-strength concrete class C60/75 and were casted 21 days after the casting of the first series. 7.4. Measurement technique

During the testing of the reinforced concrete beams the following parameters were measured:

The deflections of the beams of the series B, C, D and E during the testing period in period of 400 days, and at testing of every beam of the series A, B, C, D, E and F to action of short-term load until failure. The measurement of the deflections is done with deflection-meters STOPANI, Italy with accuracy of 1/100mm

Strains in the concrete at level of tension reinforcement, at level of neutral axis, at level of compression reinforcement and at the top of the concrete beams are measured in period of 400 days with mechanical deflection meter type “Hugenberger”, Switzerland with base of 250mm. At testing to action of short-term loads until failure, the strain except with the mechanical deflection meter are also measured with strain gauges type “KYOWA”, Japan and measurement bridge type “Hottinger Baldvin-HBM”, Germany.

The deformation concrete characteristics of shrinkage and creep during the time are tested with control specimens prisms with dimensions 12/12/36cm measured


with mechanical deformation meter type “Hugenberger”, Switzerland with base of 250mm.

The width of the crack is measured with microscope for measuring of the crack with accuracy of 0.02mm.

The forces are measured with dynamometer with accuracy of 5kg.

7.5. Loading method and testing of the elements

In the Laboratory of the Faculty of Civil Engineering, Skopje all reinforced concrete beams and control specimens cured during the testing were placed, in same environmental conditions.

The permanent and the variable load are applied through gravity levers, whereupon for determined level of the loads increase of the initial force of the lever for 16.8 times, was established which is transferred to the beam (figure 7.7). For determined level of loading before the start of the beams testing, the forces which are transferred to the beam are measured with dynamometer.

Figure 7.7 Gravity lever For the series of beams “B”, the permanent load on level of FG was applied with gravity load (beams from the previous testing, which were previously measured with dynamometer and to the required value of the load are complemented with concrete control specimens-cubes, were used).

For the other series of beams the applying of the load is with the gravity levers for the action of the permanent load, and the variable load is applied with lowering and rising with hand secondary lever in ratio 4:1.

The beams are system of simply supported beam which is loaded with two concentrated forces which are on distance of 100cm from the supports and on mutual distance of 80cm (figure 7.8).

At testing to action of short-term loads until failure, for applying of loading hydraulic jack is used with capacity of 100kN.

For reinforced concrete beams, during the testing the following measuring points were defined:


For measurement of the strains 20 points were determined, set along section height on the upper edge in the concrete, in the level of compression reinforcement, level of approximately determined neutral axis and on the level of tension reinforcement. In this way measurement of the strains was done in three cross-sections. One of them is in the middle of the span and the other two cross-sections are 25cm on left and right side from the section in the middle of the span. The position of the strain measuring points is shown on figure 7.8.

The measuring of time-dependant deformations of reinforced concrete beams was done only in the middle of the beam span. At testing when action of short-term load until failure was applied, the deformations were measured in the middle of the span and in two sections that are on 40cm left and right side from the middle of the span. The position of the strain measuring points is shown on figure 7.8.

Also the strains were measured by strain gauges which were set in the middle section of the beam, the strains in the concrete in the upper edge of the section, the strains in the tensile reinforcement and at level of the lower edge of the section the strains in the concrete were measured until the appearance of the first crack in the section.

Figure 7.8 Position of the measuring points on the reinforced concrete beam


Figure 7.9 Picture of the position of the measuring points of the reinforced concrete beam The mechanical characteristics of the concrete are tested on hydraulic press HPM3000 from ZRMK-Ljubljana, Slovenia and the universal press E40, VEB of 400kN, Germany (shown on figures 7.10, 7.11 and 7.12).

Figure 7.10 Testing of the compressive strength

Figure 7.11 Testing of the flexure tensile strength


Figure 7.12 Testing of splitting tensile strength The registration of the strains at testing of the concrete modulus of elasticity is done with strain gauges type “KYOWA”, Japan with base of 120mm and measuring bridge from “HBM”, Germany (shown on figure 7.13).

Figure 7.13 Testing of the modulus of elasticity The deformations due to autogenous shrinkage, drying shrinkage and concrete creep in the period of t=1-430 days are monitored with mechanical deformation meter “Hugenberger”, Switzerland with base of 250mm. For testing of the concrete creep steel frames were used composed of 4 powerful springs with which the initial compressive strength of the control samples during t=40-430 days was maintained (shown on figure 7.14).


Figure 7.14a) Testing of the autogenous shrinkage and drying shriknkage

Figure 7.14b) Testing of the creep of concrete The reinforcement is tested in the universal press type E40, VEB of 400kN, Germany.

The testing of the concrete and reinforcement characteristics are carried according to the valid Macedonian standards, except the testing of the autogenous shrinkage which is done according to the suggestion for testing of the autogenous shrinkage of the cement paste, mortar and concrete by the Technical committee for autogenous shrinkage of the Japanese Institute for concrete (Tazawa, 1998) shown on figure 7.15.

According to this method for concrete with maximum aggregate grain smaller than Dmax=32mm mould in form of a prism was used with dimensions 100/100/500mm. In particularly made openings in the mould, gauge plugs in the centre of the cross-section are located, and with that the measuring base is 440mm, respecting the standard recommendations every gauge plug to enter in the concrete mass minimum 30mm. The mould is specially prepared, on the bottom PTFE sheet (poly tetra fluor ethylene) is set with thickness of 1mm, and on the sides expandable polystyrene with thickness of 3mm is set. Afterwards over them polyester film with thickness 0.1mm is set in order to prevent the contact between the concrete and the mould. The method of preparation of the mould for testing of the autogenous shrinkage is shown on figure 7.15.

After the finished initial measurements, the prism is taken out of the mould, and contact gauge are set, as for the testing of drying shrinkage and in order the drying of the sample to be prevented, the prism is coated with self-scaling aluminium foil and plastic bags are set in order relative humidity of 70% to be provided. During the measuring, the prisms were taken out of the plastic bags, and after the performed measurement, they were returned in the same PVC bags. In this way also the deformations caused by the autogenous shrinkage were monitored.


a)Testing mould b)PTFE sheet set at the bottom

c)Set gauge plug, PTFE sheet and expandable polystyrene

d)Set Polyester film

7.15 Preparation of the moulds for testing of the autogenous shrinkage 7.6. Experimentally determined mechanical and deformation properties of the concrete

and the reinforcement

For establishment of the concrete characteristics for concrete class C30/37 and for high-strength concrete class C60/75 testing at age of 40-50 days were carried out, in the moment when the loading of the reinforced concrete beams at age of 400 days is done, when after the termination of the monitoring of the behaviour of the reinforced concrete beams during the time, testing of the short-term loading until failure was done.

The results of the testing of the concrete class C30/37 characteristics is shown in table 7.10, and the results of the testing of the concrete class C60/70 characteristics is shown in table 7.11.




With the testing of the modulus of elasticity the dependency of the diagram stresses-strains is determined at concrete age of 40 and 400 days, under the effect of compressive stresses. The diagrams stresses-strains are shown as mean value of testing of series of three cylinders with dimensions D/H=15/30cm. The diagram stresses-strains for the concrete class C30/37 is shown on figure 7.16, and on figure 7.17 the comparison between the experimental diagram and the diagram proposed in Eurocode-2 is given.

0

5

10

15

20

25

30

35

0 0.5 1 1.5 2 2.5

[10-3]ms

c [M

Pa

]

C30/37 t=40C30/37 EC-2 t=40

Figure 7.16 Diagrams σc-εc for concrete class C30/37 at t=40 days and according EC-2

0

5

10

15

20

25

30

35

40

0 0.5 1 1.5 2 2.5

[10-3]ms

c [M

Pa

]

C30/37 t=400C30/37 EC-2 t=400

Figure 7.17 Diagrams σc-εc for concrete CC-30, class C30/37 at t=400 days and according

EC-2

Ec=30483.2MPa

Ec=33149.6MPa


On figure 7.18 the diagram stresses-strains for concrete class C60/75 is shown at concrete age of 400 days, and on figure 7.19 the comparison between the experimental diagram stresses-strains and the proposed diagram according to Eurocode-2 is shown.

From the diagram of the dependency between stresses-strains at testing of the compressive short-term load, at testing of the modulus of elasticity can be noticed good match of the dependencies obtained experimentally and the dependency stresses-strains which is given in Eurocode-2.

In the same way the diagrams stresses-strains for the high-strength concrete class C60/75 for concrete age of 40 days are given and the comparison with the proposed diagram stresses-strains according to the model of Eurocode-2 (figure 7.20) is given for concrete age of 400 days (figure 7.20).

0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3

[10-3]ms

c [M

Pa

]

C60/75 t=40C60/75 EC-2 t=40

Figure 7.18 Diagrams σc-εc for concrete class C60/75 at t=40 days and according to EC-2 Also at high-strength concrete good match of the experimentally obtained results and the proposed model for dependency stresses-strains according to Eurocode-2 can be noticed, except maybe small variation at measurements which are close to the failure, when it should be measured the biggest value of the strain which is maybe result of the testing method, because compression force is applied, and not with testing with controlled deformation.

At concrete age of 400 days at high-strength concrete CC-70, i.e. class C60/75 almost ideal match of the experimental diagram stresses-strains and the proposed diagram according to Eurocode-2 can be noticed, which is a result of the bigger stiffness of this concrete and the possibility for obtaining the strains at maximum applied stress.

From the testing of the dependency stresses-strains good match of the experimental results and the proposal given in Eurocode-2 at concrete age of 40 and 400 days can be noticed for the two types of concrete, ordinary and high-strength concrete.

On the following diagram on figure 8.19 the dependency stresses-strains for concrete class C30/37 and concrete class C60/75 is shown, from which can be noticed that the concrete age between 40 and 400 days has bigger effect at ordinary concrete, while at high-strength concrete the concrete age practically doesn’t have any influence on the shape of the

Ec=39470.9MPa


diagram, except the initial bigger stiffness before the appearance of the micro-cracks, which contributes to have higher value of the elasticity modulus.

0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3

[10-3]ms

c [M

Pa

]

C60/75 t=400C60/75 EC-2 t=400

Figure 7.19 Diagrams σc-εc for concrete class C6-/75 at t=400 days and according to EC-2

0

10

20

30

40

50

60

70

80

0 0.5 1 1.5 2 2.5

[10-3]ms

c [M

Pa

] C30/37 t=40d

C30/37 t=400d

C60/75 t=40d

C60/75 t=400d

Figure 7.20 Diagrams σc-εc for concrete CC-30, class C30/37 at t=40 and t=400 days and for concrete class C60/75 at t=40 and t=400 days

The development of the deformations from the autogenous shrinkage of the concrete, drying shrinkage and concrete creep for ordinary concrete class C30/37 and high-strength concrete class C60/75 are shown on the diagrams through the dependency stress-time, for period of monitoring of the deformations of 430 days.

Ec=41228.6MPa


The strains due to deformation characteristics of the concrete are measured in conditions of constant environmental conditions of relative mean humidity of the environment RH=63% and mean temperature T=17°C.

With the measurement of the strains from the autogenous shrinkage is started from the beginning of the concrete setting, and the strains from the measurement of the drying shrinkage is started after 1 day, when the control samples are released from the mould. The beginning of the setting for the concrete C30/37, which is tested with the previous testing, is 6 hours and 10 minutes, for high-strength concrete C60/75 the beginning of the concrete setting is 8 hours and 40 minutes.

At testing of the deformations due to concrete creep, in the first 40 to 50 days the deformations from the concrete shrinkage were monitored, and then compressive stress with value of 10 MPa was applied, which is on the level of the service stresses at action of the permanent and variable load. Afterwards the deformation from the concrete creep is monitored in the following 400 days. In the same way, i.e. on the same level of compressive stresses the control samples for monitoring of the creep deformations of the high-strength concrete class C60/75 were also exposed. The results of the concrete creep testing except through the measured strains are given also with calculation of the creep coefficient as ratio between the strain of the concrete creep and the instantaneous elastic strain at applying of the compressive stress.

The development of the deformations from the autogenous shrinkage, drying shrinkage and creep of the ordinary concrete class C30/37 is shown through the dependency strain-time on figure 7.21 7.22 7.23 and 7.24.

000.00E+00

050.00E-06

100.00E-06

150.00E-06

200.00E-06

250.00E-06

300.00E-06

350.00E-06

0 40 80 120 160 200 240 280 320 360 400 440

t [days]

Au

tog

eno

us

shri

nka

ge ε a

s [1

0-6]μ

s

AS-01-C30/37

AS-02-C30/37

Figure 7.21 Diagrams εas-t of the autogenous shrinkage of the concrete class C30/37 The deformations of the autogenous shrinkage and the drying shrinkage, on the separated samples, which are shown on figure 7.21 and 7.22 doesn’t deviate much between each other which gives the possibility comparison to be made with certain models for prediction of this values, for example according to the model in Eurocode-2 and the model of the Japanese society of civil engineers from 2002, and in the same time can serve as information which types of values are obtained during the experimental testing.

Also during the testing of the deformations due to concrete creep, very small difference between the testing results of the two samples can be noticed and on the same way


comparison can be made between the above mentioned models for prediction of concrete creep.

000.00E+00

100.00E-06

200.00E-06

300.00E-06

400.00E-06

500.00E-06

600.00E-06

700.00E-06

0 40 80 120 160 200 240 280 320 360 400 440

t [days]

Dry

ing

sh

rin

kag

e ε d

s [1

0-6]μ

s

DS-01-C30/37DS-02-C30/37DS-03-C30/37

Figure 7.22 Diagrams εds-t of the drying shrinkage of concrete class C30/37

000000.00E+00

000250.00E-06

000500.00E-06

000750.00E-06

001000.00E-06

001250.00E-06

001500.00E-06

001750.00E-06

0 40 80 120 160 200 240 280 320 360 400 440

t [days]

Dry

ing

cre

ep ε

cc [1

0-6]μ

s

DS-CC-01-C30/37

DS-CC-02-C30/37

Figure 7.23 Diagrams εcc-t from the creep of the concrete class C30/27


000000.00E+00

000250.00E-06

000500.00E-06

000750.00E-06

001000.00E-06

001250.00E-06

001500.00E-06

001750.00E-06

0 40 80 120 160 200 240 280 320 360 400 440

t [days]

Def

orm

atio

n ε

[10-6

]μs

CC-C30/37DS-C30/37AS-C30/37

Figure 7.24 Diagrams εas-t, εds-t and εcc-t of the autogenous shrinkage, drying shrinkage and

creep of the concrete class C30/37

On the diagram shown on figure 7.24, on one summarizing diagram the concrete deformations caused by the autogenous shrinkage, drying shrinkage and concrete creep and their relation can be seen. In the total strain, the strain of the autogenous shrinkage participates with 7.8%, instantaneous strain 19.7%, strain from drying shrinkage 37.0% and creep strain 35.5%.

The development of the deformations from the autogenous shrinkage, drying shrinkage and creep of the high-strength concrete class C60/75 is shown through the dependency strain-time on figure 7.25, 7.26, 7.27 and 7.28.

