higher order finite element analysis of shear a

HIGHER ORDER FINITE ELEMENT ANALYSIS OF SHEAR

WALLS FOR STATIC AND DYNAMIC LOADS

by

RAMAKRISHNA NARAYANASWAMI, B.Sc. In C.E., M. Tech. In C.E,

A DISSERTATION

IN

CIVIL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for

the Degree of

DOCTOR OF PHILOSOPHY

Approved

August, 1971

H C ^ -^ \ 'T AC

T3 197/ f\}o,7f

ACKNOWLEDGMENTS

I am deeply indebted to Dr. C. V. Girija Vallabhan for his

guidance and counseling during this investigation. I also wish to

express my deep appreciation to Dr. Ernst W. Kiesling for serving as

Chairman of the Advisory Committee and for his guidance and encour

agement throughout my graduate studies at Texas Tech. I am also

grateful to Dr. Kishor C. Mehta, Dr. Wayne T. Ford and Dr. Donald H.

Helmers for their helpful criticisms and valuable suggestions.

ii

TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS ii

LIST OF TABLES vii

LIST OF FIGURES viii

LIST OF SYMBOLS xi

I. INTRODUCTION 1

Definition of the Shear Wall Problem 1

Review of Previous Research 2

Conventional Methods of Analysis 2

Finite Element Method 5

Scope of Present Investigation 7

II. THE FINITE ELEMENT METHOD 9

The 16-Degree Freedom Rectangular Element 13

Details of the Element 13

Strains 15

Stresses 16

Equivalent Nodal Forces 17

Basis for the Construction of Element Stiffness Matrix 17

Calculation of the Element Stiffness Matrix 20

Distributed Loads 22

Concentrated Normal Load 22

Uniformly Distributed Normal Surface Force . . . . 24

Linearly Varying Normal Surface Force 28

iii

iv

Symmetric Parabolic Normal Surface Force 29

Shear Surface Force 30

Other Loading Conditions 32

The Assemblage Stiffness Matrix of the Continuum. . . 33

Properties of the Stiffness Matrix 33

Computer Program and Solution of Matrix Equation. . . 34

III. ANALYSIS OF SHEAR WALLS FOR STATIC LOADING 35

Assessment of Accuracy 35

Necessity of Error Analysis 36

A Beam Problem 37

8-Degree Freedom Rectangular Element 37

16-Degree Freedom Rectangular Element 40

Comparison of 8-Degree Freedom and 16-Degree

Freedom Element 40

The Shear Wall Problem 46

Equal Walls 46

Comparison of Results 51

Unequal Walls 56

Discretization Errors in Shear Wall Problems 57

Thermal Stresses 66

Beam Problem 67

Shear Wall Problem 68

IV. PARAMETRIC STUDY OF SHEAR WALLS 75

Non-Dimensional Parameters 77

Non-Dimensional Curves 78

Curves for Displacement 78

v

Curves for Stresses 81

Curves for Moment in Lintel Beams 88

Axial Force in Lintel Beams 88

V. ANALYSIS OF SHEAR WALLS FOR DYNAMIC LOADS 91

Equations of Motion 92

Stiffness Matrix 94

Mass Matrices 94

Damping Matrix 95

Response of Structures to Earthquake 96

Step-by-Step Formulation of Equations and Solution. . 98

Vibration Problems 98

Earthquake Response 101

Example Problems 101

Beam Problem 101

Example 1 101

Example 2 102

Shear Wall Problem 108

Effect of Damping 109

Effect of Thickness of Lintel Beam on

Fundamental Period of Shear Wall 109

Effect of Floor Loads in the Fundamental Period 120

Earthquake Response 123

General Observations 123

VI. CONCLUSIONS AND RECOMMENDATIONS 129

Conclusions 129

Recommendations for Further Study 130

vi

LIST OF REFERENCES 132

APPENDICES , 136

A. 6-d.o.f., 8 d.o.f. and 12 d.o.f. Finite Elements . . 137

B. Results of Shear Wall Analysis 145

vii

LIST OF TABLES

Table Page

Q DQd(vol] for 16 d.o.f. Rectangular Element. . . . 21

3.1 Beam Analysis-8 d.o.f. Rectangular Element 39

3.2 Beam Analysis-16 d.o.f. Rectangular Element 43

3.3 Forces and Displacements in Lintel Beams 52

3.4 Comparison of Results of Shear Wall Analysis (Equal Walls) 53

3.5 Comparison of Results of Shear Wall Analysis (Unequal Walls) 59

3.6 Discretization Errors in Shear Wall Problems 64

3.7 Thermal Stresses in Beam 69

3.8 Uniform Temperature in Shear Wall 71

3.9 Differential Heating of Walls 73

5.1 Vibration Analysis of Beam 103

5.2 Earthquake Data-Olympia, April 13, 1949

(30.2727 Seconds Duration) 105

5.3 Simply Supported Beam-Earthquake Analysis 106

5.4 Vibration Analysis of Shear Wall-8 d.o.f.

Rectangular Element 110 5.5 Vibration Analysis of Shear Wall-16 d.o.f.

Rectangular Element Ill

5.6 Vibration Analysis of Shear Wall With Damping . . . . 116

5.7 Variation of Fundamental Period with Thickness of

Lintel Beam 121 5.8 Variation of Fundamental Period with Floor Mass

Factor, y 121

5.9 Earthquake Analysis of Shear Wall 125

viii

LIST OF FIGURES

Figure Page

2.1 Typical 16 d.o.f. Rectangular Element 12

2.2 Normal Load Distribution on the Element 23

2.3 Shear Load Distribution on the Element 31

2.4 Other Loading Conditions 32

3.1 Simply Supported Beam 38

3.2 Interaction Curves for Errors-8 d.o.f. Element-

7 Digit Floating Point Arithmetic 41

3.3 Interaction Curves for Errors-8 d.o.f. Element-15 Digit Floating Point Arithmetic 42

3.4 Interaction Curves for Errors-16 d.o.f. Element-7 Digit Floating Point Arithmetic 44

3.5 Interaction Curves for Errors-16 d.o.f. Element-

15 Digit Floating Point Arithmetic 45

10-Story Shear Wall with Equal Walls 47

Finite Element Idealization pf 10-Story Shear Wall. . 47

Displacement Characteristics of Shear Wall 48

Lateral Deflection of Left Side of Wall 49

Beam Forces and Deformations 50

10-Story Shear Wall with Unequal Walls 58

Finite Element Idealization of 10-Story Shear Wall. . 58

Different Finite Element Idealizations of 10-Story Shear Wall 61

Fixed Beam: Temperature Stresses 67

Shear Wall: Uniform Temperature 70

Plan of 6-story Structure 72

3.

3 .

3 .

3.

3.

3.

3.

6

7

8

9

10

,11

,12

3.13 to

3.18

3.

3.

3.

,19

.20

.21

4.

4.

4.

4.

4.

4.

4.

4.

4,

4.

4.

4.

5.

5.

5.

5.

5.

5.

5.

,1

.2

.3

.4

.5

.6

.7

.8

.9

.10

.11

.12

.1

,2

,3

,4

5

6

7

ix

Shear Wall 76

Finite Element Idealization 76

10-story Shear Wall - Log TT VS Log TT 79

Shear Wall Parametric Curves - Log TT VS Log TT . . 80

10-story Shear Wall - TT. VS Log TT, 82 2 ° 4

Shear Wall Parametric Curves - TT_ vs Log TT, . . . . 83 2 4

10-Story Shear Wall - Log TT vs TT 84

Shear Wall Parametric Curves - Log TT vs TT . . . . 85

10-story Shear Wall - TT VS TT 86

Shear Wall Parametric Curves - TT vs TT 87

10-story Shear Wall - Log TT vs Log TT 89 J 6

Shear Wall Parametric Curves - Log TT vs Log TT . . 90 3 6

Beam Subjected to Static and Dynamic Loadings . . . . 91

Earthquake Excitation of Building Frame 96

Simple Beam for Vibration Analysis 101

Forced Vibration-Beam Problem 104

Earthquake Analysis-Beam Problem-Olympia Earthquake-April, 1949 107

Shear Wall for Dynamic Analysis 108

Forced Vibration of Shear Wall-8 d.o.f. Finite Element Analysis-Case (i) 112

5.8 Forced Vibration of Shear Wall-8 d.o.f. Finite Element Analysis-Case (ii) 113

5.9 Forced Vibration of Shear Wall-16 d.o.f. Finite Element Analysis-Case (i) 114

5.10 Forced Vibration of Shear Wall-16 d.o.f. Finite Element Analysis-Case (ii) 115

5.11 Vibration Analysis, with Damping, of Shear Wall a - 0.002 117

X

5.12 Vibration Analysis, with Damping, of Shear Wall

a - 0.005 118

5.13 Model Shear Wall 119

5.14 Fundamental Period vs Depth of Lintel Beam 122

5.15 Fundamental Period vs Gamma 124

5.16 Earthquake Analysis of Shear Wall-Maximum Displacement vs Time 126

5.17 Earthquake Analysis of Shear Wall-Stress at Base of Wall vs Time 127

xi

LIST OF SYMBOLS

a,b: Sides of the rectangular finite element.

B: Width of the shear wall.

[B]: Matrix of co-ordinates of the nodes of the finite element.

[C]: Damping matrix.

d: Depth of the lintel beams.

[D]: Elasticity matrix relating stress and strain.

E: Modulus of elasticity of the material of the finite element

E ,E, : Modulus of elasticity of the material of wall and lintel beam respectively.

[F]: Global force vector,

i" F : Component of force in the direction of u displacement at

node i of the element.

V

F : Component of force in the direction of v displacement at node i of the element.

{f}: Vector of nodal forces due to external load.

{f} : Vector of nodal forces due to initial strain, e o

{f} : Vector of nodal forces due to body forces. P

H: Height of shear wall.

h: Height of story.

I ,1 : Moment of inertia of wall and beam, respectively, with respect to centroidal axes of bending.

[K]: Assemblage stiffness matrix of the continuum.

[k]: Stiffness matrix of the finite element in the local coordinate system of the element.

[k]: Stiffness matrix of the element in the global coordinate system of the continuum.

M , : Moment at A of the lintel beam, AB. ab

^ a

[M]

{P}

[Q]

xii

Moment at B of the lintel beam* AB

Mass matrix.

Vector of body force components.

Matrix relating total strain, e, and constant coefficients, a, of the element.

q: Intensity of load.

{R }: Vector of amplitudes.

{R(t)}: Time dependent load vector.

r: Relative motion with respect to the ground.

r : Ground motion, g

r : Total motion.

{r}: Displacement vector of the finite element.

{r},{r}5 First and second derivative respectively, with respect to time, of displacement.

s: Span of lintel beam.

t: Thickness of the finite element.

[T]: Matrix of transformation relating the global co-ordinate system of the continuum and the local co-ordinate system of the element.

u: Function used to represent translatory motion in the x-direetion at every point within and on the boundary of the finite element.

u : Translatory motion in x-direetion at node i of the finite element.

{U}: Global displacement vector.

v: Function used to represent translatory motion in the y-direetion at every point within and on the boundary of the finite element.

V : Translatory motion in the y-direction at node i of the 1

finite element.

V: Potential energy of the applied loads.

xiii

X: Body force component in the x-direetion.

Y: Body force component in the y-direetion.

a: Coefficient of thermal expansion.

Vector of coefficients in displacement function.

Parameter in the Newmark-6 parameter method.

Relative lateral displacement between the ends A and B of beam Afi.

Time interval.

Vector of total strain.

Vector of strain caused by applied loads.

Vector of initial strain.

{o}: Vector of stress in the element.

v: Poisson's ratio of the material of the finite element.

•: Total potential energy.

6: Temperature rise.

e ,e, : Slope at the ends of the beam AB.

10 : Circular frequency of vibration of mode n.

"r

"2 =

11,: 3 s

{ a } :

B :

A:

At:

( G ) :

it }: a

's>=

T T , :

E l s w w

W' h - d

h

n^d

Maximum displacement in the coupled shear wall 4" Maximum displacement in the solid shear wall

Maximum stress in wall in the coupled shear wall ^5* Maximum stress in the solid shear wall

Maximum of the moments in the lintel beams 6* Maximum moment in the solid shear wall

CHAPTER I

INTRODUCTION

Definition of the Shear Wall Problem

The shear wall structure is now widely accepted as a rational

and economic form of multi-story construction. Recent years have

seen throughout the world a rapid increase in the number of multi

story structures, for both commercial and residential purposes. This

rapid growth has intensified the need for greater knowledge of the

structural behavior of shear walls and the complex interactions be

tween the walls, floor slabs and frames. Attempts to apply more

sophisticated and accurate method of analysis have usually been

hindered by the large amount of computation involved; however, this

obstacle is being gradually overcome by the availability and appli

cation of larger electronic computers with more relevant programs.

As buildings increase in height, it becomes more important to

ensure adequate lateral stiffness to resist loads which may arise

due to wind, seismic or perhaps even blast effects. This stiffness

may be achieved in various ways. In framed structures, it may be

obtained by bracing members, by the rigidity of the joints, or by

infilling the frame with shear resistant panels. An obvious simpli

fication of the latter is shear wall construction, in which the rel

atively high in-plane stiffness of the walls, both external and

internal, is employed to resist the lateral forces. The floor slabs,

which are also extremely stiff in their own plane, serve not only to

collect and distribute the lateral forces to the walls, but, by a

complex structural interaction with the walls, increase the lateral

stiffness of the building. Box-type structures, formed from groups

of walls surrounding lift shaft and stair wells, are also very effi

cient in providing lateral bracing.

Currently, the most common method of providing the required

rigidity uses the internal and external walls, which are necessary

in any event for obvious functional reasons. These walls normally

contain openings for doors, windows and corridors. It is also often

possible to construct two similar co-planar walls connected by beams

on each floor level. For tall buildings with large wind loads, it

is reasonable to take this connection into account. This arrange

ment constitutes, for design purposes, what is known as coupled shear

walls, or inter-connected shear walls.

The growing research effort in the analysis and design of shear

walls has resulted in a large number of research papers. Some of

these use the classical methods of indeterminate structural analysis

(References 1 to 11); others use numerical techniques such as finite

difference and finite element methods (References 12 to 17). In an

attempt to provide a summary of the current knowledge relating to

shear wall structures, a brief review of the relevant research papers

is given in the following paragraphs.

Review of Previous Research

Conventional Methods of Analysis

Several papers have been published concerning the approximate

solution of coupled shear walls. Most of these are based on the idea

of replacing a large number of discrete members by a continuous medium

for solution of structural problems. First suggested by Southwell

in connection with the analysis of stresses in aeroplane wheels, the

concept was used by Chitty (1) in the analysis of a cantilever com

posed of a number of parallel beams interconnected by cross beams.

In Chitty's analysis, the cross beams are replaced by a continuous

medium which can transmit actions similar to those transmitted by

the beams. Assuming that the medium, which would correspond to the

web of an equivalent beam, will apply a continuous load and a con

tinuous moment of varying intensity on each flange, a differential

equation for flexure is set up. Solving the equation, expressions

for shear, moment, slope, and deflection are obtained.

Chitty and Wan (2) applied the results to the analysis of building

frames subjected to wind loading. Their investigations considered

a number of cantilever frames, including those with columns of dif

ferent stiffness, with beeims of different stiffnesses, and multibay

structures.

Beck (3) presented an approximate method of analysis wherein a

continuous system replaced the discontinuous frame system. Beginning

with five floors, the accuracy of the results was found to be suf

ficient for practical application and the accuracy was found to in

crease with larger number of floors.

Rosman (4) presented a simple, approximate analysis for various

types of shear walls widely used in engineering practice. The con

tinuous system method was used and the integral shear forces in the

continuous connections of individual walls were chosen as the stat

ically redundant functions. Deformations due to bending moment, the

contribution of normal forces in the walls and shear forces in the

connecting beams were taken into account.

One limitation of Rosman's method is an assumption that points

of contraflexure occur at mid span of the connecting besims. This

assumption is admissible if the cross-sections of individual walls

are much greater than the cross-sections of the connecting beams.

Mcleod (5) has pointed out the conditions under which Rosman's method

will not give accurate results.

