Thesis on Concrete Structural Design for Sustainability (Optimising Structural Form) 2013

Concrete Structural Design for Sustainability in

Residential and Small Commercial Buildings

“How can ‘structural form’ contribute to better solutions?”

Department of Civil Engineering

Prepared for:

Prof Mark Alexander

Mr Vernon Collis

Prepared by:

Koketšo Moyaba (MYBKOK001)

Course:

CIV4044S

Submission Date:

11 November 2013

i

Plagiarism Declaration

I know that plagiarism is wrong. Plagiarism is to use another’s work and to pretend that it

is one’s own.

I have used the Harvard convention for citation and referencing. Each contribution to,

and quotation in, this thesis from the work or works of other people has been attributed,

and has been cited and referenced.

This thesis is my own work.

I have not allowed, and will not allow, anyone to copy my work with the intension of

passing it off as his or her own work.

Student Number Name Signature

MYBKOK001 Moyaba, Koketšo

ii

Abstract

Concrete structures have been designed throughout history by paying attention to their form.

Before the invention of reinforced concrete, it was imperative to design a concrete structure that

took mainly if not entirely compressive loads. This was done to avoid the relative weakness of

earlier building materials, including unreinforced concrete under tensile loads. These form-active

concrete structures are potentially the most efficient concrete structural components with regard

to their load carrying capacity in relation to their weight. Form-active design is more complex

since it requires an understanding of the shape the concrete structure would take under a

particular load if it had no bending stiffness, i.e. if it were to behave like a cable. Due to the high

structural efficiency, form-active structures play an important role with regard to sustainability.

They use less material to achieve higher load carrying capacities, and therefore they reduce the

use of natural resources.

This thesis focuses on concrete floor slabs in residential buildings and small commercial

buildings. The aim is to study the behaviour of these slabs under loading and analyse how

structural form considerations can lead to better design solutions. These structural components

were chosen due to their significant contribution to the construction of both residential and small

commercial buildings. However, this choice imposes certain restrictions in the freedom of

designing form-active structures; i.e. dimensional restriction such as depth of the slab, flatness of

the slab on the top surface, etc. These restrictions are important since they determine whether the

slab will be able to correctly perform its function.

The analytical methodology involved designing a set of traditional solid slabs and improving

their designs according to material and cost optimisations. These slabs were then compared with

form active slabs which were designed to take the parabolic shape of their bending moment

diagrams. From an analysis of the designs, it was found that form active slabs are approximately

two times more efficient than the traditional solid slab. This efficiency is with regard to load

carrying capacity and the amount of material used, both concrete and steel. The reduction in the

concrete used for a form-active slab with the same span as a traditional solid slab, can be

approximately 60% and that of steel can be approximately 70%. These values are a clear

indication that the consideration of structural form can lead to better and sustainable solutions.

One way spanning slabs can be applied to both residential and small commercial buildings since

these buildings are commonly constructed as post-and-beam structures, with discontinuous

joints. Furthermore these buildings have significantly lower and more predictable loads than

other types of buildings. This helps with regard to avoiding failure by unexpected excessive

point loads. Larger commercial and institutional buildings such as malls, hospitals, etc. can also

adopt the application of one way spanning form-active slabs. This is because of the common

attribute most of these buildings have, which is large hallways and corridors, which are suitable

application areas for these types of slabs. This thesis has in this regard successfully shown the

economical and sustainable advantages of concrete structural design through the consideration of

structural form by investigating the special case of one way form-active slabs.

iii

Acknowledgements

I would like to extend my gratitude to the following people who were helpful in the compiling of

this thesis:

Prof Mark Alexander:

For the excellent supervision throughout the entire project, and equally important I appreciate

the guidance which kept me on track in all respects of the research; this includes problem

solving methodologies, thorough checking on progress and the writing of this document, and

useful insight on concrete properties related to the problem statement posed by this thesis, thank

you.

Mr Vernon Collis, PrEng:

Thanks are due for providing insight on the topic of form-active structures and pointing to the

excellent resources from which the bulk of this thesis is based. Moreover, I appreciate the

guidance with regard to the practical design of form-active slabs, including that of structures that

comes with both experience and passion, ke a leboga.

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Table of Contents

Plagiarism Declaration i

Abstract ii

Acknowledgements iii

Table of Contents iv

List of Figures vii

List of Tables viii

List of Equations viii

Glossary of terms and abbreviations ix

1 Introduction 1

1.1 Background 1

1.2 Problem Statement 2

1.3 Scope of Thesis 2

2 Motivation for Study 4

2.1 Introduction 4

2.2 Advances in Concrete Slab Design in the Last Century 4

2.2.1 Hollow Core Slabs 4

2.2.2 Bubble Deck Slabs 4

2.2.3 Holedeck Slabs 5

2.3 Project Proposal 8

2.3.1 Objectives of Thesis 8

2.3.2 Expected Outcome of Research 8

2.3.3 Strategy 8

2.3.4 Computer Aided Modelling 9

2.4 Closure 9

3 Literature Review – General Overview 10

3.1 Introduction 10

3.2 Properties of Concrete 10

3.3 Concrete Structures throughout History 11

3.4 Traditional Concrete Slab Design Process 14

3.4.1 Ultimate Limit State Design 14

3.4.2 Serviceability Design 15

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3.5 Concrete Slab Design for Suspended Slabs 16

3.5.1 Inherent Inefficiencies in Solid Slab Design 17

3.5.2 Advantages of Solid Slab Design 18

3.6 Concrete Slab Design for ‘Slab-on-Grade’ 19

3.7 Sustainable Design 20

3.8 Closure - Main Findings from Literature 21

4 Design of Form-Active Structures 22

4.1 Introduction 22

4.2 Rigidity of the Element 22

4.3 Influence of Form-Active Shape 23

4.4 Influence of Load Pattern 24

4.4.1 Point Loads 24

4.4.2 Uniformly distributed Loads 24

4.5 Degree of Form-Active Elements 25

4.6 Structural Efficiency 27

4.7 Restrictions in Designing Form-active Slabs 27

4.7.1 Comparison of Continuous and Discontinuous Structures 27

4.8 Closure 29

5 Analytical Methodology 30

5.1 Introduction 30

5.2 Choice of Load Analysis Method 30

5.3 Traditional Solid Slab Design 31

5.3.1 Optimisation of the Solid Slab Design 32

5.3.2 Structural Efficiencies 34

5.4 Form-Active Slab Design 34

5.4.1 Roof Slab Design 35

5.4.2 Floor Slab Design 37

5.4.3 Comparison of Form-active Shapes 37

5.4.4 Structural Efficiencies 38

5.5 Closure 38

6 Discussion of Results 39

6.1 Introduction 39

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6.2 Comparison of Structural Efficiencies 39

6.3 The Effect of Increasing the Concrete Crushing Strength fcu 41

6.4 Comparisons from FEA Models 42

6.4.1 Setting up of Abaqus Models 42

6.4.2 Discussion of FEA Results 44

6.4.3 Failure Modes of Form-active Slabs 48

6.5 Closure 49

7 Conclusions 50

7.1 Specific Conclusions 50

7.2 General Conclusions 52

8 References 53

Appendices 55

Appendix A - Solid Slab Design Calculations 56

Appendix B - Load Analysis 61

Appendix C - Optimisation of Solid Slabs, Slab A and Slab D 65

Appendix D - Form-Active Slab Design 68

Appendix E - Drawings 72

vii

List of Figures

Figure 1: Bubble deck slab (Source: Structural Engineers, Sustainability and Leed®) ............................................... 5

Figure 2: Holedeck system at Logytel R+D building (Source: Holedeck the lean structure, online) ........................... 6

Figure 3: Rib form holedeck (Holedeck, the lean structure) ......................................................................................... 6

Figure 4: Structural efficiency chart, Macdonald ......................................................................................................... 7

Figure 5: Inside the Pantheon, Italy, Rome, (Source: Giovanni Paolo Pannini, Left), Cross-section and Plan of the

Pantheon (Right) ......................................................................................................................................................... 12

Figure 6:1935 Zarzuela Hippodrome by Eduardo Torroja (Source: International Database and Gallery of Structures,

Online) ........................................................................................................................................................................ 13

Figure 8: Actual and equivalent stress distribution at failure. (Source: SABS 0100-1: 4.3.3.3) ................................ 15

Figure 7: Cracked RC beam section, showing qualitatively the amount of concrete being ignored. .......................... 17

Figure 9: Tensile form-active shapes, (Top) and equivalent compressive form-active shapes, (Bottom) (Source

MacDonald) ................................................................................................................................................................ 24

Figure 10: Bending moment parabola plotted with arc passing through the same points ........................................... 27

Figure 11: Material optimisation graph of steel vs. concrete, Moyaba ....................................................................... 33

Figure 12: Cost optimisation graph, Moyaba ............................................................................................................. 34

Figure 13: Form-active slab stress distribution, Moyaba ............................................................................................ 36

Figure 14: Form-active floor slab ............................................................................................................................... 37

Figure 15: Example of form-active slab showing cross sectional areas for efficiency calculations, Moyaba ............ 38

Figure 16: Comparison of required concrete in Solid and Form-active slabs for different spans, Moyaba ................ 40

Figure 17: Comparison of required steel in Solid and Form-active slabs for different spans, Moyaba ...................... 41

Figure 18: Dimensions used for modelling the 1m strip form-active slab, Moyaba ................................................... 42

Figure 19: Solid Slab A model showing loading and end supports (left); Form-active Slab A showing loading and

end supports (right), Moyaba ...................................................................................................................................... 43

Figure 20: Von Misses contour plot for the 6m span solid slab fixed at both ends, Moyaba ..................................... 45

Figure 21: Element group of the mid-span of the solid slab in figure 20, Moyaba .................................................... 45

Figure 22: Isometric top view of form-active slab showing the von Mises contour plot, Moyaba ............................. 46

Figure 23: Top view of form-active slab showing the von Mises contour plot, Moyaba ........................................... 46

Figure 24: Isometric bottom view of the form-active slab showing the von Mises contour plot, Moyaba ................. 47

Figure 25: Cross section of the mid-span of the form-active slab showing the von Mises stresses, Moyaba ............. 47

Figure 26: Detail showing correct support connection and lateral support connection in form-active slabs, Collis .. 49

viii

List of Tables

Table 1: Comparison of Bending Moment Parabola and Arc of a Circle ................................................................... 26

Table 2: Comparison of Continuous and Discontinuous Structures, Moyaba, ref. MacDonald ................................. 28

Table 3: Typical span/depth ratios, (Cobb, 2004) ....................................................................................................... 31

Table 4: Summary of solid slab design (Appendix A), Moyaba................................................................................. 39

Table 5: Summary of form-active roof slab design (Appendix E), Moyaba ............................................................... 39

Table 6: Material reduction obtained by designing form-active slabs, Moyaba ......................................................... 40

List of Equations

Equation 1: Ultimate load equation, SANS 0100-1 .................................................................................................... 14

Equation 2: General equation of a parabola ................................................................................................................ 26

Equation 3: Implicit equation of a circle with its centre on the y-axis........................................................................ 26

Equation 4: Explicit equation of circle with respect to y ............................................................................................ 26

Equation 5: Structural Efficiency ............................................................................................................................... 27

Equation 6: Ratio of maximum Y value to minimum Y value for axes alignment ..................................................... 33

Equation 7: Fundamental defining equation of a moment force ................................................................................. 36

Equation 8: SI Equation of concrete's elasticity modulus based on empirical values, PCI ........................................ 43

Equation 9: von Mises stress equation ........................................................................................................................ 44

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Glossary of terms and abbreviations

Form-active – Referring to a structural element in which the shape of the longitudinal axis in relation to the load

applied results in purely axial internal forces only.

Form-active shape – This is the shape that corresponds to a particular loading pattern. This shape is brought about

by loading a structural element with no bending stiffness, the shape that the element takes is called the form-active

shape since it converts all the forces generated by the loading to axial forces alone and thus fully utilising the entire

cross section equally. Therefore the forma active shape can be understood as the most efficient shape.

Form-active structure – A structure whose shape under a particular load pattern, takes the form-active shape of

that load. This structure can also be called a fully form-active structure.

Semi form-active structure – This is a structure whose shape is similar to its load’s form-active shape in certain

parts. However, due to restrictions in geometry, it cannot take the true form-active shape possibly because of the

structure’s intended purpose.

Structural efficiency – The ratio of strength to weight of a structural element, whereby the efficiency is regarded as

high if the number is high.

Improved section/ ‘slab’ – These are structural sections or slab with an improved cross section and/or longitudinal

profile. This is done by distributing the material to locations where they are optimised.

Structural form – This is the shape or form that a structure will take once designed or constructed. It is directly

related to how efficient the structure will be, and this efficiency will depend on how similar the structural form is to

the form-active shape.

Stiffness – The stiffness of a structural element is the ability to resist certain types of forces without excessive

deformation, i.e. bending stiffness is the ability to resist bending type forces without excessive deformation of the

element.

Post-and-beam structures – These are structure with a typical framework configuration comprising of beams and

load bearing walls.

Fixity – The fixity of a structural configuration is the way in which a member is supported or connected to other.

The fixity can be pinned, fixed or hinged, i.e. fixity with lateral restraint, rigid fixity with rotation and lateral

restraint, connection with no moment transfer respectively.

Rigidity – This is ability of a structure or structural element to resist motion

FEA – Finite element analysis, this is the analysis of small elements of a structure to obtain the stress distributions

under applied loading

Von Mises Stresses – These are the resultant stresses of the principal and shear stresses of an element

1

Moyaba: Concrete design for sustainability (Structural form)

Introduction

1 Introduction

1.1 Background

According to the Royal Society of Chemistry, concrete is the single most widely used

material in the world (RSC, 2008). The journal continues to explain that because of

concrete’s widespread use, it comes without surprise that concrete also has a large carbon

footprint. Quantitatively the paper explains that the production of concrete globally, accounts

for about 5% of the annual global Carbon dioxide production from human activity. That is

mainly because of the vast quantities being used yearly. A largely significant part of the

concrete being produced is in architectural applications, be it residential buildings,

Commercial, and other Civil Engineering related structures.

Analysing various applications simultaneously might be a futile effort as patterns are less

clear to establish and the efforts in creating sustainable designs/construction may be diverted

and scattered. A focus has to be chosen wisely. Of all the applications of concrete, a common

one is small scale buildings, both residential and small commercial buildings. In South

Africa, over 50% of all cement sold goes to these fields of applications. Components made

from concrete fall strictly within the expertise of Structural Engineering. These components

are usually concrete columns, beams and slabs.

Concrete slabs account for a substantial amount of concrete volume with regard to concrete

components in residential and small scale commercial buildings, i.e. both ground floor slabs

and suspended floor slabs in multi-storey buildings. For residential and small commercial

buildings, the slab geometry is usually flat. The flat geometry coupled with the loading

generally being applied perpendicular to the slabs, results in internal bending forces causing

an uneven stress distribution within the cross section. As a result of the bending loads, some

parts of the cross section are under-stressed, thus the material is inefficiently used since not

all the weight of the concrete is fully utilised. This is discussed in detail in section 3 under

preliminary literature review when looking at the stress block diagram under a bending type

load.

Reinforcing of concrete components is done because concrete is strong under compression

and weak under tension. Since the focus in traditional slab design does not lie manly on the

longitudinal geometry of the slab, significant tensile forces arise. In South Africa, concrete

slabs are usually designed according to the code as flat slabs or slabs with mass reduction

incorporated into the design. The differences in design procedures in various countries or

regions of countries are benign. They are mainly imposed by climates with regard to concrete

durability, safety factors and other analysis procedures; however, the concepts remain related

and aren’t mostly concerned with creating form-active slabs.

Taking into account the structural form in slabs not only looks at the geometry of the slab but

also the shape which it would take if it had no rigidity under a particular loading pattern, that

is in essence being a form-active slab, and that is where this research is focused (Macdonald,

2002, pp. 37-46)

2


Introduction

1.2 Problem Statement

The problem concrete slab design in residential and small commercial buildings faces is the

standardised design procedure since it does not take account of the form-active shape a slab

would take under its service or ultimate loading scenario. It is noteworthy to appreciate the

constraints posed on slabs in residential and small commercial buildings with regard to the

form/shape they may take. However, given today’s regressing state of the environment, the

design needs to incorporate sustainability. To do so, materials being used should be designed

to act in their strongest nature for efficiency, and for concrete that strength is in compression,

hence the focus of this thesis being on structural from in slabs.

