Paper " STRUT-AND-TIE MODEL AND 3-D NONLINEAR FINITE ELEMENT ANALYSIS FOR THE PREDICTION OF THE BEHAVIOR OF RC FSHALLOW AND DEEP BEAMS WITH OPENINGS

Salah E. El-Metwally/et al/Engineering Research Journal 141 (March 2014) C50 – C70

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STRUT-AND-TIE MODEL AND 3-D NONLINEAR FINITE ELEMENT

ANALYSIS FOR THE PREDICTION OF THE BEHAVIOR OF RC

FSHALLOW AND DEEP BEAMS WITH OPENINGS

Waleed E. El-Demerdash1, Salah E. El-Metwally

2, Mohamed E. El-Zoughiby

3, Ahmed A. Ghaleb

4

1Teaching Assistant MET Academy, 2Prof. of Structural Concrete, 3Associate Prof., 4Associate Prof.

Structural Engineering Department, Mansoura University, El-Mansoura, Egypt

ABSTRACT: The Strut-and-Tie Model, STM, has been widely applied for the design of non-

flexural and deep members in reinforced concrete structures. In this paper, strut-and-tie models for

selected (shallow and deep) beams with openings, have been suggested based on the available

experimental results of; crack patterns, modes of failure, and internal stresses trajectors obtained

from elastic finite element analysis. The proposed STM approach is, then, applied to one group of

simple shallow beams and one group of simple deep beams tested experimentally. In addition, a

three-dimensional nonlinear finite element analysis using ANSYS 12.0 computer program has

been employed for two selected (shallow and deep) beams which were analyzed using the STM

method. Some of the important factors affecting the behavior of reinforced concrete beams

(named: concrete compressive and tensile strength, span to depth ratio, shear span to depth ratio,

physical and mechanical properties of horizontal, vertical web reinforcement and main steel,

loading position, opening dimensions and location) are investigated throughout a parametric study

with the aid of the nonlinear finite element analysis. With such analysis, results of cracking

patterns, deflections, failure mode and strain and stress distributions, that cannot be determined

using the strut-and-tie model, are obtained. A comparison of the finite element results with test

results and STM results has been carried out.

Keywords: Strut-and-Tie model; Finite element method; Shallow and deep beams; Openings;

Normal strength concrete; High strength concrete.

1. INTRODUCTION

As per the ACI 318M-11 code [2], each shear span (av) of the beam in Fig. 1a, where av < 2h, is a

D-region. If two D-regions overlap or meet as shown in Fig. 1b, they can be considered as a single

D-region for design purposes. The maximum length-to-depth ratio of such a D-region would

approximately equal 2. Thus, the smallest angle between the strut and the tie in a D-region is arctan

(1/2) = 26.5 degrees, rounded to 25 degrees. If there is a B-region between the D-regions in a shear

span, as shown in Fig. 1c, the strength of the shear span is governed by the strength of the B-region

if the B- and D-regions have similar geometry and reinforcement. This is because the shear

strength of a B-region is less than that of a comparable D-region.

Figure 1 Description of deep and slender beams (ACI 318-2011).


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In practice, transverse openings in reinforced concrete beams are essential for different

usages such as utility services. Including transverse openings in the web of a reinforced concrete

beam is associated with, not only a sudden change in the dimensions of the cross section of the

beam, but also the corners of the opening would be subjected to stress concentration and it is

possible to induce transverse cracks in the beam. Also, it can reduce the stiffness, which leads to

excessive deformations and considerable distribution of forces and internal moments in a

continuous beam.

In this paper, two types of reinforced concrete (shallow and deep) beams with openings are

studied. Some of the existing design codes; e.g., the ACI 318M-11 code [2] and the Egyptian Code

(2007) [4], define a beam to be deep when the span-to-overall member depth ratio (L/h) equals 4 or

less, or the shear span-to-overall member depth ratio (a/h) is equal or less than 2 and span-to-depth

ratio (L/d) equals 4 or less, or the shear span-to-depth ratio (a/d) is equal or less than 2,

respectively. Because of its proportions, the strength of a deep beam is usually controlled by shear,

rather than by flexure, provided that normal amounts of longitudinal reinforcement are used. On

the other hand, the shear strength of deep beams is significantly greater than that predicted using

expressions developed for shallow (ordinary) beams.

The strut-and-tie model has proved to be a useful and consistent method for the analysis

and design of structural concrete, specially, D-regions; therefore, the method is utilized to carry out

this investigation. In addition, the finite element package (ANSYS 12) is used to perform a 3-D

nonlinear finite element analysis of the selected (shallow and deep) beams which had been tested

experimentally. This finite element analysis, on one hand, is used to check the output results

obtained from the strut-and-tie model and completes, on the other hand, the understanding of the

behavior of the considered reinforced concrete (shallow and deep) beams. The results from both

the strut-and-tie model and the finite element analysis are compared with the experimental data.