000.00E+00

050.00E-06

100.00E-06

150.00E-06

200.00E-06

250.00E-06

300.00E-06

350.00E-06

0 40 80 120 160 200 240 280 320 360 400 440

t [days]

Au

tog

eno

us

shri

nka

ge ε a

s [1

0-6]μ

s

AS-03-C60/75AS-04-C60/75

Figure 7.25 Diagrams εas-t of the autogenous shrinkage of concrete class C60/75


000.00E+00

100.00E-06

200.00E-06

300.00E-06

400.00E-06

500.00E-06

600.00E-06

700.00E-06

0 40 80 120 160 200 240 280 320 360 400 440

t [days]

Dry

ing

sh

rin

kag

e ε d

s [1

0-6]μ

s

DS-04-C60/75DS-05-C60/75DS-06-C60/75

Figure 7.26 Diagrams εds-t from the drying shrinkage for concrete class C60/75

000000.00E+00

000250.00E-06

000500.00E-06

000750.00E-06

001000.00E-06

001250.00E-06

001500.00E-06

001750.00E-06

0 40 80 120 160 200 240 280 320 360 400 440

t [days]

Dry

ing

cre

ep ε

cc [1

0-6]μ

s

DS-CC-03-C60/75DS-CC-04-C60/75DS-CC-05-C60/75

Figure 7.27 Diagrams εcc-t from the creep of the concrete class C60/75 In the diagram shown on figure 7.28 can be noticed that in the total strain, the strain from the autogenous shrinkage participates with 25.2%, drying shrinkage strain 40.1%, momentary strain 20.4% and creep strain 14.3%. In ratio to the ordinary concrete can be noticed that in the total strain of the high-strength concrete bigger participation has the strain from the autogenous shrinkage which is 25.2% comparing to the ordinary concrete in which is 7.8%. We can say that at ordinary and high-strength concrete the momentary strain and the drying shrinkage strain have approximately same participation in the total strain, while the creep strain is much smaller at high-strength concrete 14.3%, and at ordinary concrete is 35.5%.


000000.00E+00

000250.00E-06

000500.00E-06

000750.00E-06

001000.00E-06

001250.00E-06

001500.00E-06

001750.00E-06

0 40 80 120 160 200 240 280 320 360 400 440

t [days]

Def

orm

atio

n ε

cc

[10-6

]μs

AS-C60/75DS-C60/75DS-CC-C60/75

Figure 7.28 Diagrams εas-t, εds-t and εcc-t of the autogenous shrinkage, drying shrinkage and creep of the concrete class C60/75

This appoints to the fact that the high-strength concrete due to the low water-cement factor has bigger strain from the autogenous shrinkage which can’t be neglected, as for the ordinary concrete, and due to the bigger stiffness for the same level of compressive stresses has smaller deformations from the concrete creep comparing to the ordinary concrete.

On the following diagrams shown on figure 7.29, 7.30 and 7.31 the comparisons between certain strain of the autogenous shrinkage, drying shrinkage and strains from the concrete creep between the ordinary concrete and the high-strength concrete are shown.

000000.00E+00

000250.00E-06

000500.00E-06

000750.00E-06

001000.00E-06

001250.00E-06

001500.00E-06

001750.00E-06

0 40 80 120 160 200 240 280 320 360 400 440

t [days]

Au

tog

eno

us

shri

nka

ge

εa

s [1

0-6]μ

s AS-C30/37

AS-C60/75

Figure 7.29 Diagrams εas-t from the autogenous shrinkage of concrete C30/37 and C60/75


000000.00E+00

000250.00E-06

000500.00E-06

000750.00E-06

001000.00E-06

001250.00E-06

001500.00E-06

001750.00E-06

0 40 80 120 160 200 240 280 320 360 400 440

t [days]

Dry

ing

sh

rin

kag

e ε

ds

[10-6

]μs

DS-C30/37

DS-C60/75

Figure 7.30 Diagram εds-t from the drying shrinkage of concrete C30/37 and C60/75

000000.00E+00

000250.00E-06

000500.00E-06

000750.00E-06

001000.00E-06

001250.00E-06

001500.00E-06

001750.00E-06

0 40 80 120 160 200 240 280 320 360 400 440

t [days]

Dry

ing

cre

ep ε

cc [1

0-6]μ

s

DS-CC-C30/37

DS-CC-C60/75

Figure 7.31 Diagram εcc-t from creep of the concrete C30/37 and C60/75 From this comparative diagrams shown on figure 7.29, 7.30 and 7.31 can be noticed that the high-strength concrete has for 2.35 times bigger strain of the autogenous shrinkage than the ordinary concrete, then for 21.6% smaller value of the drying shrinkage comparing to the ordinary concrete and for 71% smaller creep strain comparing to the ordinary concrete. In ratio to the total strain from the concrete shrinkage as a sum of the autogenous shrinkage and drying shrinkage the ordinary concrete and the high-strength concrete have approximately equal values i.e. the total shrinkage strain of the ordinary concrete is smaller than the shrinkage strain of the high-strength concrete for 5.3%.


In the following table 7.12 the results from the experimental testing of the autogenous shrinkage deformations, drying shrinkage deformations and concrete creep are shown and their comparison with the models for prediction of the appropriate deformation according to Eurocode-2 and the Japanese society of civil engineers from 2002 for ordinary concrete CC-30, class C30/37, and in table 7.13 for the high-strength concrete CC70, class C60/75 are shown.


Deformation characteristics Experimental

results According to

EUROCODE-2 According to JSCE-

2002

CC-30, class C30/37

Autogenous shrinkage strain as [10-6]

112.4 60.3 47.2

Instantenous strain el [10-6] 284.5 / /

Drying shrinkage strain ds [10-6]

535.0 572.2 552.4

Creep strain cc [10-6]

514.0 / 875.7

Total strain c [10-6]

1445.9 / /

Creep coefficient (t,t0)

1.820 2.100 /


Deformation characteristics Experimental

results According to

EUROCODE-2 According to JSCE-

2002

CC-70, class C60/75

Autogenous shrinkage strain as [10-6]

263.7 168.8 247.0

Instantenous strain el [10-6] 213.0 / /

Drying shrinkage strain ds [10-6]

419.3 401.1 505.9

Creep strain cc [10-6]

149.0 / 431.9

Total strain c [10-6]

1045.0 / /

Creep coefficient (t,t0)

0.703 0.970 /

For estimation of the comparative results during the prediction of the values of the appropriate strains, it should be known that according to Eurocode-2 allowed deviation is ±20%, and according to JSCE-2002 allowed deviation is ±20%.

In ratio to the prediction models of the autogenous shrinkage strains, can be noticed that the experimental results are little higher for the ordinary concrete, which can appear due to the relatively bigger cement quantity of 360kg/m3 and due to the smaller relative humidity of the environment where the samples were cured, which is Rh=63%, which is smaller than the necessary minimal relative humidity of RH=70%, for this kind of testing. The other results of the drying shrinkage strains, the experimental results and the results of the predicted values are in the framework of the allowed deviations. At high-strength concrete, at shrinkage strains exist bigger differences in the value of the experimental results and the value of the prediction models, which can be, due to the fact that this type of concrete is under the effect of stresses of 0.15fck, and the prediction models are for stresses effect to 0.4 fck. The results from the experimental testing of the shrinkage strain of the ordinary concrete and the


predicted values are in the framework of the allowed limits which can be due to the fact that at this type of concrete the compressive stresses are with size of 0.30 fck.

The testing of the reinforcement 10 and 12 is carried for obtaining of the basic mechanical characteristics: yield strength, tensile strength and modulus of elasticity through the one-axial tensile testing. For every profile and reinforcement diameter testing on 10 control test which are taken at reinforcing of the reinforced concrete beams was done.

The results from the testing are shown in table 7.14, and the diagrams of dependency stresses-elongations are on figure 7.32. Table 7.14 Mechanical characteristics of the reinforcement

Reinforcement type

Diameter

[mm]

Yield strength

f0.2 [MPa]

Tensile strength

fm [MPa]

Modulus of elasticity

Ea [MPa]

400/500-2 8 562 660 200000

400/500-2 12 466 719 200200

The results from the reinforcement testing have shown that the reinforcement meets the valid conditions, and according to the yield strength it can fit into the standards of EUROCODE-2 where the yield strength is between 400-600MPa.

Figure 7.32 Diagram force-elongations for reinforcement 10 and 12

7 Experimental research of the variable load effect on time-dependant behavior of concrete 105

7.7. Analysis of the results from experimental research

In the experimental tests of reinforced concrete beams, many results are obtain for the deflections, crack widths and strains in the concrete and reinforcement, which with respect to the loading method and time of testing can be divided in:

Data obtained at testing with short-term load to failure at concrete age of 40 and 400 days

Data obtained with monitoring of behaviour of reinforced concrete beams under the effect of long-term load in period of 360 days

Data obtained with monitoring of behaviour of reinforced concrete beams under the effect of long-term permanent load and repeated variable load

7.7.1. Time-dependant deflections

The effect of the long-term loads to the behaviour of reinforced concrete beams is determined through monitoring of the development of deflections during the time in the period of t=40-400 days. The tests were doing on beams loaded with different loading histories with long-term permanent load and effect of repeated variable load.

The series “B” beams are loaded with long-term permanent load at level FG=(2x4)kN chosen so that in the section of the beams cracks doesn’t appear.

The series “C” beams are loaded with long-term permanent load at level FG+Q/2=(2x7.6)kN which at beams made of ordinary concrete causes cracks appearance, and at beams made of high-strength concrete doesn’t causes cracks appearance. The level of the load in this series of beams have been chosen in order to represent sum of long-term load FG and 50% of the repeated variable load FQ replaced as quasi-permanent load.

The series of beams “D” and “E” are loaded with long-term permanent load FG and with repeated variable load FQ that changes in cycles loading/unloading by 24 hours for the series “D” beams and in cycles by 48 hours at series “E” beams. Under the effect of this load cracks have appeared in the section of all beams independently of the concrete quality. Deflections of the long-term effect of the load at series “B” beams

The development of the deflections during the time in the middle of the span of the beams B1, B2, B3 and B4 are showing on figure 7.33 with the diagram deflection (a) – time (t). The values of the deflection in the moment of loading at concrete age of t=40 days and the final value of the measured deflection during the time at concrete age t=400 days, are given in table 7.15.


Beam B1 B2 B3 B4

a(t0=40) [mm] 0.56 0.68 0.30 0.27

a(t=400) [mm] 1.67 2.09 0.69 0.81

For the same level of loading of the permanent load FG=(2x4)kN can be noticed that the deflections during the time are smaller at beams made of high-strength concrete B3 and B4 with respect to the beams made of ordinary concrete B1 and B2. The beams made of high-strength concrete have for 60.1% smaller deflection during the time at concrete age t=400 days with respect to the beams made of ordinary concrete. For comparison at concrete age of 40 days the high-strength concrete has for 29% bigger elasticity modulus with respect to the elasticity modulus of the ordinary concrete, which contributes for smaller measured initial


deflections, but to the deflections during the time the concrete creep affects the most, which additionally increases the difference in the deflections.

0 50 100 150 200 250 300 350 4000.0

0.5

1.0

1.5

2.0

2.5

3.0

F F

a

Beam: B1 Beam: B2 Beam: B3 Beam: B4

De

flect

ion

a (

mm

)

Time t (days)

Figure 7.33 Development of the deflections during the time for series “B” beams

The concrete creep for this level of the load takes place at stresses b=0.071fck for the ordinary concrete and b=0.016fck for the high-strength concrete, which contributes for different development of the deflections during the time at beams made of ordinary and high-strength concrete. At beams made of ordinary concrete the deflection during the time is for 3.03 times bigger with respect to the initial deflection, and this ratio at beams made of high-strength concrete is 2.63 times bigger deflection with respect to the initial deflection. Deflections of long-term effect of the load at series “C” beams

The development of the deflections during the time in the middle of the span of the beams C1, C2, C3 and C4 are showing on figure 7.34. The values of the deflection in the moment of loading at concrete age t=40 days and the final value of the deflection during the time at concrete age t=400 days, are given in table 7.16.


Beam C1 C2 C3 C4

a(t0=40) [mm] 2.47 2.35 1.15 0.75

a(t=400) [mm] 4.89 4.73 1.80 1.35

Characteristic for this series of beams is that at their loading with permanent load FG+Q/2=(2x7.6)kN, to crack appearance comes in the beams C1 and C2 made of ordinary concrete, and at beams C3 and C4 made of high-strength concrete there is no crack appearance. This affects directly the measured deflections during the time, because the beams C1 and C2 have degraded stiffness due to the development of the crack width.


0 50 100 150 200 250 300 350 4000

1

2

3

4

5

6

7

F F

a

Beam: C1 Beam: C2 Beam: C3 Beam: C4

De

flect

ion

a (

mm

)

Time t (days)

Figure 7.34 Development of the deflections during the time for series “C” beams

Therefore, the deflections during the time are much smaller at beams made of high-strength concrete C3 and C4 with respect to the beams made of ordinary concrete C1 and C2. The beams made of high-strength concrete have for 67.3% smaller deflection during the time at concrete age t=400 days with respect to the beams made of ordinary concrete. This difference is much bigger in ratio to the differences that exist between the elasticity modulus of the high-strength and ordinary concrete, only 24%, and therefore great influence to the final deflections has the concrete creep which takes place at stresses b=0.284fck for the ordinary concrete and b=0.098fck for the high-strength concrete. The development of the deflections during the time at beams made of ordinary and high-strength concrete is different due to the concrete creep. At beams made of ordinary concrete the deflection during the time is for 2 times bigger with respect to the initial deflection, and this ratio at beams made of high-strength concrete is 1.66. Deflections of long-term effect of the load at series “D” beams

The values of the deflection in the moment of loading at concrete age t=40 days at level of load FG and at level of load FG+Q as well as the values of the measured deflection during the time at concrete age t=400 days for the same levels of loads are given in table 7.17. Table 7.17 Measured values of the deflection at t0=40 days and during the time t=400 days

Beam D1 D2 D3 D4

aG(t0=40) [mm] 0.58 1.04 0.50 0.58

aG(t=400) [mm] 4.34 5.44 2.57 2.89

aG+Q(t0=40) [mm] 3.53 4.67 2.68 2.80

aG+Q(t=400) [mm] 5.88 6.80 3.73 3.96


The development of the deflections during the time in the middle of the span of the beams is showing separately for the beam D1 made of ordinary concrete on figure 7.35 and for beams D3 made of high-strength concrete on figure 7.36, in order to be obtained idea for the shape of the diagram.

0 50 100 150 200 250 300 350 4000

1

2

3

4

5

6

7

8

9

10

F F

a

Beam: D1

De

flect

ion

a (

mm

)

Time t (days)


0 50 100 150 200 250 300 350 4000

1

2

3

4

5

6

7

8

9

10

F F

a

Beam: D3

De

flect

ion

a (

mm

)

Time t (days)



The values of the deflection in the moment of loading at concrete age t=40 days at level of load FG and at level of load FG+Q as well as the values of the measured deflection during the time at concrete age t=400 days for the same levels of loads are given in table 7.17.

Characteristic for this series of beams is that at their level of loading with load FG+Q=(2x11.6)kN comes to crack appearance independently of the concrete quality.

Due to the higher mechanical properties of the high-strength concrete, the deflections during the time at load FG+Q are smaller at beams made of high-strength concrete D3 and D4 with respect to the beams made of ordinary concrete D1 and D2. The beams made of high-strength concrete have for 39.4% smaller deflection during the time at concrete age t=400 days with respect to the beams made of ordinary concrete at level of the load FG+Q. The difference at level of the deflection during the time at concrete age of 400 days FG is 44.2% smaller deflection at beams made of high-strength concrete with respect to the beams made of ordinary concrete.

This difference is much bigger with respect to the differences that exist between the elasticity modulus of high-strength and ordinary concrete that is 24%. Therefore great influence to the final deflections has the degradation of the stiffness of the section due to the crack appearance, whose number is bigger at beams made of ordinary concrete; and the concrete creep which takes place at stresses b=0.386fck for the ordinary concrete and b=0.20fck for high-strength concrete.

