Dr Rafiq LUSAS Courseworklawrencetaylor.info/wp-content/uploads/2016/07/... · The modelling of crack widths has also been found to be very useful for serviceability limits in industry

Abstract The objective of the work was to analyse a 2D concrete beam using non-linear analysis. This report has produced four parametric studies altering the positioning of the point load, the depth of the beam, the length of the beam, the amount of rebar in the beam and the thickness of the beam. The work has then been verified using approximate analysis techniques and compared against a 3D model. It has been found that non-linear analysis gives a closer representation to what occurs in the real world as the beam is loaded. Verification techniques using stress and deflection have worked well while the beam is reacting linearly to an increase in loading. However, the techniques begin to diverge from the true result as the beams behave non-linearly. It has also been found that the non-linear models react very differently to linear models under extreme loading and are therefore vital when analysing large loading conditions. The modelling of crack widths has also been found to be very useful for serviceability limits in industry.

Group Members:

Oliver Harris – Student ID: 10346136

Lawrence Taylor – Student ID: 10395608

Yu Tam – Student ID: 10511595

Daniel Wilkinson – Student ID: 10342869

DR RAFIQ LUSAS

COURSEWORK Module Code: MATH511

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Table of Contents 1.0 Introduction ...................................................................................................................................... 3

2.0 Literature Review .............................................................................................................................. 4

2.1 Failure Modes of Concrete Beams ................................................................................................ 4

2.2 Crack Control of Concrete Structures ........................................................................................... 4

2.3 Linear and Nonlinear Analysis ....................................................................................................... 5

2.4 Summary of Literature Search ...................................................................................................... 5

3.0 Methodology ..................................................................................................................................... 6

3.1 Model Assumptions: ..................................................................................................................... 7

4.0 Validation .......................................................................................................................................... 7

4.1 Equilibrium .................................................................................................................................... 7

4.2 Symmetry ...................................................................................................................................... 7

4.3 Supports ........................................................................................................................................ 7

5.0 Verification ........................................................................................................................................ 8

5.1 Approximate method .................................................................................................................... 8

5.2 Visual inspection of deflected shape ............................................................................................ 9

6.0 Observations ................................................................................................................................... 12

7.0 Parametric Studies .......................................................................................................................... 13

7.1 The Position of the Point Load – Lawrence Taylor (LT) ............................................................... 13

7.1.1 Beam Verification ................................................................................................................. 16

7.2 Beam Depth and Width – Daniel Wilkinson (DW) ...................................................................... 18

Changing the depth of beam ............................................................................................................ 19

Changing the length of the beam ..................................................................................................... 22


7.3 Size of Steel Reinforcement Bars and Compression Reinforcement – Oliver Harris (OH) .......... 24

Compression reinforcement ............................................................................................................. 26


7.4 Changing Beam Thickness– Yu Tam (YT) ..................................................................................... 29


8.0 Discussion ........................................................................................................................................ 33

8.1 Elastic to Plastic ........................................................................................................................... 33

8.2 Cracking ....................................................................................................................................... 33

8.3 Deflection .................................................................................................................................... 34

8.4 Model Discussion & Limitations .................................................................................................. 35

8.5 Assumptions ................................................................................................................................ 35

8.5 3D Model ..................................................................................................................................... 36

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8.6 Real World Applications .............................................................................................................. 37

8.7 Verification of the Models .......................................................................................................... 38

9.0 Conclusion ....................................................................................................................................... 39

References ............................................................................................................................................ 40

Appendices ............................................................................................................................................ 42

Appendix A - Series of Contour Plots of Stress and Crack Paths as Load Increases ......................... 42

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1.0 Introduction

Reinforced concrete beams take advantage of composite material properties utilising the high

compressive strength of concrete and the high tensile strength and ductile properties of steel

reinforcement. As the concrete around them deform, the tensile stresses are taken in by the steel

beams whilst the compressive stresses are resisted by the concrete; using each material property to

an advantage. This behaviour of the reinforced concrete occurs in both elastic and plastic modes

which makes the calculations of stress patterns and deflections particularly difficult to predict

accurately. Therefore, codes of practice use either elastic or plastic approaches for calculations of

the reinforced concrete beams and most calculations assume that the behaviour is elastic and that

cracks do not have an effect on the overall strength of the beam.

In this report, Finite Element Analysis (FEA) using LUSAS (software) has been undertaken to produce a nonlinear analysis of a simply supported reinforced concrete beam in a number of configurations through parametric studies. The beam model used is presented below in Figure 1 - The concrete beam model. To undertake a parametric study, five separate variables have been changed and the effects on the concrete beam have been studied and reported. The following characteristics of the beam have been changed in the following ways: (1) the width and the height of the reinforced concrete beam (2) the size of rebar in the beam (3) the thickness of the beam (4) the positioning of the point load along the beam.

Presented below are the aims and objectives of the parametric study:

Aim:

To investigate non-linear analysis of a concrete beam through a series of parametric studies in a Finite Element Analysis package and how the distribution of stresses is affected in order to discover the benefit of using non-linear analysis in building analysis.

Objectives:

Produce a thorough literature search encompassing all current knowledge of non-linear analysis of concrete beams.

Complete four parametric studies undertaken by individual members of the group of changing the beam geometry, reinforcement sizes, position of point load and load magnitude.

Use approximate analysis techniques to verify the FEA computer outputs.

Discuss observations of the FEA and the effectiveness of the use of models.

Discuss real world applications of techniques incorporating non-linear analysis of concrete beams

Figure 1 - The concrete beam model

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2.0 Literature Review 2.1 Failure Modes of Concrete Beams Common forms of failure modes in concrete beams include compression, flexural and shear failure.

It is important to understand the concept of these failure modes so that the strength and durability

of a concrete beam may be enhanced by improved design. The failure patterns in the beam can be

used to assess how the beam failed e.g. flexural, shear etc. and how this type of failure may be

avoided. To assess the type of failure, the positioning and angle of the cracks in the beams can be

examined; Figure 2 below highlights the difference between shear and flexural cracking.

Shown in Figure 2, diagonal cracking indicates shear failure of the beam member and vertical cracks

along the base of the beam indicate flexural failure as the tensile forces exceed those of the

concrete (as the steel yields). This is noted by Richardson (no date) who also states that cracks due

to shear forces form where tension stresses due to shear are greatest e.g. near the supports of a

simply supported beam. According to Assakkaf (no date), shear failure of a beam is more difficult to

predict than flexural failure. If a beam does not have properly designed shear reinforcement, then it

may collapse suddenly and without warning. The analysis of the speed of failure is also noted by Al-

Nasra (2013) who states that “Since shear failure is frequently sudden with little or no advanced

warning, the design for shear must ensure that the shear strength for every member in the structure

exceeds the flexural strength”. The model to be investigated in this report does not include any

shear reinforcement and therefore this type of failure must be observed carefully if incorporated

into a real world design scenario. Not shown in Figure 2 are negative flexural cracking over the

supports on the top of the beam and also as vertical cracks (Pirro, 2012).

2.2 Crack Control of Concrete Structures A major design implication for reinforced concrete beams compared with those of steel beams is the

potential for cracking to occur once the tensile stress in the beam exceeds that of the concrete itself.

