Structural Behaviour of Segmental Arch Structures

School of Engineering and Information Technology

An Undergraduate Thesis submitted in partial fulfilment of the requirements for

the degree of

Bachelor of Engineering - Civil

Structural Behaviour of Segmental Arch Structures

By

Yianni Lolias

Student No. s213259

May 2014

Supervisor

Prof. David Lilley


Charles Darwin University

Co-Supervisor

Robert Wolff


Charles Darwin University

Acknowledgments

I would like to thank my supervisor, Professor David Lilley, for his professional guidance and

assistance throughout the duration of this Thesis. His engineering experience has helped me

direct my study and draw conclusions based on the physical tests and results obtained.

Special thanks must go to my father, Manuel Lolias, for his help regarding supply of various

materials and assistance with the construction of arch formwork/testing equipment.

Special thanks to my colleague Kenneth Kadirgamar, regardless of having his own thesis to

complete, he made time to assist me in all practical experiments which were conducted.

Thanks must also go to the technical officers of Charles Darwin University, Brendan Von

Gerhardt and Hemangi Surti, who were always able to assist in running lab experiments.

I would also like to thank Holcim Australia for the kind supply of concrete aggregate, and Spiros

Welding for the willingness to fabricate and provide the steel loading rig components.

Thesis Abstract

Student Name: Yianni Lolias

Supervisor: Prof. David Lilley

Thesis Topic: Structural Behaviour of Segmental Arch Structures

Arch structures have been utilized through the ages, beginning in the ancient civilizations of

Greece, Egypt and Rome, to present day with their common use in bridges. Arches are well

known for the ability to carry loads spanning large areas.

This thesis aims to determine key structural behaviours of segmented arch structures under

concentrated loading by conducting experiments and developing finite element models.

Segmented arch refers to the traditional masonry construction using voussoir sections (stone

blocks). Three segmented arch scale models have been constructed using concrete in attempt to

compare finite element computer models to structural behaviour experienced in a practical

experiment.

A number of finite element models have being developed using the SAP2000 program. Each

model loaded in the same way as the concrete model, analysed, and compared. Structural

behaviours considered include internal stress, deflections and predicted location of failure points.

Finite element arch models which are under the influence of stressing tendons (acting as post-

tension cables) have also been developed and compared to a scale post-tensioned concrete model.

The results from finite element models and scale concrete models will give an insight to the level

of difficulty and accuracy to which the physical behaviour can be modelled. The comparison will

provide an understanding of the general behaviour of arches under concentrated loading.

Through this study, it has been found when comparing arch attributes such as the shape and rise

to span ratio, that there is a number of key behavioural differences between the arches. These are

as follows:

Parabolic arches deflect less in the vertical direction than corresponding circular arches

for the same rise and span, for almost all given loading cases.

Most predicted values of shear force were found to be less in parabolic arches when

compared to corresponding circular arches, and the difference between these

corresponding values was found to increase with increasing rise to span ratio.

It was found that the amount of tension experienced due to the dead load of an arch could

be minimized if not removed completely by using a parabolic arch.

The post-tensioned concrete arch which was constructed and tested, withstood a substantially

greater load than the solid voussoir arch before failure. The addition of a tension member to

produce compressive stresses in the concrete is therefore very effective in increasing the

structures loading capacity when concentrated loads are applied.

Table of Contents

1. Introduction ............................................................................................................................. 11

1.1 Background ........................................................................................................................ 11

1.2 Objectives and Scope ......................................................................................................... 12

1.3 Organization of Thesis ....................................................................................................... 12

2. Literature Review .................................................................................................................... 14

2.1 Concrete and Masonry Arches ......................................................................................... 14

2.2 Finite Element Modelling (SAP2000) ............................................................................... 16

2.3 Flexi-Arch (Flexible Concrete Arch) ................................................................................ 16

3. Design of Concrete Mix ........................................................................................................... 20

3.1 Mix Design .......................................................................................................................... 20

3.2 Sieve Test/Particle Distribution ........................................................................................ 21

3.3 Concrete Cylinder Compression Tests ............................................................................ 22

3.4 Use of Concrete Properties in SAP2000 and Microstran Computer Models ............... 24

4. Arch Modelling ........................................................................................................................ 26

4.1 Design .................................................................................................................................. 26

4.2 Comparison of shape ......................................................................................................... 27

4.3 Microstran Model Loading ............................................................................................... 29

4.4 Analysis of Fixed Support Microstran Models ............................................................... 30

4.4.1 Analysis of Maximum Vertical Deflection ................................................................... 34

4.4.2 Axial Force Comparison ............................................................................................... 35

4.4.3 Shear Force Comparison ............................................................................................... 35

4.4.4 Comparison of Bending Moments ................................................................................ 36

4.4.5 Line of Thrust Analysis ................................................................................................. 36

4.5 Comparison of Microstran Pinned and Fixed Results ................................................... 38

4.6 Finite Element Models – SAP2000 ................................................................................... 44

4.6.1 Model Layout ................................................................................................................ 44

4.6.2 Loading Applied ............................................................................................................ 47

4.6.3 Deflections/Deflected Shape ......................................................................................... 47

4.6.4 Expected Location of Failure ........................................................................................ 49

4.6.5 SAP2000 Model Issues and Improvements .................................................................. 51

5. Physical Testing ....................................................................................................................... 52

5.1 Design and Construction of Arch Model Formwork ...................................................... 52

5.2 Plywood Inner Form .......................................................................................................... 54

5.3 Loading Rig ........................................................................................................................ 57

5.4 The Concrete Arches ......................................................................................................... 59

................................................................................................................................................... 59

5.4.1 Pouring Method for Solid and Post-tensioning Voussoirs ............................................ 60

5.4.2 Concrete Voussoir Appearance ..................................................................................... 61

5.4.3 Concrete Pouring Issues ................................................................................................ 62

5.5 Arch Erection ..................................................................................................................... 63

5.5.1 Method of Erection........................................................................................................ 63

5.5.2 Arch Erection Issues...................................................................................................... 65

5.6 Loading Results for Concrete Arches .............................................................................. 66

5.6.1 Solid Voussoir Arch Loading ........................................................................................ 67

5.6.2 Post-tensioned Arch Loading ........................................................................................ 71

6. Comparison of Finite Element Models to Concrete Models ................................................ 80

7. Recommendations for Further Research and Improvements ............................................. 84

8. Summary and Conclusions ..................................................................................................... 86

Reference List .............................................................................................................................. 88

Appendix A – Concrete Design .................................................................................................. 91

Concrete Mix Design Form ..................................................................................................... 91

Concrete Design Calculations and Explanations .................................................................. 92

Appendix B – Microstran Arch Model Example – Circular 0.35m Rise ............................... 94

Appendix C – Example Microstran Model Analysis Report – Circular 0.35m rise, Dead

Load Case ............................................................................................................. 96

Appendix D – Example Calculation of Eccentricity for Circular and Parabolic 0.35m Rise

Arch, Dead Load Case ....................................................................................... 102

Appendix E – Strain Energy Method Deflection Calculation Check ................................... 108

Appendix F – Scale Model Arch Formwork Design ............................................................... 112

Appendix G – Arch Inner Form Calculations ........................................................................ 115

Appendix H – Loading Rig Calculations and Capacity Checks ............................................ 118

Initial Design Calculations: ................................................................................................... 118

Capacity Check and Failure Mode Analysis: ...................................................................... 125

Structural Behaviour of Segmental Arch Structures 6

List of Tables

TABLE 1: MATERIAL FOR SELECTED CONCRETE MIX TABLE 2: MIX PROPERTIES OF THE DESIGNED CONCRETE TABLE 3: SIEVE DISTRIBUTION TEST RESULTS TABLE 4: CONCRETE CYLINDER COMPRESSION TEST RESULTS TABLE 5: 0.35M RISE – CIRCULAR ARCH – MAXIMUM STRUCTURAL BEHAVIOURS TABLE 6: 0.35M RISE – PARABOLIC ARCH – MAXIMUM STRUCTURAL BEHAVIOURS TABLE 7: 0.6M RISE – CIRCULAR ARCH – MAXIMUM STRUCTURAL BEHAVIOURS TABLE 8: 0.6M RISE – PARABOLIC ARCH – MAXIMUM STRUCTURAL BEHAVIOURS TABLE 9: 1M RISE – CIRCULAR ARCH – MAXIMUM STRUCTURAL BEHAVIOURS TABLE 10: 1M RISE – PARABOLIC ARCH – MAXIMUM STRUCTURAL BEHAVIOURS TABLE 11: PERCENTAGE DIFFERENCE IN MAXIMUM VERTICAL DEFLECTION BETWEEN 0.35M RISE

ARCHES – FIXED SUPPORTS TABLE 12: PERCENTAGE DIFFERENCE IN MAXIMUM VERTICAL DEFLECTION BETWEEN 0.6M RISE

ARCHES – FIXED SUPPORTS TABLE 13: PERCENTAGE DIFFERENCE IN MAXIMUM VERTICAL DEFLECTION BETWEEN 1M RISE

ARCHES – FIXED SUPPORTS TABLE 14: PERCENTAGE DIFFERENCE IN MAXIMUM SHEAR FORCE BETWEEN CIRCULAR AND

PARABOLIC ARCHES – FIXED SUPPORTS TABLE 15: THRUST LINE COMPARISON FOR PINNED SUPPORT ARCHES UNDER DEAD LOAD TABLE 16: 0.35M RISE – CIRCULAR ARCH PINNED SUPPORTS – MAXIMUM STRUCTURAL

BEHAVIOURS TABLE 17: 0.35M RISE – PARABOLIC ARCH PINNED SUPPORTS – MAXIMUM STRUCTURAL

BEHAVIOURS TABLE 18: 0.6M RISE – CIRCULAR ARCH PINNED SUPPORTS – MAXIMUM STRUCTURAL

BEHAVIOURS TABLE 19: 0.6M RISE – PARABOLIC ARCH PINNED SUPPORTS – MAXIMUM STRUCTURAL

BEHAVIOURS TABLE 20: 1M RISE – CIRCULAR ARCH PINNED SUPPORTS – MAXIMUM STRUCTURAL BEHAVIOURS TABLE 21: 1M RISE – PARABOLIC ARCH PINNED SUPPORTS – MAXIMUM STRUCTURAL

BEHAVIOURS TABLE 22: PERCENTAGE DIFFERENCE IN MAXIMUM VERTICAL DEFLECTION BETWEEN 0.35M RISE

ARCHES – PINNED SUPPORTS TABLE 23: PERCENTAGE DIFFERENCE IN MAXIMUM VERTICAL DEFLECTION BETWEEN 0.6M RISE

ARCHES – PINNED SUPPORTS TABLE 24: PERCENTAGE DIFFERENCE IN MAXIMUM VERTICAL DEFLECTION BETWEEN 1M RISE

ARCHES – PINNED SUPPORTS TABLE 25: PERCENTAGE DIFFERENCE IN MAXIMUM SHEAR FORCE BETWEEN CIRCULAR AND

PARABOLIC ARCHES – PINNED SUPPORTS TABLE 26: DEFLECTION OBTAINED FOR VARYING SURFACE PRESSURES AT MID-SPAN – POST-

TENSIONED FEM TABLE 27: PRESSURE VS DEFLECTION AT MID-SPAN - POST-TENSIONED ARCH


TABLE 28: PRESSURE CONVERSION TO LOAD TABLE 29: SUMMARY COMPARISON BETWEEN FEMS AND CONCRETE ARCHES TABLE 30: 0.35M RISE ARCH – LINE OF THRUST POSITIONING COMPARISON – FIXED SUPPORTS TABLE 31: 0.6M RISE ARCH – LINE OF THRUST POSITIONING COMPARISON – FIXED SUPPORTS TABLE 32: 1M RISE ARCH – LINE OF THRUST POSITIONING COMPARISON – FIXED SUPPORTS


List of Figures

FIGURE 1: COMPARISON BETWEEN THE CONSTRUCTION OF A TRUE (VOUSSOIR) ARCH AND A CORBEL

ARCH (TRUE ARCH VS. CORBEL ARCH FIGURE, 2013, HTTP://WWW.ESSENTIAL-

HUMANITIES.NET/SUPPLEMENTARY-ART-ARTICLES/CORBELLING/#.UJW8IBYKBIS)

FIGURE 2: FALSE ARCHES CONSTRUCTED FROM LARGE STONES (TRIANGULAR ARCH FIGURE, 2011,

HTTP://ARCHITECHSTOK.WORDPRESS.COM/2011/05/19/TYPES-OF-ARCHES/)

FIGURE 3: ARCH CENTRING (CENTRING USED TO BUILD THE ARCH OF A STONE BRIDGE IN THE 19TH

CENTURY PICTURE, 2009,

HTTP://COUNTERLIGHTSRANTSANDBLATHER1.BLOGSPOT.COM.AU/2009_01_01_ARCHIVE.HTM

L)

FIGURE 4: FLEXI-ARCH CONSTRUCTION (TWO FORMS OF ARCH CONSTRUCTION FIGURE, 2007,

HTTP://PDFS.FINDTHENEEDLE.CO.UK/14181..PDF)

FIGURE 5: HONEYCOMBING ON CONCRETE TEST CYLINDERS

FIGURE 6: FAILURE OF 28-DAY CONCRETE CYLINDERS

FIGURE 7: SAP2000 – 0.6M RISE ARCH – 17 ANGULAR DIVISIONS

FIGURE 8: SAP2000 - 0.6M RISE ARCH – 2 RADIAL, 2 DEPTH AND 17 ANGULAR DIVISIONS

FIGURE 9: SOLID ARCH VOUSSOIR - FEM DEFLECTION – DEAD LOAD AND SURFACE PRESSURE OF

9×10-4 KN/MM

2

FIGURE 10: POST-TENSIONED ARCH – FEM DEFLECTED SHAPE - DEAD LOAD, MID-SPAN SURFACE

PRESSURE AND POST-TENSIONING

FIGURE 11: INTERNAL STRESS - SOLID FINITE ELEMENT ARCH

FIGURE 12: STEEL FORM

FIGURE 13: SECTION VIEW OF STEEL FORM

FIGURE 14: CONSTRUCTION OF INNER FORM

FIGURE 15: SIDE VIEW OF INNER FORM

FIGURE 16: INNER FORM AND STEEL FORM

FIGURE 17: LOADING RIG COMPONENTS

FIGURE 18: LOADING RIG SETUP FOR MID-SPAN LOADING

FIGURE 19: VOUSSOIRS FOR ARCH ONE AND TWO

FIGURE 20: VOUSSOIR FOR ARCH THREE – POST-TENSIONING

FIGURE 21: PVC 'T-JOINT' FOR POST-TENSIONING AND BLEED HOLE

FIGURE 22: MOST SEVERE CASE OF HONEYCOMBING IN THIRD ARCH VOUSSOIRS

FIGURE 23: ARCH POSITIONED IN BASE PFC HORIZONTALLY

FIGURE 24: SIDE VIEW OF ARCH POSITIONED IN BASE PFC HORIZONTALLY

FIGURE 25: ERECTED ARCH

FIGURE 26: ARCH APEX SLIGHTLY OUT OF PLANE DUE TO LIFTING PROCEDURE

FIGURE 27: HYDRAULIC JACK CYLINDER POSITIONING

FIGURE 28: HYDRAULIC JACK CYLINDER POSITIONING (SIDE VIEW)

FIGURE 29: SOLID VOUSSOIR ARCH FAILURE INITIATED

FIGURE 30: SOLID VOUSSOIR ARCH FAILURE

FIGURE 31: SOLID VOUSSOIR ARCH COLLAPSE

FIGURE 32: IRREGULAR VOUSSOIR SURFACE APPARENT IN ERECTED ARCH

FIGURE 33: LOAD INDICATING WASHER UNEVEN COMPRESSION

FIGURE 34: (A) ARCH PRIOR TO POST-TENSIONING, (B) ARCH AFTER POST-TENSIONING


FIGURE 35: ADJUSTMENT TO TIMBER SUPPORTS FOR POST-TENSIONED ARCH LOADING

FIGURE 36: ERECT POST-TENSIONED ARCH

FIGURE 37: POST-TENSIONED ARCH KEYSTONE AFTER FAILURE

FIGURE 38: POST-TENSIONED ARCH KEYSTONE TOP SURFACE AFTER FAILURE

FIGURE 39: DEFORMATION IN TOP PFC PLATE AFTER LOADING OF POST-TENSIONED ARCH

FIGURE 40: MICROSTRAN - 0.35M RISE CIRCULAR ARCH NODE POSITIONING

FIGURE 41: MICROSTRAN - 0.35M RISE CIRCULAR ARCH MEMBER POSITIONING

FIGURE 42: 0.35M RISE CIRCULAR ARCH - LOAD CASE 1 - DEAD LOAD - (1) LOADING, (2)

DEFLECTED SHAPE, (3) BENDING MOMENT DIAGRAM, (4) SHEAR FORCE DIAGRAM

FIGURE 43: 0.35M RISE CIRCULAR ARCH - LOAD CASE 2 - MID-SPAN CONCENTRATED LOAD - (1)

LOADING, (2) EXPECTED DEFLECTED SHAPE, (3) BENDING MOMENT DIAGRAM, (4) SHEAR

FORCE DIAGRAM

FIGURE 44: 0.35M RISE CIRCULAR ARCH - LOAD CASE 3 – THIRD-SPAN POINT CONCENTRATED

LOAD - (1) LOADING, (2) EXPECTED DEFLECTED SHAPE, (3) BENDING MOMENT DIAGRAM, (4)

SHEAR FORCE DIAGRAM


List of Graphs

GRAPH 1: PERCENTAGE PASSING – FINE AGGREGATES

GRAPH 2: CIRCULAR AND PARABOLIC ARCHES 0.35M RISE

GRAPH 3: CIRCULAR AND PARABOLIC ARCHES 0.6M RISE

GRAPH 4: CIRCULAR AND PARABOLIC ARCHES 1M RISE

GRAPH 5: PRESSURE VS MID-SPAN DEFLECTION – POST-TENSIONED FEM

GRAPH 6: PRESSURE VS MID-SPAN DEFLECTION – CONCRETE POST-TENSIONED MODEL

GRAPH 7: MID-SPAN DEFLECTION COMPARISON – POST-TENSIONED FEM AND CONCRETE ARCH


1. Introduction

1.1 Background

Arch structures have been used in construction since the ancient civilizations of Greece and

Rome. These civilizations utilized arches in order to span large areas and carry loads which could

otherwise not be possible with a column and beam approach. However the arch was also utilized

for its aesthetic contribution to structures, not just its structural advantages.

The voussoir arch or the True Arch is considered as one of the fundamental elements in the

conquest of space, which was one of the Roman architectural contributions. It is known that the

Romans frequently adopted construction techniques from the civilizations in which they

conquered, as a result, they considered the Greeks to be the inventors of the voussoir arch as the

Greeks were ahead of the Romans regarding the technique of vaulting (Adam, J, 1994).

The arch is designed in order to produce a system which transports the applied loads to supports

primarily through compression stresses in the arch, eliminating the possibility of tensile stresses

occurring within the chosen material. This is achieved, to some degree, through design of the

arch shape to match as closely as possible to the line of thrust within the arch.

The ‘middle third rule’ has been regarded by some nineteenth and twentieth century engineers as

being of prime importance in the arch design. This rule states that the line of thrust of an arch

(thrust line which is to be expected under the applied loading) should not fall outside the middle

third of an arch cross-section (assuming a square or rectangular shaped cross-section). If the

applied load stays within this ‘core’ middle third, then the stresses experienced within the section

will be compressive (Heyman, J, 1982, p.23). This is of importance, as tensile stresses in a

voussoir arch will cause the individual voussoirs to separate (concrete and masonry also have

very little tensile strength). When the line of thrust exceeds past the middle third, not only will

tensile stresses become present, but the compressive stress on one side of the cross-section at the

location of the exceeding thrust line will also increase. This is due to the reduction of contacting

area in which the load is passed from one voussoir to another when tensile stress begins to

separate voussoirs. This would obviously indicate that the most effective solution is to design an

arch whose middle third section curvature closely mimics that of its line of thrust. This thesis

however, aims to create arch models and observe the effects of the line of thrust positioning,

rather than designing the arches to suite the line of thrust.


1.2 Objectives and Scope

The primary objectives of this Thesis were to:

Investigate the effects different arch attributes (such as shape, rise to span ratio and post-

stressing reinforcement) have on the structural behaviour of the arch, such as

deflections/displacements, internal stresses, bending moment and failure locations.

Investigate the structural behaviour of a segmented (voussoir) arch under self-weight and

point loading, comparing a scale model in post-tensioned concrete to finite element

models (FEMs). This would indicate the accuracy of modelling physical attributes

through finite element analysis, as well as difficulties which could be encountered.

The structural behaviours have been modelled using Microstran software and SAP2000 (which is

a finite element analysis program by Computers and Structures Inc).

1.3 Organization of Thesis

This Thesis comprises eight main chapters. An overview of each chapter can be seen below.

Chapter 1 gives an overview introduction to the topic background as well as the Thesis scope and

objectives.

