Report Body New

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Chapter 1

INTRODUCTION

1.1 GENERAL INTRODUCTION

Beams with large depths in relation to spans are called deep beams. As per

Clause 29 of IS456: 2000 [1], a simply supported beam is classified as deep when the

ratio of its effective span L to overall depth D is less than 2. Continuous beams are

considered as deep when the ratio L/D is less than 2.5. The effective span is defined

as the centre-to-centre distance between the supports or 1.15 times the clear span

whichever is less. According to ACI-318:2008 [2] the deep beam is defined as the

ratio of effective span to depth is less than or equal to four. RC deep beams have

many useful applications in building structures such as transfer girders, wall footings,

foundation pile caps, floor diaphragms, and shear walls. Particularly, the use of deep

beams at the lower levels in tall buildings for both residential and commercial

purposes has increased rapidly because of their convenience and economic efficiency.

Deep beams have depth comparable to its span so the volume of the beam is very high

as compared to the floor beams. ACI 318 [2] provides design method for deep beams.

Strut and tie model is commonly used for the design of deep beams.

Strut is compression member and tie is tension member assumed inside the

deep beam. Design of these member in such a way that compressive force in the strut

never exceeds the permissible compressive force in concrete and tensile force never

exceed the permissible tensile force in reinforcement. Selection of strut and tie model

for deep beams need knowledge about the compression region and tension region

inside it. Concept of strut and tie model was obtained from the steel design concept.

For example jib of crane and large truss work for roofs. Here the truss member in

compression and tension region designs separately. In the deep beam tension zone and

compression zone is designed using strut and tie model. The Strut-and-Tie model

(STM) approach evolves as one of the most useful design methods for shear critical

structures. Strut-and-tie modelling (STM) is an approach used to design discontinuity

regions (D-regions) in reinforced and pre-stressed concrete structures. A STM reduces

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complex states of stress within a D-region of a reinforced or pre-stressed concrete

member into a truss comprised of simple, uniaxial stress paths. Each uniaxial stress

path is considered a member of the STM. Members of the STM subjected to tensile

stresses are called ties and represent the location where reinforcement should be

placed. STM members subjected to compression are called struts. The intersection

points of struts and ties are called nodes. Knowing the forces acting on the boundaries

of the STM, the forces in each of the truss members can be determined using basic

truss theory. With the forces in each strut and tie determined from basic statics, the

resulting stresses within the elements themselves must be compared with specification

permissible values. Since a STM is comprised of elements in uniaxial tension or

compression, appropriate reinforcement must be provided. Through the use of this

approach, an estimation of strength of a structural element can be made and the

element appropriately detailed. Unlike the sectional methods of design, the strut-and–

tie method does not lend itself to a cook book approach and therefore requires the

application of engineering judgment.

Forces in the strut and tie are determined by equilibrium equation. This

procedure becomes cumbersome when the STM is complex. It will occur in the beam

column joint, continuous deep beam etc. Computer Aided Strut and Tie (CAST) is the

software used for finding the forces in member. It is only an existing windows-based

application for STM with rich of graphical user interfaces. CAST has been developed

by the University of Illinois since 2001.It was used to serve students and practicing

engineers to grasp the concept and to try various idealized strut-and-tie models with

ease. CAST components comprise of nodes and elements. Elements represent struts or

ties meeting at nodes. For particular cases, stabilizer is necessary to make the model

numerically stable. The stabilizer is a strut with zero force. Three types of boundary

conditions can be applied at exterior nodes: plate, point loads, and supports. Body

force and support can also be applied to interior nodes within D-region. The model

must be restrained to prevent rigid body movement. In CAST, positive value of forces

is point outward from node and vice versa. Since the direction of point loads on the

boundary will follow the axis of the strut, no element connectivity is allowed on

boundary.

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Deep beam designed by strut and tie is heavy beam. Literature review shows

that its volume can be decreased without affecting its performance. The process of

arranging the material inside a body for resisting external load is called optimisation.

Extensive research has been undertaken in the field of topology optimization,

particularly in the past few decades. Largely due to the critical importance on weight

savings in aerospace structures, the topology optimization methods have been

developed primarily for applications in elastic isotropic and composite laminate

material. On the contrary, weight saving is not such a critical factor in civil

engineering design problems, and perhaps because of this, topology optimization for

these applications have been studied with less enthusiasm.

Most of structural optimization methods for RC are based on cost optimization

where the cost is assumed to be directly proportional to the amount of material. Thus

the objective function is equivalent to that of the fully stressed design, which is to

minimize the structure‟s volume or mass. Gradient-based methods such as Lagrangian

methods have been applied to sizing problems of reinforcement in a concrete beam.

Optimization of shape of concrete sections as well as sizing of reinforcement has been

formulated as a non-linear mathematical programming problem and solved for

optimal concrete section geometry, reinforcement amount and locations and for

location of the neutral axis. These methods however, do not alter the arrangements of

struts nor ties and hence offer a limited design improvement.

Structural optimization is one application of optimization. Here the purpose is

to find the optimal material distribution according to some given demands of a

structure. Some common functions to minimize are the mass, displacement or the

compliance (strain energy). This problem is most often subject to some constraints,

for example constraints on the mass or on the size of the component. The problem of

structural optimization can, be separated in three different areas: sizing optimization,

shape optimization and topology optimization

Sizing optimization is the simplest form of structural optimization. The shape

of the structure is known and the objective is to optimize the structure by adjusting

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sizes of the components. Here the design variables are the sizes of the structural

elements. Topology optimization is a mathematical approach that optimizes material

layout within a given design space, for a given set of loads and boundary conditions

such that the resulting layout meets a prescribed set of performance targets. Using

topology optimization, engineers can find the best concept design that meets the

design requirements.

Shape optimization is part of the field of optimal control theory. The typical

problem is to find the shape which is optimal in that it minimizes a certain cost

functional while satisfying given constraints. In many cases, the functional being

solved depends on the solution of a given partial differential equation defined on the

variable domain.

Topology optimization is a mathematical approach that optimizes material

layout within a given design space, for a given set of loads and boundary conditions

such that the resulting layout meets a prescribed set of performance targets. Using

topology optimization, engineers can find the best concept design that meets the

design requirements. TOP OPT, SOLID THINKING, OPTISTRUT, TOSCA,

GENESIS, ANSYS etc. are software that used for topology optimisation. Topology

optimisation with the help of ANSYS is used for optimisation of deep beam for this

project. ANSYS is complex software that gives all the necessary details for this

project. ANSYS is simulation software developed by United States. Fluid mechanics,

heat transfer, structural, electromagnetic are the some field of ANSYS. Structural

design of modern vehicle part is carried out in ANSYS. In the field of civil

engineering ANSYS analysis is very difficult. ANSYS analysis is based on finite

element method. The accuracy of the entire project is depends on the size of the

element. Finer mesh analysis is more accurate and vice versa. Finer mesh analysis of

large building very complex and need super computers. These disadvantages make the

low popularity of software in civil engineering.

Deep beam with span 7m is modelled and optimised in ANSYS. Design of

deep beam was done based on ACI: 318 2008 and IS 456: 2000. Both the design

yield approximate same result. ANSYS 12.1 with civil FEM is used for this project.

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ANSYS can per form 100 number of iteration for optimisation. Solution stops only

when results converge. Optimisation is possible only with element modelled using

solid 95 elements. This is unable to crack or crush. In order to obtain crack pattern

and deflection beam models repeated using solid 65 elements. Result of optimisation

shows new beam without decreasing performance. In this project volume is optimised

50% without loss of strength. For validation of the project an experimental study is

conducted. Due to the unavailability of modern laboratory equipment, models of deep

beam and optimised beam were tested in the laboratory. Optimised beam and deep

beam modelled in ANSYS using SOLID 65 element for determining deflection and

crack pattern. Deflection, cracking and ultimate load bearing capacity was checked

and compared with experimental results. Universal testing machine (UTM) with

maximum load capacity of 600kN was used for loading purpose. Dial gauge is used to

find deflection at mid span.

1.2 SOFTWARE USED FOR PROJECT

Analysis and optimisation of the project is carried out in Computer Aided

Strut and Tie and ANSYS respectively. Computer Aided Strut and Tie (CAST) used

for creating strut and tie model for a deep beam. On analysing the model we obtain

the compressive force in strut and tensile force in tie. Much software that is available

for topology optimisation is explained in above paragraph. Selection of software

depends on the nature of the problem. Topology optimisation is very common in

mechanical engineering field for the optimisation of vehicle parts. In civil engineering

reinforced concrete is a composite material. It can be modelled using ANSYS.

Topology optimisation of composite material can be done in ANSYS. ANSYS

analysis spread over the field of fluid mechanics, structural design, electromagnetic

problem, heat transfer problem etc. Here ANSYS software is used for optimisation of

deep beam. Degree of freedom in three directions, stress intensity, von Mises stress

field, crack pattern, stress in steel, topology optimisation density plot etc. can be

obtained from ANSYS.

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1.3 ORGANISATION OF THESIS

Chapter 1 gives introduction to this project with previous study made in this

category. All the ideas about the topics and software are used in this project depicted

in introduction. Details of software used at various at portion of project are also

described. Details of experimental programme such as type of experiment,

instruments used are discussed in this section.

Chapter 2 deals with all the past study details on this topic. All the research‟s

in this topic from last 20 years included in literature reviews. Each and every paper

was studied well and all the core details such as purpose of study and method of study

was consolidated and explained in these section. Literature review was then

concluded with importance of present study. This section contains main objective of

the project. Methodology of this project is provided in last section of this topic for

explain the details of project work.

Chapter 3 is core of this paper that is topology optimisation theory and

procedure. Different type of optimisation, definition, importance of each types,

mathematical explanation, why topology optimisation, different software available for

topology optimisation, why ANSYS etc. are explained in this section. Topology

optimisation by ANSYS software is done in this chapter.

Chapter 4 gives idea about conventional design of deep beam with ACI 318

2008 code. Design based on IS 456 2000 was done in this chapter. A comparison of

this to were provided there. All terms used in the code like strut, tie, nodal zone, etc.

were explained in this section. Beam analysed using computer aided strut and tie

(CAST) and designed using ACI and IS methods. Detailing is also provided for both

the design methods for easy assessment of section details.

Chapter 5 deals with experimental investigation part of the project. Here

design analysis and details of test specimen is described. As in every lab tests such as

test on aggregates and cube test details are explained in this chapter. Test set ups and

modelling explained with the help of figures.