Coull and Choudhury (6,7) have advanced a solution, based on

Rosman's theory, suitable for design office calculations. General

graphical solutions are given for a set of two coupled walls subjected

to uniform lateral load or to a triangular variation of lateral load.

Barnard and Schwaighofer (8) have attempted to reduce the amount

of computation required by using a simple approximate form of distri

bution of the shear forces in the connecting beams, using a combina

tion of a straight line and parabola to approximate the true curve.

Gould (9) and Khan and Sbarounis (10) have investigated the

interaction of shear walls and frames. Gould suggests a method of

proportioning the external load between the shear walls and the frames;

Khan and Sbarounis use a numerical technique to supply design charts

useful for design office calculations.

Gurfinkel (11) has generalized the concept of cantilever moment

distribution to the total structure. The method is suitable for

analysis coupled shear walls subjected to lateral loads and of

Vierendeel girders under transverse loads.

Though some yield reasonably accurate results, all conventional

methods of solution have some inherent weaknesses and/or limitations.

Specifically, it is difficult, if not impossible, to incorporate the

following points in the analysis of shear walls by the conventional

methods.

(i) the stress concentration at the corner junctions of lintel

(connecting) beams and walls.

(ii) the effect of different elastic constant for walls and

lintel beams, or for different portions of the structure.

(iii) changes in cross-section of the wall (sudden or gradually

varying)

(iv) variation in the thickness of the wall—either uniform

variation or sudden changes in thickness at different

levels of wall.

Finite Element Method

The finite element method removes many of these limitations from

the solution of shear wall problems. By using finer elements near

points where stress concentrations occur, a complete picture of the

stress pattern at such places can be obtained. Points (ii), (iii),

and (iv) can be very easily incorporated into the computer program.

The ease with which such problems can be solved adds greatly to the

versatility of the finite element method over conventional methods.

Girija Vallabhan (12, 13) has used constant strain triangular

and first order rectangular elements for analysis of shear walls.

The functions chosen to define uniquely the state of displacement

within the element are

u - a^ + a^x + a^y + a^xy (1.1)

V - a^ + a^x + a^y + oigXy. (1.2)

These elements give accurate results for problems where the

displacement pattern in the continuum is similar to those given

above; but for the large class of problems where the displacement

pattern is that of quadratic or cubic polynomials, the first order

elements give only approximate results.

Oakberg and Weaver (14) used three types of elements: the

8-degree freedom element as interior elements, the 12-degree freedom

element as edge elements, and the lO-degree freedom element as tran

sitional elements. They have also mentioned that they are investi

gating a refined finite element for shear wall models.

Mcleod (15) has used a 12-degree freedom element. The displace

ment functions used are

2 2 u « A + A_x + A^y + A^2^y "•• A,y + ^IQ^^ (1*3)

2 2 V = A_ -»- AgX + A2y + A^xy + A-y + A x y (1.4)

These elements cannot take advantage of the symmetry or anti

symmetry of a given structural problem. Though this may not seem to

be a serious drawback of this element, in practice it often turns

out to be so. This is because finite element solutions are highly

susceptible to errors due to round-off; hence, for a given discreti

zation, the solution obtained by considering the symmetry or anti

symmetry of the given structural problem, (by which the unknown dis

placement parameters will be reduced), is always to be preferred.

Sarrazin (16) has used a rectangular element with 24 degrees of

freedom; the element has one node in each corner and six degrees of

freedom per node. He concluded that the lateral flexibility of shear

walls could be determined with acceptable accuracy by a finite element

discretization and that the same could be said about the stress dis

tributions away from singularity points. But for the study of stress

concentrations, he has found, the number of finite elements should

be increased near the singularity point.

Franklin (17) has developed an analytical procedure which utilizes

quadrilateral linear strain finite elements (12-deg freedom), special

frame elements, axial force rod elements, and bi-directional tie

link elements to study two-dimensional reinforced concrete frames

with attached shear panels subjected to large lateral forces. He

has reported that the analytical procedure has successfully predicted

the complete structural behavior of these frames through cracking to

yield.

Scope of Present Investigation

This dissertation represents an attempt to advance the current

knowledge of the structural behavior of shear walls. A computer

program for a 16-degree freedom rectangular element for analysis of

coupled shear wall structures has been developed. To develop some

8

error characteristics for this rectangular element, the program is

used to solve a beam problem for which a mathematically exact theory

of elasticity solution is available. From this study, the error

propagation in the final result due to different discretizations,

when solving structural problems using the 16-degree freedom element,

was achieved. Based on this study, six to eight different discreti

zations have been chosen for a shear wall problem. The results from

this analysis are compared with other finite element and conventional

methods of analysis of shear walls.

A parametric study to investigate the effects of the various

parameters on the structural behavior of shear walls is conducted.

Curves showing the influence of the dimensional and non-dimensional

parameters on the structural action of coupled shear walls are pre

sented.

Following the development of a static analysis, the concept is

extended to obtain the dynamic properties of a general shear wall

structure. The natural period of vibration of a 6-story shear wall

structure has been calculated. Techniques are developed to obtain

the response of a shear wall structure to an arbitrary ground accel

eration history, such as that produced by a strong earthquake. The

influence of mass concentrations at floor levels, and the depth of

the lintel beam, on the natural period of vibration of the shear wall

is shown.

CHAPTER II

THE FINITE ELEMENT METHOD

The finite element method is essentially a generalization of

standard structural analysis procedures which permit the calculation

of stresses and deflections in two- and three-dimensional structures

by techniques similar to those used in the analysis of ordinary framed

structures. In this method, the structure is assumed to consist of

a finite number of elements, interconnected at a finite number of

Joints or nodal points. All of the material properties of the origi

nal system are retained in the individual elements. The method provides

a unified approach by which any type of structural configuration may

be analysed.

The formulation of the matrix transformation theory of struc

tures (18, 19) is perhaps the immediate background for the develop

ment of the finite element method, for, although there is no theoret

ical necessity of utilizing matrix methods in a finite element analy

sis, they provide the most practical means for organizing the compu

tations. The basic concept of the finite element method, and of

matrix structural analysis methods in general, is that every struc

ture may be considered to be an assemblage of individual structural

components or elements. It is this characteristic which distinguishes

a structural system from a continuum. If, therefore, an elastic

continuum can be idealized as an assemblage of appropriately shaped

two- or three-dimensional elements, the continuum can be analyzed

10

by standard methods of structural analysis. The approximation that

is introduced in the finite element method is thus one of a physical

nature; a modified structural system is substituted for the actual

continuum. This, as a rule, implies approximations either to the

compatibility conditions or to the equilibrium conditions or to both.

Thus, we have the so-called compatible models, which assume continuous

displacement functions inside each element, maintaining the displace

ment compatibility along interelement boundaries; the equilibrium models,

which assume equilibrating stresses inside the element maintaining

equilibrium of boundary tractions; the so-called hybrid methods, which

assume stresses inside the elements and compatible displacements

along the boundaries (hybrid I) or displacements inside the element

and equilibrating tractions along the boundaries (hybrid II); and,

finally, the mixed methods, based on Reissener's principle, which

assume continuous displacements and stresses inside the element,

maintaining the displacement compatibility along the boundaries. (20)

The approach used herein is that of the compatible (or displace

ment) models. The various steps in the formulation can be summarized

as follows

(a) The continuum is separated by imaginary lines or surfaces

into a number of 'finite elements'.

(b) The elements are assumed to be interconnected at a discrete

number of nodal points situated on their boundaries. The

displacements of these nodal points will be the basic un

knowns of the problem.

(c) A function (or functions) is chosen to define uniquely the

11

state of displacement within each 'finite element' in terms

of its nodal displacements.

(d) The displacement functions now define uniquely the state

of strain within an element in terms of the nodal displace

ments. These strains, together with any initial strains

and the elastic properties of the material will define the

state of stress throughout the element and, hence, also on

its boundaries.

(e) A system of forces concentrated at the nodes and equili

brating the boundary stresses and any distributed loads is

then determined, resulting in a stiffness relationship of

the form

[K]{U} = {F} (2.1)

in which, [K] « stiffness of the assemblage of finite ele

ments of the continuum; {U} = undetermined displacements

at the nodal points; {F} = the vector of forces concentrated

at the nodes.

(f) The final step consists of solving for the unknown displace

ments.

The intention here is to study the use of plane stress triangular

and rectangular elements (first order and higher order)* for the

*Where the assumed displacement functions for the finite element yield linear variation of displacements along the boundary of the element, a first order element results; where the displacement variation along the boundary of the element is given by a polynomial of degree 2 or above, a higher order element results.

12

analysis of structures, with special reference to the analysis of

shear wall problems in multi-story structures. The 6-degree freedom

constant strain triangular element, the 8-degree freedom linear

strain rectangular element, the 12-degree freedom linear strain

triangular element and the 16-degree freedom quadratic strain rec

tangular element are discussed herein. All the above elements,

except the 16-degree freedom element, have been discussed in detail

elsewhere (20, 21, 22, 23, 24, 25, 26); a brief outline of the dis

placement shapes used, and the stiffness matrix calculation for these

elements is given in Appendix A. The 16-degree freedom element has

been used by Ergatoudis (27), but details have not been presented of

the distributed loading that has to be used with these elements.

The derivation of the element stiffness and the distributed loads

for the 16-degree freedom element are discussed in detail in the

following pages.

FIGURE 2.1 TYPICAL 16-DEGREE FREEDOM RECT. ELEMENT

13

The 16-degree Freedom Rectangular Element

Details of the Element

A typical finite element, 'e', is defined by nodes i, j, k, 1,

m, n, o and p, as shown in Figure 2.1. For convenience, the nodes

j, 1, m and o are chosen at the middle of the sides, though they can

be chosen anywhere along the sides without altering the formulation.

The 16-degrees of freedom for the element are obtained by providing

two translatory motions for each node, viz the translatory motion in

the x-direction, denoted by 'u' and in the y-direction, denoted by

» , . • v'. Though it is possible to choose the x- and y-axes for a specific

element different from the x- and y-axes of the continuum as a whole,

considerable ease in calculations results if the same set of x- and

y-axes are chosen.

The 16 components of the element displacements are listed as a

vector

r 's u.

{r}

u

(2.2)

u

The displacements within the element must be uniquely defined

14

by these 16 values. The displacement shape chosen consists of two

polynomials which exhibit parabolic variations along the sides and

include two cubic terms in addition to all quadratics

2 2 2 2 u - a^ + ^2'' "*" ^3^ "*" 4^ "*" °'5' "*" 6^ "*" °'7' "*" °'8^

(2.3)

" • ^ 9 "*" °'l0'' ^ °'liy " °'l2'' ^ ''u'^^ ^ ""l^ " °'l5'' y " °'l6'' ^

The constant coefficients a can be determined in terms of the un

known displacements u., v., u., v.,..., u , v by writing the nodal 1 1 J J P P

coordinates for the appropriate nodal displacements. For example.

"i " "i + "2"! * "3^1 •" "4*1 * "i^î * "eî "• " r V i "" "sî^'i

(2.4)

Uj - a^ + a^Xj + Cjy^ + a x ^ + OjX^y^ + a y ^ + a^x^^^ + OgX^y^^

and so on up to u . P

Similarly, we have

(2.5)

and so on up to v . P

Substituting Equations 2.4 and 2.5 into Equation 2.2 and expres

sing in Matrix notation, we have

{r} = [B]{a} (2.6)

in which {r} is a 16 x 1 column vector of undetermined displacements

at the nodes of the element,

15

[B] is a 16 X 16 matrix of co-ordinates of the nodes of the

element

{a} is a 16 X 1 column vector of constant coefficients for the

element.

Since [B] is non-singular, we can write (a) in terms of {r} as

{a} - [B]"^{r}. (2.7)

Strains

The total strain at any point within the element can be defined

by its three components which contribute to internal work.

(e) y

6u/6x

6v/6y

5u 6y_ 6y 6x

(2.8)

Performing the differentiation on u and v of Equation 2.3, we have

it)

0 1 0 2 x y 0 2 x y y 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 0 X 2y X 2xy

2 0 0 1 0 X 2y X 2xy 0 1 0 2x y 0 2xy y J

a.

a,

a,

a

a,

a,

a.

8 >

a

a

a

a

a

10

11

12

13

14

"15

(2 .9 )

16

or, in matrix notation

it) - [Q]{a}. (2.10)

Substituting for {a} from Equation 2.7, we have

{e} - [Q][B]"-^{r}. (2.11)

In general, the material within the element boundaries may be

subjected to initial strains such as may be due to temperature changes,

shrinkage, crystal growth, and so on. If such strains are denoted

by e , then the strain caused by the loads, (e }, will be the dif-" o a

ference between the total and initial strains, i.e..

{e } = {E} - {e }. a o

(2.12)

Stresses

Assuming general elastic behavior, the relationship between

stresses and strains will be linear and of the form

{a} = [D]({e} - {e }) o

(2.13)

in which [D] is the elasticity matrix. For the case of plane stress

in an isotropic material (28),

[D] = 1 - V

V 1

0 0

0

0

(1 - V)

in which E = Modulus of Elasticity of the material

(2.14)

V = Poisson's ratio of the material.

17

Equivalent nodal forces

Let

{f} - < (2.15)

define the nodal forces which are statically equivalent to the bound

ary stresses and distributed loads on the element. The components

u v F, and F. correspond to the directions of u and v displacements of

the element at node i. The body force components in the direction

of the u and v displacements are denoted by X and Y and is expressed

in vector notation by {p}, i.e., {p} = { }. (2.16)

Basis for the construction of the element stiffness matrix

The basis for the finite element procedure lies in its equiva

lence to the minimization of the total potential energy of the sys

tem in terms of a prescribed displacement field. If this displace

ment field is defined in a suitable way, then convergence to the

correct result must occur. The process is thus equivalent to the

well-known Ritz procedure (29).

18

The total potential energy, •, of any displacement state is defined

as

" - / ' -• - U - / {"}'^{p}d(vol) - {r}' {f} (2.17)

in which U is defined as

}d(vol). (2.18) lJ'\p./{c

Substituting for {e } and {a} from Equations (2.12) and (2.13) into tt

Equation (2.18),

.-if ({£> - {£ })^[D]({e} - (e })d(vol). (2.19) o o

Rearranging,

" - /

({e}'^[D]{e} - {G}'^[D]{e^} - {e^}'^[D]{e}

+ {e }^[D]{e })d(vol). (2.20) o o

Since the [D] matrix is symmetric, we have

{e}^[D]{e } = {e }' [D]{e}. (2.21) o o

Substituting Equation (2.21) into Equation (2.20), we get

- i / = 4 / ({e}'^[D]{e} - 2{e}'^[D]{e } + {e }'^[D] { e ^ } ) d ( v o l ) . (2 .22) ^ ' o o o

S u b s t i t u t i n g for {e} from Equat ion ( 2 . 1 1 ) ,

- 2{r}'^([B]"^) '^[Q]'^[D]{e^} + {e^} '^[D]{e^})d(vol) . (2 .23)

= 4 / ( { r}^( [B] ^ ^ [ Q ] ^ [ D ] [ Q ] [ B ] " ^ r }

19

Writing Equation (2.3) in matrix notation, we have

u n - [AKa} (2.24)

in which

2 2 2 2 - , l x y x x y y x y x y 0 0 0 0 0 0 0 0 , /9 9«;>,

I A J " L 2 2 2 2 ^^'^^)

0 0 0 0 0 0 0 0 I x y x x y y x y x y

Substituting for {a} from Equation (2.7) into Equation (2.24),

we have

{"} - [A]([B]"^{r}. (2.26)

The principle of minimum potential energy states that, of all

geometrically compatible displacement states, those which also sat

isfy the force-balance conditions give stationary values to the total

potential energy *.