Chapter 2 contains the Thesis literature review. This entails advances in the use of similar arch

systems and tests completed relating to use of these in applications such as drain culverts and in

general where soil (backfill) pressures are expected on the arch.

In Chapter 3, the process undertaken to produce a concrete mix design is detailed, as well as sieve

size distributions and concrete testing completed to enable an accurate representation of concrete

properties for use in Microstran/SAP2000 modelling.


Chapter 4 describes method followed to produce line and solid element models in the Microstran

and SAP2000 programs respectively, along with analysis and comparison of results.

In Chapter 5, the physical testing of the arches is detailed; from the design and fabrication/

construction of the two concrete arch formworks, to the load testing of the concrete arches.

The predicted results from the finite element models are compared to those obtained from loading

of the concrete arches in Chapter 6.

In Chapter 7, recommendations are given following the findings of the study for areas of further

research and improvement.

The final section, Chapter 8, contains a summary and conclusion of the experiments, models

created and results obtained.


2. Literature Review

2.1 Concrete and Masonry Arches

Brick arches have not been found prior to the Middle Kingdom (between 2050BC and 1650BC)

but are known of as far back as the first Dynasty of Egypt. There are generally two kinds of brick

arches, one whose voussoirs are standard bricks, and the other constructed with bricks designed

specifically for the intended application. Those constructed with standard bricks are the earliest

form of arches, however no evidence has been found of the use of these structures in large

spanning applications (no more than a few meters) (Clarke & Engelbach 1990)

The method of arching using brick voussoirs (true arch) is rarely observed in Egyptian

monuments. It required that a form of support is constructed in order to place the bricks in their

required locations; which is referred to as centring. More commonly seen in Egyptian

construction is the use of false arches. Rather than using friction between bricks of a true arch,

the arch would be cut out of previously placed stone slabs. Two longer slab sections could be cut

to form a triangular arch, or bricks laid horizontally, bridging out from either side of an opening

until both sides met in the centre forming an arch (corbelled). This did not require the use of

centring. Images of the different arching structures can be seen below in Figures 1 and 2, and

arch centring in Figure 3.

Figure 1: Comparison between the construction of a true (voussoir) arch and a corbel arch (True Arch vs.

Corbel Arch figure, 2013, http://www.essential-humanities.net/supplementary-art-

articles/corbelling/#.Ujw8IbykBIs)

http://www.essential-humanities.net/supplementary-art-articles/corbelling/#.Ujw8IbykBIs



Figure 2: False arches constructed from large stones (Triangular Arch figure, 2011,

http://architechstok.wordpress.com/2011/05/19/types-of-arches/)

Figure 3: Arch Centring (Centring used to build the arch of a stone bridge in the 19th century picture, 2009,

http://counterlightsrantsandblather1.blogspot.com.au/2009_01_01_archive.html)

http://architechstok.wordpress.com/2011/05/19/types-of-arches/

http://counterlightsrantsandblather1.blogspot.com.au/2009_01_01_archive.html


The voussoir arch is more commonly seen in Roman architecture than that of Egyptian. This

form of arch can either use bricks/blocks cut specifically for their intended purpose to fit a

required shape, or standard bricks. In either method, stability comes from the axial force and

friction between voussoirs.

2.2 Finite Element Modelling (SAP2000)

Finite element models have been developed to predict structural behaviour of an arch which will

be used for comparison to the constructed concrete arches. Moaveni (2003) explains the basic

computational processes involved in finite element analysis, which are as follows:

Pre-processing Phase

1. Divide the problem into a number of nodes and elements.

2. An approximated continuous function is used to represent the element.

3. Development of the equations for the element.

4. Assembly of all elements to represent the entire problem and construct the global stiffness

matrix.

5. Application of initial and boundary conditions, and application of selected loading cases.

Solution Phase

6. Solving equations (linear and non-linear simultaneously) to obtain results at node

positions, such as deflections/displacements or temperatures in the case of heat transfer

problems.

7. Obtaining other desired information such as internal stresses.

2.3 Flexi-Arch (Flexible Concrete Arch)

Queens University of Belfast (QUB) developed a flexible concrete arch system for use in small

bridging applications. This flexible concrete arch has been constructed using a number of precast

voussoirs made of high strength concrete. This system was created to address concerns associated

with the annual expenditure on the maintenance and strengthening of minor masonry structures in

Northern Ireland, of which the majority were masonry arches (Taylor et al. 2007).


Flexi-arch has been designed and created as a system, by which it can be transported to the site

location as a flat strip, lifted into its final arching shape and placed on its pre-prepared concrete

foundations. A key design feature is the integrated polymeric grid reinforcement which is

designed to carry the load of the arch during the lifting procedure. The polymeric reinforcement

is placed between the upper faces of the voussoirs and an insitu screed cast over the voussoirs,

which can be seen in figure 4 below.

Figure 4: flexi-arch construction (Two forms of arch construction figure, 2007,

http://pdfs.findtheneedle.co.uk/14181..pdf)

The Flexi-arch system avoids the common disadvantage involved with the use of steel reinforced

concrete structures. Due to the reinforcement used for lifting of the arch being a polymeric grid,

the possibility of steel corrosion can be avoided, reducing the maintenance cost of the structure.

These arches are currently being used in applications for highway traffic bridging of up to fifteen

metre spans (newsedge101’s channel 2010).

Bourke, Taylor, Robinson and Long (2010) explain experiments which were conducted in a

laboratory on scale models of the FlexiArch system, and the results obtained. Three arch models

were constructed using concrete voussoirs with a pre-determined geometry, which was obtained

using the span and height required for a circular shaped arch. The formwork for pouring of the

individual voussoirs was made using ‘Lexan’ plastic. Each of the 23 voussoirs which made up

one arch had a depth of 66mm, a length of 333mm and resulted in an arch spanning 1.67m with a

rise of 0.67m.

The three arches constructed (or three arch rings) each had the same span and rise, however the

backfill used in the testing of the arches varied for arches 2 and 3. Arch one and three were made

http://pdfs.findtheneedle.co.uk/14181..pdf


of hollow-core voussoirs, whereas arch two was made of solid voussoirs. Each arch was

constructed with a polymeric reinforcing, and a 13mm top screed connecting each voussoir.

Crack inducers were notched into the screed above each voussoir joint to control the cracking

which would be experienced during lifting of the arch.

After curing of the screed, the arches were lifted to form their finished shape three times. Each

time the strain was measured in the polymeric reinforcement which was carrying the self-weight

of the arch. It was found that the polymeric reinforcing was performing well below its capacity.

Each arch was fitted with deflection transducers and vibrating wire strain gauges under the arch

ring, located at mid-span, third points and near the abutments (Bourke et al. 2010). Backfill

material was inserted over the arch and loading was applied through a 150mm wide plate at the

third point (1/3 span from the abutment).

It was found that each arch failure occurred initially in the backfill material used as the steel

loading plate applied pressure. Deformation of the arch at the location of the failed backfill

material and the opposite third point were subsequent with increased load. A summary of the

results found were as follows.

Arch 1 made with hollow voussoirs and backfilled with 6mm aggregate showed considerable

shear failure in the material as it was not well graded. As load increased hinge formation occurred

as an opening in the voussoir blocks became larger at the positioning of the load. Deflection at

loading point was found to be 28mm for a load of 22.6kN.

Arch 2 made with solid voussoirs was backfilled with Type 3 GSB material, showed that the well

graded material was stronger when compacted and as a result the distribution of the load to the

arch through the fill material was improved. Less penetration of the loading plate was observed.

Deflections were noticeably less than in arch 1. It was found that it was impossible to break the

arch as the internal polymeric reinforcing held the voussoirs together. Three hinges were visible

during loading at the third point. A deflection of 22mm was reached at a load of 34.1kN.

Arch 3 was again backfilled with Type 3 GSB material, but constructed with hollow voussoirs.

This arch showed similar behaviour to arch 2 as the backfill contributes largely to the behaviour

of the arch. 16mm deflection at loading point was experienced at a load of 25.4kN.


Non-linear finite element models were developed to compare a third-scale model with varying

strengths of backfill. The results of the analysis proved to resemble closely to the model

experiments. The backfill material failed first through punching shear followed by deformation

leading to failure of the arch bridge.


3. Design of Concrete Mix

3.1 Mix Design

A concrete mix has been designed to be used in the arch model. This mix was designed as high

strength 40MPa at 28 days. The process followed to obtain water, cement, fine and coarse

aggregate ratios is detailed in the document, ‘Design of Normal Concrete Mixes’ written by

Teychenne, D et al (1988). The materials selected for the concrete mix can be seen in Table 1

below.

Table 1: Material for Selected Concrete Mix

Fine Aggregates Howard Springs Sand (supplied by Holcim Australia)

Coarse Aggregates 10mm crushed aggregate (supplied by Holcim Australia)

Cement General Purpose (GP) Cement

Note: 10mm aggregate was used as it was the preferred size for pouring concrete compression test cylinders.

The completed concrete mix design form and calculations can be viewed in Appendix A but a

brief overview of the result obtained is as follows in Table 2 below.

Table 2: Mix Properties of the Designed Concrete

28 day design compressive strength 40MPa

Mean Target Strength 53.12MPa

Water/Cement Ratio 0.47

Water Content 250kg/m3

Cement Content 530kg/m3

Fine Aggregate Content 755kg/m3

Coarse Aggregate Content 815kg/m3

As part of the concrete mix design process, it was necessary to determine the amount of the

selected fines aggregates passing a 600µm sieve, to find the appropriate fine aggregate content. In

order to accomplish this a sieve test was completed on the fine aggregates.


3.2 Sieve Test/Particle Distribution

A sieve distribution test of the fine aggregates being used in the concrete mix was undertaken in

order to determine the content required. The basic method followed is as detailed below:

1. The weight of each sieve required was measured on a scale (in this case 4.75mm to 75µm

sized sieves for fines).

2. Sample of fines weighed (503g was used in this test).

3. Fines were carefully placed into the stack of sieves and vibrated on the sieve machine for

20min.

4. Once sieving was complete, combined mass of sieves and soil retained was recorded.

5. Percentage passing each sieve was then calculated.

The results of the test are as follows in Table 3.

Table 3: Sieve Distribution Test Results

Sieve Size Sand Weight (excluding sieve

weights) (g)

Percentage Passing (%)

4.75mm 1 99.8

2.36mm 10 97.8

1.18mm 41 89.7

600µm 192 51.5

300µm 217 8.4

150µm 39 0.6

75µm 2 0.2

Base Plate 1 0

Total 503 -

The result can also be seen below in Graph 1


Graph 1: Percentage Passing – Fine Aggregates

As the total weight of sand after the sieving was found to be 503g, it is clear that no fines were

lost during the testing procedure. The key percentage required is that passing the 600µm sieve.

Due to the result obtained, a percentage passing of 52% was used in the concrete mix design.

3.3 Concrete Cylinder Compression Tests

Compression tests were conducted in order to accurately gauge the strength of the concrete,

which was to be used in the planned concrete model. It is important that the correct concrete

strength be used in the FEMs to ensure that the comparison between these and a model made of

concrete is as accurate as possible.

Using the designed concrete mix, six cylinders were poured, which allowed two for 7-day tests,

two for 14-day tests and two for 28-day tests. All cylinders were poured in two roughly equal

layers and rod compacted 25 times on each layer.

0

20

40

60

80

100

120

4.75mm 2.36mm 1.18mm 600µm 300µm 150µm 75µm Base Plate

Per

sen

tage

Pas

sin

g (%

)

Sieve Sizes

Persentage Passing - Fine Aggregates


The concrete produced was found to be highly workable which was due to the slump of 60-

180mm used in the design.

The finished surface of all cylinders, after the cylinder moulds were removed, showed little

honeycombing. Images of these cylinders can be seen below in Figure 5 (right side image shows

the worst case of honeycombing).

Figure 5: Honeycombing on concrete test cylinders

Images of these cylinders after 28-day testing showing examples of the failure which occurred

can be seen in Figure 6.

Figure 6: Failure of 28-day concrete cylinders

The results of the compression tested cylinders can be seen in Table 4 below.


Table 4: Concrete Cylinder Compression Test Results

Test Day Force Applied at Failure (kN) Compressive Stress/Strength (MPa)

(Area of cylinder = 7853.98mm2)

Average

Stress/Compressive

Strength (MPa)

Cylinder 1 Cylinder 2 Cylinder 1 Cylinder 2 -

7 338 336 43.04 42.78 42.91

14 391 378 49.78 48.13 48.95

28 412 413 52.46 52.58 52.52

As can be seen in Table 4, the 28 day compressive stress was found to be 52.52MPa. This is a 1%

difference to the target mean compressive stress at which the concrete was designed (53.12MPa).

Therefore, the compressive strength of 52.52MPa has been used in the development of FEMs

representing the arch structures.

3.4 Use of Concrete Properties in SAP2000 and Microstran Computer Models

In order to produce FEMs which could represent the physical model as accurately as possible, the

concrete properties which have been designed and tested were used in the analysis. For this

reason, the following specific properties have been altered:

Compressive Strength: The compressive strength of the material has been input as 52.52MPa

in the SAP2000 models. The Microstran models were analysed with a pre-set material class of a

50MPa concrete.

Density: The density of the material was changed from 2450kg/m3 on the Microstran models, to

2350kg/m3 to correspond to the designed concrete. When the concrete arches were poured, three

cylinders were also poured to check the density of the concrete. A more accurate density of

2280kg/m3 was calculated and used in the SAP2000 models.

Young’s Modulus of Elasticity: The modulus of elasticity has been changed to a value of

33.597GPa. This has been calculated using the equation:


𝐸𝑐 = 𝜌1.5[0.024√𝑓𝑐𝑚𝑖 + 0.12], which is used when 𝑓𝑐𝑚𝑖 > 40𝑀𝑃𝑎 where 𝑓𝑐𝑚𝑖 is the mean

target cylinder strength of 52.52MPa.

Other material properties were left as the pre-set concrete values, or standard 50MPa concrete

approximations.


4. Arch Modelling

4.1 Design

Concrete is a nonlinear material, however it is an accepted international practice to assume the

initial modulus of concrete to be the modulus of elasticity (Loo, Y & Chowdhury, S 2010).

The assumption of concrete behaving as a linear elastic material is not accurate. However, for the

purpose of this study, it will be modelled as such. Once a certain stress is reached, micro-cracking

will occur in the concrete. Micro-cracking is the formation of fine cracks that are not visible to

the naked eye, which will alter the concrete behaviour. Micro-cracks can also form with the

natural hydration process of cement (Concrete Construction, 2009). Larger cracks will also form

in the concrete prior to failure of the arch which do not occur in the elastic region of the material

behaviour. Assuming linear-elastic behaviour in the development of FEMs will however allow a

comparison between a physical concrete model and a linear-elastic approximation.

The consideration of backfill material has not been included in this study. The investigation is

regarding the structural behaviour of the arch structure itself under the influence of different

loads, and not surrounding materials or supports.

One of the objectives in this Thesis was to compare the structural behaviour of different arch

shapes with varying rise to span ratios. In considering the design of FEMs, it was important to

realise that a range of variables is required in order to effectively compare behavioural

differences. It was decided that comparison of line element models would be between six

different arches (all having a span of 2 metres and a constant cross-section of 150mm x 150mm)

which are listed below.

Circular arch with a rise of 0.35m

Circular arch with a rise of 0.6m

Circular arch with a rise of 1m (semicircle)

Parabolic arch with a rise of 0.35m

Parabolic arch with a rise of 0.6m

Parabolic arch with a rise of 1m

These models have been compared using the Microstran software. In order to relate the behaviour

of these arches at corresponding locations, each node on the model has been set at regular x-value


(span) intervals. A node is set at every 0.05 metres, meaning that in each model there are a total

of 41 nodes and 40 members. Ensuring that the nodes are positioned at intervals corresponding to

the same location on all arches allows for solid comparison. Each arch was modelled with fixed

supports. Example images of the arch node and member positioning can be seen in Appendix B,

Figures 40 and 41 for one arch type. These models were later analysed with pinned supports as it

was apparent that a fixed support would have been difficult to produce for the concrete arches

which were constructed.

4.2 Comparison of shape

Prior to producing the line element models in Microstran and FEMs in SAP2000, the arch shapes

were compared, concentrating on the difference in the vertical positioning between circular and

parabolic. For this, the position of comparable points were organised as mentioned for the nodes

in section 4.1 above. The x-position kept constant and the y-coordinate calculated for circular and

parabolic. The findings can be seen below in Graphs 2, 3 and 4.

Graph 2: Circular and Parabolic Arches 0.35m Rise

It can be seen from graph 2 that there is not a significant difference, but actually significant

similarity between the two arches at a low rise of 0.35m. The largest percentage difference in the

vertical distance between these circular and parabolic arches, with respect to the circular arch was

found to be 5.4% at node points 3, 4, 39 and 38.

0

0.1

0.2

0.3

0.4

-1.5 -1 -0.5 0 0.5 1 1.5

Ve

rtic

al P

osi

tio

n (

m)

Horizontal Position (m)

Arch Shape Comparison - 0.35m Rise

Parabola

Circular


Graph 3: Circular and Parabolic Arches 0.6m Rise

From graph 3 above, it can be seen that as the rise of the arch has increased, the difference in the

vertical position between the circular and parabolic arches has also increased. The largest

percentage difference in vertical positioning was found to be 30.7% at node points 2 and 40.

Graph 4: Circular and Parabolic Arches 1m Rise

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-1.5 -1 -0.5 0 0.5 1 1.5

Ve

rtic

al P

osi

tio

n (

m)

Horrizontal Position (m)

Arch Shape Comparison - 0.6m Rise

Parabola

Circular

0

0.2

0.4

0.6

0.8

1

1.2

-1.5 -1 -0.5 0 0.5 1 1.5

Ve

rtic

al P

osi

tio

n (

m)

Horizontal Position (m)

Arch Shape Comparison - 1m Rise

Parabola

Circular


As the rise of the arch has increased, the difference in the vertical positioning between the two

arch shapes has increased further. The largest percentage difference in the vertical positioning

being 68.8% at node points 4 and 38.

It has been established that with increasing rise to span ratio comes increasing differences

between circular and parabolic arches. Therefor it should also follow that the lower rising arches

should behave almost completely in compression under the action of dead load, whereas tension

should occur in the arches with higher rises. This is due to the fact that the line of thrust produced

by dead load occurs in the shape of a parabola with the corresponding rise and span. Therefore,

the closer the circular and parabolic arches are, the closer the line of thrust (due to dead load)

should be to staying in the middle third of the circular arch section for that particular rise. These

findings are discussed further in the analysis of results in Section 4.4.5.

4.3 Microstran Model Loading

In choosing the positioning of the loads to be placed on the arch ring, it was important to consider

the worst position for a point load, producing the largest axial forces, shear forces, bending

moments and deflections. For this reason, two concentrated loads have been chosen to observe

the effects of load positioning as well as arch shape and rise. The loads used are listed below. An

example of the loading applied to an arch can be seen in Appendix B Figures 42(1), 43(1) and

44(1).

Self-weight dead load – uniformly distributed load (UDL) applied along arch ring (found

using the density of the design concrete)

Mid-span – concentrated load (CL) (5kN)

Third-span point – CL (5kN)

Figures 43, 44 and 45 in Appendix B also illustrate the expected deflected shapes, bending

moment diagrams and shear force diagrams for the range of loading conditions applied to the

example arch.


4.4 Analysis of Fixed Support Microstran Models

A total of eighteen cases have been analysed using the Microstran software. One example report

generated as an output in the analysis can be seen in Appendix C with maximum structural

behaviours highlighted. The maximum values of structural behaviour for each of the eighteen

cases can be seen tabulated below from Tables 5 to 10. To ensure that there were no major errors

in the developed models, a manual calculation of deflection using strain energy was completed

(which can be seen in Appendix E) and compared to one of the Microstran models (with altered

supporting conditions to match the assumptions in the calculation). A 6% difference was found

between the manual calculation and the result obtained from the Microstran software which was

found to be acceptable. Appendix E also contains the calculation for determining Young’s

Modulus of Elasticity and the second moment of area for the section.