The focus is to investigate how the structural form of ground floor and suspended slabs could

be altered to help them experience almost entirely compressive forces as opposed to tensile

forces, thus allowing for reduction of the concrete strengths. The investigation will be

reserved for suspended floor slabs because of the relative urgency with regard to higher

bending stresses. Slabs-on-grade are continuously supported on the ground they are cast on

and thus minimal bending stresses occur thus they are thinner than suspended floors and

generally require less reinforcing, mostly just for crack control.

Before looking at structural form in suspended floor slabs for residential and small

commercial buildings, other tools for improving structural efficiency need to be looked at so

as to be able to reasonably compare efficiencies, these are covered in the preliminary

literature review. The methods researched are all mass reduction methods or improved

sections; they are concerned with either improving the profile section of a slab or the cross

section of the slab. These methods are commonly used and they have the structural advantage

of maintaining an overall shape similar to that of a flat slab which allows it to carry out its

duty.

With regard to structural form in suspended floor slabs, there are some restrictions, the

obvious ones being the shape itself. Since the main purpose of the slab is to carry furniture,

people, etc. it is generally required to be flat on top, and again a height restriction is also

obvious with regard to depth. Compressive form-active structures also pose a unique

challenge. These structures are rigid and thus with changing load patterns or magnitude they

cannot adapt the new required form-active shape of the new load and thus bending stresses

will develop in the slabs. These challenges are to be addressed in this thesis.

1.3 Scope of Thesis

Given the broad applications of concrete as a building material, this thesis will not look at all

applications of concrete. The chosen focus area will be residential and small commercial

buildings’ ground and suspended floor slabs, particularly suspended floor slabs. This is

because within the mentioned buildings themselves, floor slabs account for a significant

amount of concrete used. Moreover, residential and small commercial buildings are a

significant contributor to all construction works, locally and also globally. Therefore it is a

fruitful basis for investigation to consider the issues of sustainable designs with regard to the

abovementioned alone and disregarding the design of other concrete structural components.

3


Introduction

Sustainability is often a vague term, with different meanings attached to it, even within the

Built Environment itself. For the purposes of this thesis, the following definition will be used:

“Meet[ing] the needs of the present generation without compromising the ability of the future

generations to meet their needs” (Brundtland & World Commision on Enviroment and

Development, 1987). The level at which the use of concrete is used ought to be one that sees

concretes carbon footprint being reduced and so is the extraction of the raw materials for

production of concrete.

This thesis will base its study on existing methods of the design of concrete structures. It will

take a purely theoretical approach on the investigation since the methods being investigated

already have an extensive backing of empirical data. The thesis aims at finding out how the

current methods of design in the concrete slabs mentioned earlier could be augmented by

other innovative methods observed throughout history, i.e. Structural form.

An outline of the scope of this thesis is as follows:

Conducting an overview study of form-active structures throughout recorded

History, mostly in architecture

A study of different types of floor slabs used in practise, particularly ‘improved’ slabs

with a higher structural efficiency and placing them on a scale in comparison to the

traditional solid slab and from active structures

(Preliminary Literature Review)

A comprehensive desk study of form-active structures, with particular reference to

suspended floor slabs in residential and small commercial buildings. The study is to

look at the limitations of imposing structural form on these types of concrete floor

slabs

An overview study of the relationship of form-active structures and the bending

moment diagram. This will extend to a vital analysis of the similarities of the bending

moment diagram to the shape of a parabola and catenary and which of the two shapes

is suitable to adopt to represent a form-active shape and when it is suitable

(Continued Literature Review)

Finally, a set of solid slabs (representative to typical dimensions residential

buildings’ floor slabs or small commercial buildings will have) are to be designed.

The load carrying capacities of each of these slabs are to be calculated and their

structural efficiencies determined by taking the ratio of their calculated strengths to

their particular weights. From these efficiencies, an attempt at obtaining a pattern will

be made. From the findings of the research on form-active structures, catenary curves,

parabolas and bending moment diagrams, an attempt in imposing some of these

shapes on the abovementioned slabs. The shapes are to be added to the solid slabs as

negative volumes to reduce the overall mass of the slabs in order to minimise the

internal bending stresses within the slabs. All the slabs are to be designed as one way

spanning to better understand their behaviour from first principles. Using finite

element analysis, a comparison and discussion of the results from the designs is to be

carried out from where conclusions will be drawn.

4


Motivation for Study

2 Motivation for Study

2.1 Introduction

As mentioned under the introduction in the previous section, concrete is the world’s most

widely used building material (RSC, 2008). The work that goes into constructing residential

and small commercial buildings is immense, and so is the concrete utilised. In the last half

century, residential and small commercial buildings have been designed and constructed

using some inefficient components with regard to their structural form, in particular ground

and suspended floor slabs.

The following sub-sections show previous innovations in concrete floor slab design, these

innovations were established in the previous century and they serve as an example of the

success of designing more efficient concrete slabs. These examples will serve as motivation

for research in improving concrete slabs by changing their structural form. Thereafter the

project proposal will be outlined.

2.2 Advances in Concrete Slab Design in the Last Century

2.2.1 Hollow Core Slabs

Simply put, these slabs have one directional hollows. Structurally they can be idealised as a

series of I-beams or T-beams. “In the mid-20th

century, the voided or hollow core floor

system was created to reduce the high weight-to-strength ratio of typical concrete systems.

This concept removes and/or replaces concrete from the center of the slab, where it is less

useful, with a lighter material in order to decrease the dead weight of the concrete floor.

However, these hollow cavities significantly decrease the slabs resistance to shear and fire,

thus reducing its structural integrity” (Lai, 2010, p. 7; Anastas & Zimmerman, 2003)

2.2.2 Bubble Deck Slabs

These slabs are also voided slabs. The difference relative to the above mentioned hollow core

slabs is that the voids are not one directional, they are spheres. Based on the tests done, these

types of slabs distribute the forces optimally in two directions as opposed to the one

directional hollow core slabs. In essence they negative in more or less the same way that solid

slabs behave, only with the added advantage that they are lighter. The hollow spheres used

have no added negative impact to the strength of the slabs (Călin, Gînţu, & Dascălu, 2009).

As noted in the previous section that one of the short fallings of the hollow core slab system

was the significant reduction in features such as shear resistance and fire resistance. Lai

explained that this caused an overall deterioration in the slabs structural integrity. However,

with regard to the Bubbkedeck® system, and according to the paper on Summary of Tests

and Studies Done Abroad on the Bubble Deck System: “The tests reveal that the shear

strength is even higher than presupposed. This indicates a positive influence of the balls.

Furthermore, the practical experience shows a positive effect in the process of concreting –

the balls cause an effect similar to plastification additives” (Călin, Gînţu, & Dascălu, 2009).

5



Figure 1: Bubble deck slab (Source: Structural Engineers, Sustainability and Leed®)

2.2.3 Holedeck Slabs

This slab system mimics that of a coffer/waffle slab. The innovation is in the reduction of

excess concrete within a traditional waffle slab. According to an e-portfolio on the holedeck

system, the system can achieve greater spans between supports due to its structural form.

Furthermore, the level of service is very high since the services are easily housed within the

depth of the slab itself. So this removes the need of suspended ceilings to hide services or

even HVAC systems. – (Holedeck the lean structure, online)

A comprehensive list of advantages this system offers is outlined in the abovementioned e-

portfolio for the holedeck system. These advantages include:

A reduction in built height

Less concrete consumption since passive concrete is removed

Total building’s weight reduction, giving a better structural performance

Reduction of building elements, e.g. no need for suspended ceilings, etc.

Reduction of implementation costs

With regard to the structural performance, the portfolio continues to explain that the lean

matrix structure improves the slab’s stiffness and the light weight also helps improve its

seismic performance, this is because in the event of a seismic event, a building with a lesser

total weight has less participation and is less vulnerable to collapse due to the horizontal

loadings of such a scenario. “Transmission of horizontal loads: the waffle slabs, like the rest

of bidirectional slabs, are more effective for horizontal requests than a unidirectional one.

Bending stiffness: the relative mass inertia of the section compared to the same area of a

concrete slab is 18 times bigger. Horizontal forces by earthquake: compared to a conventional

waffle slab Holedeck reduces its weight by around 15%. Therefore earthquake strengths are

reduced by approximately 8 to 10%.” (Holedeck®, 2013)

The following figures show the effect of slab geometry on the structural properties of slabs. It

can be seen how similar cross sectional areas, equivalent to same amount of material, can

drastically improve the structural integrity of a member if intelligent geometries are chosen.

Bubbledeck System

These are flat slabs similar to the hollow core slabs,

except that the hollow cores are not unidirectional but

are bidirectional. They are lightweight and because of

that they can span longer lengths than the conventional

solid flat slabs. Because of their inherent light weight

they have a lesser mass participation during seismic

events and are thus also structural efficient in that regard.

(Klane, 2007)

6



Figure 2: Holedeck system at Logytel R+D building (Source: Holedeck the lean structure, online)

Figure 3: Rib form holedeck (Holedeck, the lean structure)

The figure on the following page shows where the three discussed slabs fall on an efficiency

chart, and this position shows that more research into structural form can give better results.

Holedeck System®

This type of slab is quite similar to the

coffered/waffle slab. The difference however, is that

the holedeck has an improved longitudinal profile

section mimicking a truss. This further reduces the

total mass of the slab and uses an intricate lean

structure similar to the internal porous morphology

of bones. This slab type is probably more suitable in

larger buildings like parking lots because its larger

depth can be fully exploited by using the spaces in-

between for service pipes such as air-conditioning,

lighting, fire water pipes, etc.

(Holedeck®, 2013)

Mass Reduction and Stiffening of the

Holedeck Slab

This image shows the technique used in the design

of holedeck slabs. The image shows cross sections

of equal area but having different bending

stiffness. This is similar to the classic example of I

beams performing better under transverse loads

than solid rectangular sections.

This cross section of the slab is called the

improved section since the concrete is placed at

positions where it will be utilised. The improved

efficiency of these types of slabs allows them to

span greater lengths with much less material.

A clear restriction with this method however, is

seen on the rib form holedeck, the overall depth of

the slab goes to the vicinity of 500mm. This large

depth is very useful for other slab applications

such as parking garages as the space in between

the ribs may be used for service ducts, however for

the purposes of residential floors and small

commercial buildings’ floors these concrete depths

will be a questionable use of space

(Holedeck®, 2013)

This thesis will look at how form-active shapes

can be applied to solid slabs.

7



Figure 4: Structural efficiency chart, Macdonald

Structural

Efficiency Chart

This table gives a

qualitative comparison

of similar structural

elements with different

types of structural

efficiencies due to their

configurations or forms.

The interest lies on the

non-form-active slabs

shown. It can be seen

that they are not the

most structurally

efficient even though

they are improved.

(Macdonald, 2002)

This thesis will look at

how form-active shapes

can be applied to solid

slabs.

8



2.3 Project Proposal

2.3.1 Objectives of Thesis

This research is aimed at investigating the design of concrete structures which are focused

more on the structural form of the infrastructure/member being designed. The is to try and

study how the same principles or slight variations thereof could be applied to ground and

suspended floor slabs in residential and small commercial buildings. The above will thus

cover the following:

A review of the mechanical properties of low strength concrete

A study of the timeline of concrete, focusing on form-active structures

A review of widely used design methods and results of ground and suspended

concrete floor slabs in residential and small commercial buildings

Contrasting and possibly merging of design techniques of form-active structures with

that of the latter and furthermore overlaying them with successful historical examples

2.3.2 Expected Outcome of Research

This research is aimed at identifying different structural forms that a residential or small

commercial building floor slab can assume to have a better structural performance than an

ordinary solid slab. Given the advances that have already been made with the examples

given; hollow core, bubble deck and holedeck slabs, the expected outcome of this research is

not to find another similar floor slab system.

This research will firstly seek to compare and quantitate the three mentioned slab systems to

one another in the literature review. Secondly, it will analyse and quantitate different forms a

slab can take and therefore compare the resulting structural performances with the traditional

solid slabs’ structural performance. The latter will be with strict adherence to the

serviceability limit state of slabs. Third and lastly in terms of sustainability, this research

expects to find out whether reduction in material is more economical or the reduction in

concrete strength, in concrete slab design.

2.3.3 Strategy

In depth research on the three mentioned floor slab systems will be necessary to obtain more

data which will be used to compare the systems quantitatively to one another. This will help

determine which system is more sustainable and which is more economical. The comparison

of these systems comes as an appreciation that they are innovative systems which are closely

and directly related to this thesis because if their concepts. Moreover, it is not the aim of this

thesis to come up with a new system altogether, the aim is to see how we can optimise

concrete slabs for sustainable developments and if there are existing methods in place, this

will serve as ratification having critically looked at them.

9



2.3.4 Computer Aided Modelling

The use of computer software will be important to this thesis. No lab work is expected for the

purposes discussed thus far. The models are to be created and analysed while varying their

structural forms. The modelling here will be of a solid slab given shape modifications, and

with every different shape/form, the depth of the concrete will be varied to be able to

generate a report showing the impacts of structural form on the amount of material used. The

resulting behaviours are to be documented and reported in a way that allows comparing and

contrasting. From a discussion of these results with sustainability in mind, a conclusion will

be drawn.

2.4 Closure

The three examples of slabs discussed in this section all successfully reduce the amount of

concrete needed in their construction. The positive outcomes of optimising geometry are not

only with regard to sustainable design, they also improve the load carrying capacity of the

slabs, their spanning capabilities and in some cases such as the holedeck slab serviceability is

greatly increased because of the framework type structure which allows service ducts to be

easily installed and maintained (Holedeck®, 2013). The discussed slabs were improved by

mass reduction techniques.

Mass reduction techniques are useful and provide sound solutions; nonetheless they do not

produce the most structural efficient elements, as shown in figure 4. A more efficient element

takes advantage of the form which the structure as a whole assumes; therefore an in-depth

study on the behaviour of form-active structures is relevant to investigate how the structural

form of a slab can be manipulated to increase its structural efficiency, i.e. reducing the

amount of weight a slab needs in order to carry a given amount of loading.

Implementing structural form in the floor slab design answers the question of how structural

form can be used to contribute to a better solution. With respect to residential and small

commercial buildings, there are implicit restrictions in the freedom of designing with full

consideration of structural form, most obvious is the requirement that residential and small

commercial building floor slabs need to be flat on the top side, thus restricting the freedom of

adopting a true form-active shape. Another restriction is with regard to the slab depth; floor-

ceiling heights for these types of buildings typically range between 2.6m -3.0m, and thus it

would be a questionable use of space to have a slab taking up an equivalent amount of depth.

Designing compressive form-active structures is a less exact procedure compared to tensile

structures, i.e. tensile structures with no bending stiffness can re-adjust their shape to assume

new form-active shapes that accommodate the new load’s change, the converse does not hold

for compressive structures, particularly concrete. Since concrete becomes solid once it has

set, it cannot adopt a new form-active shape should the loading pattern change, thus the form-

active shape chosen may not remain truly the form-active shape. These and more constraints

are to be looked at into detail in this thesis with the aim of finding out ways of how form may

give better solutions given all its constraints in the chosen application of residential and small

commercial buildings’ floor slabs.

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3 Literature Review – General Overview

3.1 Introduction

This section will look at the general properties of concrete which will be important in the

manipulating of its form in the designing of form-active slabs. A summary timeline of case

studies of form-active structures designed throughout history is included, the purpose of the

timeline is to highlight the success of using structural form as a design guideline to help

concrete perform in its optimum form.

The traditional design of solid concrete slabs will be reviewed to look at its weaknesses and

strengths in order to see where improvements can be focused on, and also which attributes of

the design need to be preserved. This review of the traditional design process will be

followed by a brief sub-section on sustainability which will show the importance of

considering sustainability in infrastructure design.

3.2 Properties of Concrete

Concrete has numerous basic qualities; these are described briefly by journal from the

Cement Concrete Aggregates Australia as follows:

Four main properties of concrete

Workability

Cohesiveness

Strength and

Durability

Concrete has three different states (each state having different properties)

Plastic

Setting

Hardening

(CCAA, Concrete Basics: A guide to concrete practice 4th Ed, 2004, p. 6)

For the purposes of this thesis, the concern will be with regard to concrete properties of

strength and durability in the hardened state.

This thesis will concentrate on the hardened state of concrete. The reason behind that is

because the investigation lies in the hardened form that the concrete would assume once it

has been cast. An important attribute to notice however is the fact that concretes workability

diminishes as it hardens, meaning that a particular chosen form cannot be remoulded into

another once the concrete has set.