To minimize the total potential energy <t» with respect to the

nodal displacements treated as variable parameters, we establish a

system of equations of the type

^(^) . 0 (2 27^

Substituting for U from Equation (2.23) and for { } from Equa

tion (2.26) into Equation (2.17) and performing the above differen

tiation, we have

/([B]"^'^[Q]'^[D][Q]([B]"^d(vol){r} - y*([B]"S' [Q]' [D]{e }d(vol)

-fa.,-' "• ) fA] {p}cl(vol) - {f} = 0. (2.28)

20

This is rewritten in the familiar form

[k]{r} + (f}^ + if} - if) (2.29) o P

in which [k] is the stiffness matrix of the element given by

[k] -J([B]"S'^[Q]'^[D][Q]([B]"Sd(vol). (2.29a)

The terms {f} , {f} and {f} are the nodal forces due to initial ^ P o *

strain, body forces and externally applied forces.

In the absence of initial strains and body forces. Equation

(2.29) reduces to

[k]{r} - {f}. (2.30)

Calculation of the Element Stiffness Matrix

Since the matrix [B] is independent of the thickness 't' and

-1 T -1 the dimensions 'a' and 'b' of the rectangle, the ([B] ) and [B]

terms can be taken outside the range of integration. Thus, the stif

fness matrix [k] of the element becomes

'•' </ ' [k] = ([B] " ) ^ ( I [Q]'^{D}[Q]d(vol))[B] ^ (2.31)

For the case of the plane stress 16-degree freedom element in

an isotropic material, the elements of the matrix ( j [Q] D [Q]d(vol))

are tabulated in Table 2.1. From the nodes of the element, and the

coordinates of the nodal point, the matrix [B] is generated in the

computing machine. The matrix ( f [Q] [D][Q]d(vol)) tabulated in

Table 2.1 is input into the computer. The [B] is evaluated in the

machine. In calculating the inverse, the 16 x 16 matrix is rear-

21

O <N A

U N , 0 *^ "O «N m

<^ H N 9 t^ O A "9 *§ IB "9 "S « «S <N N "^ M •»

t " "*-& < 0 -o -' s. •o « «d

CM 1 | m r T H (« « > r I i «M « «n « >«

I

I Si

o

o >

u U U (N u

u . 0 <N JO r\ ja N O > * f i o - < t » M O o H ~ "

«s

N

? S t

O «M - *

<M

o m ' 2 ^ « i ' * "S 2 « c s eg

•o •a -o , 0 «M 9 m « 9 to «9 «

0 0 < N 0 C M - * P M > » 0 C « 4 0 « *

O U O

u

.fi o CM ,a o o

0 0 0 0 0 0 0 0 0

CO ( S

^ ^ •O <N CN PM * " ' ^

O i A ^ 4 < M . O • « < N J O

0

0

0

0

J O

« p,4

0

(VI

• 0 PX

cd

•0 ,0 P>4

T3 4

0

p / »

« •*

0

•a pM

px

>»

•0 •n

t CM

• C>4

•0 .A « Pvl

PM 09 +

PM X>

• 0 >ff

9 •¥• •C P^

^cT « « PM • *

•0 Pvl

Ji • «

O O TJ

O ^

n

»

U

Is O (0

s "g 9

O Ok

M U

M B (0 4J U 0) u

t M 0

» «t • 0 Tl 10

,, x>

* 10

u 01 l4

<*•*

0

u « 4) a M u •r4

.C 4J

,, «J

.A I

22

ranged and partitioned into four 8 x 8 submatrices, in which the two

off-diagonal sub-matrices are null matrices and the diagonal sub-

matrices are identical. Hence, the inverse of only one 8 x 8 matrix

need be determined. The stiffness matrix [k] of each element is

evaluated in the machine by obtaining the matrix product of the three

matrices, viz., ([B]'" )' , ( f [Q]' [D] [Q]d(vol)), and ([B]""^).

Distributed Loads

Corresponding to the 16 generalized displacements of the ele

ment (2 per node), we have an appropriate generalized force vector,

which also has 16 components (2 per node). This force vector also

known as the distributed load for the element can be evaluated by

taking the partial derivative with respect to the generalized dis

placements of the potential energy of the given loads while under

going the displacements.

Concentrated Normal Load

A concentrated load of magnitude P is shown acting normally to

the longitudinal side at node 'o ', in Figure 2.2(a)(i). Since the

potential energy of the load P while undergoing the generalised dis

placements consists of only one term, viz., the product of P and the

displacement in the direction of P, it is clear that the partial

derivative with respect to the generalised displacement will give the

same load P at node 'o' as the distributed load. This is shown in

Figure 2.2(b) (i). The elements of the generalised force vector are

given by

23

r—•

<> >

g ( Q — ^ — e — ^

lb—^«

&t

<)- ; :

^ ' ^ O JD •D o

UJ

LiJ - J UJ

- i

iP O - Q

o o

"8 Q

Z

&r-—1

§ko"^

<

? 0

> © -

j (

*

1

• \

'

{!")

—i

T3

£ 3 .O ^ ^

t D

is

vole

n 3 or UJ

o u

3 go or \ -co Q

Q < O

cr o

a.

) —

E

— © -

JC

~J)

—i

( ) 3

CO CVJ

LU

CD

24

{f} -

0

0

0

0

0

0

0

0 y

0

0

0

0

0

p

0

0

(2.32)

No loss of generality occurs when considering the load in the

y-direction at node 'o'. If, instead, the load had been, say, in

the x-direction at node i (normal to the short side), the corres-

u ponding distributed load would be F = P. In other words, the

appropriate element in the generalised force vector is equal to the

applied concentrated load.

If, however, the applied load was in a direction parallel to

the sides of the element, i.e., in the x-direction at node '0' and

in the y-direction at node 'it', the load is producing the effect of

shear on the element. Such cases are treated in a later paragraph

titled 'Shear Load'.

Uniformly Distributed Normal Surface Force

A uniformly distributed surface force of intensity q per unit

25

length is shown in Figure 2.2(a)(ii). The load is shown acting

normally to the face of the element defined by the nodes n, o and p.

The potential energy, V, of the applied load while undergoing the

generalised displacements is equal to the double integral of the

load intensity times the displacement in its direction over the area

of the element.

i.e., V - / Q / Q q V dx dy. (2.33)

Substituting for v from equations (2.3), we have

+ a,,xy^) dx dy. (2.34) io

Along the side defined by the nodes n, o and p, we have y = b.

Hence, performing the integration along the side defined by the nodes

n, o and p.

+ a,,xb^) dx. (2.35) 16

Performing the integration along dx and substituting the limits,

2 ^3 ^2^ 2

V = qia^a + a^Q f" + ot ba + a^^ 3" " 3 V "" 4 ^

-^15'T'^ie^' ''-'''

This equation can be re-arranged as

26

V - q(a^a2a^ ... a^^)

i.e., V = q{a} 0

0

a b /2

0

0

0

0

0

0

0

0

a

U ab

3 a

3

a b/2

ab

a\/3

a b

(2.37)

(2.38)

Substituting for the vector of a's from Equation 2.7, we have

27

V - qirl^dBj'S"^

0

0

2u2 a b

(2.39)

Hence, the distributed load (or, the generalised force vector)

3{r}

3{r) (q{r}^([B]"S^

q ( ( B ] ' V

<

OO

OO

OO

OO

ab

a /3

a^b/2

ab2

a \ / 3

a b /2

(2.40)

(2.41)

(2.42)

28

— 1 T On substituting the values of ([B] ) , the elements of the

generalised force vector for the element shown in Figure 2.2(a)(ii)

with sides *a* and 'b' become

{f} -

0

0

0

0

0

0

0

0 L (2.43)

0

0

0

q a/6

0

4q a/6

0

q a/6

Linearly Varying Normal Surface Force

A normal surface force varying linearly from zero at one end to

q at the other end of the face of the element defined by the nodes

n, o and p is shown in Figure 2.2(a)(iii). The intensity of the

load at a distance x from the node n is q —. The potential energy Cl

of the applied load while undergoing the generalised displacement is,

therefore.

«/ O •' O q — V dx dy. a

(2.44)

29

Proceeding similarly to that of the preceding case, we can

evaluate V and hence the generalised force vector. The generalised

force vector is found to be

0

0

0

0

0

0

0

{f} - « 0 V (2.45)

0

0

0

0

0

q a/3

0

q a/6

Symmetric Parabolic Normal Surface Force

A normal surface force varying parabolically from zero at the

ends to a maximum of 'q' at the middle of the face of the element

d«»! li>rr« hy tho n.doR n, o and p is shown in Figure 2.2(a)(iv). The

intensity of the load at a distance x from the node n is —5- x(a - x), a

The potential energy of the applied load while undergoing the gener

alised displacement is, therefore.

' • / : /

b 4£ o 2 a

x(a - x) V dx dy. (2.46)

Proceeding in an analogous manner to that of the preceding cases,

30

we can evaluate V and hence, the generalised force vector. It is

found to be

{f} -<

0

0

0

0

0

0

0

0 L (2.47)

0

0

0

q a/15

0

8q a/15

0

q a/15

Shear Surface Force

Corresponding to the four different types of normal surface

forces shown in Figures 2.2(a)(i) through 2.2(a)(iv), are shown

shear surface forces in Figures 2.3(a)(i) through 2.3(a)(iv). The

equivalent distributed loads for these cases are shown in Figures

2.3(b)(i) through 2.3(b)(iv). The magnitudes of these generalised

force vectors can be determined in an identical manner as those for

normal loads. The expression for potential energy in these cases is

V = ^ / ^ q u dx dy. (2.48)

The only difference in the case of the shear loads from that of the

normal loads lies in the appropriate displacement vector to be used

31

o|iQ» O Q

SN4"

&isl

o<?

Z LU

LU - I LU

LU

(iO

3 .

It

> a <

> (

i O (

e O a

o ^

^ o <>

a o a

8

o a>

CO

1 Q. <

&^r

sw

- • — «

():=

® O d

&^* O O €>

o IT

^

4 — e *

o^

o o

0)

Q

c o >

o e — 1 6

Q I -

c/)

Q

Q

Q:

i3 X cr> ro cvi

<>3

6 o 4

•^t < > w

o o ^

LxJ Q:

32

in forming the double integral for V. Since the form of the dis

placement shape function used for u and v is identical, the components

of the distributed load have the same magnitude as in the case of

normal loads. The sense of the loads, however, does correspond to

the applied shear load.

Other Loading Conditions

PARABOLA

(l) (ii) (iii)

RGURE 2.4 OTHER LOADING CONDITIONS

Some of the other loading conditions that are normally encountered

in civil engineering practice are shown in Figure 2.4. These can

be treated as combinations of the loading conditions already dis

cussed. For example, the case of the inclined load can be treated

as the sum of its component normal and shear loads. The trapezoidal

loading shown in Figure 2.4(ii) can be considered as the sum of a

uniformly distributed load of intensity q.. , and the linearly varying

load (with zero ordinate at left end and (q ~ Qi) ^t the right end).

The loading of Figure 2.4(iii) may be treated as the sum of uniformly

distributed load, linearly varying load and a symmetric parabolic

load.

33

Generalised forces for other loading conditions may be derived

using the techniques presented herein.

The Assemblage Stiffness Matrix of the Continuum

In general, the element stiffness matrix can be evaluated using

a local co-ordinate system different from the global co-ordinate

system of the entire continuum. The stiffness of the element in the

global co-ordinate system is obtained by means of an appropriate

transformation. The general form of this transformation is

[k] - [T]" [k] [T] (2.49)

in which [T] is a matrix of transformation relating the global co

ordinate system of the continuum and the local co-ordinate system of

the element.

An alternate approach consists of evaluating directly the ele

ment stiffness matrix in the global co-ordinate system. This method

is adopted in the computer program developed herein.

Once the element stiffness matrix is available in the global

co-ordinate system, the assemblage stiffness matrix of the continuum

is easily formed. This is obtained simply by appropriate addition

of the stiffness matrices of all the elements of the continuum.

Properties of the Stiffness Matrix

The assemblage stiffness matrix is symmetric and banded. The

symmetry property follows from the Betti-Maxwell theorem (30). Since

the displacement of any one node point is affected only by the stiff

ness of the elements of which the nodal point in question forms a

34

part, it becomes clear that the assemblage stiffness matrix will have

only few coupling terms. By suitably ordering the nodal points, the

assemblage stiffness matrix can be converted into a banded matrix.

Computer Program and Solution of Matrix Equation

The assemblage stiffness matrix is formed in blocks and stored

on disks. Only the upper half of the banded matrix is stored in the

machine because of its symmetry. The matrix in the present form is

singular. When the appropriate boundary conditions are introduced,

the matrix equation (2.1) connecting the assemblage stiffness matrix

[K], global displacement vector (U) and the global force vector {F}

can be solved for {U}. The Gaussian elimination procedure (31) is

used for the solution. Having obtained the global displacement

vector, the strain and stresses at every point in the continuum is

evaluated from Equations (2.11) and (2.13).

CHAPTER III

ANALYSIS OF SHEAR WALLS FOR STATIC LOADING

Stress distribution in shear walls coupled with lintel beams

presents a boundary value problem in elasticity (28), assuming linear

material properties for walls and beams. It may be approximated as

a plane-stress boundary value problem in a multiply connected region.

The finite element method is a convenient numerical technique for

the solution of such problems.

In this chapter, the 16-degree freedom element is used in a

finite element analysis of a 10-story shear wall. The results from

this analysis are then compared with the solutions obtained using

conventional methods and the 6-degree, 8-degree and 12-degree freedom

finite elements.

Assessment of Accuracy

Due to the rectangular geometry of shear walls, automatic mesh

generation techniques can be employed for the solution of shear wall

problems. Thus, for the case of triangular elements or rectangular

elements, the nodes of the element and the co-ordinates of the nodes

can be easily generated in the machine. However, since the rectangular

elements have greater degrees of freedom than triangular elements for

the higher order elements employed herein, it is advantageous to use

the former for shear wall problems. Much work has been done by

Argyris (19), Felippa (24) and others in assessing the accuracy of

35

36

constant, linear and quadratic strain triangular finite elements.

However, data regarding the error involved in finite element solu

tions due to the discretization of the continuum and the round-off

generated in the computing machine are not available for rectangular

finite elements.

Necessity of Error Analysis

There is no doubt that the finite element method, in the limit

of subdivision, yields exact solutions to plane elasticity problems.

In practice, however, the finite element method always yields an

approximate solution to an elasticity problem, due to the difficulty

of employing such a fine mesh. For an assumed displacement function

with compatible elements, the displacements obtained are always under

estimated (32). More often, the analyst is interested to know 'a

priori' the error involved in the solution of any problem with a

given element subdivision. But the error in the finite element solu

tion depends primarily on the type of problem itself. One way by

which this error can be estimated is by comparing the results for

different element subdivisions on problems with known exact solutions

If the exact solution is that of a uniform stress field in the

continuum, then whatever be the element subdivision, the finite ele

ment solution will coincide exactly with the exact one. But, for

other stress distributions, the finite element solution is found to

contain errors. This error is dependent on many factors such as the

fineness of element subdivision, the relative dimensions of the sides

of the rectangular element, and the number of digits carried in the

37

digital computer. It is apparent that the larger the number of ele

ments, the closer will be the discretization approximation to the

exact problem; on the contrary, it is evident that the larger the

number of unknowns, the greater will be the error due to round-off.

Such behavior in the analyses was observed while solving shear wall

problems using different rectangular elements and element subdivisions

Since there is no exact solution for shear wall problems against

which the results from different finite element formulations can be

compared, a simple beam problem was chosen to illustrate the error

characteristics.

A Beam Problem

A uniform beam of rectangular cross section carrying a uniformly

distributed load, and simply supported at the ends by parabolically

distributed shear stress is analysed. The beam is shown in Figure

3.1(a), and has a span-depth ratio of ten. The exact solution for

such a problem is known (28). The finite element idealization of

the beam is given in Figure 3.1(b).