Table 5: 0.35m Rise – Circular Arch – Maximum Structural Behaviours

Load Case Maximum Deflection Maximum Axial Force Maximum Shear Force Maximum Bending Moment

x-direction

(mm) Location

y-direction

(mm) Location

Axial Force

(N) Location

Shear Force

(N) Location

Bending Moment

(Nmm) Location

Dead Load 0.000223

node 13(+), 29(-)

[inward] -0.003009 node 21 -861.08 node 1, 41 97.15

node 1(-),

41(+) -24646.20

node 1,

41

Mid-span CL 0.001877

node 6(-), 36(+)

[outward] -0.036081 node 21 -6239.46 member 8, 33 2408.22

member 20(-),

21(+) 620540.13 node 21

Third-span CL 0.010020 node 13 -0.034640 node 14 -5799.61 member 2 -2592.32

member 13 (at

node 14) 652247.75 node 14

Table 6: 0.35m Rise – Parabolic Arch – Maximum Structural Behaviours


x-direction

(mm) Location

y-direction

(mm) Location

Axial Force

(N) Location

Shear Force

(N) Location

Bending Moment

(Nmm) Location

Dead Load 0.0002 node 13(+), 29(-) -0.0029 node 21 -858.93 node 1, 41 93.80

node 1(-),

41(+) -28097.54

node 1,

41

Mid-span CL 0.0017 node 7(-), 35(+) -0.0342 node 21 -6248.15 member 8, 33 2396.54

member 20(-),

21(+) 595512.75 mode 21

Third-span CL 0.0099 node 13 -0.0343 node 14 -5773.98 member 1 -2500.69 member 13 649616.75 node 14


Table 7: 0.6m Rise – Circular Arch – Maximum Structural Behaviours


x-direction

(mm) Location

y-direction

(mm) Location

Axial Force

(N) Location

Shear Force

(N) Location

Bending Moment

(Nmm) Location

Dead Load 0.0001

node 4(-), 5(-),

37(+), 38(+) -0.0020 node 21 -782.42 node 1, 41 99.61

node 2(+),

40(-) 11683.38 node 21


member 20(-),

21(+) 622175.69 node 21


Table 8: 0.6m Rise – Parabolic Arch – Maximum Structural Behaviours


x-direction

(mm) Location

y-direction

(mm) Location

Axial Force

(N) Location

Shear Force

(N) Location

Bending Moment

(Nmm) Location

Dead Load 0.0002 node 12(+), 30(-) -0.0015 node 21 -766.73 node 1, 41 64.88

node 1(-),

41(+) -15075.42

node 1,

41


member 20(-),

21(+) 535439.06 node 21

Third-span CL 0.0174 node 13 -0.0291 node 14 -4672.13 member 1 2229.09 member 14 621412.31 node 14


Table 9: 1m Rise – Circular Arch – Maximum Structural Behaviours


x-direction

(mm) Location

y-direction

(mm) Location

Axial Force

(N) Location

Shear Force

(N) Location

Bending Moment

(Nmm) Location

Dead Load 0.0014 node 4(-), 38(+) -0.0038 node 21 -854.79 node 1, 41 213.51

node 2(+),

40(-) 46658.03

node 1,

41


member 20(-),

21(+) 773881.13 node 21


Table 10: 1m Rise – Parabolic Arch – Maximum Structural Behaviours


x-direction

(mm) Location

y-direction

(mm) Location

Axial Force

(N) Location

Shear Force

(N) Location

Bending Moment

(Nmm) Location

Dead Load 0.0004 node 9(+), 31(-) -0.0011

node 12,

13, 29,

30

-827.21 node 1, 41 68.83 node 1(-),

41(+) -15679.62

node 1,

41

Mid-span CL 0.0103 node 8(-), 34(+) -0.0200 node 21 -3324.94 member 9, 32 2387.41 member 20(-),

21(+) 534472.25 node 21

Third-span CL 0.0341 node 13 -0.0315 node 13 -4088.25 member 1 2058.52 member 14 617509.94 node 14


4.4.1 Analysis of Maximum Vertical Deflection

When comparing the maximum values of predicted vertical deflection due to the different loading

cases, it was found that the difference in shape had a large impact on the displacement. The

circular arches presented maximum deflections which were consistently greater than those of the

corresponding parabolic (with the same rise) for the three loads applied. However not only do the

parabolic arches perform with less deflection, but it was also determined that as the rise to span

ratio of the arches increased, the difference in deflection between corresponding circular and

parabolic arches also increased. This can be seen in the data presented in Tables 11, 12 and 13

below.

Table 11: Percentage difference in maximum vertical deflection between 0.35m rise arches – Fixed Supports

0.35m Arch Deflections (mm) Percentage Differences in Vertical

Deflection (%) Circular Parabolic

Dead Load -0.0030 -0.0029 4.0

Mid-Span Load -0.0361 -0.0342 5.1

Third-span Load -0.0346 -0.0343 1.1

Table 12: Percentage difference in maximum vertical deflection between 0.6m rise arches – Fixed Supports

0.6m Arch Deflections Percentage Differences in Vertical


Dead Load -0.0020 -0.001 27.2

Mid-Span Load -0.0301 -0.022 25.4

Third-span Load -0.0316 -0.0291 8.1

Table 13: Percentage difference in maximum vertical deflection between 1m rise arches – Fixed Supports

1m Arch Deflections Percentage Differences in Vertical


Dead Load -0.0038 -0.0011 70.6

Mid-Span Load -0.0496 -0.0200 59.7

Third-span Load -0.0455 -0.0315 30.8

The tables above show that the difference in vertical deflections between circular and parabolic

arches consistently increase with increasing rise. The most variation is generally found in the

dead load and mid-span loading cases.


For all concentrated load cases, the maximum deflection occurred at or within one node of the

loading point.

4.4.2 Axial Force Comparison

Examining Table 5 through 10 above, it is can be seen that as the rise of the arch increases, the

maximum axial forces present in the members decrease. There is however, no significant

difference in the axial forces present in the corresponding circular and parabolic arches. It was

found that, with respect to the difference between loading conditions, the third-span load

produced the largest axial force in both 1 metre rise arches and both 0.6 metre rise arches. This

cannot be said for those of 0.35 metre rise, with maximum values occurring in the mid-span load

case for both circular and parabolic.

4.4.3 Shear Force Comparison

It was found that there was one significant pattern in the shear force results. Shear force values

experienced were found to be less in parabolic arches when compared to corresponding circular

arches, and the difference between these values for corresponding arches was found to increase

with increasing rise. Table 14 below shows the percentage difference in shear force between the

circular and parabolic arches for the varying rises.

Table 14: Percentage difference in maximum shear force between circular and parabolic arches – Fixed

Supports

Percentage difference in maximum shear force (%)

0.35m rise 0.6m rise 1m rise

Dead load 3.5 34.9 67.8

Mid-span load 0.5 1.2 2.3

Third-span Load 3.5 11.1 21.7

It can be seen from Table 14 above, that even though the change in maximum shear force

(between the circular and parabolic) increases with increasing rise, for all load cases, the dead

load case shows this most significantly. This shows that an effective way to decrease the shear

force in the arch is to use a parabolic arch that is the required rise and span. However, the

difference present in the lowest rising arches (0.35m) are not significant. It therefore follows that


for the lowest rising circular arches, it would perhaps not be efficient to use a parabolic arch for

such a small change in shear force; as it is more difficult to design and construct due to the

different angled sections needed. A circular arch can be designed and built with one size of

symmetrical voussoirs.

4.4.4 Comparison of Bending Moments

The case which produced the smallest of the maximum bending moments was the circular 0.6

metre rise arch with an applied dead load; which occurred at node 21 (mid-span). The largest of

the maximum bending moments was found to be in the circular arch with 1 metre rise under

concentrated load of 5kN at mid-span. This bending moment occurred at mid-span.

For most cases, the maximum bending moments for all arch cases under the influence of dead

load was found to be at the supports (node 1 and 41 symmetrically, due to the fixed supports).

The only exception to this was found to be the circular 0.6 meter rise arch.

The 0.6m rise arch has been found to experience the least deflection and bending moment of all

three circular arches. This is also the circular arch which experienced the least amount of tensile

stress accumulating in the dead load case (this will be explained below in Section 4.4.5). For this

reason, the 0.6 metre rise arch seems to be behaving as the most effective structure in supporting

the loads.

4.4.5 Line of Thrust Analysis

The bending moment and the axial forces at each node have been combined to produce the axial

force positioned at a calculated eccentricity, which is achieved by dividing the value of bending

moment by the value of axial force. The eccentricity of the force at each node represents the line

of thrust through the arch for the selected load case. The position of the theoretical line of thrust

can be used to determine the state of stress in the arch assuming linear-elasticity. If the line of

thrust is within the middle third of the cross section, the arch will behave entirely in compression

within that section. If it falls out the middle third then tension will become present, and at the

locations where the thrust line lies outside of the section completely, failure is more likely to

occur due to increased tensile and compressive stresses on either side of the arch ring.


This has been applied to all arch and load cases. The only significant result was found to be

comparison of the dead load cases. As the line of thrust due to the dead load of the arch is in the

shape of a parabola, it should follow that the closer in shape the circular arches are to the

parabolic arches, less tension should be developed; and that the parabolic arches should remain

essentially in compression as the line of thrust will closely follow the middle third. The results of

this analysis can be seen in Appendix D (Tables 30, 31 and 32), where a comparison has been set

between circular and parabolic arches of the same rise under the influence of dead load.

It can be seen in Tables 31 and 32 predominantly that the line of thrust positioning has an

influence on whether tension is developed or failure is likely to occur. It has been established

earlier in Section 4.2 that as the rise to span ratio of the arches increases, the difference in shape

between circular and parabolic arches for the same span and rise also increases. This results in

tension developing over more of the arch for higher rises. This was observed in the results, as for

the 1 metre rise circular arch, the eccentricity in most of the arch was outside of the middle third

(resulting in tension), and at the mid-span point the thrust line was at an eccentricity greater than

75mm (outside of the section), implying failure will begin in this area first due to the higher

stresses. As the rise is lessened, to a 0.6 metre circular arch, the tension developed is much less

due to the shape being closer to a parabola.

It can also be seen from Tables 30, 31 and 32 that for the parabolic arches of 0.6 metre rise and 1

metre rise, no tensile stresses were present as the line of thrust did not leave the middle third.

This shows that if the arch can follow the line of thrust for the certain load, the presence of tensile

stresses can be avoided.

The 0.35 metre rise circular and parabolic arches did however accumulate tensile stresses at the

same location (supports). This is due to the large similarity in shape between 0.35 metre circular

and parabolic. The reason tensile stresses are present in these two arches is due to the supports

being fixed, causing large bending moments at these locations, resulting in large eccentricities.

To prove this, the same calculations for eccentricity have been completed for identical arches and

dead loading, with the only difference being pinned supports rather than fixed. The result showed

that which was expected, the bending moments at the support became zero and no tension was

developed in either of the 0.35 metre rise arches. Results can be seen summarised in table 15

below.


Table 15: Thrust line comparison for pinned support arches under dead load

Line of thrust outside the middle third

(Tension occurrence)

Line of thrust outside the arch section

(Indicating possible failure point)

0.35m circular No No

0.35m parabolic No No

0.6m circular Yes No

0.6m parabolic No No

1m circular Yes Yes

1m parabolic No No

This shows that when the circular arch shape is closest to the parabolic, the line of thrust will stay

closer to the middle third of the section, and arches modelled as parabolic can eliminate any

tensile stress occurring due to dead load.

4.5 Comparison of Microstran Pinned and Fixed Results

It was found, when considering the practical experiments of the concrete arches that it would

have been difficult to produce a fixed support when load testing these arches. For this reason each

of the Microstran models were analysed with pinned supports and the following results (Tables

16 to 25) were obtained.


Table 16: 0.35m Rise – Circular Arch Pinned Supports – Maximum Structural Behaviours


x-direction

(mm) Location

y-direction

(mm) Location

Axial Force

(N) Location

Shear Force

(N) Location

Bending Moment

(Nmm) Location

Dead Load 0.0003

Node 10(+), 32(-)

[inward] -0.0029 Node 21 -933.54 Node 1, 41 48.75

Node 1(-),

41(+) 9917.0 Node 21

Mid-span CL 0.0028 Node 6(-), 36(+) -0.0367 Node 21 -5921.86

Node 7, 8, 34,

35 2413.80

Node 20(-),

21(-/+), 22(+) 642574.5 Node 21

Third-span CL 0.0210 Node 12 -0.0491 Node 13 -5719.99 Node 2, 3 2531.54 Node 14, 15 774839.3 Node 14

Table 17: 0.35m Rise – Parabolic Arch Pinned Supports – Maximum Structural Behaviours


x-direction

(mm) Location

y-direction

(mm) Location

Axial Force

(N) Location

Shear Force

(N) Location

Bending Moment

(Nmm) Location

Dead Load 0.0003 Node 10(+), 32(-) -0.0028 Node 21 -941.97 1, 41 37.83

Node 1(-),

41(+) 6791.4

Node

13, 29


Node 7, 8, 34,

35 2401.47

Node 20(-),

21(-/+), 22(+) 613420.3 Node 21



Table 18: 0.6m Rise – Circular Arch Pinned Supports – Maximum Structural Behaviours


x-direction

(mm) Location

y-direction

(mm) Location

Axial Force

(N) Location

Shear Force

(N) Location

Bending Moment

(Nmm) Location

Dead Load 0.0002

Node 4(-), 5(-),

37(+), 38(+) -0.0022 Node 21 -776.00 Node 1, 41 88.77

Node 2(+),

40(-) 13207.0 Node 21


Node 6, 7, 35,

36 2433.32

Node 20(-),

21(-/+), 22(+) 704153.4 Node 21

Third-span CL 0.0409 Node 32 -0.0471 Node 13 -4256.60 Node 3, 4 -2324.94 Node 13, 14 779283.1 Node 14

Table 19: 0.6m Rise – Parabolic Arch Pinned Supports – Maximum Structural Behaviours


x-direction

(mm) Location

y-direction

(mm) Location

Axial Force

(N) Location

Shear Force

(N) Location

Bending Moment

(Nmm) Location

Dead Load 0.0004 Node 8(+), 34(-) -0.0013

Node

19, 23 -787.63 Node 1, 41 40.43

Node 1(-),

41(+) 5941.1

Node 7,

35


Node 7, 8, 34,

35 2403.53

Node 20(-),

21(-/+), 22(+) 592280.9 Node 21



Table 20: 1m Rise – Circular Arch Pinned Supports – Maximum Structural Behaviours


x-direction

(mm) Location

y-direction

(mm) Location

Axial Force

(N) Location

Shear Force

(N) Location

Bending Moment

(Nmm) Location

Dead Load 0.0030 Node 3(-), 39(+) -0.0058 Node 21 -845.42 Node 1, 41 -154.97

Node 2(+),

40(-) -46399.0

Node 3,

39

Mid-span CL -0.0367 Node 4(-), 38(+) -0.0714 Node 21 -2960.44

Node 4, 5, 37,

38 2459.52

Node 20(-),

21(-/+), 22(+) 911354.7 Node 21

Third-span CL 0.0953 Node 35 -0.0680 Node 14 -3651.45 Node 2, 3 -2605.70 Node 13, 14 887713.0 Node 14

Table 21: 1m Rise – Parabolic Arch Pinned Supports – Maximum Structural Behaviours


x-direction

(mm) Location

y-direction

(mm) Location

Axial Force

(N) Location

Shear Force

(N) Location

Bending Moment

(Nmm) Location

Dead Load 0.0009 Node 7(+), 35(-) -0.0012

Node 9,

33 -836.71 Node 1, 41 50.31

Node 1(-),

41(+) 10579.9

Node 6,

36


Node 7, 8, 34,

35 2402.43

Node 20(-),

21(-/+), 22(+) 608632.8 Node 21



Table 22: Percentage difference in maximum vertical deflection between 0.35m rise arches – Pinned Supports

0.35m Arch Deflections (mm)

Percentage Differences in Vertical Deflection (%)

Circular Parabolic

Dead Load -0.0029 -0.0028 3.8

Mid-Span Load -0.0367 -0.0346 5.6

Third-span Load -0.0491 -0.0492 -0.1

Table 23: Percentage difference in maximum vertical deflection between 0.6m rise arches – Pinned Supports

0.6m Arch Deflections


Circular Parabolic

Dead Load -0.0022 -0.0013 41.0

Mid-Span Load -0.0367 -0.0253 31.3

Third-span Load -0.0471 -0.0452 4.1

Table 24: Percentage difference in maximum vertical deflection between 1m rise arches – Pinned Supports

1m Arch Deflections


Circular Parabolic

Dead Load -0.0058 -0.0012 79.0

Mid-Span Load -0.0714 -0.0247 65.5

Third-span Load -0.0680 -0.0509 25.1


Table 25: Percentage difference in maximum shear force between circular and parabolic arches – Pinned

Supports

Percentage Difference in Maximum Absolute Shear Force (%)

0.35m rise 0.6m rise 1m rise

Dead Load 22.4 54.5 132.5

Mid-span Load 0.5 1.2 2.3

Third-span Load -3.4 -8.1 13.1

The results show similar patterns to those explained for fixed supports with regards to

comparison between loading cases and arch shape to stresses, deflections and bending moments.

However one result did change, and this can be seen in Table 25 above. The shear Forces present

in the arch do not follow the same pattern with pinned supports. For the third-span load case, the

0.35 metre and 0.6 metre parabolic arches experienced a higher maximum shear stress than the

corresponding circular arches (which can be seen as the negative number).

There were some significant differences between the data obtained with pinned supports and

fixed support arches. Comparing these shows that there was no major difference in axial or shear

forces. However maximum vertical deflections estimated for pinned supports, were found to

increase by approximately 41.8-61.6% for the third-span load situation for all arch cases. This is

expected due to the pinned supports now allowing the arch more degrees of freedom (rotation),

and when the arch is loaded un-symmetrically, this rotation at the supports will occur on a larger

scale (rather than when load is applied symmetrically at mid-span).

It was also found that as the rise of the arch increased the percentage difference between the

vertical deflection for pinned supports and fixed supports also increased (meaning that the largest

deflection differences were found for the 1 metre rise). The percentage difference was

consistently greater for parabolic arches when compared to the corresponding circular arches.

For mid-span and third-span load conditions, pinned supports produced a larger maximum

bending moment in all arch cases when compared to the corresponding load and arch cases for

fixed supports. The maximum bending moments estimated for the dead load cases were found to

be, in most cases, significantly less for arches with pinned supports. This was expected as the

large bending moments were at the location of the fixed support, and there is zero bending

moment at a pinned support. For concentrated loading, the maximum bending moment was at the

location of the load in all cases.


From Tables 22 to 25 above, it can be seen that the greatest variations between the circular and

parabolic arches occurred for the application of the dead load. This would indicate that selecting

a parabolic arch shape to reduce the shear forces, deflections and bending moments would yield a

more effective structure. However it is also noted that the dead load may not cause the arch to

develop internal stresses and deflections to the same magnitude that applied loads would.

Meaning that the behavioural benefit of using a parabolic arch in place of a circular arch when

concentrated loads are applied may not out-weigh the complications of producing the parabolic

shape. If however the more-realistic situation of backfill material, and uniformly distributed loads

are considered to be applied to the arch, the use of the parabola could prove more efficient in

reducing the development of tension.

4.6 Finite Element Models – SAP2000

In attempt to accurately model the concrete arch, three dimensional FEMs have been created and

analysed using the SAP2000 software. When analysing models in SAP2000 software which

consist of solid block elements, the best result is obtained when angles on the inside corners of

the element are near to 900, and the aspect ratio of the element is not too large (ratio of longest to

shortest block sides) (Computers and Structures Inc. 2009). As a result, one of the aims during

the development of these FEMs was to keep the element shapes close to cubic.

4.6.1 Model Layout

All arches were generated using a fixed radius, meaning circular arches would be developed;

however the number of elements can be input by specifying a number of divisions in all three

dimensions. These are depth, radial (or height for the set out of these models) and angular

divisions. The angular divisions dictate the number of flat top surfaces, or number of voussoirs

there will be. However this was required to change when the divisions for the other two

dimensions were altered if all elements were to be roughly cubic.

The first arch considered was a seventeen voussoir arch, which had an appearance identical to the

concrete arches constructed. This consisted of seventeen angular divisions, one radial and one

depth. This can be seen below in Figure 7.


Figure 7: SAP2000 – 0.6m rise arch – 17 angular divisions

The second configuration considered was that of an increased number of element for the post-

tensioned model. This was due to the requirement for a node at the centre of the cross-section,

which would allow the placement of post-tensioning. However as the number of elements was

increased, in order to keep the element shapes from elongating, the angular divisions would need

to be increased. This was not done, as the number of voussoirs required was fixed at seventeen.

These elongated elements may produce some minor inaccuracies in the result. However these are

only preliminary models, and there are components of these models which need to be improved

before an accurate comparison can be made between them and the concrete arches. Figure 8

shows a FEM of the post-tensioned arch with two radial, two depth and seventeen angular

divisions.


Figure 8: SAP2000 - 0.6m rise arch – 2 radial, 2 depth and 17 angular divisions

Due to time restraints on this thesis, and issues regarding the availability of the SAP2000

software, it was very difficult to produce models which closely represented the concrete arches

tested. One major difference between the constructed concrete arches and the FEMs were the

joint connections. The voussoir joints of the concrete arches constructed transfer load through

axial force and friction from one voussoir to another. Whereas the FEMs produced consider the

arch as a continuous structure (allowing bending moments to be transferred through connections).

The exclusion of a friction joint between each voussoir has greatly impacted the accuracy of the

FEM predictions. Another component found to be difficult to model was the post-tensioning; and

due to tension measuring inaccuracies (further explained in Section 5.6.2), the load placed

through the concrete arch was not known. For this reason, the post-tensioned FEM has been set

with the desired 20kN load placed through each voussoir, from either support. This may not have

produced accurate data approximations, but provided an understanding of the estimated stress

patterns and deflections which were expected.