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1.4 SUMMARY

Introduction to the project topics is discussed in this chapter. Various software

used for analysis and two design method of Indian standard and ACI method were

explained. Details of experimental investigation also described. Details of each

chapter in this project are explained in organisation of thesis section.

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Chapter 2

LITERATURE RIEVEW

2.1 INTRODUCTION

Optimisation in concrete beam is used for finding the optimum reinforcement,

removing the inefficient material, optimum layout of strut and tie reduce the cost of

structure etc. optimisation on concrete structure is difficulty because the composite

nature of reinforced concrete. Yet numbers of research were found for literature

review in reinforced concrete beams.

Choi and Kwak [3] presented a simple and effective algorithm for cost

optimization of rectangular beams and columns of RC frames by using a direct search

method according to the American (ACI 318-89) and Korean Codes. Design sections

for concrete elements are selected from some predetermined discrete sections which

have been accepted as suitable sections. The cost function included the costs of

concrete, reinforcement and formwork. The optimization of the entire structure was

accomplished through optimization of the individual members. A simple beam and a

10-storey RC building with two spans in each direction have been optimized as two

examples.

Chakrabarty [4] applied the geometric programming to find minimum cost

design of rectangular singly reinforced concrete beams. The area of tensile steel

reinforcement, the depth and width of the beam were the design variables. The

objective function was the cost of unit length of beam including the cost of tensile

reinforcement, concrete and formwork. He considered the strength constraints but

ignored the ductility and side constrains in his formulation. He did not introduce any

specific design code requirements in the constraint equations.

Moharrami and Grierson [5] studied minimum cost of RC building

frameworks. The width, depth and area of longitudinal reinforcement of member

sections are taken as the design variables. They used OC method to minimize the total

cost of building including the costs of concrete reinforcement and formwork, subject

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to constraints on strength and stiffness to the American Code (ACI 318-89). The

columns have rectangular cross-sections and beams can be rectangular, L or T shapes.

Although in practice the design variables of the problem take discrete values, in this

approach they have been assumed continuous.

Das Gupta et al. [6] considered minimum cost design of RC frames in

accordance with the British Code (BS 8110). The objective function is the total costs

of concrete, steel, and formwork. Design variables are width and depth of the cross-

section of each member and area of reinforcement in specific sections of members.

The problem has been transferred into an equivalent unconstraint minimization

problem using the exterior penalty function method of sequential unconstraint

minimization technique.

Khaleel et al. [7] made a comprehensive study on the optimization of simply

supported partially pre-stressed concrete girders under multiple strength and

serviceability criteria is presented, Using sequential quadratic programming a set of

optimal geometrical dimensions, amounts of pre-stressing and non pre-stressing steel,

and spacing between shear reinforcements are obtained. Results point to the need for

non pre-stressing steel to obtain economical designs; this study automates the design

of partially pre-stressed concrete girders and provides needed design solutions to

problems which are of importance to practicing engineers.

Adamu et al. [8] used continuum-type optimality criteria (COC) approach for

economic design of RC singly reinforced beams with rectangular cross section to the

European Code (CEB). The cost function consists of the cost of concrete, longitudinal

reinforcement and formwork of the beam. The design variables are the width,

effective depth, and steel ratio of a cross-section. The optimality criteria are derived

using the calculus of variation on an augmented Lagrangian function. An iterative

procedure based on optimality criteria is applied to optimize design of a RC beam

with fixed support at one end and simple support at the other end with variable depth

and width along its length.

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Al-Salloum and Siddiqi [9] presented minimum cost design of singly

reinforced rectangular concrete beams based on the American Code (ACI 318- 89)

provisions. The cost function contains the cost of concrete, bending reinforcement and

formwork of the unit length of the beam. They presented a closed-form solution for

optimal depth and steel ratio of a beam with given width and design moment in terms

of material costs and strength parameters.

Zielinski et al. [10] investigated optimum design of RC short-tied rectangular

columns under biaxial bending based on the Canadian Code (CAN3-A23.3- M84).

The cost components of the objective function are the cost of concrete, longitudinal

reinforcement and formwork. Design variables are the cross sectional dimensions of

the column and the area of tension and compression bars or, alternatively, the number

of reinforcement along the width and depth of a column cross-section. Then the

problem is solved by the Powell‟s method.

Fadaee and Grierson [11] presented the optimal design of three-dimensional

RC skeletal structures with rectangular section members subjected to biaxial

moments, biaxial shears, and axial loads to the American Code (ACI 318-95). They

used OC method to minimize the cost of concrete, steel and formwork for the

structure. The design variables are concrete cross-sectional dimensions and the area of

longitudinal bars in member sections. In another paper [38] considered optimization

of three-dimensional RC structures having shear walls, which can be subjected to pure

shear. The design variables for shear walls are the thickness of the wall, the area of

vertical reinforcement, horizontal distance between the vertical stirrups, the area of

horizontal reinforcement, vertical space between the horizontal stirrups and the area

of vertical flexural reinforcement. They used the method for design optimization of a

one-bay and one-storey space framework.

Balling and Yao [12] studied optimization of three-dimensional RC structures

with rectangular T or L-shape beams according to the ACI Code (ACI 318-89) by two

different methods. The first method is a bi-level method that optimizes concrete cross-

section dimensions in one level and the number, diameter and topology of

reinforcement in another level. The second method represents the number, diameter

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and topology of reinforcement through a single design variable, which is the total area

of steel. In this method, the area of steel and the cross-sectional dimensions of

members are optimized simultaneously. They concluded that the optimum costs

obtained from these two methods are very close to each other. Based on this

conclusion; they proposed a simplified method for cost optimization of space frames.

The cost function consisted of material, fabrication, and placement costs of the

concrete and reinforcement.

Aguilar et al. [13] was conducted study for determining the shear capacity of

deep beam based on ACI 318 2008. They tested four reinforced concrete deep beams.

The behaviour of the deep beams is described in terms of cracking pattern, load-

versus-deflection response, failure mode, and strains in steel reinforcement and

concrete. The test specimens were designed with two different approaches, which

consisted of: 1) the procedure described in Sections 10.7 and 11.8 of the ACI 318-99

Code (ACI Committee 318 1999); and 2) the Strut-and-Tie Method given in

Appendix A of the ACI 318-02 Building Code. The experimental failure load of each

specimen is compared with the load capacities calculated using the procedures given

in the ACI 318-99 Code, and Appendix A of the ACI 318-02 Building Code.

Barakat et al. [14] presented a general approach to the single objective

reliability based optimum (SORBO) design of pre-stressed concrete beams (PCB).

Several limit states were considered, including permissible tensile and compressive

stresses at both initial and final stages, pre-stressing losses, ultimate shear strength,

ultimate flexural strength, cracking moment, crack width, and the immediate

deflection and the final long term deflection. The results consist of the initial and final

pre-stressing forces, Pre-stressing losses, immediate and final deflections, and upper

and lower bounds on the parabolic tendon profile.

Sahab et al. [15] studied the cost optimisation of reinforced concrete flat slab

buildings according to the British Code of Practice (BS8110). The objective function

was the total cost of the building including the cost of floors, columns and

foundations. The cost of each structural element covers that of material and labour for

reinforcement, concrete and formwork. Cost optimisation for three reinforced

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concrete flat slab buildings is illustrated and the results of the optimum and

conventional design procedures are compared.

Govindaraj and Ramasamy [16] presented the application of Genetic

Algorithms for the optimum detailed design of reinforced concrete continuous beams

based on Indian Standard specifications. The produced optimum design satisfies the

strength, serviceability, ductility, durability and other constraints related to good

design and detailing practice. The optimum design results are compared with those in

the available literature. An example problem is illustrated and the results are

presented. It was concluded that the proposed optimum design model yields rational,

reliable, economical and practical designs.

Liang [17] presented the performance-based optimization of strut-and-tie

models in reinforced concrete beam-column connections. The performance-based

optimization (PBO) technique was employed to investigate optimal strut-and-tie

models in reinforced concrete beam-column connections. Developing strut-and-tie

models in reinforced concrete beam-column connections was treated as a topology

optimization problem. The optimal strut-and-tie model in a concrete beam column

connection was generated by gradually removing inefficient finite elements from the

connection in an optimization process. Optimal strut-and-tie models for the design

and detailing of opening knee joints, exterior beam-column connections and interior

beam-column connections are presented. He concluded PBO technology was an

effective tool for the design and detailing of structural concrete particularly the D-

regions such as beam-column connections.

Liang et al. [18] conducted topology optimization of strut-and-tie models in

non-flexural reinforced concrete members by using the Evolutionary Structural

Optimization (ESO) method for plane stress continuum structures with displacement

constraints. It was shown that the proposed procedure can effectively generate optimal

topologies of strut-and-tie models in non-flexural reinforced concrete members such

as deep beams and corbels. It was concluded that on removing inefficient materials

from the concrete member, the strut and-tie model within the member was gradually

evolved towards an optimum.

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Guerra and Kiousis [19] conducted study for achieve optimal design of

reinforced concrete (RC) structures. Optimal sizing and reinforcing for beam and

column members in multi-bay and multi-storey RC structures incorporates optimal

stiffness correlation among all structural members and results in cost savings over

typical-practice design solutions. He incorporated material and labour costs for

forming and placing concrete and steel as a function of member size. He fined

MATLAB‟s (The math works.) sequential quadratic programming algorithm is highly

efficient and accurate for the design optimization of RC structures.

Guanand and Doh [20] developed strut-and-tie models of a total of fourteen

concrete deep beams with varying size and location of web openings using a topology

optimisation approach. By systematically eliminating inefficient materials from an

over-designed discretized domain, the load transfer mechanism in deep beams was

progressively characterised by the residual part of the structure. He used topology

optimisation method with stress and displacement constraints are employed with the

aim of maximising the material efficiency and overall stiffness of deep beams. The

method was used successfully to automatically produce the strut-and-tie models for a

total of 14 concrete deep beams with varying size and location of web openings. He

concluded that optimal topology (strut-and-tie model) was a fully stressed design and

accurately represents the actual stress path and the load transfer mechanism.

Mohammed and Ismail [21] was presented design optimization of structural

concrete beams using Genetic Algorithm (GA) technique. Two types of structural

beams were studied, namely: simple reinforced concrete beams and simple pre-

stressed concrete beams. These beams were designed according to the requirements of

the ACI 318-05 code. The cost function comprises the cost of concrete and the cost of

reinforcement. He concluded that the GA optimizer does a remarkable effort on

minimizing the expensive material in the objective function of the numerical

examples. This effort was devoted to reduce the amount of this material since it has

the higher percentage of the total cost. Therefore, he concluded that the GA search

and optimization technique was powerful and intelligent.