It is well known that the tensile strength of concrete is low (approximately 10% that of its

compressive strength) and should therefore be taken as zero during design. According to ‘Reinforced

Concrete’ (2016), cracking is hard to prevent and in fact there is micro cracking present in most

concrete beams. This cracking can lead to moisture penetration which will undermine the

reinforcement of the beam causing rusting and potentially spalling and damage to finishes. It should

be noted that flexural cracks may be related to longitudinal splitting cracks. This relationship is based

on splitting cracks allowing moisture to reach the steel pieces in the concrete and corrode them,

reducing their ability to resist flexure cracks. Reduction in resistance may cause additional flexural

cracks (Giuriani, 1998). Indeed, Eurocodes allow for cracking in concrete although reinforcement

spacing is limited which enables the cracking to be minimised. According to Brooker (2006), when

there are no specific requirements, crack widths in a beam should be limited to 0.3mm and under

Figure 2 - Highlighting the difference in crack patterns of shear and flexural failure (Paul, 2015).

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certain criteria 0.4mm. This guidance is also noted by CIRIA (2014) which states that cracking is not a

defect, but the basis of reinforced concrete design, as reinforcement is used to control the crack

widths. It also notes that problems may occur when the cracks become too large (e.g. allowing water

ingress) making the structure unsuitable for use. CIRIA (2014) also provides design guidance to how

cracking should be limited: 0.3mm for durability, 0.05 to 0.2mm for water retaining structures and

0.3mm or greater for appearances. It’s worth noting that these measurements are for a guide

pertaining to ‘Early-age thermal crack control in concrete’, related to the entire lifespan of the

concrete used.

2.3 Linear and Nonlinear Analysis When designing structures, it is common to analyse them using linear elastic analysis methods i.e. a

concrete beam tested using Finite Element Analysis will produce reasonable results of how the beam

will behave under normal loading conditions. The material will not permanently deform and

therefore the assumptions used by the computer package will be correct. However, should the

concrete beam (or other structure) be loaded beyond its yield strength, linear FEA analysis will no

longer provide accurate results – the deflection is no longer linearly proportional to the applied

loading (Waters, 2012). Non-linear elastic analysis, as opposed to linear elastic analysis, takes into

account a number of factors that produce a safer and more cost effective estimation of structures

behaviour. Some of these features include nonlinear stress-strain behaviour, tensile cracking and

compression crushing failures and temperature-dependent creep strains (Bathe et al., 1989). Non-

linear analysis takes into account the propagation of cracks through the concrete, assuming that it is

non-elastic. This is unlike linear analysis which works on the assumption that the concrete is

completely elastic and therefore cracks do not form.

Mansur and Tan (1999) state that because the beam behaves in a nonlinear way at higher loads,

there will be a redistribution of forces and moments compared to results obtained from a purely

linear elastic analysis. Mansur and Tan (1999) go on to note that using only linear elastic analysis has

been found to give reasonable results for design purposes and that the effects of force redistribution

have been recognised by codes of practice, and are utilised in those design methods and checks by

performing redistribution of forces and moments.

To model a material using nonlinear analysis, LUSAS adopts the Newton-Raphson Method iterative

procedure in which a linear response is first estimated and then corrections occur to eliminate out of

balance forces. These corrections are referenced against the convergence criterion selected in the

initial setup of the problem (LUSAS, 2015). This is not the only recognised method for nonlinear

analysis. The modified Newton-Raphson method may also be adopted however the use of this

method may present more problems as the procedure may be more likely to diverge than converge

resulting in a failure of the method. Because of the usefulness of nonlinear analysis in design, this

report will utilise it to analyse the beams and compare these results against elastic finite element

analysis using LUSAS.

2.4 Summary of Literature Search To conclude, the report will pay particular attention to crack patterns that form in the reinforced

concrete beam as the load is incrementally increased. Attention will be made to these cracks to

understand the mode of failure. Crack control of the beams as highlighted in the literature search

above is also important to the serviceability of the structure. Note will be taken as to how large the

cracks are becoming in the beams.

The aims and objectives of this report have been detailed in Section 1.

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3.0 Methodology A nonlinear concrete beam analysis was carried out using LUSAS software to model a reinforced concrete beam with a point load in 2D. The beam to be modelled was, 1650mm long, width of 150mm, 300mm deep and with a bar area of 400mm². The methodology follows the procedure suggested in the LUSAS applications manual example (2016) with beam dimensions, loading, supports and mesh summarised in Figure 3.

Figure 3 - Beam dimensions and attributes, source: LUSAS (2016)

Once the geometry of the beam was defined, a mesh was applied to the model. For the reinforcement a bar element line mesh was used and for the concrete of the beam plane stress elements was applied as a surface. The mesh density was uniform up to the point load and then graded at 2 to 1 ratio towards the edge of the beam. Due to the model being in 2D, additional geometric properties were required to represent the beam in 3D. For the bars an area of 400mm2 was defined and for the beam a thickness of 150mm was applied. For the material attributes the reinforcement bars were set as steel with nonlinear properties applied. For the beam, concrete was used with the plastic model type of smoothed multi crack (model 102). The beam support at the bottom left hand side (node 1) was simply supported in the y direction and at the centre line of the beam, due to symmetry, was horizontally restrained in the x direction. A point load of -1N was applied 450mm from the centre of the beam. Figure 4 shows the supports and loading applied to the model of the beam:

Figure 4 - Beam supports and loading

Nonlinear controls were defined within the software to set the iteration process and properties of the load cases. Therefore, the initial load case had a point load of 5kN with subsequent iterations with increments of 2kN. The nonlinear analysis was to be terminated when the beams deflection at the centre of the beam became greater than 3mm. All parametric studies were based on this initial model of the concrete beam. For each parametric study, certain aspects of the beam will be changed and analysed. From these analyses, the crack propagation, the stress distribution and the effect that carrying out non-linear analysis compared with linear analysis has on the accuracy of the stress results.

7

3.1 Model Assumptions: The superposition of nodal degrees of freedom assumes that the concrete and

reinforcement are perfectly bonded.

Self-weight of beam negligible compared to point load.

Effects of shear reinforcement (shear links) can be ignored.

The deflection of 3mm is measured from the top right hand node of each model.

4.0 Validation

4.1 Equilibrium Using the LUSAS print wizard tool to find the reaction component in the global axis, it was found that

the only reaction in the y-direction (Fy) was at node 1. This was expected due to the concrete beam

being modelled as a half model in 2D with only node 1 being supported in the y-direction. The print

wizard tool was utilised for each load case checking that downward forces were equal to the

reactions at node 1, this is shown in table 1. This shows that the model is in equilibrium as the sum

of the applied load is equal to the sum of reactions.

Table 1 - LUSAS print wizard reaction results in Fy at node 1 for each applied load.

Point Load Applied (kN)

Reaction at node 1 - Fy

(kN)

5 5

7 7

9 9

11 11

13 13

15 15

4.2 Symmetry The concrete beam has symmetrical loading and support conditions along the centre line of the

beam, therefore a half model was used to represent the beam. To verify that the half model was

representing the full beam model, a stress contour plot of beam was mirrored along the centre line

as seen in figure 5. This shows that the maximum tension and compression occurs at the centre of

the beam which is to be expected as the supports are at the edge of the beam with the loading

occurring towards the centre. Therefore it is appropriate to use a half model and verifies that the

central support conditions applied to the model are correct.

Figure 5 - Half model with plane stress contours mirrored

4.3 Supports The support conditions used for the concrete beam model are simply supported at the edges in the y

direction. To model the support conditions, node 1 is restrained in translation in the y direction and

is free to translate and rotate in all other directions. The model is also restrained in translation in the

x-axis along the length of the right hand edge of the model which is a centreline. This represents the

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centre of the beam as well as the simply supported edge as seen in figure 6. This shows that the

models restraints correspond to the structural support conditions and so correctly models them.

Figure 6 - Support conditions applied to model of concrete beam

5.0 Verification

5.1 Approximate method The model can be verified using the bending stress equation for approximate analysis. Figure 7

shows the line diagram of the beam and below are the calculations for the moment at the centre of

the beam that is then converted into a stress which is then compared to the LUSAS result.