2. THE APPROACH FOR DEVELOPING A STM FOR BEAM WITH OPENINGS

The approach for developing a Strut-and-Tie Model STM for a whole beam with opening can be

described in the following consequence:

Use the equilibrium conditions for the whole beam to find the external reactions:

Find the “load path “, i.e. the flow of the external loads from their acting positions to the

supports”. The load path principle results in the position and the orientation of the main tension

and compression stresses taking into account the resulting transverse stresses due to load path

deviation.

Following the load path, the STM can be constructed:

For additional verification, a stress analysis can be performed using linear elastic finite element,

which produces the stress trajectories (tension and compression). It is also possible to consider

the nonlinear behavior and the cracking of concrete in the finite element analysis. Following the

stress trajectories for both tension and compression stresses, the position and orientation of struts

and ties can be traced; thus, the STM is constructed. Generally, the strut and tie directions should

be within ±15oof the direction of the compressive and tensile stress trajectories, respectively.

Check the external and internal equilibrium of the model. In the first, the equilibrium conditions

of the applied loads with the external reactions for the whole structure or the part around the

opening are fulfilled. The later is satisfied through the fulfillment of equilibrium conditions at

each node. Through the last step the member forces are calculated.

Diagonal struts are generally oriented parallel to the expected axis of cracking.

Struts must not cross or overlap each other; otherwise, the overlapping parts of the struts would

be overstressed. The widths of struts are chosen to carry the forces in the struts using the

effective strength of the concrete in the struts.


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Ties are permitted to cross struts or other ties.

If photographs of test specimens are available, the crack pattern may assist in selecting the best

strut-and-tie model, the location of the struts fall between cracks. Struts should not cross cracks.

The model with the least number of shortest ties is likely the best.

The angle, θ, between the axis of any strut and any tie entering a single node shall not be taken as

less than 25 degrees.

3. STRENGTH LIMITS OF STRUT-AND-TIE MODEL′S COMPONENTS

For the proposed strut-and-tie model, the strengths of ties, concrete struts and nodal zones are as in

the following.

Reinforced Ties

In this study, the contribution of tensile strength of a concrete tie is ignored and normally tie

forces are carried by reinforcement. The tie cross-section is constant along its length and is

obtained from the tie force and the yield stress of steel. The nominal strength of a tie shall be

taken as

where is the cross section of area of steel and is the yield stress of steel.

The width of the tie is to be determined to satisfy safety for compressive stresses at nodes.

Depending on the distribution of the tie reinforcement, the effective tie width may vary between

the following values but with an upper limit given afterwards.

In case of using one row of bars without sufficient development length beyond the nodal

zones (Fig. 2a):

In case of using one row of bars and providing sufficient development length beyond the

nodal zones for a distance not less than , where is the concrete cover (Fig. 2b):

where is the bar diameter.

In case of using more than one row of bars and providing sufficient development length

beyond the nodal zones for a distance not less than , where is the concrete cover

(Fig. 2c):

( )

where is the number of bars and the clear distance between bars.

) ) ) ( )

Figure 2 The width of the tie used to determine the dimensions of the node.

The upper limit is established as the width corresponding to the width in a hydrostatic nodal

zone, calculated as

( ⁄ )


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where is the applicable effective compressive strength of a nodal zone and is computed from

[2,4] as

or

The stands for a cylinder concrete compressive strength and for a cube concrete

compressive strength, is the effectiveness factor for nodal zones, and is the breadth of the

beam.

Concrete Struts

The shape of a strut is highly dependent upon the force path from which the strut arises and the

details of any tension reinforcement connected to the tie. As discussed by Schlaich and Schäfer

[6], there are three major geometric shape classes for struts: prismatic, bottle-shaped, and

compression fan, as shown in Fig.3. Prismatic struts are the most basic type of struts, and they

are typically used to model the compressive stress block of a beam element as shown in Fig. 3a.

Bottle-shaped struts are formed when the geometric conditions at the end of the struts are well

defined, but the rest of the strut is not confined to a specific portion of the structural element. The

geometric conditions at the ends of bottle-shaped struts are typically determined by the details of

bearing pads and/or the reinforcement details of any adjoined steel. The best way to visualize a

bottle-shaped strut is to imagine forces dispersing as they move away from the ends of the strut as

shown in Fig. 3b. The bulging stress trajectories cause transverse tensile stresses to form in the

strut which can lead to longitudinal cracking of the strut. Appropriate crack control reinforcement

should always be placed across bottle-shaped struts to avoid premature failure. The last major type

of struts is the compression fan, which is formed when stresses flow from a large area to a much

smaller area. Compression fans are assumed to have negligible curvature and, therefore, they do

not develop transverse tensile stresses. The simplest example of a compression fan is a strut that

carries a uniformly distributed load to a support reaction in a deep beam as shown in Fig. 3c.

Figure 3 Geometric shapes of struts.