The type of concrete chosen for this study was normal density concrete with compressive

crushing strength of 20 – 15MPa. This is a common concrete mix for relatively small scale

construction works such as the ones chosen for this research, i.e. residential buildings and

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Literature Review

small commercial buildings. Even though the concrete mix is not of high strength, it has been

found that certain advantageous properties may still be retained in lower strength concrete

mixes. Usually, concrete with increased compressive crushing strength also has improved

strength in tension, shear resistance and also resistance to abrasion and wear. Conversely,

decreasing the compressive crushing strength reduces the mentioned strength properties.

However properties such as resistance to weathering, abrasion and wear may be retained even

in lower strength concretes (Wilby, 1977). It remains that when designing a structural

element using relatively lower strength concrete, the chosen form should accommodate all

the resistance that was lost by lowering the strength such as removing all internal tension

stresses and also accommodating shear.

The concept that durability of concrete can be retained, while lowering the concrete’s

crushing strength, or even improved is backed by the research being done by BASF and Dhir

2008, Director of the Concrete Technology Group at the University of Dundee UK. The

findings from the research show that reducing cement levels can enhance the concretes

durability. It directly follows that reducing cement levels is in effect reducing the bonding

strength and consequently the crushing strength. The components that make up concrete are,

aggregate, sand and cement, and in all the three major components, cement is the one

component that introduces the pores in the mix. Thus, reducing the level of this component

reduces the pores that would otherwise form. This leads to the reduction of all associated

durability weaknesses in concrete (RSC, 2008).

Concrete is a versatile building material. It can be cast into many different forms. For that

reason, the flexibility to design with focus on the form of the structure is attainable. Since

concrete is strong in compression and about ten orders of magnitudes weaker in tension, it

would be efficient to cast the concrete member in a form that transmits mostly compressive

forces and avoids tensile forces. This focus on structural form will be further elaborated

under the next section which is ‘Form-active Structures’.

3.3 Concrete Structures throughout History

Concrete as a building material dates back hundreds of years. There is scepticism among

researchers on when the use of concrete as a building material started. As explained by a

paper from The University of Memphis, referencing R.E. Shaeffer’s book, "Reinforced

Concrete: Preliminary Design for Architects and Builders": “Many researchers believe that

the first use of a truly cementitious binding agent (as opposed to the ordinary lime commonly

used in ancient mortars) occurred in southern Italy in about the second century B.C.” –

(History on Concrete, Online).

Taking note that the history of concrete structures is quite broad in itself, this will not be the

focus in this thesis. Looking at the history is mainly aimed at trying to notice patterns in

design techniques evident in the as built structures.

To outline some of the significant feats in concrete design and construction throughout the

years a timeline would be necessary. Some of the events from a timeline as laid out by the

Auburn University’s College of Architecture, Design and Construction is as follows:

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Literature Review

118-35AD – The Pantheon: This building is to date, the building with the largest

unreinforced concrete dome with a diameter of 43m and an oculus/skylight of

diameter 8.9m - (Historical Timeline of Concrete, Online)

The cross sections of this building have been ‘improved’ by using improvements such

as coffers or voids (Macdonald, 2002). These improvements are in-fact mass

redistribution techniques that give the structure or structural elements more structural

efficiencies.

Figure 5: Inside the Pantheon, Italy, Rome, (Source: Giovanni Paolo Pannini, Left), Cross-section and

Plan of the Pantheon (Right)

1871-1875 – “William E. Ward builds the first landmark building in reinforced

concrete in Port Chester, NY. Designed by Architect Robert Mook” - (Historical

Timeline of Concrete, Online)

In the 1900’s the use of reinforced concrete was becoming widespread, the

focus being on reinforced concrete and not so much on the form the concrete could

take to account for the loads. The design was driven mainly by the buildings purpose

rather than how an optimum shape could be adjusted to fit into the buildings purpose,

in a sense the art that went into buildings such as the Pantheon was lost.

1927 – “Eugene Freyssinet develops successful pre-stressed concrete” - (Historical

Timeline of Concrete, Online). Again this goes to show where the shift in thinking

amongst designers was headed, towards finding ways of reducing the concrete’s

weaknesses by adding steel in these regions of weakness, a process as explained

earlier as reinforcing. The focus was not on optimising the shape of the concrete

members.

1930 - “Eduardo Torroja, designed the first thin shelled roof at Algeciras” –

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(Historical Timeline of Concrete, Online). A paradigm shift in the design of structures

was taking place. Designers were realising the potential concrete structures may have

if the structural form is altered accordingly.

Figure 6:1935 Zarzuela Hippodrome by Eduardo Torroja (Source: International Database and Gallery of

Structures, Online)

At around this time, the 1930’s and a few years after, the design of thin shell

structures was gaining popularity. But due to lack of confidence amongst designers

given the relative newness of the field, or perhaps even paradigm shifts in

architecture, the technique didn’t propagate as the now common method of reinforced

concrete structures.

At the present moment, the design of form-active structures is now better researched,

however it still doesn’t account for a very significant part of the constructed

infrastructure.

With regard to the topic at hand, it can be seen how versatile concrete as a material is.

Different methods of using it in construction have been briefly discussed. Examples that

show that its shape can be optimised to carry loads have been shown in a very passing

manner. However, taking from history, one can learn that one of the crucial properties of

concrete is that shapes such as domes and shells, can help reduce the need of extra

reinforcement as in the Pantheon, or even reduce the amount of concrete as in the Zarzuela

hippodrome and many other examples of its likes. These tie back to the question of slabs and

the focus on incorporating slab geometry into design in order to enable the slabs to not

require the nominal concrete crushing strengths they currently do, mainly for residential and

small commercial buildings?

This question will be better broken down under the section ‘Concrete Slab Design’ as it can

requires a deeper understanding of what goes into the designs of concrete slabs, or any other

concrete member for that matter.

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3.4 Traditional Concrete Slab Design Process

3.4.1 Ultimate Limit State Design

This method of design considers the worst loading case scenario that could be imposed on a

structure during its assumed lifetime. The scenario is called the ultimate limit state (ULS). It

includes numerous safety factors, some of which are embedded in the design equations.

These safety factors are applied on the dead load of the structure, which is the structures own

self weight and secondly the factors are applied on the live/variable loading that will be

applied on the structure. The latter is the weight that the structures primary function is to

support.

(1)

Equation 1: Ultimate load equation, SANS 0100-1

Where n = load, Gn = Dead load, Qn = Live load, SANS 0100-1.

Based on the heavily factored ultimate load, giving the ultimate design moments, the section

is designed using a series of design equations which give the ‘optimum’ cross section, the

allowable span and the required minimum steel reinforcing bars required. Emphasis here is

on the ‘optimum’ cross section.

As discussed notably a few times earlier in this paper, concrete is significantly weaker in

tension, more precisely, to the order of roughly 10 times. Thus, a concrete with crushing

strength of 30MPa will most likely have about 3MPa tensile strength. As mentioned in

section 3.3.1, one of the inherent deficiencies in the design of RC structures which carry

transverse loads, is that they are designed to crack in the tension zones. This allowance is to

trigger the tension steel reinforcing to carry the tensile forces. Altogether, that renders the

concrete in the tension zone obsolete. However it is no coincidence that the concrete in the

tension zone carries no load, it is one of the fundamental assumptions in this design

procedure.

This fundamental assumption carries the bulk of the inefficiency in the design process of

slabs. If the concrete adds no structural worth except for the dead load, it would be an

excellent improvement in the design to safely remove this concrete. This reduction will result

in the reduction of the dead load. In most cases the dead load of residential floor slabs and

small commercial buildings’ floor slabs account for over 50% of the total carried load.

Therefore this particular improvement will most likely improve the structural efficiency of

these slabs.

In the design process we assume that the concrete in the tension zone carries no weight

whatsoever. Hence the stresses below the neutral axis are only represented by a line (fy/γm)

which is due to the stresses in the reinforcing bar. See figure 7 on the following page. This

assumption begs the question of why then do we include this excess concrete which serves no

structural performance role in the first place? Noting that it is not as easy as removing all the

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tension concrete since the tension forces will simply redistribute themselves in the remaining

concrete, leading to an looped problem with no solution.

Figure 7: Actual and equivalent stress distribution at failure. (Source: SABS 0100-1: 4.3.3.3)

This particular problem has led to innovations such as the hollow core slabs, Bubble deck

slabs and Holedeck slabs, all of which will be reviewed in section 3.5. This research however

seeks to employ a different approach which considers changing the entire structures form.

3.4.2 Serviceability Design

This part of the design of RC structures seeks to ensure that the structure performs all its

functions during its lifetime adequately. Deflections have to be minimised, the structure has

to be stable, and cracks have to be controlled and so on. These are all encompassed within the

SLS, the serviceability limit state.

If the structural form of the entire structure is to be looked at as a whole, it ought to be

appreciated that serviceability limitations will be imposed. Some structural forms are

structurally superior to others, but these forms may easily defeat the purposes of the structure

if utilised. An example may be taken from a piece of A4 paper or a similar sized paper.

Holding the paper horizontally from one end alone, the paper will surely collapse. However,

adding a curvature to the paper will allow it to remain horizontal and not collapse while still

being held at one end.

A similar scenario may be researched for slabs, and similar results may be realised. Be that as

it may, it would be poor design to make a slab that is heavily curved because its purpose

amongst others is to hold furniture and other household appliances, all of which need flat

horizontal surfaces to rest on. There is also a level of comfort that needs to be maintained,

people are more likely to be comfortable in a building with relatively flat slabs as opposed to

curvy slabs. All of this factors fall directly under the SLS design and thus in further research

the considerations of SLS design need to be prudently followed.

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3.5 Concrete Slab Design for Suspended Slabs

Design of reinforced concrete (RC) structures is a well-documented procedure. As a result

many countries have their own set of standards which serve as design guidelines for RC

structures. As mentioned earlier in the introduction, the procedures differ from country to

country and even amongst different states in some places. This is due to different climatic

conditions which affect the durability of the concrete such as corrosive environments. Other

factors that influence the differences in the design standards are preferences in calculation

methods. However, these differences will not be accounted for in this thesis since the aim is

not to scrutinise the available standards but to investigate how the form of concrete slabs

could be optimised to allow for a possible reduction in the concrete strength, or more

commonly the amount of materials used.

The design procedure of a concrete structure i.e. member sizing can be summarised as

follows; the member’s cross section is determined by calculating adequate areas of concrete

and steel, the steel being important for the tension forces generated due to the moment within

the cross section. This definition encompasses the methodology followed in the design of RC

slabs. A section of a slab is taken, which would usually be treated like a beam, and from that

section an external moment is calculated form the given loading. This method of slab design

assumes that all concrete in the tension zone does no work in carrying the tensile forces,

moreover this concrete is often more than half the total concrete. Clearly this is a major

unsustainable use of material.

For this reason that the design procedures focus on the member’s cross section, it is therefore

a region of interest where some innovations looked at for weaknesses in the current

traditional methods of design. Successful attempts on how they could reduce excess concrete

and only have it where it is absolutely needed were made. This has resulted in shedding of

significant amounts dead loads. Consequently that has successfully improved even the

structural performances of members which have been designed with these considerations.

As a prerequisite to the design of RC structures, the loads to be carried by the structure ought

to be calculated. Again, there are many methods available to deal with this. These methods

will not be discussed here as they are a topic by themselves. Nonetheless, it is noteworthy

that the methods to be assumed in this paper are for elastic analysis of slabs since slabs fail in

either crushing or by shear failure as opposed to yielding. This means that the design

procedure will concentrate on elastic design of RC slabs and not the elastic design.

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3.5.1 Inherent Inefficiencies in Solid Slab Design

Since reinforced concrete elements are designed for either axial stresses, shear stresses or

bending stresses it is necessary to understand the influences of each case. For beams and/or

slabs, the loads are usually applied transverse to the element, this causing both bending and

shear stresses within the cross section; this puts the concrete in a position where it cannot act

efficiently without assistance. The bending moments cause a distribution of both compression

and tension in the cross section, thus putting a significant amount of concrete under tension

where it is naturally weaker. Most commonly the assistance of the concrete in tension is

provided by reinforcing steel, however a major deficiency in this method is that traditionally

the concrete under tension is left there and adds no structural worth besides adding more

weight while the steel is designed to carry all the tensile forces.

In the design of RC slabs or beams, it is assumed that for serviceability design the section is

uncracked while it cracks under ultimate loading. According to how the design of the beam or

slab was done there are mainly three ways that it could fail; sudden failure, failure with prior

warning and combined steel and concrete failure. The traditional RC design of these elements

has the advantage of having a control over how the element fails should the load exceed the

ultimate one of a concrete member. Nonetheless the design does not focus on mitigating the

weaknesses that arise due to carrying a transverse load; for the most part it focuses on

strengthening the weaknesses that arise because of the transverse loads but it seldom focuses

on removing these weaknesses. A unique solution to this will be discussed in the analytical

methodology section which converts the moment into forces by introducing a lever arm

outside the cross section.

In the calculation of ultimate limit state (ULS) loads as discussed in 3.4.1, the dead load has a

lesser partial factor that the live loading, it seldom happens that the live load exceeds the self-

weight of the beam or slab, see equation (1). That is the cause of an inherited low efficiency.

Figure 8: Cracked RC beam section, showing qualitatively the amount of concrete being ignored.

(Quimby B, Reinforced Concrete Design)

The design procedure for RC slabs and beams is rigorous and proven to work, however it has

significant disadvantages in terms of the material it requires for the element. In addition to the

material required, a bulk of the material adds no structural value to the element thus making it

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inefficient. Thus Improvements of the concrete’s strength are likely to have no major impact

on structures designed with this underlying assumption of the ignored cracked concrete. Due

to the amount of concrete utilised for caring the compression forces, increasing the concrete’s

crushing strength has a reduced improving effect, particularly to beams and slabs. As can be

seen in the image above, the concrete that carries the compressive load is indicated, this

concrete accounts for less than half the section. Therefore improvements on the concrete’s

crushing strength are lost in the concrete that carries no load.

3.5.2 Advantages of Solid Slab Design

Introducing improvements to the traditional slab design ought to consider both the

weaknesses of the traditional methods as well as its advantages. The importance of including

the advantages is to ensure that improving the design or making it more efficient in this case

does not remove the favourable attributes of the traditional methods. Thus any improvement

should in effect build on the current positive attributes.

The advantages carried by the traditional slab design as briefly mentioned in the previous

section is that of the flexibility in deciding the failure modes. Depending on the designer’s

preference, the beam/slab can be designed as follows;

Over reinforced section

This type of design is generally not preferred since the failure mode is sudden and thus puts

the people using the structure in danger. The concrete reaches crushing strength and ruptures

early before the steel could reach its yield strength. There is therefore no prior warning for

impeding failure.

Under reinforced section

This is probably a safer design since failure is noticeable before it actually happens. The steel

reaches yield strength before the concrete crushes thus large deformations will be visible thus

giving clear signs of impending failure.

Balanced section

This is a less practical design because predicting the equivalent behaviours of both the

concrete and steel simultaneously is not an exact procedure.

These advantages are carried by the traditional design procedures of reinforced concrete slabs

or beams. They are of a high importance since they give a measure of control in the

behaviour of the constructed structure and consequently give a certain guarantee on the

structure serving its purpose to a measurable degree of safety. This is important to consider in

this thesis because in trying to consider structural form in the design of slabs, a degree of

safety needs to be considered. Therefore the mentioned advantages of the solid slab design

need to be considered in weighing the pros and cons of the traditional slab design to those of

form-active slabs.

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3.6 Concrete Slab Design for ‘Slab-on-Grade’

The design of ground floor slabs (Slabs-on-grade) is considered in this thesis for a measure of

completeness as the problem of inefficient structural members includes most if not all

structural elements. Slabs-on-grade however are a unique exception since their support

conditions are continuous. The focus in the design of slabs on grade is usually on the grade

that they are cast on, making this founding conditions string enough is usually a first priority

and depending on how strong the grade is, the concrete utilised may be of a lower crushing

strength.

Slabs on grade also support immense transverse loads, however the transverse loads can be

assumed to be axial loads, this is because the ground is continuous under the slab and thus all

internal forces generated will most likely be of the axial kind, making the slabs performance

efficient since the slab will be in full compression. However this assumption may be

misleading due to the dynamic nature of the ground. Issues such as differential settlement

caused by cyclic movements between dry and wet seasons may create zone where the slabs

acts as a suspended slab, which would pose a significant danger should the slab be of a lower

crushing strength, nonetheless the failure of the slab may not be as fatal as that of the

suspended slab due lower potential energy. This danger due to the dynamic nature of the

ground may well be considered a topic for the strength of the grade.