Va = Vb = 5kN

𝐼 =𝑏𝑑3

12=

150 × 3003

12= 337,500,000𝑚𝑚4

Maximum moment @ centre of beam:

𝑀 = 𝑉𝑎 × 1.65 − 5 × 0.45 = 6𝑘𝑁𝑚

𝜎 =𝑀𝑌

𝐼=

6 × 150 𝑥 106

337500000= 2.667𝑁/𝑚𝑚2

This value is then compared to the maximum magnitude of plane stress at the centre of the beam

using a 2D graph cut along centre line as seen in figure 8. The profile of the plane stress at the centre

of the beam is linear with depth as predicted and with the largest stress of 2536kN/m2 and

2362kN/m2 occurring at the top and bottom of the beam respectively. Comparing the hand

calculation to the LUSAS stress plot shows excellent correlation. The validation was carried out on all

load cases and the results are summarised in table 2.

1.2m 0.45m

3.3m

5kN 5kN

0.15m

0.3m

Figure 7 - Approximate analysis of Concrete beam

9

Table 2 - Summary of Validation


Hand Stress Calculation N/mm2

LUSAS Result N/mm2

Percentage difference

Comment

5 2.667 2.536 4.9% Excellent

Correlation

7 3.733 3.556 4.7% Excellent

Correlation

9 4.800 4.725 1.6% Excellent

Correlation

11 5.867 6.368 8.5% Good

Correlation

13 6.933 8.766 26.4% Poor

Correlation

15 8.000 12.684 58.6% Poor

Correlation

Table 2 shows that the hand calculation have very good correlation with the LUSAS results for the

smaller loads applied to the beam. Whereas for the larger loads applied to the beam the correlation

between the hand calculations and LUSAS results becomes poor. This was to be expected as the

hand calculations assume that the material behaves elastically which, at smaller loads, the beam

primarily does. On the other hand, under larger loads, the beam begins to behave elastically as well

as plastically hence the variation with the hand calculations as they do not account for the plastic

behaviour of the beam. The model of the beam behaves as expected with excellent correlation for

the lower load factors therefore the model can be treated as valid.

Figure 8 - Cut at centre of beam showing profile of plane stress with respect to depth

5.2 Visual inspection of deflected shape The deflected shape of the beam is expected to be curved with the largest displacement at the

centre of the beam. The deflected shape can be seen in figure 9, and shows that the deflection

increases towards the centre of the beam, as expected.

-3000

-2000

-1000

0

1000

2000

3000

0 50 100 150 200 250 300

Pla

ne

Str

ess

(kN

/m2

)

Depth of Beam (mm)

Plane Stress vs. Beam Depth

10

Figure 9 - Deflected shape of beam half model

The magnitude of deflection has been calculated by using the bending deflection formula for a beam

as seen in figure 10 and the equations below.

Figure 10 - Line drawing of beam (Rafiq, 2016)

E = 42 x103 N/mm2

I = 337.5 x106 mm4

𝛿 =𝑊𝐿3

6𝐸𝐼[3𝑎

4𝐿−

𝑎3

𝐿3]

𝛿 =5000 × 33003

6 × 42 × 103 × 337.5 × 106[3 × 1200

4 × 3300−

12003

33003] = 0.47𝑚𝑚

A hand calculation was carried out for each load case and then compared to the LUSAS result which

is summarised in table 3.

Table 3 - Summary of deflection validation


Hand Deflection Calculation (mm)

LUSAS Result (mm)

Percentage difference

Comment

5 0.47 0.45 4.3% Excellent

Correlation

7 0.66 0.63 4.5% Excellent

Correlation

9 0.85 0.83 3.5% Excellent

Correlation

11 1.04 1.15 10.6% Good

Correlation

13 1.23 1.69 37.4% Poor

Correlation

15 1.42 3.09 117.6% No Correlation

As for the plane stress validation, the hand deflection calculations do not account for the plastic

behaviour of the beam and this is why there is no correlation between the methods for the point

load applied of 15kN. The hand calculations do not account for the reinforcement properties of the

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bars and therefore only gives good results for where the beam behaves elastically. Generally the

magnitude of deflection as well as the deflected shape suggests that the model is valid.

For the parametric studies, each member of the group has undertaken a short verification process

mimicking the approximate methods in Sections 5.1 and 5.2.

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6.0 Observations Figure 11 shows the profile of plane stresses for each load cases cut at the centre of the beam. From

this it can be seen how the stresses distribute as the load is increased and the cracks begin to form.

At 5kN the distribution of stresses is linear and suggests that the stress is spread evenly between the

top and bottom of the beam. As the load case increases, the maximum tension stresses moves

towards the centre of the beam as cracks form; the compression stress increases steadily at the top

of the beam. At 15kN, the distribution is highly non-linear with little stress at the bottom of the

beam, significant amounts of stress in tension is distributed at the centre.

Figure 11 - Profile of plane stresses

Attached in the Appendix A are a series of contour plots of stress as well as crack paths across the

beam. The Appendix shows that cracks propagate from the bottom surface of the centre of the

beam and as the load increases the crack propagates towards the top of the beam. For larger loads

cracks begin to appear towards the supports. From the contour plots it can be seen that for beams

with large amounts of cracking the stress redistributes towards the centre of the beam.

The cracks occurring in the concrete beam have small widths as seen in Figure 12 which shows the

contours of the crack widths, the maximum crack width being 0.11mm at the bottom of the beam.

Figure 12 - Crack widths of standard beam

0

50

100

150

200

250

300

-15 -10 -5 0 5

De

pth

of

Be

am (

mm

)

Plane Stress (N/mm2)

5kN

7kN

9kN

11kN

13kN

15kN

13

7.0 Parametric Studies

7.1 The Position of the Point Load – Lawrence Taylor (LT) The parametric study performed by LT involved the movement of the point load subjected on the

beam. All other variables have been kept the same as the original model. The purpose was to see

how this would affect the stress distribution and cracking profile of the beam. It is assumed that

more cracks will be produced when the point load is applied in position 1 compared with position 5,

as the load applied will increase further than that of position 5 to produce a deflection of 3mm. To

test this hypothesis five FEA models have been produced by varying the point at which the point

load is along the beam at 225mm intervals starting from the central point on the beam Position 5 to

the left hand side of the beam Position 1 (Figure 13).

As the point load positioning was changed, the mesh sizing on the right hand side of the point load

also changed. Each 225mm change in position from the centre of the beam also increased the

number of mesh intervals by two. This was done to increase the accuracy of the results as otherwise

the meshing may have become too coarse to provide reasonable results. The graded mesh utilised

on the left hand side of the point load was not changed, as the area it was calculating the results was

shortening thereby creating more accurate results.

Figure 14 below illustrates the Load vs. Deflection diagrams of the beams modelled. It can be seen

that the further away from the centre of the beam that the point load is applied, the larger the

necessary load must be applied to result in the same deflection of 3mm. This observation is

expected and helps to validate that the model is working as expected. A surprising observation from

Figure 14 also suggests that all beam models behave relatively linear until 1mm of deflection is

surpassed. After 1mm of deflection has been surpassed, the beams begin to deflect more readily

compared to the increase in loading. Fewer incremental increases in the load (set at 2000N) are

required to increase deflection as the point load is moved towards the centre of the beam. Indeed,

Figure 13 – Beam with point load positions 1 to 5.

Position 1

Position 2

Position 3

Position 4

Position 5

14

position 5 where the point load is at the centre of the beam requires only two increases to produce

the same 3mm deflection compared to position 1 where the model requires 4/5 increases in loading

to produce the same result.