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Nagarajan and Madhavan [22] developed strut and tie models for simply

supported deep beams using topology optimization. The design of deep beams using

topology optimization was illustrated using an example and was compared with

available code recommendations. STM can be easily developed by using topology

optimization and it can be used to design deep beams subjected to any type of loading.

His conclusion was area of steel calculated using STM is less than that required

according to IS 456: 2000 recommendations.

Boualem and Fedghouche [23] conducted structural design optimization of

reinforced concrete T-beams under ultimate design loads. An analytical approach of

the problem based on a minimum cost design criterion and a reduced number of

design variables, was developed. It was shown, among other things, that the problem

formulation can be cast into a nonlinear mathematical programming format. Typical

examples were presented to illustrate the applicability of the formulation in

accordance with the current French design code BAEL-99. The results were then

confronted to design solutions of reinforced concrete T-beams derived from current

design practice. The optimal solutions show clearly that significant savings can be

made in the predicted amounts and consequently in the absolute costs of the

construction materials to be used.

Siradech and Benjapon [24] were used topology optimization as a tool to

determine the optimal reinforcement for reinforced concrete beam. The topology

optimization process was based on a unit finite element cell with layers of concrete

and steel. The thickness of the reinforced steel layer of this unit cell was then adjusted

when the concrete layer could not carry the tensile or compressive stress. At the same

time, unit cells which carried very low stress were eliminated. The process was

performed iteratively to create a topology of reinforced concrete beam which satisfied

design conditions.

Alankar and Chaudhary [25] aims to develop a Genetic Algorithm (GA)

based model to minimize the cost of single span steel concrete composite bridges with

precast decks. The cross-sectional dimensions have been considered as decision

variables in the present optimum design model along with the number of days for

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which the precast slabs should be placed in a casting yard. The model formulation

accounts for the cost of concrete, steel beam and slab storage in the casting yard. The

cost optimization problem has then been formulated and implemented using

MATLAB by employing GA which is computationally efficient in solving such types

of complex optimization problems.

Kaveh and Sabzi [26] applied big bang-big Crunch algorithm for optimal

design of reinforced concrete planar frames under the gravity and lateral loads.

Optimization was based on ACI 318-08 code. Columns were assumed to resist axial

loads and bending moments, while beams resist only bending moments. The main aim

was to minimize the cost of material and construction of the reinforced concrete

frames under the applied loads such that the strength requirements of the ACI 318

code are fulfilled. In this paper, the big bang - big Crunch algorithm was proven able

to find optimal design of 2D reinforced concrete frames. BB-BC was applied for the

first time to this kind of discrete optimization problems. Numerical results

demonstrate the feasibility and efficiency of the proposed approach.

Bhalchandra and Adsul [27] conducted optimum design of simply supported

doubly reinforced beams with uniformly distributed and concentrated load has been

done by incorporating actual self-weight of beam, parabolic stress block, and

moment-equilibrium and serviceability constraints besides other constraints. The

principal design objective was to minimize the total cost of a structure. The result of

his study was genetic algorithm optimization technique showed a cost that was less

than the cost obtained from the generalized reduced gradient technique and interior

point optimization technique.

Amir [28] studied topology optimisation procedure for reinforced concrete

structure. Here he explained the procedure for designing beam using strut and tie

model. FEA method was used for topology optimisation. His major conclusion was

that Load-bearing per unit weight improved by over 20% for a given range of

displacements and topology optimisation was also used for finding the optimal layout

of the reinforcement in beams.

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Gahtani et al. [29] was concerned with the cost minimization of pre-stressed

concrete beams using a special differential evolution-based technique. The optimum

design is posed as single-objective optimization problem in presence of constraints

formulated in accordance with the current European building code. This paper

investigated the application of the constrained differential evolution algorithm (ICDE)

for the optimum design of pre-stressed concrete beams. Ultimate and serviceability

limit states from the current European building code have been considered, and the

manufacturing cost has been assumed as objective function to be minimized.

Adib [30] presented optimization of thin walled column elements under axial

load, optimum design of wing structure, minimum cost design of grid floor, elevated

water tank staging and minimum weight design of sheet-stringer panels. Genetic

algorithm was used for optimisation purpose. His major conclusions were stress

constraint was the only active constraint at the optimum point and the optimal design

is sensitive to the way in which this constraint is defined. And the optimal design was

sensitive to the degree of correlation between member stresses.

2.2 COMMENTS ON LITERATURE REVIEW

Literature review has found that though there are many papers on topology

optimisation techniques, there is little research being done on its application in

reinforced concrete design and verifying these methods through physical testing. It

was observed that these optimisation techniques are well developed methods that have

been verified through vigorous numerical analysis. Topology optimisation using FEM

software ANSYS is a highly accurate method. Here is a possibility of optimisation

using ANSYS. Here used ANSYS software for topology optimisation. Yet, physical

testing is essential as it has been well documented that physical behaviour of

reinforced concrete is hard to model using numerical modelling.

2.3 OBJECTIVES

The aims and specific objectives of this project are as follows:

Review of strut and tie model for concrete deep beam.

Analysis of deep beam using Computer Aided Strut and Tie (CAST) software.

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Study of ANSYS software for topology optimisation.

To implement topology optimisation for concrete deep beam.

To be able to provide reinforced concrete designers a simple and effective

method of reducing the volume of deep beam without reducing its

performance.

2.4 METHODOLOGY

The methodology used to complete this dissertation is described in the following

steps:

1. Review of strut and tie model for design of deep beams.

2. Design the deep beam using ACI 318 2008 design approach.

3. Design the deep beam using IS 456 2000 design approach.

4. Research background information relating to topology optimisation.

5. Using ANSYS software to determine the optimum layout and design using

strut-and-tie method.

6. Concrete mix design for M25 mix.

7. Cube test for determining concrete compressive strength.

8. Prepare samples of deep beam and optimised beam.

9. Perform three point bending test on both beams and compare the result.

2.5 SUMMARY

In this chapter previous studies made in field optimisation, strut and tie model,

optimisation using genetic algorithm etc. were explained in literature review.

Comments on literature review were included for understanding the potential of the

problem. This chapter also explained the objectives and methodology of the project.

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Chapter 3

TOPOLOGY OPTIMISATION IN DEEP BEAM

3.1 INTRODUCTION

This chapter explains about basics of topology optimisation. Optimisation

methods, Software used for topology optimisation, basics of topology optimisation by

ANSYS, beam modelling in ANSYS, different element used for modelling concrete

and steel and its property etc. are explained in the following sections. Modelled

images from ANSYS are also provided.

3.2 OPTIMISATION

Topology optimization is perhaps the most difficult of all three types of

structural optimization. The optimization is performed by determining the optimal

topology of the structure. The design variables control the topology of the design.

Optimization therefore occurs through the determination of design variable values

which correspond to the component topology providing optimal structural behaviour.

While it is easy to control a structure‟s shape and size as the design variables are the

coordinates of the boundary (shape optimization) or the physical dimensions (size

optimization), it is difficult to control the topology of the structure. In this problem,

the design domain is created by assembling a large number of basic elements or

building blocks. By beginning with a set of building block representing the maximum

allowable region (region in space which the structure may occupy) each block is

allowed to either exist or vanish from the design domain, a unique design is evolved.

For example in the topology optimization of a cantilever plate, the plate is discretised

into small rectangular elements (building blocks), where each element is controlled by

design variables which can vary continuously between 0 and 1. When a particular

design variable has a value of 0, it is considered to be a hole, likewise, when a design

variable has a value of 1, it is considered to be fully material. The elements with

intermediate values are considered materials of intermediate densities. The

development of topological optimization can be attributed to [31]. They presented a

homogenization based optimization approach of topology optimization. They

19

assumed that the structure is formed by a set of non-homogenous elements which are

composed of solid and void regions and obtained optimal design under volume

constraint through optimization process. In their method, the regions with dense cells

are defined as structural shape, and those with void cells are areas of unnecessary

material. The maximization of the integral stiffness of a structure composed of one or

two isotropic materials of large stiffness using the homogenization technique was

discussed by [32]. Application of Genetic algorithm for topology optimization was

made by [33]. Given structure‟s boundary conditions and allowable design domain, a

discretised design domain is created. The genetic algorithm then generates an optimal

structure topology by evolving a population of chromosomes, where each

chromosome, after mapping into the design domain creates a potentially-optimal

structure topology. [34] computed the effective properties of strong and weak

materials. It is shown that when 4-noded quadrilateral elements are used, the resulting

topology consists of artificially high stiffness material which is difficult to

manufacture. This material appears in specified manner and is known as the checker

board pattern due to alternate solid and void elements. [35] Investigated a continuous

topology optimization framework based on hybrid combinations of classical Reuss

and Voigt (stiff) mixing rules. To avoid checker boarding instabilities, the continuous

topology optimization formulation is coupled with a novel spatial filtering procedure.

[36] Summarized the current knowledge about numerical instabilities such as

checkerboards, mesh-dependence and local minima occurring in applications of the

topology optimization method. The checkerboard problem refers to the formation of

regions of alternating solid and void elements ordered in a checkerboard-like fashion.