Another factor that may cause slabs on grade to have similar properties as suspended floor

slabs is the membrane effect that the slab has on the ground immediately below it. This

membrane effect simply creates a seal for all the dampness in the grade below the slab. The

water in the grade will no longer be able to evaporate, thus causing a swelling in the ground.

The problem arises because the grade at the edges is able to let off some of the dampness thus

shrinks during dry seasons while the middle part remains swollen. The effect that this

scenario has is that the slab will then be equivalent to a cantilever slab thus requiring more

bending stiffness than it was designed to have (CCAA, Guide to Residential Floors, 2003).

In the event of external loading, slabs on grade are not always designed to be structurally

active. This is explained better by the Concrete Steel Reinforcement Institute’s (CSRI’s)

report on reinforcing in slabs on grades. One of the advantages of having steel reinforcing or

welded wire mesh is gaining structural strength after cracking during overloading. “When

overloading occurs, such that the cracking moment limit of the concrete slab has been

exceeded, structural cracks may occur. The steel will then act as structural reinforcement and

provide moment capacity according to normal, cracked-section, reinforced concrete theory.

This concept may also be intentionally used in the original design concept of the slab; that is,

designing the slab to have structurally-active reinforcement under externally applied

loadings.” (CRSI, 1998)

The above implies that trying to optimise structural form for this slab application may yield

results that may not be significant enough. In the design of suspended slabs, we look to the

procedure of the design to try look for ways to optimise the geometry. With slabs-on-grade

not so much attention is given to designs for flexure because it is minimal.

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3.7 Sustainable Design

Sustainable design with regard to concrete construction can be achieved in a number of ways.

More commonly this can be achieved by physical reduction in material used. This is

characteristic to the three slab systems discussed earlier. Sustainability in all the three cases

was mainly derived from the reduction of material used. This of course triggers a trickledown

effect whereby other sustainability milestones are achieved such as better thermal insulation

and high levels of services.

On the other hand, significantly reducing concrete crushing strengths from the nominal 20-

25MPa strengths might bring out another level of sustainability as it is directly related to

reductions in cement that goes into the concrete mixture, consequently reducing the total

carbon footprint directly from slab constructions in residential and small commercial

buildings.

This thesis is concerned with both the reduction of material used and the reduction of the

amount of cement in slabs. Designing concrete slabs using structural form implies that the

slabs are designed to act almost entirely in compression. When concrete acts in compression,

the effect of lowering or increasing the concrete’s crushing strength, i.e. decreasing or

increasing the cement content in the mixture, has a direct and sensitive effect on the

performance of the concrete. As mentioned earlier under section 3.5.1, changing the cement

content for the traditional slab has a reduced effect due to the ignored concrete, for the form-

active slab however the effects are expected to be high. In simpler terms, form-active slabs

are aimed at using lesser material, and also being able to perform at lower crushing strengths

because of their ability to act in compression.

As underlined by Anastas and Zimmerman in The 12 Principles of Green Engineering;

Principle 5 explains that for sustainability a product that is designed ought to be output pulled

rather than input pushed. This is directly related to the structural efficiency of slabs.

Traditional slabs have a low efficiency below one, this shows that the weight carried by the

slabs are mostly from their own weights as opposed to the live loads they are designed to

carry. The implementing of structural form will help make the slabs more output pulled by

making them have much lesser weight than the traditional solid slabs carrying an equivalent

live load, thus taking their structural efficiencies higher. Principle 8 explains that design for

unnecessary capacity or capability ought to be avoided, the so called “one size fits all”

designs (Anastas & Zimmerman, 2003). This principle is directly related to the design

process of form-active structures as a whole and can serve as relevant motivation in pursuing

such designs to attain sustainable development. Design of form-active structures comprises of

the study of the load to be carried and analysing what shape the structure requires for the

most efficient way of carrying that load, i.e. the solution is a tailored solution specifically

meant for that particular loading scenario, moreover the solution considers the type of

material that will be used to construct the structure, i.e. tensile material with no bending

stiffness or rigid material with bending stiffness as in concrete. The implementation of

structural form in the design of slabs can thus be said to be in line with sustainable

development as it follows the relevant guidelines as highlighted.

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3.8 Closure - Main Findings from Literature

The investigation of ways to optimise the structural form of slabs is a fairly new field that has

been a topic of research in the last century, basing its methods on other form-active structures

such as bridges and larger structures. However, the design of traditional slabs is a well-

established craft. Ground-breaking innovations have already been made, all of which have

one eminent feature in common, and that is the removing of ‘passive’ concrete from the slab

itself. The focus has been in the internal structure or cross section of the slab rather than the

slab as a whole. This has been a fruitful focal point as it has yielded successful results.

Further research could be focused on how the structure as a whole can be in a form which is

structurally more superior to the traditional solid slab. This is aimed at finding out if at all the

overall shapes of slabs can be altered to enhance their structural performance and allow lower

concrete strengths to be used. Still, this has to be with close adherence to serviceability limits

of slabs in general as their function is the governing factor on whether they will be able to

perform their duties.

With regard to slabs-on-grade, there is little room for optimisation since the strength of these

slabs is lower than that of suspended slabs. This is due to the fact that slabs-on-grade seldom

experience high deflections or moments as they sit directly on a well compacted subgrade

which directly receives the loading the slab experiences. For this reason, this thesis will not

focus on slabs on grade. The focus will be on roof slabs and suspended floor slabs as these

slabs experience larger transverse loads and will benefit more from increased efficiencies.

From the review on the traditional solid slab design it was found that the major factor that

gives the traditional solid slab its inefficiency is the ignored cracked concrete when designing

the solid slabs. This concrete that carries no load is where the recent slab innovations are

focusing on for mass reduction. The recent slab innovations, i.e. hollow core slabs, hollow bi-

directional slabs and improved waffle slabs have an increased efficiency when compared to

the traditional solid slab, however, they are still not entirely efficient on a scale which

compares types of structures. This motivates this thesis’s proposal to investigate structural

form as a design guideline for designing residential and small commercial building concrete

slabs, i.e. form-active structures were shown to potentially be the most efficient structures.

With regard to sustainability an overview of the impact of changing the concrete’s crushing

strengths on the traditional slabs showed that because of the underlying assumption of

ignoring the cracked concrete, the effects are reduced. For form-active slabs however, the

effects of changing the concrete’s crushing strength are significant since the concrete is

almost entirely in compression. Therefore designing the slabs by considering structural form,

the effects of the concrete’s crushing strength can be augmented and hence lower strengths

may be used. Furthermore, a higher structural efficiency can be achieved, thus using less

material to carry more load. These characteristics of designing form-active structures are

integral to sustainable development, this is shown in The 12 Principles of Green Engineering

by Anastas and Zimmerman.

22


Form-Active Structures

4 Design of Form-Active Structures

4.1 Introduction

The design of form-active structures requires a critical analysis of the load to be carried and

the form/shape that that particular load requires for an efficient performance. Simply put, this

type of design requires a regressive technique which analyses the type of load and the form

the load would generate on a zero stiffness member. This shape is then translated to the

structures form. This section is aimed at outlining the procedure for designing these structures

so that the procedure can be applied to residential and small commercial buildings’ roof and

suspended floor slabs.

The formation of a form-active structure has to include three major factors that determine the

success or failure in designing such a structure. These factors are; rigidity, element

shape/form-active shape and load pattern. Depending on these three major determining

factors, the shape can then be deemed either non form-active, semi form-active or fully form-

active. This section will explain each of these three factors and how they influence whether or

not a structure is form-active, this analogy is based on the book Structure and Architecture by

MacDonald A.J. alongside the case studies from the books Why Buildings Stand Up and

Why Buildings Fall Down by Salvadori et.al.

4.2 Rigidity of the Element

The rigidity of a structural element has a major bearing on what type of loads that structure

can support but more importantly how that structure will support the load. Non rigid

structural elements have the capacity to change their shape in order to support imposed loads.

A classic example is the string; it has no bending resistance and furthermore cannot support

compressive loads. A string can support tensile forces alone. The method in which the string

will support tensile loads depends on its support conditions; clearly a string supported on one

end alone will not change its shape under loading except of course for the elongation that

may occur. However, a string supported on more than one support and under transverse

loading will assume a shape that will allow it to convert the load into internal forces similar

to that of a string supported on one end under loading. This is one of the key concepts of

form-active structures i.e. checking what shape a member with no bending stiffness assumes

under a load with a certain magnitude characteristic (point loads or uniformly distributed

loads).

Tensile members with no bending stiffness are a special case because they are able to change

their shapes with changing load magnitudes and pattern assuming no failure at this point.

Compression members on the other hand pose a notable challenge since they are rigid; this

means they cannot readily change their shapes due to a changed load pattern or magnitude.

The solution for this is taking the mirror image of a tensile member under the same loading

and creating a similar shape for the compression member (Macdonald, 2002). This is to be

covered in more detail under section 5.3.

23



The rigidity of the element therefore plays the first important role in the design of form-active

structure as it gives an indication whether the element will be able to adjust itself to dynamic

loads or not. Compressive member such as concrete suspended floor slabs require taking the

mirror shape of a tensile member under the same kind of loading. A cause for caution is to

notice potential failure problems with compressive form-active structures; they cannot re-

adjust themselves and thus are more susceptible to failure due to excessive unpredicted load

patterns not similar to the design load patterns and magnitudes. For the same reason,

compressive form-active structure cannot be called true form-active structures where there is

dynamic loading expected since they cannot re-adjust (Macdonald, 2002).

4.3 Influence of Form-Active Shape

As mentioned in section, 4.2, form-active structures take the shape of that allows them to

convert the loading into axial internal forces. This is the main reason why form-active

structures are likely to be the most efficient structures. With regard to the problem statement

of this statement being the application of form on residential concrete floor slabs and

commercial floor slabs, the internal forces are flexure and shear. With flexure, there is a

distribution of tensile forces and compressive forces in a linear fashion. This linear

distribution leaves a large amount of concrete being under stressed and this is the root of the

inefficiencies in such slabs. It follows from this that an efficient form of internal stress is

axial stress. This is because all material in the cross section is equally utilised, thus all

material give full account of their structural worth.

Because of the flexibility in non-rigid members, they automatically assume a shape that

converts the imposed loads into equivalent internal axial stresses only since that is all they

can support, i.e. axial stresses. For rigid elements, usually strong in compression like

concrete, the shape a non-rigid element would take under the same load pattern and

magnitude is flipped or mirrored and applied to that rigid element (Macdonald, 2002). This

concept of modelling a self-supporting arch is also supported by Rousseau and Saint-Aubin

in their chapter in Calculus of Variations (Rousseau & Saint-Aubin, 2008, pp. 483-486). The

arch converts the imposed loading on the rigid element into axial compressive stresses within

its cross section, and again due to the uniform manner in which the stresses are then

distributed, the structure becomes efficient. The image on the next page summarises how

form-active shapes are formed in tensile non-rigid elements and how they can be applied to

compressive rigid elements that carry the same loading pattern and magnitudes.

24



Figure 9: Tensile form-active shapes, (Top) and equivalent compressive form-active shapes, (Bottom)

(Source MacDonald)

4.4 Influence of Load Pattern

The two previous sections explained how the rigidity determines the type of form-active

structure, tensile or compressive, whereby the advantages of tensile structures were

highlighted and the disadvantages of compressive/rigid structures were discussed. They also

explained how this rigidity translates to the creation of form-active shapes; non-rigid/tensile

shapes are flexible enough to re-adjust to always remain form-active assuming no failure,

whereas compressive/rigid structures have to take the form-active shape of a tensile element

under similar loading patterns and magnitudes. This section deals with the loading pattern

and magnitude; this is the last major key in determining whether a structure will be form-

active or not. The load patterns are divided into point loads and uniformly distributed loads,

both of which the intensities/magnitudes of the loads are important.

4.4.1 Point Loads

As can be seen in figure 9 in the previous page, point loads give distinctly different form-

active shapes when being compared to uniformly distributed loads. Due to the point loads

concentrated load it creates taught areas in the non-rigid member in straight lines. For the

application in residential and small commercial buildings’ slabs these point loads are not the

design load patterns; however their intensities need to be monitored to avoid failure in these

slabs since they usually designed for uniformly distributed loads.

4.4.2 Uniformly distributed Loads

Non-rigid elements under uniformly distributed loads take the shape of a parabola or

catenary. In the case were the self-weight of the non-rigid member is significant a catenary

shape is formed and in cases were the live load, in this case the UDL overshadows the self-

weight of the element the shape of a parabola is assumed. (Rousseau & Saint-Aubin, 2008)

25



For application on slabs parabolas are thus the more accurate representation of the form-

active shape. This particular dependency of form-active shapes on the type of loading is

related to the bending moment diagrams that would result from these types of loading. In

relation to figure 9 again, it is evident that the form-active shape on the left is representative

of the bending moment diagram that would be plotted for such a loading arrangement,

consequently the loading magnitude as well. It follows logically that the middle form-active

shape is related to the bending moment diagram that would be plotted for the two point loads

with the different magnitudes, the same applies for the form-active shape on the right. Since

the latter is a shape resulting from a uniformly distributed load, the resulting bending moment

diagram follows the shape of a parabola, see appendix B for the equation of a bending

moment diagram resulting from a UDL. This supports the reasoning for taking a parabola as a

representation of a form-active shape in the design of form-active slabs under UDL’s.

The importance in the form-active shape being directly related to the loads bending moment

diagram will be discussed in the Analytical Methodology section whereby the design of the

form-active slabs will be discussed.

4.5 Degree of Form-Active Elements

According to how closely related the element shape is to the form-active shape, the element

can be categorised in one of the following three categories;

Non Form-Active Structures

These elements have no relationship with the form-active shape. This is characteristic of the

traditional solid slab and the discussed slabs in sections 2.2.1 -2.2.3. The slabs discussed in

sections 2.2.1 – 2.2.3 also take the overall shape of the solid slab. However, these slabs are

more efficient than the solid slab because they have improves section either in cross section

or longitudinal section. The improved section means the concrete that adds no structural

worth was remove, leaving the structure being lighter and thus more efficient.

Semi Form-Active Structures

These structures have overall shapes closely related to the true form-active shapes but do not

have the exact form-active shape. They are more efficient than elements with improved

sections because not only do they have excess concrete removed but also their form is

structurally active.

Form-active concrete slabs and other concrete structures are likely to fall under this category

simply because they cannot readjust their design shapes once the concrete has set. This is a

downside since dynamic live loading is inevitable, therefore in the design of these structures

for UDL’s, point loads should be considered so as to allow for the behaviour of the structure

under the possible loading it was not designed for.

Fully Form-Active Structures/Form-Active Structures

These structures maintain the true form-active shape of that particular load pattern and

magnitude. These are typically non-rigid elements. Some historical structures have certain

26



elements in their structure being fully form-active structures. More related to the topic of

concrete slabs is the case study of the Pantheon in Rome mentioned in section 3.3. The

Pantheon’s form-active shape was a semi-circle. This meant that there was no necessity for

lateral supports since the load transmition at the supports would be vertical and thus have no

lateral components. The use of a circle can also be implemented in the design of form-active

slabs, this is because the differences in a parabola derived from the bending moment diagram

and an arc of a circle passing through the same point is negligible.

This similarity of the form-active parabola shape in relation to a circle will be used in the

design of form-active slabs mainly because of the convenience it brings in terms the

practicality in setting out of form work. Since the differences are significantly minute, to a

few millimetres they can be ignored for practicality. The following table was derived from

the analysis of a form-active parabola generated from a bending moment diagram of a 6 m

one way spanning solid slab and was compared to an arc passing through the same x and y

intercepts. The generation of this form-active parabola is to be discussed in the following

design section, 5 and the results are again found in appendix D.

Table 1: Comparison of Bending Moment Parabola and Arc of a Circle

Point Position Parabola Circle % Diff.

x1 -3000 mm 0.000 0.000 0.00

x2 -2400 mm 0.216 0.222 2.59

x3 -1800 mm 0.384 0.389 1.42

x4 -1200 mm 0.504 0.507 0.62

x5 -600 mm 0.576 0.577 0.15

x6 0 mm 0.600 0.600 0.00

x7 600 mm 0.576 0.577 0.15

x8 1200 mm 0.504 0.507 0.62

x9 1800 mm 0.384 0.389 1.42

x10 2400 mm 0.216 0.222 2.59

x11 3000 mm 0.000 0.000 0.00

(2)

Equation 2: General equation of a parabola

(3)

Equation 3: Implicit equation of a circle with its centre on the y-axis

(4)

Equation 4: Explicit equation of circle with respect to y

27



Figure 10: Bending moment parabola plotted with arc passing through the same points

4.6 Structural Efficiency

Structural efficiency is the measure of how much load a structural element can carry in

relation to its own self weight (Macdonald, 2002). An element which carries more load than

its own self-weight can be considered as more efficient than the structure who’s self-weight is

more than the load it carries. This measure of efficiency is a numerical one obtained by

taking the ration of the load by the self-weight of the element.

