Finite Element Program in Autocad Vba

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FINITE ELEMENT PROGRAM IN AUTOCAD VBA

FOR WINKLER FOUNDATION

by

Name Roll No.

1. Jhaman Lalchandani 94 CE 31

Supervised by

Prof. Mumtaz Ali Memon

Mr. Rasool Bux Mahar

Department of Civil Engineering

Mehran University of Engineering and Technology, Jamshoro

Submitted in partial fulfillment of the requirements for the degree of

Bachelor of Civil Engineering

July, 2000


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DDEEDDIICCAATTIIOONN

This work is dedicated to Leibniz, Gottfried Wilhelm, (1646-1716), German philosopher,

mathematician, and statesman, regarded as one of the supreme intellects of the 17th

century.

Leibniz was considered a universal genius by his contemporaries. His work encompasses

not only mathematics and philosophy but also theology, law, diplomacy, politics, history,

philology, and physics.


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CCEERRTTIIFFIICCAATTEE

This is to certify that the work presented in this thesis titled ‘FINITE

ELEMENT PROGRAM IN AUTOCAD VBA FOR WINKLER FOUNDATION’ is

entirely written by the following students themselves under the supervision of Prof.

Mumtaz Ali Memon and Mr. Rasool Bux Mahar.

Name Roll No.

1. Haibat Khan Abro 94 CE 60

2. Jhaman Lalchandani 94 CE 31

Thesis Supervisors External Examiner

Chairman

Department of Civil Engineering

Date :________________


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ACKNOWLEDGEMENTS

The authors wish to express their gratitude to Mr. M. Soomar Khatri, Chairman, Dept. of

Civil Engineering, for the help and cooperation he rendered at every stage of the project.

The authors are indebted to Prof. Mumtaz Ali Memon and Mr. Rasool Bux Mahar, the

thesis supervisors, for the advice and help in preparing the thesis.

The authors acknowledge, with thanks, the time spared and the help and expert advice

given by Dr. Ghulam Hussain Sidiqui.

In the last, but not the least, the authors are extremely grateful to Mr. Parvaiz Ahmed

Shaikh, Mr. Fazlullah Solangi, and Mr. Mohammad Muhajir for lending their books for a

considerable length of time.


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Abstract

In conventional method for design of combined footings, the soil-structure interaction isusually neglected because it requires a lot of calculations. Instead, the footings are designed

by assuming that they are rigid. However, the demerit of this assumption is that thecomputed bending moments are higher than actual, which means that larger amounts ofreinforcement steel are to be provided than required. This soil-structure interactionphenomena, was first studied by Winkler (1867) by assuming it as a beam (structure) onelastic springs (soil) placed continuously underneath it. Even though, this ‘WinklerFoundation Model’ gives much better approximation of soil pressure, it did not gain muchpopularity because the problem becomes statically indeterminate and requires lots ofcalculations.

The authors have developed a finite element program BEAMDEZ in AutoCAD VBA, which is based on Winkler Foundation Model and computes design parameters (bendingmoment, shear, soil pressure, deflection, rotation, etc at all node points in the discretized

footing) and draws shear, bending moment, soil pressure, and deflection diagrams in AutoCAD. The program also considers soil-structure separation and elastic range of soil. Itis illustrated through examples that bending moments obtained by this method are smallerthan those obtained by conventional method and therefore the reinforcement design requiresabout 30% less steel than by conventional method.


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TABLE OF CONTENTS

LIST OF SYMBOLS X

INTRODUCTION 1

1.1 BACKGROUND 1

1.2 AIM 21.3 FOOTINGS 31.4 DESIGN OF FOOTINGS 41.4.1 Design of Combined Footing by Conventional (Rigid) Method 51.4.2 Design of Combined Footing by Finite element method 5

1.5 NUMERICAL SOLUTIONS 61.6 FINITE ELEMENT METHOD 81.7 HISTORICAL COMMENTS 9

FINITE ELEMENT METHOD 10

2.1 BACKGROUND OF THE FINITE ELEMENT METHOD 102.2 MATRIX METHODS (STIFFNESS/DISPLACEMENT METHOD) 112.3 STIFFNESS AND FLEXIBILITY METHODS OF MATRIX A NALYSIS 12

2.4 STIFFNESS METHOD 122.5 A NALYSIS OF CONTINUUM STRUCTURES 13

2.6 CLASSICAL A NALYSIS OF SOLIDS 132.7 FINITE ELEMENT A NALYSIS OF SOLIDS 142.8 Fundamental requirements 15

2.9 FINITE ELEMENT CONCEPTS 152.9.1 Elements 152.9.2 Type and Number of Elements 16

2.9.3 Equilibrium and Degrees of Freedom 162.9.4 External Loads 16

2.9.5 Accuracy of Solution 172.9.6 Necessity of Computer Application 172.9.7 Continuity of Displacements 17

2.9.8 Plotting of Results 17

2.10 APPLICABILITY TO DIFFERENT FIELDS 172.11 GENERAL STEPS IN FINITE ELEMENT METHOD 182.12 MATRIX OPERATIONS 192.12.1 Transpose of Matrix 19

2.12.2 Matrix Multiplication 192.12.3 Symmetric Matrices 202.12.4 Band Matrix 202.12.5 Identity Matrix 21

2.12.6 Inverse Matrix 21


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2.13 ADVANTAGES OF FINITE ELEMENT METHOD 212.13.1 Better Approximations 22

2.13.2 Trial Solutions 222.13.3 Boundary Conditions 222.13.4 Material Properties 232.13.5 Non Homogenous Continua 23

2.13.6 Systematic Generality 232.14 LIMITATIONS OF FINITE ELEMENT METHOD 242.14.1 Validity of Results 242.14.2 Computer Requirements 242.14.3 Discretizing the Continuum and Data Input 252.14.4 Interpretation of Results 25

COMBINED FOOTINGS 26

3.1 COMBINED FOOTING 26

3.2 PURPOSE OF COMBINED FOOTINGS 26

3.2.1 Columns Near Property Line 263.2.2 Closely Spaced Columns 273.2.3 Poor Soil 273.2.4 Differential Settlement 28

3.3 SHAPE OF COMBINED FOOTINGS 283.3.1 Rectangular Combined Footing 293.3.2 Trapezoidal Footing 293.3.3 Strap Footing 303.3.4 Strip, Grid, And Mat Foundation 31

3.4 DESIGN OF R ECTANGULAR COMBINED FOOTINGS 313.4.1 Footing Dimensions 32

3.4.2 Rounding of Dimensions 32

3.4.3 Shear and Moment Computations 323.4.4 Depth of Footing 32

3.4.5 Reinforcement Design 333.4.5.1 Main Reinforcement 33

3.4.5.2 Transverse Reinforcement 333.4.5.3 Shear Reinforcement 343.5 DESIGN STEPS 34

3.6 DEMERITS OF R IGID DESIGN 353.6.1 Assumptions of Rigid Design 35

3.7 BEAM ON ELASTIC FOUNDATION 36

BEAM ON ELASTIC FOUNDATION 37

4.1 ELASTIC FOUNDATION 37

4.2 WINKLER MODEL 374.3 LIMITATIONS OF WINKLER MODEL 384.4 OTHER MODELS 384.5 MODULUS OF SUBGRADE R EACTION 39

4.6 PREVIOUS EFFORTS BY RESEARCHERS TO EVALUATE THE VALUE OF K 414.7 NEW RESEARCH FOR EVALUATION OF THE VALUE OF K 42


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4.7.1 Augmentation of k 424.7.2 Vallabhan’s Iterative Procedures 43

4.7.3 Non-Dimensional Parameters For k 434.8 DISPLACEMENT METHOD 434.9 BASIC DIFFERENTIAL EQUATION 434.10 GENERAL SOLUTION OF THE DIFFERENTIAL EQUATION 45

4.11 BOUNDARY CONDITIONS OF AN U NLOADED MEMBER 484.12 STIFFNESS MATRIX OF A MEMBER ON ELASTIC FOUNDATION 494.13 APPLICATION OF THE FINITE ELEMENT METHOD 674.14 GENERAL R ELATIONS 674.15 DEVELOPMENT OF THE ELEMENT A MATRIX 714.16 DEVELOPMENT OF THE ELEMENT B (OR AT) MATRIX 72