The strength of concrete in compression stress fields depends to a great extent on the multi-axial

state of stress and on the disturbances from cracks and reinforcement. The effective compressive

strength of the concrete in a strut may be obtained from [2, 4]:

or 0.67

where is the effectiveness factor for concrete struts, which takes into account the stress

conditions, strut geometry and the angle of cracking surrounding the strut. The value of

according to the ACI 318M-11 code [2] is adopted in this investigation, Table 1.


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Table 1 ACI-Code values of coefficient for struts

λ is a modification factor to account for the use of lightweight concrete. λ = 0.85 for sand-

lightweight concrete and 0.75 for all-lightweight concrete and λ = 1.0 for normal weight concrete.

The nominal compressive strength of a strut without longitudinal reinforcement shall be

taken the smaller value of:

at the two ends of the strut, where is the cross-sectional area at one end of the strut, and is

the smaller of:

The effective compressive strength of the concrete in the strut.

The effective compressive strength of the concrete in the nodal zone.

The design of struts shall be based on

where is the strength reduction factor. In another form

( )

Nodal Zones

The compressive strength of concrete of the nodal zone depends on many factors including the

tensile straining from intersecting ties, confinement provided by compressive reactions and

confinement provided by transverse reinforcement. To distinguish between the different straining

and confinement conditions for nodal zones, it is helpful to identify these zones as follows, Fig. 4:

C-C-C nodal zone bounded by compression struts only (hydrostatic node);

C-C-T nodal zone bounded by compression struts and one tension tie;

C-T-T nodal zone bounded by compression strut and two tension ties; and

T-T-T nodal zone bounded by tension ties only.

Figure 4 Classification of nodes.

Strut condition

A strut with constant cross-section along its length. 1.0

For struts located such that the width of the midsection of the strut is larger than the

width at the nodes (bottle-shaped struts):

a) With reinforcement normal to the center-line of the strut to resist the

transversal tensile force.

b) Without reinforcement normal to the center-line of the strut

0.75

0.60λ

For struts in tension members, or the tension flanges of members. 0.40

For all other cases. 0.60λ


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The effective compressive strength of concrete in a nodal zone can be obtained from [2, 4]:

or

where is the effectiveness factor of a nodal zone and it is assumed as given in Table 2 according

to the ACI 318M-11 code [2]. The nominal compressive strength of a nodal zone, , shall be

where is the effective compressive strength of concrete in the nodal zone and is the smaller

of:

The area of the face of the nodal zone on which acts, taken perpendicular to the line

of action of the strut force .

The area of a section through the nodal zone, taken perpendicular to the line of action of

the resultant force on the section.

Table 2 ACI 318M-11Codevalues of coefficient for nodes

Nodal zone

Compression-Compression-Compression, C-C-C 1.00

Compression-Compression-Tension, C-C-T 0.80

Compression-Tension-Tension, C-T-T* 0.60

Tension-Tension-Tension, T-T-T 0.40

*In nodal zones anchoring two or more ties with the presence of one strut

In smeared nodes, where the deviation of forces may be smeared or spread over some length, the

check of stress is often not critical and it is only required to check the anchorage of the reinforcing

bars. On the other hand, singular or concentrated nodes have to be carefully checked.

4. ORDINARY BEAMS WITH OPENINGS

To illustrate how to model and analyze reinforced ordinary beams with openings using a strut-and-

tie method and nonlinear finite element analysis (ANSYS 12) [3], the Beam group C tested by

Abdalla et al. [1] are utilized.

4.1 Verification Beam of Group C for STM approach

Figure 5b shows a simple reinforced concrete ordinary beam with rectangular opening

(100×300mm), with two top point loads along with the proposed strut-and-tie model. The model

has thirty-five compression struts S1 to S35, thirty-eight tension ties T1 to T38, and fifty-four nodes

N1 to N54. Two external top point loads are applied at nodes N39. The tension ties T29 to T38

represent the main longitudinal reinforcement and the vertical reinforcement is represented by the

other ties.

Numerical Scheme for the Verification Beam:

Input data:

Beam size: h = 250mm, d = 210mm, b = 100mm, and b1 = 50mm b2 = 100mm.

Shear span-to-depth ratio: a = 670 mm, and (a/d) = (670/210) = 3.2

Materials: = 52MPa, = 400MPa, = 240MPa, and As = 410 (314.16 mm

2), and Asv

=50.27mm2 per stirrup (one-leg).


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(a) Concrete dimensions and location of opening.

(b) Details of the proposed strut-and-tie model.

(c) Strut labels for the proposed strut-and-tie model.

(d) Tie labels for the proposed strut-and-tie model.

Figure 5 Beam Group C with rectangular opening (100×300mm) [1].


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(e) Node labels for the proposed strut-and-tie model.

(f) Nodal Zone N1.

Figure 5 Cont.

Numerical Scheme for the Verification Beam:

Input data:

Beam size: h = 250mm, d = 210mm, b = 100mm, and b1 = 50mm b2 = 100mm.