8-degree Freedom Rectangular Element

The error in the solution depends mainly on the number "n" of

rectangular elements, the a/b ratio (ratio of sides) of the rectan

gular element and the number of digits carried in the computer. The

percentage error in the maximum vertical deflection is tabulated in

Table 3.1. The interaction curves for the error characteristics from

several solutions with different a/b ratios and numbers of elements,

when seven-digit floating point arithmetic is used in the computer,

38

PARABOUC BOUNDARY SHEAR TRACTION

UNIFORMLY DISTRIBUTED L O A O l K / f t

^

CNJ

1 \ \ i * 1 1 \ 1 1 1

20

a) BEAM- DIMENSIONS AND LOADING

-iir HINGED

^ J?^ ON ROLLERS

b) FINITE ELEMENT IDEALIZATION

FIGURE 3.1 SIMPLY SUPPORTED BEAM

39

Table 3.1

Beam Analysis-8 d.o.f. Rectangular Element

Serial No.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

a

10.0 5.0 3.33 2.50 2.00 1.666 5.0 2.5 1.667 1.25 0.833 4.0 2.0 1.333 1.00 0.666 1.0 0.5 2.0 1.0 0.6667 0.50 1.0 0.5 0.333 0.25 0.80 0.40 0.2666 0.50 0.25 0.1667

b

1.0 0.5 0.333 0.25 0.20 0.1666 1.0 0.5 0.333 0.25 0.1667 1.0 0.5 0.333 0.25 0.1666 0.333 0.1667 1.0 0.5 0.333 0.25 1.0 0.5 0.333 0.25 1.0 0.50 0.333 1.00 0.50 0.333

a/b

10 10 10 10 10 10 5 5 5 5 5 4 4 4 4 4 3 3 2 2 2 2 1 1 1 1 0.80 0.80 0.80 0.50 0.50 0.50

n

4 16 36 64 100 144 8 32 72 128 288 10 40 90 160 360 120 480 20 80 180 320 40 160 360 640 50 200 450 80 320 720

First Order Element

7-Digit Carrying Machine

Max. Displacement

0.2320 0.8660 1.4500 1.8799 2.0656 2.1749 0.8620 1.9140 2.2861 2.2664 1.8063 1.1219 2.2122 2.5283 2.4385 1.9533 2.6794 1.8096 2.2247 2.7600 2.6989 2.2804 2.8700 3.3464 4.1109 7.142 2.9726 3.4527 4.4460 3.0885 3.5296 5.6007

%

Error

96.80 72.80 54.41 41.15


Max. Displacement

0.2320 0.8660 1.4753 1.9370

35.33 2.2597 31.91 73.00 40.30 28.40 29.04 43.45 64.88 30.74 20.85 23.66 38.85 16.12 43.35 30.35 13.47 15.50 28.61 10.08 4.70 29.20 124.00 6.95 8.09 39.30 3.31 10.49 75.33

2.4829 0.8620 1.9330 2.4797 2.7499 2.9805 1.1232 2.2535 2.6818 2.8940

*

2.8915

2.2327 2.8847 3.0490 3.1107 2.8505 3.1004 3.1426

2.9421 3.1284 3.1640 3.0610 3.1573

%

Error

96.80 72.80 53.81 1 39.36 29.26 22.27 73.00 39.50 22.36 13.91 6.69 64.83 29.45 16.04 9.47

9.47

30.10 9.69 4.55 2.61 10.76 2.94 1.22

7.89 2.07 0.95 4.17 1.09

*Computer core storage not sufficient to solve this.

40

are shown in Figure 3.2. Using double precision in the calculations

(15-digit floating point arithmetic), the errors are calculated for

the same set of problems. The corresponding error characteristics

are shown in Figure 3.3. Comparing Figures 3.2 and 3.3, it is clear

that large errors result from round-off in the machine. One should

not conclude from the graphs that for a/b ratios less than one, the

error will always be small, since the small errors in such cases are

only a characteristic of this beam problem.

16-degree Freedom Element

The 16-degree freedom rectangular elements were used to solve

the same beam problem. The results are shown in Table 3.2. The

errors in this case were considerably lower than for the case of

8-degree freedom elements. The interaction curves for the error

characteristics are shown in Figures 3.4 and 3.5.

Comparison of 8-degree Freedom and 16-degree Freedom Element

From a comparison of Tables 3.1 and 3.2 and Figures 3.2 through

3.5, the following conclusions are reached.

1. The 16-degree freedom element gives a better representation

of the true stress and displacement than would be obtained

with the same number of nodes using a much finer subdivision

into the 8-degree freedom elements. For example, with Just

one element (total 8 nodes) on the 20' x 2' beam, the 16-

degree freedom element gave a maximum error in the central

displacement of only 20.5%, whereas with four elements

(total of 9 nodes), the 8-degree freedom element gave an error

4J

z Ui Z Ul u < (0

s

UJ

o

UJ

< z UJ u Q: UJ 0 .

o/b

10.0

5 0

4 0

3.0

2 0

1 0

0.8

0.5

SYMBOL

X

o D

A ^

®

El A

NUMBER OF ELEMENTS (LOG SCALE)

FIGURE 3.2 INTERACTION CURVE--FIRST ORDER RECTANGULAR ELEMENT -- 7- DIGIT CARRYING COMPUTING MACHJNE

42

a/b

100

5.0

4.0

2.0

1.0

o.e 0.5

SYMBOL

X

O D ^

9 a A


FIGURE 3.3 INTERACTION CURVE - - FIRST ORDER RECTANGULAR ELEMENT 15-DlGlT CARRYING COMPUTING MACHINE

43

Table 3.2

Beam Analyi

Serial No.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

a

20.0

10.0

6.66

5.0

10.0

5.0

3.33

2.5

4.0

2.0

1.333

1.0

2.0

1.0

0.666

0.5

1.0

0.5

0.333

0.25

jis - :

b

2.0

1.0

0.66

0.5

2.0

1.0

0.66

0.5

2.0

1.0

0.666

0.50

2.0

1.0

0.666

0.5

2.0

1.0

0.666

0.50

16 degree of freedom i

a/b

10

10

10

10

5

5

5

5

2

2

2

2

1

1

1

1

0.5

0.5

0.5

0.5

n

1

4

9

16

2

8

18

32

5

20

45

80

10

40

90

160

20

80

180

320

Hit

rectangu!

jher Ord«


Max. Displacement

2.5030

2.9690

2.9440

2.0880

2.7800

3.3400

5.1627

19.1336

3.1300

3.1950

3.6040

2.8816

3.1737

2.9290

2.6732

2.2200

3.3215

3.9312

5.4400

— — *

%

Error

20.50

7.06

7.88

34.80

12.80

4.78

61.68

498.97

2.02

0.01

12.80

10.00

0.65

8.30

16.32

30.50

3.98

23.07

70.50

Lar element

*r Element


Max. Displacement

2.55

2.78

3.07

3.14

2.76

3.14

3.18

3.19

3.15

3.19

3.19

3.19

3.18

3.19

3.19

3.18

3.19

%

Error

19.80

13.10

3.98

1.57

13.40

1.69

0.39

0.13

1.39

0.08

0.20

0.50

0.37

0.02

0.06

0.31

0.02

*Computer core storage not sufficient to solve this

44

UJ

S UJ o < - I

X

I

tr UJ

UJ

<

Z UJ

o (C UJ 0 .

a / b

10 O

5.0

2.0

1.0

0 .5

SYMBOL

X

o •

o A


FIGURE 3.4 INTERACTION CURVE - HIGHER ORDER RECTANGULAR ELEMENT - 7 DIGIT CARRYING MACHINE

45

z UJ 2 UJ o < - i a. tn o 2 3 2 X <

a: o cr Q: UJ

< z UJ o Q: UJ

o / b

10

5

2

SYMBOL

X

O

ft


FIGURE 3.5 INTERACTION CURVE - HIGHER ORDER RECTANGULAR ELEMENT-15 DIGIT CARRYING COMPUTING MACHINE

46

of 96.8%. Similar differences in errors for the 2 different

types of elements occur for other cases.

2. Users of the finite element method are cautioned about the

presence of high round-off errors in finite element solu

tions, especially if the number of digits carried in the

computing machine is small (such as the 7-dlglt carrying

machines).

3. Using the 16-degree freedom element and double precision

on the computer (15 digits carried in the calculations)

the error computed in the maximum displacement is found to

be very small (of the order of 2% and less) for the usual

discretization patterns.

The Shear Wall Problem

Equal Walls

A 10-story shear wall structure with two equal walls connected

by lintel beams is shown in Figure 3.6. The shear wall is subjected

to a wind load of 1 kip per sq. ft. for the total height. The 16-

degree freedom element is used to analyse the structure. The finite

element idealization is shown in Figure 3.7.

The 0 , 0 , and a stresses in each element were calculated. X y xy

It was found that the normal stresses a in the vertical direction y

were more critical than the others in the wall. The maximum a stress y

was found to be 45.05 kips/sq. ft. (tension). The displacement

characteristics of shear walls are shown in Figure 3.8 and the lateral

deflection of the left side of the wall is shown in Figure 3.9.

47

S s

SB S

<> J>

Sc

8S =

UJ

n^ UJ '< N

M^uj _ <

S 9

o IZ

,88S'I

. 0 0 2 1

48

u 001 FT

^^r' 1

^EE^'-!

~1 I I I I

f^T^

^ -5»- ^

L-__-^- -3

t :a

FIGURE 3.8 DISPLACEMENT CHARACTERISTICS OF SHEAR WALL

49

1200

100.0

80.0

(FT

.

, 60 .0

WA

LI

^

H

o 40.0 ui X

20.0

BASE 0.0

•

/

/

/

/

/ /

/ /

/

/

/

• /

/

/

/

/

1 1

/

/ /

/

1 1

OOI 0.02 0 0 3 0.04

DISPLACEMENT (FT.)

FIGURE 3.9 LATERAL DISPLACEMENT OF LEFT SIDE OF WALL

50

In designing coupled shear walls, the structural engineer is

concerned with the accurate determination of the bending moments,

shear forces and axial forces acting on the lintel beams. The bending

moments in each of the lintel beams were computed from the values of

the average slopes and the average vertical displacements at the

ends of the beams. For any beam, AB, (Figure 3.10) the bending moments

^ab *^^ a *^ ^"^^ ^ *^^ ® ^^^ given by the slope deflection equa

tions (30) ,

FIGURE 3.10 BEAM FORCES AND DEFORMATIONS

M , - ^ (26 + e, + 1^) ab l a b )c

(3 .1 )

«.a = ¥ ( - . - a - f > (3 .2 )

in which 6 and 6, are the slopes at the ends of the beam, A is the a b

EI relative lateral displacement between the ends, and — is the relative

51

stiffness of the beam. The values 6 , 9 , , and A, for the lintel a b

beams, were computed from the nodal displacements of the ends of the

beam. These values together with the bending moments M , and M, »a'

at the ends of each of the 10 lintel beams, are given in Table 3.3.

Comparison of Results

With the development of higher order finite elements, a high

degree of sophistication has been achieved in the finite element

€Uialysis of structures. Not only is the practical designer presently

at a loss to understand how the results from the finite element

formulations compare with those of conventional methods, but no data

is available to compare the results of different finite element solu

tions. To provide this comparison, the shear wall problem of Fig

ure 3.6 is solved by the following methods.

(i) 6-degree freedom finite element

(ii) 8-degree freedom finite element

(iii) 12-degree freedom finite element

(iv) Using the curves supplied by Coull and Choudhury, based

on Rosman's method (6)

(v) The Generalised Cantilever moment distribution (11).

The results from these five methods and the 16-degree freedom

elements discussed earlier are shown in Table 3.4.

As indicated in an earlier paragraph, there is no exact solution

for the shear wall problem against which the results of the six dif

ferent methods can be compared. Rosman's method of analysis is

widely used by practising engineers to solve shear wall problems.

Table 3.3

52

Forces and Displacements in Lintel Beams

Lintel

Beam

(1)

1

2

3

4

5

6

7

8

9

10

16-degree freedom higher order rectangular finite element method of analysis

6 , in a radians

X 10"^

(2)

0.843

1.447

1.834

2.097

2.256

2.336

2.358

2.343

2.318

2.252

6^, in

radians

X lO"^

(3)

0.804

1.413

1.816

2.089

2.254

2.336

2.360

2.348

2.328

2.323

A, in

feet

X lO""

(4)

1.651

2.866

3.717

4.275

4.602

4.754

4.784

4.745

4.680

4.616

\b' '-kip-feet

(5)

16.55

28.68

36.87

42.32

45.57

47.13

47.51

47.17

46.61

22.94

^ a ' ^" kip-feet

(6)

16.42

28.57

36.81

42.29

45.56

47.13

47.51

47.19

46.64

23.06

53

0)

(d

(0 3

00 •H OB

^ •

en Oi

r-i

.o fd H

i H (0 3 M < 0)

JS w U4

o n 4J r H 3 00 OJ

06

U-t

o c o (D

(0

o

0 •o o « V4 -ri OB 0) 4J •H > 3 r j OJ ^

QJ U QJ U d C a 09 0) eg 5 •H

O O S Q

(d 3 QB ,X3 0) i H "O > r H 3 Wi 3 O 3 O (C O O O

i H CO NO 1

. •o r*« a> M-l O Csl i H o

• as 1 0) so O OB R

• i H 4J i H OB T ) (0 a CO 4J

U -r^ U 0 vO 0 O 0 0)

•H H a n a

(0 vO 1 . »0 fO CJ Q

UH 0 ro iH O • SB N 0) (H

O OB R • i H 4J i H OB

"O CO C CO 4J *j -ri u a

eg 0 O O tt) M H a n a

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m CO r». CM «*

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sj -

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<*

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NO

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m NO

m «n sf

m r*. CO

m »*

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CM CM

0 0

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P^ >3-

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r>. <r NS-

m -3-

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m NO CM

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NO >a-

m i H

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t

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r m r>. «H CM

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54

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tt) iH .o (0 H

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a a i « q> <« 0 •H O O s O

(0 3 OB ^ tt) r H » 0 > rH 3 (<« 3 O 3 O ^ CJ O O

r H CO NO 1

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^ H O.H a

r H (0 so 1

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• Z R Q) i H O 00 R

» T-i U t-i (0 • O flJ C CO 4J

4J - H 4J C CM O O O tt)

rH H a n a

r H CO r^ 1

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U-l Z R tt) CO OB R

O r H 4J r H 00 • CO C CO 4J

• O 4J • H 4J C O O O tt)

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tt)

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c tt) 4J J 3 r H *J 0 0 i H «4H

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CO

o • •>* S t

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m S t

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<<-»» CJ v « /

o i n •

ON CM

/ - N

CJ V—/

0 0 r>.

• NO CM

73 C 0)

4J 4-> 4 3 r- l 43 00 r H 00

• H CO "H Cd :» u

•

sr r H

ft.)

N - /

c O

• H i J O tt)

i H 14H tt)

Q

a 2 a • H X CO

2

S t m NO t-i

•

o

m o m m CM

o • o

NO CM NO NO so eg o • o

ON CM r-i

m CM

o • o

0 0 ON so i H

o • o

ON

o ON ON rH r-\ O

•

o

(4-1

o ^ PU r H O CO

H >

•

m T H

55

One of the assumptions made in this method is that the top lintel

beam has one-half the stiffness of the remaining beams (which are

all of equal stiffness). The dimensions of the shear wall problem

of Figure 3.6 were chosen to satisfy this condition. No more assump

tions involving the dimensions of the shear wall were made in any of

the six methods. Thus, in choosing the dimensions, care was exer

cised to ensure that errors on this count would be minimized. The

differences in the results are therefore due to the structural sim

plification assumed in the methods.

From the comparison, the following conclusions are drawn.

(1) The values of maximum deflection, stresses on the wall at

the base and the moments on the lintel beam (except the top beam)

agree closely for the three methods, viz, 12-degree freedom element,

16-degree freedom element and the curves given by Coull and Choudhury.

The latter method, based on the Rosman's theory, is widely accepted

as giving fairly accurate results for such symmetric coupled shear

walls (with the top lintel beam having one-half the stiffness of the

remaining equally stiff beams). In the absence of an exact solution,

the close agreement of the results from the 12-degree and 16-degree

freedom elements with that of Rosman's method indicate that these

methods could be used to solve such shear wall problems accurately.

(2) The Generalised Cantilever Moment Distribution, used by

Gurfinkel to solve shear wall problems, highly overestimates the

maximum deflection, stresses at base of wall and lintel beam moments.

(3) For the usual discretization patterns the 6-degree and 8-

degree freedom elements do not give a good representation of the struc-

56

tural behavior of shear walls; a discretization of the continuum

using these elements underestimates the maximum values of stresses

and deflection. Care has been taken to reduce the errors due to

round-off by using 15-digit floating point arithmetic in the calcu

lations; the difference in the final results is thus due largely to

thfi inability of these elements to represent correctly the inter

action between the stiff walls and the flexible lintel beams.