The mesh-dependence problem refers to obtaining qualitatively different solutions for

different mesh-sizes or discretization. A local minimum refers to the problem of

obtaining different solutions to the same discretised problem when choosing different

algorithmic parameters. A web-based interface for a topology optimization program

was presented by [37]. The program is available over World Wide Web. The paper

discusses implementation issues and educational aspects as well as statistics and

experience with the program. [38] Studied a level-set method for numerical shape

optimization of elastic structures. The approach combines the level-set algorithm with

20

the classical shape gradient. Although this method is not specifically designed for

topology optimization, it can easily handle topology changes for a very large class of

objective functions. [35] Presented a node-based design variable implementation for

continuum structural topology optimization in a finite element framework and

explored its properties in the context of solving a number of different design

examples. Since the implementation ensures C0 continuity of design variables, it is

immune to element wise checker boarding instabilities that are a concern with

element-based design variables. The objective of maximizing the Eigen frequency of

vibrating structures for avoiding the resonance condition was considered by [38]. This

can also be achieved by maximizing the gap between two consecutive frequencies of

the given order. Different approaches are considered and discussed for topology

optimization involving simple and multiple Eigen frequencies of linearly elastic

structures without damping. The mathematical formulations of these topology

optimization problems and several illustrative results are presented. [36] Suggested a

new way to solve pressure load problems in topology optimization. Using a mixed

displacement–pressure formulation for the underlying finite element problem, we

define the void phase to be an incompressible hydrostatic fluid. [8] Evaluated and

compared the established numerical methods of structural topology optimization that

have reached the stage of application in industrial software. [40] presented a new

penalization scheme for the SIMP method. One advantage of the present method is

the linear density stiffness relationship which has advantage for self-weight or Eigen

frequency problem. The topology optimization problem is solved through derived

Optimality criterion method (OC), which is also introduced in the paper. [18]

presented a stochastic direct search method for topology optimization of continuum

structures. In a systematic approach requiring repeated evaluations of the objective

function, the element exchange method (EEM) eliminates the less influential solid

elements by switching them into void elements and converts the more influential void

elements into solid resulting in an optimal 0–1 topology as the solution converges. For

compliance minimization problems, the element strain energy is used as the principal

criterion for element exchange operation. [28] Obtained topologically optimal

configuration of sheet metal brackets using Optimality Criterion approach through

21

commercially available finite element solver ANSYS and obtained compliance versus

iterations plots for various aspect ratio structures (brackets) under different boundary

conditions.

Topology optimization is a mathematical approach that optimises material

layout within a given design space, for a given set of loads and boundary conditions

such that the resulting layout meets a prescribed set of performance targets.

Mathematically one can pose a generic problem as follows:

Min ∫ ( )

(3.1)

Subjected to

1. * +

2. Design constraints

3. Governing differential equations

Objective function ( ∫ ( )

)

This is the goal of the optimization study which is to be minimised over the

selection field. For example, one would want to minimise the compliance of a

structure to increase structural stiffness.

Design space ( )

Design space is the allowable volume within which the design can exist.

Assembly and packaging requirements, human and tool accessibility are some of the

factors that need to be considered in identifying this space. With the definition of the

design space, regions or components in the model that cannot be modified during the

course of the optimisation are considered as non-design regions.

The Discrete Selection Field (ρ)

This is the field over which the discrete optimisation is to be performed. It

selects or deselects a point on the design space to further the design objective. By

selection it has to take the value, 1, and by de-selection it has to take the value, 0.

22

Design constraints

These are design criteria that need to satisfy. These could include material

availability constraints, displacement constraints, Design for flexure, Minimum

spacing between bars, and Maximum spacing between bars, Maximum and minimum

reinforcement ratios, Design for shear Deflection control etc.

Governing Differential Equation

This is the one that governs the physics of the structure to be built. For example

the elasticity equation in the case of stiff structures would be the governing

differential equation.

3.3 OPTIMISATION METHODS

Optimisation is widely uses in aerospace industry, automobile industry, etc. in

civil engineering especially structural engineering it creates new challenges. [17]

developed mat lab 99 line code for optimisation of concrete structure. It is modified to

88 line and 104 line codes and is known as modified mat lab codes.

Software such as ANSYS, OPTISTRUCT (Altair), TOSCA (Fe-design),

GENESIS, ABAQUS, etc. is available for topology optimisation. The software can

run direct topology optimisation by imputing the material property, loads and support

condition.

In this project topology optimisation is done in ANSYS software. ANSYS 12.1

with CIVIL FEM is used for optimisation of deep beams.

3.3.1 Topology optimization using ANSYS

The goal of topological optimization is to find the best use of material for a

body such that an objective criterion (i.e. global stiffness, natural frequency, etc.)

attains a maximum or minimum value subject to given constraints (i.e. volume

reduction). In this work, maximization of static stiffness has been considered. This

can also be stated as the problem of minimization of compliance of the structure.

Compliance is a form of work done on the structure by the applied load. Lesser

23

compliance means lesser work is done by the load on the structure, which results in

lesser energy is stored in the structure which in turn, means that the structure is stiffer.

Mathematically,

Compliance =∫

∫

∑ (3.2)

Where, u = Displacement field

f = Distributed body force (gravity load etc.)

= Point load on ith node

= ith displacement degree of freedom

t = Traction force

S = Surface area of the continuum

V = Volume of the continuum

ANSYS employs gradient based methods of topology optimization, in which

the design variables are continuous in nature and not discrete. These types of methods

require a penalization scheme for evolving true, material and void topologies. SIMP

(Solid Isotropic Material with Penalization) is a most commonly penalization scheme,

and is explained in the next section.

The SIMP method

The SIMP stands for Solid Isotropic Material with Penalization method. This

is the penalization scheme or the power law through which is the basis for evolution

of a 0-1 topology in gradient based methods.

In the SIMP method, each finite element (formed due to meshing in ANSYS)

is given an additional property of pseudo-density, where 0< , which alters the

stiffness properties of the material.

(3.3)

Where,

= Density of the jth element

= Density of the base material

24

= Pseudo-density of the jth element

This Pseudo-density of each finite element serves as the design variables for

the topology optimization problem. The stiffness of jth

element depends on its

Pseudo-density in such a way that,

(3.4)

Where, = Stiffness of the base material

= Penalization power

For, = 0, which means no material exists

For, = 1, which means that material exists

In SIMP is taken to be greater than 1 so that intermediate densities are

unfavourable in the sense that that the stiffness obtained is small as compared with the

volume of the material. In other words, specifying a value of higher than 1 makes it

uneconomical to have intermediate densities in the optimal design.

As a matter of fact, for problems where the volume constraint is active, results

shows that optimization does actually result in such designs if one chooses

sufficiently large (in order to achieve complete 0-1 designs, is usually

required).

In ANSYS, the standard formulation of topology optimisation problem defines

the problem as minimising the structural stiffness and maximising the fundamental

frequency while satisfying a constraint on volume of the structure. Another problem is

the maximisation of natural frequency of the structure subjected to dynamic loading,

while satisfying a constraint on the volume of the structure.

The objective function (function which is to be minimized in topology

optimization) is generally the compliance of the structure. A constraint on usable

volume is applied on the structure. As the volume reduces, the structure‟s stiffness

also reduces. So the volume constraint is of opposing nature. The Compliance of a

discretised finite element is given by,

25

C (x) = (3.5)

The force vector (which is a function of the design variable ) is given by

K (x) u=F (3.6)

Therefore, C(x) can be written as,

C(x) = ∑ ( ) (3.7)

Subjected to ∑ V0 (3.8)

A lower bound on the design variables has been applied to avoid singularity in

the stiffness matrix

3.4 SPECIMEN GEOMETRY AND BOUNDARY CONDITIONS

In the present investigation, specimen geometry and boundary conditions

applied have been used as shown in the Fig.3.1.

Model

A simply supported deep beam of dimensions 700cmx36cmx15cm is shown in

Fig.3.1. The beam is acted upon by two concentrated load of 200kN each as shown in

figure.

233.3cm 233.3cm 233.3cm

200kN 200kN

700cm

Fig.3.1 Deep beam model

26

3.5 MODELLING IN ANSYS

ANSYS provides different solid element for modelling. Each of them is

different in property. Beam modelling in ANSYS based on the requirement of the

project. SOLID 65 can capable of crushing in compression and cracking in tension. It

is used for concrete modelling. Crack and crush at ultimate load can be obtained from

the SOLID 65. Main objective of this project is topology optimization. SOLID 65

is not supported for topology optimization. So another element SOLID 95 is used for

modelling the deep beam for optimization. SOLID 95 unable to crack and crush at

ultimate loads. In the initial modelling of deep beam is done using SOLID 95 for

topology optimization. For obtaining crack pattern and ultimate loads deep beam is

again modelled in SOLID 65. Similarly optimized beam is modelled in SOLID 65 for

obtain crack and crush results. Details of SOLID 65 and SOLID 95 are explained in

next section.

Reinforcement can be modelled in different element such as beam (BEAM

188) and link (2D spar 8). These two types are not supported for topology

optimization. Discrete reinforcement is provided in solid 95 elements for topology

optimization. In this type of reinforcing tension, compression and shear reinforcement

is different. For detection of crack and crush reinforcement is modelled using BEAM

188 in SOLID 65. Both type reinforcing is explained below.

Solid65

SOLID65 (Fig.3.2) is used for the 3-D modelling of solids with or without

reinforcing bars (rebar). The solid is capable of cracking in tension and crushing in

compression. In concrete applications, for example, the solid capability of the element

may be used to model the concrete while the rebar capability is available for

modelling reinforcement behaviour. Other cases for which the element is also

applicable would be reinforced composites (such as fiberglass), and geological

materials (such as rock). The element is defined by eight nodes having three degrees

of freedom at each node: translations in the nodal x, y, and z directions. Up to three

different rebar specifications may be defined.

27

The concrete element is similar to the SOLID45 (3-D Structural Solid)

element with the addition of special cracking and crushing capabilities. The most

important aspect of this element is the treatment of nonlinear material properties. The

concrete is capable of cracking (in three orthogonal directions), crushing, plastic

deformation, and creep. The rebar are capable of tension and compression, but not

shear. They are also capable of plastic deformation and creep

Fig.3.2 SOLID65 element geometry

Solid95

SOLID95 (Fig.3.3) is a higher-order version of the 3-D 8-node solid element

SOLID45. It can tolerate irregular shapes without as much loss of accuracy. SOLID95

elements have compatible displacement shapes and are well suited to model curved

boundaries. The element is defined by 20 nodes having three degrees of freedom per

node: translations in the nodal x, y, and z directions. The element may have any

spatial orientation. SOLID95 has plasticity, creep, stress stiffening, large deflection,

and large strain capabilities.

Reinf264

The element is suitable for simulating reinforcing fibers with arbitrary

orientations (Fig.3.4). Each fiber is modelled separately as a spar that has only

28

Fig.3.3. SOLID 95 element geometry

uniaxial stiffness. We can specify multiple reinforcing fibres in one REINF264

element. The nodal locations, degrees of freedom, and connectivity of the REINF264

element are identical to those of the base element. REINF264 used with standard 3-D

link, beam, shell and solid elements.

REINF264 has plasticity, stress stiffening, creep, large deflection, and large

strain capabilities

Fig.3.4 REINF264 element geometry

29

Beam 188

BEAM188 (Fig.3.5) is suitable for analysing slender to moderately thick beam

structures. The element is based on Timoshenko beam theory which includes shear-

deformation effects. The element provides options for unrestrained warping and

restrained warping of cross-sections.