(5)

Equation 5: Structural Efficiency

4.7 Restrictions in Designing Form-active Slabs

The designing of form-active structures is entirely dependent on the form the structure will

take, i.e. the form-active shape. Thus for some structural components restrictions will be

obvious in incorporating structural form freely. An example of such structural elements is the

floor slab typically found in residential buildings and small commercial buildings.

In South Africa, the most commonly used method of design and construction for residential

buildings and small commercial buildings, is the load bearing wall method. This method uses

a rectangular grid of load bearing walls usually perpendicular to one another. These grids are

similar in multi storey buildings of this type and ought to maintain the same plan section to

allow for continuity in the load bearing walls. This system is also known as the post and

beam system.

The post and beam system is characterised by the discontinuous joint connections. The

discontinuity of the joints implies that there is no moment transfer between joints. This

assumption simplifies both the design and construction of such structures. The following

section summarises the advantages and disadvantages of using continuous or discontinuous

systems. Furthermore, this comparison will give the restrictions imposed by the systems.

4.7.1 Comparison of Continuous and Discontinuous Structures

The comparison of continuous or discontinuous structures is necessary as it determine the

restrictions posed to the design of form-active structures by each of the two categories.

Clearly continuous structures are more flexible with regard to the leeway a designer has when

0.00 m

0.20 m

0.40 m

0.60 m

0.80 m

-3 m -2 m -1 m 0 m 1 m 2 m 3 m

Slab A - Parabola

Slab A - Circle

28



designing form-active structures, however, the current design and construction methods

popular in the design of residential buildings and small commercial buildings is the post and

beam method which is based on the discontinuous joints theory. This method is widely used

because of its economic benefits. Furthermore, designing for discontinuous joints implies

designing with extreme maximum bending moments at mid-span; this ultimately means

designing for the worst possible case of the ultimate dead and live load. The following table

shows the comparison of continuous and discontinuous structures.

Table 2: Comparison of Continuous and Discontinuous Structures, Moyaba, ref. MacDonald

Continuous Structures Discontinuous Structures

i. These structures contain more than the

minimum required support conditions

and are thus usually statically

indeterminate

ii. They have lower internal forces

iii. Due to the fixity conditions, smaller

elements are required to carry the same

load an equivalent

discontinuous structure would need

iv. They are usually more efficient

v. They give more freedom to the designer

with regard to structural form

i. Mostly statically determinate

ii. They are easy to design and construct

iii. Their flexibility helps with issues like

differential settlements, the likelihood of

large cracks forming is lower than that

of continuous structures due to the

rigidness

iv. These structures are economical

v. Due to the basic configuration,

elementary bending moment forces are

generated which are easier to analyse

and thus more viable

Most residential and small commercial buildings in South Africa are designed and

constructed according to the load bearing wall configuration. This structural configuration is

also a post and beam type configuration and thus the fixity assumption is that it is pinned, i.e.

no moment transfer occurs through joints. This type of configuration has a defining

characteristic of being relatively basic and with less complexity, thus practical in a wide

range of applications.

However, the simplicity of the post-and-beam/load bearing wall configuration comes with a

number of restrictions. This configuration uses a basic rectilinear arrangement, and as

mentioned before the arrangement has to be repeated on each floor in multi-storey buildings.

This geometric arrangement thus restricts the designer from changing the overall arrangement

in the aim of imposing efficient structural forms. Nonetheless, the advantage of this basic

arrangement and joint type is that the bending moments generated in the structural elements

29



are the elementary type which require less analysing time and thus easier to design for

(Macdonald, 2002).

In designing form-active structural elements for post-and-beam structures, the complete

structure will not be form-active. However, the elements that are essentially the structures

building blocks are designed to be form-active.

4.8 Closure

Having assessed the design requirements of form-active structures alongside the properties of

concrete as discussed in the earlier chapters a few reflections can be made;

Due to the rigidity of concrete, the concrete needs to mimic a mirrored shape of a

tensile structure under the same loading pattern and magnitude. From the literature

reviewed and analysis carried out as shown in appendix D, this shape is equivalent to

the resulting bending moment diagram under the particular load.

The design of form-active slabs for residential and small commercial buildings can

only go to the extent of being semi form-active. This is mostly because of

serviceability restrictions, i.e. the purpose of the slabs and partially due to the fact that

concrete is a rigid material during its lifetime and thus cannot readily re-adjust its

shape to accommodate changing load patterns and magnitudes.

Unlike the shape taken up by a cable under its own weight, i.e. catenary, the bending

moment diagram takes the form of a parabola. However, when compared with an arc

of a circle passing through the same points, the parabola has little difference to the

arc. Therefore the arc of a circle can be used to estimate this bending moment diagram

parabola when designing form-active slabs.

In order to keep the design concise and practical, the chosen types of structures for

these slabs were discontinuous structures, i.e. post-and-beam structures. This is also

due to the fact that these structures are the commonly designed structures in South

Africa for residential and small commercial buildings. For the same reason, the slab

design is to focus on one-way spanning slabs.

These key points are to serve as the boundary conditions in the design of the form-active

slabs in the following section, the form-active slabs are to be contrasted with equivalent one

way spanning slabs of the same span to allow for a thorough comparison of the two types of

slabs.

30


Analytical Methodology

5 Analytical Methodology

5.1 Introduction

This section is concerned with the application of the concepts discussed in section 4 to design

form-active slabs in order to compare them with the equivalent traditional solid slabs. The

solid slab, is to be enhanced according to steel and concrete optimisation, i.e. material and

cost optimisation. The improvement of the traditional solid slab is done in order to compare

the form-active slab with a practical and rigorously designed slab. Both the design of the

traditional as well as the form-active slab are to be done as components for post-and-beam

structures, thus they are to be designed as discontinuous with pin supports at both ends. As

mentioned in the previous section, the choice of the post-and-beam structure as a design

boundary was driven by the widespread use of this structural arrangement both in design and

construction in South Africa.

The comparison of these slabs will be with regard to their structural efficiencies,

encompassing both the concrete and steel utilised for the two types of slabs. This section’s

aim is thus to provide quantitative results to be used to support the predicted properties of

form-active slabs.

5.2 Choice of Load Analysis Method

The chosen load analysis method assumes concrete in the elastic zones. This is because

concrete does not yield, it crushes. The chosen slab strip to be designed, both as a solid slab

according to SANS 0100-1 and form-active slab design techniques was a 1m strip of a one

way spanning slab. Using the 1m strip simplifies the application of this strip to the entire area

of the slab since it is a unit. The strip was chosen to be one way spanning. This was to avoid

analysis methods such as load apportioning in two way spanning slabs as that may divert the

purpose mentioned earlier of finding out what difference it makes to design a slab as form-

active rather than a solid flat slab. Thus, simplicity in load analysis was needed for a clearer

observation on what applying different form has.

One way spanning slabs simply supported on either ends give an elementary loading

arrangement which results in simple bending moment diagrams of which if correctly

designed they can be easily applied on site. Choosing complex loading arrangements and

support conditions such as continuous supports will increase the complexity of the resulting

bending moment diagram, and as the relationship of the bending moment diagram with the

form-active shape was discussed in the previous section, the form-active shape will also have

to be a complex one.

While it is possible to design complex form-active structural elements from complex bending

moment diagrams, it would not be practical since the constructed structure has to take that

shape to a reasonable precision. For this reason, a simple loading arrangement was chosen

and its bending moment diagram was used for both the traditional solid slab design and the

form-active slab design, See Appendix B.

31



5.3 Traditional Solid Slab Design

For the purposes of all calculations to follow four types of slabs were chosen. These slabs

were chosen for their dimensions which represent typical values in South Africa for Floor

Slabs in residential buildings and small commercial buildings, See Appendix A for detailed

calculations for minimum area of steel required for these slabs;

Slab A = ly x 6m, depth = 214 mm

Slab B = ly x 5m, depth = 192 mm

Slab C = ly x 4m, depth = 154 mm

Slab D = ly x 3m, depth = 125 mm

These slabs are all one-way spanning. This is again done to keep all coming calculations in

the design considering form-active shapes simplified and easy to grasp in terms of the

concepts discussed in section 5. 1m wide strips where chosen on each slab and were designed

as beams according to SANS 0100-1: 4.3.3.4.1, see appendix A. For each of the 1m strips in

each slab, the depth was determined by the ratios given in the code as shown below. A

general observation to make from the table and the depths obtained from it as shown above is

that the portion of the weight of the slab increases with increasing span. This is important to

not with regard to structural efficiencies because it shows that structural efficiency decreases

with an increasing span, this is because the required depth increases with increasing span,

however the live load remains the same.

Table 3: Typical span/depth ratios, (Cobb, 2004)

Element Typical Spans

(m)

Overall depth or thickness (mm)

Simply Supported Continuous Cantilever

One way spanning slabs

Two way spanning slabs

Flat slabs

Close centre ribbed slabs

(ribs at 600 mm c/c)

Coffered slabs

(ribs at 900-1500 mm c/c)

Post tensioned flat slabs

5-6

6-11

4-8

6-14

8-14

9-10

L/22-30

L/24-35

L/27

L/23

L/15-20

L/35-40

L/28-36

L/34-40

L/36

L/31

L/19-24

L/38-45

L/7-10

-

L/7-10

L/9

L/7

L/10-12

Rectangular beams

(width > 250 mm)

Flanged beams

3-10

5-15

L/12

L/10

L/15

L/12

L/6

L/6

Columns 2.5-8 H/10-20 H/10-20 H/10

Walls 2-4 H/45 H/45 H/15-18

Retaining walls 2-8 - - H/10-14

NOTE: 125mm is nominally the minimum concrete floor thickness for fire resistance.

32



The slabs were deigned as under reinforced sections assuming that the tension concrete has

cracked thus doing no further work. The results obtained for these slabs are as follows, see

appendix A for the calculations and load analysis which will also include form-active slabs.

For these slabs, the focus area was the amount of material that is required, mainly concrete.

Therefore shear reinforcement was not designed for, in relation to shear floor slab, because of

their relative depth in comparison to RC beams are small, and thus shear failure is less likely

unless it is in the form of punching shear around column supports. These kinds of failures

however, are not part of the scope of this thesis. The focus is on showing how application of

structural form in slabs can help with reduction of material and thus improving efficiency.

5.3.1 Optimisation of the Solid Slab Design

The design of traditional solid slabs can be optimised in and of itself. This optimisation may

be according to material optimisation or even cost optimisation, see Appendix C. Material

optimisation is purely the proper balance of how much concrete is used and how much steel

is then required for the reinforcing. With cost optimisation, the cost of each material, i.e.

concrete and steel has to be considered and furthermore they have to be put in contrast with

the total cost. The optimisation of flat slab design is important so that in comparing the

efficiencies between form-active slabs and flat slabs, rigorous designs are necessary in both

cases so as to better reflect on which is the better.

Material optimisation is done with a direct influence from the chosen span/depth ratio. The

range chosen for these was obtained from Table 3 as shown in the previous page, with the

range being l/22-30 for one way spanning slabs, there is a range of possibilities that the depth

can take. Clearly, choosing a lower span/depth ratio will result in a deeper depth and a higher

ratio will give a shallower depth. However, the different depths have an effect in the amount

of reinforcing required, which is in turn linked to the rebar layout.

Due to the practical considerations related to slab design such as crack control, the amount

and layout of steel provided is of importance. Regardless of the required bending

reinforcement steel, the layout has to conform to the required crack control layout. It would

therefore be an improper use of material to design a deep solid slab with the aim of reducing

the reinforcement steel, when it does not meet crack control layout requirements. This could

be avoided by designing a shallower slab with more reinforcing which simultaneously

satisfies the spacing or layout requirements for crack control.

The graph shown in figure 12 on the following page was obtained by using the described

optimisation technique described above on a one way solid slab with the short span of 3 m.

The graph indicates how increasing the effective depth of the slab requires more concrete and

less steel and how the converse also holds. Furthermore this graph shows the efficient

intersection point were both graphs meet indicating a suitable effective depth, from which an

l/d ratio can be calculated by using the design span. As a measure of accuracy the scales of

the two vertical axes were kept constant, i.e. area of steel and volume of concrete used.

33



Ensuring the accuracy of the intersection point of these two graphs was done by aligning the

primary axis with the secondary axis. The alignment was done by keeping the ratios of the

maximum y value to the minimum y value for both the primary and secondary axis equal, i.e.

(6)

Equation 6: Ratio of maximum Y value to minimum Y value for axes alignment

In order to align the tick marks the increments on the secondary axis were calculated by a

similar factor to that of the primary axis. These alignments avoided distorting the theoretical

intersection of the two graphs because they are in different ranges of value, i.e. area and

volume. Hence, after ensuring the accurate intersection point the efficient effective depth was

extrapolated on the horizontal axis and this point represents the most efficient combination of

the two materials.

Figure 11: Material optimisation graph of steel vs. concrete, Moyaba

Cost optimisation was also considered as it has a major weighing in industry. In all material

combinations concrete is counts for the most volume, however, since steel is more expensive

than concrete, the cost efficiency calculation is necessary in determining the most optimum

combination.

Designing for a deeper slab reduces the bending reinforcing requirement; this implies that as

the concrete’s total cost increases as the steels total cost decreases and vice-versa when

designing for a shallow slab. Nonetheless a balance needs to be determined because the true

determining factors are how sensitive the total cost is to the change in either the concrete cost

or steel cost. Figure 13 below shows the relationship the concrete cost, steel cost and total

cost have with increasing effective depth of the 3m one way spanning solid slab.

0.21

0.25

0.29

0.34

0.38

0.42

125

150

175

200

225

250

100 110 120 130 140

Vo

lum

e o

f C

on

crete

w U

sed

(m

3)

Are

a o

f S

teel

Use

d (

mm

2)

Effective Depth (mm)

Steel Used

Concrete Used

34



Figure 12: Cost optimisation graph, Moyaba

Using both the material and cost optimisation graphs, a value of the effective depth required

for an efficient combination was extrapolated. This effective depth was in turn used for

calculating the span/depth ratio to be used for the design of the slabs as discussed in section

6.2. The span/depth ratio was nonetheless within the range specified by the values in Table 3.

5.3.2 Structural Efficiencies

Using equation 5 from section 4.6 as shown below, the structural efficiencies of the slabs

were calculated.

(5)

The load used was the factored load from equation 1in section 3.4.1, i.e. the load was 1.6 x

Gn. The live load (Gn) here was considered to be 3 kN/m2, this is more than the 2 kN/m

2

required by the code. This was done to illustrate the performance of form-active structures at

higher loads. The calculated structural efficiencies are plotted against the slabs’ span.

5.4 Form-Active Slab Design

In the design of form-active slabs, the load analysis results obtained for the solid slabs were

used, see appendix D. Anticipating that the form-active slabs are to be lighter than the

traditional concrete floor slabs, the loading used for the solid slabs will thus be an over-

estimate of the true loading the form-active slabs are to experience. This was left as it is since

it was a conservative assumption of load, and thus a safer one in the initial trials of designing

form-active slabs.

The design of these slabs had to adhere to the restrictions posed by the post-and-beam

configuration used in residential and small commercial buildings as mentioned in the earlier

chapters. The standing out restrictions being the purpose of these slabs gave a distinction in

0

20

40

60

80

100

120

140

160

180

100 110 120 130 140

Co

st (

R)


Cost Optimisation

Steel

Concrete

Total

35



the type of slabs to design, thus two types of suspended concrete floors were distinguished,

i.e. the roof slab and the floor slab.

5.4.1 Roof Slab Design

This type of concrete slab has less restriction when it comes to its form-active design, unlike

floor slabs the roof slab can fully adopt the shape of its bending moment diagram since it

service purpose does not require it to be flat on top, unless aesthetically specified. An added

advantage of having roof slabs take the shape of their bending moment diagram is that the

shape will help with drainage issues on the roof.

The design process as tabulated in Appendix D was done as follows: (This procedure was

applied to form-active slabs A, B, C and D of equal spans to the traditional solid slabs A, B,

C and D)

1. From the load analysis spreadsheet, the bending moment diagram of each slab, A to D

was obtained. From each bending moment diagram, a factor of the maximum moment

of each slab with its span was obtained, i.e. Span/Mmax

2. The values of these factors are to be used for translating the maximum moment

to the maximum lever arm of the proposed form-active shape for each slab, i.e.

maximum lever arm = (Mmax factor) x (Span), a single factor is to be chosen for

application on all spans.