The stress diagrams were analysed to investigate where and how the principal stresses formed in

the beams. The cracking profiles are also plotted to investigate how different formations in stress

affect the cracking in the beam models. Below in Figure 15 is the diagram relating to Position 1 when

the beam has reached a deflection of 3mm (the furthest point load from the centre of the beam).

Figures 16 and 17 show the output for Positions 3 and 5.

Figure 14 - Graph comparing load vs. deflection of the five beam models produced.

Figure 15 - Figure showing the stresses and cracks produced in the beam (Position 1)

Figure 16 - Figure showing the stresses and cracks produced in the beam (Position 3).

15

From Figures 15, 16 and 17 it can be seen that as the point load is moved closer towards the centre

of the beam, the cracking that is occurring in the beams is reducing. The angle of cracking in Figure

15 is far more pronounced than the cracking occurring in Figure 17. As discussed in the literature

review (Section 2.2), these angled cracks suggest shear failure forming in the beam as the point load

is moved closer to the support. Of course, this observation is highly important due to the fact that

the beams modelled have no shear links, thereby reducing the capacity of the beam in shear.

It is clear to see from a design point of view that if a beams deflection limit is designed as 3mm (or

whatever the specification outlines), care should be taken to position the point load as close to the

centre of the beam as possible to reduce the number of cracks produced for serviceability limits. The

client may not be pleased if the concrete beams produce too many obvious cracks. It can also be

seen from Figure 15 that the compressive stress produced by the models is not excessive and will

not create compressive cracks in the beam. However, it should be noted that this is an unrealistic

beam model as the point load is being applied on a knife edge. In reality the point load would be

applied over a larger area thereby reducing the stress produced.

Below the severity of the cracks are analysed (Figures 18, 19 and 20), as this will dictate

serviceability requirements in industry (specifically relating to crack control).

Figure 17 - Figure showing the stresses and cracks produced in the beam (Position 5).

Figure 18 - Figure showing the crack width output (Position 1).


16

Table 4 - Table highlighting crack widths in models.

Beam Model Crack Width

(mm)

Position 1 0.0922

Position 2 0.1148

Position 3 0.0795

Position 4 0.1408

Position 5 0.1698

It has been observed when comparing the crack widths, that the closer to the centre of the beam

that the point load is moved the greater the increase in crack width (Table 4). This is an important

observation as it is the opposite of what would be expected from the above observations. Therefore,

although less cracks form the closer the point load is to the centre of the beam, the larger the cracks

become. It should be noted at this point that the cracks are still far below those of the crack control

requirements (highlighted in Section 2.2).

7.1.1 Beam Verification

The same verification method has been used as shown in Section 5.0. Results will be discussed in

Section 8.7.

Table 5 - Table showing verification of FEA results vs. elastic hand calculations.

Model Point Load (kN) Hand Calc Stress (N/mm2) LUSAS Stress (N/mm2) Comment

Position 1 9 3.00 2.85 Excellent

Position 1 23 7.67 11.75 Poor







Position 5 9 3.30 8.28 Bad

Position 5 13 4.76 18.66 Bad


17

Table 6- Deflection Hand Calculations

Model Point Load (kN) Hand Calc Deflection (mm) LUSAS Deflection (mm) Comment


Position 1 23 1.54 3.50 OK







Position 5 9 0.48 1.04 OK

Position 5 13 0.69 3.07 Bad

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7.2 Beam Depth and Width – Daniel Wilkinson (DW) The parametric study carried out by DBW required changing the dimensions of the concrete beam

but keeping the reinforcement bars the same size and in the same location. The purpose of this

parametric study was to determine the effect of changing the dimension on the stress distribution

throughout the beam with relation to the location of the reinforcement bars. The point load is

maintained in the same location on the parametric beams by ratios of length either side of the point

load. This ensures that the point load is acting in the same place, proportionally, for each of the

beams to make sure that comparative analysis is relevant to the report.

Figure 21 – 500mm deep beam

It was decided that changing both the length and the depth of the beam would be suitable for

demonstrating the effects of dimension on the non-linear analysis of the concrete beam. With this

criterion in mind, it was decided that investigating a beam depth of 500mm (Figure 21) and 100mm

(Figure 22) would be beneficial to see the difference in stress distribution in relation to the original

beam of depth 300mm.

Figure 22 – 100mm deep beam

This is to incorporate a wide range of results in order to accurately determine how cracks propagate

throughout the concrete and how the concrete responds under non-linear analysis. In addition to

this, still maintaining some reality as far as design is concerned, it was decided that a beam of depth

500mm would portray real world scenarios better than an extreme depth of 1000mm. Although

1000mm would demonstrate the effects of stress and crack propagation in a deep beam

environment, it was felt that it was unlikely to occur in real life as considerably more reinforcement

would be needed for a beam of that size in the real world.

In addition to the depth of beam being changed, the effects of changing the length of the beam on

the stress distribution and crack propagation would also be explored. Similarly, to the depth models,

reality was still observed and so the maximum length of beam, bearing in mind that the model is a

symmetrical half model, is a 10m span. Although this might be rare in practice, it proves as an

example of how the non-linear analysis uses the deflection as one of the input criteria in the analysis

and, with the beam being much longer and therefore has a larger deflection, how this then affects

the outcome result.

19

Changing the depth of beam

Figure 23 – 500mm Beam contour plot

The 500mm deep beam model, shown above (Figure 23) at the loading factor 43kN (the load factor

needed to reach a deflection of 3mm or over), shows the stress distribution and the crack

propagation in the concrete. At this point the beam has cracked all the way through; the cracks have

propagated from the bottom of the beam to the top. To the right of the point load, it can be seen

that the concrete beam is experiencing compression as the concrete is cracking at the bottom

forcing the concrete at top of the beam together. When compared to Figure 2, the model is showing

a mix between vertical flexural cracks and incline flexural-shear cracks. This arises from the tensile

forces in the concrete becoming larger than the tensile strength of the concrete. The crack width can

be seen below in Figure 24; the largest crack width is 0.52mm compared with the 0.11mm crack

width of the original beam, a significant increase. As detailed in the literature review, this sized crack

is far above the limits imposed by the Eurocodes and standards of 0.3mm-0.4mm.

Table 7 - Table showing crack widths in models

Beam Model Crack Width

(mm)

100mm 0.0786

300mm 0.0795

500mm 0.2904

L = 10m 0.0748

Figure 24 - 500mm crack widths

Figure 25 below shows the stress at the bottom of the beam, the level of the reinforcement,

throughout a selection of the loading factors. Loading factors 1kN-21kN exhibit similar loading stress

to the original beam, this was to be expected in the first instance of loading. However, when loading

levels 33kN-43kN are reached, the maximum stresses in the bottom of the beam move along the

beam towards the left support, dramatically decreasing and the moving into slight compression

20

around 1200mm. As found in the literature search, issues can occur at the supports in beams as the

stresses move from the centre of the beam towards to outer edges. As shown in the validation of

the original beam, the initial loading causes the beam to deflect elastically, but as the loading

increases, the beam begins to deflect plastically and so will not return back to its original position

when the loading is lifted.

Figure 25 – 500mm beam stress patterns

Figure 26 – 500mm beam stress patterns

The large peak in the Figure 26 above can be attributed to the increase in compression stress in the

top of the beam due to the failure of the concrete below due to cracking, as seen in Figure 23. The

top and bottom areas of the beam are compared here and it can be seen that where there are low

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

0 200 400 600 800 1000 1200 1400 1600 1800

Stre

ss (

N/m

m²)

Length (m)

Graph to show stress through bottom of 500mm beam at different loading factors

43kN

37kN

33kN

21kN

11kN

7kN

1kN

-20

-15

-10

-5

0

5

0 200 400 600 800 1000 1200 1400 1600 1800

Stre

ss N

/mm

²)

Distance (m)

Graph to show stress through 500mm beam at Top and Bottom

Bottom

Top

21

stresses in the bottom of the beam due to the concrete failing, larger peaks in stresses appear in the

top of the beam.