Shear span-to-depth ratio: a = 670 mm, and (a/d) = (670/210) = 3.2

Materials: = 52MPa, = 400MPa, = 240MPa, and As = 410 (314.16 mm

2), and Asv

=50.27mm2 per stirrup (one-leg).

The internal lever arm, Ld :

The term a1 (height of node N1) can be computed from

( )

where is the number of steel layers, is the longitudinal steel diameter, is the clear

concrete cover.

( )

The width of strut is assumed equal to ( ) and can be computed from equation

and thus Ld= h – 0.5 (a

1+ a

2) = 195.79mm.


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Table 3 Calculated member forces for the strut-and-tie model

Model

Label

Force

kN T or C

Model

Label

Force

kN T or C

Model

Label

Force

kN T or C

Model

Label

Force

kN

T or

C

1 550555 C 20 55555 C 4 50555 T 23 26.45 T

2 55555 C 21 50555 C 5 55555 T 24 27.84 T

3 50550 C 22 55555 C 6 50554 T 25 28.41 T

4 50555 C 23 55555 C 7 05555 T 26 30.04 T

5 40555 C 24 55555 C 8 50555 T 27 31.05 T

6 55555 C 25 55555 C 9 55555 T 55 55555 T

7 45555 C 26 55555 C 10 55555 T 54 05555 T

8 50555 C 27 55555 C 11 55555 T 55 55555 T

9 45554 C 28 55555 C 12 54555 T 55 55550 T

10 00550 C 29 54555 C 13 55555 T 55 50555 T

11 55550 C 30 55555 C 14 55555 T 55 55555 T

12 55555 C 31 55555 C 15 05555 T 55 55555 T

13 55545 C 32 55555 C 16 55550 T 50 55555 T

14 55555 C 33 05550 C 17 55555 T 55 54555 T

15 05550 C 34 50550 C 18 55555 T 55 55555 T

16 55555 C 35 554555 C 19 55555 T 55 55555 T

17 55550 C 1 50555 T 20 50555 T - - -

18 55555 C 2 55555 T 21 55555 T - - -

19 55555 C 3 55555 T 22 55555 T - - -

Table 4 Summary of concrete struts calculations

Model

Label

MPa

Strut

width

Max.

strut

capacity

Actual

Strut

force

Okay Model

Label

MPa

Strut

width

Max.

strut

capacity

Actual

Strut

force

Okay

1 44.20 92.91 410.66 550555 yes 20 44.20 20.00 88.40 50555 yes

2 44.20 20.00 88.40 55555 yes 21 44.20 20.00 88.40 55555 yes

3 44.20 28.43 125.66 50550 yes 22 44.20 20.00 88.40 55555 yes

4 44.20 20.00 88.40 50555 yes 23 44.20 20.00 88.40 26.45 yes

5 44.20 28.43 125.66 40555 yes 24 44.20 99.35 88.40 27.84 yes

6 44.20 20.00 88.40 55555 yes 25 44.20 20.00 88.40 28.41 yes

7 44.20 20.00 88.40 45555 No 26 44.20 20.00 88.40 30.04 yes

8 44.20 20.00 88.40 50555 yes 27 44.20 20.00 88.40 31.05 yes

9 44.20 28.43 125.66 45554 yes 28 44.20 20.00 88.40 55555 yes

10 44.20 20.00 88.40 00550 yes 29 44.20 20.00 88.40 05555 yes

11 44.20 28.43 125.66 55550 yes 30 44.20 20.00 88.40 55555 yes

12 44.20 20.00 88.40 55555 yes 31 44.20 20.00 88.40 55550 yes

13 44.20 28.43 125.66 55545 yes 32 44.20 20.00 88.40 50555 yes

14 44.20 28.43 125.66 55555 yes 33 44.20 20.00 88.40 55555 yes

15 44.20 28.43 125.66 05550 yes 34 44.20 20.00 88.40 55555 yes

16 44.20 28.43 125.66 55555 yes 35 44.20 20.00 88.40 55555 yes

17 44.20 20.00 88.40 55550 yes 35 44.20 20.00 88.40 54555 yes

18 44.20 28.43 125.66 55555 yes 35 44.20 20.00 88.40 55555 yes

19 44.20 20.00 88.40 55555 yes 35 44.20 20.00 88.40 55555 yes


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Width of struts:

The widths of struts S1 to S35 are calculated based on bearing plates and the widths of the ties are

shown in Table 4. STM forces:

The forces in all member calculated form static are shown in Table 3. A summary of concrete struts

calculations is given in Table 4.

Finally, kN and kN

A summary of concrete node calculations is given in Table 5.