The element is a linear, quadratic, or cubic two-node beam element in 3-D.

BEAM188 has six or seven degrees of freedom at each node. These include

translations in the x, y, and z directions and rotations about the x, y, and z directions.

A seventh degree of freedom (warping magnitude) is optional. This element is well-

suited for linear, large rotation, and/or large strain nonlinear applications. The element

includes stress stiffness terms, by default, in any analysis with large deflection. The

provided stress-stiffness terms enable the elements to analyse flexural, lateral, and

torsional stability problems (using eigenvalue buckling, or collapse studies with arc

length methods or nonlinear stabilization).

Elasticity, plasticity, creep and other nonlinear material models are supported.

A cross-section associated with this element type can be a built-up section referencing

more than one material.

Fig.3.5 BEAM 188 element geometry

30

3.5.1 Material properties of Concrete

Open shear transfer coefficient =0-1

Closed shear transfer coefficient =0

Uniaxial cracking stress =1.6MPa

Uniaxial crushing stress =25MPa

Modulus of elasticity = 25000

Poisons ratio =0.21

3.5.2 Material properties of Reinforcement

Modulus of elasticity = 25000MPa

Poisons ratio =0.21

Reinforcement modelled using REINF 264 is given in Fig.3.6

Fig.3.6.Reinforcement modelled using ANSYS

Reinforcement with concrete model is shown in Fig.3.7.

31

Fig.3.7. Beam modelled using ANSYS

3.6 SUMMARY

This chapter explained about of topology optimisation programme.

Optimisation methods, Software used for topology optimisation, basics of topology

optimisation by ANSYS, beam modelling in ANSYS, different element used for

modelling concrete and steel and its property are explained. Modelled images beam

and reinforcement from ANSYS are provided in this chapter.

32

Chapter 4

CONVENTIONAL DESIGN OF DEEP BEAM

4.1 INTRODUCTION

Design of concrete structure design can be done referring any standard codes.

Many papers were published for different design approach. ACI318:2008 [1] and

IS456:2000 [2] is used as design codes. Strut and tie method of design is adopted in

ACI 318: 2008 while IS 456: 2000 uses method of shear wall design. Here deep beam

designed using both methods.

4.2 STRUT AND TIE METHOD OF DESIGN

Reinforced concrete beam theory is based on equilibrium and the constitutive

behaviour of the materials, steel and concrete. Particularly important is the

assumption that strain varies linearly through the depth of a member and that, as a

result plane sections remain plane. St. Venant‟s principle validated this assumption by

stating that strains around load or member cross section discontinuity vary in an

approximately linear fashion at distance greater than or equal to the greatest cross

sectional dimension „h‟ from the point of load application as shown in Fig. 4.1 and

Fig.4.2.

At points closer than the distance „h‟ to discontinuous load or member

dimensions, St Venant‟s principle is not applicable. Reinforced concrete structures

can be divided into regions where beam theory is valid and regions where

discontinuities affect member behaviour. A region where beam theory is valid is

referred to as B-regions and a region with discontinuities is referred to as D-regions.

When the concrete is elastic and un-cracked, the stresses in D-regions can be

determined using finite element analysis and elastic theory. After concrete cracks the

strain field is disrupted and internal forces are redistributed. The internal force can be

represented by a statically determinate truss known as the strut-and-tie model, which

allows the complex problem to be simplified. Fig.4.3 shows examples of strut and-tie

model in typical reinforced concrete members.

33

Fig.4.1.Geometric discontinuity Fig.4.2.Loading discontinuity

34

Fige.4.3 Examples of strut and-tie model

4.3 DEFINITIONS

The following terms are used in this section

B-region – A portion of a structure in which the Bernoulli-Euler assumption

that plane sections remain plane can be applied.

Discontinuity – An abrupt change in member‟s geometry or loading.

D-region – The portion of a member within a distance equal to the member

depth h from a force discontinuity or a geometry discontinuity. In D-regions

Bernoulli- Euler‟s assumption is not valid after the concrete cracks.

Node – A point in a strut-and-tie model where the axes of the struts, ties and

concentrated forces acting on the joint intersect.

35

Nodal zone – The volume of concrete surrounding a node that transfers strut-

and-tie forces through the node.

Strut – A compressive member in a strut-and-tie model. A strut represents the

resultant of a parallel or fan-shaped compressive field.

Bottle-shaped strut – A strut that is wider at mid-length than at its ends.

Strut-and-tie model – A truss model of a structural member, made up of

struts and ties connected at nodes that are capable of transforming the factored

loads to the supports.

Tie – A tension member in a strut-and-tie model.

4.4 KEY COMPONENTS OF STRUT-AND-TIE MODELS

Strut-and-tie modelling is considered the basic tool in the design and detailing

of structural concrete under bending, shear and torsion. The designer specifies a load

path and then designs and details the structure such that this load path is sufficiently

strong to carry the applied loads. The loads applied to the structural concrete member

are transferred through a set of compressive stress fields that are distributed and

interconnected by tension ties. The compression stress fields are idealised using

compression members called struts while tensile stress fields are idealised using

tension members called ties. Tension ties can be reinforcing steel bars or pre-stressed

tendons or concrete in tension. Concrete‟s tensile strength is considerably less than its

compressive strength and normally concrete‟s tensile resistance is ignored.

Struts

A strut is an internal compression member. It may have a prismatic, fan or

bottle shape as shown in Fig.4.4. Prismatic shape is an idealised representation of fan

or bottle shaped struts. The dimensions of the cross section of the strut are established

by the contact area between the strut and the nodal zone.

Bottle shaped struts are wider at the centre than the ends and as the

compression zone spreads along the length of bottle shaped struts, tensile stresses

perpendicular to the axis of the strut may cause longitudinal cracking. For simplicity

in design, bottle shaped struts are idealised as having linearly tapered ends and

36

Fig.4.4 Types of concrete struts and related stress fields.

uniform centre sections as shown in Fig.4.5.The capacity of the struts is proportional

to the concrete compressive strength and it is affected by the lateral stresses in bottle

shaped struts. Because of longitudinal splitting, bottled shaped struts are weaker than

prismatic struts, even though they possess a larger cross section at mid-length.

Prismatic stress field Fan stress field (no bursting forces)

Bursting forces

Bottle stress field

37

Fig.4.5 Bottle-shaped strut.

Ties

A tie is a tension member in a strut-and-tie model. The ties consist of either

steel bar or a pre-stressed tendon. For design purpose, it is assumed that the concrete

within the tie does not carry any tensile force. Concrete does assist in reducing tie

deformation at service load.

Nodes

Nodes are points within strut-and-tie models where the axis of struts, ties and

concentrated loads intersect. For equilibrium, at least three forces must act on a node.

Nodes are defined by the sign of forces acting at it. Therefore, a CCC node resists

three compressive forces; a CCT resists two compressive forces and a one tensile

force. There can be multiple forces acting at a node but care must be taken to ensure

there is room for anchorage of tie reinforcements. Fig.4.6 illustrates some common

node classifications.

Both tensile and compressive forces place nodes in compression because

tensile forces are treated as if they pass through the node and apply compression in

the anchorage face. There are two types of nodes, non-hydrostatic and hydrostatic

Crack

Strut

Tie

Width used to compute Ac

38

nodes. A node is hydrostatic if all members are at right angles to the adjacent node

angle other than right angle, the node is non-hydrostatic as shown in Fig.4.7 (a) and

Fig.4.7 (b).

Fig.4.6. Classification of nodes.

Fig.4.7. Node types

(a) C-C-C node

(b) C-C-T node

(c) C-T-T node (d) T-T-T node

(a) Hydrostatic node (b) Non-hydrostatic node

39

Advantages of Using Strut-and-Tie Modelling

1. The designer can easily idealise the flow of internal forces in a structural

concrete member.

2. The influence of shear and moment can be accounted for simultaneously and

directly in one model.

3. The designer can give special attentions to the potential weak spots indicated

by the strut-and-tie model.

4. It offers a unified, rational and safe design procedure for structural concrete.

Limitations of Strut-and-Tie Modelling

Strut-and-tie modelling is good for structures at overload, that is, after

extensive cracking and large deformations have occurred. It is not suited to

representing transitional behaviour when the structure is changing from un-cracked to

the fully cracked condition. The strut-and-tie model is a conservative design approach

which means that it is almost always over designed.

The strut-and-tie modelling offers the designer the flexibility to focus on

performance design while also providing a safe design. Different performance criteria

may be achieved with strut-and-tie modelling, however, the ultimate failure mode and

load cannot be predicted by strut-and-tie modelling.

4.5 DESIGN OF DEEP BEAM BASED ON ACI 318 2008

Here deep beam with span=7m and depth=3.6m is designed as per ACI 318

2008. Concrete grade assumed is M25 with two point load of 2000kN as shown in

Fig.4.8.

4.5.1 Analysis using Computer aided strut and tie (CAST)

Computer aided strut and tie (CAST) is software for analysis of deep beam

using strut and tie method (STM).CAST utilizes a single interface for creation or

modification of strut-and-tie models, truss analysis, selection of reinforcing steel and

capacity checks of the struts and nodes.

40

CAST has been developed by the University of Illinois since 2001. It was used

to serve students and practicing engineers to grasp the concept and to try various

idealized strut-and-tie models with ease.

CAST components comprise of nodes and elements. Elements represent struts

or ties meeting at nodes. For particular cases, stabilizer is necessary to make the

model numerically stable. Three types of boundary conditions can be applied at

exterior nodes: plate, point loads, and supports. Body force and support can also be

applied to interior nodes within D-region. The model must be restrained to prevent

rigid body movement. In CAST, positive value of forces is point outward from node

and vice versa. Since the direction of point loads on the boundary will follow the axis

of the strut, no element connectivity is allowed on boundary.

Strut and tie model is modelled in cast software. Analysis is done in cast

software for determining the load on strut, tie and its nature. Result of CAST analysis

is shown in Fig.4.8

Fig.4.8. CAST analysis result

4.5.2 Structural design

12345As per ACI 318 2008 nominal shear strength (Vn) for deep beams shall not

exceed 10 √ (ACI 318 Cl 11.7.2) (4.1)

41

And ≤Φ (4.2)

Where

=shear strength.