The failure phenomena can be explored in Figure 27. This figure shows that the beams with greater

thicknesses fail more suddenly than those with smaller depths. The loading at the 100mm beam

depth shows that the deflection increases gradually with the increase of the loading force. However,

if this is compared to beams of a greater depth, it becomes clear that as the depth increases, the less

the loading needs to change to incite a larger change in the deflection. At this point it is clear that

the steel reinforcement is no longer adequte to ensure that the beam does not suddenly fail. This

shows that as the beam size increases, the reinforcement must be increased as well.

Figure 27 – Deflection against depth

When the same effects are studied in the 100mm depth beam, whilst obviosuly requiring less load to

reach the deflection limit of 3mm, it exhibits the same stress patterns as the 500mm beam. The

crack propagation is similar to the deeper beam, however the cracks are far more spread out

showing dangerous structural weaknesses dow the entire length of the beam. However, in this case

there is far less inclined flexural cracking than in the 500mm beam.

Figure 28 – 100mm Contour plot

Unlike the 500mm beam, the cracks in this beam do not reach the top of the beam once taken to a

deflection of 3mm. Compared to the 500mm beam, the compression stresses in the top of the beam

are larger but there is no surge in compression stresses after the point load, the stresses simply stay

somewhat uniform after this point. This is true in both the bottom and the top of the beam, and

0.00E+00

1.00E+04

2.00E+04

3.00E+04

4.00E+04

5.00E+04

6.00E+04

7.00E+04

8.00E+04

9.00E+04

0 0.5 1 1.5 2 2.5 3 3.5

Load

(N

)

Deflection (mm)

Graph to show load against deflection for varying depth

Original

500mm

750mm

1000mm

22

unlike the 500mm beam the stresses are not affected by the cracks to the greater extent that the

deeper beam was subjected to.

Figure 29 – 100mm stress distribution

Changing the length of the beam In addition to changing the depth dimension, the length of the beam was also changed to see the

cracking effect on a longer beam; it was decided that a beam of 10m overall should be tested (5m

half model).

Figure 30 – 100mm contour plot (Altered termination limit)

As the beam was much longer than the original, the force it took to reach the 3mm deflection was

much less so the effect of changing the maximum deflection was explored. When changing the

maximum deflection to say -20mm, the cracks in the beam still did not form all the way through the

beam (Figure 30), unlike the models where the depth of the beam is changed.

It was then decided that to see the propagation of cracks throughout the beam that the maximum

deflection could be changed to -100mm instead of -3mm. Obviously unrealistic but it demonstrated

that the non-linear analysis is showing the distribution of stresses and the propagation of cracks at

failure and this time the cracks form all the way through the beam (Figure 31).

-16

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

0 200 400 600 800 1000 1200 1400 1600 1800

Stre

ss N

/mm

²

Distance (m)

Graph to show stresses in Top and Bottom of 100mm beam

Bottom

Top

23

Figure 31 – 100mm crack propagation at -100mm deflection

7.2.1 Beam Verification Table 8 - DW Beam Stress Verification

Table 9 - Beam Deflection Verification


Depth = 100mm 9 BEAM FAIL BEAM FAIL BEAM FAIL

Depth = 100mm 4.081 19.60 16.25 Good

Depth = 300mm 9 4.80 4.73 Excellent

Depth = 300mm 15 8.00 12.68 Poor



L = 3.3m 9 4.80 4.73 Excellent

L = 3.3m 15 8.00 12.68 Poor

L = 10m 9 11.70 16.54 Good

L = 10m 11 14.30 19.05 Good


Depth = 100mm 9 BEAM FAIL BEAM FAIL BEAM FAIL

Depth = 100mm 4.081 10.46 10.24 Excellent


Depth = 300mm 15 1.42 3.09 Poor


Depth = 500mm 43 0.88 3.07 Poor

L = 3.3m 9 0.85 0.83 Excellent

L = 3.3m 15 1.42 3.09 Poor

L = 10m 9 20.57 60.65 Terrible

L = 10m 11 25.14 107.44 Terrible

24

7.3 Size of Steel Reinforcement Bars and Compression Reinforcement – Oliver Harris

(OH) For design purposes, reinforcement bars for beams are specified by number of bars and the nominal

bar diameter in mm. Standard nominal bar diameters available in the UK range from 6mm to 50mm.

The tension reinforcement bar size is determined in EC2 by using equation 1 to find the area of steel

required and then the number of bars and size can be selected from a manufacturer that have an

equal or larger area of steel than what is required.

𝐴𝑠 =𝑀

𝑓𝑦𝑑𝑍 Equation (1)

As = Area of tension reinforcement required M= Moment

Fyd = Design yield strength of steel Z = Lever Arm

This parametric study aims to investigate how varying the bar size effects the non-linear behaviour

of the concrete beam. A small selection of bars with a range of diameters and cross-sectional areas

were selected to be modelled, as seen in table 7. In practice most beam reinforcement bars

diameters are 8mm – 16mm this range is covered within the parametric study as well as more

extreme cases for beams with very large loads that could require up to 40mm bars. It is expected

that as bar size increase that the effect of cracking is reduced for equivalent loads.

Table 10 - Total cross-sectional area of 2 bars (British Standards Institute, 2005).

Bar Size Cross-sectional

area of bars

6mm 56

10mm 157

16mm 402

25mm 982

40mm 2513

Figure 32 - Parametric study of bar diameter size and how it affects the relationship between Load and deflection

0

5

10

15

20

25

30

35

40

0.00 1.00 2.00 3.00 4.00

Tota

l Lo

ad (

kN)

Deflection (mm)

Graph Showing Load vs. Deflection

40mm Bars

25mm Bars

16mm Bars

10mm Bars

6mm Bars

25

The parametric study shows the impact that the size of the reinforcement bars has on the deflection

of a concrete beam, as seen in figure 32. For a concrete beam as the bar size increased the load

factor required to reach the limit of 3mm deflection increased, as expected.

The beam with 6mm bar did not reach 3mm deflection as it fails before the beam deflects at the

centre by 3mm. This is suggested from figure 33, where it is evident that the cracks have propagated

through the entire height of the beam. This leads to the contour plot being obscure results as the

beam has cracked in to two and has failed. Additionally, the deflection for the last load factor of

78mm is very large and it a result of the beam breaking, this result was excluded from graph 2 as it is

irrelevant due to the beam failing.

Figure 33 - Concrete Beam with 6mm bars, crack pattern shows that the entire beam has cracked.

From figures 34 & 35, the contours of the crack widths of the final 2 load factors shows that the

critical crack has propagated from bottom to top of the beam. The pen-ultimate load factor shows

that the crack width at the bottom of the beam is 0.11mm in magnitude and converges to 0 towards

the centre of the beam. Whereas for the final load factor the crack width at the bottom is 10.2mm

and converges to 0.5mm at the top of the beam and displays a clear crack throughout the beam

hence the beam has failed.

Figure 34

26

Figure 35

For the smaller bars of 6mm and 10mm the limit of 6mm was met within relatively few iteration,

therefore the models were solved again with a reduced load step from 2kN to 0.5kN, this lead to

greater accuracy in the deflection against load graph as seen in graph 2.

Figure 36

Interesting for larger bars cracks on the compression side of the beam begin to propagate. This is

evident for the 40mm diameter bars with loads of 30kN and greater are applied cracking starts to

occur on the compression side of the beam and develops further as seen in figure 36. This suggests

for large concrete beams with significant loading that compression reinforcement could be required.

