Dr Rafiq Report:

Shear Wall with Opening

STAD505 - Lab Report

Abstract The aim of this report was to use LUSAS Modeller to complete analysis of a shear wall with openings and study the effect of moving the openings had on the stress generated in the connecting beams and to determine the maximum stress value. The maximum stress value found in the link beams were at the second floor level and gave a value of -3346kN/m². Also studied is how effective the approximate analysis methods are for verifying the structures. Parametric studies were conducted by changing the thickness of the interconnecting beams, positioning the holes in a configuration for a stair case and positioning the openings in a vertical line but moving the openings in a horizontal direction. It was found that the approximate analysis methods work well to determine the stresses generated in most cases but when moving the positioning of the openings horizontally no correlation was found.

Daniel Wilkinson (Student ID: 10342869)

Yu Tam (Student ID: 10511595)

Lawrence Taylor (Student ID: 10395608)

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Contents 1.0 Introduction ..................................................................................................................... 2

2.0 Literature Review ............................................................................................................ 2

2.2 Computer Modelling of Shear Walls ............................................................................. 3

2.3 Shear Wall Approximate Analysis ................................................................................ 3

2.5 Summary of Literature ................................................................................................. 4

3.0 Assumptions ................................................................................................................... 5

4.0 Analytical Models ............................................................................................................ 5

4.1 Basic Shear Wall Model ............................................................................................... 5

4.2 Daniel Wilkinson Model (DW) ...................................................................................... 6

4.3 Yu Tam Model (YT) ..................................................................................................... 6

4.4 Lawrence Taylor Model (LT) ........................................................................................ 7

5.0 Validation of Analytical Models ........................................................................................ 8

5.1 Frame Model ............................................................................................................... 8

5.2 Plane Stress Model ..................................................................................................... 8

6.0 Results of Plane Stress Models ....................................................................................... 9

7.0 Verification of computer results ..................................................................................... 10

7.1 Frame model & Free Body Diagram method of verification ........................................ 10

7.2 Wall-Beam Junction Models ...................................................................................... 12

8.0 Discussions and observations ....................................................................................... 14

8.1 Maximum Stress Observations .................................................................................. 15

8.2 Stress changes in the Wall-Beam Junction Models ................................................... 15

8.3 Placement of shear wall openings to reduce maximum developed stress in structures

........................................................................................................................................ 16

8.4 Mode of deformation of shear walls ........................................................................... 21

8.5 Comparison of plane stress models, frame models and Wall-Beam Junction models 21

8.6 Stiffness of the shear wall models.............................................................................. 22

9.0 Summary of design implications .................................................................................... 22

10.0 Conclusions ................................................................................................................ 23

References ......................................................................................................................... 25

Appendices ......................................................................................................................... 26

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1.0 Introduction

Shear walls are an integral part of a buildings lateral support system. The aim of the

wall structure is to provide greater stiffness than the buildings columns by providing a

much larger second moment of area. The problem to be studied is of a shear wall

with six rectangular openings placed throughout the height of the structure. The

report models the openings at different positions in the structure and investigates

into where the stresses develop and at the magnitude of the corresponding stresses.

Also to be studied is the reliability of different approximate methods.

Four separate experiments with openings in different configurations in a shear wall

were carried out using LUSAS (a finite element analysis package) using a plane

stress element type. The positioning of the openings was changed in each

experiment to look at how much this movement affected the magnitude of the

stresses and the positioning of the stresses in the structure. Approximate analysis

using a frame model was then performed to verify the computer results. Also

performed on the models were smaller partial models of the wall-beam junctions to

help verify and to prove that partial models can be used to adequately design a

shear wall structure, even with limited computational power. Section 2 contains a

literature review, Section 3 describes all assumptions used in the modelling process

and approximate analysis, Section 4 presents the different structures to be analysed,

and Section 5 explains how the structure was validated. Section 6 presents the

results of the plane stress analyses of the models and Section 7 presents the results

of the verification of the models. Section 8 includes the discussion and observations

from the modelling and verification procedure. Section 9 includes a summary of the

design implications found from the report and Section 10 concludes the report.

Appendix A includes a CD containing all LUSAS models created for the report.

2.0 Literature Review

Developing lateral restraint in a building is essential if it is to withstand any form of

lateral force whether it is from wind loading, earthquake loading, impact loading or

blast loading etc. There are currently three main methods of providing such restraint

in a building: bracing a steel framed building, providing fixed connections in a steel

frame building and using shear walls to provide stiffness to a concrete structure.

Shear walls can provide a substantial amount of stiffness in a structure, and as noted

by Rafiq (2015a) will mean that most of the lateral force applied to the exterior faces

of the building will be transmitted into the shear walls as they have a much greater

stiffness than that of the surrounding structure.

Shear walls with openings behave differently to those without opening. Shear walls

without openings behave more like a cantilevered beam during deformation whereas

a shear wall with openings will deform under shear deformation (Constructor, no

date). It is suggested that a shear wall with openings depends on the way the two

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halves of a shear wall are connected i.e. either pinned or fixed together as this will

affect the type of deformation of the structure. And as noted by MacLeod (1977) the

design of shear wall with openings using a computer programme is relatively straight

forward but checking the resultant output using approximate analysis methods can

be more challenging to complete with sufficient accuracy.