In order to understand the development of the deflections during the time under the effect of permanent load FG and variable load FQ on figure 7.37 and 7.38 the development of the deflection during the time for 20 days after the loading for beams D1 made of ordinary concrete is shown and for beam D3 made of high-strength concrete.

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

10

3.53 3.70

3.66 3.82

3.76 3.92

3.87 4.00

3.9

8

4.09

4.01 4.11

4.07 4.17

4.12 4.22

4.19 4.2

8

2.24

2.18 2.

41

2.35 2.49

2.45 2.

65

2.59

2.66

2.59 2.70

2.65 2.76

2.70 2.82

2.76 2.87

0.58

0.64

0.67

0.68

F F

a

Beam: D1

De

flect

ion

a (

mm

)

Time t (days)

Figure 7.37 Development of the deflections for beam “D1” in 20 days after loading

On the diagrams shown on figure 7.37 and 7.38, the phases of the cycles loading/unloading with the variable load FQ can be seeing. At beginning, the beams have been loaded with permanent load, FG that does not cause appearance of cracks in the section. In the following four days due to the influence of shrinkage and creep of the concrete, their initial momentary deflection was increased. When on the beam, besides the permanent load FG also variable


load FQ has been applied, in the beams comes to crack appearance, decrease of the stiffness of the section due to the increase of the deflection with momentary initial deflection of this load. In the following 24 hours due to the influence of the concrete creep under the effect of the variable load, the initial momentary deflection was increased. After 24 hours after the unloading of the variable load comes to momentary elastic deflection to certain level and during the next 24 hours when only the permanent load acts due to the reversible creep there have been further decrease of the deflection.

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

10

2.68 2.77

2.75 2.85

2.82

2.89

2.85

2.92

2.88

2.94

2.90

2.96

2.93

2.98

2.94

3.00

2.97

1.68

1.63 1.73

1.69

1.76

1.72

1.78

1.74

1.79

1.75

1.82

1.78

1.83

1.79

1.85

1.82

0.5

0.52

0.53

0.53

0.54

F F

a

Beam: D3

De

flect

ion

a (

mm

)

Time t (days)

Figure 7.38 Development of the deflections for beam “D3” in 20 days after loading

0 50 100 150 200 250 300 350 4000

1

2

3

4

5

6

7

8

9

10

F F

a

Beams: D1 and D2 Beams: D3 and D4

De

flect

ion

a (

mm

)

Time t (days)

Figure 7.39 Development of the deflections during the time for series of beams “D” presented

with mean value of the beams D1 and D2 and for beams D3 and D4


At applying of the variable load the momentary deflection with respect to the deflection of the permanent load is increased for approximately 5 times, at beams made of ordinary concrete 5.19 times, and at beams made of high-strength concrete for 4.96 times. Although the ratio is equal, the beams made of high-strength concrete due to the higher elasticity modulus have smaller initial values of the deflections shown on figure 7.39.

It can be noticing that the increase of the deflections is more intensive when the concrete has smaller age at loading. At beams made of ordinary concrete subjected to permanent load FG and variable load FQ at concrete age t=40 days, the instantaneous deflection is 56.8% of the total deflection at age of 400 days. For the considered loading history, contribution of the creep in the development of the total deflection is 43.2%. For the beams made of high-strength concrete at same comparison, the momentary deflection is 71% of the total deflection, and the contribution of the deflection from the concrete creep in the total deflection is 29% at concrete age of 400 days. Deflections of the long-term effect of the load at series “E” beams

The development of the deflections during the time in the middle of the span is shown separately for the beam E1 made of ordinary concrete on figure 7.40, and for the beam E3 made of high-strength concrete on figure 7.41.

The values of the deflection in the moment of loading at concrete age t=40 days at level of load FG and at level of load FG+Q,as well as the values of the measured deflection during the time at concrete age t=400 days for the same levels of load are given in table 7.18.

The behaviour of the reinforced concrete beams of series “E” is approximately equal to the behaviour of the series “D” beams.

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F F

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Beam: E1

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F F

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Time t (days)



Beam E1 E2 E3 E4

aG(t0=40) [mm] 0.86 0.64 0.47 0.46

aG(t=400) [mm] 5.28 5.02 2.76 2.91

aG+Q(t0=40) [mm] 4.46 4.06 2.37 2.39

aG+Q(t=400) [mm] 6.77 6.54 3.88 4.08

Characteristic for this series of beams is that at level of loading with load FG+Q=(2x11.6)kN comes to crack appearance independently of the concrete quality.

Also at this series of beams due to the higher mechanical characteristics of the high-strength concrete, the deflections during the time at load FG+Q are smaller at beams made of high-strength concrete E3 and E4 comparing to the beams made of ordinary concrete E1 and E2. Beams made of high-strength concrete have for 40.2% smaller deflection during the time at concrete age t=400 days comparing to the beams made of ordinary concrete for level of load FG+Q. The difference at level of the deflection during the time at concrete age of 400 days at level of load FG is 45% smaller deflection at beams made of high-strength concrete comparing to the beams made of ordinary concrete.

This difference is much bigger comparing to the difference that exists between the elasticity modulus of the high-strength and the ordinary concrete, which is 24%. Due to that, great influence to the final deflections has the degradation of the stiffness of the section due to the cracks appearance, whose number is bigger at beams made of ordinary concrete; and the concrete creep which takes place at stresses b=0.386fck for ordinary concrete and b=0.20fck for high-strength concrete.


In order to understand the development of the deflections during the time, under the effect of permanent load FG and variable load, on figure 7.42 and 7.43 the development of the deflection during the time for 20 days after the loading is shown, for the beam E1 made of ordinary concrete and for the beam E3 made of high-strength concrete.

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7.23

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5.15

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5.59

2.86

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Time t (days)



As well as at series “D” beams under the effect of the concrete creep, the momentary deflections at applying of the variable load FQ added to the permanent load FG have been increased for time duration of the variable load of the beam. At unloading besides the momentary elastic deflection, the deflection has decreased under the effect of the reversible creep while only the permanent load acts.

At applying of the variable load the momentary deflection comparing to the deflection of permanent load, is increased for approximately 5.25 times, at beams made of ordinary concrete 5.72 times, and at beams made of high-strength concrete for 4.78. Although the ratio is equal, the beams made of high-strength concrete due to the higher elasticity modulus, have smaller initial values of the deflections in ratio to the beams made of ordinary concrete, which can be seeing from the diagram shown on figure 7.44.

It can be noticing that the increase of the deflections is more intensive when the concrete has smaller age at loading. At beams made of ordinary concrete subjected to permanent load FG and variable load FQ at concrete age t=40 days, the momentary deflections is 58% of the total deflection at concrete age of 400 days. For the considered loading history, the contribution of the creep in the development of the total deflection is 42%. For the beams made of high-strength concrete at same comparison the momentary deflections is 54.8% of the total deflection, and the contribution of the deflection of the concrete creep in the total strain is 45.2% at concrete age of 400 days.

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F F

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Beams: E1 and E2 Beams: E3 and E4

Def

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a (m

m)

Time t (days)

Figure 7.44 Development of the deflections during the time for series of beams “E” presented

with mean vale of the beams E1 and E2 and for the beams E3 and E4


Comparison of the deflections of the long-term load effect for series of beams “B” and “C”

The analysis of the effect of long-term permanent load is done based on the measured deflections at series “B” and “C” beams, where at series “B” beams the level of the load is FG=(2x4)kN and at series “C” beams the level of the load is FG+Q/2=(2x7.8)kN. The beams have been monitoring in period of 360 days. The deflections during the time in the middle of the span of the beams, expressed as mean value, are showing with the dependency diagrams deflection (a) – time (t) on figures 7.45 and 7.46.

Due to the difference in the level of the load, some differences appear in the behaviour of the reinforced concrete beams of these two series. Characteristic is that, under the effect of the load at beams made of ordinary concrete, from series “B” cracks does not appear, and at beams from series “C” comes to appearance and development of the cracks alongside the beam, which affects significantly the behaviour of the beams.

The series “B” beams have smaller initial deflections in the moment of loading, expressed as mean value of B1 and B2 and is 0.62mm, and at series, “C” beams the initial deflection, expressed as mean value of C1 and C2 is 2.41mm. The initial deflection in the moment of loading at series “B” beams is for 74.3% smaller comparing to the initial deflection at series “C” beams, because at series “B” beams there is no cracks appearance. The mean final value of the deflections at series “B” beams is 1.88mm, and at series, “C” beams 4.81mm. The final deflection at series “B” beams is smaller for 60.9% comparing to the series “C” beams. Due to the effects of the concrete creep, the increase of the initial deflection at series “B” beams is 1.26mm or 67% comparing to the final measured deflection, and at series ”C” beams the increase of the deflection is 2.40mm or 50% comparing to the final deflection.

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F F

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Beams: B1 and B2 Beams: C1 and C2

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a (

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Time t (days)

Figure 7.45 Development of the deflections during the time for series of beams “B” and “C”

made of ordinary concrete The deflection due to concrete creep is bigger at series “C” beams for 90% comparing to the deflection due to concrete creep at series “B” beams, which is due to the intensity of the stresses at which creep takes place. At series “B” beams the stresses at which creep takes


place are smaller b=0.071fck and at series “C” beams are b=0.284fck. The difference between the level of force FG+Q/2=(2x7.8)kN and the level FG=(2x4)kN is 95%.

The analysis of the results shows that the participation of the deflection of the concrete creep in the total deflection expressed in percentage is bigger at lower level of the load, such as at series “B” beams; and at higher level of the load, its participation is smaller, series “C” beams, in the final deflection of the reinforced concrete beams.

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F F

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Beams: B3 and B4 Beams: C3 and C4

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a (

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Time t (days)

Figure 7.46 Development of the deflections during the time for series of beams “B” and “C”

made of high-strength concrete

For beams made of high-strength concrete, for series “B” beams and series “C” beams, characteristic is that under the effect of the permanent load there is no cracks appearance although they are exposed to different level of load. The series “B” beams due to the lower level of the load have smaller initial deflection expressed as mean value of the beams B3 and B4, and they are 0.285mm; and at series “C” beams is measured mean initial value of 0.95mm. The beams made of high-strength concrete, series “B” beams, have for 70% smaller deflection comparing to the series “C” beams. The final measured deflection for the beams made of high-strength concrete from series “B” is 0.75mm, and at series, “C” beams 1.575mm. The difference in the measured final deflections is 52.4% smaller deflections at series “B” beams comparing to the series “C” beams. The increase of the initial deflection at series “B” beams made of high-strength concrete is 0.465mm i.e. 62% of the final deflection, and at series “C” 0.625mm i.e. 39.7%. The deflection of the concrete creep is for 34.3% bigger at series “C” beams comparing to the deflection of the concrete creep at series “B” beams, which is due to the intensity of the stresses at which creep takes place. At series “B” beams the stresses at which takes place the creep are smaller b=0.016fck, and at series “C” beams are b=0.098fck.

Due to the influence of the concrete creep, also at beams made of high-strength concrete can be noticed that the participation of the deflections due to concrete creep in the total strain is bigger at lower levels of load because has smaller initial deflections comparing to the bigger initial deflections at higher level of the load.

Due to the higher mechanical characteristics, the high-strength concrete has smaller initial deflections and smaller final deflections.


Comparison of the deflections of the long-term effect of the permanent load and the effect of the variable load for the series of beams “C”, “D” and “E”

The analysis of the influence of the permanent and variable load to the behaviour of the reinforced concrete beams is doing with the experimental results obtained with measurement of the deflections in the middle of the span for the series of beams “D” and “E”. At these series of beams, the series “D” beams are beams which are loaded with permanent load FG=(2x4)kN and variable load FQ=(2x7.6)kN which changes in cycles of loading/unloading by 24 hours. Series of beams “E” is fully loaded with same load as series of beams “D” except that the variable load is applying in cycles of loading/unloading by 48 hours. The deflections of the beams have been monitoring during 360 days. The deflections during the time in the middle of the span of the beams, expressed as mean value, are showing with the dependency diagrams deflection (a) - time (t) on figures 7.47 and 7.48.

Because subject of studying in the doctoral dissertation is the behaviour of the reinforced concrete beams under the effect of permanent and variable load, in order to determine the participation of the variable load as quasi-permanent load, in this analysis are concluded also the series “C” beams.

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F F

a

Beams: C1 and C2 Beams: D1 and D2 Beams: E1 and E2

Def

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a (m

m)

Time t (days)

Figure 7.47 Development of the deflection during the time for series of beams “C”, “D” and

“E” made of ordinary concrete The series of beams “C” is chosen in the experimental tests so that the effect of the permanent and variable load is replaced with the effect of the quasi-permanent load FG+Q/2 which consists of effect of the permanent load and a part of the variable load determined with assumption of 2=0.50.

The analysis of the results is given in the following table 7.19. In the series of beams “C”, because the method of applying of the load with full size is in one moment, at these beams comes to appearance and development of the cracks in the section of the beams, immediately after the loading. At series of beams “D” and “E” at applying of the permanent load FG does not come to cracks appearance, but at applying also of the permanent load when acts the force FG+Q=FG+FQ comes to appearance and development of


cracks. During the monitoring of the behavior of the reinforced concrete beams, the section of the beams is with cracks and enables comparison of the behavior at effect of different loading histories.

Table 7.19 Comparison of the measured deflections for the beams of series “C”, “D” and “E” made of ordinary concrete

Deflection Series of beams made of ordinary concrete

C(1-2) D(1-2) E(1-2)

aG(40) [mm] / 0.83 0.77

aG+Q/2(40) [mm] 2.41 / /

aG(400) [mm] / 4.89 5.15

aG+Q/2(400) [mm] 4.81 / /

aG+Q(40) [mm] / 3.60 3.86

aG+Q(400) [mm] / 6.34 6.66

From the results shown in the table can be noticed that at level of load FG, in the moment of loading at concrete age of 40 days, at series of beams “D” and “E” we have approximately equal deflections whose difference is only 7.8%.

At the level of the long-term permanent load, at concrete age of 400 days, can be noticed that the deflections at all series of beams are approximately equal and their difference is 5.3% which is due to the different loading history. Greatest are the deflections at beams of series “E” which has the most unfavorable loading history because the permanent load stays longer on the beam during 48 hours, unlike the loading history at beams of series “D” where the variable loads stays 24 hours. Therefore the deflections of the series “E” beams have the tendency to show bigger deflections comparing to the series “D” beams. The values of the measured deflections are approximately equal, with insignificant differences; because at series “E” beams we have longer time of unloading 48 hours, which contributes for development of bigger elastic deflections comparing to the elastic deflections at series “D” beams.

The comparison of the beams of series “D” and “E” and the series “C” at level of effect of the permanent load, shows that all beams have approximately equal deflection, which means that the effect of the variable load can be replaced with quasi-permanent load, approximately with coefficient 2=0.50, with certain required modifications and defining of the precise coefficient for reaching the required deflection. The series of beams “D” at level of the permanent load there is 1.6% bigger deflection than the series of beams “C”. The same is noticed also at series of beams “E” which at level of the permanent load have for 7.1% bigger deflection comparing to the deflection at series of beams “C”.

The measured deflections at level of permanent and variable load, at concrete age of 40 and 400 days, can be noticed that the deflections are for 7.2% bigger at series of beams “E” comparing to the series of beams “D” at concrete age of 40 days, and 5.0% at concrete age of 400 days. The deflections of the concrete creep at series of beams “D” and “E” are bigger comparing to the deflections ue to creep at series of beams “C” due to the higher level of stresses because of the loading history. The compressive stresses in the concrete in series “C” are b=0.284fck and at series of beams “D” and “E” b=0.386fck.