4.6.2 Loading Applied

Initially, the arches were planned to be loaded and modelled for mid-span and third-span loads,

however due to time constraints, only the mid-span loading has been completed. The behaviour

of the two arches were analysed under the influence of the following loading:

- Dead load (self-weight).

- Mid-span surface pressure (placed on top of the keystone).

- Tensile load through the arch via steel tendon (only for post-tensioned arch).

For the FEM which is intended to approximate the behaviour of the solid voussoir arch, the

surface-pressure was applied as the failure load of the concrete arch (which is discussed in

Section 5.6.1). However the pressure loading for the post-tensioned FEM was varied in-order to

compare deflections expected to that found during the concrete arch loading.

4.6.3 Deflections/Deflected Shape

Figure 9: Solid arch voussoir - FEM deflection – dead load and surface pressure of 9×10-4 kN/mm2

Figure 9 above displays the deflected shape expected when the arch is loaded at mid-span, to a

pressure of 9 × 10−4𝑘𝑁/𝑚𝑚2. This is the pressure at which the solid concrete voussoir arch

failed during the load test. The approximated deflection obtained at mid-span was equal to

0.1379mm downward.


Figure 10: Post-tensioned arch – FEM deflected shape - dead load, mid-span surface pressure and post-

tensioning

Figure 10 above shows the expected deflected shape for the post-tensioned FEM. It is, as

expected, the same general shape as that above in Figure 9, however the values of deflection

obtained were recorded for varying surface pressure loadings. This can be seen in Table 26

below.

Table 26: Deflection Obtained for Varying Surface Pressures at Mid-span – Post-tensioned FEM

Pressure (MPa) Deflection Mid-span (mm downward)

0.0 0.00

0.9 0.19

3.0 0.50

6.0 0.94

9.0 1.39

12.0 1.83

15.0 2.27

16.5 (Failure occurred in concrete model) 2.50

Note: The pressure values were chosen based on the pressures applied in the concrete arch

loading, which can be seen in Section 5.6.2. The relationship between the data in Table 26 can be

seen below on Graph 5.


Graph 5: Pressure Vs Mid-span Deflection – Post-tensioned FEM

This data has been used later in Chapter 6, for comparison purposes between FEMs and concrete

arch loading results.

4.6.4 Expected Location of Failure

In order to gauge the failure mode of each arch, the internal stresses estimated by the FEMs were

analysed. As it was understood that the results obtained from these preliminary models would not

be accurate to that of the loaded concrete arches, these were used to observe the stress patterns

more than the actual values of stress. This showed where the greatest compressive and tensile

stresses would occur for a given load.

Figure 11 below displays the expected stress pattern with an applied dead load and mid-span

surface pressure of 9 × 10−4𝑘𝑁/𝑚𝑚2 on the solid arch model. The maximum compressive and

tensile stresses estimated were 1.935MPa at the top surface of the keystone, and 1.204MPa at the

bottom of the keystone respectively. The FEM also estimates tensile stresses to develop near to

the quarter-span points on the upper arch surface, to a maximum of 0.38MPa. This indicated that

if the arch supports are restrained, and the structure remains in-plane during loading, failure in the

0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10 12 14 16 18

Def

lect

ion

(m

m)

Applied Pressure (MPa)

Applied Pressure Vs Predicted Deflection at Mid-span - Post-tensioned Arch Loading


arch would occur due to the separation of the voussoirs where tension is expected. As the arch is

not post-tensioned, tensile stresses will cause voussoir joints to separate.

Figure 11: Internal stress - Solid finite element arch

The post-tensioned FEM showed a near identical stress pattern to that in Figure 11, as expected.

However, stress values were different due to the addition of compressive stresses caused by post-

tensioning, and the increased applied loading. When the arch is modelled to the failure load of the

concrete arch, the maximum compressive and tensile stresses expected in the arch were found to

be 68.64MPa at the top surface of the keystone, and 44.81MPa at the bottom of the keystone

respectively.

For the post-tensioned arch, the failure mode is expected to differ to that of the solid voussoir

arch. The locations where tension is expected would begin to separate, however due to the

addition of post-tensioning reinforcement, not only has a compressive stress been placed on the

arch section, but the steel will resist the tensile stresses experienced once this compression is

surpassed. This would lead to failure in the area of maximum compression, as a stress of

68.64MPa is greater than the compressive strength of the concrete at 52.52MPa.


4.6.5 SAP2000 Model Issues and Improvements

Initial intentions were to develop each of the FEMs consisting of friction joints between each

voussoir as previously explained. As this was not achieved, in order to improve the accuracy of

these models, the voussoir joints would need to be altered to prevent the fixed connection

between one voussoir and another ensuring the arch will essentially behave as a sixteen hinge

arch.

The post-tensioning applied to the FEM arch does not behaves in a manner which closely

approximates the behaviour of the tested concrete arch. The tensioning force was applied

separately to each voussoir. If further attempts were made to model this, it would be ideal to

apply this load at each support, and ensure the tensioning member follows the centreline of the

arch section. The losses in the member load could then be modelled, as there is expected to be a

reduction in the load applied to the voussoirs due to friction; the further the voussoir is from the

application of the load, the higher the loss in that voussoir will be. This would indicate that the

load present in the tensioning member within the keystone would be the minimum due to the

friction caused from all other voussoirs.


5. Physical Testing

5.1 Design and Construction of Arch Model Formwork

The formwork which was used to create the arch voussoirs was designed and fabricated,

completely made of steel and consists of a channel between two parallel flanged channel (PFC)

sections. Both sections are bolted to a flat bottom plate in order to create a beam formwork for a

rectangular section. The size of the section can be varied by adjusting one PFC into a secondary

row of bolt holes. The full length of the formwork is 4m and the maximum section size capable

of being poured is 180x200mm (width x depth)

The detailed drawings of the formwork can be seen in Appendix F. This formwork was not

designed to accommodate any significant loads, but to be durable and versatile. Figure 12 and 13

show images of the completed steel form.

In order to pour the arch voussoirs for the proposed arch model, a separate inner plywood form

has been constructed to be situated inside of the steel formwork. This inner form is explained

further in Section 5.2.


Figure 12: Steel Form

Figure 13: Section view of steel form


5.2 Plywood Inner Form

In order to produce voussoirs which were symmetrical and to the design size, a form needed to be

constructed and placed inside of the larger steel form to enable the pouring of arch blocks. This

inner form was made of plywood and all joints were fixed with screws. Figures 14 and 15 below

show images of one half of the completed formwork and Figure 16 shows half of the inner form

being placed inside of the larger steel form.

Figure 14: Construction of Inner Form


Figure 15: Side view of Inner Form

Figure 16: Inner Form and Steel Form


During construction of the form, the accuracy to which the plywood was required to be cut was a

major importance to ensure equally sized and symmetrical blocks. For this reason, all wood was

cut using a table saw where the angle of the blade can be altered. Even though a high level of

accuracy was achieved with regards to the plywood angles, the table saw can only be set to one

decimal place of a degree. For this reason angles could not be perfect, therefore there were slight

discrepancies between the design form and the completed formwork.

It was also considered that, due to the setup of the plywood trapezoids, there may have been

slight movements of the tops of the trapezoids (causing angles of the respective voussoirs to

change) during pouring under the weight of wet concrete. However these trapezoids were found

to be rigid when all screws were drilled. However, to reduce any possible movement, a thin strip

of plywood was screwed to all trapezoid top surfaces and ends butted to the steel formwork. This

can be seen in Figure 14, 15 and 16 above.

Calculations completed in order to construct this form can be seen in Appendix G.


5.3 Loading Rig

To load test each of the arches, a 10 tonne hydraulic jack was used. In order to place the load in

the desired location on the arches, a loading rig needed to be constructed to hold the hydraulic

jack in place. As each arch was initially intended to be load tested at mid-span and at third-span,

two sets of steel components were constructed in order to re-locate the load. These steel

components are shown below in Figure 17. However due to time restraints, the loading was only

completed at the mid-span location.

Figure 17: Loading Rig components

Initially the stabilizing rig (base PFC) was required to prevent the supports of the arch from

separating; as the desired failure mode is compression of the concrete voussoirs, not outward

movement of the supports due to horizontal thrusts developed. The base PFC was also intended

to keep the arch stable and in-plane during the loading procedure, but when the complications

associated with the loading method of the arches arose, the PFC base became part of the larger

loading rig, while keeping its usefulness as a means of support restriction.

Even though loading was only carry out at mid-span, the loading rig has been designed to secure

the system for third-span load tests. It has been constructed with a base parallel flanged channel

section (180PFC75) which is connected to a top securing plate (made of the same PFC section)

by four steel straps with varying lengths. The steel straps are 75mm in width and 6m thickness.


The loading rig setup can be seen below in Figure 18 for the mid-span loading.

Figure 18: Loading Rig setup for mid-span loading

The loading rig was designed with the intention of loading the arch with a ten tonne hydraulic

jack. However it was difficult to estimate the load which the concrete arch would actually

withstand before failure. With this in mind, the material specified and fabricated was to a higher

standard ensuring that the rig would not be damaged if a slightly higher load was required to

break the arch. The rig was designed using the smallest possible steel grade for each standard

component. This was due to the fact that the strength of the steel being used to fabricate the

components was initially unknown. Once the rig was fabricated, and the steel grades were

known, capacity design checks were completed to determine the extra loading capacity due to the

increased steel grade.

All calculations and design capacity checks for this loading rig can be seen in Appendix H.


5.4 The Concrete Arches

Three sets of concrete arch voussoirs were poured in total, the first two being solid concrete, and

the third being intended for observing the effects of post-tensioning. For this reason, PVC pipe

was placed in all voussoirs of the third arch. This is to ensure a steel rod could be placed through

the arch and tensioned. The voussoirs for the two types of arches can be seen in Figures 19 and

20 below. All three arches were designed to have the same span and rise of 2 metres and 0.6

metres respectively.

Figure 19: Voussoirs for arch one and two

Figure 20: Voussoir for arch three – post-tensioning


5.4.1 Pouring Method for Solid and Post-tensioning Voussoirs

The following steps were completed in order to produce the concrete voussoirs for arch one and

two.

The plywood inner form was first placed inside the steel form.

Concrete fine and coarse aggregates, cement and water were mixed in the designed

proportions.

Inside of the form (voussoir voids) sprayed with form oil to ensure no sticking of concrete

to formwork.

Each block poured and vibrated 3-4 times, for a total of 17 voussoirs for each arch.

Damp hessian sacks were placed over poured concrete to ensure concrete was kept moist.

Hessian sacks were propped from wet concrete with plastic chairs.

Plastic placed over hessian to keep moisture from evaporating.

Blocks were left in form for 2 days, with hessian sacks being wet each day ensuring

concrete is kept from drying out.

After two days, all blocks were taken out of the formwork and placed in clean water to

cure.

The process which was followed to pour the third set of arch voussoirs was almost identical to

that of the solid voussoirs, however the following steps were also taken to produce the post-

tensioning ducts using PVC pipes in each voussoir.


30mm PVC pipes were cut to the correct length with angled ends to suite the slope of the

plywood form. These PVC pipes were set in the centre of each voussoir, and held in place

while concrete was placed into each mould, and while vibrating was completed.

For this third arch, two bleed holes have been created with a ‘T’ shaped PVC joint in the

first and last blocks. This was to allow the void through the arch to be filled with grout

once the tensioning rod is tightened. This T shaped joint can be seen below in Figure 21

prior to pouring. However it was later decided that grouting the void was not necessary, as

the effect of the bond between reinforcing and concrete was not intended to be studied.

Figure 21: PVC 'T-joint' for post-tensioning and bleed hole

5.4.2 Concrete Voussoir Appearance

Once the concrete voussoirs were removed from the formwork, it was apparent that

honeycombing was present on the surface of some sections. This was primarily a concern for the

third arch due to the PVC pipes being situated at voussoir centres. This is further explained in

Section 5.4.3 Concrete Pouring Issues.


The thin strip of plywood placed across the top surface of the voussoirs prevented a smooth

finish, however this is not an issue as appearance was not a major concern, as long as the load

could be applied evenly to the top surface.

5.4.3 Concrete Pouring Issues

The third arch in particular was difficult to pour. Cutting the PVC pipes at an accurate angle was

challenging, and for most, the PVC needed to be held in place while concrete was poured and

vibrated. Finding a more effective way to secure the PVC inside the formwork for pouring,

without having to dismantle the form when extracting the voussoirs would have been ideal if

more arches were to be poured.

One of the main issues was the honeycombing present on the surface of the voussoirs for the third

arch. Due to the location and the fragile positioning of the PVC pipe (required to hold it in place

to ensure it did not move), in conjunction with the restricting area caused by the thin strip of

plywood, it was difficult to vibrate the concrete in the lower corners of the voussoirs. Figure 22

below shows cases of this honeycombing.

Figure 22: Most severe case of honeycombing in third arch voussoirs


5.5 Arch Erection

One of the issues faced during the planning of this thesis was the method of how to erect the arch

which was not post-tensioned. If each voussoir was to be placed individually, a form of centring

would be required, such as discussed earlier in Section 2.1 and seen in Figure 3.

5.5.1 Method of Erection

To avoid the construction and later dismantling of arch centring, each arch was erected by first

laying all voussoirs in place horizontally on a large sheet of plywood, then lifting the plywood

from the apex of the arch and rotating about the support points. Figures 23 to 25 shows this

process. As the end support voussoir faces do not finish horizontally, timber support wedges were

cut to the required angle for effective load transfer to the PFC base.

Figure 23: Arch positioned in base PFC horizontally


Figure 24: Side View of arch positioned in base PFC horizontally

Figure 25: Erected arch


5.5.2 Arch Erection Issues

Using this process in order to set the arch upright lead to other issues. As the arch was required to

be placed inside the flanges of the loading rig base, the voussoirs needed to be lifted while

situated in the PFC. This meant two other complications arose, one was that another thin 10mm

sheet of plywood (almost the same thickness as the flanges of the PFC) was required so that when

erected, the voussoirs were not excessively offset from the two support voussoir (this can be seen

in Figures 24 and 25 as the brown colour ply). The second complication was that due to the

weight of the base PFC, it was very difficult to ensure that the arch and base would fall in place at

the same time. When the base fell first, it caused some of the arch voussoirs to become slightly

offset, to the point where the apex of the arch is out of plane. This can be seen in Figure 26

below. However it is also understood that this appearance could also be due to surfaces of the

voussoirs and timber support wedges not being perfectly flat/level.

This alignment was improved by clamping the base PFC to the bottom plywood, allowing both

ply and base to rotate at the same time when the arch was lifted into its erect position.

Figure 26: Arch apex slightly out of plane due to lifting procedure


These issues were faced due to the large weight of the components being used and the weight of

the arch.

5.6 Loading Results for Concrete Arches

Loading of the solid voussoir arch and the post-tensioned arch was carried out using a ten tonne

hydraulic jack, which was secured on the underside of the top PFC component of the Loading

Rig. This arrangement can be seen in Figures 27 and 28 below.

Figure 27: Hydraulic Jack Cylinder Positioning


Figure 28: Hydraulic Jack Cylinder Positioning (side view)

The pump shown in Figure 28 above was used to exert the load onto the arch, and a pressure

gauge was utilised to approximate the pressure exerted onto the keystone voussoir. Both the solid

and post-tensioned arches we loaded at the mid-span points and the following sections describe

steps taken and observations made.

5.6.1 Solid Voussoir Arch Loading

Once the arch had been erected, and hydraulic jack secured, the load was steadily applied to the

top surface of the keystone. When the solid voussoir arch was loaded, deformation became

apparent almost immediately. The voussoirs on the underside of the keystone, and top surface at

the quarter-span point began to separate at a pressure of approximately 9 bar, which converts to

0.9MPa. When attempting to increase the load, deformation continued in these same locations,

however no further pressure was held by the arch. Failure of the solid voussoir arch occurred at

0.9MPa applied to the top surface of the keystone (mid-span).


Once the concentrated load produced a tensile stress in the arch which overcame the compressive

stresses due to the dead load, the voussoirs began to separate at those locations. This would not

be the case if loaded with a uniformly distributed load, as the increase in compressive stresses on

the top surface would act opposing the arch from deforming outwards at the third-span points.

This would essentially hold the arch together until these compressive stresses are overcome by

tensile stresses, or the voussoirs fail in compression. However as a concentrated load was applied,

these opposing forces were not present, allowing the voussoirs to separate at a much lower load.

The failure of the arch can be seen below in Figures 29 to 31.

Figure 29: Solid Voussoir Arch Failure Initiated


Figure 30: Solid Voussoir Arch Failure

Figure 31: Solid Voussoir Arch Collapse


It was apparent that when the arch began deforming, the movement at the quarter-span point only

occurred on one side of the arch. As the arch was symmetrical, and load positioned at mid-span,

it should follow that the deformation at the quarter-span points should have occurred on both

sides symmetrically. This unsymmetrical deformation can be seen in Figure 31 above.

The possible causes for this were considered and two factors arose; irregular surface of voussoirs

and of timber supports. It was noticed that the surfaces of the individual arch voussoirs were not

perfectly smooth and therefore it was difficult to ensure that all joints were flush. At one joint

location, it was noticed when the arch was erected that the two voussoirs were only contacting at

one point. This can be seen in Figure 32 below.

Figure 32: Irregular voussoir surface apparent in erected arch

This room for movement in some joints may have contributed to the unsymmetrical deformation.

Due to difficulties regarding the sanding of the timber supports, these were not perfectly level


surfaces. This could have also contributed to the deformed shape, as the two supports were not

identical, leading to one support possibly causing rotation before the other.

It was an initial intention to measure the deflection at mid-span during the loading of this arch,

however when test loading, the movement was found to be excessively large for a very small

load, and it would not have been ideal to attempt to measure this with the hasty collapse of the

arch.

5.6.2 Post-tensioned Arch Loading

The first step in setting up the post-tensioned arch for loading was to pass the member to be

tensioned through each voussoir. This was completed with all voussoirs laid horizontally. The

tension member used was a 12mm threaded galvanised bar, tightened with high tensile snug-lock

nuts. In attempt to gauge the tensile force placed on the bar, and there for the compressive stress

through the arch, load indicating washers (or direct tension indicators) were utilised which were

rated to 20kN. However due to availability issues, 16mm washers were used. This created

complications due to the threaded bar having a diameter of 12mm, and voussoir duct opening

diameter being 30mm. This meant that a number of washers were required on each supporting

surface to transfer the load to the block. With the increased number of washers, slippage began to

occur between washers when tightened, and as a consequence the load indicating washer was not

compressed correctly. This slippage occurred due to the angle at which the bar protruded the arch

ends. All washers were forced in one direction during tightening, which caused the load

indicating washer to compress unevenly. This can be seen in Figure 33 below.


Figure 33: Load indicating Washer Uneven Compression

Due to this issue with the load indicating washer, the post-tensioning load placed on the arch

could not be correctly measured.

Once the threaded bar was tensioned, the arch was pulled into shape. Figure 34 (a) and (b) show

the initial situation with no tension on the bar, and the post tensioned arch respectively.


(a) (b)

Figure 34: (a) Arch prior to post-tensioning, (b) Arch after post-tensioning

Once the arch was tensioned, the excess threaded bar was removed such that the assembly of nut

and washers could nest inside of the timber supports (once a hole had been drilled into these

supports). The adjustment which needed to be made to the support can be seen in Figure 35

below.


Figure 35: Adjustment to timber supports for post-tensioned arch loading

The arch was erected using a gantry crane and placed onto the supports in the base PFC; which is

shown in Figure 36 below.


Figure 36: Erect post-tensioned arch

As it was expected this arch would withstand higher loads than the solid voussoir arch, the mid-

span deflection was measures at regular loading intervals. The results for deflection vs load

obtained can be seen below in Table 27.

Table 27: Pressure Vs Deflection at Mid-Span - Post-tensioned Arch

Pressure (MPa) Deflection Mid-span (mm downward)

0.0 0.00

0.9 (after loading rig adjustment/movement due to load) 1.12

3.0 2.75

6.0 3.56

9.0 4.18

12.0 4.97

15.0 5.37

16.5 (Failure occurred) Unknown

This data can be seen plotted on Graph 6 below. It was not possible to record the deflection prior

to failure, and therefore the data was only plotted to 15MPa.


Graph 6: Pressure Vs Mid-span Deflection

This shows the expected relationship between deflection and the pressure applied at mid-span.

The two concrete arch loadings differed in many ways. The three important aspects are as

follows:

Load Capacity

Deformed Shape

Failure Mode

In terms of concentrated load capacity, the post-tensioned arch greatly surpassed the solid

voussoir arch with pressures of 16.5MPa and 0.9MPa respectively. This is over eighteen times

the capacity. The deformed shape of the post-tensioned arch was not noticeable until measured,

which cannot be said for the solid voussoir arch which showed large deflections almost

immediately after pressure was applied.

Once the post-tensioned arch reached its failure load, the keystone failed in compression, where a

crack propagated horizontally, in the upper portion of the voussoir, from the right side to the left.