Table 5 Summary of effective concrete node calculation

Model

Label

Type βn Surrounding

Forces, kN C/T

Available

width (mm)

MPa

Max.

capacity, kN

Actual

force, kN Okay

1 CCT

0.80 550555 C 92.91 35.36 328.53 550555 yes

0.80 55555 T 80.00 35.36 282.88 55555 yes

0.80 55550 C 50.00 35.36 176.80 55550 yes

54 CCC

1.00 55550 C 100.0 44.20 555555 55550 yes

1.00 55555 C 28.43 44.20 550555 55555 yes

1.00 55555 C 99.35 44.20 554555 55555 yes

1.00 554555 C 20.00 44.20 55555 554555 No

Therefore, the nominal shear force is

kN and

⁄

4.2 Nonlinear Finite Element Analysis of the Verification Beam

Finite Element Results

The computer package (ANSYS-12) [3] has been used to carry out the nonlinear finite element

analysis of all beams in this study. From the finite element results of this specimen, Fig. 6, at about

20 percent of the ultimate load, the first vertical flexural cracks were formed in the region of the

maximum bending moment. At about 40 percent of the ultimate load, a sudden major inclined

tension crack was formed almost in the middle part of the shear span. In the mean time, the cracks

propagated above the openings to the point load, and below the opening to the supports. With

further increase in the applied load, the existing vertical flexural and inclined shear cracks were

formed parallel to the original inclined cracks in the shear span, as shown in Fig. 6g. At about 97

percent of the ultimate load, cracks at the corner of opening to point load and support, increased

and failure occurred in the opening region. The finite element results for the first flexural and

diagonal cracking load and ultimate load of beam with rectangular openings are 14.86kN, 33.40kN

and 42kN, respectively. Fig. 6 shows the output of “ANSYS” program figures of the beam. For

this beam with rectangular openings the load path is deviated around the opening, the concrete

stresses are forced to deviate through a narrow load path, and as a result the stress redistribution

increases as shown in Fig. 6c. Increasing the concrete strength tends to increase the load capacity.

The higher compressive stresses exist at nodal zones (point loads and supports), while a reduction


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in the compressive stresses takes place in the inclined struts joining the loading points and

supports. This reduction is due to the diagonal cracks and the web opening in the load path. Comparison of the Results

The experimental ultimate failure load Vu,Exp [1] and the predicted failure load of the tested beam

Vu,FEM are 41kN and 42kN, respectively. The mean value of the ratio Vu,Exp to Vu,FEM for this

ordinary beam is 0.98, which demonstrates that the nonlinear finite element model provides

accurate prediction of the ultimate load of ordinary beams with openings. Clear that the adopted

nonlinear finite element model provides useful tool in understanding the behavior of simple

ordinary beams with openings.

(c) Vector plots of principal stresses. (d) 1st cracks for flexure at load 14.86kN.

(e) Flexure cracks pattern. (f) 1st Cracks for shear at load 33.40kN.

(g) Diagonal cracks pattern. (h) Cracks pattern at failure load 42kN.

Figure 6 Output of “ANSYS” program for beam group C [1].

5. DEEP BEAMS WITH OPENINGS

To illustrate how to model and analyze reinforced deep beams with openings using the strut-and-

tie method and nonlinear finite element analysis, the Beam DSON3 Group A tested by El-Azab

[5] is selected. 5.1 Verification Beam DSON3 of Group A for STM approach

Figures 7a and 7b show a simple reinforced concrete deep beam with a rectangular opening, with

one top point load along with the proposed of refined strut-and-tie model, Fig. 7c, and a simplified

strut-and-tie model, Fig. 7d. The simplified model has five compression struts S1 to S5, five tension

(a) Applied loads and boundary

conditions. (b) Reinforcement configurations.


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ties T1 to T5, and six nodes N1 to N6, and the point load is applied at node N4. The tension ties T1

and T2 represent the main longitudinal reinforcement and the vertical and horizontal

reinforcements are represented by ties T3 and T4.

(a) Concrete dimensions and location of opening.

(b) Details of reinforcement.

(c) Details of the proposed refined strut-and-tie model using inclined ties.

Figure 7 Beam DSON3 group A [5].

Sh =

10

0m

m

Sv = 200mm


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(d) Details of the proposed simplified strut-and-tie model using inclined ties.

Figure 7 Cont.

Numerical Scheme for the Verification Beam DSON3:

Input data:

Materials: = 30.45MPa, = 410MPa (Ø16), = 244.5MPa (Ø6), = 260.2MPa (Ø8),

=

410 and As = 416 (804.25 mm2), where

is the cylinder compressive strength of concrete,

is yield stress of longitudinal steel, is yield stress of vertical stirrups, is yield stress of

horizontal stirrups is the area of secondary steel, and As is the area of main steel.

For simplify visualization of strut widths and geometry of nodes in the verification example for

this beam the simplified strut-and-tie model shown in Fig. 7d is used.

Width of struts: The strut widths were determined by developing a realistic geometry of the struts

as they extend from the nodes shown in Fig. 8.

Figure 8 Visualization of strut widths.


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The term a1 (height of node N1) can be computed from

( )

where is the number of steel layers, is the longitudinal steel diameter, is the clear

concrete cover.

( )

STM forces:

The forces in all members are determined from statics and their magnitudes in kN are as indicated

in Table 6. The struts, ties, and nodes are labeled in the Fig. 7d.

kN

is the nominal strength capacity of the tie when reaching its yield strength.