The deflections due to concrete creep at series of beams “D” and “E” are approximately equal i.e. 2.74mm and 2.80mm appropriately. These series of beams have for 16.7% bigger deflection due to concrete creep comparing to the deflection due to creep at series “C”.

Characteristic at series of beams “D” and “E” is the participation of the deflection due to creep in the total deflection, which is smaller comparing to the series of beams “B” and “C”. The participation of the deflection due to creep in the total deflection at level of load under the effect of permanent and variable load, at series of beams “D” is 43.2%, and at series of beams “E” is 42%. For comparison, the participation of the deflection due to creep in the total deflection, at series of beams “B” is 67%, and at series of beams “C” is 50%.

At beams made of high-strength concrete, can’t be made real comparison between the series “C” and the series of beams “D” and “E”, because at beams of series “C” there is no cracks appearance.

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F F

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Beams: C3 and C4 Beams: D3 and D4 Beams: E3 and E4

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Time t (days)

Figure 7.48 Development of the deflections during the time for series of beams “C”, “D” and

“E” made of high-strength concrete

The analysis of the results for the beams made of high-strength concrete is shown on figure 7.48 and in the following table 7.20.

At the level of the long-term permanent load, at age of 400 days, can be noticed that the deflections at all series of beams are approximately equal and the difference between them is for 4%, which is due to the different loading history. Greatest are the deflections at beams of series “E” which has the most unfavorable loading history because the permanent load stays longer on the beam during 48 hours, unlike the loading history at beams of series “D” where the variable loads stays 24 hours.

The measured deflections at level of the effect of permanent and variable load, at concrete age of 400 days, are for 34% bigger at series of beams “E”, comparing to the series of beams “D”.

The deflections due to concrete creep at series of beams “D” and “E” are taking place at compressive stresses b=0.146fck.


Table 7.20 Comparison of the measured deflections for beams of series “C”, “D” and “E” made of high-strength concrete

Deflection Series of beams made of high-strength concrete

C(3-4) D(3-4) E(3-4)

aG(40) [mm] / 0.55 0.47

aG+Q/2(40) [mm] 0.95 / /

aG(400) [mm] / 2.73 2.84

aG+Q/2(400) [mm] 1.58 / /

aG+Q(40) [mm] / 2.62 2.18

aG+Q(400) [mm] / 3.85 3.98

The deflections due to concrete creep at series of beams “D” and “E” are approximately equal i.e. 1.23mm and 1.80mm appropriately, which is due to the loading history, which is more unfavorable for series of beams “E”.

From the analysis of the results for the predicted level of loading, appears that the variable load in order to be replaced with quasi-permanent load, requires bigger participation of the variable in the quasi-permanent load, in order appearance and development of cracks to be caused in the section, and in that way the behavior of the reinforced concrete beams with the given loading history to be compared. According to the analysis of the results, the value of the coefficient of participation 2 for determination of the quasi-permanent load should be closer to 1, i.e. 2≈1.


7.7.2 Time-dependant cracks

During the monitoring of the reinforced concrete beams in the period of t=400 days the development of the cracks widths is measured at beams where comes to cracks appearance depending on the loading history.

Cracks depending on the loading history appeared at beams from series “C”, only at beams made of ordinary concrete for the effect of the long-term permanent load FG and 50% of the variable load, which acts as long-term load FQ/2 ,with total load FG+FQ/2=(2x4)+(2x7.8)kN and at beams from series “D” and “E” at beams made of ordinary and high-strength concrete. At beams from series “D” and “E” loading history consists of long-term permanent load FG=(2x4)kN and effect of repeated variable load FQ=(2x7.6)kN at which the cycles of loading/unloading are changing on 24 and 48 hours appropriately for each series of beams. The intensity of the load is chosen in a way to have cracks appearance at these beams at applying of the variable load.

At testing, development of the cracks width is measured on all cracks which have appeared in the middle part of the beam between the two concentrated forces at span of 80cm.

At beams made of ordinary concrete, immediately after the applying of the determined load all cracks were formed, their width was measured during the whole time and hasn’t come to appearance of new cracks in the period of monitoring of the beams t=40-400 days. At beams made of high-strength concrete for series of beams “D” and “E” after the applying of the variable load there was cracks appearance, and in the period of monitoring of the beams have come to appearance of another new crack. The test results are shown as mean value of the development of the crack width during the time, appropriately for beams from series “C” on the diagram shown on figure 7.49, for series “D” beams on the diagrams on figure 7.50, 7.51, 7.52 and 7.53; and for series ”E” beams on the diagrams on figure 7.54, 7.55, 7.56 and 7.57.

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Cra

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Beam C1

Beam C2

Figure 7.49 Diagram of development of the crack width w – time t for beams C1 and C2


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The analysis of the development of the crack width during the time is given in table 7.21. Table 7.21 Development of the crack width during the time

Beam Crack width w[mm]

wG+Q/2(t=0) wG+Q/2(t=400)

C1 0.1210 0.2699

C2 0.1594 0.2866

0.1402 0.2783

wG+Q(t=40) wG(t=40) wG+Q(t=400) wG(t=400)

D1 0.1713 0.0765 0.2342 0.1374

D2 0.1542 0.0625 0.2182 0.1171

0.1628 0.0695 0.2262 0.1273

D3 0.1021 0.0545 0.1196 0.0546

D4 0.0735 0.0407 0.0838 0.0407

0.0878 0.0476 0.1017 0.0477

wG+Q(t=40) wG(t=40) wG+Q(t=400) wG(t=400)

E1 0.1459 0.0766 0.1746 0.1270

E2 0.1689 0.1211 0.2276 0.1579

0.1574 0.0989 0.2011 0.1425

E3 0.0828 0.0560 0.1174 0.0863

E4 0.0737 0.0547 0.1176 0.0807

0.0783 0.0554 0.1175 0.0835

From the results shown in table 7.21 can be concluded that the development of the crack width during the time is greatest when long-term permanent load acts, as at series “C” beams, where under the effect of the load FG+Q/2 we have increase of the crack width during the time for approximately two times comparing to the initial deformation. At beams made of ordinary concrete, at beams from series “C” greatest crack widths were measured during the time comparing to the other series of beams “D” and “E” due to the long-term effect of the permanent load and part of the variable load, and also the effect of the higher level of the constant stress in the section of the beam.

At beams from series “D” made of ordinary concrete, the development of the crack width has been monitored at level of the long-term permanent load FG and at level of the load when acts the long-term permanent load FG and the repeated load FQ. On the same way also the beams from series “E” have been monitored.


Can be noticed that this loading history has bigger influence to the development of the crack width at level of the permanent long-term load where there is an increase of the crack width during the time for 83.2% comparing to the initial crack width at beams made of ordinary concrete. At level of the permanent and variable load the increase of the crack width during the time is 38.9% bigger comparing to the initial value of the crack width. This shows that to crack widths, besides the effect of the permanent load, great influence has also the effect of the repeated variable load. For the same series of beams, at beams made of high-strength concrete the development of the crack width during the time has increase of the crack width, during the time, for 15.8% comparing to the initial at level of the effect of the permanent and variable load. At the level of the long-term permanent load practically there is no increase of the initial crack width. The beams made of high-strength concrete have for 55.0% smaller crack widths during the time comparing to the beams made of ordinary concrete, which is due to the higher mechanical characteristics of the high-strength concrete, and with that are smaller the intensity of the stresses in the cross-section of the beams made of high-strength concrete.

The same development of the crack width during the time is noticed at beams from series “E”, but with certain retreats because at beams from series “E” during the time appears one more crack at beams made of ordinary concrete and two cracks at beams made of high-strength concrete during the considered period, and they show certain differences in the behaviour of the reinforced concrete elements of series “D”. Also this loading history has bigger influence to the development of the crack width at level of the long-term permanent load where we have increase of the crack width during the time for 44.1% comparing to the initial crack width. At level of the effect of the permanent and variable load the increase of the crack width during the time is 27.8% comparing to the initial value of the crack width. For the same series of beams, at beams made of high-strength concrete the development of the crack width during the time has increase of the crack width for 50.1% comparing to the initial at level of the effect of the permanent and variable load. At level of the long-term permanent load has practically same increase of 50.7%. The beams made of high-strength concrete have for 41.6% smaller crack width during the time comparing to the beams made of ordinary concrete, which is due to the higher mechanical characteristics of the high-strength concrete, and with that also the intensity of the stresses are smaller in the cross-section of the beams made of high-strength concrete.

Even if it is expected at beams from series “E” made of ordinary concrete to have bigger development of the crack width comparing to the beams from series “D”, this isn’t the case, due to the later appearance of new cracks during the time, which affect the final development of the crack width during the time and the cracks which already appeared.


7.7.3. Time-dependant strains in the concrete

At testing of short-term loads to failure in the cross-section of the reinforced concrete beams the strains are measured in the middle at the compression edge in the concrete, at level of the compression and tensile reinforcement and at control level, the level of the neutral axis. According to the concrete age at testing of the reinforced concrete beams, the measured strains can be divided in two groups:

Strains that are measured on the reinforced concrete beams of series “A” which are tested to short-term loads to failure at concrete age of t=40 days

Strains that are measured at reinforced concrete beams of the series “B”, “C”, “D”, “E” and “F” which were tested to short-term loads to failure at concrete age of t=400 days. Because these series of beams, besides the series “F” beams, were exposed to the effect of determined loading history in the period t=40-400 days before the testing to failure, the measurement of the strains can be divided in two parts:

1. Strains which are measured in the moment of applying of the loading history at concrete age of t=40 days up to the moment of testing with short-term loads to failure at concrete age of t=400 days, i.e. change of the strains during the time.

2. Strains at testing of short-term loads to failure at concrete age of t=400 days, after the finished monitoring of the beams during the time, which presents initial state at testing to failure.

The measuring of the strains in cross-section for reinforced concrete beams of series “A” at concrete age t=40 days and series “F” at concrete age of t=400 days, are beginning from zero condition when on the beams acts only the self-weight of the beam.

At testing to failure for the series “B” beams zero condition is when acts only the self-weight of the beam, even if previously in the period t=40-400 they are loaded with effect of the long-term permanent load FG, because doesn’t exist possibility the testing to failure to continue from the zero condition when load FG is acting.

For the beams of series “C”, “D” and “E” which in the period t=40-400 are exposed to certain loading history, the effect of permanent load FG and variable load FQ at testing to failure continues from the level of the permanent load FG.

When the short-term load is applied, one cycle of loading to the level of the exploitation load has been done (effect of permanent and variable load) and unloading to the level of the permanent load, and after that repeated loading is done and increase of the force up to reaching of physical failure of the beam.

The measuring of the strains is done in three cross-sections alongside the beam, in the middle of the span of the beam, and on the left and on the right of the section in the middle of the span at distance of 25cm. The strains are measured with mechanical deformation meter with base of 250mm and with measuring tapes.

From the results of the strains measuring only the results in the section in the middle of the span of the beam are shown. Strains of the long-term effect of the load for series “B” beams

Characteristic for the series “B” beams is that during the period t=40-400 days they are exposed to loading history which consists of effect of long-term permanent load with value FG. This load doesn’t cause crack appearance in the section in the period of monitoring of the behaviour of the beams. Because it wasn’t possible, at concrete age of t=400 days, from the value of the load FG to be continued with the testing to failure, the beams were first unloaded and then tested to failure.


The measured strains of the series “B” beams, at the monitoring of the behaviour of the determined loading history and the development of the strains during the time, and at testing to failure at concrete age t=400 days, for the beams made of ordinary concrete class C30/37, B1 and B2 are shown on figure 7.58, 7.59, 7.60 and 7.61, and for the beams made of high-strength concrete class C60/75, B3 and B4 are shown on figure 7.62, 7.63, 7.64 and 7.65.

2 -2

0

14

28

A2A1

B4B3

B2B1

D20

Beam B1

D10

D18D15

D12

D8D5

D2-0

.01

2

-0.4

72

0

-0.4

42

t=0 day t=50 days t=300 days t=10 days t=100 days t=330 days

0.11

20.

08

-0.2

58-0

.216

Hei

ght

(cm

)

0.05

20.

056

-0.1

28-0

.07

Strain (10-3)

Figure 7.58 Measured strains during the time of the effect of the permanent load FG for beam

B1 at concrete age t=40-400 days

2 -2

0

14

28

A2A1

B4B3

B2B1

D20

Beam B2

D10

D18D15

D12

D8D5

D2

0.02

8

-0.4

94

0.02

-0.4

72


0.12

0.09

6

-0.2

7-0

.242

Hei

ght

(cm

)

0.02

0.06

-0.1

22-0

.046

Strain (10-3)




2 -2

0

14

28

A2A1

B4B3

B2B1

D20

Beam B3

D10

D18D15

D12

D8D5

D2

0.02

-0.1

72

0.02

-0.1

58


0.09

20.

064

-0.0

86-0

.088

Hei

ght

(cm

)

0.03

20.

06

-0.0

74-0

.03

Strain (10-3)



2 -2

0

14

28

A2A1

B4B3

B2B1

D20

Beam B4

D10

D18D15

D12

D8D5

D2

0.01

2

-0.1

46

0.00

4

-0.1

32


0.10

40.

06

-0.0

74-0

.066

Hei

ght

(cm

)

0.03

60.

056

-0.0

48-0

.01

Strain (10-3)




The measured strains in the moment of appearance of the design load are shown in table 7.22.

Table 7.22 Measured strains in the section of the series “B” beams at monitoring of the strains during the time under the effect of long-term permanent load FG=(2x4)kN

Element Concrete

age

[days]

Measured strains

Concrete

b[‰]

Tension reinforcement

a[‰]

B1 t=40 -0.070 0.052

t=400 -0.472 -0.012

B2 t=40 -0.046 0.020

t=400 -0.494 0.028

B3 t=40 -0.030 0.032

t=400 -0.172 0.020

B4 t=40 -0.010 0.036

t=400 -0.146 0.012


Table 7.23 Measured strains in the section of the series “B” beams at testing of short-term loads to failure

Element Concrete age

[days]

Measured strains Force

Concrete

b[‰]


a[‰]

Force at theoretical failure

Ft[kN]

Force at physical failure

Fu[kN]

B1 t=400 -2.332 7.922 2x26 2x31.1

B2 t=400 -3.048 9.408 2x25 2x28.2

B3 t=400 -1.652 9.912 2x25 2x30.1

B4 t=400 -1.380 9.082 2x26 2x31.6

In the moment of the theoretical failure the measured strains in the compression edge of the concrete in the section are between b=2.332 3.048‰ at beams made of ordinary concrete, and at beams made of high-strength concrete b=1.6521.380‰. Can be noticed that the beams made of high-strength concrete have for 43.6% smaller value of the strain at the compression edge of the concrete, comparing to the beams made of ordinary concrete, which is due to the higher mechanical characteristics of the high-strength concrete.

The strain in the reinforcement in the moment of the theoretical failure for the beams made of ordinary concrete is in the limits between a=7.922 9.408‰, and for the beams made of high-strength concrete is in the limits between a=9.912 9.082‰ and the difference


between them is insignificant 8.8% which is due to the fact that the section of the beam is reinforced with same percentage of reinforcing.

The force at which the theoretical failure appears is identical for the beams made of ordinary concrete and for the beams made of high-strength concrete Ft=2x25.5kN, because the section of the element is designed so that to failure comes through the reinforcement and independently of the concrete quality, the theoretical failure depends only on the percentage of reinforcing of the section.