2.2 Computer Modelling of Shear Walls

Today using a Finite Element Analysis (FEA) package, a detailed model of a shear

wall may be produced using different element types that may include: plate, shell,

solid and plane stress elements. According to Chen and Kabeyasawa (2000, p. 3)

“Reinforced concrete wall panel is usually subjected to in-plane shear moment and

axial force”. Plane stress analysis of a structure limits the mathematical model to

normal stress and shear stresses that are in plane with the structure, and any forces

perpendicular to the plane are assumed to be zero and that the load(s) applied are

over the thickness of the plate (University of California, no date). This type of

analysis also assumes the thickness of the element to be much thinner than that of

the other two-dimension. Because plane stress analysis only deals with stresses in

the plane of the object, it may not produce real world results. Oztorun, Citipitiogluand

and Akkas (1998) conduct a study into shear walls using a model with six degrees of

freedom, so that the wall may bend out of plane allowing an analysis combining

bending and plane stress in the context of a building being modelled. However, the

problem to be studied is of a shear wall without any connecting superstructure, so a

plane stress element type will be chosen as the shear wall will not be pushed out of

plane during the experiments. The plane stress analysis chosen also works under

the assumption of elastic analysis i.e. an increase in force on the structure increases

the deformation of the structure linearly (Rafiq, 2015b). Unfortunately, this

assumption may not be totally correct as forces perpendicular to the plane will be

produced when the shear wall is in the building. However, once factors of safety

have been applied to the design, the structure will be more than capable of resisting

these stresses which have not been modelled in the computer programme. The

programme also assumes the concrete used within the building is homogeneous, but

due to the nature of concrete this assumption is not correct.

2.3 Shear Wall Approximate Analysis

FEA packages as highlighted above can give very detailed analyses of the stress

being produced within a structure. However, due to the complexity of these computer

packages it is a very real prospect that the engineer may make mistakes. The

complexity of how forces are transmitted through a shear wall structure with

openings are apparent in an article by MacLeod and Hosny (1976) in which it is

shown that applying a vertical load atop of the walls does not transmit the load

equally to the supports at the base of the structure. The process of analysing a shear

wall with openings is therefore a complex process.

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Indeed, MacLeod (1977) states that the current methods of approximate analysis do

not take into account the shear forces that are applied through the beams. He

therefore uses a new approach that takes these extra forces into consideration.

According to MacLeod (1977) there are only a couple of generally accepted methods

of approximate analysis of shear wall systems with openings but only the frame

method is a practical solution. This correlates with the findings of Rafiq (2015b) who

uses the technique to establish good correlation between a plane stress model and

MacLeod’s approximate frame analysis approach. Oztorun, Citipitiogluand and

Akkas (1998) question the usefulness of the current approximate analysis methods

of laterally loaded shear walls as they are perceived to not be ‘within acceptable

limits’. The paper did also note that MacLeod’s (1977) approach is similar to some

other methods. It should be noted however that Oztorun, Citipitiogluand and Akkas

(1998) do not put forward a more accurate approximate analysis solution to the

problem of laterally loaded shear walls with openings.

As described above, MacLeod and Green (1973) and MacLeod (1977) present a

method of approximate analysis of a shear wall with openings by analysing it as a

frame structure. They note that previous studies looking at equations analysing the

stresses created within the shear wall are not accurate when openings are applied or

if the openings are not symmetrical in the structure. Their method creates a frame

model of the structure and applies rigid joints, this is then analysed to produce much

more accurate results over a greater range of model types. Because it is a relatively

straight forward model to implement (reducing the chance of error from the engineer)

and that it gives reasonable results, the frame method will be used as the main

approximate analysis method within this report.

2.5 Summary of Literature

To summarise, the laboratory experiments will focus on the movement of openings in

a shear wall structure to perform parametric studies into the effect that the

movement has into the deflection of the structure (serviceability) and the stresses

developed in the structure. Approximate methods of analysis will also be used to

verify that the models are behaving as the computer output suggests. The

approximate method to be utilised will be the frame method as this is a relatively

simple technique to analyse the shear wall. The computer modelling method to be

used will be a plane stress element created in a finite element analysis package.

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3.0 Assumptions

For the computer models the following values have been applied:

The structure of the concrete is homogeneous throughout.

The programme is using linear elastic analysis. The choice of loading applied

to the shear wall model does not matter as the model assumes linear elastic

analysis so the response of the structure under deflection will increase linearly

with the applied loading.

Only forces applied along the thickness of the shear wall exist in the building

structure.

There is no moment created in the shear wall.

For the approximate analysis frame models:

The points of contra flexure are in the centre of the beams and columns.

The rigid arms are much stiffer than those of the walls.

Cladding in the structure will be attached to the beams and floor slabs not to

the columns, so the frames have point loads acting on them.

4.0 Analytical Models

Each student undertook a parametric study of a basic shear wall model. The

characteristics for each individual model are presented below:

Thickness of the shear wall = 300mm

Lateral loading applied to the structure is a face load = 87.302kN/m²

The supports are pinned (as this is a plane stress analysis)

The mesh divisions (unless otherwise stated) are 4 per metre

The properties of the concrete used are:

o Young’s modulus = 30x106N/m²

o Poisson’s ratio = 0.2

o Density = 2.4 (2,400kg/m³)

o Thermal expansion = 10x10-6

The models presented below picked as it was decided that in real world application

the configurations outlined below will be used in conjunction with stair cores, and

doors within a building.