The high compressive stress caused the top surface of the keystone to shatter. This can be seen

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14 16 18

Def

lect

ion

(m

m)

Applied Pressure (MPa)

Applied Pressure Vs Deflection at Mid-span - Post-tensioned Arch Loading


below in Figures 37 and 38. Even after failure of the keystone, the arch did not collapse. It

remained secure and (apart from the keystone) did not show any signs of structural damage. The

threaded bar held the arch in place after failure occurred.

Figure 37: Post-tensioned arch keystone after failure

Figure 38: Post-tensioned arch keystone top surface after failure


As the solid voussoir arch failed due to tensile stresses causing separation in the voussoirs, it can

easily be seen that the post-tensioning bar was successful in applying a compressive stress to the

arch. The tensile stresses which developed due to the concentrated load needed to overcome the

compressive stresses applied before the arch began to separate. However, even after these

compressive stresses were overcome, increased tensile stresses were then transferred to and

carried by the threaded bar reinforcing. This lead to concrete failure in compression.

Due to the fact that compressive and tensile stresses are developed in the arch at different

locations for any given applied load, the addition of post-tensioning (causing an initial

compression of the arch) would have decreased the capacity of the concrete which was in

compression. However it allowed the arch to withstand all tensile stresses which were present and

increased the failure load dramatically.

In future, this could be modelled, and the tensile stress which is expected in the arch could be

found for a certain load. The post-tensioning could then be designed around this expected stress

in order to produce the most efficient system without applying excessive compression. This was

not completed in this thesis.

The reason the post-tensioned arch failed in compression at the top of the keystone was due to the

combination of compressive stresses in this location. Under the arches own self weight, the top of

the keystone was in compression, while to bottom was in tension. This in conjunction with the

applied compressive pressure on the top surface (loading), and the compression of the arch due to

post-tensioning has lead the top portion of this voussoir to develop the greatest compressive

stress and therefore fail first. It was also noted that the arch could have been re-loaded, as even

after failure of the keystone, the system remained structurally sound. It would have withstood

further increases in load before catastrophic failure occurred.

As the post-tensioned arch held a substantially greater load than the solid voussoir arch before

failure, the addition of a tension member to counter tensile stresses is very effective where

concentrated loads are expected to be applied.


After the loading was complete, it was realised that the cylinder securing PFC had deformed

significantly. The 6mm base was bent upward, and as a result the flanges had rotated due to the

load from the steel strips. This deformation can be seen in Figure 39 below.

Figure 39: Deformation in top PFC plate after loading of post-tensioned arch

During the design of the loading rig, the secondary bending of the base of this PFC section was

not considered, which is why the unexpected deformation at this point occurred. As this

deformation was caused by the upward load from the hydraulic jack, if further loading was to be

undertaken, it is recommended that the PFC securing plate be altered, with a thicker base to resist

the stress experienced.


6. Comparison of Finite Element Models to Concrete Models

It is understood that there are considerable differences between the finite element arch models

developed and concrete arches constructed, with these differences causing large discrepancies in

results obtained. To gauge the extent of the inaccuracies, the following sections compare the

results obtained with regards to deflections, failure load and failure mode.

Table 28 below shows the conversion of pressures applied, to loads in kN and kg, using the

loading jack contacting area of:

𝐴 = 𝜋 × 𝑟2 = 𝜋 × (0.0175)2 = 9.6211 × 10−4𝑚2, where the diameter of the circular jack end

is 35mm

Table 28: Pressure conversion to Load

Pressure (Bar) Pressure (MPa) Force (kN) Load (kg)

0 0.0 0.0 0.0

9 0.9 0.9 88.3

30 3.0 2.9 292.2

60 6.0 5.8 588.5

90 9.0 8.7 882.7

120 12.0 11.5 1176.9

150 15.0 14.4 1471.2

165 16.5 15.9 1618.3

Table 29 below summarises the comparison of mid-span deflection, for both arch types at the

failure loads.


Table 29: Summary Comparison between FEMs and Concrete Arches

Arch Type Concrete

model /Finite

Element

Failure

Pressure

(MPa)

Failure

Load

(kN)

Downward Mid-

span Deflection at

failure load (mm)

Failure Mode

Observed/

Expected

Failure Location

Solid

Voussoir

Concrete 0.9MPa 0.87kN

Unable to be

measured (expected

to be above 10mm)

Voussoir

separation due to

tensile stresses

Underside of

keystone, top

surface at quarter-

span point

Finite

Element

Loaded

according to

failure of

concrete arch

- 0.14mm

Stress patterns

indicate voussoir

separation due to

tensile stresses

Underside of

keystone, top

surface at quarter-

span point

Post-

tensioned

Concrete 16.5MPa 15.88kN

Unknown, nearest

deflection 5.37mm at

15MPa (14.432kN)

Compressive

failure of

concrete

Top surface of

keystone

Finite

Element

Loaded

according to

failure of

concrete arch

- 2.50mm

Stress patterns

indicate

compressive

failure of

concrete or

failure in tension

Top surface of

keystone

(compressive),

Underside of the

keystone (tensile)

It can be seen above, that if it is assumed the voussoirs of the post-tension finite element

approximation do not separate, due to the pre-loaded compressive stress, and the tensile strength

of the threaded bar, all FEMs suggest a failure mode and location identical to that observed for

the corresponding concrete arch.

When the failure pressures were applied to the solid voussoir arch model, they produced stresses

in the arch which were well below the compressive and tensile strength of the concrete. The

concrete arch failed at this load due to the separation of voussoirs under a tensile stress, not

tensile failure of the concrete itself. As the FEM was modelled as a continuous arch, it could not

be used to approximate the failure load of the concrete arch or the deflection experienced

(evidence of this in Table 29 above as a large difference in the deflections was found); however

stress patterns in the FEM allowed an expected failure location to be determined.

The Post-tensioned concrete arch failed at a load of 15.875kN applied to the top surface of the

keystone. Applying this failure load to the keystone of the post-tensioned FEM produced a more

comparable result.

In order to compare the deflection experienced, the FEM was analysed at a range of pressures

which corresponded to those applied to the concrete arch during loading (which is shown in

section 4.6.3 above). Graph 7 below shows the deflections measured from the concrete arch


loading, and the deflections predicted by the FEMs (using the data from Tables 26, 27 and 28),

plotted against the applied load in kilo-newtons (kN).

Graph 7: Mid-span Deflection Comparison – Post-tensioned FEM and Concrete arch

Although the FEM deflections predicted are nearer to those measured for the post-tensioned arch

when compared to the solid voussoir arch, a considerable difference is still observed. It is

expected that due to the arch failing in compression, rather than separation of the voussoirs, the

error of modelling the voussoir joints as fixed rather than friction reduced slightly. However the

fact that the deflections experienced during the loading of the concrete arch were up to 3.14mm

greater than those predicted by the FEM shows that the modelled connections do influence the

deflection results significantly.

When the failure load of the post-tensioned arch was applied to the FEM, The expected stress

‘pattern’ in the arch was almost identical to that of the solid voussoir arch regarding which areas

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14 16 18

Do

wn

war

d D

efle

ctio

n (

mm

)

Load Applied (kN)

Applied Load Vs Deflection at Mid-span - Post-tensioned FEM and concrete arch

Concrete Arch FEM


were expected to develop the tension and maximum compression. These stresses were obviously

increased due to the increased failure load. As an initial compressive stress was placed through

the arch, all predicted compressive stresses were increased, and tensile stresses decreased. The

FEM therefore showed evidence that the arch would be more likely to fail in compression at the

keystone, due to the increase in maximum compressive stresses present at this location. The load

placed on the FEM (that which caused failure in the concrete arch) resulted in a predicted

maximum compressive stress of 68.64MPa and a maximum tensile stress of 44.81MPa. This

shows that at the load applied, the FEM was beyond concrete compression failure as the

compressive strength of the concrete was 52.52MPa. This may have occurred as a result of

increased compressive stress due to a possible excessive post-tensioning load, however when the

FEM was analysed without this load, the compressive stresses predicted at the top of the keystone

was 67.61MPa, which still exceeded the concrete strength.

A factor which contributed to the failure of the concrete arch keystone is concentrated stresses in

the joints. As the load was applied, and tension was developed, the voussoirs would have

separated (to a much lesser extent than the solid voussoir arch, due to the post-tensioning). This

separation due to tension would decrease the contacting area between voussoirs, therefore

increasing compressive stresses. As the maximum compression and maximum tension are

developed at the top and bottom of the keystone respectively, the effect was greatest at this

location, causing failure.

The degree to which an accurate comparison can be made is limited due to the following two

issues/differences:

Voussoir connecting joints were not modelled accurately to represent friction joints,

instead a continuous arch was modelled.

The post-tensioned load placed through the concrete arch was not able to be accurately

measured.

It is expected that the error between the FEM and concrete arch model deflections could be

considerably reduced if further studies were undertaken to improve the FEM with regards to the

two aspects mentioned.


7. Recommendations for Further Research and Improvements

Throughout this thesis, the objectives of observing and analysing the behaviour to segmented

arches, and comparing a physical concrete model to finite element approximations have been the

main concern. One aspect of the research conducted which could be further explored is the

development of finite element models. Therefore the following recommendations for further

research are made:

Improving the finite element models produced by changing the element joint connections,

allowing each voussoir to transmit only axial and frictional forces. Removing this

fixed/continuous element connection will ensure that the arch is behaving as a structure

constructed with individual voussoirs.

Further explore the possible methods of applying post-tensioning to an arch structure. By

applying the tensioning load from each of the arch supports rather than loading in each

voussoir. This can lead to incorporating the friction losses in the post-tensioning member.

In terms of the physical testing completed on the concrete arches, the following recommendations

are made for improvements which could be implemented and further studies which could be

completed:

It is recommended that if further physical testing be completed, construction of smaller

arch models be considered (approximately 100mm cubes); as it was found that transport,

erection and loading were difficult aspects due to the large weight of each of the physical

arches.

It is recommended that improvements be considered for the method of post-tensioning for

the concrete arch. Use of a flexible steel cable, rather than the threaded rod, will ensure

that tension can be applied with minimal slippage. Issues were faced regarding the

tightening of the threaded bar as there was a significant difference between the diameter

of the voussoir PVC conduit and the tensioning member. Consideration of the size of the

voussoir void should only occur once the tension member thickness is known.

Method of pouring post-tensioned concrete voussoirs could be improved with regards to

placement of the PVC conduit. As this was held in place during the concrete pour,

honeycombing was apparent on the finished surface due to insufficient vibration. If the


inner formwork could be improved to allow the fixing of PVC pipes without having to

dismantle the form to extract the voussoirs, the concrete could be sufficiently vibrated

without the concern of unintentionally relocating the conduit.

It was difficult to produce good quality flat supports using the timber wedges, and it is

recommended that if further testing is to be completed, these supports should be

constructed from concrete (which can be moulded) or fabricated in steel. This will

improve stability and reduce the separation of voussoirs due to uneven supports.

Investigate the behaviour of the concrete arches when the location of the concentrated

load is varied along the span. It is recommended that uniformly distributed loads and

moving loads are considered.

Further research conducted on voussoir arches will lead to a better understanding of the structures

under a variety of loading conditions, and possibly lead to the investigation of their use in small

bridging applications.


8. Summary and Conclusions

This undergraduate thesis provided an introductory study into the behaviour of voussoir arches.

Groundwork has been completed for the topic and possible areas of further research have been

highlighted.

To complete the objectives of this thesis, laboratory experiments have been conducted and linear

elastic line element/finite element models were analysed to determine the differences in structural

behaviour which arise due to changes in arch shape, rise to span ratio and load positioning.

Through this study, it has been found when comparing these arch attributes, that there is a

number of key behavioural differences between the arches. These are as follows:

Parabolic arches deflect less in the vertical direction than corresponding circular arches

for the same rise and span, for almost all given loading cases.

Most predicted values of shear force were found to be less in parabolic arches when

compared to corresponding circular arches, and the difference between these

corresponding values was found to increase with increasing rise to span ratio.

It was found that the amount of tension experienced due to the dead load of an arch could

be minimized if not removed completely by using a parabolic arch.

It has been established that the greatest variations between the circular and parabolic arches occur

during the application of the dead load or uniformly distributed load. This would indicate that

using a parabolic arch shape to reduce the values of shear forces, deflections and bending

moments would yield a somewhat more efficient structure. If a more-realistic situation of backfill

material, and uniformly distributed loads are applied to the arch, the use of the parabola could

prove more effective in reducing the development of tension.

This thesis has explored the construction of concrete arches and issues associated with their

physical testing. A post-tensioned concrete arch has been constructed and preliminary finite

element models developed to explore the modelling of physical attributes through finite element

analysis, as well as difficulties which could be encountered. Further development of the finite


element models needs to be undertaken before an accurate comparison can be made between

them and the physical concrete arch model.

The post-tensioned concrete arch withstood a substantially greater load than the solid voussoir

arch before failure. The addition of a tension member to produce compressive stresses in the

concrete is therefore very effective in increasing the structures loading capacity when

concentrated loads are applied.

The study of the behaviour of voussoir arches will lead to a better understanding of the structures

under a variety of loading conditions, and possibly lead to the investigation of their use in small

bridging applications.


Reference List

Adam, J, 1994, Roman Building – Materials and Techniques, First Edition, B.T. Batsford Ltd,

London, England

Bourke, J, Taylor, S, Robinson, D & Long, A 2010, Analysis of a flexible concrete arch, Queen’s

University, Belfast, UK, viewed 22nd September 2013, < http://www.arch-

bridges.com/paper2010/pdf/16-Analysis%20of%20a%20flexible%20concrete%20arch.pdf>

Centring used to build the arch of a bridge in the 19th century picture, 2009, digital image,

viewed 20th September 2013,

<http://counterlightsrantsandblather1.blogspot.com.au/2009_01_01_archive.html>

Clarke, S & Engelbach, R 1990, Ancient Egyptian Construction and Architecture, Dover

Publications Inc., New York, USA

Computers and Structures Inc., 2009, CSI Analysis Reference Manual For SAP2000, ETABS, and

SAFE, Computers and Structures Inc., Berkeley, California, USA, pg. 21

Concrete Construction 2009, The Relevance of Microcracking, viewed 2nd October 2013,

<http://www.concreteconstruction.net/internet/the-relevance-of-microcracking.aspx>

Gorenc, B, Tinyou, R & Syam, A 2012, Steel Designers’ Handbook, 8th edn, NewSouth

Publishing, Sydney NSW.

http://www.arch-bridges.com/paper2010/pdf/16-Analysis%20of%20a%20flexible%20concrete%20arch.pdf

http://www.arch-bridges.com/paper2010/pdf/16-Analysis%20of%20a%20flexible%20concrete%20arch.pdf

http://counterlightsrantsandblather1.blogspot.com.au/2009_01_01_archive.html

http://www.concreteconstruction.net/internet/the-relevance-of-microcracking.aspx


Heyman, J 1982, The Masonry Arch, First Edition, Ellis Horwood Limited, West Sussex,

England

Loo, Y & Chowdhury, S 2010, Reinforced & Prestressed Concrete – Analysis and Design with

emphasis on application of AS3600-2009, Third Edition, Cambridge University Press, New York

Moaveni, S 2003, Finite Element Analysis, Second Edition, Pearson Education, Inc. Upper

Saddle River, New Jersey

newsedge101’s channel 2010, Construction – Macrete FelxiArch Bridge System, YouTube video,

viewed 22nd September 2013, < http://www.youtube.com/watch?v=pQbBxY3BmSI>

Taylor, S, Long, A, Robinson, D, Rankin, B, Gupta, A, Kirkpatrick, J & Hogg, I 2007,

Development of a Flexible Concrete Arch, Queens University of Belfast (QUB) & Macrete

Ireland, viewed 22nd September 2013, < http://pdfs.findtheneedle.co.uk/14181..pdf>

Teychenne, D, Franklin, R & Erntroy, H, 1988, Design of Normal Concrete Mixes, Second

Edition, Construction Research Communications Ltd, Rosebery Avenue London

Triangular Arch figure, 2011, digital image, viewed 20 September 2013,

<http://architechstok.wordpress.com/2011/05/19/types-of-arches/>

True Arch vs. Corbel Arch figure, 2013, digital image, viewed 20 September 2013,

<http://www.essential-humanities.net/supplementary-art-articles/corbelling/#.Ujw8IbykBIs>

http://www.youtube.com/watch?v=pQbBxY3BmSI


http://architechstok.wordpress.com/2011/05/19/types-of-arches/



Two forms of arch construction figure, digital image, 2007, viewed 22nd September 2013,

<http://pdfs.findtheneedle.co.uk/14181..pdf>



Appendix A – Concrete Design

Concrete Mix Design Form


Concrete Design Calculations and Explanations



Appendix B – Microstran Arch Model Example – Circular 0.35m

Rise

Figure 41: Microstran - 0.35m rise circular arch member positioning

(1) (2)

(3) (4)

Figure 42: 0.35m rise circular arch - Load Case 1 - Dead Load - (1) Loading, (2) Deflected Shape, (3) Bending

Moment Diagram, (4) Shear Force Diagram

X

Y

Z

theta: 270 phi: 0

1

2

3

4

5

67

89

1011

1213

14 15 16 17 18 19 20 21 22 23 24 25 26 27 2829

3031

3233

3435

36

37

38

39

40

41

X

Y

Z

theta: 270 phi: 0

1

2

3

4

56

78

910

1112

1314 15 16 17 18 19 20 21 22 23 24 25 26 27

2829

3031

3233

3435

36

37

38

39

40

X

Y

Z

theta: 270 phi: 0

0.519

0.519

0.519

0.519

0.5190.519

0.5190.519

0.5190.519

0.5190.519

0.5190.5190.5190.5190.5190.5190.5190.5190.5190.5190.5190.5190.5190.5190.519

0.5190.519

0.5190.519

0.5190.519

0.5190.519

0.519

0.519

0.519

0.519

0.519

X

Y

Z

theta: 270 phi: 0

X

Y

Z

theta: 270 phi: 0X

Y

Z

theta: 270 phi: 0

Figure 40: Microstran - 0.35m rise circular arch node positioning


(1) (2)

(3) (4)

Figure 43: 0.35m rise circular arch - Load Case 2 - Mid-span Concentrated Load - (1) Loading, (2) Expected

Deflected Shape, (3) Bending Moment Diagram, (4) Shear Force Diagram

(1) (2)

(3) (4)

Figure 44: 0.35m rise circular arch - Load Case 3 – Third-span Point Concentrated Load - (1) Loading, (2)

Expected Deflected Shape, (3) Bending Moment Diagram, (4) Shear Force Diagram

X

Y

Z

theta: 270 phi: 0

5000

X

Y

Z

theta: 270 phi: 0

X

Y

Z

theta: 270 phi: 0

X

Y

Z

theta: 270 phi: 0

X

Y

Z

theta: 270 phi: 0

5000

X

Y

Z

theta: 270 phi: 0

X

Y

Z

theta: 270 phi: 0

X

Y

Z

theta: 270 phi: 0


Appendix C – Example Microstran Model Analysis Report –

Circular 0.35m rise, Dead Load Case

== I N P U T / A N A L Y S I S R E P O R T ==

Job: 0.35m rise Circular

Title: 0.35m rise Circular

Type: Plane frame

Date: 12 Oct 2013

Time: 5:37 PM

Nodes ............................. 41

Members ........................... 40

Spring supports ................... 0

Sections .......................... 1

Materials ......................... 1

Primary load cases ................ 4

Combination load cases ............ 0

Analysis: Linear elastic

== L O A D C A S E S ==

Case Type Analysis Title

1 P L Dead Load

Analysis Types:

S - Skipped (not analysed)