3. The factors obtained ranged from 0.10 to 0.3, to keep the lever arms to a reasonable

size, the chosen factors were 0.10 to 0.15, i.e. l/10 –l/15

4. From equation 2 in 5.4, equation of a parabola, it can be seen that to obtain an explicit

equation of a parabola, three known points are needed in order to calculate the value

of a, b, and c. Two points are known immediately for each slab and these are the

support coordinates which will be represented as the x-intercepts on a Cartesian plane

with rectangular coordinates. Therefore each slab has points [-x, 0] and [+x, 0], the

third point is the calculated maximum lever arm as explained earlier, this will be the

y-intercept, [0, -y].

Equation of parabola: (2)

5. An automated spreadsheet was created which does the above calculations and thus

obtains a function used to plot a form-active shape for each slab, A-B related to its

own bending moment diagram.

6. The values of the parabolas were then calculated at increments of the span/10, i.e.

increment of 600mm for Slab A and 300mm for slab D. The same calculations as in

step 4-5 were done for a circle since the differences were insignificant as discussed in

section 5.4.

7. Since the form-active shapes were not semi circles, their reactions at the supports are

bound to have both vertical and horizontal components, bringing about the need for

lateral support. This is to be provided by steel bars.

36



8. The steps 1-7 give a general set up of that the form-active structure is to take; they

also show how the forces are to be distributed in the complete structural element. See

image below:

Figure 13: Form-active slab stress distribution, Moyaba

9. Due to the curved shape in the concrete member, the forces are converted to axial

compression forces, and since the concrete member meets the support at an angle to

the vertical, there will be outward forces at the support. These outward forces will

thus be taken up by a tension member being a steel rod. The figure above explains

how the forces are transmitted in the concrete through to the supports. Because of the

internal forces now being axial, the concrete and steel will work at their optimum

performance and thus less material will be needed.

10. The calculation of the axial forces is a basic calculation which derives the forces at

mid-span from the maximum bending moment. Recall that the fundamental definition

of a bending moment is as follows:

Moment force = Force x Lever arm. (7)

Equation 7: Fundamental defining equation of a moment force

It then follows that form the lever arm calculated in step 2-3 can be used together with

the maximum bending moment at mid-span for each slab to calculate the resultant

axial force in either the tension member or compression member shown in figure 12.

The equilibrium condition can also be invoked, taking moments about the tension

zone: (Compression force) x (Lever arm) = Mmax

Therefore (Compression force) = (Mmax) / (Lever arm), and vice versa for

obtaining the tension force.

11. Using the definition of stress, Stress = Force/Area, the minimum required area for

supporting the calculated force in step 10 can be calculated, this applies for both the

compression member and the tension member whereby the stress used for calculating

minimum concrete are is the concrete crushing strength, fcu, and for steel, fy.

37



12. The minimum area of steel required will be directly used to read of from the bar size

selection chart, this minimum area will have to be applied every 1m strip of the slab

due to the chosen 1m design strip. However, for the concrete the square root of the

area was used to obtain the concrete depth required.

5.4.2 Floor Slab Design

The design of suspended floor slabs has to adhere to the service limitations of having a flat

top. The design procedure will be similar to the one described above. To create a flat top, the

voids may be created towards the support. These voids are to be covered with screed, and

possibly mesh for the screed’s robustness. This is shown on figure 14 below.

Other methods may include the use of timber floors on the top side of the form-active slab.

These timber floors will be supported at the supports of the slab and at mid-span.

Figure 14: Form-active floor slab

5.4.3 Comparison of Form-active Shapes

The form-active shapes considered for the design of concrete slabs in this thesis were the

parabola and the catenary. The catenary as explained in section 5, occurs on a chain or a

tension member like a cable on a suspension bridge under the uniformly distributed load of

its own weight. This implies that a catenary will develop when its self-weight is significant.

For the application on floor slabs, a cable spanning the same distances as the chosen slabs

will have a negligible self-weight when compared to the ULS loads the slabs are to be

designed to carry. This analogy was encountered on a study of catenaries whereby a common

case study is the example of the cable on a suspension bridge, since the deck of the bridge

overshadows the self-weight of the catenary, the resulting from active shape becomes a

parabola. For this reason a parabola was chosen as the preferred form-active shape, this was

supported by the shape of the bending moment diagram of a simply supported 1m strip taking

a form of a parabola, see Appendix D.

The parabola was compared with an arc of a circle passing through the same three points, i.e.

the two supports on either end of the span and the position of the maximum lever arm.

38



Showing negligible differences to the parabola, it was established that in the producing of

construction drawings of form-active slabs, radii of these arcs can be used on conjunction

with the height of the lever arms on the chosen increment along the span.

5.4.4 Structural Efficiencies

The structural efficiencies of these slabs were calculated in the same way as those of the solid

slabs. The areas of the cross sections of the slabs were obtained from AutoCAD after having

drawn the section according to its calculated radius and the heights of lever arms per

increment used, see figure 15 below and Appendix E for Slab A, B and C. The image below

shows an example of a designed form-active slab showing the cross sectional areas used to

calculate the volumes of the concrete. The image shows the inclusion of shear blocks at the

support, this is to cater for shear as it is largest at the support.

Figure 15: Example of form-active slab showing cross sectional areas for efficiency calculations, Moyaba

5.5 Closure

The results for the outlined methodologies in this section form a basis for a comparison in the

discussion of the results to follow, the detailed calculations described in this section are

presented in their respective appendices. The method of designing form-active slabs was

found to be simpler and straightforward when compared to the traditional design process.

This simplicity is due to the fact that the transverse loads are split into compression and

tension in the design of form-active slabs, thus each member (compression and tension)

requires a simple calculation of a minimum area required to safely carry its portion of the

load from the maximum bending moment.

39


Results

6 Discussion of Results

6.1 Introduction

This section deals with the comparison of the results obtained after having gone through the

calculations described in section 5. This comparison is to be done by looking at the compiled

tables showing the efficiencies of each slab. A further discussion is to be made on the failure

methods of the form-active slabs; this will not cover the failure modes of solid slabs as these

have already been discussed in the earlier sections.

The closure of this section will be a comparison done using finite element analysis models

(FEM’s) created in Abaqus FEA. This was done in order to show in detail the performance of

form-active slabs in contrast with that of solid slabs. The models will be able to show the

stress distributions in cross sections and throughout the entire structure, thus, showing where

the structures are likely to fail and how they handle the generated stresses. As an added

discussion, deflections in both solid slabs and form-active slabs will be compared.

6.2 Comparison of Structural Efficiencies

As it was expected from the literature review and study on form-active structures, form-active

structures are more efficient than other types of structures. Having compared the solid slabs

which were designed by choosing the most efficient concrete and steel material combination

and cost optimisation, it was still found that the form-active slabs were more efficient than

the solid slabs. The following tables show the design values of the designed slabs and their

calculated efficiencies in the end. All loads carried by these slabs were resolved into the

equivalent concentrated force so that the efficiencies were obtained from total forces and not

line loads, hence the representation of the forces in kN.

Table 4: Summary of solid slab design (Appendix A), Moyaba

Property Slab A Slab B Slab C Slab D

Area of Steel Req.

Live Load

Weight of Conc.

Efficiency

599 mm2

4.8 kN

42.30 kN

0.16

432 mm2

4.8 kN

29.38 kN

0.21

308 mm2

4.8 kN

18.80 kN

0.33

192 mm2

4.8 kN

10.58 kN

0.54

Table 5: Summary of form-active roof slab design (Appendix E), Moyaba

Property Slab A Slab B Slab C Slab D

Area of Steel Req.

Live Load

Weight of Conc.

Efficiency

177 mm2

4.8 kN

14.0 kN

0.34

139 mm2

4.8 kN

11.0 kN

0.44

99 mm2

4.8 kN

6.8 kN

0.71

68 mm2

4.8 kN

5.0 kN

0.96

40


Results

From the efficiencies shown in table 4 and 5 it can be noted that the efficiencies are

approximately doubled in the form-active slabs. This is tribute the fact that in the form-active

slabs, the concrete in the cross section does all the work to carry the compressive forces and

there is no excess concrete without structural duties. In the solid slabs, the concrete in the

tension zone can be more than half the cross section and it has no structural role in carrying

the load, thus it adds to the dead load but not to the load carrying capacity thus reducing the

structural performance of the slab.

It can also be noted that the required steel for tensile forces is drastically reduced in the form-

active slabs; therefore there is a reduction in both the concrete required and in the steel

required. These reductions are shown in table 6 below.

Table 6: Material reduction obtained by designing form-active slabs, Moyaba

Spans Flat Slab Form-active % Reduction

(mm) Conc. Vol.

(m3)

Req. Steel

(mm2)

Conc. Vol.

(m3)

Req. Steel

(mm2)

Conc. Vol.

(m3)

Req. Steel

(m2)

6000 1.29 599

0.60 177 53.7% 70.5%

5000 0.96 432

0.47 139 51.3% 67.9%

4000 0.62 308

0.29 99 53.0% 67.7%

3000 0.38 192

0.21 68 43.3% 64.7%

A set of graphs showing patterns from these tables were produced, these graphs show a

general trend that with decreasing span, less material is used, i.e. both concrete and steel. The

latter is an obvious case, the main purpose of the graphs were to show a comparison of the

trends of the solid slab and the form-active slab. It can be seen that regardless of the span,

there is still a major reduction in the material required when designing form-active slab.

Moreover, this is strengthened by the overall increased structural efficiency, i.e. load carrying

capacity of the form-active slabs.

Figure 16: Comparison of required concrete in Solid and Form-active slabs for different spans, Moyaba

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

6000 5000 4000 3000

Min

imu

m r

eq

uir

ed c

on

c.

vo

l. (

m3)

Span (mm)

Solid Slab

Form Active Slab

41


Results

Figure 17: Comparison of required steel in Solid and Form-active slabs for different spans, Moyaba

From the results shown, it can be deduced that there is an overall improvement in structural

efficiency and material reduction, both in concrete and steel.

6.3 The Effect of Increasing the Concrete Crushing Strength fcu

As mentioned in the literature review section 3.7, Sustainable Design, it was necessary to

investigate the effect of increasing or decreasing the concrete’s crushing strength as it is

directly linked to the sustainability of concrete as a material itself.

For solid slab design it was shown in section 3.3.1 Inherent Inefficiencies in Solid Slab

Design; that increasing the crushing strength of concrete does not have a major impact to the

load carrying capacity of solid slabs, (or similar structural components like beams). This is

because most of the concrete does no structural work, i.e. the concrete in the tension zone is

assumed to carry no load. Therefore it holds that improving a part that will be assumed to do

no structural work is pointless.

With form-active slabs however, it was found that using concrete strengths as low as 15 MPa

gave safe designs. Furthermore the increasing of the concrete crushing strength significantly

changes the amount of concrete required to a lesser amount. This sensitivity to the concrete’s

crushing strength is as a result of the underlying design equation of designing form-active

structures, i.e. Stress = Force/Area whereby the stress is the concrete crushing strength. The

required minimum cross sectional area for concrete is directly related to the fcu, thus making

any change to the crushing strength noticeable in the results obtained.

0

100

200

300

400

500

600

700

6000 5000 4000 3000

Min

imu

m r

eq

uir

ed s

teel

are

a/m

(mm

2)

Span (mm)

Solid Slab

Form Active Slab

42


Results

6.4 Comparisons from FEA Models

Finite element models were created for the purpose of checking how the form-active slabs

fail. Failure modes of the slid slabs are covered in their design process and thus are well

established and can be accounted for, however. The failure modes of form-active structures

are also well documented, however, because of the way the forces are distributed in them a

common mistake is assuming that this distribution is even throughout the structure, and it is

not.

As established in section 4.3 Influence of load pattern, the form-active shape is a direct result

of the load pattern and magnitude. Thus a load eccentricity is a potential failure hazard as it

will change the stress distribution in the structure. The FEA models help in showing were

potential hazards are that might need safety reinforcing.

6.4.1 Setting up of Abaqus Models

The slabs used for the finite element analysis were the 6m span solid slab and the 6m span

form-active slab. In the case of the form-active slab, the dimensions we measured from the

drawings produced in AutoCAD 2013. The dimensions for the solid slab were a simple

rectangle of effective depth (d) = 214 mm and strip of 1000 mm. All dimensions and units

were kept as mm, N, and MPa (N/mm2), this was done because Abaqus does not have any

particular units and thus requires the consistency of the user with their chosen units.

Figure 18: Dimensions used for modelling the 1m strip form-active slab, Moyaba

The material properties for concrete were calculated for a concrete crushing strength of 25

MPa as this was the strength used for the previous calculations.

43


Results

The Young’s Modulus of concrete was obtained as follows:

Equation 8: SI Equation of concrete's elasticity modulus based on empirical values, PCI

(8)

The Poisson’s ration for the concrete was taken as 0.2. These material properties were applied

to both the solid slab model and the form-active slab model. This allowed the models to

mimic the behaviour of the slabs under loading conditions as they would within the material’s

elastic range. Reinforcing steel was not modelled in this case because the aim of the

modelling is to show were the structures reach yield point with regard to concrete.

The fixity of the models was modelled as fixed in both sides. This was done in order to model

them as close to the real life scenario as possible, even though the slabs were designed as pin

supported, there will be a degree of fixity in the constructed product. Designing them as

having pinned support is nonetheless a safe design because it results in larger members since

the resulting moment is higher than with fixed support members.

The loading of the models was defined as a pressure. This pressure was calculated from the

line loads used in the design of these slabs, see appendix B. The pressure was converted to

MPa, i.e. Line load = 11 kN/m,

Pressure = 0.011N/mm2 to remain consistent with the chosen set of units.

A single static analysis was done, which means that the initial conditions is the slab models

with their fixed supports under no loading, this is followed by increments of loading until

they reach the ultimate load, ULS.

Figure 19: Solid Slab A model showing loading and end supports (left); Form-active Slab A showing

loading and end supports (right), Moyaba

Having set up both models, a deflection analysis was created and submitted to obtain the

redistribution of stresses in both models. For the solid slab, the distribution can be predicted

to be the same as the design assumes, i.e. hogging moments at the supports and a sagging

moment at mid-span. For the form-active slab, the stress distributions in the cross section are

expected to be increasing from the neutral axis towards the extreme tension fibres and the

extreme compression fibres in a linear manner. For the form-active slab, the stress

distribution in the cross section is expected to be only in compression and not tension.

44


Results

6.4.2 Discussion of FEA Results

The results from the FEA models under full ULS loads are discussed below. These results are

contour plots of stress distributions. Theses stresses are essentially a resultant stress of all the

principal normal stresses together with the shear stresses.

Abaqus uses the von Mises stresses as these resultant stresses, or alternatively one could view

the stresses separately, but for the purposes of this analysis the total stresses were needed in

order to compare them with the maximum yield strength of concrete, 25MPa and its tensile

strength which is ca. 2.5MPa.

The following equation is used in Abaqus in order to compute the von Mises stresses:

Equation 9: von Mises stress equation

(9)

This is in effect the resultant stress of all the principal and shear stresses acting on an

element, and can thus be compared to the yield stresses in order to check whether a structure

fails and where it fails if at all it does. From viewing the contour plots, caution zones can be

determined where reinforcing might be needed as a measure of conservativeness in the form-

active structures. Furthermore, deflections in solid slabs can be compared to those of form-

active slabs at equal magnification factors.

The following figures show the contour plots of the von Mises stresses on the solid slab

model and on a cross section of the mid-span of the solid slab. This is then followed by the

contour plots of the von Mises stresses in a form-active slab. For the form-active slab, three

different views were inspected; the top view, isometric top view and the isometric bottom

view. This was done to show the caution zones in the slab that may need reinforcing. The

mid-span cross section of the form-active slab was also plotted to compare it with that of the

solid slab. Finally a comparison of the deflections of the solid slab and form-active slab are

shown at equal magnification factors.

45


Results

Figure 20: Von Misses contour plot for the 6m span solid slab fixed at both ends, Moyaba

This figure shows that the maximum stresses are found nearest the supports and also at mid-

span. It can also be seen that the expected stress distribution in cross section is that of the

traditional stress block diagram with increasing tension or compression stresses increasing

from the neutral axis in either direction, this is also shown in figure 21 below which shows

the stress distributions at an isolated element group from mid-span of the above figure.

Figure 21: Element group of the mid-span of the solid slab in figure 20, Moyaba

This figure shows clearly the expected stress distribution increasing from the neutral axis to

the extreme compression fibre on the top face and increasing to the extreme tension fibre at

the bottom.