Table 6 Calculated member forces for the strut-and-tie model

Model

Label

Force

kN T or C

Model

Label

Force

kN T or C

S1 64.14 C T1 61.59 T

S2 58.04 C T2 4.27 T

S3 75.42 C T3 46.86 T

S4 48.44 C T4 37.04 T

S5 31.34 C T5 22.46 T

Finally, kN

Checking of stress limits:

a. Concrete Struts:

Knowing that = 30.45MPa, the term (

) will be:

MPa, for Strut S

j (j = 2 to 5)

MPa, for Strut S

j (j = 1)

Maximum strut capacity,

. Table 7 summarizes the calculations performed for

each of the struts.

Table 7 Summary of concrete strut calculation

Model

Label

βs

MPa Strut width

Max. strut

capacity (kN)

Actual Strut

capacity (kN) Okay

1 0.60 15.53 52.00 64.60 64.14 yes

2 1.00 25.88 53.00 109.73 58.04 yes

3 1.00 25.88 43.00 89.03 75.42 yes

4 1.00 25.88 42.00 86.96 48.44 yes

5 1.00 25.88 52.00 107.66 31.34 yes

b. Nodes:

The capacity of a node is calculated by finding the product of the limiting compressive stress in the

node region and the cross sectional area of the member at the node interface. Table 8 summarizes

the calculations performed for effective nodes N1, N3 and N4; maximum node capacity, and

, which is safe. Thus,


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kN, ⁄

Using vertical and horizontal ties, an alternative refined strut-and-tie model of Beam DSON3 is

shown in Fig. 9 whereas an alternative simplified model is shown in Fig. 10.


Model

Label

Type βn

Surrounding

Forces

kN

C/T

Available

width

(mm)

MPa

Max.

capacity

kN

Actual

capacity

kN

Okay

1 CCT

0.80 64.14 C 94.00 20.71 155.71 64.14 yes

0.80 64.00 C 100.00 20.71 165.68 64.00 yes

0.80 4.27 T 80.00 20.71 132.54 4.27 yes

4 CCC

1.00 75.42 C 43.00 25.88 89.03 75.42 yes

1.00 48.44 C 42.00 25.88 86.96 48.44 yes

1.00 128.0 C 100.0 25.88 207.04 128.0 yes

3 CTT

0.60 31.34 C 52.00 15.53 63.77 31.34 yes

0.60 46.86 T 55.00 15.53 68.33 46.86 yes

0.60 4.27 T 69.00 15.53 85.73 4.27 yes

0.60 61.59 T 69.00 15.53 85.73 61.59 yes

Figure 9 Alternative refined model of Beam DSON3 using vertical and horizontal ties.

For the alternative simplified model in Fig. 10, upon carrying out the calculations of the model,

kN, ⁄

Checking of stress limits:

a. Concrete Struts:

Knowing that = 30.45MPa, the term (

) will be:

MPa, for Strut S

j (j = 1, 3,4, 5, 7 and 8)

MPa, for Strut S

j (j = 2, and 5)


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Table 10 summarizes the calculations performed for each of the struts; maximum strut capacity,

Figure 10 Alternative simplified model of Beam DSON3 using vertical and horizontal ties.

Table 9 Calculated member forces for proposed simplified strut-and-tie model

Model

Label

Force

kN T or C

Model

Label

Force

kN T or C

S1 66.29 C T1 72.39 T

S2 75.25 C T2 13.00 T

S3 91.81 C T3 27.00 T

S4 88.11 C T4 17.33 T

S5 37.58 C T5 53.27 T

S6 43.00 C T6 6.47 T

S7 65.27 C T7 6.47 T

S8 12.83 C -- -- --

Figure 11 Visualization of strut widths.

T2

T1 T4

T3

S6

T5

T6

S1

S2

S4

S5

N1

N2

N3

N4 N5

N7 N6

N8 N9

S3

S7 S8 T3

T5 T7

N10


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Table 10 Summary of concrete strut calculation

Model

Label βs

MPa Strut width

Max. strut

capacity (kN)

Actual Strut

capacity (kN) Okay

1 1.00 25.88 113.0 233.96 66.29 yes

2 0.60 15.53 62.00 77.03 75.25 yes

3 1.00 25.88 59.00 122.15 91.81 yes

4 1.00 25.88 52.00 107.66 88.11 yes

5 0.60 15.53 36.00 44.73 37.58 yes

6 1.00 25.88 47.00 97.31 43.00 yes

7 1.00 25.88 51.00 105.59 65.27 yes

8 1.00 25.88 62.00 128.36 12.83 yes

b. Nodes:

The capacity of a node is calculated by finding the product of the limiting compressive stress in the

node region and the cross sectional area of the member at the node interface. Table 11 summarizes

the calculations performed for nodes N1, N5 and N8; maximum node capacity, and

, which is safe for the model in Fig. 10, ⁄

.