4.1 Basic Shear Wall Model

A basic model was first created to which the parametric studies can be compared.

This model has openings in the middle of the shear wall and allows for the

conception of parametric studies both in the form of changing the position of the

openings and the size of the openings.

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4.2 Daniel Wilkinson Model (DW)

The first parametric study focused on the position of the shear wall openings. This

design allows for a stair core adjacent to the shear wall, and was considered to be a

practically relevant layout for shear wall openings. For the parametric study, the

openings were moved from opposing sides at 0.5m intervals to map the maximum

stresses throughout the shear wall. This lead to the modelling of the following

positions of the shear wall openings:

These models were then reversed to give the stresses of the openings on the

opposite side of the shear wall. These models provide a full spectrum of results

across the shear wall and so the pattern of stresses as a result of the position of the

openings may be found and an optimum spacing and position for the shear wall

openings may be found.

4.3 Yu Tam Model (YT)

The second parametric study allows for a column of shear wall openings (such as

might be found in a building where the lift and stair core are both at one end of the

building but on either side of the shear wall). It was decided that the same number of

Figure 2: DW parametric study models.

Figure 1: Basic shear wall model with dimensions and FEA model.

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models would be made as the previous parametric study and spacing of 0.25m at

the very edge and 0.5m intervals from then on.

As with the previous parametric study, these models were then mirrored to obtain the

stress values from the other side of the centreline to allow for a full spectrum of

stresses across the shear wall.

4.4 Lawrence Taylor Model (LT)

The parametric study below experiments with the size of the openings in the shear

wall. Five models have been produced whereby the openings were both increased in

height (decreasing the size of the interconnecting beams) and decreased in height to

study the effect that this will have on the stresses produced in the structure and the

maximum deflection of the structure.

Figure 3: YT parametric study models.

Figure 4: LT parametric study models.

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5.0 Validation of Analytical Models

5.1 Frame Model

The basic frame model of the shear wall was created using the rigid arm method.

The rigid arm method was chosen as this gave a more accurate representation of

the stresses involved in the shear wall as discussed by McLeod (1977).

The rigid arm model used stiff connections between the columns and beams,

simulating the presence of a solid shear wall to give a much more accurate result

when determining the shear force in the connecting beams. This was achieved by

increasing the depth of the element from (factor of 20) 3m to 60m whilst keeping the

thickness of 300mm. Fixed supports were used in the wire frame as the model is 2D.

The loading on the side of the frame was implemented at the connecting beam

nodes. The assumption made that the shear wall would be subject to 87.302kN/m2 of

wind loading and therefore point loads of 100kN and 50kN were applied to the frame

models to simulate this.

5.2 Plane Stress Model

The full model was constructed using plane stress plate elements to allow for the

validation of the approximate method using the rigid arm method and for the visual

representation of the stress. The model used 4 divisions per metre in order to create

a mesh that would give satisfactory results but not require a large degree of

computational space. The frame model will be validated by hand in the following

sections to ensure that the models are working correctly in LUSAS. This frame

model can then be used to validate the plane stress model.

Figure 5: Frame model showing concentrated loading.

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6.0 Results of Plane Stress Models

Table 1: Results of the basic plane stress model:

Table 2: Results of DW’s parametric study:

Table 3: Results of YT’s parametric study:

Max. Stress,

σ (kN/m²)

Min. Stress,

σ (kN/m²)

Max. Deflection,

δ (mm)

Level at which Max.

Stress Occurs

2951 -3346 3.114 2nd Floor Beam

Basic Model - Plane Stress Model

Distance

from LHS

Max. Stress,

σ (kN/m²)

Min. Stress,

σ (kN/m²)

Max. Deflection

(x-axis), δ (mm)

Level at which Max.

Stress Occurs

0.25m 1653 -2677 4.39 1st Floor

0.5m 1337 -1878 3.57 1st Floor

1m 1068 -1635 2.86 1st Floor

1.5m 931 -1467 2.51 2nd Floor

2m 1007 -1471 2.40 1st Floor

2.5m 2291 -2420 2.67 2nd Floor

3m 2951 -3346 3.08 2nd Floor

3.5m 2142 -2598 2.68 2nd Floor

4m 1205 -1281 2.41 2nd Floor

4.5m 1203 -1246 2.52 1st Floor

5m 1384 -1495 2.86 1st Floor

5.5m 1637 -2000 3.57 1st Floor

6.5m 2015 -2712 4.39 1st Floor

Daniel Wilkinson - Plane Stress Model

Distance

from LHS

Max. Stress,

σ (kN/m²)

Min. Stress,

σ (kN/m²)

Max. Deflection

(x-axis), δ (mm)

Level at which Max.