The 12-degree freedom triangular element and the 16-degree free

dom rectangular element use a second degree polynomial as the dis

placement function. The triangular elements have the advantage of

fitting into any geometry, whereas the rectangular elements make use

of 2 additional quadratic terms in the polynomial for displacement

function. Since shear wall problems have rectangular geometry, no

specific advantage accrues to the triangular elements (12-degree

freedom) over the rectangular elements (16-degree freedom) in this

case. Hence, only the 16-degree freedom rectangular element is used

for shear wall problems hereinafter.

Unequal Walls

Since close agreement was obtained by the 16-degree freedom

elements and Rosman's method for a coupled shear wall with equal

walls, a comparison of the two methods is now attempted for the case

of a coupled shear wall with unequal walls. Mcleod (5) has shown

that the analysis of shear walls by the continuous connection medium

theory used in Rosman's method can show significant error when the

bending stiffness of a wall section, which is flexible in comparison

57

with the adjacent section, approaches that of the connecting beams.

Such a shear wall problem is shown in Figure 3.11. This problem is

solved by the 16-degree freedom element and by using the design curves

given by Coull and Choudhury based on Rosman's method. The finite

element idealisation is shown in Figure 3.12. The results are pre

sented in Table 3.5.

From Table 3.5, we find that differences of the order of 5.0%

to 50.OX occur in the values of stresses on wall, moments on beams

and maximum deflection (with one case of 178.0% difference in the

moment on the top lintel beam). There is no exact solution to this

problem against which these results can be compared. However, in

view of the proven nature of the errors in the continuous connection

medium theory for such problems, the differences in the results by

the two methods is attributed to the assumptions made in the continuous

connection medium theory. The 16-degree freedom element does not

make any such questionable assumptions for this problem. The results

by this method, therefore, are accepted as being fairly accurate.

Discretization Errors in Shear Wall Problems

It is obvious that the subdivision pattern of the continuum

into finite elements must play an important role in the convergence

process. But, the analyst is limited in the degree of fineness for

division of the continuum since the number of elements will affect

the round-off errors generated in the computing machine. There is,

thus, an optimum discretization of the continuum, not necessarily the

finest subdivision, for which the combined errors due to discretiza-

58

o

y-

— - — - — , ,

—

._

— • - -

:EI!^:^^ — *

«0

m < ) <

• V

> - « H CM (

• - • H

> — ( I

r

C k

^\

4

- « ^

LU

UJ 3

iZ 9

ro

,vi ,06 .ce r

DDDDnnnnc CM

fl I M t i n i M H I t t t l i m t T T T T T ^

/ / / / / / / / / /

„0-.06

i o in

i UJ g

<

rd UJ Q:

O

59

Table 3.5

Comparison of Results of Shear Wall Analysis (Unequal Walls)

SI. No.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

Location

Bendii

Beam 1

Beam 2

Beam 3

Beam 4

Beam 5

Beam 6

Beam 7

Beam 8

Beam 9

Beam 10

Stress

Left Wall left end

Left Wall right end

Right Wall left end

Right Wall right end

Top of Wall

16-d.o.f. Higher Order Element

ig moment in Lintel

2.948

3.243

3.4138

3.6078

3.5487

3.2113

2.6882

2.1652

1.7940

0.930

es on the Wall at B

18.30 (ten)

23.30 (comp)

3.58 (comp)

17.40 (comp)

Maximum Deflecti

0.00335

Curves by Coull and Choudhury

beams (kip -

2.128

3.433

4.063

4.255

3.995

3.738

3.375

2.858

2.700

2.587

ase (kips/sq

16.42 (T)

15.44 (C)

8.66 (C)

10.94 (C)

on in ft.

0.00215

Difference: Expressed as a % on Results ty 16-d.o.f. Element

- ft)

+28.0

- 5.85

-19.00

-18.00

-12.62

-19.40

-25.50

-32.10

-50.50

-178.00

. ft.)

+10.3

+33.7

-141.8

+37.0

35.8

60

tion and round-off will be a minimum. This optimum discretization

depends primarily on the problem itself.

To arrive at some guidelines for discretizing the shear wall

problems for 'best* results, seven different structural idealizations

of the shear wall problem of Figure 3.6 were analysed by the 16-

degree freedom element. The maximum displacement in the wall was

selected as the criterion on which the comparison is made. The struc

tural idealizations are shown in Figures 3.7 and 3.13 through 3.18:

the results are presented in Table 3.6.

A careful examination of the results leads to the following

conclusions.

(1) A reasonably accurate value of the maximum deflection is

0.026 ft. Cases 3 through 7, with 2 or more elements on the lintel

beam, show the maximum deflection to be of this magnitude, whereas

Cases 1 and 2 underestimate it. For all such cases of coupled shear

walls, where 2 stiff walls are connected by flexible lintel beams,

at least 2 elements are needed in the lintel beams for correctly

representing the structural interaction between the wall and the

beams.

(2) The number of elements comprising the walls does not have

an appreciable effect on the discretization of the continuum.

(3) Choosing a very fine mesh with a large number of nodal

points and elements may have an undesired effect on the final results

by inducing large errors due to round-off. This tendency is seen in

Case 6, where the maximum deflection shows an increase from those of

Cases 3, 4, 5 and 7, even though 15-digit floating point arithmetic

61

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66

was used in the computer.

Thermal Stresses

One of the causes of initial stresses in a body is non-uniform

heating. With rising temperature, the elements of a body expand.

Such an expansion cannot proceed freely in a continuous body; so

stresses due to the heating are developed. The thermal stresses so

induced are of great practical Importance; in some cases, these

stresses may even be the governing criterion for design.

Simpler problems of thermal stress can be treated similarly to

boundary force problems of the type already considered. In other

words, the change in the temperature of the structure is converted

into an equivalent thermal load. If {e } represent the initial

strains due to temperature, we may write

{ej =< V (3.3)

xy

Although this initial strain may, in general, depend on the

position within the element, it will normally be defined by an average,

constant, value. For the case of plane stress in an isotropic mate

rial, with a coefficient of thermal expansion a, comprising an ele

ment subjected to a temperature rise 0,

67

a 6

{e } - J a e (3.4)

since no shear strains are caused by thermal dilatation.

With no other external forces on the structure, and in the absence

of body forces. Equation 2.29 reduces to

[k]{r} - -{f} (3.5)

in which {f} is the nodal force due to initial strain, o

Proceeding in an identical manner to problems involving external

loads, the assemblage stiffness matrix and the global force vector

are formed. The matrix equation (2.1) is then solved for the global

displacement vector. From the global displacement vector, the

strain in the element {e} can be found from Equation 2.11. The

stresses can then be found from Equation 2.13.

Beam Problem

E = 5 x lO' IbtAqft. l/-Oi*> 0 » I x IO'por F

/

JQ:

U

c ToT"

"or

20' ] J< 3

FIGURE 3.19 FIXED BEAM TEMPERATURE STRESSES

68

The beam shown in Figure 3.19 is subjected to (i) a uniform

temperature rise through 100' F and (ii) a linear variation of tem

perature from 100' F at the top of the beam to zero at the bottom of

the beam, with no variation of temperature along the length. These

cases have an exact solution (28). The results from the finite ele

ment solution and the exact solution are shown in Table 3.7. The

close agreement in the two cases attests to the correctness of the

method adopted.

Shear Wall Problem

A six-story shear wall, shown in Figure 3.20, is subjected to

an Increase of temperature of 100' F. The deflected form of the

shear wall due to this temperature change is shown in dotted lines

on the figure. The maximum stress at the base of the wall and the

maximum moment in the lintel beams are given in Table 3.8.

Of greater concern to the designer is the case of non-uniform

heating of the walls. One such design problem is posed in Figure

3.21. The exposed wall is at a higher temperature than the interior

wall. For purposes of illustration, the difference in temperature

between the walls is taken as 100'. The results of the analysis are

shown in Table 3.9.

The following conclusions are made with regard to temperature

stresses in shear walls for the material properties used herein.

(1) The magnitude of the maximum deflection due to a uniform

increase of 100' F is approximately equal to that of a uniform lateral

load of 0.7 kips per ft. height of wall. Of course, the sense of

69

Table 3.7

Iten

Stress

Stress

Stress

Stress

Stress

Stress

Stress

Stress

Stress

Stress

I

at

at

at

at

at

at

at

at

at

at

A

B

C

D

E

Cat

A

B

C

D

E

Thermal Stresses in Be

Finite Element Solution

•am

Theory of Elasticity Solu

Case (i) Uniform Temperature

50,000 Ibs./sq. ft.

50,000 Ibs./sq. ft.

50,000 Ibs./sq. ft.

50,000 Ibs./sq. ft.

50,000 Ibs./sq. ft.

50,000 Ibs./sq.

50,000 Ibs./sq.

50,000 Ibs./sq.

50,000 Ibs./sq.

50,000 Ibs./sq.

le (ii) Linear Variation of Temperature

50,500 Ibs./sq. ft.

37,675 Ibs./sq. ft.

25,250 Ibs./sq. ft.

12,625 Ibs./sq. ft.

0.0

50,000 Ibs./sq.

37,500 Ibs./sq.

25,000 Ibs./sq.

12,500 Ibs./sq.

0.0

tion

ft.

ft.

ft.

ft.

ft.

ft.

ft.

ft.

ft.

70

I

I

I

U--

1 I I — I

1 I

I r" r' I 1 1

•M

I

^Wx^^^^\

CVJ

esj

. E = 5 K I 0 * Ibi/fqft.

' V- 0.15

/ O ' lKlO'*p«r®F

/ /

/

20-0 20-0" I 20-0' *

FIGURE 3.20 SHEAR WALL UNIFORM TEMPERATURE

71

Table 3.8

Location

Lintel Beams

Beam 1

Beam 2

Beam 3

Beam 4

Beam 5

Beam 6

Uniform Temperature in i

8-d.o.f. element Total Nodal polnt8-244 Total Element8-192

Shear Force (kips)

0.01

0.01

0.03

0.036

0.045

0.048

Axial Force (kips)

117.00 (comp.)

57.18 (comp.)

19.30 (comp.)

0.74 (comp.)

8.38 (tension)

7.33 (tension)

Bending Moment (kip-ft.)

2.299

1.705

0.85

0.36

0.18

0.11

Shear Wall

16-d.o.f. element Total Nodal points-168 Total Elements-36

Shear Force (kips)

0.16

0.15

0.18

0.25

0.29

0.38

Axial Force (kips)

107.98 (comp.)

57.12 (comp.)

19.22 (comp.)

1.16 (comp.)

6.20 (tension)

8.48 (tension)


1.23

1.71

0.85

0.35

0.14

0.03

Stresses on Wall at Base (kips/sq. ft.)

Left end left wall

Right end left wall

Left end right wall

Right end right wall

16.90 (comp.)

11.60 (tension)

11.60 (tension)

16.90 (comp.)

38.11 (comp.)

24.98 (tension)

25.03 (tension)

38.12 (comp.)

Maximum Deflection (ft.)

Top of wall 0.00314 0.003133

72

3 I

a I E w O

«^ 'c 3

C

TO »

o

o

V

20

-0

b 1

o CSJ

20

-0

.

> ..o-.o.^ * (I

nte

rio

r w

all

(low

er

tem

p.)

beam

at

leve

l

tem

p)

• - 1 , 1 I 1

if 11 M —

u) 5

LU Q:

h-O 3 Q: I -

>-cr o h-

iD

U. O

< _ l Q.

cvi ro

. 0 - 0 ^

73

Table 3.9

Differential Heating of Walls

Location

Lintel Beams

Beam 1

Beam 2

Beam 3

Beam 4

Beam 5

Beam 6

Left end left wall

Right end left wall

Left end right wal

Right end right wal

Top of wa

8-d.o.f. element Total Nodal polnts*244 Total Elements-192

Shear Force (kips)

17.79

28.60

34.60

37.40

38.80

34.2

Stres

1

1

ill

Axial Force (kips)

62.14 (comp.)

30.50 (comp.)

10.20 (comp.)

0.70 (comp.)

3.40 (tension)

4.80 (tension)


48.80

78.65

94.78

102.50

106.36

94.03

ses on Wall at Base (

111.00 (comp.)

86.3 (tension)

73.8 (comp.)

92.8 (tension)

Maximum Deflection

0.0374

16-d.o.f. element Total Nodal points=168 Total Elements-36

Shear Force (kips)

37.00

64.53

80.22

88.96

93.31

93.27

icips/sq.

Axial Force (kips)

46.60 (conp.)

24.40 (comp.)

7.72 (comp.)

0.69 (comp.)

2.04 (tension)

2.16 (tension)


61.81

121.90

151.55

167.95

176.00

175.85

ft.)

211.58 (comp.)

190.65 (tension)

164.21 (comp.)

171.07 (tension)

(ft.)

0.050127

74

displacement is different. Due to lateral load, an antisymmetric

bending occurs, whereas due to uniform rise (or fall) of temperature,

a symmetric bending of the shear wall results.

(2) The maximum axial compressive stress in the lintel beams

is 375 pounds per square inch and occurs on the bottom lintel beam.

This leads to the observation that buckling of lintel beams is a

possibility for some combination of values of span of lintel beam,

its thickness, width and height of wall, and temperature rise in the

wall.

(3) The magnitude, as well as the direction, of the maximum

displacement for the case of non-uniform heating of walls shown in

Figure 3.21 is equivalent to a lateral load of 11.90 kips per ft.

height of wall acting from right to left. In other words, other

factors remaining the same, a non-uniform heating of one wall alone

produces about 17 times as much displacement as for the case of uni

form heating. This in turn produces high tensile and compressive

stresses in the wall, which calls for additional reinforcement.

(4) The close agreement between the results, shown in Table 3.8,

of the 8- and 16-degrees of freedom finite element analyses of the

shear wall for the case of uniform temperature calls for an explana

tion in view of the earlier observation that the 8-degree of freedom

elements are not able to correctly represent the structural inter

action between the stiff walls and the flexible lintel beams. The

answer to this apparent contradiction lies in the fact that the

lintel beams are not subjected to any appreciable bending due to uni

form temperature, and hence the first order finite elements are able

to represent satisfactorily the resulting behavior of the shear wall.

CHAPTER IV

PARAMETRIC STUDY OF SHEAR WALLS

To determine the influence of different parameters on the over

all structural behavior of shear walls, a parametric study was con

ducted. Non-dimensional curves involving displacement and stresses

in walls and bending moment in lintel beams are drawn. These non-

dimensional curves are useful to the practising engineer, in that

they enable him to determine the values of moments, stresses and

displacement in a coupled shear wall.

The analysis is done on 10-story, 8-story, 6-story and 4-story

shear walls with equal walls; the dimensions of the 10-story shear

wall are sho%im in Figure 4.1. The thickness, t, of the shear wall

is one foot. A uniform lateral load of 1 kip per foot height is

applied externally. The same cross-section is adopted for the 8-story,

6-story and 4-story shear walls; the overall height H is the only

dimension that changes in these cases. The 10-story shear wall

problems were solved using the idealization shown in Figure 4.2; the

8-, 6-, and 4-story shear wall problems are solved using the ideali

zation shown in Figure 3.15. By varying the span and depth of the

lintel beams, a total of 125 problems are analyzed. The overall

width of the shear wall B and the story height h are kept constant

for all the problems.

The boundary forces were evaluated from the calculated dis

placements. These forces together with the externally applied load

75

76

I 8

e» »

c; 3

- 

UJ Q: 3 CD U .

77

should satisfy the principles of statics. However, the results in

30 of the problems, where the walls and lintel beams were flexible,

did not satisfy static equilibrium conditions. The errors in these

cases are attributed to the discretization of the continuum and

round-off in the computing machine. The results of these problems

were discarded; results from the remaining 95 problems are shown in

Tables 7.2 through 7.10 in Appendix B.

Non-Dimensional Parameters

In the design of shear walls, the engineer is primarily inter

ested in the values of the maximum displacement and maximum stress

in the walls, and in the values of the axial force and bending moment

in the lintel beams. Some important non-dimensional parameters, and

other terms used, are defined below.