L - Linear

N - Non-linear

== N O D E C O O R D I N A T E S ==

Node X Y Z Restraint

mm mm mm

1 -1000.000 0.000 0.000 111111

2 -950.000 33.400 0.000 000000

3 -900.000 68.900 0.000 000000

4 -850.000 101.500 0.000 000000

5 -800.000 131.600 0.000 000000

6 -750.000 159.300 0.000 000000

7 -700.000 184.800 0.000 000000

8 -650.000 208.000 0.000 000000

9 -600.000 229.200 0.000 000000

10 -550.000 248.500 0.000 000000

11 -500.000 265.900 0.000 000000

12 -450.000 281.400 0.000 000000

13 -400.000 295.200 0.000 000000

14 -350.000 307.200 0.000 000000

15 -300.000 317.600 0.000 000000

16 -250.000 326.400 0.000 000000

17 -200.000 333.400 0.000 000000

18 -150.000 339.000 0.000 000000

19 -100.000 342.900 0.000 000000

20 -50.000 345.200 0.000 000000

21 0.000 346.000 0.000 000000

22 50.000 345.200 0.000 000000

23 100.000 342.900 0.000 000000

24 150.000 339.000 0.000 000000

25 200.000 333.400 0.000 000000

26 250.000 326.400 0.000 000000

27 300.000 317.600 0.000 000000

28 350.000 307.200 0.000 000000


29 400.000 295.200 0.000 000000

30 450.000 281.400 0.000 000000

31 500.000 265.900 0.000 000000

32 550.000 248.500 0.000 000000

33 600.000 229.200 0.000 000000

34 650.000 208.000 0.000 000000

35 700.000 184.800 0.000 000000

36 750.000 159.300 0.000 000000

37 800.000 131.600 0.000 000000

38 850.000 101.500 0.000 000000

39 900.000 68.900 0.000 000000

40 950.000 33.400 0.000 000000

41 1000.000 0.000 0.000 111111

== M E M B E R D E F I N I T I O N ==

Member A B C Prop Matl Rel-A Rel-B Length

mm

1 1 2 Y 1 1 000000 000000 60.130

2 2 3 Y 1 1 000000 000000 61.321

3 3 4 Y 1 1 000000 000000 59.689

4 4 5 Y 1 1 000000 000000 58.361

5 5 6 Y 1 1 000000 000000 57.160

6 6 7 Y 1 1 000000 000000 56.127

7 7 8 Y 1 1 000000 000000 55.120

8 8 9 Y 1 1 000000 000000 54.309

9 9 10 Y 1 1 000000 000000 53.596

10 10 11 Y 1 1 000000 000000 52.941

11 11 12 Y 1 1 000000 000000 52.347

12 12 13 Y 1 1 000000 000000 51.869

13 13 14 Y 1 1 000000 000000 51.420

14 14 15 Y 1 1 000000 000000 51.070

15 15 16 Y 1 1 000000 000000 50.768

16 16 17 Y 1 1 000000 000000 50.488

17 17 18 Y 1 1 000000 000000 50.313

18 18 19 Y 1 1 000000 000000 50.152

19 19 20 Y 1 1 000000 000000 50.053

20 20 21 Y 1 1 000000 000000 50.006

21 21 22 Y 1 1 000000 000000 50.006

22 22 23 Y 1 1 000000 000000 50.053

23 23 24 Y 1 1 000000 000000 50.152

24 24 25 Y 1 1 000000 000000 50.313

25 25 26 Y 1 1 000000 000000 50.488

26 26 27 Y 1 1 000000 000000 50.768

27 27 28 Y 1 1 000000 000000 51.070

28 28 29 Y 1 1 000000 000000 51.420

29 29 30 Y 1 1 000000 000000 51.869

30 30 31 Y 1 1 000000 000000 52.347

31 31 32 Y 1 1 000000 000000 52.941

32 32 33 Y 1 1 000000 000000 53.596

33 33 34 Y 1 1 000000 000000 54.309

34 34 35 Y 1 1 000000 000000 55.120

35 35 36 Y 1 1 000000 000000 56.127

36 36 37 Y 1 1 000000 000000 57.160

37 37 38 Y 1 1 000000 000000 58.361

38 38 39 Y 1 1 000000 000000 59.689

39 39 40 Y 1 1 000000 000000 61.321

40 40 41 Y 1 1 000000 000000 60.130

== S T A N D A R D S H A P E S ==

Section Shape Name Comment D1/D4 D2/D5 D3/D6

1 RECT 0.35CIRCULAR comment 150.000 150.000

Dimension codes:

RECT - D1=D D2=B

== S E C T I O N P R O P E R T I E S ==


Section Ax Ay Az J Iy Iz fact

mm2 mm2 mm2 mm4 mm4 mm4

1 2.250E+04 0.000E+00 0.000E+00 7.138E+07 4.219E+07 4.219E+07 1.000

== M A T E R I A L P R O P E R T I E S ==

Material E u Density Alpha

N/mm2 t/mm3 /deg C

1 3.360E+04 0.2000 2.350E-09 1.170E-05

== T A B L E O F Q U A N T I T I E S ==

MATERIAL 1

Section Name Length Mass Comment

mm tonne

1 0.35CIRCULAR 2154.480 0.114 comment

---------- ----------

2154.480 0.114

== C O N D I T I O N N U M B E R ==

Maximum condition number: 6.400E+00 at node: 39 DOFN: 6

== A P P L I E D L O A D I N G ==

CASE 1: Dead Load

-- Member Loads --

Member Form T A S F1 X1 F2 X2

1 UNIF FY GL -0.519

2 UNIF FY GL -0.519

3 UNIF FY GL -0.519

4 UNIF FY GL -0.519

5 UNIF FY GL -0.519

6 UNIF FY GL -0.519

7 UNIF FY GL -0.519

8 UNIF FY GL -0.519

9 UNIF FY GL -0.519

10 UNIF FY GL -0.519
































-- Sum of Applied Loads (Global Axes): --

FX: 0.000 FY: -1118.175 FZ: 0.000

Moments about the global origin:

MX: 0.000 MY: 0.000 MZ: 0.000

== N O D E D I S P L A C E M E N T S ==

CASE 1: Dead Load

Node X-Disp Y-Disp Z-Disp X-Rotn Y-Rotn Z-Rotn

mm mm mm rad rad rad

1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

2 -0.000040 -0.000062 0.000000 0.000000 0.000000 -0.000001

3 -0.000048 -0.000168 0.000000 0.000000 0.000000 -0.000002

4 -0.000034 -0.000307 0.000000 0.000000 0.000000 -0.000002

5 -0.000006 -0.000474 0.000000 0.000000 0.000000 -0.000003

6 0.000031 -0.000663 0.000000 0.000000 0.000000 -0.000003

7 0.000072 -0.000868 0.000000 0.000000 0.000000 -0.000004

8 0.000112 -0.001084 0.000000 0.000000 0.000000 -0.000004

9 0.000148 -0.001307 0.000000 0.000000 0.000000 -0.000004

10 0.000179 -0.001530 0.000000 0.000000 0.000000 -0.000004

11 0.000203 -0.001750 0.000000 0.000000 0.000000 -0.000004

12 0.000217 -0.001962 0.000000 0.000000 0.000000 -0.000004

13 0.000223 -0.002163 0.000000 0.000000 0.000000 -0.000004

14 0.000219 -0.002348 0.000000 0.000000 0.000000 -0.000003

15 0.000207 -0.002515 0.000000 0.000000 0.000000 -0.000003

16 0.000186 -0.002661 0.000000 0.000000 0.000000 -0.000003

17 0.000158 -0.002784 0.000000 0.000000 0.000000 -0.000002

18 0.000124 -0.002881 0.000000 0.000000 0.000000 -0.000002

19 0.000086 -0.002952 0.000000 0.000000 0.000000 -0.000001

20 0.000044 -0.002995 0.000000 0.000000 0.000000 -0.000001

21 0.000000 -0.003009 0.000000 0.000000 0.000000 0.000000

22 -0.000044 -0.002995 0.000000 0.000000 0.000000 0.000001

23 -0.000086 -0.002952 0.000000 0.000000 0.000000 0.000001

24 -0.000124 -0.002881 0.000000 0.000000 0.000000 0.000002

25 -0.000158 -0.002784 0.000000 0.000000 0.000000 0.000002

26 -0.000186 -0.002661 0.000000 0.000000 0.000000 0.000003

27 -0.000207 -0.002515 0.000000 0.000000 0.000000 0.000003

28 -0.000219 -0.002348 0.000000 0.000000 0.000000 0.000003

29 -0.000223 -0.002163 0.000000 0.000000 0.000000 0.000004

30 -0.000217 -0.001962 0.000000 0.000000 0.000000 0.000004

31 -0.000203 -0.001750 0.000000 0.000000 0.000000 0.000004

32 -0.000179 -0.001530 0.000000 0.000000 0.000000 0.000004

33 -0.000148 -0.001307 0.000000 0.000000 0.000000 0.000004

34 -0.000112 -0.001084 0.000000 0.000000 0.000000 0.000004

35 -0.000072 -0.000868 0.000000 0.000000 0.000000 0.000004

36 -0.000031 -0.000663 0.000000 0.000000 0.000000 0.000003

37 0.000006 -0.000474 0.000000 0.000000 0.000000 0.000003

38 0.000034 -0.000307 0.000000 0.000000 0.000000 0.000002

39 0.000048 -0.000168 0.000000 0.000000 0.000000 0.000002

40 0.000040 -0.000062 0.000000 0.000000 0.000000 0.000001

41 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

== M E M B E R F O R C E S ==

CASE 1: Dead Load

Member Node Axial Shear-y Shear-z Torque Moment-y Moment-z

N N N Nmm Nmm Nmm

1 1 -861.0859 -97.1480 0.0000 0.0000 0.0000-24646.1953

2 -843.7513 -71.1980 0.0000 0.0000 0.0000-19584.9102


2 2 -845.4366 -47.1417 0.0000 0.0000 0.0000-19584.9102

3 -827.0121 -21.1917 0.0000 0.0000 0.0000-17489.7754

3 3 -825.5234 -53.9377 0.0000 0.0000 0.0000-17489.7754

4 -808.6039 -27.9877 0.0000 0.0000 0.0000-15044.7588

4 4 -807.0789 -56.9851 0.0000 0.0000 0.0000-15044.7588

5 -791.4570 -31.0351 0.0000 0.0000 0.0000-12476.2861

5 5 -789.8284 -59.4854 0.0000 0.0000 0.0000-12476.2861

6 -775.4521 -33.5354 0.0000 0.0000 0.0000 -9817.7441

6 6 -773.8463 -60.1033 0.0000 0.0000 0.0000 -9817.7441

7 -760.6118 -34.1533 0.0000 0.0000 0.0000 -7172.5679

7 7 -758.8166 -62.4031 0.0000 0.0000 0.0000 -7172.5679

8 -746.7758 -36.4531 0.0000 0.0000 0.0000 -4448.0776

8 8 -745.1413 -61.3793 0.0000 0.0000 0.0000 -4448.0776

9 -734.1385 -35.4293 0.0000 0.0000 0.0000 -1819.3020

9 9 -732.5910 -59.3712 0.0000 0.0000 0.0000 -1819.3020

10 -722.5743 -33.4212 0.0000 0.0000 0.0000 667.3307

10 10 -721.0502 -57.5952 0.0000 0.0000 0.0000 667.3307

11 -712.0196 -31.6452 0.0000 0.0000 0.0000 3029.5757

11 11 -710.5164 -56.0343 0.0000 0.0000 0.0000 3029.5757

12 -702.4719 -30.0843 0.0000 0.0000 0.0000 5283.6187

12 12 -701.1858 -52.0601 0.0000 0.0000 0.0000 5283.6187

13 -694.0236 -26.1101 0.0000 0.0000 0.0000 7310.9429

13 13 -692.7473 -49.5148 0.0000 0.0000 0.0000 7310.9429

14 -686.5193 -23.5648 0.0000 0.0000 0.0000 9189.8145

14 14 -685.4827 -44.4683 0.0000 0.0000 0.0000 9189.8145

15 -680.0851 -18.5183 0.0000 0.0000 0.0000 10798.1846

15 15 -679.1899 -39.4938 0.0000 0.0000 0.0000 10798.1846

16 -674.6227 -13.5438 0.0000 0.0000 0.0000 12144.5029

16 16 -673.7312 -37.2230 0.0000 0.0000 0.0000 12144.5029

17 -670.0982 -11.2730 0.0000 0.0000 0.0000 13368.7285

17 17 -669.5330 -29.7348 0.0000 0.0000 0.0000 13368.7285

18 -666.6266 -3.7848 0.0000 0.0000 0.0000 14211.9561

18 18 -666.1208 -26.2390 0.0000 0.0000 0.0000 14211.9561

19 -664.0967 -0.2890 0.0000 0.0000 0.0000 14877.1719

19 19 -663.7502 -21.4529 0.0000 0.0000 0.0000 14877.1719

20 -662.5565 4.4971 0.0000 0.0000 0.0000 15301.5166

20 20 -662.3937 -15.3585 0.0000 0.0000 0.0000 15301.5166

21 -661.9785 10.5915 0.0000 0.0000 0.0000 15420.7070

21 21 -661.9785 -10.5915 0.0000 0.0000 0.0000 15420.7070

22 -662.3937 15.3585 0.0000 0.0000 0.0000 15301.5166

22 22 -662.5565 -4.4971 0.0000 0.0000 0.0000 15301.5166

23 -663.7502 21.4529 0.0000 0.0000 0.0000 14877.1719

23 23 -664.0967 0.2890 0.0000 0.0000 0.0000 14877.1719

24 -666.1208 26.2390 0.0000 0.0000 0.0000 14211.9561

24 24 -666.6266 3.7848 0.0000 0.0000 0.0000 14211.9561

25 -669.5330 29.7348 0.0000 0.0000 0.0000 13368.7285

25 25 -670.0982 11.2730 0.0000 0.0000 0.0000 13368.7285

26 -673.7312 37.2230 0.0000 0.0000 0.0000 12144.5029

26 26 -674.6227 13.5438 0.0000 0.0000 0.0000 12144.5029

27 -679.1899 39.4938 0.0000 0.0000 0.0000 10798.1846

27 27 -680.0851 18.5183 0.0000 0.0000 0.0000 10798.1846

28 -685.4827 44.4683 0.0000 0.0000 0.0000 9189.8145

28 28 -686.5193 23.5648 0.0000 0.0000 0.0000 9189.8145

29 -692.7473 49.5148 0.0000 0.0000 0.0000 7310.9429

29 29 -694.0236 26.1101 0.0000 0.0000 0.0000 7310.9429

30 -701.1858 52.0601 0.0000 0.0000 0.0000 5283.6187

30 30 -702.4719 30.0843 0.0000 0.0000 0.0000 5283.6187

31 -710.5164 56.0343 0.0000 0.0000 0.0000 3029.5757

31 31 -712.0196 31.6452 0.0000 0.0000 0.0000 3029.5757

32 -721.0502 57.5952 0.0000 0.0000 0.0000 667.3307

32 32 -722.5743 33.4212 0.0000 0.0000 0.0000 667.3307

33 -732.5910 59.3712 0.0000 0.0000 0.0000 -1819.3020

33 33 -734.1385 35.4293 0.0000 0.0000 0.0000 -1819.3020

34 -745.1413 61.3793 0.0000 0.0000 0.0000 -4448.0776

34 34 -746.7758 36.4531 0.0000 0.0000 0.0000 -4448.0776

35 -758.8166 62.4031 0.0000 0.0000 0.0000 -7172.5679

35 35 -760.6118 34.1533 0.0000 0.0000 0.0000 -7172.5679

36 -773.8463 60.1033 0.0000 0.0000 0.0000 -9817.7441

36 36 -775.4521 33.5354 0.0000 0.0000 0.0000 -9817.7441

37 -789.8284 59.4854 0.0000 0.0000 0.0000-12476.2861

37 37 -791.4570 31.0351 0.0000 0.0000 0.0000-12476.2861

38 -807.0789 56.9851 0.0000 0.0000 0.0000-15044.7588


38 38 -808.6039 27.9877 0.0000 0.0000 0.0000-15044.7588

39 -825.5234 53.9377 0.0000 0.0000 0.0000-17489.7754

39 39 -827.0121 21.1917 0.0000 0.0000 0.0000-17489.7754

40 -845.4366 47.1417 0.0000 0.0000 0.0000-19584.9102

40 40 -843.7513 71.1980 0.0000 0.0000 0.0000-19584.9102

41 -861.0859 97.1480 0.0000 0.0000 0.0000-24646.1953

Positive Forces (Member Axes):

Axial - Tension Shear - End A sagging

Torque - Right-hand twist Moment - Sagging

== S U P P O R T R E A C T I O N S ==

CASE 1: Dead Load

Node Force-X Force-Y Force-Z Moment-X Moment-Y Moment-Z

N N N Nmm Nmm Nmm

1 662.0632 559.0876 0.0000 0.0000 0.0000 24646.1953

41 -662.0632 559.0876 0.0000 0.0000 0.0000-24646.1953

SUM: 0.0000 1118.1752 0.0000 (all nodes)

Max. residual: 5.824E-09 at DOFN: 54

(Reactions act on structure in positive global axis directions.)


Appendix D – Example Calculation of Eccentricity for Circular and

Parabolic 0.35m Rise Arch, Dead Load Case Note: cells highlighted in orange are locations of tensile stress occurring and those highlighted in

red are the location of possible failure points (where thrust line leaves the arch section).

Table 30: 0.35m Rise Arch – Line of Thrust Positioning Comparison – Fixed Supports

0.35m Rise Arch Circular Parabolic

member node Axial Force

Moment

Eccentricity of

Axial Load Axial Force Moment

Eccentricity of

Axial Load

N Nmm mm N Nmm mm

1 1 -861.09 -24646.20 28.62 -858.93 -28097.54 32.71

2 -843.75 -19584.91 23.21 -841.44 -23224.33 27.60

2 2 -845.44 -19584.91 23.17 -839.59 -23224.33 27.66

3 -827.01 -17489.78 21.15 -822.99 -18781.99 22.82

3 3 -825.52 -17489.78 21.19 -821.22 -18781.99 22.87

4 -808.60 -15044.76 18.61 -805.50 -14746.76 18.31

4 4 -807.08 -15044.76 18.64 -803.83 -14746.76 18.35

5 -791.46 -12476.29 15.76 -788.98 -11095.76 14.06

5 5 -789.83 -12476.29 15.80 -787.30 -11095.76 14.09

6 -775.45 -9817.74 12.66 -773.39 -7740.51 10.01

6 6 -773.85 -9817.74 12.69 -771.87 -7740.51 10.03

7 -760.61 -7172.57 9.43 -758.85 -4725.53 6.23

7 7 -758.82 -7172.57 9.45 -757.33 -4725.53 6.24

8 -746.78 -4448.08 5.96 -745.24 -1964.48 2.64

8 8 -745.14 -4448.08 5.97 -743.89 -1964.48 2.64

9 -734.14 -1819.30 2.48 -732.68 495.86 -0.68

9 9 -732.59 -1819.30 2.48 -731.44 495.86 -0.68

10 -722.57 667.33 -0.92 -721.11 2672.97 -3.71

10 10 -721.05 667.33 -0.93 -719.98 2672.97 -3.71

11 -712.02 3029.58 -4.25 -710.53 4583.16 -6.45

11 11 -710.52 3029.58 -4.26 -709.43 4583.16 -6.46

12 -702.47 5283.62 -7.52 -700.92 6307.90 -9.00

12 12 -701.19 5283.62 -7.54 -699.96 6307.90 -9.01

13 -694.02 7310.94 -10.53 -692.34 7795.39 -11.26

13 13 -692.75 7310.94 -10.55 -691.50 7795.39 -11.27

14 -686.52 9189.81 -13.39 -684.75 9058.07 -13.23

14 14 -685.48 9189.81 -13.41 -684.03 9058.07 -13.24

15 -680.09 10798.18 -15.88 -678.17 10107.04 -14.90

15 15 -679.19 10798.18 -15.90 -677.50 10107.04 -14.92

16 -674.62 12144.50 -18.00 -672.57 11018.24 -16.38

16 16 -673.73 12144.50 -18.03 -672.04 11018.24 -16.40

17 -670.10 13368.73 -19.95 -668.00 11734.19 -17.57

17 17 -669.53 13368.73 -19.97 -667.54 11734.19 -17.58

18 -666.63 14211.96 -21.32 -664.42 12327.85 -18.55

18 18 -666.12 14211.96 -21.34 -664.09 12327.85 -18.56


19 -664.10 14877.17 -22.40 -661.86 12738.69 -19.25

19 19 -663.75 14877.17 -22.41 -661.64 12738.69 -19.25

20 -662.56 15301.52 -23.09 -660.29 12970.46 -19.64

20 20 -662.39 15301.52 -23.10 -660.19 12970.46 -19.65

21 -661.98 15420.71 -23.29 -659.73 13025.47 -19.74

21 21 -661.98 15420.71 -23.29 -659.73 13025.47 -19.74

22 -662.39 15301.52 -23.10 -660.19 12970.46 -19.65

22 22 -662.56 15301.52 -23.09 -660.29 12970.46 -19.64

23 -663.75 14877.17 -22.41 -661.64 12738.69 -19.25

23 23 -664.10 14877.17 -22.40 -661.86 12738.69 -19.25

24 -666.12 14211.96 -21.34 -664.09 12327.85 -18.56

24 24 -666.63 14211.96 -21.32 -664.42 12327.85 -18.55

25 -669.53 13368.73 -19.97 -667.54 11734.19 -17.58

25 25 -670.10 13368.73 -19.95 -668.00 11734.19 -17.57

26 -673.73 12144.50 -18.03 -672.04 11018.24 -16.40

26 26 -674.62 12144.50 -18.00 -672.57 11018.24 -16.38

27 -679.19 10798.18 -15.90 -677.50 10107.04 -14.92

27 27 -680.09 10798.18 -15.88 -678.17 10107.04 -14.90

28 -685.48 9189.81 -13.41 -684.03 9058.07 -13.24

28 28 -686.52 9189.81 -13.39 -684.75 9058.07 -13.23

29 -692.75 7310.94 -10.55 -691.50 7795.39 -11.27

29 29 -694.02 7310.94 -10.53 -692.34 7795.39 -11.26

30 -701.19 5283.62 -7.54 -699.96 6307.90 -9.01

30 30 -702.47 5283.62 -7.52 -700.92 6307.90 -9.00

31 -710.52 3029.58 -4.26 -709.43 4583.16 -6.46

31 31 -712.02 3029.58 -4.25 -710.53 4583.16 -6.45

32 -721.05 667.33 -0.93 -719.98 2672.97 -3.71

32 32 -722.57 667.33 -0.92 -721.11 2672.97 -3.71

33 -732.59 -1819.30 2.48 -731.44 495.86 -0.68

33 33 -734.14 -1819.30 2.48 -732.68 495.86 -0.68

34 -745.14 -4448.08 5.97 -743.89 -1964.48 2.64

34 34 -746.78 -4448.08 5.96 -745.24 -1964.48 2.64

35 -758.82 -7172.57 9.45 -757.33 -4725.53 6.24

35 35 -760.61 -7172.57 9.43 -758.85 -4725.53 6.23

36 -773.85 -9817.74 12.69 -771.87 -7740.51 10.03

36 36 -775.45 -9817.74 12.66 -773.39 -7740.51 10.01

37 -789.83 -12476.29 15.80 -787.30 -11095.76 14.09

37 37 -791.46 -12476.29 15.76 -788.98 -11095.76 14.06

38 -807.08 -15044.76 18.64 -803.83 -14746.76 18.35

38 38 -808.60 -15044.76 18.61 -805.50 -14746.76 18.31

39 -825.52 -17489.78 21.19 -821.22 -18781.99 22.87

39 39 -827.01 -17489.78 21.15 -822.99 -18781.99 22.82

40 -845.44 -19584.91 23.17 -839.59 -23224.33 27.66

40 40 -843.75 -19584.91 23.21 -841.44 -23224.33 27.60

41 -861.09 -24646.20 28.62 -858.93 -28097.54 32.71


Table 31: 0.6m Rise Arch – Line of Thrust Positioning Comparison – Fixed Supports