The contribution of the higher mechanical characteristics can be noticed at achieving of the force of the physical failure. It has been registered for 4.1% higher force of physical failure at beams made of high-strength concrete comparing to the force of physical failure at beams made of ordinary concrete.

The measured strains, from the effect of the long-term permanent load during the time, in the period of 360 days (concrete age t=400 days), at the compression edge of the concrete in the section for the beams made of ordinary concrete is in the limits between b=0.472 0.494‰ and at beams made of high-strength concrete in the limits between b=0.146 0.172‰. The participation of this strain in the total strain at failure is 18% at beams made of ordinary concrete and 10.5% at beams made of high-strength concrete.

Can be noticed that at testing of the beams to failure, at series “B” beams, great influence has the loading history in the period of 360 days (concrete age t=400 days) which causes initial strains at the compression edge of the concrete, and also the measured strains from the shrinkage of the concrete before applying of the load have great influence.

The measured strains at the level of the tension reinforcement, from the effect of the long-term permanent load during the time, are very small and practically they don’t have influence to the final strains at testing to failure of the beams, because this load in the period of loading doesn’t cause cracks appearance during the entire period of monitoring of the beams. Strains from the long-term effect of the load for series “C” beams

Characteristic for the series “C” beams is that during the period t=40-400 days they are exposed to loading history which consists of effect of the long-term permanent load with value FG and part of the variable load FQ/2 which acts as long-term permanent load with total value FG+FQ/2=FG+Q/2=(2x7.8)kN.

The loading history for series “C” beams is selected, to act as long-term permanent load with value FG+Q/2 in order data to be obtained with part of the effect of the variable load can be replaced and can act as quasi-permanent load.

This load causes cracks appearance at beams made of ordinary concrete, and doesn’t cause cracks appearance at beams made of high-strength concrete at monitoring of the behaviour of the beams during the period t=40-400 days.

At concrete age of t=400 days the beams are unloaded to the level of the load FG and then testing of short-term loads to failure was performed.

The measured strains at series “C” beams, at monitoring of the behaviour at certain loading history and the development of the strains during the time, and the testing to failure at concrete age t=40-400 days, for the beams made of ordinary concrete class C30/37, C1 and C2 are shown on figure 7.62, 7.63, 7.64 and 7.65, and for the beams made of high-strength concrete class C60/75, C3 and C4 are shown on figure 7.66, 7.67, 7.68 and 7.69.


2 -2

0

14

28

A2A1

B4B3

B2B1

D20

Beam C1

D10

D18D15

D12

D8D5

D2

0.8

-0.9

0.92

8

-0.9

68

0.93

2

-0.9

34


0.52

60.

486

-0.6

72-0

.622

Hei

ght

(cm

)

0.23

40.

378

-0.4

7-0

.322

Strain (10-3)

Figure 7.62 Measured strains during the time from the effect of the permanent load FG+Q/2 for

beam C1 at concrete age t=40-400 days

2 -2

0

14

28

A2A1

B4B3

B2B1

D20

Beam C2

D10

D18D15

D12

D8D5

D2

0.72

6

-0.7

78

0.89

-0.8

74

0.89

2

-0.8

46


0.89

8

0.85

6

-0.6

1-0

.566

Hei

ght

(cm

)

0.54

4

0.74

4

-0.4

18

-0.2

54

Strain (10-3)




2 -2

0

14

28

A2A1

B4B3

B2B1

D20

Beam C3

D10

D18D15

D12

D8D5

D2

0.27

6

-0.2

9

0.22

-0.2

82

0.15

4

-0.2

88


0.18

60.

168

-0.1

82-0

.186

Hei

ght

(cm

)

0.10

60.

142

-0.1

54-0

.09

Strain (10-3)



2 -2

0

14

28

A2A1

B4B3

B2B1

D20

Beam C4

D10

D18D15

D12

D8D5

D2

0.07

4

-0.2

08

0.10

8

-0.2

4

0.08

6

-0.2

48


0.14

40.

128

-0.1

56-0

.16

Hei

ght

(cm

)

0.08

6

0.11

-0.1

36-0

.086

Strain (10-3)





Table 7.24 Measured strains in the section of the series “C” beams at monitoring of the strains during the time under the effect of the long-term permanent load FG+Q/2=(2x7.8)kN.


[days]

Measured strains

Concrete

b[‰]


a[‰]

C1 t=40 -0.322 0.234

t=400 -0.900 0.800

C2 t=40 -0.254 0.544

t=400 -0.778 0.890

C3 t=40 -0.090 0.106

t=400 -0.290 0.276

C4 t=40 -0.086 0.086

t=400 -0.208 0.108


Table 7.25 Measured strains in the section of the series “C” beams at testing of short-term loads to failure


[days]


Concrete

b[‰]


a[‰]


Ft[kN]


Fu[kN]

C1 t=400 -2.960 8.592 2x25 2x28.2

C2 t=400 -2.848 4.940 2x25 2x27.6

C3 t=400 -1.712 9.240 2x26 2x30.8

C4 t=400 -1.770 10.142 2x26 2x32.8

In the moment of the theoretical failure the measured strains at the compression edge of the concrete are between b=2.848 2.960‰ at beams made of ordinary concrete, and at beams made of high-strength concrete b=1.7121.770‰. Can be noticed that the beams made of high-strength concrete have for 40% smaller value of the strain in the compression edge of the concrete comparing to the beams made of ordinary concrete, which is due to the higher mechanical characteristics of the high-strength concrete.

The strain in the reinforcement, in the moment of the theoretical failure, for the beams made of ordinary concrete is in the limits a=4.940 8.592‰ and for the beams made of high-strength concrete in the limits between a=9.240 10.142‰ and the difference between them


is insignificant and is 12.8%, which is due to the fact that the section of the beam is reinforced with the same percentage of reinforcement.

The force at which the theoretical failure appears is approximately identical for the beams made of ordinary concrete and the beams made of high-strength concrete, i.e. Ft=2x25kN and Ft=2x26kN appropriately, because the section of the element is designed so that the failure appears through the reinforcement and independently of the concrete quality, the theoretical failure depends only on the reinforcing percentage of the section.

The contribution of the higher mechanical characteristics of the high-strength concrete can be noticed at achieving of the force of the physical failure. It has been registered for 14% higher force of the physical failure at beams made of high-strength concrete comparing to the force of the physical failure at the beams made of ordinary concrete.

The measured strains, from the effect of the long-term permanent load during the time in the period of 360 days (concrete age t=400 days), at the compression edge of the section for the beams made of ordinary concrete are in the limits between b=0.778 0.900‰ and at beams made of high-strength concrete in the limits b=0.208 0.290‰. The participation of this strain in the total strain at failure is 28.9% at beams made of ordinary concrete and 14.3% at beams made of high-strength concrete.

The measured strains, from the effect of the long-term permanent load during the time in the period of 360 days (concrete age t=400 days), of the tension reinforcement for the beams made of ordinary concrete are in the limits between a=0.800 0.890‰ and at beams made of high-strength concrete in the limits a=0.108 0.276‰.. The participation of this strain in the total strain at failure is 9.8% at beams made of ordinary concrete and 2% at beams made of high-strength concrete.

At measuring of the strains can be noticed that the loading history at series “C” beams has greater influence comparing to the loading history at series “B” beams in the period of 360 days (concrete age t=400 days) which are causing greater initial strains in the compression edge of the concrete and at the level of the tension reinforcement due to the cracks appearance in the section of the beams made of ordinary concrete.

The measured strains at the level of the compression edge and at the level of the tension reinforcement, from the effect of the permanent load and part of the variable load which acts as permanent load during the time, are much smaller at beams made of high-strength concrete and they have very small influence to the final strains at testing to failure of the beams, because this load in the period of loading doesn’t cause cracks appearance. Strains from the long-term effect of the load for series “D” beams

Characteristic for the series “D” beams is that during the period t=40-400 days they are exposed to loading history which consists of effect of the long-term permanent load with value FG and variable load FQ which changes in cycles of loading and unloading by 24 hours and has value FG±FQ=(2x4)kN±(2x7.6)kN.

This load causes cracks appearance at beams made of ordinary and high-strength concrete at applying of the variable load at concrete age of t=40 days.

At concrete age of t=400 days, the beams are unloaded to the level of the load FG and then the testing of short-term loads to failure was performed.

The measured strains of the series “D” beams, at monitoring of the behaviour of the determined loading history and the development of the strains during the time and the testing to failure at concrete age t=400 days, for the beams made of ordinary concrete class C30/37, D1 and D2 are shown on figure 7.66 and 7.67 and for the beams made of high-strength concrete class C60/75, D3 and D4 are shown on figure 7.68 and 7.69.


2 -2

0

14

28

2 -2 2 -2

A2A1

B4B3

B2B1

D20

Beam D1

D10

D18D15

D12

D8D5

D2

G G+Q G G+Q

1.35

0.86

8

-0.3

04

-0.5

02

Hei

ght

(cm

)

0.06

6

1.18

8

-0.4

4

-0.0

98

Strain (10-3)

t=0 day t=10 days

Strain (10-3)

-0.3

82-0

.56

0.97

1.46

t=50 days

Strain (10-3)

-0.5

32-0

.72

1.06

61.

602

2 -2

0

14

28

2 -2 2 -2

A2A1

B4B3

B2B1

D20

Beam D1

D10

D18D15

D12

D8D5

D2

G+Q G

Hei

ght

(cm

)

1.64

4

1.10

2

-0.5

42-0

.762

Strain (10-3)

t=100 days t=300 days

Strain (10-3)

-0.8

08-1

.016

1.10

6

1.67

8

t=330 days

Strain (10-3)

-0.8

28-1

.038

1.08

6

1.68

6

Figure 7.66 Measured strains during the time from the effect of the permanent load FG and variable load FQ for beam D1 at concrete age t=40-400 days


2 -2

0

14

28

2 -2 2 -2

A2A1

B4B3

B2B1

D20

Beam D2

D10

D18D15

D12

D8D5

D2

G G+Q G G+Q

0.83

8

0.55

-0.3

56

-0.5

52

Hei

ght

(cm

)

0.09

4

0.77

4

-0.4

9

-0.1

14

Strain (10-3)

t=0 day t=10 days

Strain (10-3)

-0.4

28

-0.6

08

0.59

40.

904

t=50 days

Strain (10-3)

-0.5

84

-0.7

7

0.60

60.

954

2 -2

0

14

28

2 -2 2 -2

A2A1

B4B3

B2B1

D20

Beam D2

D10

D18D15

D12

D8D5

D2

G+Q G

Hei

ght

(cm

)

0.97

0.63

4

-0.6

16-0

.812

Strain (10-3)


Strain (10-3)

-0.8

5-1

.02

0.53

80.

832

t=330 days

Strain (10-3)

-0.8

52-1

.032

0.51

60.

822



2 -2

0

14

28

2 -2 2 -2

A2A1

B4B3

B2B1

D20

Beam D3

D10

D18D15

D12

D8D5

D2

G G+Q G G+Q

0.73

2

0.48

4

-0.1

78

-0.3

36

Hei

ght

(cm

)

0.04

8

0.68

6

-0.3

12

-0.0

62

Strain (10-3)

t=0 day t=10 days

Strain (10-3)

-0.2

14

-0.3

68

0.49

80.

796

t=50 days

Strain (10-3)

-0.2

32

-0.3

92

0.5

0.82

2

2 -2

0

14

28

2 -2 2 -2

A2A1

B4B3

B2B1

D20

Beam D3

D10

D18D15

D12

D8D5

D2

G+Q G

Hei

ght

(cm

)

0.84

60.

548

-0.2

38-0

.396

Strain (10-3)


Strain (10-3)

-0.3

56-0

.506

0.5

0.80

8

t=330 days

Strain (10-3)

-0.3

88-0

.54

0.49

80.

806



2 -20

14

28

2 -2 2 -2

A2A1

B4B3

B2B1

D20

Beam D4

D10

D18D15

D12

D8D5

D2

G G+Q G G+Q

1.31

2

0.87

2

-0.2

08

-0.3

8

Hei

ght

(cm

)

0.05

6

1.23

4

-0.3

52

-0.0

6

Strain (10-3)

t=0 day t=10 days

Strain (10-3)

-0.2

54

-0.4

08

0.94

21.

388

t=50 days

Strain (10-3)

-0.2

84

-0.4

46

0.97

21.

45

2 -2

0

14

28

2 -2 2 -2

A2A1

B4B3

B2B1

D20

Beam D4

D10

D18D15

D12

D8D5

D2

G+Q G

Hei

ght

(cm

)

1.46

61.

01

-0.2

96-0

.442

Strain (10-3)


Strain (10-3)

-0.3

94-0

.556

1.02

1.50

6

t=330 days

Strain (10-3)

-0.3

88-0

.582

1.03

21.

5




Table 7.26 Measured strains in the section of the series “D” beams at monitoring of the strains during the time under the effect of the permanent long-term load FG±FQ/2=(2x4)kN±(2x7.6)kN.

Element Load

F [kN]

Concrete age

[days]

Measured strains

Concrete

b[‰]


a[‰]

D1

FG=(2x4)kN t=40 -0.098 0.066

t=400 -0.828 1.086

FG±FQ=(2x4)±(2x7.6)kN t=40 -0.440 1.188

t=400 -1.038 1.686

D2

FG=(2x4)kN t=40 -0.114 0.094

t=400 -0.852 0.516

FG±FQ=(2x4)±(2x7.6)kN t=40 -0.490 0.774

t=400 -1.032 0.822

D3

FG=(2x4)kN t=40 -0.062 0.048

t=400 -0.388 0.498

FG±FQ=(2x4)±(2x7.6)kN t=40 -0.312 0.686

t=400 -0.540 0.806

D4

FG=(2x4)kN t=40 -0.060 0.056

t=400 -0.388 1.032

FG±FQ=(2x4)±(2x7.6)kN t=40 -0.352 1.234

t=400 -0.582 1.500


Table 7.27 Measured strains in the section of the series “D” beams at testing of short-term loads to failure


[days]


Concrete

b[‰]


a[‰]


Ft[kN]


Fu[kN]

D1 t=400 -2.470 11.116 2x24 2x29.2

D2 t=400 -2.398 6.742 2x26 2x28.2

D3 t=400 -1.886 7.846 2x27 2x34.7

D4 t=400 -1.810 9.780 2x27 2x32.8


In the moment of the theoretical failure the measured strains at the compression edge of the concrete in the section are between b=2.398 2.470‰ at beams made of ordinary concrete, and at beams made of high-strength concrete b=1.8101.886‰. Can be noticed that the beams made of high-strength concrete have for 24.1% smaller value of the strain in the compression edge of the concrete comparing to the beams made of ordinary concrete, which is due to the higher mechanical characteristics of the high-strength concrete.

The strain in the reinforcement in the moment of the theoretical failure for the beams made of ordinary concrete is in the limits between a=6.742 11.116‰, and for the beams made of high-strength concrete is in the limits between a=7.846 9.780‰, and the difference between them is insignificant, 1.3%, which is due to the fact that the section of the beam is reinforced with same reinforcing percentage.

The force at which the theoretical failure appears is approximately identical for the beams made of ordinary and high-strength concrete Ft=2x25kN and Ft=2x27kN appropriately, because the section of the element is also designed so that the failure appears through the reinforcement and independently on the concrete quality, the theoretical failure depends only on the reinforcing percentage of the section.

The contribution of the higher mechanical characteristics of the high-strength concrete can be noticed at achieving of the force of the physical failure. It has been registered for 17.8% higher force of the physical failure at beams made of high-strength concrete comparing to the force of the physical failure at the beams made of ordinary concrete.