A solid shear wall of the same overall dimensions as that of a

given coupled shear wall is defined as a shear wall with breadth B,

height H and thickness t. Non-dimensional parameters denoted by TT

through n. are defined as follows

Relative wall stiffness 1 Relative lintel beam stiffness

E I • -^LJi X -r. where E and I denote the modulus of elasticity E^I^ h> W W

of the material of wall and moment of inertia of

wall respectively; E^, I, the corresponding values

for lintel beams.

TT

, (h - d) ^2 h

78

d TT_ - IT X —

3 1 s

Maximum displacement in the coupled shear wall 4 Maximum displacement in the solid shear wall

Maximum stress in wall in the coupled shear wall 5 Maximum stress in the solid shear wall

Maximum of the moments in the lintel beams 6 Maximum moment in the solid shear wall

The maximum displacement and maximum stress in the solid shear

wall are evaluated by treating the solid shear wall as a vertical

cantilever and applying the elementary bending theory. For the

dimensions of the shear walls analysed this is not correct; however,

since the purpose here is to arrive at a suitable non-dimensional

parameter, use of these characteristic values is justifiable.

Non-Dimensional Curves

Ten different curves in five categories involving the displace

ment and stresses in the walls and moment in the lintel beams are drawn

in Figures 4.3 through 4.12. In each category, one curve shows the

characteristics for the 10-story shear wall and the other shows the

characteristics for the 10-, 8-, 6- and 4-8tory shear walls.

Curves for displacement

(1) IT- vs IT,

The plot of log TT against log TT is shown in Figures 4.3 and

4.4. For smaller number of stories, a linear relationship between

log TT and log TT, was observed for small values of -. For values

of TT greater than about 200, the points tend to lie on a curve. As

79

S/B

SY

MB

OL

do d d d c> d

• X o • *( » <

If Ctfi M

C

o o o

o

Q

o b

 -a:

i I

O

ro

UJ (T 3

U.

(3nVDS 9 0 1 ) •'iZ.

80

<s cn

BCL

2 ^

^s d d

M O

333

167

0167

d 6 d

a * ^

^

II II c

o d o

< o <o CD Q

O Q

I I . • o d o

- 1 - ^ o

3

8

UJ >

3 O

O Q: I -UJ

< cr < a.

<

Q: <

UJ

C/)

UJ Q: 3 O Li_

(3nvDS sxn) tiz.

81

the number of stories increases, the linear relationship is seen for

higher values of TT also. However, the tendency of the points to

lie on a curve was noticed in this case for low values of TT and —. 1 B

(11) TT2 vs TT

The plot of TT against log TT, , for different values of "I is

shown in Figures 4.5 and 4.6. For low values of |-, the relationship

of TT and log TT is given by a curve of very small curvature. For

higher values of —, the curvature increases. However, for the 4-

story shear wall, the curvature seems to decrease with increasing values

°* B-

Curves for Stresses

(1) TT VS TT

The plot of log TT against TT is shown in Figures 4.7 and 4.8.

For the 10-story shear wall, a linear relationship between log TT

s s and TT is seen for small values of —. For higher values of —, the

J O D

points lie on a curve. Similar behavior is seen for the 8-story, 6-

story and 4-story shear walls.

(ii) TT_ vs TT

The plot of TT- against TT. for different values of — is shown in

Figures 4.9 and 4.10. For small values of —, the relationship be

tween TT- and TT^ is given by a curve of very small curvature. For

s higher values of — the curvature increases.

B

82

100

o tn

Z. 10

Symbol

• X

O

a

w 6

L.

S/B 0.80 0667 0.50 0333 OI67 0.10 0.0167

r\- number of stories

0.00 020 0.40 0.60 080 1.00

TTo

FIGURE 4.5 10-STORY SHEAR WALL - TTZ vs LOG 7r4

83

Syntel X o a it-

A

S/B 0.887 0.50 0.333 0 167

0.0167 n= numoer of stories

100 •

o u V)

2

\? 10 .

0.00 O20 0.40 060 0.80 1.00

TTn

FIGURE 4.6 SHEAR WALL PARAMETRIC CURVES ng VS. 7r4 (LOG)

84

S/B

SY

MB

OL

0.8

0

0.6

67

0.5

0

0.33

3

0.1

67

O

.K)

0.0<

67

w >^ O o * • <

o

11

c

Q O O O

2 <

8

o d

m

>

3 UJ I

<

Q: < UJ X cn

o I -cn

I

o

UJ a: CD

o o o CO

—I—

q

=ji

85

S/B

SY

MB

OL

0667

0.50

0.

333

0.

167

OJ0

I67

H o a * <

1* C « l II c

I

o

UJ

CD O

O - d

N"

Q

cn UJ > cr 3 O

o ct h-UJ <

i cc < UJ X en

oo

o OJ

O o o

UJ a: 3

'JL

86

20.0

16.0.

12.0-

m

8.0-

4.0

0 0 0.0 0.20 0.40 0.60

SytxixA

o

X

o

o

«

o

A

S/B

0.80

0.667

0.50

0.333

OI67

0.10

0.0167 n - number of

stories

0.80 LOO

TT,

FIGURE 4.9 10-STORY SHEAR WALL - v^ VS. TTg

87

12

O

% • •

n

6

4 '

2 .

SYMBOJ «

0

a 4fc

A

S/B 0.667 0 5 0 0.333 0J67

0.0167 n = numoer of stories

A A—nslO

0.2 0/4 0.6 O.e 1.0

TTo

FIGURE 4.10 SHEAR WALL PARAMETRIC CURVES

TTg VS 7%

88

Curves for Moment in Lintel Beams

(i) TT vs 1T

The plot of log TT against log TT for different values of — is J O D

shown in Figures 4.11 and 4.12. The relationship between log TT and

log TT is almost linear.

Axial Force in Lintel Beams

The axial force in all the lintel beams was approximately the

same, except those in the top and bottom lintel beams. These beams

carried a lesser load than others. The sum of the axial forces found

in the lintel beams was of the order of 40 to 60 per cent of the

total lateral load. As the number of stories increases, the number

of lintel beams also Increases. Consequently, the magnitude of the

axial force carried by each lintel beam, expressed as a percentage

of the total lateral load, decreases. For the 8-story shear wall,

the maximum axial force in the lintel beams was of the order of 6 to

9 per cent of the total lateral load. Since the value of the axial

force was almost the same for different values of —, curves showing

the variation of axial force have not been drawn. The values of the

axial forces in all the lintel beams for the 8-story, 6-story and 4-

story shear walls are shown in Tables 7.8 through 7.10.

89

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Onvos 9on) ^Ji

CHAPTER V

ANALYSIS OF SHEAR WALLS FOR DYNAMIC LOADS

In a structural dynamics problem, the loading and all aspects

of the structural response (deflections, internal forces, stresses,

etc.) vary with timei hence a separate solution must be obtained for

each instant of time during the entire history of interest. A dynamic

analysis thus entails a greater expenditure of effort than a static

analysis of the same structure.

The basic feature of a dynamic problem may be recognised with

reference to the simple beam structure shown in Figure 5.1.

Pit)

Looding 1

Inertia Forces f^lXft)

Moment Diogrom

0) STATIC b) DYNAMIC

FIGURE 5.1 BEAM SUBJECTED TO STATIC AND DYNAMIC LOADING

When the beam is subjected to a static load, the internal forces

resisting the load may be evaluated by simple statics, and from these

the resulting stresses and deflections may be found. If the same

load were applied dynamically, the time-varying deflections would

91

92

involve accelerations, and by D'Alembert's principle, these would

engender inertia forces resisting the motion as shown in Figure 5.1b.

Thus the beam may be considered to be subjected to two loadings: the

external loading P(t) that causes the motion and the inertia forces

f (t) that resist its acceleration. The magnitude of these forces

depends upon the rate of loading of the structure and on its stiff

ness and mass characteristics. The basic difficulty of a dynamic

analysis results from the fact that the deflections which lead to

the development of the inertia forces are themselves influenced by

these inertia forces. To break this closed cycle of cause and effect,

the problem must be formulated in terms of differential equations—

expressing the inertia forces in terms of the time derivatives of the

structural deflections.

Equations of Motion

In dynamics, the counterpart of the minimum potential energy

theorem is Hamilton's principle (33). The equation of motion, or

dynamic equilibrium, for a structure subjected to a time-dependent

load {R(t)}, is given by, (34, 35, 36)

[M]{r} + [C]{r} -»- [K]{r} = {R(t)} (5.1)

where

[M], [C], and [K] are the mass, damping and stiffness matrices

respectively

{r} is the displacement of the structure with respect to the

ground or base

93

{r} and (r) the first and second derivative, respectively, of

the displacement of the structure with respect to time.

In the case of free vibrations where R(t) - 0 and an undamped

response is considered. Equation 5.1 reduces to the following form

[M]{r} + [K]{r} - {0}. (5.2)

The response of a linear elastic structure is simplified by the fact

that the motion is simple harmonic, i.e.

{r } - {R }e ''nt. (5.3) n n

Hence, we have,

{r } - - u)^ {r } n - 1, 2, 3, ... N (5.4) n n n

n where N is the total number of degrees of freedom of the system, co

are the undamped natural frequencies, and {r } are the associated

vibration modes. The equation of motion in free vibrations may now

be written in the following form

- u) [M]{R } + [K]{R } = 0. (5.5) n n n

The equation is in the form

[A]{x} = X[B]{x} (5.6)

which represents the classic linear algebraic eigenvalue problem in

applied mathematics.

i

Equation 5.5 may generally be solved by any one of the following

two processes. The first approach is to directly integrate the equa-

94

tion of motion by a numerical process and the second method is to

superimpose individual modal responses. The numerical method is used

in this investigation.

Stiffness Matrix

Stiffness matrix is evaluated as explained in Chapter II.

Mass Matrices

There are two possible forms that the mass matrix of the struc

ture may assume, the "consistent mass" matrix and the "lumped mass"

matrix.

The components of the consistent mass matrix (37) represent con

sistent inertial forces at the nodal points and are associated with

the assumed displacement functions. They can be evaluated, using

energy concepts, in an analogous manner to that of the distributed

loads discussed in Chapter II. This approach leads to a mass matrix

with coupling of the degrees of freedom associated with each finite

element. The construction of this matrix is similar to standard

stiffness matrix synthesis techniques. Mass coefficients are com

puted for individual elements of the structure and are combined by sim

ple superposition to obtain a mass matrix for the complete structural

system. The consistent mass method gives an upper bound to the fre

quencies of free vibration.

In the lumped mass technique, the mass of the structure is assumed

"lumped" at the nodal points. This approach has the following two

main advantages:

(1) The matrix is diagonal.

95

(2) It is easy to form. For a rectangular element with 4 nodes

at the corners, the mass at each node - ~- (5.7)

where p - Density of the material of the element

A - Element Area

h - Element Thickness.

Felippa (24), as well as several other investigators, studying

finite element methods for dynamic analysis, have shown that the

results obtainable using consistent mass systems are not so accurate

as those of lumped mass approaches.

In this investigation, the lumped mass approach is used.

Damping Matrix

One of the difficulties encountered in forced vibration analysis

of structures, (Equation 5.1) is that of estimating accurately the

amount of damping in the structure to be erected. Model tests can

at best give an approximate value of the damping.

In normal mode analyses, it is assumed that damping is such that

it uncouples the normal modes. This is equivalent to assuming that

the damping matrix [C] is of the general form (38)

N-1

I Jl-0

[C] = [M] Y a ([M]"^K]) (5.8)

in which a » arbitrary constants. If only the first two terms of

this general expression are included, then, the simplified damping

matrix takes the form

[C] = a^ [M] + a^ [K]. (5.9)

96

It is usually assumed that the damping matrix is proportional

to either the mass matrix or the stiffness matrix or a linear com

bination of both (39). A damping matrix proportional to the stiff

ness matrix will result in increasing damping in higher modes. Since

this is found to be the case in test results on some prototypes,

this approach of using

[C] - a[K] (5.10)

is adopted in this investigation. Two different values of a, viz.,

0.002 and 0.005, to correspond to the realistic damping existing in

shear walls, have been used herein.

Response of Structures to Earthquake

X

UJ u z UJ

UJ

^t'Ttt

Relotive motion

^ KWKXXKW^^XWV

rg = grourtd motion

FIGURE 5.2 EARTHQUAKE EXCITATION OF BUILDING FRAME

The dynamic problem of Figure 5.1(b) d i f f e r s from the earthquake

e x c i t a t i o n problem in tha t a dynamic load i s shown applied d i r e c t l y

97

to the structure. In the earthquake problem, the excitation is pro

vided by the motion rg(t), introduced at the support of the struc

ture, as shown in Figure 5.2: there is no external loading (i.e.,

P - 0).

The inertia force depends on the total acceleration of the mass

r^, which includes a component relative to the base r plus the accel

eration of the base r g

i.e. r - r + r^. (5.11)

Thus, the inertia force - [M]{r } (5.12)

- [M]{r} + [M]{r }. (5.13)

Hence, writing the equation of motion, i.e.. Equation 5.1, (for the

case of no external loading), we have

[M]{r} + [M]{r } + [C]{r} + [K]{r} = 0 (5.14)

or.

[M]{r} + [C]{r} + [K]{r} - -[M]{r }. (5.15) g

Equation 5.15 is now in the same form as Equation 5.1, with

{R(t)} - -[M]{'rg}. (5.16)

The item -[M]{r } represents an effective load resulting from

the ground motion, and thus, it is apparent that the earthquake input

is exactly equivalent to a dynamic load equal to the product of the

ground acceleration and mass of the structure. The negative sign

98

merely indicates that the effective load opposes the direction of

ground acceleration.

Step-by-Step Formulation of Equation and Solution

Vibration Problems

The Newmark-6 formulation (40) is used for solving Equation 5.1.

In this formulation, Newmark introduces a parameter 0 which can be

changed to suit the requirements of the problem at hand. The dif

ferential equation 5.1 is solved step by step with the aid of the

following equations.

V l " n - („ + Vl> ".17)

2

n+i n n z n n+i n

The net effect of 3 is to change the form of the variation of

acceleration during the time interval At. By letting 6 = 0 , the

acceleration is constant and equal to r during each time interval.

If g = 1/8, the acceleration is constant at r from the beginning

and then changes to r ., at the middle of the interval. The value

° n+1 6 = 1/6 assumes that the acceleration changes linearly from r to

r ,; 6 = 1/4 corresponds to assuming that the acceleration remains n+1

constant at an average value of (r + r ) / 2 . From the point of

view of stability, it has been shown that, if the damping is uniform,

and positive (or zero), the Newmark-6 method is stable for any size

of time step, provided 6 I 1/^ (^D• In the procedure followed here

in, therefore, a value of 6 » 1/A is used.

99

Substituting for r^^^ and r^^^ from Equations 5.17 and 5.l8,

into Equation 5.1, and using a value for 6-1/4, we have

(5.19)

+ IK] ((r„} > .t{r^} + Al! f-, , L^ <^^^^,) . (^(^),

Rearranging terms, we have

([M] + ^ [C] + ^ (K]) {r^^j} -

(5.20)

{R(t)) - (C]({r„) + M c;^}) . [K]({r_ } + it {-r^} + {r^}).

The following substitutions are now made

[GJ - [M] + ^ [C] + ^ [K] (5.21)

(b) - (r^) +A|{r^} (522)

2 {d} . {r^} + At {r^} + {r^}. (5.23)

From Equation 5.10,[C] - a[K] in which a < < 1. Equation 5.20 can

now be written as

[G]{r _ } - {R(t)} - [C]{b} - [K]{d}. (5.24)

If the stiffness and mass matrices are evaluated, the damping

matrix (equal to a times the stiffness matrix) and the [G] matrix

become known. The right hand side of Equation 5.24 is known if all

the information at the beginning of a time interval is available.