Circular Parabolic

member node Axial Force Moment Eccentricity Axial Force Moment Eccentricity

N Nmm mm N Nmm mm

1 1 -782.42 6037.78 -7.72 -766.73 -15075.42 19.66

2 -738.62 -2461.82 3.33 -736.37 -11080.85 15.05

2 2 -744.21 -2461.82 3.31 -735.11 -11080.85 15.07

3 -707.44 -7092.58 10.03 -706.30 -7707.51 10.91

3 3 -710.40 -7092.58 9.98 -705.14 -7707.51 10.93

4 -678.80 -9262.86 13.65 -677.89 -4897.58 7.22

4 4 -679.99 -9262.86 13.62 -676.84 -4897.58 7.24

5 -652.39 -9844.67 15.09 -651.15 -2594.70 3.98

5 5 -652.43 -9844.67 15.09 -650.22 -2594.70 3.99

6 -628.07 -9374.33 14.93 -626.08 -744.14 1.19

6 6 -627.34 -9374.33 14.94 -625.27 -744.14 1.19

7 -605.72 -8205.38 13.55 -602.69 707.10 -1.17

7 7 -604.48 -8205.38 13.57 -602.01 707.10 -1.17

8 -585.22 -6586.44 11.25 -580.99 1810.09 -3.12

8 8 -583.67 -6586.44 11.28 -580.43 1810.09 -3.12

9 -566.50 -4692.74 8.28 -560.97 2613.79 -4.66

9 9 -564.76 -4692.74 8.31 -560.53 2613.79 -4.66

10 -549.47 -2653.72 4.83 -542.63 3164.88 -5.83

10 10 -547.66 -2653.72 4.85 -542.32 3164.88 -5.84

11 -534.08 -576.62 1.08 -525.97 3507.54 -6.67

11 11 -532.24 -576.62 1.08 -525.77 3507.54 -6.67

12 -520.26 1475.28 -2.84 -510.98 3683.22 -7.21

12 12 -518.50 1475.28 -2.85 -510.89 3683.22 -7.21

13 -507.99 3429.88 -6.75 -497.66 3730.44 -7.50

13 13 -506.33 3429.88 -6.77 -497.66 3730.44 -7.50

14 -497.23 5243.97 -10.55 -485.98 3684.53 -7.58

14 14 -495.71 5243.97 -10.58 -486.05 3684.53 -7.58

15 -487.93 6876.45 -14.09 -475.93 3577.39 -7.52

15 15 -486.58 6876.45 -14.13 -476.05 3577.39 -7.51

16 -480.09 8302.60 -17.29 -467.49 3437.27 -7.35

16 16 -478.93 8302.60 -17.34 -467.64 3437.27 -7.35

17 -473.67 9497.96 -20.05 -460.64 3288.53 -7.14

17 17 -472.73 9497.96 -20.09 -460.79 3288.53 -7.14

18 -468.67 10444.65 -22.29 -455.34 3151.44 -6.92

18 18 -467.95 10444.65 -22.32 -455.48 3151.44 -6.92

19 -465.07 11130.12 -23.93 -451.59 3041.96 -6.74

19 19 -464.58 11130.12 -23.96 -451.70 3041.96 -6.73

20 -462.86 11544.69 -24.94 -449.36 2971.63 -6.61

20 20 -462.62 11544.69 -24.96 -449.42 2971.63 -6.61

21 -462.04 11683.38 -25.29 -448.64 2947.41 -6.57


21 21 -462.04 11683.38 -25.29 -448.64 2947.41 -6.57

22 -462.62 11544.69 -24.96 -449.42 2971.63 -6.61

22 22 -462.86 11544.69 -24.94 -449.36 2971.63 -6.61

23 -464.58 11130.12 -23.96 -451.70 3041.96 -6.73

23 23 -465.07 11130.12 -23.93 -451.59 3041.96 -6.74

24 -467.95 10444.65 -22.32 -455.48 3151.44 -6.92

24 24 -468.67 10444.65 -22.29 -455.34 3151.44 -6.92

25 -472.73 9497.96 -20.09 -460.79 3288.53 -7.14

25 25 -473.67 9497.96 -20.05 -460.64 3288.53 -7.14

26 -478.93 8302.60 -17.34 -467.64 3437.27 -7.35

26 26 -480.09 8302.60 -17.29 -467.49 3437.27 -7.35

27 -486.58 6876.45 -14.13 -476.05 3577.39 -7.51

27 27 -487.93 6876.45 -14.09 -475.93 3577.39 -7.52

28 -495.71 5243.97 -10.58 -486.05 3684.53 -7.58

28 28 -497.23 5243.97 -10.55 -485.98 3684.53 -7.58

29 -506.33 3429.88 -6.77 -497.66 3730.44 -7.50

29 29 -507.99 3429.88 -6.75 -497.66 3730.44 -7.50

30 -518.50 1475.28 -2.85 -510.89 3683.22 -7.21

30 30 -520.26 1475.28 -2.84 -510.98 3683.22 -7.21

31 -532.24 -576.62 1.08 -525.77 3507.54 -6.67

31 31 -534.08 -576.62 1.08 -525.97 3507.54 -6.67

32 -547.66 -2653.72 4.85 -542.32 3164.88 -5.84

32 32 -549.47 -2653.72 4.83 -542.63 3164.88 -5.83

33 -564.76 -4692.74 8.31 -560.53 2613.79 -4.66

33 33 -566.50 -4692.74 8.28 -560.97 2613.79 -4.66

34 -583.67 -6586.44 11.28 -580.43 1810.09 -3.12

34 34 -585.22 -6586.44 11.25 -580.99 1810.09 -3.12

35 -604.48 -8205.38 13.57 -602.01 707.10 -1.17

35 35 -605.72 -8205.38 13.55 -602.69 707.10 -1.17

36 -627.34 -9374.33 14.94 -625.27 -744.14 1.19

36 36 -628.07 -9374.33 14.93 -626.08 -744.14 1.19

37 -652.43 -9844.67 15.09 -650.22 -2594.70 3.99

37 37 -652.39 -9844.67 15.09 -651.15 -2594.70 3.98

38 -679.99 -9262.86 13.62 -676.84 -4897.58 7.24

38 38 -678.80 -9262.86 13.65 -677.89 -4897.58 7.22

39 -710.40 -7092.58 9.98 -705.14 -7707.51 10.93

39 39 -707.44 -7092.58 10.03 -706.30 -7707.51 10.91

40 -744.21 -2461.82 3.31 -735.11 -11080.85 15.07

40 40 -738.62 -2461.82 3.33 -736.37 -11080.85 15.05

41 -782.42 6037.78 -7.72 -766.73 -15075.42 19.66


Table 32: 1m Rise Arch – Line of Thrust Positioning Comparison – Fixed Supports

Circular Parabolic

member node Axial Force Moment Eccentricity Axial Force Moment Eccentricity

N Nmm mm N Nmm mm

1 1 -854.7983 46658.03 -54.583669 -827.2112 -15679.6 18.95480018

2 -692.7405 -16757.7 24.1903945 -776.6087 -9559.57 12.30937588

2 2 -722.9429 -16757.7 23.1797919 -775.4954 -9559.57 12.3270472

3 -658.7737 -25583.9 38.8357173 -727.4879 -4644.4 6.384163778

3 3 -663.5118 -25583.9 38.5583937 -726.4887 -4644.4 6.392944446

4 -616.3399 -26994.9 43.7986879 -681.0762 -820.011 1.203992446

4 4 -616.0427 -26994.9 43.8198178 -680.2088 -820.011 1.205527773

5 -578.0415 -25241.4 43.6671414 -637.3913 2026.226 -3.17893514

5 5 -575.4348 -25241.4 43.864952 -636.6752 2026.226 -3.18251064

6 -543.5475 -21892.3 40.276792 -596.4527 4005.229 -6.71508151

6 6 -539.7526 -21892.3 40.5599706 -595.9091 4005.229 -6.72120714

7 -512.4012 -17713.7 34.5700104 -558.2816 5225.98 -9.36083152

7 7 -508.0141 -17713.7 34.8685495 -557.9327 5225.98 -9.36668527

8 -484.2491 -13143.7 27.1423352 -522.9003 5795.235 -11.0828688

8 8 -479.6151 -13143.7 27.4045821 -522.7691 5795.235 -11.0856502

9 -458.8188 -8452.21 18.4216752 -490.3316 5817.191 -11.8637905

9 9 -454.1403 -8452.21 18.6114531 -490.4401 5817.191 -11.8611659

10 -435.8923 -3812.26 8.74587989 -460.5976 5393.089 -11.7088949

10 10 -431.356 -3812.26 8.83785481 -460.9648 5393.089 -11.6995677

11 -415.3345 646.473 -1.5565117 -433.7173 4620.746 -10.6538201

11 11 -410.9912 646.473 -1.5729607 -434.3562 4620.746 -10.6381493

12 -396.9782 4845.285 -12.205418 -409.7037 3594.014 -8.7722271

12 12 -392.9517 4845.285 -12.330484 -410.6172 3594.014 -8.75271153

13 -380.7604 8714.487 -22.887063 -388.5598 2402.134 -6.18214828

13 13 -377.0685 8714.487 -23.111152 -389.7352 2402.134 -6.16350358

14 -366.5692 12211.45 -33.312805 -370.2727 1128.998 -3.04909868

14 14 -363.2841 12211.45 -33.614045 -371.6755 1128.998 -3.03759059

15 -354.3625 15296.43 -43.166065 -354.808 -147.704 0.416293601

15 15 -351.5002 15296.43 -43.51757 -356.3752 -147.704 0.414462903

16 -344.0733 17943.24 -52.14947 -342.1028 -1357.43 3.9679152

16 16 -341.6523 17943.24 -52.519009 -343.7396 -1357.43 3.949021003

17 -335.6579 20133.61 -59.982522 -332.0621 -2437.85 7.341545151

17 17 -333.7044 20133.61 -60.333659 -333.6426 -2437.85 7.306767481

18 -329.0905 21851.36 -66.399253 -324.5601 -3335.88 10.27817128

18 18 -327.6139 21851.36 -66.698523 -325.937 -3335.88 10.23475181

19 -324.3442 23086.05 -71.17764 -319.4495 -4008.85 12.54923767

19 19 -323.355 23086.05 -71.395385 -320.4734 -4008.85 12.50914335

20 -321.4036 23829.99 -74.143502 -316.5809 -4425.41 13.97876088

20 20 -320.9077 23829.99 -74.258076 -317.1283 -4425.41 13.95463193


21 -320.259 24078.49 -75.184436 -315.8307 -4566.41 14.45841585

21 21 -320.259 24078.49 -75.184436 -315.8307 -4566.41 14.45841585

22 -320.9077 23829.99 -74.258076 -317.1283 -4425.41 13.95463193

22 22 -321.4036 23829.99 -74.143502 -316.5809 -4425.41 13.97876088

23 -323.355 23086.05 -71.395385 -320.4734 -4008.85 12.50914335

23 23 -324.3442 23086.05 -71.17764 -319.4495 -4008.85 12.54923767

24 -327.6139 21851.36 -66.698523 -325.937 -3335.88 10.23475181

24 24 -329.0905 21851.36 -66.399253 -324.5601 -3335.88 10.27817128

25 -333.7044 20133.61 -60.333659 -333.6426 -2437.85 7.306767481

25 25 -335.6579 20133.61 -59.982522 -332.0621 -2437.85 7.341545151

26 -341.6523 17943.24 -52.519009 -343.7396 -1357.43 3.949021003

26 26 -344.0733 17943.24 -52.14947 -342.1028 -1357.43 3.9679152

27 -351.5002 15296.43 -43.51757 -356.3752 -147.704 0.414462903

27 27 -354.3625 15296.43 -43.166065 -354.808 -147.704 0.416293601

28 -363.2841 12211.45 -33.614045 -371.6755 1128.998 -3.03759059

28 28 -366.5692 12211.45 -33.312805 -370.2727 1128.998 -3.04909868

29 -377.0685 8714.487 -23.111152 -389.7352 2402.134 -6.16350358

29 29 -380.7604 8714.487 -22.887063 -388.5598 2402.134 -6.18214828

30 -392.9517 4845.285 -12.330484 -410.6172 3594.014 -8.75271153

30 30 -396.9782 4845.285 -12.205418 -409.7037 3594.014 -8.7722271

31 -410.9912 646.473 -1.5729607 -434.3562 4620.746 -10.6381493

31 31 -415.3345 646.473 -1.5565117 -433.7173 4620.746 -10.6538201

32 -431.356 -3812.26 8.83785481 -460.9648 5393.089 -11.6995677

32 32 -435.8923 -3812.26 8.74587989 -460.5976 5393.089 -11.7088949

33 -454.1403 -8452.21 18.6114531 -490.4401 5817.191 -11.8611659

33 33 -458.8188 -8452.21 18.4216752 -490.3316 5817.191 -11.8637905

34 -479.6151 -13143.7 27.4045821 -522.7691 5795.235 -11.0856502

34 34 -484.2491 -13143.7 27.1423352 -522.9003 5795.235 -11.0828688

35 -508.0141 -17713.7 34.8685495 -557.9327 5225.98 -9.36668527

35 35 -512.4012 -17713.7 34.5700104 -558.2816 5225.98 -9.36083152

36 -539.7526 -21892.3 40.5599706 -595.9091 4005.229 -6.72120714

36 36 -543.5475 -21892.3 40.276792 -596.4527 4005.229 -6.71508151

37 -575.4348 -25241.4 43.864952 -636.6752 2026.226 -3.18251064

37 37 -578.0415 -25241.4 43.6671414 -637.3913 2026.226 -3.17893514

38 -616.0427 -26994.9 43.8198178 -680.2088 -820.011 1.205527773

38 38 -616.3399 -26994.9 43.7986879 -681.0762 -820.011 1.203992446

39 -663.5118 -25583.9 38.5583937 -726.4887 -4644.4 6.392944446

39 39 -658.7737 -25583.9 38.8357173 -727.4879 -4644.4 6.384163778

40 -722.9429 -16757.7 23.1797919 -775.4954 -9559.57 12.3270472

40 40 -692.7405 -16757.7 24.1903945 -776.6087 -9559.57 12.30937588

41 -854.7983 46658.03 -54.583669 -827.2112 -15679.6 18.95480018


Appendix E – Strain Energy Method Deflection Calculation Check

Using Strain Energy Method:

First finding horizontal supports, assuming movement at support is zero,

∆=𝛿𝐶

𝛿𝐻= 0

C = ∫𝑀2

2𝐸𝐼

𝜃2

𝜃1

𝛿𝑠

∴𝛿𝐶

𝛿𝐻= ∫

𝑀𝛿𝑀

𝛿𝐻

𝐸𝐼

𝜃2

𝜃1

𝛿𝑠 = 0

𝑀 = −2.5(1 − 1𝑐𝑜𝑠𝜃) + 𝐻(1𝑠𝑖𝑛𝜃)

𝛿𝑀

𝛿𝐻= 𝑠𝑖𝑛𝜃

2.5kN

2.5kN

H

5kN

H

R = 1m

M

2.5kN

H

1m

𝜃

2.5kN 2.5kN


𝑀𝛿𝑀

𝛿𝐻= −2.5𝑠𝑖𝑛𝜃 + 2.5𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃 + 𝐻𝑠𝑖𝑛2𝜃

∴𝛿𝐶

𝛿𝐻=

1

𝐸𝐼∫ (−2.5𝑠𝑖𝑛𝜃 + 2.5𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃 + 𝐻𝑠𝑖𝑛2𝜃)

𝜃2

𝜃1

𝛿𝑠

However, 𝛿𝑠 = 𝑅𝛿𝜃 where R=1m

𝛿𝐶

𝛿𝐻=

1

𝐸𝐼∫ (−2.5𝑠𝑖𝑛𝜃 + 2.5𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃 + 𝐻𝑠𝑖𝑛2𝜃)

𝜋

2

0

𝑅𝛿𝜃

𝛿𝐶

𝛿𝐻=

1

𝐸𝐼[2.5𝑐𝑜𝑠𝜃 + 2.5

𝑠𝑖𝑛2𝜃

2+ 𝐻(

𝜃

2−𝑠𝑖𝑛2𝜃

4)]

0

𝜋

2

Simplifying this gives

𝛿𝐶

𝛿𝐻=

1

𝐸𝐼[2.5

sin(𝜋

2)2

2+ 𝐻(

𝜃

2−

𝑠𝑖𝑛2𝜋

2

4)]-[2.5cos(0)]

𝛿𝐶

𝛿𝐻=

1

𝐸𝐼[𝜋

4𝐻 − 1.25], However

𝛿𝐶

𝛿𝐻= 0

∴ 𝐻 =1.25(4)

𝜋= 1.59𝑘𝑁

From this, the deflection can be found at the mid span location.

H

2.5kN 2.5kN

1.59kN 1.59kN

5kN

M


Dummy Load Case

From the above two diagrams, the bending moments can be calculated

𝑀 = −2.5(1 − 𝑐𝑜𝑠𝜃) + 1.59𝑠𝑖𝑛𝜃

𝑀∗ = −0.5(1 − 𝑐𝑜𝑠𝜃) + 0.318𝑠𝑖𝑛𝜃

∆= ∫𝑀𝑀∗

𝐸𝐼

𝜃2

𝜃1

𝛿𝑠

∆=1

𝐸𝐼∫ (−2.5 + 2.5𝑐𝑜𝑠𝜃 + 1.59𝑠𝑖𝑛𝜃)(−0.5 + 0.5𝑐𝑜𝑠𝜃 + 0.318𝑠𝑖𝑛𝜃)

𝜋

2

0

𝑅𝛿𝜃

Simplifies to

∆

=1

𝐸𝐼[1.25𝜃 − 2.5𝑠𝑖𝑛𝜃 + 1.59𝑐𝑜𝑠𝜃 + 1.59

𝑠𝑖𝑛2𝜃

2+ 1.25 (

𝜃

2+𝑠𝑖𝑛2𝜃

4) + 0.506 (

𝜃

2−𝑠𝑖𝑛2𝜃

4)]

0

𝜋

2

Which further simplifies to

∆=2

𝐸𝐼[1.25𝜋

2− 2.5 +

1.59

2+1.25𝜋

4+0.506𝜋

4] − 1.59

∆=2

𝐸𝐼[0.04765]

H

0.5kN

0.318kN

0.318kN

0.5kN

1kN

M

*


Calculating second moment of area and Young’s Modulus

𝐼 =𝑏𝑑3

12

𝐼 =0.154

12= 4.21875 × 10−5𝑚4

Young’s Modulus, using equation 2.1(6) from pg. 13 of the text book, Reinforced & Prestressed

Concrete – Analysis and Design with emphasis on application of AS3600-2009 (Loo, Y &

Chowdhury, S 2010).

𝐸𝑐 = 𝜌1.5[0.024√𝑓𝑐𝑚𝑖 + 0.12]𝑓𝑜𝑟𝑓𝑐𝑚𝑖 > 40𝑀𝑃𝑎

𝐸𝑐 = (2350)1.5[0.024√53.12 + 0.12] Where 𝜌𝑤𝑎𝑠𝑓𝑜𝑢𝑛𝑑𝑡𝑜𝑏𝑒2350𝑘𝑔

𝑚3

𝐸𝑐 = 33597.46𝑀𝑃𝑎(= 33.5975 × 106𝑘𝑁

𝑚2)

Following this, substituting back in to deflection equation,

∆=2

(33.5975 × 106)(4.21875 × 10−5)[0.04765]

∆= 6.7236 × 10−5𝑚 = 0.06724𝑚𝑚

The comparison model with pinned supports experienced a deflection of 0.071419mm which is

approximately 6% difference when compared to this calculated deflection.