The measured strains, from the effect of the permanent load and the variable load during the time in the period of 360 days (concrete age t=400 days), at the compression edge of the section for the beams made of ordinary concrete at level of the load FG are in the limits between b=0.828 0.852‰ and at beams made of high-strength concrete in the limits b=0.388 0.388‰. The participation of this strain in the total strain at failure is 34.5% at beams made of ordinary concrete and 21.0% at beams made of high-strength concrete.

The measured strains, from the effect of the permanent load and the variable load during the time in the period of 360 days (concrete age t=400 days), at level of the tension reinforcement for the beams made of ordinary concrete at level of the load FG are in the limits between a=0.516 1.086‰ and at beams made of high-strength concrete in the limits a=0.498 1.032‰. The participation of this strain in the total strain at failure is 9.0% at beams made of ordinary concrete and 8.7% at beams made of high-strength concrete.

At measuring of the strains can be noticed that the loading history at series “D” beams has greater influence comparing to the loading history at series “B” and “C” beams in the period of 360 days (concrete age t=400 days) which are causing greater initial strains in the compression edge of the concrete and at the level of the tension reinforcement due to the cracks appearance in the section of the beams made of ordinary concrete and due to the bigger value of the load. Strains from the long-term effect of the load for series “E” beams

Characteristic for the series “E” beams is that during the period t=40-400 days they are exposed to loading history which consists of effect of the long-term permanent load with value FG and variable load FQ which changes in cycles of loading and unloading by 48 hours and has value FG±FQ=(2x4)kN±(2x7.6)kN.

This load causes cracks appearance at beams made of ordinary and high-strength concrete at applying of the variable load at concrete age t=40 days. At concrete age of t=400 days the beams are unloaded to the level of the load FG and then the testing of short-term loads to failure was performed.

The measured strains of the series “E” beams, at monitoring of the behaviour of the determined loading history and the development of the strains during the time and the testing


to failure at concrete age t=400 days, for the beams made of ordinary concrete class C30/37, E1 and E2 are shown on figure 7.70 and 7.71 and for the beams made of high-strength concrete class C60/75, E3 and E4 are shown on figure 7.72 and 7.73.

2 -2

0

14

28

2 -2 2 -2

A2A1

B4B3

B2B1

D20

Beam E1

D10

D18D15

D12

D8D5

D2

G G+Q G G+Q

1.01

6

0.65

4

-0.3

46

-0.5

62

Hei

ght

(cm

)

0.06

8

0.93

6

-0.4

9

-0.1

04

Strain (10-3)

t=0 day t=10 days

Strain (10-3)-0

.41

-0.5

92

0.71

41.

096

t=50 days

Strain (10-3)

-0.5

38

-0.7

4

0.73

1.12

2 -2

0

14

28

2 -2 2 -2

A2A1

B4B3

B2B1

D20

Beam E1

D10

D18D15

D12

D8D5

D2

G+Q G

He

ight

(cm

)

1.16

60.

746

-0.5

62-0

.786

Strain (10-3)


Strain (10-3)

-0.8

04-1

.004

0.70

21.

094

t=330 days

Strain (10-3)

-0.8

4-1

.044

0.68

41.

076

Figure 7.70 Measured strains during the time from the effect of the permanent load FG and

variable load FQ for beam E1 at concrete age t=40-400 days


2 -20

14

28

2 -2 2 -2

A2A1

B4B3

B2B1

D20

Beam E2

D10

D18D15

D12

D8D5

D2

G G+Q G G+Q

1.24

4

0.83

6

-0.3

56

-0.5

48

Hei

ght

(cm

)

0.07

6

1.08

4

-0.4

68

-0.1

04

Strain (10-3)

t=0 day t=10 days

Strain (10-3)

-0.4

2-0

.59

6

0.89

81.

324

t=50 days

Strain (10-3)

-0.5

5-0

.74

6

0.95

61.

42

2 -2

0

14

28

2 -2 2 -2

A2A1

B4B3

B2B1

D20

Beam E2

D10

D18D15

D12

D8D5

D2

G+Q G

Hei

ght

(cm

)

1.47

60.

998

-0.5

96-0

.79

Strain (10-3)


Strain (10-3)

-0.8

1-0

.998

0.95

1.43

6

t=330 days

Strain (10-3)

-0.8

3-1

.038

0.95

21.

428

Figure 7.71 Measured strains during the time from the effect of the permanent load FG and variable load FQ for beam E2 at concrete age t=40-400 days


2 -20

14

28

2 -2 2 -2

A2A1

B4B3

B2B1

D20

Beam E3

D10

D18D15

D12

D8D5

D2

G G+Q G G+Q

0.44

4

0.29

-0.1

3

-0.2

38

Hei

ght

(cm

)

0.05

8

0.38

-0.2

2

-0.0

56

Strain (10-3)

t=0 day t=10 days

Strain (10-3)

-0.1

48

-0.2

52

0.30

80.

45

t=50 days

Strain (10-3)

-0.2

44

-0.3

98

0.65

0.96

6

2 -2

0

14

28

2 -2 2 -2

A2A1

B4B3

B2B1

D20

Beam E3

D10

D18D15

D12

D8D5

D2

G+Q G

Hei

ght

(cm

)

1.14

60.

738

-0.2

38-0

.39

Strain (10-3)


Strain (10-3)

-0.3

62-0

.498

0.71

41.

108

t=330 days

Strain (10-3)

-0.3

88-0

.542

0.71

1.10

6



2 -20

14

28

2 -2 2 -2

A2A1

B4B3

B2B1

D20

Beam E4

D10

D18D15

D12

D8D5

D2

G G+Q G G+Q

0.48

0.30

2

-0.1

4

-0.2

66

Hei

ght

(cm

)

0.06

2

0.45

-0.2

6

-0.0

72

Strain (10-3)

t=0 day t=10 days

Strain (10-3)

-0.1

72

-0.2

78

0.33

40.

492

t=50 days

Strain (10-3)

-0.2

18

-0.3

46

0.40

60.

618

2 -2

0

14

28

2 -2 2 -2

A2A1

B4B3

B2B1

D20

Beam E4

D10

D18D15

D12

D8D5

D2

G+Q G

Hei

ght

(cm

)

0.65

40.

432

-0.2

26-0

.364

Strain (10-3)


Strain (10-3)

-0.3

2-0

.448

0.36

80.

592

t=330 days

Strain (10-3)

-0.3

36-0

.466

0.36

0.57

4




Table 7.28 Measured strains in the section of the series “E” beams at monitoring of the strains during the time under the effect of the permanent long-term load FG±FQ/2=(2x4)kN±(2x7.6)kN.

Element Load

F [kN]

Concrete age

[days]

Measured strains

Concrete

b[‰]

Tension reinforcement a[‰]

E1

FG=(2x4)kN t=40 -0.104 0.068

t=400 -0.840 0.684

FG±FQ=(2x4)±(2x7.6)kN t=40 -0.490 0.936

t=400 -1.044 1.076

E2

FG=(2x4)kN t=40 -0.104 0.076

t=400 -0.830 0.952

FG±FQ=(2x4)±(2x7.6)kN t=40 -0.468 1.084

t=400 -1.038 1.428

E3

FG=(2x4)kN t=40 -0.056 0.058

t=400 -0.388 0.710

FG±FQ=(2x4)±(2x7.6)kN t=40 -0.220 0.380

t=400 -0.542 1.106

E4

FG=(2x4)kN t=40 -0.072 0.062

t=400 -0.336 0.360

FG±FQ=(2x4)±(2x7.6)kN t=40 -0.260 0.450

t=400 -0.466 0.574


Table 7.29 Measured strains in the section of the series “E” beams at testing of short-term loads to failure


[days]


Concrete

b[‰]


a[‰]


Ft[kN]


Fu[kN]

E1 t=400 -2.872 9.744 2x26 2x29.5

E2 t=400 -2.420 8.148 2x25 2x28.9

E3 t=400 -2.166 9.772 2x27 2x33.2

E4 t=400 -1.918 8.836 2x29 2x32.9


In the moment of the theoretical failure the measured strains at the compression edge of the concrete in the section are between b=2.420 2.872‰ at beams made of ordinary concrete, and at beams made of high-strength concrete b=1.9182.166‰. Can be noticed that the beams made of high-strength concrete have for 22.8% smaller value of the strain in the compression edge of the concrete comparing to the beams made of ordinary concrete, which is due to the higher mechanical characteristics of the high-strength concrete.

The strain in the reinforcement in the moment of the theoretical failure for the beams made of ordinary concrete is in the limits between a=8.148 9.744‰, and for the beams made of high-strength concrete is in the limits between a=8.836 9.772‰ and the difference between them is insignificant, 4%, which is due to the fact that the section of the beam is reinforced with same reinforcing percentage.

The force at which the theoretical failure appears is approximately identical for the beams made of ordinary and high-strength concrete Ft=2x25.5kN and Ft=2x28kN appropriately, because the section of the element is also designed so that the failure appears through the reinforcement and independently on the concrete quality, the theoretical failure depends only on the reinforcing percentage of the section. The contribution of the higher mechanical characteristics of the high-strength concrete can be noticed at achieving of the force of the physical failure. It has been registered for 13.4% higher force of the physical failure at beams made of high-strength concrete comparing to the force of the physical failure at the beams made of ordinary concrete.

The measured strains, from the effect of the permanent load and the variable load during the time in the period of 360 days (concrete age t=400 days), at the compression edge of the section for the beams made of ordinary concrete at level of the load FG are in the limits between b=0.830 0.840‰ and at beams made of high-strength concrete in the limits b=0.336 0.388‰. The participation of this strain in the total strain at failure is 31.6% at beams made of ordinary concrete and 17.7% at beams made of high-strength concrete.

The measured strains, from the effect of the permanent load and the variable load during the time in the period of 360 days (concrete age t=400 days), in the compression edge at level of the tension reinforcement for load FG for the beams made of ordinary concrete are in the limits between a=0.684 0.952‰and at beams made of high-strength concrete in the limits a=0.360 0.710‰. The participation of this strain in the total strain at failure is 9.1% at beams made of ordinary concrete and 5.8% at beams made of high-strength concrete.

At measuring of the strains can be noticed that the loading history at series “E” beams has similar influence in ratio to the loading history at series “D” beams in the period of 360 days (concrete age t=400 days). Comparison of the measured strains at testing of long-term effect of the load for all series of beams

In the tables 7.30 and 7.31 changes in the strains during the time in the period t=40-400 days are given for the beams made of ordinary and high-strength concrete.

The analysis of the results from the measured strains during the time at concrete age of 400 days at the compression edge and at the level of the tension reinforcement in the cross-section of the beams, shows that the loading history has great influence to the development of the strains during the time, due to the size of the loading for different series of beams and the influences of the shrinkage and the creep of the concrete.

Because the more unfavourable loading history and the intensity of the load are giving bigger measured strains during the time, which is expectable, analysis at level of the long-term load FG=(2x4)kN can be done at which the beams before the testing to failure are unloaded in order to determine the influence of the loading.


Table 7.30 Change of the strains in the cross-section at beams made of ordinary concrete CC-30, class C30/37 during the time

Beam Time period b [‰] b,m [‰] a [‰] a,m [‰] F[kN]

B1 t=40-400 -0.472 -0.483

0.012 0.020

FG=(2x4)kN

B2 t=40-400 -0.494 0.028 FG=(2x4)kN

C1 t=40-400 -0.968 -0.921

0.932 0.911

FG+Q/2=(2x7.8)kN

C2 t=40-400 -0.874 0.890 FG+Q/2=(2x7.8)kN

C1 t=400 -0.900 -0.839

0.800 0.763

FG=(2x4)kN

C2 t=400 -0.778 0.726 FG=(2x4)kN

D1 t=40-400 -1.038 -1.035

1.686 1.254

FG+FQ=(2x4)kN+(2x7.6)kN

D2 t=40-400 -1.032 0.822 FG+FQ=(2x4)kN+(2x7.6)kN

D1 t=400 -0.828 -0.840

1.086 0.801

FG=(2x4)kN

D2 t=400 -0.852 0.516 FG=(2x4)kN

E1 t=40-400 -1.044 -1.041

1.076 1.252


E2 t=40-400 -1.038 1.428 FG+FQ=(2x4)kN+(2x7.6)kN

E1 t=400 -0.840 -0.835

0.684 0.818

FG=(2x4)kN

E2 t=400 -0.830 0.952 FG=(2x4)kN

Can be noticed that at this level of the load, greatest strains are measured at series of beams “D” and “E” which are exposed to the effect of the long-term permanent load and repeated variable load in cycles loading/unloading by 24 hours and 48 hours appropriately, comparing to the series “B” beams which are exposed to the effect of the long-term permanent load FG=(2x4)kN. The strains in the concrete are for 72.9% and 73.9% bigger, appropriately for the series of beams “E” and “D”, comparing to the measured strains at series “B” beams, for the beams made of ordinary concrete. Of course, the state of the stresses should be taken into consideration which at series of beams “D” and “E” are causing cracks appearance, and at series “B” beams doesn’t have cracks appearance from the effect of the load.

The strains in the concrete are also for 73.9% bigger at series “C” beams comparing to the series “B” beams, even if the load at these beams consists of long-term load which is a sum of the permanent load and 50% of the variable load which acts as permanent load.

The strains in the reinforcement for the series of beams “C”, “D” and “E” are in the limits between 0.7630.8010.818‰ even if they are exposed to different loading history, their differences are in the limits between 5-7.2% comparing to the series “C” beams. The measured strains at the level of the tension reinforcement can’t be compared with the strains at series “B” beams because at these beams doesn’t have cracks appearance and the reinforcement is not fully active in the section.



Beam Time period b [‰] b,m [‰] a [‰] a,m [‰] F[kN]

B3 t=40-400 -0.172 -0.159

0.020 0.016

FG=(2x4)kN

B4 t=40-400 -0.146 0.012 FG=(2x4)kN

C3 t=40-400 -0.290 -0.265

0.220 0.164

FG+Q/2=(2x7.8)kN

C4 t=40-400 -0.240 0.108 FG+Q/2=(2x7.8)kN

C3 t=400 -0.290 -0.249

0.276 0.175

FG=(2x4)kN

C4 t=400 -0.208 0.074 FG=(2x4)kN

D3 t=40-400 -0.540 -0.561

0.806 1.153


D4 t=40-400 -0.582 1.500 FG+FQ=(2x4)kN+(2x7.6)kN

D3 t=400 -0.388 -0.388

0.498 0.765

FG=(2x4)kN

D4 t=400 -0.388 1.032 FG=(2x4)kN

E3 t=40-400 -0.542 -0.504

1.106 0.840


E4 t=40-400 -0.466 0.574 FG+FQ=(2x4)kN+(2x7.6)kN

E3 t=400 -0.388 -0.362

0.710 0.535

FG=(2x4)kN

E4 t=400 -0.336 0.360 FG=(2x4)kN

Characteristic for the beams made of high-strength concrete is that at series of beams “B” and “C” the loading history doesn’t cause cracks appearance during the time, and at series of beams “D” and “E” we have cracks appearance. From here there are differences in the behavior of the beams from the different series.

Also as at beams made of ordinary concrete, at beams made of high-strength concrete can be noticed that, at level of the load FG greatest strains in the concrete are measured at series of beams “D” and “E” which are exposed to the effect of the long-term permanent load and repeated variable load in cycles loading/unloading by 24 hours and 48 hours appropriately, comparing to the series “B” beams which are exposed to effect of the long-term permanent load FG=(2x4)kN. The strains in the concrete are for 144% and 127.6% bigger, appropriately for the series of beams “E” and “D”, than the measured strains at series “B” beams.