=cylinder compressive strength=0.8 fck=20MPa

b=width=0.6m

d=depth 0.9 h=3.24m

Φ=capacity reduction factor=0.75

Substituting =7244.86kN

Φ× =5433.65>1973 hence ok

Effective compressive force at nodes and struts-(ACI 318 A.5.2)

Nodes A and B

= 0.85 (4.3)

Where the value of βn is given by

In nodal zones bounded by struts or bearing areas, or both βn = 1.0;

In nodal zones anchoring one tie βn = 0.80;

In nodal zones anchoring two or more ties βn = 0.60.

=0.85×0.8×0.8×20=10.88

=10.88MPa

Node C

=13.6MPa

42

Strength of strut (ACI 318.A.3)

Nominal compressive strength of strut without longitudinal reinforcement,

= × (4.4)

Where

fce= 0.85βs fc′assume bottled shaped strut βs=0.75 (4.5)

fce=0.85×0.75×0.8×20=10.2MPa

Width of strut

≤Φ =>0.75× ≥2002.8 (4.6)

≥2670.4

From equation (3.4) we can calculate area of strut

=>2670.4≥10.2×

=261803.92sqmm

W (width of strut)= 261803.92/600=436.33mm

Width of tie- (ACI318 A.4)

Fnt= Atsfy+ Atp(fse+ Δfp) , where Atpis zero for non-pre-stressed members (4.7)

Fnt= /Φ=1160kN

From equation (3.7) area of tensile reinforcement is given by,

Ats=1160×1000/415=2795.18sqmm

No of 25mm diameter bars= 6

Development of standard hooks in tension

Develop length is provided for increase the friction between bar and concrete

from equation (4.8) we get development length for the bar in tension.

43

Ld=

√ ×db 12 (4.8)

Where

,

Then Ld = 556.77

Shear reinforcement

The area of shear reinforcement perpendicular to the flexural tension

reinforcement, Av, shall not be less than 0.0025bws, and „s‟ shall not exceed the

smaller of d/5 and 12 in.(s=150mm)

= 0.002512>0.0025 ok (4.9)

The area of shear reinforcement parallel to the flexural tension reinforcement

= 0.002512>0.0025 ok

sinαi≥ 0.003α=46.39 (4.10)

Use 12mm dia stirrup at 150mm c/c

4.5.3 Detailing of deep beam designed by ACI method.

Detailing of beam is provided in Fig.4.9. Six no of 25mm diameter bar is

provided at the bottom of slab and 12mm bars are provided at 250mm c/c for shear

reinforcement on both face as shown in Fig.4.9

44

(All dimensions are in cm)

Fig.4.9 Details of deep beam designed by ACI method.

4.6 DESIGN OF DEEP BEAM BASED ON IS 456 2000

Determination of design bending moment

In a simply supported beam, the bending moment is calculated as in ordinary

beams. For a point load w, moment at mid spam

M=

(4.11)

=3500kNm

Determination of area of tension steel

The area of tension steel to carry the tension is determined by the empirical

method of assuming a value for the lever arm. IS 456 clause 29.2 [2] gives the

following value for „z‟ the lever arm length.

For simply supported beams,

60

Six no of 25mm dia bar

12mm dia @ 150mmc/c

700 60

360

45

Z=0.2(L+2D) when L/D is between 1and 2

=0.6L when L/Dis less than 1

Area of tension steel,

=Mu/fsZ (4.12)

Where

Z=2840mm

Fs=0.87fy=361N/mm2

Area of tension steel is 3413.834mm2

Provide 7 no of 25mm diameter bars

Design for shear

Critical section for shear

The critical section for maximum shear is located at distance from the base of

0.5 Lw or 0.5 Hw (whichever is less)

=1.8m from the bottom

Nominal shear stress

The nominal shear stress vw in walls is given by

vw=Vu/td (4.13)

Where

Vu= shear force due the design load

t= wall thickness

d= 0.8 x Lw

46

Lw=span

Then vw=0.60

This never exceeds 0.17 fck in limit stat design.

0.17 fck=4.5<0.60 hence ok

Design shear strength of concrete (cl 32.4.3 IS 456)

The design shear strength of concrete in walls, cw , without shear reinforcement is

taken as

cw=(3.0-Hw/Lw )K1√ for Hw/Lw<1 (4.14)

Where

k1=0.2 for limit state method

K1=0.12 for working stress method

Then cw=2.49

Design of shear reinforcement (cl 32.4.4 IS 456)

Shear reinforcement shall be provide to carry a shear equal to

Vu- cwt(0.8Lw) (4.15)

It is less than minimum required. Minimum reinforcement must be proved as

per cl.32.5 of IS 456:2000

Minimum requirement of reinforcement

Vertical reinforcement

Area of vertical steel/gross concrete area =0.0012 for bar with diameter <

16mm

Are of vertical steel=5040mm2

47

Spacing of 12mm diameter bar =155mm. it must be provide in both direction

Horizontal reinforcement

Area of horizontal steel/gross concrete area =0.002

Are of horizontal steel =4320mm2

it is greater than the steel required for

bending (3413mm2). So addition reinforcement must be provided at the top as holding

stirrup.

Provide 2 no of 25mm diameter bar at top.

Detailing as per the data is shown in Fig .4.10

4.6.1 Detailing of deep beam designed by IS 456 (2000) method.

Detailing of deep beam is shown in Fig.4.10. 7 no of 25mm diameter bars are

provided at the bottom as tie. Shear reinforcement (12mm) is provided at 155mm

centre to centre as shown in figure.

Fig.4.10. Details of deep beam designed by IS 456 (2000) method.

12mm dia @ 155mmc/c

7 no of 25mm dia bar

700

360

60

60

All dimensions are in cm

48

4.7 MODELLING IN ANSYS

Topology optimisation of deep beam is done in ANSYS. Modelling is

completed using solid 95 elements for concrete (which supports for topology

optimisation) and REINF 264 elements for reinforcement. Reinforcement modelled

using REINF 264 is given in Fig.4.11.Reinforcement with concrete model is shown in

Fig.4.12.

Fig.4.11.Reinforcement modelled using ANSYS

Fig.4.12. Beam modelled using ANSYS

49

4.8 RESULT OF OPTIMISATION IN ANSYS

Result of optimisation is given below in Fig 4.13. The different colours in this

represent material usefulness in the beam. Red region in the figure represent the

material with density is one. Which means that element must be there for resisting the

in applied load. Pure blue colour represents material with zero density, which means

there is no need of that element for resisting external load. In between blue and red

represent density between zero and one. The entire portion in blue can be removed for

fifty present of optimisation.

Stress in steel is represented in Fig 4.14. Reinforcement shows different

colours. Each colour represents the stress intensity. Red portion stands for highly

stressed steel while blue represent less stressed region. In between blue (minimum)

and red (maximum) stress state will vary.

Fig.4.13. Density plot of deep beam

50

Fig.4.14 Stress in steel for deep bam

4.8.1deflection

Deflection for deep beam calculated in ANSYS and is drawn in graph with

load in y axes and deflection in x axis (Fig.4.15). The maximum deflection obtained is

15.5mm.

Fig.4.15. Load deflection graph for deep beam

0

2

4

6

8

10

0 5 10 15 20

Lo

ad

x1

00

0 k

N

Deflection in mm

Load deflection

51

Deep beams failed by shear failure. Cracks starts at the centre and extends to

support. Fig.4.16. show the cracks of deep beam.

Fig.4.16. Crack pattern for deep beam obtained from ANSYS

4.9 SUMMARY

In this chapter design of deep beam by various approaches were discussed.

Both ACI and IS code practice. Both design yield same result. ACI method based on

the strut and tie modelling while IS code based on shear wall design method. Analysis

for calculating force in strut and tie was completed in CAST software. This software

is useful for complex strut and tie models. Detailing for both design were included in

this chapter. Topology optimisation result from ANSYS included here. Load

deflection, crack pattern (obtained from ANSYS) is included in this chapter. In order

to verify the results, laboratory test was conducted on both optimised and deep beam.

This is described in following sections.

52

Chapter 5

EXPERIMENTAL INVESTIGATION

5.1 INTRODUCTION

Due to the lack of laboratory facility, small models of deep beam and

optimised beam were tested for validation of the project. Conventional design method

used here for designing deep beam are STM based of ACI 318 2008 and IS 456 2000.

Comparison of both methods is done in last section. Design of deep beam, mix

design, cube test, test setups for three point bending test on deep beam etc. are

explained in this chapter.

5.2 DESIGN OF DEEP BEAM (ACI 318-2008)

5.2.1. Data for design: - In this design beam span depends on the span of UTM

available in the laboratory. Other assumed data as follows.

Assume fy=415MPa,

fck=25MPa

Span is 60cm and Assume depth 35 cm, (L/D<2)

Design load-150kN.

5.2.2 Analysis

Concentrated load -150kN

Self-weight =0.6×0.35×25×0.15=0.79kN

Total load =150.79kN

Analysis of deep beam

Manual analysis is done to calculate force in strut and tie. This value is then

checked by CAST analysis. Fig.5.1 shows support reaction for deep beam. Analysis is

done by equilibrium equation. Result of analysis is shown in Fig.5.2

53

150.79 kN

w/2 =75.39 w/2=75.39

Fig.5.1. Support reaction

Force in the member

Assume depth is 0.9H =31.5cm 150.79 kN

C

31.5cm

75.3 kN A B 75.39 kN

Fig.5.2 Strut and tie model.

θ =46.39deg

Member force =

=104.13kN

=104.13cos46.39=71.8kN

5.2.3 Analysis using Computer Aided Strut and Tie (CAST)

Computer Aided Strut and Tie can be used for calculation the load in strut and

tie. Fig .5.1.shows result of CAST analysis. From the figure strut and tie forces above

obtained is almost same to the CAST result.

θ 60cm

54

Fig.5.3. CAST analysis result

5.2.4 Structural design

As per ACI 318-2008 Vn for deep beams shall not exceed 10/12√ kN

(ACI 318 Cl 11.7.2) (5.1)

And ≤Φ (5.2)

Where

=shear strength.

=cylinder compressive strength=20MPa

bw=width=15cm

d=depth=28.8cm

Φ=capacity reduction factor=0.75

Substituting =176.01N

=>Φ× =120.747>75.39 hence ok.

55

Effective compressive force at nodes and struts-(ACI 318 A.5.2)

Nodes A and B

= 0.85 (5.3)

Where the value of βn is given by

In nodal zones bounded by struts or bearing areas, or both βn = 1.0;

In nodal zones anchoring one tie βn = 0.80;

In nodal zones anchoring two or more ties βn = 0.60.