4.17 DEVELOPMENT OF ELEMENT S MATRIX 724.18 DEVELOPMENT OF THE ELEMENT ESA

T AND EASA

T MATRICES 75

4.19 DEVELOPMENT OF THE GLOBAL MATRIX ASAT 76

4.20 ADDING THE NODE “SPRINGS” K TO ASAT 77

4.21 DEVELOPMENT OF P MATRIX 784.22 BEAM WEIGHT 78

4.23 BOUNDARY CONDITIONS 78

PROGRAM 80

5.1 CAD 805.2 CAE 805.3 AUTOCAD 2000 815.3.1 AutoCAD ActiveX Technology 81

5.4 R APID APPLICATION DEVELOPMENT (RAD) 825.5 VISUAL BASIC 82

5.6 VBA 83

5.7 AUTOCAD VISUAL BASIC FOR APPLICATIONS (VBA) 835.7.1 VBA implementation in AutoCAD 84

5.7.2 Strengths of AutoCAD ActiveX and VBA Together 855.7.2.1 Speed 85

5.7.2.2 Ease of Use 855.7.2.3 Windows Interoperability 855.7.2.4 Rapid Prototyping 85

5.7.2.5 Programmer Base 855.7.3 Embedded and Global VBA Projects 85

5.8 FLOW CHART OF THE PROGRAM 875.9 PROGRAM CODE 88

Code: Beamdimensions 88

Code: boundary 96Code: Elements 106

Code: Exporter 143Code: Filer 145

Code: loadmom 147Code: memberlen 153Code: saver 158

Code: starter 160


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USER GUIDE 162

6.1 I NSTALLATION 1626.2 LOADING THE PROGRAM 1626.3 R UNNING THE PROGRAM 163

6.4 E NTERING

BEAM

DATA

1646.5 DISCRETIZING THE BEAM 1656.6 E NTERING ELEMENT LENGTHS 1656.7 APPLIED LOADS AND MOMENTS 1656.8 BOUNDARY CONDITIONS 166

6.9 R ESULTS 1676.10 CURSOR ICONS 170

EXAMPLE PROBLEMS 171

Example problem 1: 171Example problem 2: 181

Example problem 3: 194Example problem 4: 209

Example problem 5: 224

APPENDICES 227

REFERENCE 230

INDEX 233


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LIST OF SYMBOLS

= Poisson’s ratio

= slope, rotation = multiplier = deflection = percentage of steel = area ratio = deformation = pressure = change in settlement or deformation = increment of contact pressureH = settlement of foundation

s = Poisson’s ratio of soil a = depth of Whitney’s block A = area Ab = area of steel bar Ag = gross cross-sectional area of concrete As = area of steelb = widthB = widthC = compression forced =depth of footingdb = dia. of barE = modulus of elasticityEI = bending rigidityEs = modulus of elasticity of soil f c ' = characteristic 28-day compressive strength of concrete f y = characteristic strength of steelI = identity matrixk s = modulus of subgrade reactionL = lengthLd = development lengthLF = load factorM = bending momentMn = nominal moment

Mu = ultimate (factored) momentN = number of elementsP = applied loadPu = ultimate loadP w = working loadq = bearing pressureq a = allowable bearing pressureQcon = constant soil pressure


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q ult = ultimate computed bearing pressureR = resultant forceS = center to center distance between columnsSF = safety factor T = tension force

UR = ultimate ratio V = shear force v c = shear stress w = load intensity w = width of columny = deflection


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C H A P T E R O N E

INTRODUCTION

1.1 Background

Conventionally the combined footings are designed by assuming that they are rigid. This

assumption allows the engineer to consider the soil pressure as linear. Then the footing

dimensions are set in such a way that the centroid of area (in plan) of footing coincides

with the resultant of column loads. This gives rise to uniformly distributed soil pressure

on the entire footing area, thus the combined footing for any number of columns can be

considered as an inverted uniformly loaded continuous beam with all reactions (column

loads) known. These assumptions render the problem as statically determinate, thus

simplifying the design process. However, the demerit of these assumptions is that higher

than actual bending moments are obtained, which implies that larger than required

amounts of steel are to be provided. This becomes uneconomical because steel is an

expensive item.

In practice, a footing cannot be made so much rigid that its bending becomes negligible

because this would require a great depth of concrete, which will again render it

uneconomical. Due to this bending, deflection is not same at all points; for example,

deflection will usually be more under the columns than at midspan between them. Since

soil pressure will be higher for greater deflections, these non-uniform deflections will

give rise to non-uniform soil pressure under the footing. This phenomena, known as

soil-structure interaction, was first studied by Winkler in 1867 by assuming it as a beam

(structure) on elastic springs (soil) placed continuously underneath it. Even though this

model (called Winkler Foundation Model) gave much better approximation of soil

pressure, it did not gain much popularity because the problem became statically

indeterminate and required a lot of calculations.


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In present age, when tools like matrix displacement method (which does not distinguish

between determinate and indeterminate structures) and finite element method has been

developed and high-speed digital computers are commonly available, the Winkler

Foundation Model can be solved easily and quickly. Still many engineers design the

combined footings by conventional method because the commercial software packages of

finite element analysis are rather too expensive.

1.2 Aim

Authors have developed a finite element program BEAMDEZ in AutoCAD VBA with

the aim that it should be easily available to anyone who requires it. This program

computes design parameters (bending moment, shear, soil pressure, deflection, rotation,

etc at all node points of the discretized footing) based on the theory of

beam-on-elastic-foundation and draws the shear, bending moment, soil pressure, and

deflection diagrams in AutoCAD. Soil reaction is modeled as springs under the footing

(Winkler Foundation Model). The program also considers soil-structure separation (soil

pressure is zeroed out at such nodes to prevent tension in soil) and elastic range of soil (a

certain deflection after which soil pressure becomes constant). Deflections and rotations

are obtained using the stiffness matrix. As the stiffness of the footing is considered, the

bending moment obtained is more realistic and lower in magnitude than the one

computed by the conventional (rigid) method. This results in economical design of the

footing because lesser amount of reinforcement steel is required.

Every effort has been taken to make the program user friendly. Data entry and editing

have been made very easy. Data files can be saved by the program in binary format and

results can be exported to text format. The shear force, bending moment, deflection,

rotation, and soil pressure diagrams can be obtained readily in AutoCAD and can be

manipulated using AutoCAD’s r ich set of tools. These diagrams can be printed easily and

saved in various file formats supported by AutoCAD for use in other programs such as

word processors, etc.


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A two column combined footing and a three column combined footing have been

designed by both the conventional (rigid) method and the finite element method to

illustrate the economy achieved by the later method.

1.3 Footings

Footings are structural members used to support columns and walls and transmit their

load to the soil. Footings act as transition-members to distribute the higher pressure of

loads coming from the superstructure to larger areas of soil in such a way that

the load bearing capacity of the soil is not exceeded,

Excessive settlement, differential settlement, or rotations are prevented, and

Adequate safety against overturning or sliding is maintained.

Types of footings:

Different types of footings may be used to support building columns or walls. The most

common types are as follows:

Figure-1.1 Types of footings

1. Continuous or Wall footings are used to support structural walls that carry loads from

other floors, or to support nonstructural walls. They have limited width and a


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continuous length under the wall. Wall footings may have one thickness, be stepped,

or have a sloped top.

2. Isolated or single footings are used to support single columns. They may be square,

rectangular, or circular. Again, the footing may be of uniform thickness, stepped, or

have a sloped top. This is one of the most economical types of footings and it is used

when columns are spaced at relatively long distances.

3. Combined footings usually support two or more columns in a line. The shape of the

footing in plan may be rectangular or trapezoidal, depending on column loads.

Combined footings are used when two columns are so close that single footings

cannot be used, or when one column is located at or near a property line.