This is reflected in practice with common size beams not requiring compression reinforcement.

Compression reinforcement A further model was investigated to see the effects of adding compression reinforcement to the

concrete beam. The compression reinforcement was positioned 25mm from the top of the beam

and modelled with the same properties of the tension reinforcement. In practice the compression

bar would be smaller than the tension bar however for this analysis using similar size bars should still

lead to general trends of how compression reinforcement affects the plane stress plot. What rebar

was used?

27

Figure 37 - Contour plot and crack plots of compression bar

Figure 38

Adding compression reinforcement made little difference to the deflection, this is expected as the

beam does not require compression reinforcement as it will fail in tension first which is confirmed by

this result. The contour plots are also similar, as the compression part of the beams reinforcement

has not failed.

Table 11 - Comparison of crack widths between models.

Beam Model

Crack Width (mm)

6mm 2.3376

10mm 0.5836

16mm 0.0795

25mm 0.0704

40mm 0.0418

0

2

4

6

8

10

12

14

16

0 0.5 1 1.5 2 2.5 3 3.5

Tota

l Lo

ad (

kN)

Deflection (mm)

Beam with No Compression Reinforcement

Beam with Compression Reinforcement

28

7.3.1 Beam Verification Table 12 - Beam Stress Verification



6mm 9 4.80 5.31 Excellent

6mm 11 5.86 48.04 Bad


10mm 13 6.93 17.60 Poor


16mm 15 8.00 12.68 Good


25mm 23 12.26 13.89 Excellent


40mm 37 19.73 15.49 Excellent

Compression 9 4.80 4.14 Excellent

Compression 17 9.06 12.36 Good



6mm 11 1.04 58.04 Terrible


10mm 13 1.23 8.70 Terrible


16mm 15 1.42 3.09 Poor





Compression 9 0.85 0.76 Excellent

Compression 17 1.61 3.98 Poor

29

7.4 Changing Beam Thickness– Yu Tam (YT) The parametric study undertaken by YT involved changing the width of the beam, therefore the

spacing of the reinforcement, and noting the effect on the cracking, deflection and stress

distributions.

Figure 39 - Different width of the concrete beam

As per the original beam model, reinforcement is provided in the lower face of the beam and has a

total cross-sectional area of 400mm2; the cover of the reinforcement is 25mm on both sides. The

effects of any shear reinforcement can be ignored as the beam does not contain shear links, a

measure that would be used to prevent the effects of shearing in design.

The effect of changing the width of the beam was deemed of interest to this report as this would

show the effects of inadequate reinforcement in the beam. This is of course because as the beam

width increases, the spacing between the reinforcement becomes greater, leaving a large area

without reinforcement. For the beam width, it was decided that investigating a 200mm and 300mm

wide beam would be beneficial to note the difference in stress distribution in relation to the original

beam of width 150mm.

Figure 40 - Geometric of the 2D beam model

30

Figure 41 - Increment Load factor 15000 (thickness = 150mm)

For comparative reasons, Figure 41 - Increment Load factor 15000 (thickness = 150mm) shows the

stresses and the cracking profile of the original beam with a width of 150mm. it can clearly be seen

that the point load, located in the middle, is causing compressive stresses in the top mid-section of

the beam and tensile forces on the outer regions and the bottom of the beam. The effects of the

cracking can clearly be seen because although they do not penetrate throughout the entire depth of

the beam, the cracks weaken the concrete, exerting compressive stresses on the concrete left intact.

Figure 42 – The graph of the beam displacement (thickness = 150mm)

Figure 42 shows the plot graph of the beam displacement for the original model. It can be seen that

as the beam is loaded, there is no sudden and destructive failure. This is expected in the original

beam as the reinforcement is certainly adequate for this size of beam. The gradual deflection shows

that even if the beam is loaded to failure, there will be warning signs such as cracking of finishes that

will warn occupants and allow them to leave quickly.

Figure 43 - Increment Load factor 23000 (thickness = 300mm)

Figure 43 - Increment Load factor 23000 (thickness = 300mm)show the beam with a thickness of

300mm instead of 100mm. The position of applied load and increment load factor is same but the

stress distribution is different to that observed in the original beam. Unlike the original beam, the

31

cracks are no longer clean and vertical in appearance, this time there are more diagonal and angled

cracks. In relation to Figure 2 - Highlighting the difference in crack patterns of shear and flexural

failure (Paul, 2015). these cracks can be identified as inclined flexural/shear cracks indicating that

the tensile forces have exceeded that of the concrete. In addition to this, the cracks have formed all

the way through the beam instead of part of the way.

Figure 44 - 300mm width crack widths

Figure 44 shows the maximum crack widths at the final loading increment of the 300mm wide beam.

The maximum crack width as seen here is 0.277mm which according to the limits found in the

literature search or between 0.3mm and 0.4mm is acceptable. However this does not mean that the

beam is safe and as the cracks have propagated all the way through the beam, the structural

integrity of the beam is now compromised. The maximum crack widths of the other beams are

summarised in the table below.

Table 14 – Crack Widths

Beam Thickness(mm)

Crack Width (mm)

150mm 0.0795

200mm 0.1365

250mm 0.1242

300mm 0.1843

32

Figure 45 - The comparison of different beam width

Figure 45 above shows the comparison between different widths of the concrete beam against the

loading factor that the beams are exposed to. This allows the observation of the deflection as the

loading is applied and therefore conclusions can be drawn as to the type of failure. From this graph it

is possible to see that as expected the deflection increases with the applied load but also that the

models all behave uniquely depending on their width. It can be seen that the original beam width of

150mm and the 100mm model both deflect gradually as the loading is applied and this is similar

throughout the loading until the 3mm termination.

However, when the beam widths of 200mm-300mm the gradual loading changes. There is now

evidence that as the loading is initially added, the deflection changes little. This is expected as the

width of the beam increases. However, this plot allows the observation of a point where the

deflection suddenly increases dramatically with little added loading. This shows the effect of the

cracks throughout the beam and that the beam has deformed plastically, causing a lot more

deflection without the need for as much added force.

7.4.1 Beam Verification Table 15 - Beam Stress Verification

0.00E+00

5.00E+03

1.00E+04

1.50E+04

2.00E+04

2.50E+04

3.00E+04

0 1 2 3 4 5

Tota

l lo

ad f

acto

r (N

)

Displacement(mm)

Load vs. Displacement

Thickness=100mm

Thickness=150mm

Thickness=200mm

Thickness=250mm

Thickness=300mm



150mm 15 8.00 12.68 Good


200mm 19 7.60 13.16 Good


250mm 21 6.72 12.81 Ok


300mm 25 6.67 13.49 Poor



150mm 15 1.42 3.09 Poor


200mm 19 1.35 3.86 Poor


250mm 21 1.20 3.36 Poor


300mm 25 1.19 4.46 Bad

33


34

8.0 Discussion

8.1 Elastic to Plastic Figure 11 - Profile of plane stresses, clearly visualises the transition between the elastic and plastic

behaviour of the beam. Where the load factor and deflection relationship is linear the beam is

behaving elastically whereas where the relationship becomes non-linear the plastic behaviour

becomes dominant. When the parametric study into the sizing of the bars was carried out (OH), the

elastic region for the bars seems to increase slightly with the increase of bar size. More notable is

that as bar size increases the plastic range appears to be able to take far greater loads. For example,

for 10mm bars where the load is increased between 9kN to 10kN the deflection increases drastically

approaching failure. Comparatively for larger bars such as the 25mm bar at 12kN and greater, the

rate of deflection slightly reduced with increase of load suggesting that the beam offers further

plastic deformation before approaching total failure. These two examples highlight how the bar size

has a direct impact on the plastic behaviour of the concrete beam.