It is therefore only necessary to know the initial conditions for the

100

start of the step-by-step analysis. The initial displacements, veloc

ities and accelerations are obviously zero at t » 0 for the case of

forced and free vibration analysis. Hence Equation 5.24 can be solved

for {' .,) which is the acceleration vector at the end of the time

interval. Velocities and displacements at the end of the time in

terval can then be evaluated using Equations 5.17 and 5.18. The main

steps in the solution are as follows:

1. Form stiffness matrix [K] and mass matrix [M].

2. Form [G]

2 [G] - [M] + (a ~ + ) [K]. (5.25)

3. Triangularise [G] for Guass-Elimination procedure.

4. For each time increment

(i) Calculate force vector as

{f} - {R(t)} - a[K]{b} - [K]{d} (5.26)

(ii) Solve for {r } by forward elimination and back-

substitution on {f} in the Guass-Elimination procedure

(iii) Evaluate {r .••} and {r ^} from Equations 5.17 and

5.18

(iv) Evaluate stresses from {r ^ }

(v) Scan stresses and displacements for maxima during the

vibration.

5. Repeat step 4 above for the next time interval.

In the computer program developed for vibration analysis, based

on the preceding formulation, the stiffness, mass and the [G] matrices

101

are stored on disk. Displacement, velocity, acceleration and the

stresses at every time step are calculated.

Earthquake Response

For evaluating the response of the structure to earthquake, a

procedure, identical to the vibration problem discussed above, is

used. The one and only difference between the two cases lies in the

evaluation of the force vector {f} of Equation 5.26. The value of

R(t) from Equation 5.16 is now substituted into Equation 5.26 to get

the force vector for the case of earthquake response

{f} = -[M]{r } -a[K]{b} - [K]{d} (5.27)

Beam Problem

Example Problems

w -Ji^'U-

M

E= l i lO* Ibe/ft^ Denelty « I 5 0 l b t / f t '

FIGURE 5.3 SIMPLE BEAM FOR VIBRATION ANALYSIS

Example 1

The dynamic response of the beam shown in Figure 5.3 is evaluated

The forcing function was a central concentrated vertical force of

102

1000 lbs. The values of the maximum displacement at every time step

are shown in Table 5.1; this is plotted in Figure 5.4. The funda

mental period of vibration of the system is calculated from the

equation, (36)

T « (5.28)

i-e-» T o 'o Fm (5.29)

V m

(5.30)

The value of the period was calculated as 0.952437 sec/cycle.

The time interval chosen for the step-by-step analysis should

be at least one-sixth to one-tenth the fundamental period (36,42).

A value of time interval. At = 0.1 sec, was used in this analysis.

From Figure 5.4 the fundamental period is found to be 0.93 sec/cycle,

slightly lower than the theoretical value. This could be due to the

finite element idealization. In the static case, there is an error

of about 15% for this idealization; however, improved values could

be obtained using a refined mesh idealization.

Example 2

The response of the beam, shown in Figure 5.3, when subjected

to the acceleration history of the Olympia earthquake of April 13,

1949, is evaluated. The acceleration history of the Olympia earth

quake for 1.0 second, is tabulated in Table 5.2. The displacement

at the central section of the beam for every time step, at time in

tervals of 0.01 seconds, is shown in Table 5.3. This is plotted in

103

Table 5.1

Vibration Analysis of Beam

Ins tant of Time (seconds)

0 .0 0 .05 0 .10 0 .15 0 .20 0 .25 0 .30 0 .35 0 .40 0 .45 0 .50 0 .55 0 .60 0 .65 0 .70 0 .75 0 .80 0 .85 0.90 0 .95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90

Maximum Displacement ( f e e t )

0 .0 0.0101 0.0380 0.0829 0.145 0.220 0 .293 0.356 0.407 0.436 0.436 0.414 0.372 0.308 0.235 0.164 0.0973 0.0436 0.0155 0.0103 0.0265 0.0702 0.132 0 .201 0.275 0.345 0.397 0.428 0.440 0.424 0.381 0.325 0.255 0.179 0.110 0.0573 0.0196 0.0064 0.0229

104

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Table 5.3

106

Instant of time / V

(sec.)

0.0

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33

Simply Supp

Displacement of central section

S 0» V

(ft.)

0.0 0.0000464 0.0002182 0.0005222 0.0009129 0.0013630 0.0018548 0.0023349 0.0026825 0.0027873 0.0026514 0.0023476 0.0019348 0.0014939 0.0011818 0.0011761 0.0015943 0.0024780 0.0037820 0.0053697 0.0070578 0.0086505 0.0099832 0.010987 0.011686 0.012173 0.012603 0.013153 0.014000 0.015236 0.016735 0.018212 0.019444 0.020301

»orted Beam - Earthquake Analysis

Instant of time (sec.)

0.34 0.35 0.36 0.37 0.38 0.39 0.40 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.50 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59 0.60 0.61 0.62 0.63 0.64 0.65 0.66 0.67

Displacement of central section (ft.)

0.020643 0.020333 0.019407 0.018145 0.016911 0.015961 0.015427 0.015315 0.015519 0.015866 0.016127 0.016044 0.015404 0.014178 0.012629 0.011150 0.0099786 0.0091586 0.0086954 0.0085892 0.0086942 0.0086633 0.0080945 0.0067477 0.0046790 0.0021975

-0.0003184 -0.0025514 -0.0042359 -0.0051658 -0.0052821 -0.0047812 -0.0041566 -0.0039931


0.68 0.69 0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.80 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00 1.01

Displacement of central section (ft.)

-0.0046380 -0.0061124

^^ • ^m ^0 ^* u k ^m T

-0.0082113 ** • ^m \^ \^ mm ^m «^

-0.010601 ** • ^^ b ^ ^ \0 ^* b

-0.012913 -0.014757 -0.015748 -0.015705 -0.014848 -0.013710 -0.012875 -0.012760 -0.013530 -0.015125 -0.017317 -0.019704 -0.021704 -0.022751 -0.022548 -0.021172 -0.019070 -0.016820 -0.014861 -0.013528 -0.013100 -0.013572 -0.014532 -0.015354 -0.015442 -0.014455 -0.012365 -0.009413 -0.006069 -0.002905

107

OavmH^OQ iN3W3D»ndSlO tVVLU-iJ OHVM 11 —

108

Figure 5 .5 .

Shear Wall Problem

•OT I

CM

C»J

E=5jO»K)*lb«/ft* DwMity I90lbt/ft '

2 0 - 0 2 0 - 0 20"-0'

FIGURE 5.6 SHEAR WALL FOR DYNAMIC ANALYSS

The response of the 6-story shear wall shown in Figure 5.6 was

studied. To obtain the response a suitable time interval is to be

chosen. This was achieved by a trial and error process. Two values of

the time interval. At = 0.01 and 0.015 were used in the analysis.

Two different forcing functions were used, viz., (i) a lateral uni

formly distributed load of 1000 lbs. per ft. for the entire height

of the wall and (ii) a concentrated lateral load of 1000 lbs. at the

109

top of the wall. For each case, the analysis was performed by dis

cretizing the continuum using the 8-degree and 16-degree freedom

elements. The results of the analysis are given in Tables 5.4 and

5.5. The maximum displacement of the wall plotted against the time

step for cases (i) and (ii) for the 8-degree freedom element and 16-

degree freedom element discretizations are shown in Figures 5.7 through

5.10.

From these figures, it may be noted that different responses

result when the 8-degree and the 16-degree freedom elements are used.

The fundamental period of the shear wall is found to be 0.14 seconds

when the 8-degree freedom elements are used and 0.17 seconds when

the 16-degree freedom elements are used. This difference is attributed

to the inability of the 8-degree freedom elements to adequately rep

resent the structural interaction between the stiff walls and the

relatively flexible lintel beams.

Effect of Damping

Table 5.6 shows the dynamic response of the shear wall shown

in Figure 5.6 to a concentrated lateral load of 1000 lbs. at the top

of the wall and with the damping coefficient a having values of 0.002

and 0.005. The results plotted are in Figures 5.11 and 5.12.

Effect of Thickness of Lintel Beam on Fundamental Period of Shear Wall

The 10-story shear wall shown in Figure 5.13 was analysed for

different lintel beam thicknesses of 0.18", 0.36", 0.54", 0.72",

0.90", 1.08", 1.26" and 1.44". The dimensions shown correspond to

that of a model shear wall of plexiglass (43). The period of vibra-

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Table 5 .6

Vibration Analysia of Shear Wall with Damping

Concentrated lateral load of 1000 Iba. applied at top of left wall; Maximum static deflection - 0.00017376

Damping Coefficient a - 0.002

Instant of Time (sec.)

0.0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

Maximum displacement (ft.)

0.0 0.0000347 0.0001131 0.0002079 0.0002883 0.0003014 0.0002452 0.0001581 0.0000812 0.0000581 0.0000996

Damping Coefficient a » 0.005

Instant of Time (sec.)

0.0 0.015 0.030 0.045 0.060 0.075 0.090 0.105 0.120 0.135 0.150 0.165 0.180 0.195

Maximum displacement (ft.)

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120

tion as a function of the depth of the connecting beams is tabulated

in Table 5.7 and is plotted in Figure 5.14. An important observation

from Figure 5.14 is that by increasing the depth of the connecting

beams, the fundamental period decreases up to a certain depth of the

lintel beam. This value of the depth of the beam, for this model shear

wall, is found to be 0.72", i.e., 40% of the height of the story.

Increasing the thickness of connecting beams beyond this value did

not show any change in the fundamental period. This is in contrast to

the static case, where an increase in thickness of lintel beams in

creases the stiffness of the structure and hence decreases the maxi

mum displacement. An explanation for the fundamental period remaining

constant for thicknesses of lintel beam 0.72" and above may be found

in the fact that, even though the stiffness of the structure increases,

the mass of the structure also increases and offsets any increase,

after a certain lintel beam thickness, in the stiffness.

Effect of Floor Loads in the Fundamental Period

More often than not, the shear walls act monolithically with

floor slabs and cross-frames. Analysis of isolated shear walls,

without taking into account the additional mass vibrating along with

the walls, therefore, does not give a complete picture of the vibra

tion characteristics of the wall. The additional mass or the effec

tive width of the floor slabs and cross frames vibrating with the

shear wall have not yet been satisfactorily evaluated. These depend

upon a variety of factors such as the type of construction, the stiff

ness of the connection, the spacing of the shear walls, the utiliza-

121

Table 5.7

Variation of Fundamental Period with Thickness of Lintel Beam

Thickness of lintel beam (inches)

0.18 0.36 0.54 0.72 0.90 1.08 1.26 1.44

Fundamental Period of Vibration (seconds)

0.00675 0.00540 0.00500 0.00475 0.00475 0.00475 0.00475 0.00475

Table 5.8

Variation of Fundamental Period with Floor Mass Factor, y

Floor loads vibrating with shear wall.

represented by constant y

1.0 10.2 19.5 38.0 65.8 93.5 140.0

Fundamental Period of Vibration (seconds)

0.0054 0.0152 0.0205 0.0290 0.0410 0.0510 0.0640

122

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123

tion of the structure, etc. For purposes of illustrating the effect

of the additional mass on the fundamental period, a parameter Y is

chosen such that the mass at the nodes of each story level is equal

to y times the masses at those nodes of the isolated shear wall.

The results of analysis are given in Table 5.8. A plot of fundamental

period versus Y. for the model shear wall of Figure 5.13, is shown

in Figure 5.15. From this figure, it follows that fundamental period

increases with y.

Earthquake Response

The response of the shear wall, shown in Figure 5.6, when sub

jected to the earthquake acceleration history of the Olympia earth

quake of April 13, 1949, is evaluated. The displacement of the left

extreme point of the top story and the stress at base at left end of

the wall for every time step up to a time of 0.32 sec, are shown

in Table 5.9. This is plotted in Figures 5.16 and 5.17. The axial

force and moments in the lintel beams were calculated at each time

step; these values were quite small during this interval and hence

they are not tabulated.

General Observations

From a study of the dynamic behavior of shear walls, the fol

lowing general observations are made.

(1) The discretizations using the 8-degree freedom elements

and 16-degree freedom elements gave different values of the funda

mental period of vibration of a 6-story shear wall, viz., 0.14 seconds

per cycle and 0.17 seconds per cycle. The deductive reasoning

124

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125

Table 5.9

Earthquake Analysis of Shear Wall

Serial No.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34


0.0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33

Maximum displacement

(ft.)

0.0 -0.0000101 -0.0000464 -0.0000981 -0.000122 -0.000076 0.0000541 0.000246 0.000436 0.000564 0.000614 0.000590 0.000503 0.000367 0.000163 -0.000133 -0.000503 -0.000871 -0.00113 -0.00118 -0.000991 -0.000554 0.0000997 0.000853 0.00151 0.00190 0.00203 0.00196 0.00168 0.00111 0.000209 -0.000906 -0.00205 -0.00304

Stress at left extreme point of left wall at base of wall (kips/ sq. ft.) + ve denotes tension

0.0 0.186 0 .491 0 .121

- 0 . 6 5 8 - 0 . 4 2 5

0.476 0.628 0.499 1.345 2.514 2 .431 1.562 1.553 1.723 0.299

-1 .760 - 2 . 7 2 1 -3 .056 - 3 . 7 2 8 - 3 . 8 8 1 -2 .026

0.878 3.109 4.910 6.864 7.824 7.255 5.783 3.675 0.317

-4 .492 -8 .227

-10 .587

126

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employed in the case of static analysis regarding the inability of

the 8-degree freedom analysis to correctly represent the structural

interaction of the stiff walls and flexible lintel beams, therefore,

seems to apply to dynamic analysis as well.

(2) Up to a certain depth of the lintel beam, increasing the depth

of the lintel beam decreases the fundamental period of vibration of

the shear wall. Further increase beyond this depth does not cause

change in the fundamental period. For the model shear wall problem

investigated, this depth is found to be 0.40 times the height of the

story.

(3) The natural period of vibration increases when the mass of

floor slabs is lumped with those of the isolated shear wall.

CHAPTER VI

CONCLUSIONS AND RECOMMENDATIONS

Conclusions

The application of higher order finite elements to the analysis

of shear wall structures for the effects of static and dynamic loading

has been described in this report. Based on this study, the following

conclusions are advanced.

(1) Users of the finite element method are cautioned about the

presence of high round-off errors in finite element solutions, espe

cially if the number of digits carried in the computing machine is

very small (such as the 7-digit carrying machine).

(2) The 16-degree freedom element gives a better representation

of the true stress and displacement than would be obtained with the

same number of nodes using a much finer subdivision into the 8-degree

freedom elements. For usual discretization patterns, the 16-degree

freedom element, with 15 digits carried in the calculations, gave

errors of only 2% or less in the computed maximum displacement of a

simple beam problem.

(3) A discretization of the shear wall structure using the 6-

degree or 8-degree freedom elements does not give a good representa

tion of the structural behavior of shear walls; the discretization

employing the 12-degree and 16-degree freedom elements does give

accurate results for the same problem.

(4) The Generalised Cantilever moment distribution, used by

Gurfinkel to solve shear wall problems, highly overestimates the

129

130

maximum deflection and stresses in the wall and the moments in the

lintel beams.

(5) Where two stiff walls are connected by flexible lintel

beams, an analysis using the 16-degree freedom elements requires at

least 2 elements in the lintel beams for a correct representation of

the structural behavior of the shear wall; the number of elements on

the wall does not appreciably affect the behavior.

(6) For the material properties used herein the differential

heating of the walls through 100** F produces 17 times as much dis

placement as a uniform heating of the walls through 100° F. Addi

tional reinforcement must be provided to resist the thermal stresses.

(7) Non-dimensional curves involving displacement and stresses

in walls and moment in lintel beams are presented. The influence of

some of the important parameters in the design of shear walls is,

for the first time, made available to the engineer.

(8) Up to a certain depth of the lintel beam, increasing the

depth of lintel beams decreases the fundamental period of vibration

of the shear wall. Further increase beyond this depth shows no

change on the fundamental period.

(9) The fundamental period of vibration of the isolated shear

vail structure increases when the mass of floor slabs is lumped on

to it.

Recommendations for Further Study

Based on the experience gained from this investigation, the

following recommendations for further study are made.

131

(1) A non-linear analysis taking into account the material

properties of concrete may be performed for the shear wall problem.

The results from this analysis may be compared with the linear,

isotropic analysis reported herein.