The strains in the concrete are also bigger for 56.6% at series of beams “C” comparing to the series “B” beams, even if the load at these beams consists of long-term load which is sum of the permanent load and 50% of the variable load which acts as permanent load.

The strains in the reinforcement for the series of beams “D” and “E” are in the limits between 0.5350.765‰. The differences are bigger and they are 43%, much bigger comparing to the beams made of ordinary concrete, but that doesn’t have influence to the number of cracks in moment of loading and crack occuring during the time in the considered period of loading.

8 Analytical solution of variable load effect to time-dependant behavior of concrete 151

8. Analytical solution of variable load effect to time-dependant behavior of concrete

For the problem of time-dependant behavior of reinforced concrete elements subjected to variable loads in time there are several methods such as: the principle of superposition, the algebraic relation like age adjusted effective elasticity modulus method and step-by-step procedure. All these methods include numerical or analytical procedures for the linear integral equations solution, then there is necessity of memorizing great number of data during the calculation, complicated preparation of the input data which in the phase of designing are unknown, such as: physical-mechanical and deformation characteristics of concrete, external environment conditions and loading history. Insisting to accurate numerical solutions doesn’t have sense. Therefore by caring out experiments and using simple models we can obtain simple solutions applicable in the practice. For this approach to the solution of defining the influences of the variable loads to the time-dependant behavior of reinforced concrete elements, several methods can be used:

Variable load effect to be taken into account with appropriate compliance function

Variable load effect to be taken by introduction of fictive age at loading

Variable load to be replaced with the effect of the quasi-permanent load, i.e. part of the variable load to act as permanent load

8.1 Procedure with quasi-permanent load

The method using quasi-permanent load is defined to calculate creep effects when serviceability limit state design should be performed using quasi-permanent combination of actions. This load combination provides possibility to calculate concrete creep effects, which are mostly related only with the effect of the long-term permanent load, but to take into consideration also the variable load effect. This is done in manner that to permanent load part of the variable load is added which acts as quasi-permanent load on structure determined by the quasi-permanent coefficient. (EN1992-1-1: Eurocode 2: Design of concrete structures, General rules and rules for buildings).

The quasi-permanent value of the variable load is obtained by the quasi-permanent coefficient of participation of the variable load 2. This coefficient depends on the category of the building and from the load history. But to determine this coefficient there is no data needed for the state of the stresses in section by permanent and variable loads. This is especially important when due to the cracks appearance in the sections at certain phases of loading and unloading, stiffness degradation appears in the concrete element.

For further analytical analysis of the experiment data first evaluation of autogenous shrinkage, drying shrinkage and creep strains was performed by using B3 Model (Bazant, Baweja 1995). This analysis will provide data to define improved B3 model compliance function, aging coefficient (t,t') and relaxation function R(t,t'). Than the previously mentioned information will be used to calculate and verify time-dependant deflections by using Age adjusted effective modulus method (AAEM method). To calculate variable load effects on time-dependant deflections on concrete elements quasi-permanent load procedure and principle of superposition will be used.

The analysis of the results, according to quasi-permanent load procedure, will be performed for the series of beams "D" and "E" to obtain time-dependant deflections under action of variable loads.

From the experimental researches, on figure 9.1, the experimental development of the deflections is shown during time for the series of beams D” made of ordinary and high-strength concrete. This series of beams were loaded with the long-term permanent load FG=(2x4)kN and repeated variable load FQ=(2x7.6)kN history in cycles of loading/unloading by 24 hours. The beams were loaded at concrete age of 40 days and the development of the deflections was monitored in the following 360 days at concrete age of 400 days.


Figure 8.1 Considered variation possibilities For the possible solution of the variable load effects on concrete elements, four solution variants were proposed shown on figure 9.1 [13]:

1) To determine quasi-permanent load from point of zero value of deflection to obtain the value of experimentally determined deflection at the level of action of the permanent load.

2) To determine quasi-permanent load from point of zero value of deflection to obtain the value of experimentally determined deflection at the level of action of the permanent and variable load.

3) To determine quasi-permanent load from point of instantaneous value of deflection at permanent load level, (obtained at the first cycle of loading/unloading by variable load) to obtain the value of experimentally determined deflection at action of the permanent load.

4) To determine quasi-permanent load from point of instantaneous value of deflection at permanent load level, (obtained at the first cycle of loading/unloading by variable load), to obtain the value of experimentally determined deflection at action of the permanent and variable load.

In order to compare the calculated quasi-permanent coefficient for the participation of the variable load as quasi-permanent load with the value of these coefficients given in Eurocode 2, first variant of the solution was analyzed.

The simplest solution of the problem is taking into consideration the variant 1 because with intensity of the load, as sum of permanent and quasi-permanent load, we can obtain initial and time-dependant deflection. Using variant 1 it can be written:

QGaQGaQGa 2t20exp,t [13] ..................................................... (8.1)


Where:

at,exp(G+Q)- experimentally obtained deflection under the effect of the long-term permanent load G and repeated variable load Q.

a0(G+2Q)- analytically obtained instantaneous deflection under the effect of long-term permanent load G and quasi-permanent load 2Q, which represents part of the variable load.

at(G+2Q)- Analytically obtained time-dependant deflection under the effect of the long-term permanent load G and quasi-permanent load 2Q, which represents part of the variable load.

This variant of the solution have one imperfection in instantaneous deflection calculation, which for the real loading history, is not correct. 8.2 Analytical analysis

According to Creep and Shrinkage Prediction Model for Analysis and Design of concrete structures: Model B3, analytical analysis was performed to verify experimental results from testing of creep, drying and autogenous shrinkage by original model and by improved estimation updating the compliance function based on test results. This analysis was performed for the both concrete classes C30/37 and C60/75. Ordinary concrete C30/37

On the following figure 8.2 diagrams of autogenos shrinkage, drying shrinkage and creep obtained from experimental results are shown for concrete class C30/37.

000000.00E+00

000250.00E-06

000500.00E-06

000750.00E-06

001000.00E-06

001250.00E-06

001500.00E-06

001750.00E-06

0 40 80 120 160 200 240 280 320 360 400 440

t [days]

Def

orm

atio

n ε

[10-6

]μs

Concrete creep-C30/37

Drying shrinkage-C30/37

Autogenous shrinkage-C30/37


Experimental results from testing of drying shrinkage, autogenous shrinkage and creep of concrete were analytically verified using Model B3 and updated B3 on the basis of long-term test. Results from analytical analysis are given in the following figures 8.3, 8.4, 8.5, 8.6, 8.7, 8.8, 8.9 and 8.10.


000000.00E+00

000250.00E-06

000500.00E-06

000750.00E-06

001000.00E-06

001250.00E-06

001500.00E-06

001750.00E-06

0 40 80 120 160 200 240 280 320 360 400 440

t [days]

Def

orm

atio

n ε

[10-6

]μs

CC-C30/37-experimental

B3-model

B3-model improved


000000.00E+00

000250.00E-06

000500.00E-06

000750.00E-06

001000.00E-06

001250.00E-06

001500.00E-06

001750.00E-06

0 40 80 120 160 200 240 280 320 360 400 440

t [days]

Def

orm

atio

n ε

[10-6

]μs


B3 model

B3-model improved

Figure 8.4 Diagram cc-t creep of concrete class C30/37: experimental results, B3-model and

B3-model improved


000000.00E+00

000250.00E-06

000500.00E-06

000750.00E-06

001000.00E-06

001250.00E-06

001500.00E-06

001750.00E-06

0 40 80 120 160 200 240 280 320 360 400 440

t [days]

Def

orm

atio

n ε

[10-6

]μs


B3-model improved

B3-model improved w ith humidity


000000.00E+00

000250.00E-06

000500.00E-06

000750.00E-06

001000.00E-06

001250.00E-06

001500.00E-06

001750.00E-06

0 40 80 120 160 200 240 280 320 360 400 440

t [days]

Def

orm

atio

n ε

[10-6

]μs


B3-model improved

B3 model improved w ith humidity



000000.00E+00

000250.00E-06

000500.00E-06

000750.00E-06

001000.00E-06

001250.00E-06

001500.00E-06

001750.00E-06

0 40 80 120 160 200 240 280 320 360 400 440

t [days]

Def

orm

atio

n ε

[10-6

]μs

DS-C30/37-experimental

B3-model

B3-model improved

Figure 8.7 Diagram ds-t drying shrinkage of concrete class C30/37: experimental results, B3-

model and B3-model improved

000000.00E+00

000250.00E-06

000500.00E-06

000750.00E-06

001000.00E-06

001250.00E-06

001500.00E-06

001750.00E-06

0 40 80 120 160 200 240 280 320 360 400 440

t [days]

Def

orm

atio

n ε

[10-6

]μs


B3-model improved


Figure 8.8 Diagram ds-t drying shrinkage of concrete class C30/37: experimental results, B3-

model and B3-model improved including real humidity


40

50

60

70

80000.E+00 050.E-06 100.E-06 150.E-06 200.E-06 250.E-06 300.E-06 350.E-06 400.E-06 450.E-06 500.E-06 550.E-06

Drying shrinkage strain [10-6]s

Hu

mid

ity

[%]


t=40 days

t=80 days

t=120 days

t=160 days

t=200 days

t=240 days


000000.00E+00

000125.00E-06

000250.00E-06

000375.00E-06

000500.00E-06

0 40 80 120 160 200 240 280 320 360 400 440

t [days]

Def

orm

atio

n ε

[10-6

]μs

AS-C30/37-experimental

B3-model

B3 model improved

Figure 8.10 Diagram as-t autogenous shrinkage of concrete class C30/37: experimental results, B3-model and B3-model improved


This analysis using B3 model shows that we have good agreement between the experimental and analytical results especially for the creep strain and creep compliance. Better results were obtained by using improved B3 model proposing coefficient p1 and p2 according to experimental results on the basis of square linear regression and including real humidity recorded during testing. For ordinary concrete C30/37, on the basis of linear regression, following values were obtained p1=0.9088 and p2=0.998 to adjust the creep compliance (figures 8.3, 8.4, 8.5 and 8.6).

For the drying shrinkage also according to B3 model an updating shrinkage prediction was carried out on the basis of experimental results to obtain scaling parameter p6. Real updating of shrinkage proposed in the model B3 is given by the procedure of using w∞(h) parameter. This parameter represent the estimate of the final relative loss for drying at given environmental relative humidity h for which should be taken additional test results for the final water loss that would occur at environmental relative humidity zero. Updating of shrinkage predictions was done only by using scaling parameter which for ordinary concrete C30/37 is p6=0.82 (figures 8.7 and 8.8)

From the analytical analysis it also can be seen that humidity has major influence on drying shrinkage and no effect on creep compliance. Variance in the humidity as it is shown in figure 8.9 causes variations of shrinkage strain. That's why instead of straight line for given constant humidity on the diagram we have irregular line that depends on the value of relative humidity. Also from the figure 8.9 it can be seen that when we have higher values of relative shrinkage slower developing or no developing of shrinkage strain was obtained. Some recovery in shrinkage strain was noticed when in longer period we have higher value of relative humidity above the mean relative humidity. In cases of lower values of relative humidity we have gradually higher value of shrinkage strain. This analytical analysis proves the fact that relative humidity has major influence on drying shrinkage.

Analysis of autogenous shrinkage using B3 model shows that proposed formula for autogenous shrinkage predictions is not valid for use neither for ordinary concrete and neither for high-strength concrete. In analysis of experimental data I have tried to use the formula proposed for calculation of autogenous shrinkage by model B3:

)t(S)h99.0()t( aaaa [28] ....................................................................... (8.2) But during the calculation there is problem when for t )h99.0()t( aaa was

obtained because 1tSa . If %80ha , as Bazant proposed in that case we have

aa 19.0)t( or with other words said it doesn't converge in aa )t( . Analytical analysis of the autogenous shrinkage was done in accordance of improvement of the proposed formula by new formula:

)t(S)h99.0()t( att

1

aaas

...................................................................... (8.3) Results from the autogenous shrinkage analysis by original B3 model and are given on figure 8.10.

The analytical analysis proves experimentally obtained results and gives enough input information for the calculation of variable load effects on time-dependant deflections.


High-strength concrete C60/75

On the following figure 8.11 diagrams of autogenos shrinkage, drying shrinkage and creep obtained from experimental results are shown for concrete class C60/75.

000000.00E+00

000250.00E-06

000500.00E-06

000750.00E-06

001000.00E-06

001250.00E-06

001500.00E-06

001750.00E-06

0 40 80 120 160 200 240 280 320 360 400 440

t [days]

Def

orm

atio

n ε

[10-6

]μs

Concrete creep-C60/75

Drying shrinkage-C60/75

Autogenous shrinkage-C60/75

Figure 8.11Diagram as-t autogenous shrinkage, ds-t drying shrinkage and cc-t creep of concrete class C60/75

Experimental results from testing of drying shrinkage, autogenous shrinkage and creep of concrete and B3 model analysis are given in the following figures 8.12, 8.13, 8.14, 8.15, 8.16, 8.17, 8.18 and 8.19.

000000.00E+00

000250.00E-06

000500.00E-06

000750.00E-06

001000.00E-06

001250.00E-06

001500.00E-06

001750.00E-06

0 40 80 120 160 200 240 280 320 360 400 440

t [days]

Def

orm

atio

n ε

[10-6

]μs


B3-model

B3-model improved

Figure 8.12 Diagram dc and cc-t creep of concrete class C60/75: experimental results,

B3-model and B3-model improved


000000.00E+00

000250.00E-06

000500.00E-06

000750.00E-06

001000.00E-06

001250.00E-06

001500.00E-06

001750.00E-06

0 40 80 120 160 200 240 280 320 360 400 440

t [days]

Def

orm

atio

n ε

[10-6

]μs


B3 model

B3-model improved


000000.00E+00

000250.00E-06

000500.00E-06

000750.00E-06

001000.00E-06

001250.00E-06

001500.00E-06

001750.00E-06

0 40 80 120 160 200 240 280 320 360 400 440

t [days]

Def

orm

atio

n ε

[10-6

]μs


B3-model improved




000000.00E+00

000250.00E-06

000500.00E-06

000750.00E-06

001000.00E-06

001250.00E-06

001500.00E-06

001750.00E-06

0 40 80 120 160 200 240 280 320 360 400 440

t [days]

Def

orm

atio

n ε

[10-6

]μs


B3-model improved

B3 model improved w ith humidity


000000.00E+00

000250.00E-06

000500.00E-06

000750.00E-06

001000.00E-06

001250.00E-06

001500.00E-06

001750.00E-06

0 40 80 120 160 200 240 280 320 360 400 440

t [days]

Def

orm

atio

n ε

[10-6

]μs


B3-model

B3-model improved



000000.00E+00

000250.00E-06

000500.00E-06

000750.00E-06

001000.00E-06

001250.00E-06

001500.00E-06

001750.00E-06

0 40 80 120 160 200 240 280 320 360 400 440

t [days]

Def

orm

atio

n ε

[10-6

]μs


B3-model improved



40

50

60

70

80000.E+00 050.E-06 100.E-06 150.E-06 200.E-06 250.E-06 300.E-06 350.E-06 400.E-06 450.E-06 500.E-06 550.E-06

Drying shrinkage strain [10-6]s

Hu

mid

ity

[%]


t=40 days

t=80 days

t=120

t=160

t=200

t=240 days



000000.00E+00

000125.00E-06

000250.00E-06

000375.00E-06

000500.00E-06

0 40 80 120 160 200 240 280 320 360 400 440

t [days]

Def

orm

atio

n ε

[10-6

]μs

AS-C60/75-experimental

B3-model

B3-model improved

Figure 8.19 Diagram as-t autogenous shrinkage of concrete class C60/75: experimental results, B3-model and B3-model improved including real humidity

Analysis of experimental results for the drying shrinkage, autogenous shrinkage and creep compliance for high-strength concrete C60/75 also shows good agreement with the analytically obtained results by B3 model (figures 8.12, 8.13, 8.14 and 8.15). This analysis also provide enough information for input parameters needed in calculation of time-dependant deflections of concrete elements subjected to variable loads using age adjusted effective modulus method.