Stress Occurs

0.25m 1120 -2051 3.746 2nd Floor Beam

0.5m 1340 -2246 3.462 2nd Floor Beam

1m 1659 -2531 3.144 2nd Floor Beam

1.5m 2057 -2872 3.054 2nd Floor Beam

2m 2448 -3182 3.046 2nd Floor Beam

2.5m 2767 -3354 3.066 2nd Floor Beam

3m 2951 -3346 3.078 2nd Floor Beam

3.5m 2959 -3165 3.072 2nd Floor Beam

4m 2788 -2849 3.055 2nd Floor Beam

4.5m 2483 -2459 3.059 2nd Floor Beam

5m 2143 -2057 3.142 2nd Floor Beam

5.5m 1855 -1728 3.438 2nd Floor Beam

5.75m 1648 -1496 3.691 2nd Floor Beam

Yu Tam - Plane Stress Model

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Table 4: Results of LT’s parametric study:

7.0 Verification of computer results

7.1 Frame model & Free Body Diagram method of verification

To verify the results of the plane stress models, several methods of verification were

employed. The first method of verification involved the use of the frame method as

laid out above. Frames were made of each of the plane stress models by applying

rigid arms to the beam supports to simulate the effect that this will have on the

beams. From this concentrated loads were applied to the frame model and the

vertical shear force in the beams was calculated. From this using the simple beam

equation (Equation 2), a value of the stress can be found by multiplying the shear

value by 1m (as all the models have openings have a width of 2m and the PoC are

assumed to be in the centre of the openings) to produce a moment. From this a

value for the stress can be calculated and used to verify the plane stress models.

Another analysis used (only on the basic model) was the Free Body Diagram method

of verification as presented below. The free body diagram was taken about the

second floor to verify the maximum shear value obtained there in the LUSAS model.

Points of contra flexure were assumed to be half way up

columns to allow from the calculation of reaction and

therefore the shear at that level. The following calculations

were carried out on the simple frame model:

−(100 × 1.75) − (100 × 5.25) − (100 × 8.75) − (100 × 12.25) + 5𝑉𝐵 = 0

5𝑉𝑏 = 2187.5𝑘𝑁

𝑉𝑏 = 437.5𝑘𝑁

−(100 × 1.75) − (100 × 5.25) − (100 × 8.75) − (100 × 12.25) − (50 × 15.75) + 5𝑉𝐵 = 0

5𝑉𝑏 = 3587.5𝑘𝑁

𝑉𝑏 = 717.5𝑘𝑁

Beam DepthMax. Stress,

σ (kN/m²)

Min. Stress,

σ (kN/m²)

Max. Deflection

(x-axis), δ (mm)

Level at which Max.

Stress Occurs

1.5m 1774 -2062 2.474 2nd Floor Beam

1.25m 2239 -2572 2.698 2nd Floor Beam

1m 2951 -3346 3.114 2nd Floor Beam

0.75m 4077 -4569 3.825 2nd Floor Beam

0.5m 6128 -6826 5.650 3rd Floor Beam

Lawrence Taylor - Plane Stress Model

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So the approximate value for shear at second floor level is:

717.5𝑘𝑁 − 437.5𝑘𝑁 = 280𝑘𝑁

This relates to a shear value determined from the model of 187.9kN of shear at the

second floor level utilising the frame method. This demonstrates, that although the

values have a discrepancy, that the values are in the correct order of magnitude.

To verify the plane stress model was working to the correct level of accuracy, the

rigid arm model was used to check the results given.

The plane stress model of the shear slab with openings gave a maximum stress

value of:

𝜎 = 3346𝑘𝑁/𝑚²

As show in the diagram below:

The shear value is obtained from the basic wire frame model and compared to the

plane stress model by converting the shear values into the relative stress in the

shear wall simulation.

Figure 6: Frame model and plane stress model showing maximum stresses.

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For the rigid arm model, the I value used for the beam simulation is as follows

(Equation 1):

𝐼 =𝑏𝑑3

12

𝐼 =0.3 × 13

12

𝐼 = 0.025𝑚4

The maximum shear was detected at the second floor level with a value of 187.9kN

when the frame was subjected to a wind loading of 1.25kN/m².

𝜎 =𝑀𝑦

𝐼

𝜎 =1 × 187.9 × 0.5

0.025

𝜎 = 3758𝑘𝑁/𝑚²

When compared to the plane stress model, the maximum value for stress is:

𝜎 = 3346𝑘𝑁/𝑚²

This shows extremely good correlation between the basic wire model and the plane

stress model.

When non rigid example of this frame is compared, the result is as follows:

𝜎 =𝑀𝑦

𝐼

𝜎 =1 × 98.39 × 0.5

0.025

𝜎 = 1968𝑘𝑁/𝑚²

This is compared to both the rigid arm value of 3758kN/m² and the plane stress

model of 3346kN/m². I can be seen that the non-rigid frame is underestimating the

stresses by approximately 50%. This supports the conclusion made by McLeod

(1977) that a rigid arm model gives a much closer approximation to the stresses

involved in a shear wall than a non-rigid model.

7.2 Wall-Beam Junction Models

Alongside the use of frame model for verification a partial plane stress model of the structure

was also created (Figure 7). This is used to both verify the full model’s maximum stress

results and to see how accurate a model of this type can be.

(1)

(2)

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The models employed used a plane stress element type with pinned supports at the top and

base of the model. The force applied on the beam was a face load corresponding to that of

the shear force found in the frame models (as described above). From this the shear force

was divided by the thickness of the shear wall and by the depth of the beam itself to provide

a face load in kN/m2.