4. Cantilever or strap footings consist of two single footings connected with a beam or a

strap and support two single columns. They are used when one footing supports an

eccentric column and the nearest adjacent footing lies at quite a distance from it. This

type replaces a combined footing and is more economical.

5. Raft or mat foundation consists of a single large footing, usually under the entire

building area, and supports the columns of the building. They are used when

the soil bearing capacity is low

column loads are heavy

single footings cannot be used

piles are not used

differential settlement must be reduced through the entire footing system.

1.4 Design of footings

The area of footing is determined according to the bearing capacity of the soil and

intensity of the applied loads. The depth of the footing and reinforcement are determined

for punching shear, beam shear, and bending moment calculations.


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1.4.1 Design of Combined Footing by Conventional (Rigid) Method

In conventional method, the basic assumption is that the footing is a rigid member so that

the soil pressure can be taken as linear. In order to make the soil pressure uniform, the

length of footing is set in such a manner that the resultant of loads coming from columns

acts at the centroid of the footing area. If the soil pressure is uniform and the resultant of

applied loads acts at the centroid of the footing then the settlements will also be uniform.

This assumption is approximately true if the soil is homogeneous and footing is rigid.

However, in actual practice, it is very difficult to make a rigid footing because the

thickness required would have to be large. Bowles suggests that the success of the

designs based on the assumption of a rigid member has probably resulted from a

combination of soil creep, concrete stress transfer, and overdesign.

Depth of the footing is obtained from two-way action or wide-beam shear (whichever is

greater). Reinforcement steel is designed using the selected depth and bending moment

diagram.

1.4.2 Design of Combined Footing by Finite element method

Because the footing cannot be made rigid in actual practice, therefore the settlements will

not be uniform or linear if the column spacing is large. For one thing, the more heavily

loaded columns will cause larger settlements, and thereby larger subgrade reactions, than

the lighter ones. Also, since the continuous strip or beam between the columns will

deflect upward relatively to the nearby columns, this means that the soil settlement, and

thereby the subgrade reaction, will be smaller midway between columns than directly at

the columns. This is shown schematically in figure 1.2. In this case, the subgrade

reaction can no longer be assumed as uniform. A reasonably accurate but fairly complex

analysis can then be made using the theory of beams on elastic foundations.

Structure/soil interaction problems may be simplified as a beam (structure) on springs

(soils), which is a one dimensional (1D) problem.

Even though Winkler had studied the beam on elastic springs in 1867 but the method was

not used in common practice because of the enormous amount of calculations involved.

Because of easy availability of computers and development of finite element procedures


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in present times, the beam-on-elastic-foundation analysis can be made easily by assuming

it as a beam on springs (or Winkler foundation) and using a computer program.

Figure-1.2 Combined Footing

Due to the overdesign involved in rigid method, current practice tends to modify the

design by beam-on-elastic-foundation analysis. Now the footing is considered as a

“beam” or flexural member. Modified moments (which tend to be lower in magnitude)

are obtained using finite element analysis using a computer program. Thus, an

economical design can be obtained as will be shown later using the finite element

program BEAMDEZ.

1.5 Numerical Solutions

Practically all phenomena in nature, whether biological, geological, or mechanical, can be

modeled with the help of laws of physics, using algebraic, differential, or integral


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equations relating various quantities of interest. To determine the characteristics of fluid

flow, finding the concentration of pollutants in sea water or in the atmosphere, stress

distribution in complex structures subjected to a variety of loads, and simulating weather

in an attempt to understand and predict the mechanics of formation of tornadoes and

thunder storms are a few examples of many important practical problems. To derive the

governing equations for these problems is not unduly difficult, but to solve these by exact

methods of analysis is a formidable task. In such cases, the numerical methods are

employed as an alternative means of finding approximate solutions. Among these the

finite-difference method, variational methods such as the Ritz and Galerkin methods, and

the finite-element method have been frequently used.

In the finite-difference approximation of a differential equation, the derivatives in theequations are replaced by difference equations, which involve the values of the solution at

discrete mesh points of the domain. The resulting discrete equations are solved, after

imposing the boundary equations for the values of the solution at the mesh points. Even

though the finite-difference method is simple in concept, it suffers from several

disadvantages. The most notable are the inaccuracy of derivatives of the approximated

solution, the difficulties in imposing the boundary conditions along non-straight

boundaries, the difficulty in accurately representing geometrically complex domains, and

the inability to employ non-uniform and nonrectangular meshes.

In the variational solution of differential equations, the differential equation is put into an

equivalent variational form, and then the approximate solution is assumed to be a

combination (c j j) of given approximation functions j. The parameters c j are

determined from the variational form. The variational methods suffer from the

disadvantage that the approximation functions for problems with arbitrary domains are

difficult to construct.

The finite element method overcomes the difficulty of the variational methods because it

provides a systematic procedure for the derivation of the approximation functions. The

method is endowed with two basic features, which account for its superiority over other

competing methods. First, a geometrically complex domain of the problem is represented


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as a collection of geometrically simple subdomains called finite elements. Second, over

each finite element the approximation functions are derived using the basic idea that any

continuous function can be represented by a linear combination of algebraic polynomials.

The approximation functions are derived using concepts from interpolation theory, and

are therefore called interpolation functions. Thus, the finite element method can be

interpreted as a piecewise application of the variational methods (e.g., Ritz and weighted-

residual methods), in which the approximation functions are algebraic polynomials and

the undetermined parameters represent the values of the solution at a finite number of

preselected points called nodes, on the boundary and in the interior of the element.

1.6 Finite Element Method

Availability of high-speed computers with large memories has enabled engineers to

employ various numerical discretization techniques for approximate solution of complex

problems. One such technique is the finite element method.

It was originally developed as a tool for structural analysis, but the theory and formulation

have been progressively so refined and generalized that the method has been applied

successfully to such other fields as heat flow, seepage, hydrodynamics, and rock

mechanics. Since the method has a broad applicability to different fields of science andthe computer code for its solution can be applied to different problems with little or no

change, the method has gained wide acceptance by designers and research engineers.

Even though the finite element method has been around for more than 40 years and is

recognized as an extremely valuable tool, many engineers still do not know how to go

about using it and very few engineers understand it. One of the main reasons for this is

that the subject has generally been surrounded by a high level of research activity.

Coupled with this is a fact that because of the amount of calculations, which the method

involves, it tended to go directly from its embryonic stage to an advanced computing

stage. There never seemed to be an intermediate stage at which it could have been

conveniently slotted into curricula of various engineering subjects, even though the basic

principles on which the method is based are taught. The three basic conditions involved


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in the finite element method are the equilibrium of forces, compatibility of displacements,

and stress-strain relationships and it essentially relies on the matrix methods and computer

programming for the solution of problems. The method is quite versatile and can be

applied to very complex problems.

The finite element method is the best approach available for the numerical analysis of

continua. The theory requires discretization of a given structure into a network of finite

elements and implementation of the analysis on a digital computer.

1.7 Historical comments

Finite element is not a novel idea to represent given domains as collections of discrete

elements. Ancient mathematicians had estimated the value of by discretizing the

circumference of a circle in small line segments making a polygon inscribed in the circle.

They computed the value of to accuracies of almost 40 significant digits by representing

the circle as a polygon of a finitely large number of sides.

In modern times the idea found a home in aircraft structural analysis, where, for example,

wings and fuselages are treated as assemblage of stringers, skins, and shear panels. In

1941, Hrenikoff [Hrenikoff, 1941] introduced the so-called framework method, in which

a plane elastic medium was represented as a collection of bars and beams. The use of

piecewise continuous functions defined over a subdomain to approximate the unknown

function dates back to the work of Courant [Courant, 1943], who used an assemblage of

triangular elements and the principle of minimum potential energy to study the St. Venant

torsion problem. Although certain key features of the finite element method can be found

in the works of Hrenikoff [Hrenikoff, 1941] and Courant [Courant, 1943], the formal

presentation of the finite element method is attributed to Argyris and Kelsey [Argyris,

1960] and to Turner, Clough, Martin, and Topp [Turner, 1956]. However, the term

“finite element” was first used by Clough [Clough, 1960]. Since its inception, the

literature on finite-element applications has grown exponentially and today there are

numerous journals, which are primarily devoted to the theory and application of the finite-

element method.