46


Results

Figure 22: Isometric top view of form-active slab showing the von Mises contour plot, Moyaba

This figure above shows the relatively constant stress distribution of the von Mises stresses in

comparison to those shown in figure 20 for the solid slab under full ULS loading. In this

particular case the form converts almost all the forces into compression as predicted and this

is due to the lever arm at mid-span. The lever arm factor used for this slab was l/10. As the

lever arm increases, the stress distribution becomes more constant and the rest of the stresses

in the slab also go in full compression. The lever arm factor for full compression will be l/2,

which is in effect a semi-circle which will have no need for lateral support except for the

prevention of failure due to deflection, hence the use of elements such as buttresses in other

instances.

Figure 23: Top view of form-active slab showing the von Mises contour plot, Moyaba

47


Results

Figure 23 shows the reduces stresses on the top face near the supports unlike the case of the

solid slab whereby the top and bottom face of the elements near the supports experience the

highest stresses and are more likely to need reinforcing.

Figure 24: Isometric bottom view of the form-active slab showing the von Mises contour plot, Moyaba

Figure 24 above shows clearly shows that the bottom face nearer the support is likely to

experience more stresses and thus may require reinforcing as a measure of safety.

Nonetheless, the maximum von Mises stresses as shown on all the form-active slab models

are 1.318 MPa, this is below the limit of 2.5 MPa with regard to tension forces the concrete

can withstand, however reinforcement in these areas would need to be provided as insurance

and also since shear is highest at the supports, see appendix B.

Figure 25: Cross section of the mid-span of the form-active slab showing the von Mises stresses, Moyaba

48


Results

Figure 25 on the previous page shows a different stress distribution to the one shown in figure

21 on page 52. Unlike the traditional stress distribution in RC slab design this stress

distribution increases from the lowest value at the bottom fibre and the highest value at the

top fibre. In this stress distribution all stresses are in compression and all the concrete has

structural value as all of it carries a portion of the load, thus accounting for its presence in the

structure. This distribution shows no need for tension reinforcing within the structure from

mid-span going towards the supports except for the case discussed in figure 24. The only

reinforcement required is therefore the crack control reinforcement. Unlike the solid slab

which requires sagging moment reinforcements at mid-span, the form-active slab tends to

have minimum stresses at mid-span on the bottom fibre.

A close comparison of the deflections in both the solid slab and form-active slab is shown in

figures 20 and 22. Figure 20 shows the deflected shape of the solid slab fixed on both ends

magnified 250 times and figure 22 shows the deflected shape of the form-active slab also

magnified 250 times. It is clear that the deflections in the solid slab are magnitudes more

apparent than those of the form-active slab. Since both the slabs span 6 m, this comparison is

valid.

6.4.3 Failure Modes of Form-active Slabs

Failure modes for form-active slabs may split into 3 main modes, these are

Failure due to eccentric loading

This mode has been described in section 5 where the design of form-active structures was

discussed. This failure occurs if an excessive concentrated is applied to the structure it may

fail as it was designed specifically for a particular load pattern in a given range of

magnitudes.

Shear Failure at supports

This failure is avoided by providing shear reinforcements near the supports particularly on

the bottom face of the slab. The cross section of the slab may also be thickened as a means of

insurance.

Failure to improper connection of the tension member at support

This failure is based on the side effects of not having concentric forces. The form-active slab

is split into the compression member which is the concrete slab itself and the tension member

which is the steel bars provided for lateral restraint. The resultant force of the compression

member should intersect with the resultant force of the tension member along the line of

action of the supporting walls weight. If these forces do not intersect at a common point a

couple will form and thus it will create a mechanism which will result in rupture of the slab at

the connection point. Figure 26 on the following page shows the connection detail as it

should be to avoid this failure and also the shear failure at the support.

49


Results

Figure 26: Detail showing correct support connection and lateral support connection in form-active slabs,

Collis

The figure shows that a proper connection is one whereby all lines of actions are intersecting

at a single point to avoid formation of a couple at the support. These have to be detailed

properly as shown in the drawings for each form-active slab, see appendix E.

The reinforcing steel shown in the form-active slab near the support is for providing extra

shear reinforcement as well as to complete the end connection of the slab. The Tension

member providing lateral support ought to adhere to the required steel are as calculated and

shown on each drawing.

6.5 Closure

The results discussed in this section, both from the design calculations described in the

methodology and attached in the appendices as well as the results from the finite element

analysis of the slabs were used to draw specific conclusions as discussed and tabulated. These

specific conclusions are the basis of the generalised conclusions discussed on the next page

under section 7, Conclusions.

50


Conclusion

7 Conclusions

7.1 Specific Conclusions

The literature review produced in this thesis has shown form-active structures in history that

were soundly designed. Moreover, it has shown that the application of form-active structures

can be applied virtually in all types of structures in different ways while appreciating the

restrictions certain applications might pose.

A set of specific conclusions can be drawn from the discussion of results given in the

previous section, these conclusions are explicitly for the traditional solid slabs A, B, C and D

alongside the corresponding form-active slabs A, B, C and D;

In all the above mentioned slabs, there are significant reductions in the amount of

concrete and steel used when designing form-active slabs. The reduction in the

concrete used ranges from 43.3% reduction in the 3 m spanning slabs to 53.7% in the

6 m spanning slabs. The increase in the reduction of concrete used as span increases is

due to the fact that in larger spans the depth is nominally higher for the traditional

solid slabs, whereas the depth in the form active slabs only increases by a lesser

fraction as span increases.

With regard to the reinforcing steel used, the reduction is higher than the concrete

reduction, and this is because form active slabs are designed to eliminate the need of

reinforcing steel within the slab. Thus, the steel is only needed as a tension member to

resist the lateral forces generated and the only steel needed in the slab is mainly for

the connection to the support (also acting as shear resistance) and for crack control.

The steel reduction ranged from 64.7% reduction in the 3m spanning slabs to 70.5%

for the 6 m spanning slabs. These values are tabulated in table 6 under section 6.2.

Having optimised the solid slab design according to material and cost optimisation, it

was concluded that form active slabs are still much less material intensive and thus

more sustainable. In addition the form active slabs A to D, are consequently more

economic cost wise.

The efficiency of the traditional solid slabs was found to be approximately half that of the

efficiency of form-active slabs in all the slabs A, B, C and D. This is indicative of the

structural integrity of the form-active slabs themselves. For a more critical and thorough

analysis of the structural performance of form-active slabs in comparison to the traditional

solid slab, the solid slab A and form-active slab A were modelled and analysed in Abaqus

FEA. This was done in order to obtain an in-depth evaluation of the slabs’ performance under

simulated loading. The choice of the slabs A was due to their large spans, thus the evaluation

was done on the worst case scenario amongst the slabs, i.e. the maximum span of 6 m.

From analysing the results from the finite element analysis of the traditional solid slab A and

the form active slab A, the performance of the form active slab was found to be higher than

that of the traditional solid slab. This performance was with regard to load carrying capacity

51


Conclusion

which is related to the calculated structural efficiencies and the performance was also with

regard to deflections.

The maximum von misses stresses generated in the solid slab were higher than the ones

generated in the form-active slab, this is suggestive that the traditional solid slab is more

likely to reach failure strength before the form-active slab reaches failure. The reason for the

differences in the stresses generated in the slabs, are due to the stress distributions. From a

close analysis of section cuts at the mid-span of both the traditional and form-active slabs, it

was shown that the form-active slab had a stress distribution of compressive stresses alone at

mid-span and this distribution remains virtually constant as we move towards either support.

This elimination of tensile stresses explains why not much reinforcing is needed in the form

active slabs. The traditional solid slab showed the expected stress distribution of increasing

tensile and compressive forces from the neutral axis, this is shown in figures 21 and 25 in

section 6.4.2.

The second performance criterion analysed was the resulting deflections. The deflections of

the traditional solid slab were within the limiting value of l/250 because the corresponding

span/depth ratios were used in designing them. However, in comparison to the deflections of

the form-active slabs under the same loading it was deduced that form active slabs have

relatively smaller deflections than the traditional solid slabs. This is because of the camber in

the form active shape generated by a uniformly distributed load, this is shown in figures 20

and 22 in comparison to the un-deflected ones in figure 19.

With regard to failure modes, the traditional solid slab’s failure modes are common modes of

failure covered in the traditional design process. The failure is due to cracking at positions of

maximum hogging and sagging moments, and excessive deflections. Shear failure for slabs is

less common due to their relative shallow depths in contrast to those of beams.

The failure modes for the form-active slabs are mostly due to the shear at the supports and

whether the connection of the tension member is done correctly or not. For this reason, shear

reinforcement is needed on the under-side of the slabs nearer the supports since this is a likely

place for tension forces to arise, see figure 24. Increasing the form-active slab thickness near

the supports can also help as insurance against shear failure, this is because the increased

thickness gives a larger area of concrete which can better resist the internal shear forces at

near the supports. The connection of the tension member should be done in such a way that

the resultant compression force in the slab coincides with the resultant force of the tension

member and that of the supporting wall at a common point. This ought to be done to avoid

the formation of a couple moment at the support which can lead to rapture of the slab at the

support.

From these findings it was determined that all the form-active slabs, A through to D were

superior to their traditional solid slab counterparts. This is with respect to structural efficiency

i.e. load carrying capacity, structural performance according to serviceability and in

deflections. The internal stress distributions in the form-active slabs are more favourable than

that of the traditional slabs and additionally the material used is significantly less.

52


Conclusion

7.2 General Conclusions

Having investigated the overall design of form-active structures it was deduced that the

important things to note are the loading arrangement, the magnitude of the loading as well as

the type of structure that is being designed, i.e. a rigid or flexible structure; flexible structures

are naturally form-active while compressive structures require the imitation of a flexible

structure’s mirrored shape under similar loading conditions. The latter is the case with the

design of form-active slabs.

Upon the critical inspection of the traditional design process of the solid slab, it was found

that a major inherent inefficiency in the design is the fact that more than half the concrete in

the cross section is designed to carry no load, thus it adds to the dead load without carrying

any load at all. This gives these slabs a relatively lower structural efficiency. For this reason,

the design of form-active slabs is justified. Since the form-active slab is almost entirely in

compression, all the concrete has structural worth because they carry a portion of the load.

Furthermore, increasing the concrete crushing strength in these slabs significantly increases

the structural efficiency of the slabs.

Generally the performance of form-active slabs is much higher than that of the traditional

solid slab, deflections are minimised, and the amount of concrete is reduced by up to 60%

while the amount of steel required is reduced by up to 70%. Failure modes in these slabs can

be avoided by careful construction methods which avoid formations of couple moments at

supports, preventing excessive point loads and providing adequate shear resistance near the

supports. In constructing these slabs more often less material will be wasted, thus moving

towards sustainability in concrete structural design.

The application of one way spanning form-active slabs can be implemented in most

residential and small commercial buildings. This can lead to a cumulative saving of material

and cost while improving the structural integrity of these structures. Larger building

structures such as hospitals and malls can benefit from the application of one-way spanning

form-active slabs; these types of buildings are not explicitly rectilinear in design, they are

made up of rectangular grids with large hallways or corridors connecting the grids. These

corridors can benefit greatly if the slabs are designed as one way spanning form-active slabs

with wooden decks on the top face for example, see appendix E. The corridors are long and

therefore the concern lies with the cross-span which is the shorter span, making them

appropriate practical applications for form-active slabs. This type of application can also help

with regard to serviceability and maintenance of service ducts such as piping and air

conditioning since these ducts can be easily accessed from the top of the slabs.

A further related research field from which the slabs designed in this thesis are a special case,

is the research on shell structures. These are structures concerned with the structural form in

more than one dimension, thus instead of one way spans, the shape becomes a 3 dimensional

shell structure such as the roof dome of the Pantheon in Rome. These research fields are

important because they help with regards to designing efficiently and making engineering

materials perform at their optimum.

53


References

8 References

Anastas, P., & Zimmerman, J. (2003). Design Through the 12 Principles Green Engineering.

Environmental Science & Technology, 95A-101A.

Brundtland, & World Commision on Enviroment and Development. (1987). Our Common Future.

Oxford University Press.

Călin, S., Gînţu, R., & Dascălu, G. (2009, February 3). Summary of Tests and Studies Done Abroad

on the Bubbledeck System. Retrieved July 04, 2013

CCAA, C. C. (2003, April). Guide to Residential Floors. Retrieved October 12, 2013, from Cement

Concrete & Aggregates Australia:

http://www.concrete.net.au/publications/pdf/Res%20Floors%20Web.pdf

CCAA, C. C. (2004, August). Concrete Basics: A guide to concrete practice 4th Ed. Retrieved July

19, 2013, from Cement Concrete & Aggregates Australia:

http://www.concrete.net.au/publications/pdf/concretebasics.pdf

Cobb, F. (2004). Structural Engineer's Pocket Book. Oxford: Elsevier Butterworth-Heinemann

publications.

CRSI, C. a. (1998). Reinforcing Steel in Slab-on-Grade. Illinois: Concrete and Reinforcing Steel

Institute.

Holedeck®. (2013). Holedeck®, The lean structure. Retrieved 07 19, 2013, from Holedeck®:

http://g.virbcdn.com/_f2/files/8b/FileItem-281116-HOLEDECKbrochure201303.pdf

Klane, D. (2007, April 29). Structural Engineers, Sustainability and Leed®. Retrieved July 16, 2013,

from http://content.asce.org/files/pdf/SEICongressStructuralengineersandLEED07Apr29.pdf

Lai, T. (2010, July 15). Structural behavior of Bubbledeck(R) slabs and their application to

lightweight bridge decks. Retrieved October 12, 2013, from dspace.mit.edu:

http://www.dspace.edu/bitstream/handle/1721.1/60774/693573130.pdf

Lárus H. Lárusson *, G. F. (2013). Prefabricated floor panels composed of fiber reinforced concrete.

Engineering Structures, 1-12.

Macdonald, A. J. (2002). Structure & Architecture - 2nd ed. Oxford: Architectural Press.

Rousseau, C., & Saint-Aubin, Y. (2008). Calculus of Variations. Retrieved October 11, 2013, from

math.berkeley.edu: http://math.berkeley.edu/~strain/170.S13/cov.pdf

RSC, R. S. (2008). The Concrete Conundrum. Retrieved July 16, 2013, from

www.chemistryworld.org: http://www.rsc.org/images/Construction_tcm18-114530.pdf

Salvadori, M. (1980). Why Buildings Stand Up. New York: W.W. Norton & Company.

Salvadori, M., & Heller, R. (1963). Structure in Architecture. New Jersey: Prentice-Hall.

Salvadori, M., & Levy, M. (1987). Why Buildings Fall Down. New York: W.W. Norton & Company.

54


References

The South African Bureau of Standards. (2000, March 31). The Structural Use of Concrete: SABS

0100-1. Pretoria, Gauteng Province, South Africa.

Wilby, C. B. (1977). Concrete for Structural Engineers. London: Boston Newness-Butterworths.