A summary of the verified results are presented in the tables below:

Table 5: Basic Model plane stress and verification results:

Table 6: DW plane stress results without inclusion of verification methods:

Wall-Beam Junction Free Body Diagram

Max. Stress,

σ (kN/m²)

Min. Stress,

σ (kN/m²)

Max. Deflection,

δ (mm)

Level at which Max.

Stress OccursShear (kN)

Max. Stress,

σ (kN/m²)

Max. Stress, σ

(kN/m²)Shear (kN)

2951 -3346 3.114 2nd Floor Beam 187.9 3758 4385 280.0

Frame Model

Basic Model

Plane Stress Model

Wall-Beam Junction

Distance

from LHS

Max. Stress,

σ (kN/m²)

Min. Stress,

σ (kN/m²)

Max. Deflection

(x-axis), δ (mm)

Level at which Max.

Stress OccursShear (kN)

Max. Stress,

σ (kN/m²)

Max. Stress, σ

(kN/m²)

0.25m 1653 -2677 4.39 1st Floor - - -

0.5m 1337 -1878 3.57 1st Floor - - -

1m 1068 -1635 2.86 1st Floor - - -

1.5m 931 -1467 2.51 2nd Floor - - -

2m 1007 -1471 2.40 1st Floor - - -

2.5m 2291 -2420 2.67 2nd Floor - - -

3m 2951 -3346 3.08 2nd Floor - - -

3.5m 2142 -2598 2.68 2nd Floor - - -

4m 1205 -1281 2.41 2nd Floor - - -

4.5m 1203 -1246 2.52 1st Floor - - -

5m 1384 -1495 2.86 1st Floor - - -

5.5m 1637 -2000 3.57 1st Floor - - -

6.5m 2015 -2712 4.39 1st Floor - - -

Frame Model

Daniel Wilkinson

Plane Stress Model

Figure 7: Wall beam connection contour plot.

14

Unfortunately, due to the layout of the openings on DW’s models, frame models and

wall beam models for verification could not be produced.

Table 7: YT plane stress and verification results:

Table 8: LT plane stress and verification results:

8.0 Discussions and observations

For the approximate analysis of a shear wall with openings in the centre, it has been

determined that using a rigid arm model is an effective way of quickly determining

the shear and so the stresses in the tie beam. There is close agreement with the

values given by the plane stress (3346kN/m²) and the rigid arm model (3758kN/m²)

suggesting that where there is a lack of computational power, a simple frame model

can be used and furthermore, it is simple enough to be able to validate by hand

using simple approximate analysis methods (as highlighted in Section 7.1). The

agreement between the free body diagram hand calculation and frame model shear

forces within the second floor tie beams differs to the extent that the hand

Wall-Beam Junction

Beam DepthMax. Stress,

σ (kN/m²)

Min. Stress,

σ (kN/m²)

Max. Deflection

(x-axis), δ (mm)

Level at which Max.

Stress OccursShear (kN)

Max. Stress,

σ (kN/m²)

Max. Stress, σ

(kN/m²)

1.5m 1774 -2062 2.474 2nd Floor Beam 216.7 1926 2405

1.25m 2239 -2572 2.698 2nd Floor Beam 206.3 2641 3138

1m 2951 -3346 3.114 2nd Floor Beam 187.9 3758 4385

0.75m 4077 -4569 3.825 2nd Floor Beam 155.4 5525 5908

0.5m 6128 -6826 5.650 3rd Floor Beam 104.6 8368 8132

Lawrence Taylor Model

Frame ModelPlane Stress Model

Wall-Beam Junction

Distance

from LHS

Max. Stress,

σ (kN/m²)

Min. Stress,

σ (kN/m²)

Max. Deflection

(x-axis), δ (mm)

Level at which Max.

Stress OccursShear (kN)

Max. Stress,

σ (kN/m²)

Max. Stress, σ

(kN/m²)

0.25m 1120 -2051 3.746 2nd Floor Beam 185.6 3712 10513

0.5m 1340 -2246 3.462 2nd Floor Beam 186.0 3720 4688

1m 1659 -2531 3.144 2nd Floor Beam 186.6 3732 4378

1.5m 2057 -2872 3.054 2nd Floor Beam 187.2 3744 4371

2m 2448 -3182 3.046 2nd Floor Beam 187.5 3750 4376

2.5m 2767 -3354 3.066 2nd Floor Beam 187.8 3756 4383

3m 2951 -3346 3.078 2nd Floor Beam 187.9 3758 4385

3.5m 2959 -3165 3.072 2nd Floor Beam 187.8 3756 4383

4m 2788 -2849 3.055 2nd Floor Beam 187.6 3752 4376

4.5m 2483 -2459 3.059 2nd Floor Beam 187.2 3744 4371

5m 2143 -2057 3.142 2nd Floor Beam 186.7 3734 4378

5.5m 1855 -1728 3.438 2nd Floor Beam 186.1 3722 4688

5.75m 1648 -1496 3.691 2nd Floor Beam 185.7 3714 10513

Frame Model

Yu Tam

Plane Stress Model

15

calculations make an overestimate of the shear, but this result shows that the

magnitude of the shear is in the right region and with values that were expected.