Then,

=0.85×0.8×0.8×20=10.88

=10.88MPa

Node C

=17MPa

Strength of strut (ACI 318.A.3)

Nominal compressive strength of strut without longitudinal reinforcement,

= × (5.4)

Where

fce= 0.85βsfc′assume bottled shaped strut βs=0.75 (5.5)

fce=0.85×0.75×0.8×25=12.75

Width of strut

≤Φ =>0.75× ≥104.13

≥138.84

56

From equation (5.4)

=138.8≥13.6×

=10205.88sqmm

W (width of strut) = 10205.88/150=68.03mm

Width of tie- (ACI318 A.4)

Fnt= Atsfy+ Atp(fse+ Δfp)Atpis zero for non-pre-stressed members (5.6)

Fnt= /Φ=95.733

Ats=95.733×1000/415=230.68sqmm

Use 2 no of 8 mm diameter bars and 2 no of 12 mm diameter bars

Development of standard hooks in tension

Develop length is provided for in crease the friction between bar and concrete

from equation (5.7) we get development length for the bar in tension.

Ld=

√ ×db (5.7)

Where, ,

Then Ld=267.25

Shear reinforcement

The area of shear reinforcement perpendicular to the flexural tension

reinforcement, Av, shall not be less than 0.0025bws, and „s‟ shall not exceed the

smaller of d/5 and 12 inch.(s =250)

= 0.00267>0.0025 hence ok

The area of shear reinforcement parallel to the flexural tension reinforcement

57

= 0.00267>0.0025 hence ok

sinαi≥ 0.003α=46.39

=>

+

sin46.39=0.0038>0.003 ok

5.2.5 Detailing

Detailing of reinforcement with spacing is given in Fig.5.4

Fig.5.4. Detailing of deep beam by strut and tie method.

5.3 DESIGN OF DEEP BEAM BASED ON IS 456 2000

Determination of design bending moment

In a simply supported beam, the bending moment is calculated as in ordinary

beams. For a point load w, moment at mid spam

M=

(5.8)

=26.25kNm

2 no of 8mm dia bar +2 no

12mm dia bar

8mm dia bar @250mmc/c in both directions

700mm

350mm

m

150mm

58

Determination of area of tension steel

The area of tension steel to carry the tension is determined by the empirical

method of assuming a value for the lever arm. Is 456 clause 29.2 [] follows the CEB

(committee euro international du beton) and gives the following value for „z‟ the lever

arm length.

For simply supported beams,

Z=0.2(L+2D) when L/D is between 1and 2 (5.9)

=0.6L when L/Dis less than 1

Area of tension steel,

Mu/fsZ (5.10)

Where

Z=280mm

fs=0.87fy=361N/mm2

Area of tension steel is 260mm2

Provide 2 no of 12 mm bar and 2 no of 8mm bar

Design for shear

Critical section for shear

The critical section for maximum shear is located at distance from the base of

0.5 Lw or 0.5 Hw

=0.175m from the bottom

Nominal shear stress

The nominal shear stress vw in walls is given by

vw=Vu/td (5.11)

59

Where

Vu= shear force due the design load

t= wall thickness

d= 0.8 x Lw

Lw=span

vw=0.89

This never exceeds 0.17 fck in limit stat design.

0.17 fck=4.5<0.89 hence ok

Design shear strength of concrete (cl 32.4.3 IS 456)

The design shear strength of concrete in walls, cw , without shear

reinforcement is taken as

cw=(3.0-Hw/Lw )K1√ for Hw/Lw<1 (5.12)

Where, k1=0.2 for limit state method

K1=0.12 for working stress method

cw=2.5

Design of shear reinforcement (cl 32.4.4 IS 456)

Shear reinforcement shall be provides to carry a shear equal to Vu- cwt

(0.8Lw). Here Vu is small so no need to provide shear reinforcement. Yet minimum

reinforcement must be proved as per cl.32.5 of IS 456

Minimum requirement of reinforcement

Vertical reinforcement

60

Area of vertical steel/gross concrete area =0.0012 for bar with diameter <

16mmAre of vertical steel=126mm2

Spacing of 8mm diameter bar =250mm

Horizontal reinforcement

Area of horizontal steel/gross concrete area =0.002

Are of horizontal steel =105mm2

Spacing of 8mm diameter bar =250mm

5.3.1 Detailing: - Details of tension reinforcement and shear reinforcement is given in

Fig.5. 5

Fig.5.5. Detailing of deep beam by IS 456 method.

5.4. MODELLING OF DEEP BEAM

DEEP beam (model 2) modelled using SOLID 95 element type.

Reinforcement is provided in discrete by using REINF 264 element type.

Reinforcement is shown in Fig.5.6.

350mm

m

8mm dia bar @300mmc/c in both directions

150mm

700mm 2 no of 8mm dia bar +2 no

12mm dia bar

700mm

61

Fig.5.6. Reinforcement modelling using REINF264 element

Deep beam with reinforcement is modelled in ANSYS as shown in Fig.5.7

Fig.5.7. Beam modelled using ANSYS

62

5.5 MODELLING OF OPTIMISED BEAM

Shape of optimised beam was determined by topology optimisation in

ANSYS. After getting density plot from ANSYS optimised beam is modelled in

ANSYS for comparing the load carrying capacity and crack pattern, optimised model

has particular shape. So it is difficult to model its shape in ANSYS. In that case it is

assumed that rectangular edge gives almost same result as that of optimised one.

Fig.5.8 shows model of rectangular modelled optimised beam.

Fig.5.8 Optimised beam modelled using ANSYS.

5.6 EXPERIMENTAL PROGRAMME

5.6.1 Specific gravity tests for aggregates

Specific gravity of aggregate is made use of in design calculations of concrete

mixes with the specific gravity of each constituent known. It weights can be converted

into solid volume and hence a theoretical yield of concrete per unit volume can be

calculated. Specific gravity of aggregate is also required to be considered when we

deal with light weight and heavy weight concrete.

63

Average specific gravity of rocks vary from 2.6 to 2.8

Specific Gravity of Fine Aggregate

Weight of empty cylinder, W1 = 3.55kg

Weight of empty cylinder + sand, W2 = 6.10kg

Weight of empty cylinder + sand+ water, W3 = 6.90kg

Weight of empty cylinder +water, W4 = 5.250kg

Volume of void (W3-W2) =0.8cm2

Volume of container (W4-W1) = 1.75kg

Weight of aggregate= (W2-W1) = 2.55kg

Bulk density= (W2-W1) / (W4-W1) = 1.485

Voids ratio= (W3-W2)/ {(W4-W1)-(W3-W2)} = 0.8421

Porosity=Vv/V = W3-W2/W4-W1 = 0.4571

Specific gravity = wt. of fine aggregate/wt. of equal volume of water= 2.736

Specific gravity of coarse aggregate

Wt. of empty container (W1) = 7.2kg

Wt. of container + aggregate (W2) = 21.5kg

Wt. of container + aggregate +water (W3) = 26.10kg

Wt. of container + water (W4) = 17.1kg

Volume of voids (Vv) = (W3-W2) = 4.6kg

Volume of container (V) = (W4-W1) =9.8kg

Wt. of aggregate = (W2-W1) =14.3kg

64

Bulk density= W2-W1/W4-W1 =1.459kg

Void ratio = Vv/V solid =W3-W2/ [(W4-W1)-(W3-W2)] =0.8846

Porosity = Vv/V = W3-W2/W4-W1 =0.469

Specific gravity = Wt. of aggregate/Wt. of equal volume of water

=W2-W1/ [(W4-W1)-(W3-W2)] =2.75

5.6.2 Concrete mix design (IS 10262:2009)

Concrete mix design is the method of correct proportioning of ingredients of

concrete, in order to optimise the above properties of concrete as per site

requirements. Concrete is an extremely versatile building material because, it can be

designed for strength ranging from M10 (10MPa) to M100 (100MPa) and workability

ranging from 0 mm slump to 150 mm slump. In all these cases the basic ingredients of

concrete are the same, but it is their relative proportioning that makes the difference.

Basic Ingredients of Concrete: -

1. Cement – It is the basic binding material in concrete.

2. Water – It hydrates cement and also makes concrete workable.

3. Coarse Aggregate – It is the basic building component of concrete.

4. Fine Aggregate – Along with cement paste it forms mortar grout and fills the

voids in the coarse aggregates.

5. Admixtures – They enhance certain properties of concrete e.g. gain of

strength, workability, setting properties, imperviousness etc.

Concrete mix design (IS 10262: 2009)

Grade of concrete =25MPa

Cement = OPC43

Maximum size of aggregate =20mm

65

Minimum cement content =320kg/m3

Maximum water cement ratio =0.45

Workability =100mm slump

Exposure condition =Medium

Degree of super vision =Good

Types of aggregate =Crushed

angular aggregate

Maximum cement content =450kg/m3

Calculation

Specific gravity of cement =3.15

Specific gravity of

1. Fine aggregate =2.736

2. Coarse aggregate =2.75

Water absorption

1. Fine aggregate =0.5%

2. Coarse aggregate =1%

Free surface moisture

1. Fine aggregate = Nil

2. Coarse aggregate =Nil

Fine aggregate conforming to zone 1 is assumed for calculation purpose

Target compressive strength =f1

ck=f ck +1.65s

=25+1.65 4

=31.6N/mm2

66

Selection of water cement ratio

Maximum water content =186kg

Water for 100mm slump =186+6/100 186 =197kg

Calculation of cement content

W-c ratio =0.5

Cement content =197/.50 =394kg/m3

Mix calculation

(a) Volume of concrete =1m3

(b) Volume of coarse aggregate =0.60

(c) Volume of fine aggregate =0.40

Mix calculation for 1m3 concrete.

Volume of concrete =1m3

Volume of cement =393/3.15 1/1000 =0.125m3

Volume of water =197/1 1/1000 =0.197m3

Volume of all in aggregate =1-.125-.197 =0.678m3

Mass of coarse aggregate =0.678 0.6 2.736 1000 =1113kg

Mass of fine aggregate =0.678 0.40 2.75 1000 =745.8kg

Mix proportion for one m3 concrete,

Cement = 394kg/m3

Water =197 kg/m3

Fine aggregate =746 kg/m3

Coarse aggregate =1113 kg/m3

W-c ratio = 0.5%

67

5.6.3 Result of cube test

Cubes are prepared and tested for checking the accuracy of the mix design. Six

cubes were casted and three of them tested in 7 days and remaining is tested 28 days.

Seven days compressive strength is 22.2MPa.

Twenty eight days compressive strength is 26.6MPa.