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C H A P T E R T W O

FINITE ELEMENT METHOD

The finite element method has gained wide acceptance by the engineering professions for

being an extremely valuable method of analysis. Its employment has allowed satisfactory

solution to be achieved for many problems, which had formerly been considered as

insoluble. Within a relatively short period, application of the method has been extended

to many fields. Many shareware and commercial software packages are available

nowadays for general and specific application.

Figure-2.1 Typical skeletal structures. (a) Continuous beam. (b)Multistory frame.

2.1 Background of the Finite Element Method

Many engineering structures are composed of a series of individual members, which are

connected together at a number of points. Such structures are called ‘skeletal’ structures,

the points at which the individual members are connected being referred to as ‘node

points’. Examples of such structures are the continuous beams, trusses, and the

multistory frames. Engineers have long appreciated that the analysis of these skeletal

Typical nodal point

Typical individualmember

Typical

individual

member

Typicalnodal

point

(a) (b)


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structures can be carried out by first considering the behavior of each individual element

independently and by then assembling the elements together in such a way that

equilibrium of forces and compatibility of displacements are satisfied at each nodal point.

An example of such a process is the analysis of a continuous beam by the slope-deflection

method where the relationship between the moments and the rotations within each

individual span of the beam is first established, the spans then being combined together

such that equilibrium of moments and compatibility of rotations are satisfied at the points

of interconnection.

However, when a structure comprised of many members, such as a continuous beam

containing many spans or a multistory frame containing many bays, is being analyzed,

this type of approach can become very laborious and can involve the solution of a largenumber of simultaneous equations. Because of this, in the past much research effort has

been devoted to developing analytical techniques, based on a physical appreciation of the

structural behavior, which would reduce the amount of work required to complete an

analysis, and would not require the direct solution of many simultaneous equations. A

prime example of such a technique is the Hardy Cross Moment Distribution Method, in

which, instead of setting up the simultaneous equations explicitly as in the slope-

deflection method, the solution is accomplished in a series of convenient steps.

With the advent of the electronic digital computer, however, engineers realized that the

solution of a large number of simultaneous equations no longer posed an insurmountable

problem and this prompted a return to fundamental methods of analysis, such as the

slope-deflection method. These methods, since they involve a number of repetitive steps,

are particularly suitable for automatic computation, and they have been formulated to take

maximum advantage of the capabilities of a digital computer. These so-called ‘matrix

methods’ for analyzing skeletal structures have been firmly established for a number of

years.

2.2 Matrix Methods (Stiffness/Displacement Method)

The finite element method was first introduced to civil engineering through attempts to

generalize the stiffness method to deal with structural continua, such as the covering


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sheeting on aircraft wings. As a result, many of the concepts and much of the

terminology of the stiffness method has been incorporated into the finite element method.

To understand the principles of the finite element method, it is helpful first to consider

briefly the basis of the stiffness method of structural analysis.

2.3 Stiffness and Flexibility Methods of Matrix Analysis

The matrix methods of structural analysis may be formulated in three different ways.

i. Stiffness (displacement) method.

ii. Flexibility (force) method.

iii. Mixed method

The stiffness and flexibility methods differ in the order in which the two basic conditions

of joint (or nodal) equilibrium and compatibility are treated. In the stiffness method, the

displacement compatibility conditions are satisfied and the equations of equilibrium set

up and solved to yield the unknown nodal displacements. In the flexibility method, the

conditions of joint equilibrium are first satisfied and the equations arising from the need

for compatibility of nodal displacements solved to yield the unknown forces in themembers. In addition to these two basic approaches, in recent years a mixed formulation

involving both approaches has been also used.

2.4 Stiffness Method

The stiffness method was derived for skeletal structures. Analysis of behavior under

imposed loading is performed in stages. First, member connections are prevented from

displacement and the loading on the fixed joints is determined from separate analyses ofeach member. Second, sets of linear simultaneous equations relating forces and

displacements at the end of each member are derived. These are called ‘local stiffness

matrices’. All of the calculations described so far are performed on isolated members, but

in the third stage the stiffness matrix of the entire structure is compiled using equilibrium

of forces on each joint and compatibility of displacements of member ends at each joint.


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Displacement boundary conditions are imposed at appropriate joints and these render the

overall stiffness matrix non-singular. Solution of the equations gives the joint

displacements and from these and the local stiffness matrices the load effects on each

member are determined. The method has a number of features that facilitate automatic

handling. Local stiffness matrices of members and joint loading from members can be

determined by procedures that need take no account of the overall structural

configuration. The overall stiffness matrix, called the global stiffness matrix, of a

structure can be assembled automatically from the local stiffness matrices of members

using only a specification of the members connected to each joint. Standard solution to

linear simultaneous equations can be used to determine joint displacements. A single

computer program can be derived to solve all skeletal structures from a description of

member loading, member properties, structural topology, and the displacement boundary

conditions.

2.5 Analysis of Continuum Structures

In addition to skeletal structures, engineers are often also concerned with the analysis of

continuum structures, such as deep beams, plates and slabs subjected to bending, dam

walls, folded-plate and shell structures, where the structural surface is continuous instead

of being composed of a number of individual components. Classical methods, such as the

classical theory of plate flexure, can be applied to the analysis of these continua, but such

methods have very limited fields of application because of the great difficulties that are

experienced when dealing with any irregularities in structural geometry or applied loading

conditions.

2.6 Classical Analysis of Solids

The classical approach for analyzing a solid requires finding a stress or displacement

function that satisfies the differential equations of equilibrium, the stress-strain

relationships, and the compatibility conditions at every point in the continuum, including

the boundaries. Because these requirements are so restrictive, very few classical solutions

have been found. Among those, the solutions are often infinite series that is practical


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calculations require truncation, leading to approximate results. Furthermore,

discretization of differential equations by the method of finite differences has the primary

disadvantage that boundary conditions are difficult to satisfy. A secondary disadvantage

is that accuracy of the results is usually poor.

Figure-2.2 Typical finite element idealizations of continua. (a)Dam wall (b) Folded plate

2.7 Finite Element Analysis of Solids

On the other hand, the finite element approach yields an appropriate analysis based upon

an assumed displacement field, a stress field, or a mixture of these within each element.

The finite element method represents the extension of matrix methods for skeletal

structures to the analysis of continuum structures. In the finite element method, the

continuum is idealized as a structure consisting of a number of fictitious individual

elements connected only at nodal points, as shown in figure-2.2. It is only in this

idealization that the method differs from the standard matrix method. The loads or other

influences cause deformations (or strains) throughout the continuum, accompanied by

internal stresses and reactions at restrained points. The primary objectives of analysis byfinite elements are to calculate approximately the stresses and deflections in a structure.

The finite element method is extremely powerful since it enables continua with complex

geometrical properties and loading conditions to be accurately analyzed. The method

Typical nodal point

finite element

Typical nodal point

Typical triangularfinite element

(a) (b)


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involves extensive computations but, because of the repetitive nature of these

computations, it is ideally suited for programming for a solution using a computer.

2.8 Fundamental requirements

Whatever the cause of the internal forces and deformations in a structure, three basic

conditions must be observed. These are:

i. the equilibrium of forces;

ii. the compatibility of displacements; and

iii. the laws of material behavior.

The first condition merely requires that the internal forces balance the external applied

loads. Although the use of this condition alone is sometimes sufficient to enable a

statically determinate problem to be solved, the conditions of compatibility and material

behavior then being automatically satisfied, for redundant structures it yields insufficient

information to enable a complete analysis to be conducted. In these circumstance the

conditions of compatibility must be invoked separately. Compatibility requires that the

deformed structure fits together, i.e. that the deformations of the members are compatible.

Before this condition can be used, it is necessary to know the relationship between load

and deformation for each component of the structure. This relationship, which in

problems of linear elasticity reduces to the use of Hooke’s Law, is the third condition.

The use of these three conditions is a fundamental requirement of any method of

structural analysis.