150m

m

150m

m


Appendix F – Scale Model Arch Formwork Design




Appendix G – Arch Inner Form Calculations

𝑅2 = (𝑅 − 0.6)2 + 12

𝑅2 = 𝑅2 − 1.2𝑅 + 0.36 + 1

1.2𝑅 = 1.36

𝑅1 = 1.133𝑚(𝑖𝑛𝑛𝑒𝑟𝑟𝑎𝑑𝑖𝑢𝑠)

𝑂𝑢𝑡𝑒𝑟𝑅𝑎𝑑𝑖𝑢𝑠

𝑅2 = 1.133 + 0.15 = 1.283𝑚𝑤ℎ𝑒𝑟𝑒0.15𝑚𝑖𝑠𝑡ℎ𝑒𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑑𝑒𝑝𝑡ℎ.

𝐹𝑖𝑛𝑑𝑖𝑛𝑔𝑎𝑛𝑔𝑙𝑒𝛼 𝑅1 − 0.6 = 1.133 − 0.6 = 0.533𝑚

tan(∝) =0.533

1

∝= 28.060

𝛽 = 90−∝= 90 − 28.06 = 61.940

2𝛽 = 123.880

𝐶 = 2𝜋𝑟𝑤ℎ𝑒𝑟𝑒𝑟𝑖𝑠𝑡𝑎𝑘𝑒𝑛𝑎𝑠𝑡ℎ𝑒𝑎𝑣𝑒𝑟𝑎𝑔𝑒𝑟𝑎𝑑𝑖𝑢𝑠.

𝑟 =𝑅1 + 𝑅2

2=1.283 + 1.133

2= 1.208𝑚

𝐹𝑢𝑙𝑙𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = 𝐶 = 2𝜋(1.208) = 7.5901𝑚

𝐿𝑒𝑛𝑔𝑡ℎ𝑜𝑓𝐴𝑟𝑐ℎ𝑆𝑒𝑔𝑚𝑒𝑛𝑡 = 7.5901𝑚 ×123.88

360= 2.61184𝑚𝑎𝑡𝑐𝑒𝑛𝑡𝑟𝑒𝑜𝑓𝑐𝑟𝑜𝑠𝑠𝑠𝑒𝑐𝑡𝑖𝑜𝑛

𝐹𝑜𝑟150𝑚𝑚𝑣𝑜𝑢𝑠𝑠𝑜𝑖𝑟𝑠 2.61184

0.15= 17.4𝑣𝑜𝑢𝑠𝑠𝑜𝑖𝑟𝑠𝑜𝑟

2.61184

17= 0.1536𝑚𝑤𝑖𝑑𝑡ℎ𝑎𝑡𝑐𝑒𝑛𝑡𝑟𝑒𝑜𝑓𝑒𝑎𝑐ℎ𝑣𝑜𝑢𝑠𝑠𝑜𝑖𝑟.

123.880

17= 7.2870𝑝𝑒𝑟𝑣𝑜𝑢𝑠𝑠𝑜𝑖𝑟


𝜎 = 𝜃𝑎𝑛𝑑𝜎 + 𝜃 + 7.287 = 1800

∴ 𝜎 = 𝜃 = 86.35650

𝑦 = 180 − 𝜎 = 93.64350 𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒𝑒𝑎𝑐ℎ𝑏𝑙𝑜𝑐𝑘𝑖𝑠𝑎𝑛𝑔𝑙𝑒𝑑𝑎𝑡93.64350

𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑖𝑛𝑔𝑣𝑜𝑢𝑠𝑠𝑜𝑖𝑟𝑡𝑜𝑝𝑎𝑛𝑑𝑏𝑜𝑡𝑡𝑜𝑚𝑙𝑒𝑛𝑔𝑡ℎ𝑠𝑢𝑠𝑖𝑛𝑔𝑜𝑢𝑡𝑒𝑟𝑎𝑛𝑑𝑖𝑛𝑛𝑒𝑟𝑟𝑎𝑑𝑖𝑖𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑙𝑒𝑦.

𝑎2 = 𝑏2 + 𝑐2 − 2𝑏𝑐 × cos(𝐴) 𝐵𝑜𝑡𝑡𝑜𝑚𝑓𝑎𝑐𝑒: 𝑥2 = 1.1332 + 1.1332 − 2(1.133)(1.133) × cos(7.287)

𝑥 = √0.020778 = 0.144𝑚

𝑇𝑜𝑝𝑓𝑎𝑐𝑒: 𝑧2 = 1.2832 + 1.2832 − 2(1.283)(1.283) × cos(7.287)

𝑧 = √0.02659 = 0.163𝑚

𝑁𝑜𝑤𝑙𝑒𝑛𝑔𝑡ℎ𝑜𝑓𝑡ℎ𝑒𝑡𝑜𝑝𝑠𝑢𝑟𝑓𝑎𝑐𝑒𝑜𝑓𝑡ℎ𝑒𝑎𝑟𝑐ℎ = 163𝑚𝑚 × 17𝑣𝑜𝑢𝑠𝑠𝑜𝑖𝑟𝑠 = 2771𝑚𝑚

𝐴𝑠𝑡ℎ𝑒𝑠𝑡𝑒𝑒𝑙𝑓𝑜𝑟𝑚𝑤𝑜𝑟𝑘𝑤𝑎𝑠𝑓𝑎𝑏𝑟𝑖𝑐𝑎𝑡𝑒𝑑𝑡𝑜𝑓𝑜𝑢𝑟𝑚𝑒𝑡𝑟𝑒𝑠: 4000 − 2771𝑚𝑚 = 1229𝑚𝑚 1229

17= 72.29𝑚𝑚

Allinnerplywoodmouldblockswereconstructedwitha72mmtopsurfaceascalculated above.Fromthis, andtheangleofthevoussoirs, thebasewidthfortheplywoodmoulds wascalculatedto91mm. Thefinisheddesigncanbeseenonthefollowingpage.



Appendix H – Loading Rig Calculations and Capacity Checks

Initial Design Calculations:

Note: The steel strips shown are on both sides of the rig (4 in total)

Mid-span Loading:

Vertical height of steel strips

750 + 477.5 + 45 +75

2= 1235𝑚𝑚

Horizontal distance for each steel strip, drilling holes 50mm outside of each welded plate of the

PFC base, where welded plates are 2600mm apart: 2600 + 50 + 50

2= 1350𝑚𝑚

Length of steel strip (bolt hole to bolt hole)

𝐿 = √(12352 + 13502 = 1830𝑚𝑚

(Image to the right)

Designing for the use of a 10T (tonne) hydraulic jack.

10𝑇 = 98.1𝑘𝑁 This means that each of the four steel strips will hold 2.5 tonne (2.5T) in the vertical direction.


Each of the bolts will be required to hold 5T on the top plate = 49.05kN

Using table 8.5(a), p.211 of the ‘Steel Designers Handbook’.

M24 bolt 4.6/s has a capacity of 64.3kN in single shear, where threads are included in the shear

plane. Therefore shear capacity satisfied if M24 type 4.6/s are used.

Check for tear-out of ply material:

𝜃 = 90 − 42.45 = 47.550

tan(47.55) =𝑥

24.525

𝑥 = 26.811𝑘𝑁

𝐹 = 𝑓𝑜𝑟𝑐𝑒𝑖𝑛𝑒𝑎𝑐ℎ𝑠𝑡𝑒𝑒𝑙𝑠𝑡𝑟𝑖𝑝 = √(24.5252 + 26.8112

= 36.336𝑘𝑁

Checking for bolt tear-out of ply material (steel strips):

𝑉𝑝 = 𝑎𝑒𝑡𝑝𝑓𝑢𝑝

𝑉∗ = ∅𝑉𝑝 or 𝑉∗

∅= 𝑉𝑝

Where ∅ = 0.9

𝑉∗ = 36.336

𝑡𝑝 = 6𝑚𝑚

𝑓𝑢𝑝 = 410𝑀𝑃𝑎 = 0.41𝑘𝑁/𝑚𝑚2

Note: During the design of these components, the grade of the

steel being used was not known, as spare steel members were

being used. For this reason the 410MPa has been assumed as

the lowest possible tensile strength for hot-rolled plates, from Table 2.3, page 27 of the ‘Steel

Designers’ Handbook’(Gorenc, B, Tinyou, R & Syam, A 2012).

∴36.336

0.9= 𝑎𝑒(6)(0.41)

𝑎𝑒 = 16.412𝑚𝑚

Therefore drill bolt holes at a distance 𝑎𝑒 =24

2+ 30 = 42𝑚𝑚 (where 24mm is the diameter of

the bolt being used) from the edge of the ply.

Setting 𝑎𝑒 = 42𝑚𝑚, rather than 16mm for excess capacity in the steel components as third-span

loading case will produce larger loads. The image below shows the steel strips used for the mid-

span loading case.


Checking capacity of steel strip in tension:

𝑁∗ ≤ ∅𝐴𝑓𝑦

Where 𝐴 = 0.075𝑚 × 0.006𝑚 = 4.5 × 10−4𝑚2

𝑁∗ ≤ 0.9 × (4.5 × 10−4) × (250 × 106) 𝑁∗ ≤ 101250𝑁 = 101.25𝑘𝑁

As 𝑁∗ = 36.336𝑘𝑁 < 101.25𝑘𝑁 Therefore tensile capacity satisfied.

Note: As the grade of the steel being used was not known, 250MPa has been assumed as the

lowest possible yield stress for hot-rolled plates, from Table 2.3, page 27 of the ‘Steel Designers’

Handbook’ (Gorenc, B, Tinyou, R & Syam, A 2012).

Checking at location where bolt holes reduce the area effected:

𝐴 = (75𝑚𝑚 − 26𝑚𝑚)(6) = 294𝑚𝑚2 Using 26mm as the holes should be greater than 24mm bolts.

𝑁∗ ≤ 0.9 × (2.94 × 10−4) × (250 × 106) 𝑁∗ ≤ 66.15𝑘𝑁 Which also satisfies tensile capacity, as 36.336𝑘𝑁 < 66.15𝑘𝑁.

Checking bolt tear-out of ply for top plate:

For top plate to secure hydraulic jack loading cylinder, using a 180PFC75 (identical to loading

rig base PFC).

Again assuming 𝑓𝑢𝑝 = 410𝑀𝑃𝑎

𝑉∗

∅= 𝑎𝑡𝑝𝑓𝑢𝑝

𝑎 =𝑉∗

∅ × 𝑡𝑝𝑓𝑢𝑝=

49.05

0.9(11)(0.41)= 12𝑚𝑚

This is satisfied as we are placing the bolt hole 45mm up from the base of the PFC,

45𝑚𝑚 > 12𝑚𝑚.


Third-span loading:

Outer radius of the arch 𝑅2 = 1.283𝑚, calculating height at third-span:

If we take 𝑥 = 0 at the centre of the arch,

𝑥2 + (𝑦 + (𝑅 − ℎ))2= 𝑅2

Re-arranging gives,

𝑦 = ℎ + √𝑅2 − 𝑥2 − 𝑅

Now, the third-span point is at 0.6667m or 2

3𝑚 from left or right support. This is at

𝑥 = −0.333𝑚.

∴ 𝑦 = 0.75 + √1.2832 − (−0.333)2 − 1.283

𝑦 = 0.706𝑚 at third-span

Setting the height to 0.7m and the hydraulic jack cylinder extension can be adjusted to account

for the difference.

𝐻𝑒𝑖𝑔ℎ𝑡𝑜𝑓𝑎𝑟𝑐ℎ𝑎𝑡𝑡ℎ𝑖𝑟𝑑𝑝𝑜𝑖𝑛𝑡 + ℎ𝑒𝑖𝑔ℎ𝑡𝑜𝑓𝑙𝑜𝑎𝑑𝑖𝑛𝑔𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟ℎ𝑎𝑙𝑓𝑒𝑥𝑡𝑒𝑛𝑑𝑒𝑑+ 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒𝑓𝑟𝑜𝑚𝑡𝑜𝑝𝑃𝐹𝐶𝑏𝑎𝑠𝑒𝑡𝑜𝑏𝑜𝑙𝑡ℎ𝑜𝑙𝑒

= 700𝑚𝑚 + 477.5𝑚𝑚 + 45𝑚𝑚 = 1222.5𝑚𝑚 from base to top plate bolt hole.

Therefore from base PFC bolt hole to top plate bolt hole:

1222.5𝑚𝑚 −75

2𝑚𝑚 = 1185𝑚𝑚


Horizontal distance from third-span point to nearest support bolt hole in

base PFC

= 0.667 + 0.29 + 0.06 = 1016.7𝑚 (shown in the images above and to

the right)

𝑐 = √11852 + 1016.72 = 1561.38𝑚𝑚 centre to centre of bolt holes.

(Seen in the image to the right)

Total length of steel plate required = 1561𝑚𝑚 + (24

2× 2) + (30 × 2) = 1645.4𝑚𝑚, which

adds length 𝑎𝑒 on both sides.

Horizontal distance from third-span point to farthest support bolt hole in base PFC

= 1.333 + 0.29 + 0.06 = 1.683𝑚

Length of steel strip, centre to centre of bolt holes= √11852 + 16832 = 2058.32𝑚𝑚

The two lengths of steel strips fabricated for the third-span loading can be seen below.


Calculating forces in steel strips:

+↑∑𝐹𝑉 = 0

5 − 𝐴(sin 49.37) − 𝐵(sin 35.15) = 0___________eq.1

+→∑𝐹𝐻 = 0

−𝐴(cos 49.37) + 𝐵(cos 35.15) = 0

𝐴 =𝐵(cos 35.15)

(cos 49.37)

𝐴 = 1.256 × 𝐵___________eq.2

Substituting eq.1 into eq.2 gives,

5 − 𝐵(1.256 sin 49.37) − 𝐵(sin 35.15) = 0

1.529 × 𝐵 = 5

𝑩 = 3.27𝑇 = 3.27 × 9.81 = 𝟑𝟐. 𝟎𝟖𝒌𝑵

∴ 𝑨 = 1.256 × 𝐵 = 1.256 × 32.08𝑘𝑁 = 𝟒𝟎. 𝟐𝟗𝒌𝑵 This is the largest load in all steel strips for both load positions.

Checking bolt tear-out of ply (strips) for third-span loading:

𝑉𝑝 = 𝑎𝑒𝑡𝑝𝑓𝑢𝑝

𝑉∗

∅= 𝑎𝑒𝑡𝑝𝑓𝑢𝑝

𝑎𝑒 =𝑉∗

∅ × 𝑡𝑝𝑓𝑢𝑝=

40.29

0.9(6)(0.41)= 18.20𝑚𝑚

42𝑚𝑚 > 18.20𝑚𝑚 ∴ 𝑚𝑖𝑛𝑖𝑚𝑢𝑚𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒𝑓𝑟𝑜𝑚𝑝𝑙𝑦𝑒𝑑𝑔𝑒𝑡𝑜𝑏𝑜𝑙𝑡𝑐𝑒𝑛𝑡𝑟𝑒𝑠𝑎𝑡𝑖𝑠𝑓𝑖𝑒𝑑


Checking tensile strength at bolt holes:

Capacity of steel strip in tension from above = 66.15𝑘𝑁.

66.15𝑘𝑁 > 40.29𝑘𝑁 ∴ 𝑆𝑎𝑡𝑖𝑠𝑓𝑖𝑒𝑑

The image below shows the PFC cross-section which was used in for the loading rig.

The image below shows the base PFC of the loading rig.


Capacity Check and Failure Mode Analysis:

As the grade of steel being used for the construction of the loading rig was unknown at the time

of design, the grade was assumed to be lower than expected. Once the actual grade of the steel

was known, the capacity of the rig was calculated. This can be seen below.

Grade of Steel Components:

Steel Strips → grade 350, thickness < 12mm → 𝑓𝑦 = 360𝑀𝑃𝑎𝑎𝑛𝑑𝑓𝑢 = 450𝑀𝑃𝑎

PFC → grade 350, thickness = 11mm → 𝑓𝑦 = 360𝑀𝑃𝑎𝑎𝑛𝑑𝑓𝑢 = 480𝑀𝑃𝑎

Continuing from above loading rig design calculations.

Mid-span Load:

Calculating the loads which should not be exceeded to ensure failure does not occur in the

loading rig.

Failure Load of Steel Strips:

Bolt tear-out of ply (steel strip): 𝑉∗

∅= 𝑎𝑡𝑝𝑓𝑢𝑝

Calculating the new design load with known steel grades,

𝑉∗ = ∅𝑎𝑡𝑝𝑓𝑢𝑝

𝑉∗ = 0.9 × 30 × 6 × 0.45

𝑉∗ = 72.9𝑘𝑁

Steel Strip in Tension:

𝑁∗ ≤ ∅𝐴𝑓𝑦

𝑁∗ ≤ 0.9 × (2.94 × 10−4) × (360 × 106) 𝑁∗ ≤ 95.256𝑘𝑁 > 𝑉∗ above

Therefore 𝑉∗ = 72.9𝑘𝑁 is the lowest load which will cause failure in the steel strips.

Bolt tear-out of PFC:

𝑉∗ = ∅𝑎𝑒𝑡𝑝𝑓𝑢𝑝

𝑉∗ = 0.9 × 32 × 11 × 0.48

𝑉∗ = 152.06𝑘𝑁

Now that the failure loads in individual members have been calculated, the applied load which

will cause these component forces can be found.


Load to cause 72.9kN in steel strip for mid-span loading (as this is the lowest load at which

failure will occur on the steel strip) (images below show the forces in the steel).

Vertical load on each side of the PFC:

2 × 49.2 = 98.4𝑘𝑁

98400𝑁 → 10030.58𝑘𝑔 = 10.03𝑇 Therefore upward force caused by the jack which would cause

failure in the steel strips for mid-span loading = 20.06T

Load to cause 152.06kN on the PFC flanges for mid-span

loading

152.06𝑘𝑁𝑒𝑖𝑡ℎ𝑒𝑟𝑠𝑖𝑑𝑒𝑜𝑓𝑡ℎ𝑒𝑃𝐹𝐶 → 304120𝑁𝑡𝑜𝑡𝑎𝑙 = 31001𝑘𝑔 = 31𝑇 Therefore 31T would need to be applied by the hydraulic jack to cause failure through bolt tear-

out of PFC flange.

Bolt failure in single shear

For M24 type 4.6/s bolts, capacity from Steel Designers Handbook (Table 8.5(a)) = 64.3kN

For M24 type 8.8/s bolts, capacity from Steel Designers Handbook (Table 8.5(a)) = 133kN

Therefore using type 4.6/s bolts in design to be safe, for the load on each bolt to reach 64.3kN


Vertical Load applied by the jack to cause failure of 4.6/s bolts in shear = 2 × 64.3𝑘𝑁

= 128.6𝑘𝑁 = 13.11𝑇

Using type 8.8/s bolts in design to check maximum load,

Vertical Load applied by the jack to cause failure of 8.8/s bolts in shear = 2 × 133𝑘𝑁 =266𝑘𝑁 = 27.1𝑇


Third-span Load:

Using the same load capacities which were calculated for the mid-span

loading case.

Bolt tear-out of ply (steel strip):

It is known that the load in member A (in the image to the right) is

greater than the load in member B. Therefore member A will fail first.

Applying the failure load to member A and calculating the vertical

load which must be applied by the jack to cause this load.

𝑉∗ = 72.9𝑘𝑁 in member A.

Calculating horizontal component of the force in member A

a = 72.9 cos(49.37) = 47.47𝑘𝑁

For equilibrium, the horizontal components of forces in members A and B must

be equal, 𝐹𝐻𝐴 = 𝐹𝐻𝐵.

Therefore, using the horizontal force, a, calculated for member A,

cos(35.15) =47.47

𝐵

𝐵 =47.47

cos(35.15)= 58.06𝑘𝑁

Now that A and B are known, calculating vertical components of forces,

𝐹𝑉𝐴 = 72.9 sin(49.37) = 55.33𝑘𝑁

𝐹𝑉𝐵 = 53.59 sin(27.64) = 30.85𝑘𝑁

Total vertical force on each side = 55.33𝑘𝑁 + 30.85𝑘𝑁 = 86.183𝑘𝑁 = 8.785𝑇

Therefore the load applied by the jack which will cause failure in member A (bolt tear-out) is

2 × 8.785𝑇 = 17.57𝑇


Bolt tear-out of base PFC:

Now that we know the ultimate tensile strength of the material

𝑎𝑒 has been assumed as the vertical distance from the centre of the bolt hole to the edge of the

PFC flange. This is conservative, as the actual distance would be larger due to the angle of the

steel strips.

𝑉 = 0.9 × 𝑎𝑒 × 𝑡𝑝 × 𝑡𝑢𝑝

𝑉 = 0.9 × 35 × 11 × 0.48 = 166.32𝑘𝑁

This is greater than the 72.9kN at which the steel strips will fail at by bolt tear-out, therefore bolt

tear-out of PFC base will not occur.

Documents

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