Model

Label

Type βn

Surrounding

Forces

kN

C/T

Available

width

(mm)

MPa

Max.

capacity

kN

Actual

capacity

kN

Okay

1 CCT

0.80 65.00 C 100.00 20.71 165.70 65.00 yes

0.80 66.29 C 113.0 20.71 187.22 66.29 yes

0.80 13.00 T 80.00 20.71 132.54 13.00 yes

5 CCC

1.00 130.0 C 100.0 25.88 207.04 130.0 yes

1.00 88.11 C 52.00 25.88 107.66 88.11 yes

1.00 37.58 C 36.00 25.88 74.53 37.58 yes

8 CCT

0.80 43.00 C 47.00 20.71 77.87 43.00 yes

0.80 12.83 C 62.00 20.71 102.72 12.83 yes

0.80 65.27 C 51.00 20.71 84.50 65.27 yes

0.80 24.92 T 51.00 20.71 84.50 24.92 yes

0.80 10.13 T 63.00 20.71 104.40 10.13 yes

Table 12 The STM results compared with test results

The model with the vertical and horizontal ties is the better than the model with inclined ties since

it gives larger capacity. Upon following the previous numerical scheme, the failure load and

No. Beam PEXP, kN Failure Mode,

Exp. PSTM, kN

Failure Mode,

STM PSTM / PEXP

1 DSON3 140 Opening Failure 130.0 Opening Failure 0.930

2 DSOH10 110 Opening Failure 98.56 Opening Failure 0.896

3 DCON3 220 Opening Failure 176.43 Opening Failure 0.801

4 DCOH2 360 Opening Failure 280.12 Opening Failure 0.778

5 DCOH8 290 Opening Failure 150.31 Opening Failure 0.518

Mean 0.785


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failure mode of all other beams Group A have been obtained as shown in Table 12. A comparison

between the results of the strut-and-tie model and the test data is given in the Table 12. The strut-

and-tie approach gives a mean value of 0.785 of the experimental ultimate load. 5.2Nonlinear Finite Element Analysis of the Verification Beam DSON3

Finite Element Results

First cracking occur at corner of opening, it could be noticed that the opening affects the beams

stress trajectories beam drastically, where zones of tension stresses are formed around the left-

upper corner of the opening (load side) and the corner on the same diagonal, so first cracking occur

at this corner of opening and is a shear crack, see Fig. 12f. Inversion in ordinary beams first crack

occurs in the constant moment region, and is a flexural crack (Vertical cracks). The cracking

pattern(s) in the beam can be obtained using the Crack/Crushing plot option in ANSYS. Vector

Mode plots must be turned on to view the cracking in the model. Beyond the first cracking, in the

non-linear region of the response, subsequent cracking occurs as more load is applied to the beam.

Cracking increases out towards the supports and the beam begins flexural cracking (Vertical

cracks)in the constant moment region, see Fig. 12g. Also, diagonal tension cracks are beginning to

form in the model, see Fig. 12h. This cracking increased after yielding of reinforcement, the

predicted and experimental cracking patterns of the beams at failure are shown in Fig. 12i, and the

occurrence of smeared cracks is indicated by short lines, whereas discrete cracks to indicate

crushed concrete is indicated by gray spots.

Generally, for this specimen at about 35 percent of the ultimate load, the first vertical

flexural cracks were formed in the region of the maximum bending moment. At about 28 percent

of the ultimate load, a sudden major inclined tension crack was formed almost in the middle part of

the shear span. With increasing the load, the inclined cracks propagated backwards until it reached

the beam bottom at the support blocks edges, as shown in Fig. 12h. In the mean time, the cracks

propagated above openings to point load, and down opening to supports. With further increase in

the applied load, the existing vertical flexural and inclined shear cracks were formed parallel to the

original inclined cracks in the shear span, as shown in Fig. 12i. At about 93 percent of the ultimate

load, cracks at the corner of opening nearest to the point load and nearest to the support to the

support and failure occurred in opening region.

Comparison of the Results

The experimental ultimate failure load Vu,Exp [5] and the predicted failure load of the beam

calculated from the finite element model Vu,FEM are 140kN and 130kN, respectively. The mean

value of the ratio Vu,FEM to Vu,Exp for this deep beam is 0.93, which demonstrates that the nonlinear

finite element model provides accurate prediction of the ultimate load of deep beams with

openings. Clear that the adopted nonlinear finite element model provides useful tool in

understanding the behavior of simple deep beams with openings.

Table 13 Comparison of ultimate loads

No. Beam Experimental Ultimate

Load, 2Vu, ExpkN

Analytical Ultimate

Load, 2Vu, FEM kN

1 DSON3 140 130.00 0.93

2 DSOH10 110 105.00 0.95

3 DCON3 220 218.58 0.99

4 DCOH2 360 345.00 0.96

5 DCOH8 290 285.00 0.98

Average 0.96


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A comparison between all the recorded experimental ultimate failure load Vu,Exp and the predicted

failure load for tested simple and continuous deep beams calculated from the finite element model

Vu,FEM are given in Table 13. The mean value of the ratio Vu,FEM to Vu,Exp for NSC and HSC deep

beams is 0.96, which demonstrates that the nonlinear finite element model provides satisfactory

prediction of the ultimate load for the tested simple and continuous NSC and HSC deep beams.