2.9 Finite Element Concepts

In order to work with the finite element method, some knowledge of matrix methods,mechanics, variational methods, and computer skills is necessary.

2.9.1 Elements

A finite element is a subregion of a discretized continuum. It is of finite size (not

infinitesimal or infinite) and usually has a simpler geometry than that of the continuum.


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The shape of elements can be line elements (spring and beam elements), triangular,

rectangular, tetrahedral, or brick elements, etc (figure-2.3). The vertices of the elements

are known as node points or simply nodes.

Figure-2.3 Types of finite elements

2.9.2 Type and Number of Elements

The type and number of elements used can be decided by the analyst depending on the

accuracy required and the number of critical locations.

2.9.3 Equilibrium and Degrees of Freedom

The number of ‘equilibrium’ equations for each node is set by specifying the number of

degrees of freedom, as is done for skeletal structures. A problem with infinite degrees of

freedom can be converted to one with a finite number by the finite element method in

order to simplify the solution process.

2.9.4 External Loads

The system of external loads acting on the actual solids must be replaced by an equivalent

system of forces acting at the node points.

AxisymmetricHexahedron (brick)Tetrahedron

QuadrilateralTriangleSpring

Beam (line)


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2.9.5 Accuracy of Solution

In general, the accuracy of the solution will be greater if the number of elements is large.

However, computer time (and cost) also increases with number of elements chosen so it is

generally wise only to use a dense concentration of elements in the critical areas of the

solid which are likely to be of particular interest.

2.9.6 Necessity of Computer Application

In any case, finite element method is a computer-oriented method that must be

implemented with appropriate digital computer programs.

Although the analysis of each individual element in the finite element method is

straightforward, the analysis of a large number of elements becomes extremely tedious.

For this reason’ finite element solutions to problems are carried out on computers and

there are many shareware and commercial software packages available for the purpose.

To some extent this has led to the current situation where many engineers are put off by

the apparent complexity of the subject and they leave it to the experts who tend to attach a

certain mystique to the subject through the use of computer jargon!

2.9.7 Continuity of Displacements

To improve the analytical model it is necessary to provide continuity of displacements,

and in some cases of displacement derivatives along finite element sides. The techniques

for deriving local stiffness matrices will later be illustrated for beam elements.

2.9.8 Plotting of Results

It is almost essential that plotting of results and input data should be provided in finite

element programs because these provide an instant visual check and errors can readily be

detected which would otherwise become very difficult.

2.10 Applicability to Different Fields

Although the original applications were in the area of solid mechanics, its usage has

spread to many other fields having similar mathematical bases. With such a powerful

technique for automatically solving skeletal structures, it is not surprising that engineers


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attempted to extend the method to problems of structural continua and later to general

field problems. At first attempts were made to derive local stiffness matrices for sheet

material in terms of forces and displacements at joints and to ‘connect’ sheets together as

though they were discrete members. Such attempts were not successful, because in an

actual structure displacements are continuous, whereas in the analytical model the sheets,

or finite elements, could have different values of displacements at points along common

edges. Application of finite element has been extended to soil and rock mechanics,

dynamics, earthquake analysis, torsion, head conduction, seepage, consolidation,

thermoelasticity, hydroelasticity, viscoelasticity, fluid mechanics, hydraulics, flow of

compressible fluids, electrical engineering, and many other fields.

2.11 General Steps in Finite Element Method

Since the assumption of displacement functions is the technique most commonly used,

the following steps suffice to describe this approach:

1. Divide the continuum into a finite number of subregions (or elements) of simple

geometry (triangles, rectangles, and so on).

2. Select key points on the elements to serve as nodes, where conditions of equilibrium

and compatibility are to be enforced.

3. Assume displacement functions within each element so that the displacements at each

generic point are dependent upon nodal values.

4. Satisfy strain-displacement and stress-strain relationships within a typical element.

5. Determine stiffness and equivalent nodal loads for a typical element using work or

energy principles.

6. Develop equilibrium equations for the nodes of the discretized continuum in terms of

the element contributions.

7. Solve these equilibrium equations for the nodal displacements.


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8. Calculate stresses at selected points within the elements.

9. Determine support reactions at restrained node if desired.

2.12 Matrix Operations

The following matrix operations/properties have been used in the solution of beam on

elastic foundation by finite element method:

2.12.1 Transpose of Matrix

If the rows and columns of a matrix are interchanged, a new matrix called the transposed

matrix is obtained. For example, if A is a (32) matrix given by

then its transpose is the (23) matrix

2.12.2 Matrix MultiplicationThe definition of matrix multiplication is such that two matrices A and B can only be

multiplied together to form their product AB when the number of columns of A is equal to

the number of rows of B. Such matrices are called conformable matrices. Suppose A is a

matrix of order (m p) with elements aik and B is a matrix of order ( p n) with elements

bik . Then their product AB is a matrix C of order (m n) with elements cik defined by

For example, if A and B are (3 2) and (2 2) matrices, respectively, given by

and

3231

2221

1211

aa

aa

aa

A

322212

312111

aaa

aaa AT

p

s

sk isik bac

1

3231

2221

1211

aa

aa

aa

A

2221

1211

bb

bb B


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then the product C = AB is a (3 2) matrix defined as

2.12.3 Symmetric Matrices

A symmetric matrix A is square and has elements aik such that aik = aki. In other words,

the elements above and below the leading diagonal are mirror images of each other. For

example, the matrix

is symmetric.

2.12.4 Band Matrix

A matrix is called a band matrix if it has all its nonzero entries on the main diagonal and

on sloping lines parallel to it (separated by sloping lines of zeros or not). Such matrices

are obtained by carefully selecting the node points. Matrix A shown below is an example

of a symmetric band matrix with bandwidth = 3.

Because matrix A is symmetric, its half bandwidth can be taken which is 2. Hence matrix

A can be represented as follows to save computer memory.

2232123221321131

2222122221221121

2212121221121111

babababa

babababa

babababa

C

2400000

4520000

0219000

0094600

0006320

0000273

0000035

A

evu s z

vd t r y

ut cq x

sr qbw

z y xwa

A


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Special methods for solution of linear equations defined by banded matrices have been

developed. The Cholesky Method is a popular method of solving with banded matrices.

2.12.5 Identity Matrix

A square matrix is known as the identity matrix if all the entries on its main diagonal are

unity and the remaining entries are zero. Identity matrix is denoted by I .

2.12.6 Inverse Matrix

If AB = I then B is known as inverse of A and denoted by A-1

.

Several methods are available for matrix inversion such as the determinant method,

forward elimination and backward substitution method, and Gauss Jordan elimination

method, etc. BEAMDEZ uses the Gauss Jordan elimination method, which works quite

satisfactorily, for the solution of beam on elastic foundation.

2.13 Advantages of Finite Element Method

Like all numerical approximations, the finite element method is based on the concept of

discretization. Nevertheless, as either a variational or a residual approach, the technique

recognizes the multidimensional continuity of the body. Not only does the idealization

portray the body as continuous but it also requires no separate interpolation process to

extend the approximate solution to every point within the continuum. Despite the fact

02

4521

94

63

27

35

A

1000

0100

0010

0001

I


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that the solution is obtained as a finite number of discrete node points, the formation of

field variable models inherently provide a solution at all other locations in the body.

2.13.1 Better Approximations

In the finite element method, a modified structural system consisting of discrete (finite)

elements is substituted for the actual continuum and thus the approximation is of a

physical nature. There need be no approximation in the mathematical analysis of this

substitute system. By contrast, in the finite difference method the exact equations of the

actual physical system are solved by approximate mathematical procedures.

2.13.2 Trial Solutions

In contrast to other variational and residual approaches, the finite element method does

not require trial solutions which must all apply to the entire multidimensional continuum.

The use of separate sub regions, or finite elements, for the separate trial solutions thus

permits a greater flexibility in considering continua of complex shapes.