8.1 Maximum Stress Observations

From the extensive modelling of the shear wall with both the rigid frame model and

plane stress with different opening configurations, it has become clear that the

maximum shear occurs on the second floor of the building, with the exception of

DW’s models where it occurs on the first and second floor, and so this is where the

reinforcement must be concentrated to avoid failure in the structure.

The location of the maximum stresses at each of the openings is concentrated

slightly in from the corner of the openings (Figure 8) suggesting that although

diagonal reinforcement will be needed in the corners of the openings to mitigate

propagation of cracks, that horizontal reinforcement is equally important around the

areas of concentrated stresses. It was also observed in almost all of the models that

the maximum stress occurred at the second floor level (also shown in the tabulated

results above). Interestingly, Table 8 highlights how closely the frame method of

verification can model the plane stress model. Highlighted in the table in blue is the

result of where the maximum stress occurs in the plane stress model and the

maximum shear in the frame model. As the maximum stress jumps to the third floor

level so does the maximum shear in the plane stress model.

8.2 Stress changes in the Wall-Beam Junction Models

In addition to the simple rigid arm frame model, a model concentrating on the beam-

wall connection can also be used to measure the stress and determine its location.

This model provided a maximum shear value of 4385kN/m² compared to the full

plane stress model which provided 3346kN/m². Again, although this value differs

from the full model, it gives an overestimate of the stresses involved in the shear wall

connections and in practice can prove very useful if there is a limit on computational

power. This is because the full model requires a very fine mesh to allow for an

accurate value for the stress and with a smaller beam-wall connection model, the

number of nodes in that mesh is decreased dramatically. However as highlighted in

Table 7, it is observed that the stresses calculated by the Wall-Beam Junction

models in the parametric study do not produce results similar to those of the full

model. The results of the Wall-Beam Junction model is accurate when YT’s model is

Figure 8: Frame stress model showing the position of maximum stresses in the models.

16

symmetrical in the vertical axis (i.e. the same as the Basic Models configuration) but

as the models openings are moved closer to the edges of the shear wall, the Wall-

Beam Junction model so not reflect the change in stresses.

Also observed in YT’s models is that the stresses at distances 0.25m and 5.75m

from the left hand side off the shear wall show major increases in the stress

developed. It is theorised that the major increase in stress is due to the decrease in

stiffness of the shear wall resulting in a major increase in the bending of the element

and the minor decrease in the shear from the frame model creating a very large

force on a small structure.

8.3 Placement of shear wall openings to reduce maximum developed

stress in structures

Although the concentration of stresses occurs at the same position in all of the

models created, the magnitude of the stresses decreases depending on the

configuration of the openings. It was found that the model with the highest stresses

was the original basic model with the openings in the centre of the shear wall. It can

be seen that as the shear wall openings move away from the centre, the maximum

stresses decrease both in YT’s and DW’s models. However, stresses do increase

again as the openings approach the outer walls. This suggests that optimum

locations for shear wall openings are off centre. However this also comes with issues

to do with analytical modelling and solutions. It was found with YT’s model that the

rigid arm model gave shear values that were similar across the entire length of the

shear wall suggesting that this type of modelling cannot be used in this situation as

the rigid arms allow little difference in shear through the connecting beam. This

investigation stemmed from the advice from Dr. Rafiq that to create rigid arm model

for DW’s configuration would prove too difficult to model. So it was decided that for

the interest of the report that it could be made for YT’s model to investigate the

limitations of this type of modelling.

For DW’s model, the configuration was designed to replicate the situation where a

stairwell might be against the shear wall on the inside and windows may be installed

in the shear wall to coincide with landings on the stairs.

17

The figure above shows the contour plot of the 1m from the LHS model. Compared

to the basic model, the stresses are shared out over a large area, but it is noted that

instead of the stresses peaking on the second floor as with the basic model, the

maximum stresses can be found at first floor level.

The variation in stresses can clearly be seen in the graph above. As the openings

move away from the centre, the stresses decrease instantly, showing that there are

more efficient configurations for the openings within the shear wall to limit stresses

and therefore cracking.

0

500

1000

1500

2000

2500

3000

3500

4000

0 1 2 3 4 5 6 7

Str

esses (

kN

/m²)

Distance from LHS (m)

Graph showing Calculated stresses in all models (DW)

Maximum Stresses

Minimum Stresses

Figure 9: Contour plot of DW’s 1m from LHS model.

Figure 10: Graph to show calculated stresses in all models (LT)

18

The model undertaken by YT moved the column of openings across the shear wall in

one block, thus showing the effect of only moving position rather than the size of the

connecting beams and shows a similar characteristic.

0

2000

4000

6000

8000

10000

12000

0 1 2 3 4 5 6 7

Str

esses (

kN

/m²)

Distance From LHS (m)

Graph showing calculated stresses in all models (YT)

Maximum Stresses

Minimum Stresses

Wall Beam Junction

Frame Model

Figure 11: Graph to show calculated stresses in all models (LT)

Figure 12: Section of stresses through the shear wall (YT)

19

The LUSAS modelling result revealed the series openings in the shear walls has an

effect on the top displacement of the buildings and is related to the opening

arrangement configuration. The position of the openings in relation to the solid shear

wall column location has an effect on the base shear distributions in the columns.