For high strength concrete C60/75, on the basis of linear regression, following values were obtained p1=1.4264 and p2=0.3319 to adjust the creep compliance in B3 model (figures 8.12, 8.13, 8.14 and 8.15).

From the analysis it can be concluded that for high-strength concrete prediction of creep compliance is also concerning about defining the drying and autogenous shrinkage as major issues of the problem. For the high-strength concrete also prediction of drying shrinkage is a problem because of instable relative humidity during testing period (figures 8.16, 8.17 and 8.18). Updating of shrinkage predictions was done only by using scaling parameter which for high strength concrete C60/75 is p6=1.0955 (figures 8.16 and 8.17)

Proposed formula for autogenous shrinkage predictions is better fitted to the experimentally obtained results for high-strength concrete than for ordinary concrete (figure 8.19).

For both concretes this analysis using model B3 provide necessary information about creep, total shrinkage, aging coefficient and relaxation function as an input parameters to carried out further analysis by age adjusting effective modulus method. Using AAEM method, variable load effects will be calculated to obtain the time-dependant deflections of concrete beams subjected to different loading histories.

On the following figures 8.20, 8.21, 8.22, 8.23, 8.24, 8.25, 8.26 8.27, 8.28 and 8.29 time-dependent behavior of ordinary concrete class C30/37 was presented through diagrams of deflections vs.time for series of beams "B", "C", "D" and "E" obtained by the experimental


investigation and analytical solution by AAEM method using input parameters from previous analysis by B3 model. Also analysis was performed using normalized deflection to obtain the increment of time-dependant deflection function.

For series "B" beams there are insignificant difference between experimental and analytical results using AAEM method. Instantaneous deflection obtained by experimental and analytical results have difference of 4.3%, and time-dependant deflections are almost the same (figure 8.20). Difference of 0.1% was calculated (figure 8.20). Normalized deflections practically give the same shape of curve for the development of time-dependant deflections (figure 8.21).

0

0.5

1

1.5

2

2.5

0 50 100 150 200 250 300 350 400

t[days]

Def

lect

ion

a[m

m]

Beam B1 and B2-experimental

Model B3

Model B3 improved

Figure 8.20 Diagram development of the deflection during time for series "B" beams concrete

class C30/37

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250 300 350 400

t[days]

No

rmal

ized

def

lect

ion

a/a

ma

x


Model B3

Model B3 improved

Figure 8.21 Diagram normalized deflection a/amax vs. time for series "B" beams concrete

class C30/37


For series "C" beams subjected to permanent load history of G+Q/2 there are insignificant difference between experimental and analytical results using AAEM method except at point of instantaneous deflection. Difference of 26.7% was recorded. On this result major influence have tensile strength of concrete at the time when cracks appear in the beam. Time-dependant deflections are almost the same (figure 8.22). Normalized deflections practically give the same shape of curve for development of time-dependant deflections (figure 8.22) and differences at determining of instantaneous deflection at time when load is applied.

0

1

2

3

4

5

6

0 50 100 150 200 250 300 350 400

t[days]

Def

lect

ion

a[m

m]

Beam C1 and C2-experimental

Model B3

Model B3 improved

Figure 8.22 Diagram development of the deflection during time for series "C" beams concrete

class C30/37

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250 300 350 400

t[days]

No

rmal

ized

def

lect

ion

a/a

ma

x

Beam C1 and C2-experiment

Model B3 (C1 and C2)

Model B3 improved (C1 and C2)



On figure 8.24 normalized deflection diagrams are given for beams of series "B" and "C" made of ordinary concrete class C30/37. Analysis of the normalized deflection diagrams for beams subjected to different loading history almost have the same behavior despite the fact that beams of series "B" have no cracks during time, but beams of series "C" have crack appearance at the moment when load was applied.

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250 300 350 400

t[days]

No

rmal

ized

def

lect

ion

a/a

ma

x


Model B3

Model B3 improved




Figure 8.24 Diagram normalized deflection a/amax vs. time for series "B" and "C" beams

concrete class C30/37 On the following figures 8.25 and 8.26 the behavior of concrete beams series "D" is presented, subjected to long-term permanent load G and repeated variable load Q in cycles of loading/unloading by 24 hours.

0

1

2

3

4

5

6

7

0 50 100 150 200 250 300 350 400

t[days]

Def

lect

ion

a[m

m]

Beam D1 and D2-experimental

Model B3

Model B3 improved

Figure 8.25 Diagram development of the deflection during time for series "D" beams concrete

class C30/37


For series "D" analytical solution was defined in a manner to replace variable load effect on deflection by using quasi-permanent load to define time-dependant deflection at level of permanent load. According to the analysis presented on the figure 8.56 best results were obtained when quasi-permanent coefficient was set to 2=0.49. This means that approximately 50% of the variable load has to be taken as quasi-permanent load. Analysis by normalized deflections also gives equal behavior of concrete is the same compared to experimental and analytical results (figure 8.26).

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250 300 350 400

t[days]

No

rmal

ized

def

lect

ion

a/a

ma

x

Beam D1 and D2-experiment

Model B3 (D1 and D2)

Model B3 improved (D1 and D2)

Figure 8.26 Diagram normalized deflection a/amax vs. time for series "D" beams concrete

class C30/37

On the figures 8.27 and 8.28 behavior of concrete beams series "E" is presented, subjected to long-term load G and repeated variable load Q in cycles of loading/unloading by 48 hours.

0

1

2

3

4

5

6

7

0 50 100 150 200 250 300 350 400

t[days]

Def

lect

ion

a[m

m]

Beam E1 and E2-experimental

Model B3

Model B3 improved

Figure 8.27 Diagram development of the deflection during time for series "E" beams concrete

class C30/37


For series "D" beams according to the diagrams presented on the figure 8.27 best results were obtained when quasi-permanent coefficient was set to 2=0.66. Analysis by normalized deflections gives equal behavior of concrete when experimental and analytical results were compared.

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250 300 350 400

t[days]

No

rmal

ized

def

lect

ion

a/a

ma

x

Beam E1 and E2-experiment

Model B3 (E1 and E2)

Model B3 improved (E1 and E2)

Figure 8.28 Diagram normalized deflection a/amax vs. time for series "E" beams concrete class C30/37

On figure 8.29 normalized diagrams were presented for beams of series "D" and "E". Despite the differences in the loading history shows practically equal time-dependant behavior of ordinary conrete class C30/37. That means that for these loading history variable load effect can be described by same constitutive law.

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250 300 350 400

t[days]

No

rmal

ized

def

lect

ion

a/a

ma

x

Beam D1 and D2-experiment

Model B3 (D1 and D2)

Model B3 improved (D1 and D2)

Beam E1 and E2-experiment

Model B3 (E1 and E2)

Model B3 improved (E1 and E2)

Figure 8.29 Diagram normalized deflection a/amax vs. time for series "D" and "E" beams concrete class C30/37


Time-dependant behavior of high-strength concrete for beams of series "B" is presented on the following figures 8.30 and 8.31. These beams are subjected to long-term permanent load G that causes no crack appearance. Remarkable differences are obtained at point of instantaneous deflection for 47% between experimentally and analytical results. At final deflection differences are negligible of 4.8%. Analysis by normalized deflections gives greater differences in the behavior at early age after loading comparing experimental and analytical results. This indicates that for time-dependant behavior of high-strength concrete some refinements are needed in the AAEM method.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 50 100 150 200 250 300 350 400

t[days]

Def

lect

ion

a[m

m]


Model B3

Model B3 improved

Figure 8.30 Diagram development of the deflection during time for series "B" beams concrete

class C60/75

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250 300 350 400

t[days]

No

rmal

ized

def

lect

ion

a/a

ma

x


Model B3 (B3 and B4)

Model B3 improved (B3 and B4)



Same behavior, as for beams "B", was obtained and for high-strength concrete beams of series "C. Time-dependant behavior of beams is presented on the figures 8.32 and 8.33 represented with diagrams deflection vs. time comparing experimental and analytical results. At this diagram we have same behaviour of high-strength concrete by experimental and analytical results.

0

0.5

1

1.5

2

2.5

3

0 50 100 150 200 250 300 350 400

t[days]

Def

lect

ion

a[m

m]

Beam C3 and C4-experimental

Model B3

Model B3 improved

Figure 8.32 Diagram development of the deflection during time for series "C" beams concrete

class C60/75

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250 300 350 400

t[days]

No

rmal

ized

def

lect

ion

a/a

ma

x






Analysis of the experimental and analytical results for the time-dependant behavior of concrete subjected to long-term permanent load G and repeated variable load Q using quasi-permanent load shows some differences in the behavior. Using 2=0.5 the analytical solution for determining variable load effect on deflection is far from true experimental behavior. We could receive approximately right solution for the time-dependant behavior if 2=1. But that means that variable load should be treated as permanent which is not true. This behavior of high-strength concrete is due to that service loads are near the value of cracking load and the beams are in so called stabilized state of cracking.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 50 100 150 200 250 300 350 400

t[days]

Def

lect

ion

a[m

m]


Model B3

Model B3 improved


class C60/75 under permanent load and quasi permanent load 2=0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 50 100 150 200 250 300 350 400

t[days]

Def

lect

ion

a[m

m]


Model B3

Model B3 improved


class C60/75 under permanent load and quasi permanent load 2=1.0


Normalized deflection diagrams presented on figures 8.36 and 8.37 for ordinary and high-strength concrete beams indicates that almost the same constitutive law can describe the time-dependant behavior for both concretes. Some differences are in determining the instantaneous deflection at time of loading and development of deflection in the early stage of loading.

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250 300 350 400

t[days]

No

rmal

ized

def

lect

ion

a/a

ma

x







Figure 8.36 Diagram normalized deflection a/amax vs. time for series "B" beams concrete

classC30/37 and C60/75

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250 300 350 400

t[days]

No

rmal

ized

def

lect

ion

a/a

ma

x







Figure 8.37 Diagram normalized deflection a/amax vs. time for series "C" beams concrete

class C30/37 and C60/75


Based on the experimental and analytical results of time-dependant deflections in order to take into account variable load effect quasi-permanent load was obtained by corresponding value of quasi-permanent coefficient 2 which are given in table 8.1. Table 8.1 Analytically determined coefficients2 for reinforced concrete beams made of

ordinary concrete for the series “D” and “E”

Series Dead load G[kN]

Variable load Q[kN]

Total load

G+Q[kN]

2 Quasi-permanent load

G+2Q [kN]

D (2x4) (2x7.6) (2x11.6) 0.49 (2x7.7)

E (2x4) (2x7.6) (2x11.6) 0.66 (2x9.0)

For the series “D” and “E” beams made of high-strength concrete, the analysis of the experimental results and the results obtained with the analytical AAEM method analysis, shows that at determination of the coefficients 2 stress states should be included in the analysis when cracks appears.

The chosen loading histories for these series of beams, which were determined in order to compare the behavior of reinforced concrete beams made of ordinary and high-strength concrete, for beams made of high-strength concrete higher mechanical characteristics enables cracks appearance practically at level of the service load. At this level due to the approximately equal moment of cracks appearance Mcr=11.6kNm and moment of the service loads Ms=12.6kNm stabilized state of cracks appears. In this way at analytical solution in order to obtain the experimentally measured deflections, the coefficient of participation for defining of the variable load as quasi-permanent load is approaching to 2=1, which is unreal in the practice. In this way is obtained that for the beams made of high-strength concrete the effect of the variable load should be replaced to act as permanent load in the beam, which leads to wrong solution of the problem.

For the elements made of high-strength concrete is necessary to define states with higher values of the stresses, i.e. to exist state of stresses which causes cracks from the effect of the load with smaller value of the value of the service load. This is a precondition for proper analysis of variable loads effect on the behavior of the reinforced concrete elements made of high-strength concrete and to determine the quasi-permanent coefficient 2.


9 Conclusions 175

9. Conclusions

From the experimental results and analytical analysis of time-dependant behavior of ordinary and high-strength reinforced concrete beams, subjected to long-term permanent load and repeated variable load, following conclusions can be obtained:

1. Actions of long-term permanent load and repeated variable load have significant influence on the time-dependant behavior of concrete beams made of ordinary and high-strength concrete.

2. Time-dependant behavior of concrete elements under action of long-term permanent load and repeated variable load depends from loading history, i.e. especially from the cycles of loading/unloading by variable load (∆t1=24h; ∆t2=48h).

3. Analytical analysis of drying shrinkage and creep using B3 model shows good agreement with experimental results using update parameters for improved estimations. On the basis of regression analysis, for ordinary concrete following values of update parameters for adjustment of the creep compliance p1 and p2 were obtained: p1=0.9088 and p2=0.998. For the improved estimation of the drying shrinkage coefficient p6 is set to p6=0.82. These values for high-strength concrete are p1=1.4264, p2=0.3319 and p6=1.0955.

4. Analysis of autogenous shrinkage strain according to B3 model gives incorrect estimations of strains and that's why new formula was proposed to obtain analytical solution using experimental results:

)t(S)h99.0()t( att

1

aaas

5. Using Age Adjusted Effective Modulus Method and principle of superposition analysis quasi-permanent coefficients 2 were obtained to determine influence of variable load on the time-dependant behavior of concrete for the control of deflections. According to experimental and analytical results, repeated variable load was replaced by quasi-permanent load defining it through the quasi-permanent coefficient 2. For the beams made of concrete class C30/37 and considered loading histories by repeated variable loads coefficient 2=0.50 was obtained for cycles of loading/unloading ∆t=24h. Coefficient 2=0.65 was obtained for cycles of loading/unloading ∆t=48h. For the same loading histories at high strength concrete beams class C60/75 coefficient 2=1 was obtained.

To improve understanding of time-dependant behavior of reinforced concrete under action of repeated variable load, in future experimental and theoretical research the following parameters should be analyzed:

Level of loading that causes higher rates of stresses and cracking state in the high-strength concrete beams subjected to long-term permanent load.

Geometry of concrete cross section and reinforcement ratio.

Different loading histories for the variable load (different time duration of the cycles loading/unloading) for certain reinforced concrete structures

Age of concrete at loading.

9 Conclusions 176

10 Literature 177

10. Literature [1] CEB Bulletin d'information No 235, Serviceability Models (Behaviour and modelling in

serviceability limit states including repeated and sustained loads, Lausanne, April 1997

[2] Sande D. Atanasovski, Influence of long-term actions to limit states of prestressed

concrete, Doctoral dissertation, Skopje, February 1987 [3] Franz-Josef Ulm, Michael Prat, Jean-Armand Calgaro, Ignacio Carol, Creep and

Shrinkage of Concrete, Hermes Science Publications, Paris, 1999-Editorial [4] M.Y.H.Bangash, Manual of numerical methods in concrete (Modelling and

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TIME-DEPENDANT BEHAVIOUR OF REINFORCED HIGH- Arangjelovski.pdf · the influence of the variable loads to time-dependant behaviour of the reinforced concrete structures. 24 reinforced