2.13.3 Boundary Conditions

Some of the most important advantages of the finite element method derive from the

techniques of introducing boundary conditions. This is another area in which the method

differs from other variational or residual approaches. Rather than requiring every trial

solution to satisfy the boundary conditions, one prescribes the conditions after obtaining

the algebraic equations for the assemblage. Since the boundary conditions do not enter

into the equations for the individual finite elements, one can use the same field variable

models for both internal and boundary elements. Moreover, the field models need not be

changed when the boundary conditions change.

The introduction of boundary conditions into the assembled equations is a relatively easy

process. It is simplified in that only the geometric boundary conditions need be

simplified in a variational approach because the natural conditions are implicitly satisfied.

No special technique or artificial devices are necessary, such as the non-centered

difference equations for fictitious external points often employed in the finite difference

method.


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2.13.4 Material Properties

The finite element method not only accommodates complex geometry and boundary

conditions, but it also has proven successful in representing various types of complicated

material properties that are difficult to incorporate into other numerical methods. For

example, formulations in solid mechanics have been devised for anisotropic, nonlinear,

hysteretic, time-dependent, or temperature-dependent behavior.

2.13.5 Non Homogenous Continua

One of the most difficult problems encountered in applying numerical procedures of

engineering analysis is the representation of non-homogenous continua. Nevertheless, the

finite element method readily accounts for non-homogeneity by the simple tactic of

assigning different properties to different elements if a refined representation of the

variation of material characteristics is desired, it is even possible to vary the properties

within an element according to a preselected polynomial pattern. For instance, it is

possible to accommodate continuous or discontinuous variations of the constitutive

parameters or of the thickness of a two-dimensional body.

2.13.6 Systematic Generality

The systematic generality of the finite element procedure makes it a powerful and

versatile tool for a wide range of problems. As a result, flexible general-purpose

computer programs can be constructed. Early examples of these programs are the several

structural analysis packages, which include a variety of element configurations and which

can be applied to several categories of structural problems. Among these packages are

STARDYNE, ASKA, STRUDL, SAP, NASTRAN, ETABS, and SAFE. Another

indicator of the generality of the method is that programs developed for one field of

engineering have been applied successfully to problems in a different field with little or

no modification.

Finally, an engineer may develop a concept of the finite element method at different

levels. It is possible to interpret the method in physical terms. On the other hand, the

method may be explained entirely in mathematical terms. The physical or intuitive nature


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of the procedure is particularly useful to the engineering student and practicing engineer.

Nevertheless, it is significant that the method has mathematical functions.

2.14 Limitations of Finite Element MethodOne limitation of the finite element method is that a few complex phenomena are not

accommodated adequately by the method at its current state of development. Some

examples of such phenomena from the realm of solid mechanics are cracking and fracture

behavior, contact problems, and bond failures of composite materials, and non-linear

material behavior with work softening. Another example is transient, unconfined seepage

problems. The numerical solution of propagation or transient problems is not satisfactory

in all respects. Many of these phenomena are presently under research, and refinements

of the method to accommodate these problems better can be expected.

2.14.1 Validity of Results

The finite element method has reached a high level of development as a solution

technique: however, the method yields realistic results only if the coefficients or material

parameters which describe the basic phenomena are available. Material non-linearity in

solid mechanics is a notable example of a field in which our understanding of the material

behavior has lagged behind the development of analytical tool. In order to exploit fully

the power of the finite element method, significant effort must be directed toward the

development of suitable constitutive laws and the evaluation of realistic coefficients in

material parameters.

2.14.2 Computer Requirements

Even the most efficient finite element computer codes require a relatively large amount of

computer memory and computations. Hence, use of the method had been limited to those

who had access to relatively large, high-speed computers. Advancements in computer

technology and easy availability of desktop computers with large memories have

alleviated this restriction to some degree. However, very complex problems, especially

continuous analysis of dynamic systems (like weather systems, virtual wind tunnels, etc)

still require nothing less than supercomputers.


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2.14.3 Discretizing the Continuum and Data Input

The most tedious aspects of the use of the finite element method are the basic processes of

subdividing the continuum and of generating error-free input data for the computer.

Although these processes may be automated to a degree, they have not been totally

accomplished by computer because some engineering judgment must be employed in the

discretization. Errors in the input data may go undetected and the erroneous results

obtained therefrom may appear acceptable.

2.14.4 Interpretation of Results

Finally, as for any approximate numerical method, the results of a finite element analysis

must be interpreted with care. We must be aware of the assumptions employed in the

formulation, the possibility of numerical difficulties, and the limitations in the material

characterizations used. A large volume of solution information is generated by a finite

element routine, but this data is worthwhile only when its generation and interpretation

are tempered by proper engineering judgment.


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C H A P T E R T H R E E

COMBINED FOOTINGS

3.1 Combined Footing

A spread footing supporting a line of two or more columns is called a combined footing .

A combined footing may have either rectangular or trapezoidal shape or be a series of

pads connected by narrow rigid beams called a strap footing.

3.2 Purpose of Combined Footings

The necessity of combined footings arises mostly in the following cases:

1. when the column is so close to an adjacent property that it is impossible to center an

isolated footing under the column,

2. when columns are spaced so closely that isolated footings are impracticable or

uneconomical,

3. when the bearing capacity of soil is low, it is advisable to combine the footings of two

or more columns instead of enlarging the isolated footings,

4. and when differential settlements are to be reduced.

3.2.1 Columns Near Property Line

When a column is located near a property line, part of the single footing might extend into

the neighboring property. Also it may not be possible to place columns at the center of a

spread footing if they are near mechanical equipment locations, or irregularly spaced. To

avoid this situation, the column may be placed on side or edge of the footing, causing

eccentric loading. This may not be possible under certain conditions, and sometimes it is

not an economical solution. A better design can be achieved by combining the footing

with the nearest internal column footing, forming a combined footing. The center of


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gravity of the combined footing coincides with the resultant of the loads on the two

columns.

3.2.2 Closely Spaced Columns

If the columns are spaced so closely that their footings overlap then a combined footing

becomes necessary. Another case is when the footings do not overlap but are so closely

spaced that providing a combined footing may prove to be more economical than isolated

footings. The shape of the combined footing may be rectangular or trapezoidal.

3.2.3 Poor Soil

Another case where combined footings become necessary is when the soil is poor and the

footing of one column overlaps the adjacent footing. If there is a row of columns and the

bearing capacity of the subsoil is low so that large bearing areas become necessary,

individual footings are replaced by continuous strip footings that support more than two

columns and usually all columns in a row. Sometimes such strips are arranged in both

directions, in which case a grid foundation is obtained. Such a foundation can be made to

develop a much larger bearing area much more economically than can be done by single

footings because the individual strips represent continuous beams whose moments are

much smaller than the cantilever moments in large single footings that project far out

from the column in all four directions. Bridge piers are also founded on very rigid

combined rectangular footings.

For still lower bearing capacities, the strips are made to merge, resulting in a mat

foundation. That is, the foundation consists of a solid reinforced concrete slab under the

entire building. In structural action, such a mat is very similar to a flat slab or a flat plate,

upside down, i.e. loaded upward by the bearing pressure and downward by the

concentrated column reactions. The mat foundation evidently develops the maximum

available bearing area under the building. If the soil’s capacity is so low that even this

large bearing area is insufficient, some form of deep foundation, such as piles or caissons,

must be used.


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3.2.4 Differential Settlement

When a column load is transmitted to the soil by the footing, the soil becomes

compressed. The amount of settlement depends on many factors, such as the type of soil,

the load intensity, the depth below ground level, and the type of footing. If different

footings of same structure have different settlements, new stresses develop in the

structure. Excessive differential settlement may lead to the damage (cracking and tilting)

of nonstructural members in the buildings, even failure of the affected parts.

Besides developing large bearing areas, another advantage of strip, grid, and mat

foundations is that their continuity and rigidity help in reducing differential settlements of

individual columns relative to each other, which may otherwise be caused by local

variations in the quality of subsoil, or other causes. For this purpose, continuous spread

foundations are frequently used in situations where the superstructure or the type of

occupancy provides unusual sensitivity to differential settlement.