The adjacent columns to the openings possess base shear bigger than those in the

columns away from openings. The results showed high values of the stresses

around the openings regardless of the arrangement system of openings. However,

the accompanying increase of stresses in the staggered system of openings is small

related to the corresponding in the other configurations of openings.

When the two configurations (DW & YT) are compared, (Graph below) YT’s model

shows a much more hyperbolic pattern but the stresses still peak at the same central

configuration as DW’s model and the basic model.

0

500

1000

1500

2000

2500

3000

3500

4000

0 1 2 3 4 5 6 7 8

Str

ess (

kN

/m²)

Position of Shear Wall Opening centres (m)

Chart to show the effect of the positioning of openings on the stress expereinced in the shear wall.

DW

YT

Figure 14: Graph showing maximum stress observed as position of openings is changed.

Figure 13: Section of stresses through one shear wall opening.

20

The difference in graphs could be a result of the in connection beam. In YT’s model,

the beam depth does not change, only its location within the shear wall whereas in

DW’s model, as the shear openings move away from each other, the connection

beam suddenly disappears and becomes a large piece of shear wall until the next

opening the next floor down. Another explanation for this phenomenon is explained

in the mode of deformation section below.

In Figure 15 above, it can be seen that in LT’s models the maximum stress occurs

when the thickness of the interconnecting beams is at its thinnest and that the

minimum stress occurs when the interconnecting beams are at their thickest. This

correlates with the theory that approximately the same amount of force needs to be

transmitted through the interconnecting beams in each model but the area that it

must be transmitted through is increasing and decreasing resulting in decreasing

and increasing stress development respectively.

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0.4 0.6 0.8 1 1.2 1.4 1.6

Str

esses (

kN

/m²)

Beam Depth (m)

Graph showing calculated stresses in all models (LT)

Maximum Stress

Minimum Stress

Wall Beam Junction

Frame Model

Figure 15: Frame stress model showing the position of maximum stresses in the models.

21

8.4 Mode of deformation of shear walls

As discussed in the previous section, changing the configuration of the openings has

a marked effect on the stresses and where they occur. Another reason for the

change in stresses the mode of deformation in the shear wall. In the figure above are

visual examples of the deformation of each of the shear wall configurations for the

YT and DW models. The configuration with the openings in the middle is

demonstrating, albeit slight, a rigid frame deformation mode whereas the two

differing configurations are showing shear wall deformation. It is theorised that the

shear deformation mode means that the shear wall acts in a similar way to a deep

beam by which stresses are more concentrated towards the bottom of the shear wall.

With rigid frame deformation, stresses are more likely to propagate further up the

shear wall. In DW’s model, this can be seen from the fact that the stresses were

concentrated around the first floor rather than the second as is the case with the

other configurations.

8.5 Comparison of plane stress models, frame models and Wall-Beam

Junction models

As noted in Eurocode 2 (British Standards Institution, 2014), the combined axial and

shear force should be taken in to account and so for a thorough analytical model, the

vertical forces from imposed and dead loads would also need to be included.

However, as the models in this report are unconnected to the rest of the structure,

axial loading either by imposed of self-weight loads has not been added. However,

as noted by McLeod and Green (1973) issues arise from including the axial force

within analytical models as this force manifests itself throughout the shear wall in an

arc formation. When the openings are centralised, the connecting beam coincides

with the maximum on this arc, but if the openings are off centre (As with YT & DW), a

secondary arc forms and the maximum of this arc is actually within the shear wall. In

this case additional terms must be included to the shear calculation to ensure that an

accurate stress value is reached. It is therefore theorised that this is the reasoning as

Figure 16: Deformation mode of the structures.

22

to why YT’s models of moving the shear wall openings from the centre line to the

edges will create inaccurate frame models.

Also discussed by MacLeod and Green (1973) is that in the frame models, the axial

deformation for the columns must be considered. In LT’s models (as shown in Figure

15) good correlation is found between all of the plane stress models and the frame

models. When compared to YT’s models (Figure 11) it can be seen that MacLeod

and Green’s (1973) frame analysis is much better at verifying the plane stress

computer results. It should however be noted that although there is not much

correlation between the verification results and the plane stress analysis, the

verification results of the frame and wall-beam junction models provide an

overestimate in all circumstances. Therefore it is found that the models are safe to

use but may not provide a realistic result.

8.6 Stiffness of the shear wall models

The plane stress models used as outlined in the assumptions in Section 3 mean that

the models are linearly elastic. It is therefore inconsequential what load is applied to

the structures, only that the loading is the same in all of the models. All of the plane

stress models used above utilised the same loading of 87.302kN/m2 on the side of

the shear wall. The stiffness of the structure does however change as the openings

in the walls are moved. Because the force on the side of the building stays the same

and only the deflection changes the stiffest shear wall configurations can be found. It

can be seen from Table 6 that DW’s minimum deflection occurs at 2m from the left

hand edge, YT’s models deflection increases as the openings are moved towards

the edges of the structure and LT’s minimum deflection is when the interconnecting

beams are at their thickest.