Indeed, this is the purpose of the steel bars in that they allow the gradual failure of the concrete

beam rather than the sudden failure that could be expected from a beam with no reinforcement.

This is further explored in the parametric study changing the dimensions of the beam. As seen in the

parametric studies, there comes a point in the sizing of the beam where the amount of

reinforcement from the unchanged steel bars becomes inadequate and so the concrete beam goes

from failing over time to failing suddenly. This is amplifying the need to change the amount and

sizing of the reinforcement as the size of the beam increases.

8.2 Cracking Non-linear analysis allows the cracks forming within the concrete to be modelled and visualised,

whereas linear analysis does not take this into account. The inclusion of the effect of cracking is to

some extent predictable and therefore can aid and be included in the design and analysis of beams.

Cracking affects the strength of the beam and can be accounted for in the calculation for the failure

load, allowing the design to avoid over engineering resulting in reducing costs.

Comparing the crack propagation diagram (DW) (Figure 23 – 500mm Beam contour plot) to the

stress graphs (Figure 25 – 500mm beam stress patterns) in the parametric study on changing the

dimensions of the concrete beam, it can be seen that the dramatic decrease in stress in the bottom

of the beam coincides with the increased number of cracks. In addition to this, when the stresses in

the bottom of the beam are compared with those at the top, as seen in Figure 26 – 500mm beam

stress patterns, a large peak in the stresses in the top of the beam coincides with the largest spread

of cracks that reach from the bottom of the beam to the top. This can be observed in the crack

diagram (Figure 23 – 500mm Beam contour plot) as the concentrated blue area to the right of the

point load, suggesting that this is the only area of the beam subjected to large stresses at a

deflection of 3mm, after the concrete below has failed. In relation to this, it could be advisable that

for a concrete beam of this size and loading that compression reinforcement is needed in the top of

the beam to ensure that it does not suddenly fail as shown in the (Figure 26 – 500mm beam stress

patterns), after the peak. In the cases where extreme studies were used such as the 500mm beam

depth, 300mm beam width and the smaller steel reinforcement, the cracks that propagated were

well over the 0.3mm-0.4mm threshold suggested in the standards.

Cracks as large as 10.2mm (as seen in the final loading stage of OH’s study) would be dangerous as

well as detrimental to the integrity of the beam. Cracks this large would allow water in which would

rust the steel possibly contributing to spalling. In addition to this, there could be issues with fire

35

safety due to the concrete being used as protection for the steel beams. In this situation the steel

could be softened and so the building would be dangerously unstable under fire conditions, which is

not acceptable.

In the parametric study carried out by YT, the maximum crack widths found were approximately

0.3mm, i.e. within the limits found in the literature search. This parametric study provided data to

show that although the dimensions of the beam and sizing of the steel bars are very important, it is

also very important as to how they are arranged in the beam. If there is too much space between

the bars, the tensile strength of the concrete is relied up too much and as a result, tensile cracking

was observed when the beams were spaced apart.

In the parametric study undertaken by LT, it is clear that the positioning of the point load is very

important when considering the deflection and therefore cracking. From Figures 15, 16 and 17 it can

be seen that as the point load is moved closer towards the centre of the beam, the less cracking

occurs in the beam. Clearly this is important from a design point of view and like the other

parametric studies it shows that if the reinforcement is inadequate to cater for off centre point

loads, then the tensile stress in the concrete will cause cracking. It was also noted that although less

cracks were modelled, the crack widths increased as the point loading moved towards the centre of

the beam.

The concrete of the beam is almost impossible to prevent from cracking and will contain thousands

of micro cracks. The size and location of cracks can be limited by increasing the amount of bars,

reducing the rebar spacing, improving curing methodology, and use of an appropriate concrete mix

design. Structures with severe cracking would concern the public this is considered within design

codes, where cracking is factored as a serviceability failure in limit state design. The Eurocodes limit

the reinforcement spacing which enables the cracking to be minimised. Additionally, cracking can

lead to moisture being able to penetrate and corrode the reinforcement resulting in the beam

failing. Non-linear analysis allows the major crack propagation to be identified and unusual cracking

to be prevented, and the optimisations in positioning the rebar to not only produce the safest design

but to also streamline it with regards to cost savings.

8.3 Deflection Fixing the limit of the deflection and the use of non-linear analysis allows for the observation of the

failure mode. Concrete without reinforcement is likely to fail suddenly as the concrete reaches a

stress of 10% of that of the compressive strength. Concrete with reinforcement will fail over time as

the steel rebar increases the tensile strength of the beam. This can be seen in the parametric studies

and where the steel reinforcement has become inadequate for the beam size (OH).

In the case of the deeper beams (DW), it can clearly be seen from the load against displacement

graph (Figure 27 – Deflection against depth) that there is a point where the deflection suddenly

increases rapidly without much additional load. Compared with the original beam where the

deflection happens gradually over the loading, the initial deflection of the beam is small as the

loading increases until it suddenly deflects to 3mm with very little additional load. This demonstrates

that with the deeper beams, there is little warning that the beam might fail considering the little

amount of deflection in the initial loadings. In practical applications this can be disastrous, but with

the use of adequate reinforcement in the top and bottom of the beam, the failure time can be

increased and so reducing the danger of the beam failing suddenly. The 500mm beam was modelled

using the same size reinforcement as the original beam which according to the discussion above is

not adequate to prevent the concrete from failing suddenly. This effect is also noted when the

width of the beam is changed. According to Figure 45, as the width increases the more sudden the

36

failure of the beam becomes. This is again an example where the steel reinforcement has become

inadequate for that size of concrete beam. In addition to this, there is an issue with spacing in the

beam. It is expected that in a beam of a large size, say the 300mm deep beam, that there would be

more reinforcement along the entire bottom face, not just in each corner. In the parametric study

carried out by LT, it was also apparent that at position 5, the deflection was gradual as the load was

added, but as the position of the point load was moved through to position 5, the point increased

deflection with only a little added load became apparent as in the other parametric studies.

8.4 Model Discussion & Limitations One issue with a 2D model of the beam is that the reinforcing bars are poorly modelled. Following

the right hand rule, at the height where the bars are in the concrete beam in the z-axis the material

is non-homogenous, i.e. the reinforcement is not a sheet but rather 2 bars. The 2D model does not

account for this as it models the bars taking the properties of the total cross sectional area as well as

using an appropriate mesh for bars, however it does not account for the numbering and positioning

of the bars in the z-axis. This oversight in the model will reduce the accuracy of the results, as the

effect of the stress distribution in the z-axis will be inaccurate. However, this effect shouldn’t be too

significant as the beams thickness is relatively small compared to the length (Figure 47). The effect of

the bars was further investigated in a 3D model in Section 8.5.

In addition to this, the crack widths that have been discussed depend greatly on Concrete Crack

Model 102. This model is based on a formula and so discrepancies between what it obtained

through the models in LUSAS and those in real life as concrete is non-homogeneous and therefore

will be different every time it is made and tested. Indeed, LUSAS (2016, pp. 132) states that the

model may display inaccurate crack widths if load increment increases are too great. It also goes on

to state that quadratic elements may decrease the accuracy of the results – the models used in this

report use quadratic instead of linear elements. This amplifies the fact that although concrete can be

successfully modelled in this way, there will still be unexpected or natural phenomena that cannot

be accounted for in the modelling.

The support conditions used in the realisation of the models are unrealistic from a real world

application perspective. In the models presented in this report, the support conditions are specified

so that they are supporting the beam on a ‘knife edge’ rather than on a lintel or large support.