If the bearing soil capacity is too much different under different footings, for example if

the footings of a building are partly on soil and partly on rock, a differential settlement

will occur. In such cases it is advisable to provide a joint between the two parts to

separate them, allowing for independent settlement.

3.3 Shape of Combined Footings

Combined footings are made rectangular, trapezoidal, or T shaped, the details of the

shape being arranged to produce coincidence of centroid of area and resultant of loads.

Figure-3.1 Rectangular Combined Footing


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3.3.1 Rectangular Combined Footing

A combined footing is usually made rectangular, if the rectangle can extend beyond each

column face to a distance required to make the center of the gravity of the column loads

coincide with the centroid of the footing area. Rectangular combined footings can be

divided into two categories: those that support only two columns (figure-3.1) and those

that support more than two (generally large number of) columns (figure-3.2).

In buildings where the allowable soil pressure is large enough for single footings to be

adequate for most columns, two-column footings are seen to become necessary in two

situations: (1) if columns are so close to the property line that single-column footings

cannot be made without projecting beyond that line, and (2) if some adjacent columns are

so close to each other that their footings would merge. Combined footing for rows ofcolumns are provided where soil is poor.

Figure-3.2 Combined Footing for row of columns.

If the footing is to support an exterior column at the property line where the projection of

the footing is limited, a rectangular shape can be used provided the interior column caries

greater load or a row of columns is supported by the continuous strip footing which is

rectangular. When the loading of the column is equal, the rectangular shape is used with

advantage.

3.3.2 Trapezoidal Footing

When the load of the external column near the property line is greater than the load of the

interior column, a trapezoidal footing (figure-3.3) is necessary to keep the centroid of the

footing in line with the resultant of the two column loads. In most other cases, a

rectangular footing may be advantageous.


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Figure-3.3 Trapezoidal and Strap Footings

3.3.3 Strap Footing

Another expedient that is used if a single footing cannot be centered under an exterior

column is to place the exterior column footing eccentrically and to connect it with the

nearest interior column footing by a beam or strap. This strap, being counterweighted by

the interior column load, resists the tilting tendency of the eccentric exterior footings and

equalizes the pressure under it. Such foundations are known as strap, cantilever, or

connected footings (figure-3.3).

Figure-3.4 Continuous Strip, Grid, and Mat Foundations


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3.3.4 Strip, Grid, And Mat Foundation

In the case of heavily loaded columns, particularly if they are to be supported on

relatively weak or uneven soils, continuous footings are resorted to. They may consist of

a continuous strip footing supporting all columns in a given row, or of two sets of such

strip footings intersecting at right angles so that they form one continuous grid foundation

(figure-3.4). For even larger loads or weaker soils the strips are made to merge, resulting

in a mat foundation (figure-3.4).

Figure-3.5 Resultant R of the applied loads should act at thecentroid of footing for uniform soil pressure.

3.4 Design of Rectangular Combined Footings

Generally, combined footings are constructed of reinforced concrete. The fundamental

assumption for the design of a rectangular combined footing is that the footing is rigid

and rests on a homogeneous soil, so as to give rise to a linear stress distribution on the

bottom of the footing. If the resultant of the soil pressure coincides with the resultant of

the loads (and center of gravity of the footing), the soil pressure is assumed to be

uniformly distributed, which greatly simplifies the calculations. This assumption is

approximately true if the soil is homogeneous and the footing is rigid. The ACI Code,

section 15.40, does not provide a detailed approach for the design of combined footings.

The design, in general, is based on an empirical approach.


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3.4.1 Footing Dimensions

Therefore, the dimensions of a combined footing are so proportioned that the center of

gravity of the area of the footing in contact with the soil lies on the line of action of the

resultant of the loads applied to the footing (figure-3.5). By doing so, the distribution of

soil pressure is assumed to be fairly uniform. If the resultant of the applied loads does not

coincide with the centroid of the bearing area, a bending moment develops. In this case,

the pressure on one side of the footing will be greater than the pressure on the other side,

causing higher settlement on one side and a possible rotation of the footing. Moreover,

the dimensions of the footing are chosen such that the allowable pressure is not exceeded.

3.4.2 Rounding of Dimensions

The conventional design method requires computing shears and moments at sufficient

locations that shear and moment diagrams can be drawn for critical values. It is also

standard practice to round the computed length and width to the nearest larger multiple of

75-mm or 3 inch. If this is done prior to computing shear and moment diagrams there

will be a closure error which depends on the amount the length is changed; thus, it is

recommended that footing dimensions be rounded as the final design step. Also, this

rounding may cause a small variation in the uniform pressure under the footing, but it can

be tolerated.

3.4.3 Shear and Moment Computations

The column loads are actually distributed over the column width but may be considered

to be concentrated point loads. This greatly simplifies the shear and moment

computations, and the values at the critical locations are the same by either method. Then

the resulting shear force and bending moment diagrams can be plotted. The maximum

bending moment should be adopted as the design value for the reinforced concrete

footing, which should also be checked for maximum shear and bond etc.

3.4.4 Depth of Footing

The depth based on the more critical of two-way action or wide-beam shear is computed.

Critical sections for two-way action and wide-beam are the same as for spread footings,

i.e., at d/2 and d, respectively, from the column face. It is common practice not to use


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shear reinforcement both for economy and to increase the rigidity. The labor costs to

bend and place the shear reinforcement are likely to far exceed the small savings in

concrete that would result from its use.

3.4.5 Reinforcement Design

With the depth selected, the flexural steel can be designed using the critical moments

from the moment diagram. Alternatively, the depth and the loading can be used in a

finite-element analysis to obtain modified moments for the flexural steel. These beam-

type members usually have both positive and negative moments, resulting in reinforcing

steel in both the top and bottom of the footing. The minimum percentage of steel should

be taken as 200/ f y since the footing is designed as a “beam” or flexural member.

3.4.5.1 Main Reinforcement

The main reinforcement in a combined footing is placed in the longitudinal direction.

The computation of the main steel is done on the assumption that the footing acts as one

way slab.

3.4.5.2 Transverse Reinforcement

If we compute the short, or transverse, direction bending moments as for a rectangular

spread footing, they will be in substantial error. This is because the soil pressure is larger

near the columns, from their stiffening effect on the footing, and lesser in the zone

between columns. That zone closest to, and approximately centered on, the column is

most effective and should be analyzed somewhat similar to the ACI Code requirement for

rectangular footings. The Code does not directly specify this effective column zone

width. Bowles suggests that for the effective zone for the transverse direction it should be

assumed that the column load is spread over a width under the column equal to the

column width plus 0.75d on each side (figure-3.6), whenever that is available. In other

words, the column load acts on a beam under the column within the footing which has a

maximum width of (w + 1.5d) and a length equal to the short side of the footing smaller

width, down to (w + 0.75d), may be used. Transverse reinforcement is provided at the

bottom of the footing in these zones (shown in figure-3.6).


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Figure-3.6 Zones for transverse reinforcement.

3.4.5.3 Shear Reinforcement

It is common practice not to use shear reinforcement both for economy and to increase the

rigidity (depth of concrete required is larger if we do not provide shear reinforcement and

due to this increased depth of concrete, footings becomes more rigid). The labor costs to

bend and place the shear reinforcement are likely to far exceed the small savings in

concrete that would result from its use.

3.5 Design Steps

The procedure of the design of a combined footing may be summarized as follows:

(a) Ascertain the loads on both columns and their distance apart. Convert loads to

ultimate and find ultimate soil pressure qult when allowable soil pressure qa is given.

(b) Locate load resultant from center of any column and then find footing dimensions L

and B so that the resultant acts at centroid of the area in order to have uniform soil

pressure that does not exceed the safe bearing capacity of soil.

(c) Calculate the maximum bending moment anywhere in the length of the footing.

(d) Select depth based on analysis for both wide beam and diagonal tension. First, the

depth is obtained by using allowable value of shear stress for wide beam, then the

value of d will be checked on diagonal (also called punching) tension shear.


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(e) Design main reinforcement steel between columns by using ultimate strength design

method.

(f) Calculate the transverse steel required under each column.

(g) Check dowel requirements of each column to footing. If dowels are not required then

provide at least four dowels of 0.005 Ag.