Clear that the adopted nonlinear finite element model provides useful tool in understanding the

behavior of simple and continuous NSC and HSC deep beams with openings.

(a) Beam.

(c) Meshing of beam and cross section.

(d) Deformed shape.

Figure 12 Output of “ANSYS” Program figures for beam DSON3 Group A [5].

(b) Applied loads and boundary conditions.

(e) Vector plots of principal stresses.


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(f) 1st Cracks for shear at load 36.7kN. (g) 1

st cracks for flexure at load 45.45kN.

(h) Diagonal cracks pattern. (i) Cracks pattern at failure loads 130kN.

Figure 12 Cont.

6 SUMMARY AND CONCLUSIONS

In this paper, the strut-and-tie approach has been used to predict the capacity of reinforced concrete

(shallow and deep) beams with openings subjected to different loading and boundary conditions.

Verification examples of shallow and deep beams that had been tested in previous by other

researchers (with different dimensions, materials, loads, web reinforcement, concrete strength and

boundary conditions) have been modeled and analyzed using the proposed strut-and-tie approach

utilizing elastic principal stress trajectories from finite element analysis. The obtained results

represent a lower bound of the beam capacity.

A 3-D nonlinear finite element analysis has been conducted in order to predict the ultimate

capacity of the aforementioned beams. The finite element predictions are very satisfactory when

compared with the test results.

Based on the proposed STM approach and for the range of studied factors, the following

conclusions can be drawn:

The strut-and-tie approach is a powerful tool to predict the ultimate strength and behavior

of reinforced concrete (ordinary or deep) beams with openings, and it gives freedom to

designer to choose the suitable model, according to the elastic principal stress trajectors

from finite element analysis and practice.

strut-and-tie model gives reasonable estimates of the load carrying capacity of the chosen

reinforced concrete beams with openings when compared with the experimental failure

loads.

Based on the analytical study using the nonlinear finite element analysis and for the range of

studied factors, the following conclusions can be drawn:

The 3-D nonlinear finite element analysis of simple Normal-and High-Strength Concrete

ordinary (shallow) and deep beams with openings yields accurate predictions of both the

ultimate load and the complete response.


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For shallow beams with openings loaded by two symmetrical point loads, the first crack

occurs in the constant moment region. It in general happens in the mid-span of beam, and it

is a flexural crack (Vertical crack). Cracking increases with increases the load in the

constant moment region, and cracking propagates towards the supports (diagonal cracks).

For deep beams with openings, the opening affects the beams stress trajectories drastically,

where zones of tension stresses are formed around the upper corners of the openings

(Nearest to the load) and around the lower corner (Nearest to the supports) on the same

diagonal. These shear crack occurs, in general in the shear-span of beam. Cracking

increases with increasing the load in the shear-span and propagates toward in mid-span

(flexural cracks, vertical cracks).

For all considered shallow beams with openings, a diagonal shear failure occurred at corner

of tension zones of opening, before the yielding of the longitudinal rebars.

For all considered deep beams with openings, a diagonal shear failure occurred on the

shear-span or at the upper corners of the openings (nearest to the load) and around the

lower corner (nearest to the supports) on the same diagonal before yielding of the

longitudinal bars.

The finite element solutions show that the increase of the concrete strength results in an

increase in the cracking strength and ultimate strength.

For ordinary beams, the finite element stress trajectories along the longitudinal

reinforcement occurred at the point of load application and at the edges of the opening. It is

found that the reinforcing bars reach their yielding stresses at these locations.

REFERENCES

1. Abdalla, H.A., A.M. Torkeya, H.A Haggagb and A.F. Abu-Amira., “ Design against

cracking at openings in reinforced concrete beams strengthened with composite

sheets”.Composite Structures, 2003, (60): 197-204.

2. American Concrete Institute, “Building Code Requirements for Reinforced Concrete”,

Detroit, ACI-381M-11, (2011).

3. ANSYS Release12.0, 2009 SAS IP, Inc.

4. Egyptian Code for The Design and Construction of Reinforced Concrete structure,

Cairo, 2007, Ministry of Housing and Development of New Communities, Cairo,

Egypt.

5. El-Azab, M. F., “Behavior of Reinforced High Strength Concrete Deep Beams with Web

Openings”, M. Sc. Thesis in Structural Engineering, Faculty of Engineering, El-Mansoura

University, 2007.

6. Schlaich, J. and Schäfer, K.“Design and Detailing of Structural Concrete Using Strut-and-

Tie Models”. The Structural Engineer. V. 69, No. 6, May-June, 1991, pp 113-125.