55


Appendices

Appendices

Appendix A - Solid Slab Design Calculations 56

Appendix B - Load Analysis 61

Appendix C - Optimisation of Solid Slabs, Slab A and Slab D 65

Appendix D - Form-Active Slab Design 68

Appendix E - Drawings 72

56


Appendix A

Appendix A - Solid Slab Design Calculations

57


Appendix A

Slab A Calculations Reference

1m Strip Design for Mmax

Loading

arrangement

wl2/8

ULS loads Depth (d) of slab = 214 mm as determined

for a design strip of 1 m and ϒc = 23.5 kN/m3

Gn = 0.214 × 1 m × 23.5 = 5.04 kN/m

Qn = 3 kN/m2 × 1 m strip = 3 kN/m

n = 1.2 Gn + 1.6 Qn = 1.2(5.04) + 1.6(3) =11 kN/m

SANS

0100-1

Clause:

4.2.2.1

Moments of

resistance

No redistribution of moments: consider K1 = 0.156

Kn Mu

bd2 fcu

48.8 106

(1000)(214)2(25)

0.043 0.156

∴ Tension reinforcing only

SANS

0100-1

Clause:

4.3.3.4.1

Area of steel

required

Since only tension reinforcement is being designed for

mm

mm

dk

dZ

204204

2044.0

043.025.05.0214

95.04.0

25.05.0

∴ use 204 mm

x d Z /0.45

24mm

Asmin Msag

0.87 fyZ

599mm2

Provide Y10 mm ∅ bars @ 130 mm c/c: 604 mm2

SANS

0100-1

Clause:

4.3.3.4

with

reference

to figure 4

0 kNm

20 kNm

40 kNm

60 kNm

0 m 2 m 4 m 6 m

Slab A

58


Appendix A

Slab B Calculations Reference


Loading

arrangement

wl2/8

ULS loads Depth (d) of slab = 192 mm as determined

for a design strip of 1 m and ϒc = 23.5 kN/m3

Gn = 0.192 × 1 m × 23.5 = 4.52 kN/m


n = 1.2 Gn + 1.6 Qn = 1.2(4.52) + 1.6(3) ≅ 10 kN/m

SANS

0100-1

Clause:

4.2.2.1

Moments of

resistance


Kn Mu

bd2 fcu

31.9 106

(1000)(192)2(25)

0.035 0.156


SANS

0100-1

Clause:

4.3.3.4.1

Area of steel

required


mm

mm

dk

dZ

183185

1834.0

035.025.05.0192

95.04.0

25.05.0

∴ use 183 mm

x d Z /0.45

21mm

Asmin Msag

0.87 fyZ

437mm2


SANS

0100-1

Clause:

4.3.3.4

with

reference

to figure 4

0 kNm

10 kNm

20 kNm

30 kNm

40 kNm

0 m 1 m 2 m 3 m 4 m 5 m

Slab B

59


Appendix A

Slab C Calculations Reference


Loading

arrangement

wl2/8

ULS loads Depth (d) of slab = 154 mm as determined for a design

strip of 1 m and ϒc = 23.5 kN/m3

Gn = 0.154 × 1 m × 23.5 = 3.62 kN/m


n = 1.2 Gn + 1.6 Qn = 1.2(3.62) + 1.6(3) ≅ 9 kN/m

SANS

0100-1

Clause:

4.2.2.1

Moments of

resistance


Kn Mu

bd2 fcu

18.2 106

(1000)(154)2(25)

0.031 0.156


SANS

0100-1

Clause:

4.3.3.4.1

Area of steel

required


Z d 0.5 0.25k

0.4

0.95d

154 0.5 0.250.031

0.4

146mm

148 146mm

∴ use 146 mm

x d Z /0.45

17mm

Asmin Msag

0.87 fyZ

312mm2


SANS

0100-1

Clause:

4.3.3.4

with

reference

to figure 4

0 kNm

5 kNm

10 kNm

15 kNm

20 kNm

0 m 1 m 2 m 3 m 4 m

Slab C

60


Appendix A

Slab D Calculations Reference


Loading

arrangement

wl2/8

ULS loads Depth (d) of slab = 125 mm as determined for a

design strip of 1 m and ϒc = 23.5 kN/m3

Gn = 0.125 × 1 m × 23.5 = 2.94 kN/m


n = 1.2 Gn + 1.6 Qn = 1.2(2.94) + 1.6(3) ≅ 8 kN/m

SANS

0100-1

Clause:

4.2.2.1

Moments of

resistance


Kn Mu

bd2 fcu

9.3106

(1000)(125)2(25)

0.024 0.156


SANS

0100-1

Clause:

4.3.3.4.1

Area of steel

required


Z d 0.5 0.25k

0.4

0.95d

125 0.5 0.250.024

0.4

119mm

122 119mm

∴ use 119 mm

x d Z /0.45

14mm

Asmin Msag

0.87 fyZ

197mm2


SANS

0100-1

Clause:

4.3.3.4

with

reference

to figure 4

0 kNm

2 kNm

4 kNm

6 kNm

8 kNm

10 kNm

0 m 1 m 2 m 3 m

Slab D

61


Appendix B

Appendix B - Load Analysis

62


Appendix B

Analysis of Loads

Slab Type

A B C D

Lx (Short Span, mm) 6 000 5 000 4 000 3 000

b (Design Strip, mm) 1 000 1 000 1 000 1 000

d (Effective Depth) 214 192 154 125

Span/depth ratios, Table 10

SANS 0100-1: 4.3.6.2.1

Normal Conc. Density,

kN/m3

23.5 23.5 23.5 23.5

Dead Load, kN/m 5.04 4.52 3.62 2.94

Live Load, kN/m2 3 3 3 3

Ultimate Load (n), kN/m 11 10 9 8

(1.2Gn+1.6Qn)

SANS 0100-1: 4.4.3

fcu 25

fy 460

Slab A

Slab B

Reactions Va, Vb 32.52857


Sear Eqn. = V-(nx)Bending Moment Eqn. = Vx-(nx

2)/2

Shear Moment

Shear Moment

x1 0 mm 32.53 kN 0.00 kNm

x1 0 mm 25.56 kN 0.00 kNm

x2 600 mm 26.02 kN 17.57 kNm

x2 500 mm 20.45 kN 11.50 kNm

x3 1200 mm 19.52 kN 31.23 kNm

x3 1000 mm 15.33 kN 20.45 kNm

x4 1800 mm 13.01 kN 40.99 kNm

x4 1500 mm 10.22 kN 26.84 kNm

x5 2400 mm 6.51 kN 46.84 kNm

x5 2000 mm 5.11 kN 30.67 kNm

x6 3000 mm 0.00 kN 48.79 kNm

x6 2500 mm 0.00 kN 31.95 kNm

x7 3600 mm -6.51 kN 46.84 kNm

x7 3000 mm -5.11 kN 30.67 kNm

x8 4200 mm -13.01 kN 40.99 kNm

x8 3500 mm -10.22 kN 26.84 kNm

x9 4800 mm -19.52 kN 31.23 kNm

x9 4000 mm -15.33 kN 20.45 kNm

x10 5400 mm -26.02 kN 17.57 kNm

x10 4500 mm -20.45 kN 11.50 kNm

x11 6000 mm -32.53 kN 0.00 kNm

x11 5000 mm -25.56 kN 0.00 kNm

63


Appendix B

Slab C

Slab D



Sear Eqn. = V-(nx)Bending Moment Eqn. = Vx-(nx

2)/2

Shear Moment

Shear Moment

x1 0 mm 18.28 kN 0.00 kNm

x1 0 mm 12.49 kN 0.00 kNm

x2 400 mm 14.62 kN 6.58 kNm

x2 300 mm 9.99 kN 3.37 kNm

x3 800 mm 10.97 kN 11.70 kNm

x3 600 mm 7.49 kN 5.99 kNm

x4 1200 mm 7.31 kN 15.35 kNm

x4 900 mm 5.00 kN 7.87 kNm

x5 1600 mm 3.66 kN 17.55 kNm

x5 1200 mm 2.50 kN 8.99 kNm

x6 2000 mm 0.00 kN 18.28 kNm

x6 1500 mm 0.00 kN 9.37 kNm

x7 2400 mm -3.66 kN 17.55 kNm

x7 1800 mm -2.50 kN 8.99 kNm

x8 2800 mm -7.31 kN 15.35 kNm

x8 2100 mm -5.00 kN 7.87 kNm

x9 3200 mm -10.97 kN 11.70 kNm

x9 2400 mm -7.49 kN 5.99 kNm

x10 3600 mm -14.62 kN 6.58 kNm

x10 2700 mm -9.99 kN 3.37 kNm

x11 4000 mm -18.28 kN 0.00 kNm

x11 3000 mm -12.49 kN 0.00 kNm

0 kNm

10 kNm

20 kNm

30 kNm

40 kNm

50 kNm

60 kNm

0 m 1 m 2 m 3 m 4 m 5 m 6 m

Bending Moment Diagrams

Slab A

Slab B

Slab C

Slab D

64


Appendix B

Determination of Parabolic Factors = lx/Mmax

Slab A lx/Mmax= 0.122969

Slab B lx/Mmax= 0.156509

Determination of Parabolic Factors = lx/Mmax

Slab C lx/Mmax= 0.218855

Slab D lx/Mmax= 0.32032

Therefore use lx/10 to lx/15.

NB: The weight of the strip of slab + the loading it carries, are much more significant than a

cable of the same span. Therefore the required form-active shape will take the form of a

parabola (similar to its bending moment diagram) as opposed to the catenary that would arise

on the cable under its own self-weight. This is similar to a scenario of a cable supporting a

deck of a suspension bridge; the cable follows the shape of a parabola.

-40 kN

-30 kN

-20 kN

-10 kN

0 kN

10 kN

20 kN

30 kN

40 kN

0 m 1 m 2 m 3 m 4 m 5 m 6 m 7 m

Shear Force Diagrams

Slab A

Slab B

Slab C

Slab D

65


Appendix C

Appendix C - Optimisation of Solid Slabs, Slab A

and Slab D

66


Appendix C

Slab A

Depth

(d) Area of

Steel (mm2)

Vol. of

Conc. (m3)

(Steel

Cost)/m (Conc.

Cost)/m Total Cost

273 mm 561.00 1.64 R 528

R 327 R 856

250 mm 561.00 1.50 R 528

R 300 R828

231 mm 595.00 1.38 R 560

R 277 R 837

214 mm 604.00 1.29 R 569

R 257 R 826

200 mm 628.00 1.2 R 592 R 240 R 832

NB: Using cost of Conc @ R200/m3,

Using cost of Steel @ R20/kg

Therefore: Use depth between 215mm to 235mm, thus l/d =28 and l/d = 26 respectively.

Choose l/d = 28

0.90

1.08

1.26

1.44

1.62

1.80

505

535

565

595

625

655

200 220 240 260

Vo

lum

e o

f C

on

crete

w U

sed

(m

3)

Are

a o

f S

teel

Use

d (

mm

2)


Material Optimisation

Steel vs. Concrete

Steel

Concrete

200

300

400

500

600

700

800

900

200 220 240 260

Co

st (

R)


Cost Optimisation

Steel

Concrete

Total

67


Appendix C

Slab D

Depth

(d) Area of

Steel (mm2)

Vol. of

Conc. (m3)

(Steel

Cost)/m (Conc.

Cost)/m Total

Cost

136 mm 188.00 0.41 R 89

R 82 R 170

125 mm 193.00 0.38 R 91

R 75 R 166

115 mm 209.00 0.35 R 98

R 69 R 168

107 mm 217.00 0.32 R 102

R 64 R 166

100 mm 226.00 0.3 R 106 R 60 R 166

NB: Using cost of Conc @ R200/m3,

Using cost of Steel @ R20/kg

Therefore: Use depth between 120 mm to 125 mm, thus l/d =24. Choose l/d = 24

0.21

0.25

0.29

0.34

0.38

0.42

125

150

175

200

225

250

100 110 120 130 140

Vo

lum

e o

f C

on

cret

ew U

sed

(m

3)

Are

a o

f S

teel

Use

d (

mm

2)


Material Optimisation

Steel vs. Concrete

Steel

Concrete

0

20

40

60

80

100

120

140

160

180

100 110 120 130 140

Co

st (

R)


Cost Optimisation

Steel

Concrete

Total

68


Appendix D

Appendix D - Form-Active Slab Design

69


Appendix D

Roof Slab Design

Chosen lever arm factor = 0.1

Slab A

Slab B

Mmax 48.79 kNm

Mmax 31.95 kNm

lever arm 0.60 m

lever arm 0.50 m

comp. force 81.32 kN


tensile force 81.32 kN


fcu 25 MPa

fcu 25 MPa

fy 460 MPa

fy 460 MPa

Required Ac 3 253 mm2


Depth 57 mm

Depth 51 mm

Required As 177 mm2

Required As 139 mm2

Efficiency 0.34

Efficiency 0.44

Eqn. of Parabola : y = ax

2 + bx + c


2 + bx + c

Circle : r2 = x

2 + (y-b)

2 or y = b+(r

2-(x-a))

1/2

Circle : r

2 = x

2 + (y-b)

2 or y = b+(r

2-(x-a))

1/2

x y

x y

Point 1 0 -3

Point 1 0 -2.5

Point 2 0 3

Point 2 0 2.5

Point 3 0.6 0

Point 3 0.5 0

a -0.07

a -0.08

b 0.00

b 0.00

c 0.60

c 0.50

b (circle) -7.20

b (circle) -6.00

r (circle) 7.80

r (circle) 6.50

Parabola Circle

Parabola Circle

x1 -3000 mm 0 0.000

x1 -2500 mm 0.000 0.000

x2 -2400 mm 0.216 0.222

x2 -2000 mm 0.180 0.185

x3 -1800 mm 0.384 0.389

x3 -1500 mm 0.320 0.325

x4 -1200 mm 0.504 0.507

x4 -1000 mm 0.420 0.423

x5 -600 mm 0.576 0.577

x5 -500 mm 0.480 0.481

x6 0 mm 0.6 0.600

x6 0 mm 0.500 0.500

x7 600 mm 0.576 0.577

x7 500 mm 0.480 0.481

x8 1200 mm 0.504 0.507

x8 1000 mm 0.420 0.423

x9 1800 mm 0.384 0.389

x9 1500 mm 0.320 0.325

x10 2400 mm 0.216 0.222

x10 2000 mm 0.180 0.185

x11 3000 mm 0 0.000

x11 2500 mm 0.000 0.000

70


Appendix D

Roof Slab Design

Chosen lever arm factor = 0.1

Slab C

Slab D

Mmax 18.28 kNm

Mmax 9.37 kNm

lever arm 0.40 m

lever arm 0.30 m





fcu 25 MPa

fcu 25 MPa

fy 460 MPa

fy 460 MPa



Depth 43 mm

Depth 35 mm

Required As 99 mm2

Required As 68 mm2

Efficiency 0.71

Efficiency 0.96


2 + bx + c


2 + bx + c

Circle : r2 = x

2 + (y-b)

2 or y = b+(r

2-(x-a))

1/2

Circle : r

2 = x

2 + (y-b)

2 or y = b+(r

2-(x-a))

1/2

x y

x y

Point 1 0 -2

Point 1 0 -1.5

Point 2 0 2

Point 2 0 1.5

Point 3 0.4 0

Point 3 0.3 0

a -0.10

a -0.13

b 0.00

b 0.00

c 0.40

c 0.30

b (circle) -4.80

b (circle) -3.60

r (circle) 5.20

r (circle) 3.90

Parabola Circle

Parabola Circle

x1 -2000 mm 0 0.000

x1 -1500 mm 0 0.000

x2 -1600 mm 0.144 0.148

x2 -1200 mm 0.108 0.111

x3 -1200 mm 0.256 0.260

x3 -900 mm 0.192 0.195

x4 -800 mm 0.336 0.338

x4 -600 mm 0.252 0.254

x5 -400 mm 0.384 0.385

x5 -300 mm 0.288 0.288

x6 0 mm 0.4 0.400

x6 0 mm 0.3 0.300

x7 400 mm 0.384 0.385

x7 300 mm 0.288 0.288

x8 800 mm 0.336 0.338

x8 600 mm 0.252 0.254

x9 1200 mm 0.256 0.260

x9 900 mm 0.192 0.195

x10 1600 mm 0.144 0.148

x10 1200 mm 0.108 0.111

x11 2000 mm 0 0.000

x11 1500 mm 0 0.000

71


Appendix D

Comparison of the parabola and circle’s arc passing through the same point using Slab A

Eqn. of Parabola : y = ax2 + bx + c

Eqn. of Circle : r2 = x

2 + (y-b)

2 or y = b+(r

2-(x-a)

2)

1/2

Point Position Parabola Circle % Diff. x1 -3000 mm 0.000 0.000 0.00 x2 -2400 mm 0.216 0.222 2.59 x3 -1800 mm 0.384 0.389 1.42 x4 -1200 mm 0.504 0.507 0.62 x5 -600 mm 0.576 0.577 0.15 x6 0 mm 0.600 0.600 0.00 x7 600 mm 0.576 0.577 0.15 x8 1200 mm 0.504 0.507 0.62 x9 1800 mm 0.384 0.389 1.42 x10 2400 mm 0.216 0.222 2.59 x11 3000 mm 0.000 0.000 0.00

NB: The Parabola is used to represent the form-active shape of the bending moment diagram.

The catenary is more suitable for a cable under its own self weight. For ease of producing

practical working drawings, the parabolas were compared to a circle passing through the

same three points indicated above, the differences were found to be insignificant. Thus, when

producing working drawings, an arc with the calculated radius may be shown, or

alternatively, key points showing the parabola's height at given intervals may be show. See

corresponding drawings in appendix E.

0.00 m

0.20 m

0.40 m

0.60 m

0.80 m

-3 m -2 m -1 m 0 m 1 m 2 m 3 m

Form-active shapes

Slab A

Slab B

Slab C

Slab D

0.00 m

0.20 m

0.40 m

0.60 m

0.80 m

-3 m -2 m -1 m 0 m 1 m 2 m 3 m

Parabola vs. Circle

Slab A - Parabola

Slab A - Circle

72


Appendix E

Appendix E - Drawings

Thesis on Concrete Structural Design for Sustainability (Optimising Structural Form) 2013

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