(h) Design steel for cantilever portion of footing.

3.6 Demerits of Rigid Design

In actual practice, it is very difficult to make a rigid footing as the depth would have to be

great; nevertheless, the assumption of a rigid member has been successfully used for

many foundation members. Bowles suggests that the success has probably resulted from

a combination of soil creep, concrete stress transfer, and overdesign. However, the design

is not economical because it requires higher amounts of steel than necessary.

3.6.1 Assumptions of Rigid Design

In general, ordinary combined footings are somewhat less rigid but their design by

conventional method is based on the assumption that they are absolutely rigid (no bendingoccurs) so that the soil pressure under the footing can be assumed as uniformly

distributed.

If columns are spaced at moderate distances and if the strip, grid, or mat foundation is

very rigid, the settlements in all portions of the foundation will be substantially the same.

But in reality, the foundation is relatively flexible and if the column spacing is large,

settlements will no longer be uniform or linear. For one thing, the more heavily loaded

columns will cause larger settlements, and thereby larger subgrade reactions, than the

lighter ones. Also, since the continuous strip or slab midway between columns will

deflect upward relative to the nearby columns, this means that the soil settlement, and

thereby the subgrade reaction, will be smaller midway between columns than directly at

the columns. This is shown schematically in figure-1.2. In this case, the subgrade


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reaction can no longer be assumed as uniform. A reasonably accurate but fairly complex

analysis can then be made using the theory of beams on elastic foundations.

3.7 Beam on Elastic FoundationBecause of the overdesign involved in the conventional (or “rigid”) method, current

practice tends to modify the design by a beam-on-elastic-foundation analysis. The latter

analysis produces smaller design moments than those obtained by the rigid method;

therefore, the design becomes economical. This method of analysis and its finite element

modeling is discussed in the next chapter.


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C H A P T E R F O U R

BEAM ON ELASTIC FOUNDATION

4.1 Elastic Foundation

A special kind of structural member is one which rests on a spongy material, termed as an

elastic foundation, which offers a resistance proportional to the transverse deflection; a

common example of this is a foundation beam resting on an elastic soil. Thus, there are

unknown transverse forces, equal to the product of the “stiffness modulus” of the

supporting material and the yet unknown transverse deflection, acting on structural

members on elastic foundation.

Usually, the treatment of structural members is confined to those under the action of

known transverse forces. In the displacement method of analysis, these transverse forces

are transmitted to both ends of each member, in the fixed condition. In case that some

member in a continuous beam or rigid frame is subjected to resistance offered by an

elastic foundation, the displacement method of analysis can still be used, provided that

expressions for the member stiffness matrix and for the fixed-end reactions and moments

due to common types of transverse loads can be found as functions of the stiffness

modulus. These required expressions will be derived in this chapter.

4.2 Winkler Model

Some soil/structure interaction problems may be simplified as a beam (structure) on

springs (soils), where the behavior of the soil is simplified by means of fictitious springs

placed continuously underneath the structure. In 1867, Winkler first studied the beam on

elastic springs. The model he developed is called the Winkler foundation model, which is

a one-dimensional problem. For analysis of beams and slabs resting on a soil medium,

engineers have been using this classical mathematical model. The springs representing


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subgrade soils may obey Hooke’s Law or be non-linear or elastic-plastic [Selvadurai,

1979].

The Winkler model has its advantages for obtaining fast solutions, sometimes analytical,

to more complicated soil/structure interaction problems.

4.3 Limitations of Winkler Model

The Winkler foundation model has two major problems: (1) no interaction between the

springs is considered; and (2) the spring constant may depend on a number of parameters,

such as stiffness and geometry of the beam, soil profile, and behavior. Terzaghi

[Terzaghi, 1955], Vesic [Vesic, 1961], and Selvadurai [Selvadurai, 1979], among others,

discussed the validation of the Winkler foundation model and suggested methods for the

estimation of spring constants. Despite the two major limitations, the Winkler foundation

model has found applications in the analysis of soil/structure interaction problems; e.g.,

footings on soils and laterally loaded piles in soils.

It is noted that the beam in the Winkler foundation model is based on the pure bending

beam theory commonly used in structural analysis.

4.4 Other Models

As a different approach, Pasternak [Pasternak, 1954] suggested a pure shearing model for

soil/structure interaction analyses. In the Pasternak model, no bending is considered, and

the settlement is totally controlled by shear deformation of the beam. The Pasternak

model was combined with Winkler model as an attempt to consider the interaction of the

springs. Geosynthetics have recently received wide application for reinforcement of

weak foundation soils or pavement [Koerner, 1990]. The Pasternak model was recently

developed to analyze the geosynthetic-reinforced granular base over soils [Madhav 1988;

Bordeau, 1989; Ghosh, 1994; Shukla, 1994; Yin, 1997 a, b).

Both the Winkler model and the Pasternak model take an extreme point of view (or

approximation) on the deformation behavior of beam. Another beam model that


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considers both bending and shearing was proposed by Timoshenko [Timoshenko, 1921]

for modeling vibration of beams. The Timoshenko beam still attracts people’s attention

in studying the static and dynamic response of beams [Wang et al., 1997]. However, less

attention is paid to modeling the Timoshenko beam on an elastic foundation.

Figure-4.1 Modulus of subgrade reaction.

4.5 Modulus of Subgrade Reaction

In analysis of foundation members, the resistance offered by the supporting material is

taken as unknown transverse forces, equal to the product of its “stiffness modulus” (called

modulus of subgrade reaction for soils) and the transverse deflection of the structural

member. It is used for continuous footings, mats, and various types of pilings. This ratio

is defined on figure-4.1 and the basic equation when using plate-load test data is

k s=q/ (4.1)

with terms identified on both figure-4.1 and figure-4.2. Plots of q versus from load tests

give curves of the type qualitatively shown in figure-4.2. If this type of curve is used to

obtain k s in the above equation, it is evident that the value depends on whether it is a

tangent or secant modulus and the location of the coordinates of q and .

P r e s s u r e ,

Deformation,

k s =

Use initial tangentor

initial secant line


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Figure-4.2 Determination of modulus of subgrade reaction fromplate load test data.

It is difficult to make a plate-load test, except for very small plates because of the reactionload required. Even with small plates of say 450, 600, and 750 mm diameter it is difficult

to obtain since the plate tends to be less than rigid so that a constant deflection across

the plate (and definition of k s) is difficult to obtain. Stacking the smaller plates concentric

with the larger ones tends to increase the rigidity, but in any case, the plot is of load

divided by plate contact area (nominal P/A) and the average measured deflection.

Figure-4.2 is a representation of k s taken as a constant up to a deflection Xmax. Beyond

Xmax the soil pressure is a constant value defined by

Qcon = k s(Xmax) (4.2)

Obviously one could divide the q- curve into several regions so that k s takes on values of

the slope in several regions however, this tends to incorporate too much refinement into

the problem since most analyses proceed on the basis of estimated values or at best an

approximate load test.

Some engineers do not like to use the concept of a modulus of subgrade reaction; rather

the use of Es (and ) in finite-element analyses is preferred. Bowles suggests that, until

the state of the art improves so that accurate values of Es can be obtained, the modulus of

subgrade reaction method is preferred for its greater ease of use and from the substantial

savings in computer computation time.

k =q

q

k depends on curve coordinates

used and is generally nonlinear q

Nonlinear Linear

X max


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There is a direct relationship between k s and E s but since one does not often have values

of E s, other approximations are useful and often quite satisfactory if the computed

deflection (directly dependent on k s) can be tolerated for any reasonable value. It has

been found that bending moments and the computed soil pressure are not very sensitive to

what is used for k s. This is because the structural member stiffness is usually 10 or more

times as great as the soil stiffness as measured by k s. Recognizing this, Bowles suggests

that the following formulae for approximating k s from the allowable bearing capacity

furnished by the geotechnical consultant should be used

k s = 12 (SF) qa (kip/ft3) (in fps units) (4.3)

k s = 40 (SF) qa (kN/m3) (in SI units) (4.4)

wher

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