5.6.4 Casting of RC deep beam

Reinforcement details

All reinforcement conforming to IS: 1786- 2006, stirrups were made from

8mm diameter bars. 12mm diameter bars are used at bottom of each specimen.

The reinforcement details of beam are Main reinforcement provided in the

beam was12 mm diameter bars of 2No‟s and 2 of No‟s 8mm diameter bars at bottom.

The stirrups are 8 diameter bars at 25 mm c/c. reinforcement were bends at an Angele

of 90 degree both end to get adequate development length. Two numbers of 8mm bar

is provided at the top of the beam for holding the stirrups.

Concrete

Ingredients and mix proportion

All the concrete used for casting beam was made in laboratory. The concrete

mix was designed to provide 28 day cube strength of 25MPa. Normal weight river

sand was used as fine aggregate and crushed granite stones were used as coarse

aggregates, with a maximum size of 40mm. ordinary OPC cement is used for

concreting. All the ingredients were batched by weight, and mixed in drum mixer.

Water added as per the mix design results. Drum mixer has capacity of one cubic feet.

Four such batches were prepared for casting beams.

Casting and curing

The mould is arranged at floor of the laboratory. The sides of the mould is

oiled well to prevent side wall of the mould from absorbing water from the concrete

and to facilitate easy removal of the specimen. The reinforcement cage were placed in

68

mould and cover between cage and form provided was 20 mm. Concrete mix

designed for M25 (1:1:2.5)The concrete contents such as cement, sand, aggregate and

water were weighed accurately and mixed .The mixing was done till uniform mix was

obtained. The concrete was placed in to the mould immediately after mixing and well

compacted. Compaction is done using tamping road. The beam is cast in vertical

direction (in the load bearing direction). The test specimens were re moulded at end

of 24 hours of casting. They were marked identifications. They are cured in water for

28 days. After 28 days of curing the specimen was dried in air and tested.

5.6.5 Test set up

Deep beam and optimised beam are shown in Fig 5.9 and Fig. 5.10

respectively. Point loads at the middle span of the beam are applied by universal

testing machine (UTM) in the laboratory (It works on the basics of Pascal‟s law). It

has maximum capacity of 600kN with maximum span of 70cm. Beam deflection at

the mid span is determined using dial gauge at the middle span as shown in the

Fig.5.11. Dial gauge has least count of .01mm. first 10 reading with each increment

of 10kN were recorded. Finally dial gauge were removed for finding ultimate load

carrying capacity.

Three point bending test was also conducted for optimised beam. Dial gauge

was set up at the middle bottom portion of the beam as shown in Fig.5.12.

Displacement corresponding to each increment of 10kN is recorded. Dial gauge was

removed after 100kN. At ultimate loading cracks were occurred. Cracks were marked

sequentially as they form. Ultimate load carry capacity recorded from UTM

Fig.5.9 Deep beam Fig.5.10 Optimised beam

69

Fig.5.11 Test set up for deep beam.

Fig.5.12 Test set up for optimised beam.

70

5.7 SUMMARY

This chapter explained the experimental investigation of this project. The

project is validated through laboratory experiments. Laboratory test for mix design,

determining compressive strength of concrete, three point bending test, etc. were done

in laboratory.

71

Chapter 6

RESULT AND DISCUSSION

6.1 INTRODUCTION

Results of this project are explained in following section. Result of cube test,

results obtained by ANSYS such as deflection, cracking load bearing capacity are

explained in this section. Result of optimisation i.e. density plot from ANSYS is

included in this chapter. Discussion based on the result is provided in the last section.

6.2. RESULT OF OPTIMISATION

Optimisation is carried out in ANSYS for both beams. Result of optimisation

is given below in Fig.6.1. In this figure there are different colours. These coliour

represent meterial usefullness in the beam. Red region in the figure represent the

material with density is one.which means that material must be there for resisting the

in applied load. Pure blue colour represent material with zero density. Which mens

there is no need of that material. In bet ween blue and red represent density between

zero and one. All the portion in blue was removed for fifty persent of optimisation.

Fig.6.1. Density plot of deep beam

72

From the figure all the reinforcement are shown in red colours. So at the mid portion

there is only reinforcement (without concrete). This is difficult to cast. Stress in the

reinforcement in Fig.6.2 shown is all the steel is fully stressed. Portion in red colour

represent highly stressed region and portion shown in the blue region is less stressed.

Partially stressed section of reinforcement can be removed.

Fig.6.2 Stress in steel for deep beam

6.3. RESULT OF BENDING TEST OF CONVENTIONAL DEEP BEAM

6.3.1 Load carrying capacity

Deep beam is designed for 150kN capacity. In bending test the result obtained

is 135kN. Optimised beam had load carrying capacity of 128.5kN.

6.3.2 Deflection

Load deflection graph for deep beam in ANSYS is compared with laboratory

result is given in Fig. 6.3. from the graph ansys deflection is low as compared to test

result. This is because of the assumption made in ANSYS software ie there is no slip

73

between concrete and steel. The result varition may be due to the qulity of concrete

and lack of good compaction. Yet shape of the curve is same in both the result.

Sudden in crease in deflection at the end region is same nature in both cases.

Fig.6.3. Comparison of deflection in ANSYS and experimental result

6.4. COMPARISON BETWEEN OPTIMISED BEAM AND DEEP BEAM

6.4.1 Ultimate load carrying capacity

Optimisation is done in fifty present volume reductions. Result obtained for

conventionally designed beam is 135kN. That of optimised beam is 128.5kN. That is

about only 5 reduction of load carrying capacity. Optimised beam has volume

reduction of 50%. When we compare to its load carrying capacity it is not a huge

reduction as compared to its volume reduction.

6.4.2 Deflection.

From the graph it is shown that deflection for the two beams is almost same.

In the initial part of the graph optimised beam (Beam 1) shows nearly equal deflection

to that of deep beam (Beam 2) (Fig.6.4). After 110kN deflection of optimised beam is

larger than that of deep beam.

0

5

10

15

20

25

30

0 2 4 6 8 10 12 14

load

x10

kN

Load deflection graph

Experimental

ANSYS model

Deflection in mm

74

Fig.6.4 comparison of Deflection for beam 1 and beam 2

6.4.3 Crack pattern

Deep beams are fails by shear failure. Cracks starts at the centre and extends

to support. Fig.6.5. show the cracks of deep beam. It is compared with cracks of

optimised beam shown in Fig.6.6. Probability of cracking is also checked in ANSYS.

Crack pattern for deep beam obtained in ANSYS is shown in Fig.6.7. And crack

pattern for optimised beam obtained in ANSYS is shown in Fig.6.8

Fig.6.5. Crack pattern for the deep beam

0

2

4

6

8

10

12

14

16

0 5 10 15 20

Load

(x

10kN

)

Deflection in mm

Load-deflection Graph

optimised beamBeam 2

Beam 1

75

Fig.6.6. Crack pattern for the optimised beam

Fig.6.7. Crack pattern for deep beam obtained from ANSYS

76

Fig.6.8. Crack pattern for optimised beam obtained from ANSYS

6.4.4 Failure mode

Deep beams are generally fails by shear failure. In this experiment deep beam

fails by shear failure. Intensity of point load is high as compared to the crushing

strength of concrete. Concrete under the load head was crushed due to this reason.

Fig.6.10. shows the crushing failure of concrete at the point of application of load.

ANSYS analysis is shown in Fig.6.9 shows theoretical formation of failure under the

load. Crushing failure of optimised beam obtained from ANSYS is shown in Fig.

6.11. The first failure of deep beam is crushing failure. Both test result and ANSYS

analysis gives the same output. First failure of optimised beam was concrete crush

under the point of application of load. Then crack starts. Order of formation of cracks

was ordered numerically as shown in Fig.6.5 for conventionally designed beam and

Fig.6.6 for optimised beam. Formation of first crack obtained from ANSYS for the

deep beam and optimised beam is shown in Fig.6.7 and Fig.6.8. This result shows that

deep beam fails by shear failure.

77

Fig.6.9. Concrete crushing at point of application of load obtained from

ANSYS

Fig.6.10. Concrete crushed at the point of application of load

78

Fig.6.11. crushing failure of optimised beam obtained from ANSYS

6.5 DISCUSSIONS

Topology optimisation is best method of weight saving of concrete structure

without reducing its performance. But it‟s now emerging field of structural

engineering. Here our objective is to compare the performance of optimised beam and

conventionally designed beam. If comparisons show there is no reduction in

performance then we can conclude that topology is best method of weight saving in

concrete structures. But load carrying capacity of optimised beam is slightly less than

conventionally designed beam. The reason may be due to the accuracy of optimisation

or poor construction practice. There is probability of poor construction practice when

casting of optimises beam because of its shape. Compaction of optimised beam was

not as expected due to its shape. Recasting of optimised beam with new technique

with proper compaction may improve its property.

79

6.6 SUMMARY

In this chapter various results of this project were explained and compared.

Comparison between optimised beam and conventionally designed beam were

compared for its load carrying capacity, deflection, and crack pattern. Comparison of

results between ANSYS and test specimen were compared. Comparison of load

carrying capacity, deflection was explained with the help of graph. Finally a small

discussion based on the result obtained was done.

80

Chapter 7

CONCLUSION AND SCOPE FOR FURTHER WORK

7.1 CONCLUSION

The main conclusions drawn from the current research can be summarised as

follows:

1. Deflection and ultimate load bearing capacity of deep beam obtained from

finite element (FE) analysis is comparable to that of experimental value and

this shows that ANSYS software can be used for the analysis of deep beams

2. The variation in ultimate load carrying capacity of optimised beam is marginal

and is only 5% less than that of deep beam without optimisation. There for

topology optimisation method can be effectively used for deep beams.

3. Deflection profile for deep beam and optimised beam obtained in finite

element analysis were nearly equal.

4. Crack pattern of beams obtained in finite element analysis is almost similar to

that obtained in experimental programme.

7.2 SCOPE FOR FURTHER WORK

1. In this study optimised beam layout is obtained from FE analysis using

ANSYS. Additional research can be carried out by changing the software to

MAT LAB, SOLID THINKING, and TOP OPT etc.

2. Compaction is difficult in optimised beam due to non-uniform shape of the

beam and its formwork. So modification of mould or compaction method can

be considered for additional research.

3. In this experimental research small size beams were cast and tested, large size

beam can be considered for further research.

81

REFERENCES

1. IS 456:2000, (2002)“plain and reinforced concrete code of practice” forth

revision.

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