9.0 Summary of design implications

Being able to determine where maximum stresses present themselves allows for the

design of reinforcement to prevent cracking. As with the openings in the shear walls,

it has been determined that cracking is most likely to occur just next to the corners of

shear wall openings and as such, adequate diagonal reinforcement would need to be

defined for that area as well as horizontal reinforcement. The study has involved a

shear wall with six openings placed in various positions and it can be seen that the

most stress occurs in the bottom third of the shear wall. The engineer must therefore

look at the lower portion of the wall to find the maximum stress for design of the wall

section.

As written in the literature review a plane stress model can only model forces acting

along the thickness of the plate. It cannot therefore model the forces acting

perpendicular to the plane which would occur when the shear wall is attached to a

structure. The engineer must therefore apply factors of safety to account for these

23

unknown stresses which may develop in the structure. However, this is out of the

scope of this report.

It is of paramount importance that the deflection of the structure is known for

serviceability requirements. The models show how much of an impact the positioning

of the openings can change this. The deflection and therefore the stiffness of the

structure increases as the thickness of the beams increase. The deflection increases

as the openings are moved to the edge of the structure and DW’s model shows the

least deflection occurs when the openings are only slightly moved apart.

The approximate analysis methods are generally very good at predicting the

response of the structures but not in all situations. It is therefore a pitfall of the

approximate analysis method because an engineer using the frame and wall-beam

junction models may not understand that the frame models usefulness declines as a

vertical line of shear walls is move closure to the edges.

Eurocode 2 (British Standards Institution, 2014) does not give any advice or

instructions as to the design of shear walls with openings and how to analyse the

structures using approximate analysis. This is therefore a limitation of the code of

practice.

10.0 Conclusions

In conclusion, the maximum stress in the link beams for the central openings in the

shear wall is -3346kN/m². This value was found to occur at second floor level and

just next to the corner of the opening. It can be concluded from this data that when

designing a shear wall, extra attention to detail is needed in the specification of

reinforcement for the area surrounding the shear wall openings.

The parametric study set out to determine what effect the configuration and size of

the shear wall openings had on the stresses experienced within the shear wall. It has

been determined that these factors have a marked effect on the maximum stresses

in the wall and it has been shown that there are more efficient ways to layout the

openings in the shear wall than the original with openings down the centre. Both

DW’s and YT’s models show a decrease in stresses when the openings are moved

off centre and LT’s model shows that stresses increase as the link beam depth

decreases. This conclusion being supported by the theory that the same magnitude

of force is being taken through these link beams, but there is less area for it to act

through, and as the equation: 𝜎 =𝑀𝑦

𝐼 shows, the smaller the area of the link beam,

the larger the stresses experienced are.

It has been seen that there are difficulties in (approximate analysis) modelling some

of the more complex shear wall models such as DW’s but the basic model with the

centralised openings can be easily modelled and validated using hand calculations.

24

A number of ways of conserving computational space (Frame model, Wall-Beam

connection) have been explored in this report and have proved effective in the

modelling of the central shear wall configuration/size but become less effective when

used in conjunction with changing the position of the shear wall openings. Also

analysed was the effect that the positioning of the openings has on the maximum

deflection of the shear walls.

25

References

British Standards Institution (2014) BS EN 1992-1-1:2004: Eurocode 2: Design of

concrete structures. London: British Standards Institution.

Chen, S. and Kabeyasawa, T. (2000) ‘Modelling of Reinforced Concrete Shear Wall

for Nonlinear Analysis’, 12th World Conference on Earthquake Engineering, New

Zealand National Society for Earthquake Engineering, Auckland, 30 January to 4

February 2000. IIT KANPUR [Online]. Available at:

www.iitk.ac.in/nicee/wcee/article/1596.pdf (Accessed: 24 October 2015).

MacLeod, I. (1977) ‘Structural analysis of wall systems’, The Structural Engineer,

55(11), pp. 487-495.

MacLeod, I. and Green, D. (1973) ‘Frame idealization for shear wall support

systems’, The Structural Engineer, 51(2), pp. 71-75.

MacLeod, I. and Hosny, H. (1976) ‘The distribution of vertical load in shear wall

buildings’, The Structural Engineer, 54(2), pp. 67-71.

Oztorun, N. K., Citipitioglu, E. and Akkas, N. (1998) ‘Three-dimensional finite

element analysis of shear wall buildings’, Computers & Structures, 68(2), pp. 41-55

[Online]. Available at:

http://www.sciencedirect.com/science/article/pii/S0045794998000200 (Accessed: 25

October 2015).

Rafiq, Y. (2015a) Distribution of lateral load to walls and frames [Lecture to MEng

and MSc Civil Engineering Year 4], STAD505: Advanced Structural Engineering.

Plymouth University. 28 September.

Rafiq, Y. (2015b) Deep Beam and Shear Wall with Opening [PowerPoint

presentation]. STAD505: Advanced Structural Engineering. Available at:

https://dle.plymouth.ac.uk/ (Accessed: 24 October 2015).

University of California, Santa Barbara (no date) Engineering. Available at:

www.engineering.ucsb.edu/~hpscicom/projects/stress/introge.pdf (Accessed: 23

October 2015).

26

Appendices

Appendix A – CD containing all models used within the report.