Obviously this increases the stresses at the support for the beam as there is a smaller area through

which the load is transferred. In addition to this, the loading used in the models is also unrealistic as

it is unlikely that there will be a single point load on a beam in a real world application. To increase

the accuracy of the models, it would therefore be beneficial to increase the area in which the

supports and point load has in contact with the models. It can be seen in all models that large

compressive stresses were produced because of the point load acting on a very small surface area.

Indeed, problems arose during verification for stress that the models must be analysed using the

‘Graph Through 2D’ approach to analyse the maximum stress occurring at the centre of the beam

instead of taking the maximum stress overall. This is due to the large stresses obtained under the

point loads.

8.5 Assumptions The superposition of nodal degrees of freedom assumes that the concrete and reinforcement are

perfectly bonded which in the real world is an unlikely case, as dust, poor ribbing on the rebar and

other foreign bodies can cause an imperfect bond between the concrete and the rebar. This creates

a degree of error in the models, as it is modelling the optimum conditions that are otherwise

impossible to replicate in real life.

37

Linear analysis assumes a linear relationship between stress and strain, material properties are

constant and deformations are small as to allow for plane sections to remain plane. Whereas

nonlinear analysis allows for nonlinear stress-strain relationships as well as modelling of structures

that undergo large deformation. Other minor assumptions made include the exclusion of the self-

weight of the beam which is negligible compared to the point load. As discussed in the literature

review, shear links have not been modelled and may have helped with shear failure in the beams.

8.5 3D Model In reality a beam is a 3D object and thus using 2D models could poorly represent how an actual

beam behaves. The models in this report are represented in 2D with the thickness of the beam being

applied as an attribute to the model. To ensure that a 2D model represents the behaviour of the 3D

model well, a 3D model of the beam was constructed in LUSAS. The model had the same geometry,

material properties, supports and total load. The load was split into 2 point loads located 25mm

from the edge of the beam, offering a slightly more realistic loading case compared to a single

central point load. The beams support edge was applied to the line at the bottom edge of the beam.

The model was run, and a contour plot for the final load case with crack paths was produced as seen

in Figure 46 - 3D ModelError! Reference source not found.. The cracking profile appears similar for

the 3D model as for the 2D model. The cracks occur uniformly across the thickness of the model; this

suggests a weakness with both 2D and 3D models. The cracking in an actual beam would be more

random and less uniform due to the material properties of the beam varying slightly throughout the

thickness of the beam. The model assumes that the material properties are isotropic across the

thickness of the beam resulting in a slight difference between the model and an actual beam.

Figure 46 - 3D Model

The deflection profile with respect to the load case of the 3D model was compared to the 2D model,

as seen in Figure 47 - 2D and 3D model comparison of deflection profile. This found that the profiles

were very similar and suggests that the 2D model is an acceptable form to investigate beam

cracking. This is likely due to the beam thickness is small relative to the length therefore the

thickness of the beam can be applied as an attribute to the model of a 2D beam.

38

Figure 47 - 2D and 3D model comparison of deflection profile

8.6 Real World Applications The ability to identify the location of the worst deflection and the propagation of cracks in a

concrete beam is a very important asset. There are countless applications for concrete beams

externally as well as internally and if there is excessive deflection resulting in crack formation, then it

can let water in and can begin to corrode the rebar leading to spalling and the damage of finishes.

Obviously it would be unreasonable to suggest that through non-linear analysis all cracks should be

prevented as there are micro cracks forming in concrete beams even below the Ultimate Limit State.

Non-linear analysis can be used to identify areas badly affected by cracking and allows extra

reinforcement or stronger concrete to be specified to remove the possibility of larger cracks forming

in the beam. The prevention of large cracks is paramount to protect the rebar from the elements but

also to prevent damage to the exterior finishes. Clients will want to be reassured that although the

most efficient and cost effective beams will be used in construction, there is no compromise to the

finish of the building once the beams are loaded (i.e. serviceability).

As this is an iterative procedure to solve the non-linear equations, increasing the number of

increments to be performed by limiting the maximum change in load factor will produce more points

on the load vs. displacement graph to produce results closer to the true behaviour of the beam. As

shown below however (Figure 48), this change in loading vs. deflection only changes slightly and is

therefore not something to be concerned about (LT). As shown in the literature review, because the

process is iterative using the Newton-Raphson method, getting better results requires the use of

using smaller time steps.

0

2

4

6

8

10

12

14

16

18

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Tota

l Lo

ad (

kN)

Deflection (mm)

Graph showing difference between 2D and 3D beam modelling

2D Beam 3D Beam

39

Figure 48 – Graph displaying smaller iterations (500N) vs. larger iterations (2000N)

8.7 Verification of the Models

As discussed above, the non-linear nature of the models produces differences between them and

equivalent linear models. As seen above in Sections 7.1.1, 7.2.1, 7.3.1 & 7.4.1 verification using both

the maximum stresses developed and maximum deflection developed in the non-linear models is

compared against elastic analysis methods (examples of calculations shown in Section 5.0). All

models have been verified at two stages, a 9kN load and the maximum load applied to the beam to

produce 3mm of deflection. A 9kN load was chosen as it was deemed to be in the linear region of all

load vs. deflection diagrams. It has been found that the 9kN loading gives very good verification to

the actual non-linear analysis models, due to the linear nature of the models in this region. As the

point load is further increased, the output from the verification method can vary greatly with the

observed output. It is therefore a conclusion of this report that this method should be used with

caution. Although the output from the models may vary with the observed results at the 3mm limit,

the 9kN verification process can help verify that the assumptions used in the model and model setup

match the actual beam to be modelled. It should also be noted that the elastic verification method

for stress does not model the rebar itself. When comparing deflection vs. model deflection the same

output of good correlation with the model was found at 9kN and poor correlation at the final load

increment.

40

9.0 Conclusion Linear analysis is used in most structural analysis as a way of simply determining the resultant forces

in a structure and its behaviour, making it easy to quickly make calculations to check how a building

or beam might react to certain loading or support conditions. However, its simplicity is due to

certain assumptions such as elastic behaviour that, in the real world, cannot be fully translated to a

building structure. This is where non-linear analysis can be utilised as it takes into account the

cracking within the concrete and how it then affects the strength of the concrete and the structures

ability to withstand forces after failure.

It has been found that one of the uses of non-linear analysis allows the sudden failure in the

concrete beam to be observed in detail as there are separate events that need to be identified that

all add to the sudden failure such as crack propagation. Although the deflection is contributory to

the overall failure, the deflection is also due in part to the cracks that form in the concrete beam as

the beam is loaded. In linear analysis, the cracks and crack propagation are not taken into account

and it assumes that the beam is elastic. This is an important aspect in the considerations for design

as it must be ensured that, as cracks decrease the beams ability to take the stresses and loads, that

major cracks should be prevented from forming.

Modelling of the beams in 2D has been found to be sufficient when comparing the results against 3D

models and verification techniques used are only applicable when the models behaviour is still

linear. At the 3mm limit, the verification process is generally no longer reliable to verify the models.

It has been seen that changing the size of the reinforcement effects the displacement of the beam

but also has an effect on the propagation of cracks and the distribution of stresses. For large bars,

cracks begin to propagate in compression at the top of the beam as opposed to small bars in which

the cracks propagate upwards. Changing the depth of the beam increases the cracks formed. Moving

the point load towards the centre of the beam increases the crack widths and increasing the

thickness of the beams also increases the crack widths obtained.

41

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43

Appendices

Appendix A - Series of Contour Plots of Stress and Crack Paths as Load Increases

Load = 5kN - Contours and crack pattern of plane stress SX





44


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Dr Rafiq LUSAS Courseworklawrencetaylor.info/wp-content/uploads/2016/07/... · The modelling of crack widths has also been found to be very useful for serviceability limits in industry