(2) The lower modes of vibration of the shear wall may be

evaluated by solving the general eigenvalue problem.

LIST OF REFERENCES

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2. Chitty, L., and Wan, W. Y., "Tall Building Structures Under Wind Load," Proceedings, 7th International Congress for Applied Mechanics, V. 1, 1948, pp. 254-268.

3. Beck, Hubert, "Contribution to the Analysis of Coupled Shear Walls," ACI Journal, Proceedings, V. 59, August, 1962, pp. 1055-1070.

4. Rosman, Riko, "Approximate Analysis of Shear Walls Subject to Lateral Loads," ACI Journal, Proceedings, V. Gl, June, 1964, pp. 717-733.

5. Mcleod, I. A., "Connected Shear Walls of Unequal Width," Journal, American Concrete Institute, Detroit, Michigan, Vol. 67, May, 1970, pp. 408-412.

6. Coull, Alexander and Choudhury, J. R., "Stresses and Deflections in Coupled Shear Walls," ACI Journal, Proceedings V. 64, February, 1967, pp. 65-72.

7. Coull, Alexander and Choudhury, J. R., "Analysis of Coupled Shear Walls," ACI Journal, Proceedings, V. 64, September, 1967, pp. 587-593.

8. Barnard, P. R. and Schwaighofer, J., "The Interaction of Shear Walls Connected Solely through Slabs," Proceedings, Symposium on Tall Buildings (University of Southampton, April, 1966), Pergamon Press, Oxford, England, 1967, pp. 157-180.

9. Gould, Phillip L., "Inter-action of Shear Wall-Frame Systems in Multi-Story Buildings," ACI Journal, Proceedings V. 62, January, 1965, pp. 45-70.

10. Khan, Fazlur R., and Sbarounis, John A., "Interaction of Shear Walls and Frames," Journal of the Str. Div., ASCE, Proceedings, June, 1964, pp. 285-335.

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132

133

13. Girija Vallabhan, C. V., "Analysis of Shear Walls by Finite Element Method," Proceedings of the Symposium on "Application of Finite Element Method in Civil Engineering," Vanderbilt University, Nashville, Tennessee, 1969.

14. Oakberg, R. G., and Weaver, W. Jr., "Analysis of Frames with Shear Walls by Finite Elements," Proceedings of the Symposium on "Application of Finite Element Methods in Civil Engineering," Vanderbilt University, Nashville, Tennessee, 1969.

15. Macleod, I. A., "New Rectangular Finite Element for Shear Wall Analysis," Journal, Structural Division, ASCE, Vol. 90, March, 1969, pp. 309-409.

16. Sarrazin, M. A., "A Higher Order Rectangular Finite Element and Its Application to the Analysis of Shear Walls," Submitted to the Massachusetts Institute of Technology, 1968.

17. Franklin, H. A., "Nonlinear Analysis of Reinforced Concrete Frames and Panels," Office of Research Services, University of California, Berkeley, California, March, 1970.

18. Langefors, B. , "Analysis of Elastic Structures by Matrix Transformation, with Special Regard to Semi-monocoque Structures," Journal Aeronautical Science, 19, No. 7 (1952).

19. Argyris, J. H. , "Energy Theorems and Structural Analysis," Butterworths Scientific Publications, London, 1960 (Reprinted from Aircraft Engineering, 1954-55).

20. Plan, T. H. H., "Lecture Notes for the Special Summer Program 1.59S on Finite Element Methods in Solid Mechanics," M.I.T. June, 1968.

21. Turner, M. L. , Clough, R. W. , Martin, H. C , and Topp, L. V., "Stiffness and Deflection Analysis of Complex Structures," Journal of Aero. Sciences, Vol. 23, September, 1956.

22. Clough, R. W., "The Finite Element Method in Structural Mechanics," Chapter 7 of "Stress Analysis", edited by 0. C. Zienkiewicz and G. S. Hollister, John Wiley and Sons, 1965.

23. Wilson, E. L., "Finite Element Analysis of Two-Dimensional Structures," SESM Report No. 63-2, University of California, Berkeley, June, 1963.

24. Felippa, C. A., "Refined Finite Element Analysis of Linear and Non-linear Two-Dimensional Structures," SESM Report, No. 66-22, University of California, Berkeley, October, 1966.

134

25. Argyris, J. H., "Triangular Elements with Linearly Varying Strain for the Matrix Displacement Method," Journal Royal Aeronautical Society Technical Note, 69, pp. 711-713, October, 1965.

26. Zienkiewiez, 0. C , and Cheung, Y. K., The Finite Element Method in Structural and Continuum Mechanics. McGraw-Hill Publishing Company, London, 1967.

27. Ergatoudis, J., "Quadrilateral elements in Plane Analysis," M.Sc. Thesis, University of Wales, Swansea, 1966.

28. Timoshenko, S., and Goodier, J. H., Theory of Elasticity, 2nd ed. McGraw Hill Book Company, Inc., New York, 1951.

29. Crandall, Stephen H., Engineering Analysis. McGraw Hill Book Company, Inc., New York, 1956.

30. Norris, C. H., and Wilbur, J. B., Elementary Structural Analysis, McGraw Hill Book Company, Inc., New York, 1960.

31. Hildebrand, F. B., Introduction to Numerical Analysis« McGraw Hill Book Company, Inc., New York, 1956.

32. Veubeke, Fraejis De B.,"Displacement and Equilibrium Models in the Finite Element Method,"Chapter 9 of Stress Analysis, ed. 0. C. Zienkiewiez and G. S. Hollister, John Wiley and Sons, Inc., 1965.

33. Fung, Y. C., Foundations of Solid Mechanics, Prentice-Hall, Inc., Englewood Cliffs, New Jersey.

34. Carr, Athol J., "A Refined Finite Element Analysis of thin Shell Structures including Dynamic loadings," Report No. SESM 67-9, University of California, Berkeley, California.

35. Wiegel, Robert L., Earthquake Engineering, Prentice-Hall, Inc. Englewood Cliffs, New Jersey.

36. Biggs, John M. , Introduction to Structural Dynamics, McGraw-Hill Book Company, New York, 1964.

37. Archer, J. S., "Consistent Mass Matrix for Distributed Mass Systems," Journal of the Structural Division, ASCE, Vol. 89, August, 1963, pp. 161-173.

38. Tahbildar, Umesh Chandra and Tottenham, Hugh, "Earthquake Response of Arch Dams," Journal of the Structural Division, ASCE, November, 1970, pp. 2321-2336.

39. Przemieniecki, J. S., Theory of Matrix Structural Analysis, McGraw-Hill Book Company, New York, 1968.

135

40. Wang, Ping-Chun, Numerical and Matrix Method in Structural Mechanics, John Wiley and Sons, Inc., New York, 1966.

41. McNeal, R. H., and McCormick, C. W., "The NASTRAN Computer Program for Structural Analysis," Society of Automotive Engineers, Inc., New York.

42. Blume, John A., Newmark, Nathan M., and Corning, Leo H., Design of Multi-story Reinforced Concrete Buildings for Earthquake Motions, published by Portland Cement Association, 33 West Grand Avenue, Chicago 10, Illinois.

43. Tso, War K., and Chan, Ho-Bong, "Dynamic Analysis of Plane Coupled Shear Walls," Journal of the Engineering Mechanics Division, ASCE, pp. 33-48, February, 1971.

APPENDIX

A. 6 d.o.f., 8 d.o.f. and 12 d.o.f. Finite Elements,

B. Results of Shear Wall Analysis.

136

APPENDIX A

The 6-degree Freedom Element

Figure 7.1

The displacement functions used to represent the displacement

at any point within and on the boundary of the element are

u = a. + a-x + a-y

V = a, + a_x + a,y. 4 5 6

(7.1)

(7.2)

The [B] matrix of Equation 2.6 becomes

[B] =

1

0

1

0

1

0

) 0

j ^j 0

0

0

1

0

1

0

1

0 0

j j 0

(7.3)

The [Q] matrix of Equation 2.10 reduces to

137

138

[Q]

0

0 0 0 0 0 1

0 0 1 0 1 »J (7.4)

The [D] matrix of Equation 2.14 remains the same. The stiffness

matrix, [k] of Equation 2.29a, for this element, is evaluated in the

machine. The [B]"^ andjQV<i(vol) matrices, which are input into

the computing machine, are shown in Equations (7.5) and (7.7)

[B] -1

2 AREA

^^jVVj> °

(x^Yj-x^y^)

(x^Yj-x.y^)

(yj-y,)

(y,-y,)

(y,-yj)

0

0

(yj-y,)

(y,"y,)

(y.-yj)

(V'^j)

(x^-x^)

(x -x^)

0

0

0

0

0

0

(V^j)

(x^-x^)

(x.-x^)J

(7.5)

in which

AREA = Area of the triangle ijk and is given by

iS'i^^j" y j (7.6)

/

.39

Q DQd(vol) - AREA x THICKNESS x

0

1-v'

0

Ev

1-v'

0

0

E 2(l+v)

0

E 2(l+v)

0

0

E 2(l+v)

0

E 2(l+v)

0

Ev

1-v

0

1-v'

(7.7)

The 8-degree Freedom Rectangular Element

Figure 7.2


within and on the boundary of the element are

u = a + a x + a.y + a.xy (7.8)

V » a^ + a,x + a^y + o'gXy (7.9)

140

[B]

A (

1

0

1

0

1

0

1

0

JL cqi

" i

0

^J 0

\

0

^1

0

i iac io i

^1

0

^j

0

^k

0

h 0

a i.b t

" i ^ i

0

' ' j^'j

0

W 0

x^y^

0

>ecomes

0

1

0

1

0

1

0

1

0

\

0

" j

0

\

0

^1

0

^ i

0

' ' j

0

^k

0

^1

0

" l ^ l

0

"j^ ' j

0

"k^k

0

" l ^ l

(7 .10)

The [Ql m a t r i x of Equat ion 2.10 reduces to

[Ql

O l O y O O O o "

O O O O O O l x

O O l x O l O y

(7 .11)

The [D] m a t r i x of Equat ion 2.14 remains the same. The s t i f f

ness m a t r i x , [k] of Equat ion 2.29a, for t h i s e lement , i s shown in

Equat ion 7 .12 .

The 12-degree Freedom T r i a n g u l a r Element

• • ,

Figure 7.3

141

Ui I

r

' " 0 0

1

f*.

> 1

r-i

.cl^ 1

• >

*n 1

1-4 0 0

^ • >

r-i Xx'

• •A ia|>c

1

1 *> . 0 | v 0

/^ t - l i

> «n

0 0

/^ 1

^ «»/ {•A K '

« 8 M

+ | « ji\ri

1

/^ p

+ -^ N . ^

00

« J9 l fi 1

^ > 1

r-i

J5 1

1 GO

1

1

*^ > 1-1 ^ x

1 <o JCM

. O M

• H

i CO s . '

0 0

/ • ^

> 1

s.^

|<a .OJvO

1

1-° iQJvO

/^s •>

r% 1 0 0

N 1 ^

l" <o|«^ +

^ • ^

^ 1

. -1

»^ 1 (0

Xi |vO

r>

00

-

<"N

> 1

1

1(0

1 + 00 P H |

/ • ^

1 ri

ICN ( B M

1

i « .o|«

1

> + r-t

00

1

'• •>

1

^ 1-

«|«o

+ |<g

J3|<*t

«|R 1

/^ ? I

.-• (0

x> CM i H

•> 1

i H 00

!•<) («|>C

1

1 ri

1 (B fM

. 0 | i - t

1

^-s •>

+ p-l 0 0

l'<a • In + /"N

•>

1 p-l

^^ 1 •« ^ | N O

> 1

«-l

^^

0 0

/•-s •>

1 r^ • ^

<M

+ 1(0

*0|C0

1

•>

+ r-l 00

1

<!-» •>

r-i s.^

!•<> (0|vO

+ 1 <Q

.o|c^

)

IO|sO

+ ^^ ? 1

r-l

1

^^ 1 >

0 0

(0|<*>

+ > 1

t - l «—' 1(0

X>|vO

i H 0 0

^s

">

(o|S

+ A9.

1 (0|co

+ ^ s

1

1 (0 .o|sO

^|m

ot: H

w

W5

4J

M (0 M

3 § iH a 9 « O O

M

« t - l 00

c (0 4J

u <u u

IM

o (A (U

•o TH (A

• • J3

« rt

oo c 10 i J

u 01

u VM

o (0 (0 (U

c it u • H

£ i J

,, 4J

142


at any point within and on the boundary of the element are

2 2 u • a, -•- a-X + a.y + a.x + a_xy + a,y (7.13)

1 2 3 4 5 6'

V - a- + a^x + a^y -•- a.^x + a..xy + a <,y 7 8 10 11 12-

(7.14)

The [B] matrix of Equation 2.6 becomes

[B]

1

0

1

0

1

0

1

0

1

0

1

0

__

\

0

' ' l

0

" j

0

X m

0

\

0

X n

0

^ i

0

^ 1

0

^ j

0

^m

0

^k

0

^ n

0

2

0

2

0

2 X .

J 0

2 X

m

0

2 \

0

2 X

n

0

x^y^

0

x^y^

0

" j ^ j

0

X y m- m

0

Vk 0

X y n"^n

0

2 ^ i

0

2 ^ 1

0

2

0

2 ^m

0

2 ^k

0

2 ^n

0

0

1

0

1

0

1

0

1

0

1

0

1

0

\

0

\

0

'^j

0

X m

0

^k

0

X n

0

^ i

0

^ 1

0

^ j

0

^m

0

^k

0

^n

0

2 \

0

2 \

0

2

" j

0

2 X

m 0

2 ^k

0

2 X

n

0

x^y^

0

^1^1

0

x j y .

0

X y m- m

0

Vk 0

X y n-^n

• "

0

2 ^ i

0

2 ^ 1

0

2

^ j

0

2 ^m

0

2 ^k

0

2 ^ n ^

The [Q] matrix of Equation 2.10 reduces to

[Q] =

0 1 0 2x y 0

0 0 0 0 0 0

0 0 1 0 X 2y

0 0 0 0 0 0

0 0 1 0 X 2y

0 1 0 2x y 0

(7.15)

(7.16)

The [D] matrix of Equation 2.14 remains the same. The stiffness

143

matrix, [k] of Equation 2.29a, for this element, is not explicitly

evaluated herein. The matrixJQ^DQd(Vol) is calculated. This matrix,

which is shown in Table 7.1, is input to the computer. From the co

ordinates of the nodal points of the element, the [B] matrix is

formed in the computer and its inverse evaluated. The stiffness

matrix for this element is now obtained as the matrix product of

-1 T r T -1

([B] ) ,JQ DQd(Vol), and [B] . Evaluation of this matrix product

is also done in the computing machine. To reduce errors due to

round-off in the calculated value of the stiffness matrix, the fol

lowing steps are taken.

(i) The [B] matrix is rearranged by making the rows 1, 3,

5, 7, 9 and 11 as the first 6 rows and the rows 2, 4, 6, 8, 10 and

12 as the next 6 rows. It is now partitioned into 4 six-by-six sub-

matrices, where the diagonal sub-matrices are the same and the off-

diagonal sub-matrices are null matrices. Thus, the inverse of only

one six-by-six matrix is evaluated. The inverse of the twelve-by-

twelve [B] matrix is now obtained by an appropriate transformation.

(ii) Double precision is used in forming the inverse and the matrix product.

The distributed loads to be used with this element can be eval

uated on similar lines to those of Equations 2.33 through 2.50.

144

V r-i .O

CO H

C 0)

s 0)

iH U VI CO

r-i

o 60

i H 8 O

•o 0) 0) Wl

bb, 0) 0) >» 60

I CN

0) J3

V4

O

o

o

o

o

o

o

Vi

o* o

C/5 PL, O

CN

Vi PC u CN

cn <y

f o >•»'

o

CO on n CN

o

o

o

o

o

o

o

Vi cr T) CN

Vi or

+dR

)S

2

(U » '

to oi >*

CO CO CO

^ or o-> O O O O C M - ^ O O O ^

CO

o o o o o o o o c o

CO CO O O T 3 0 0 0 0 T 3

o o o o o o o

CO t o or pcJ

O O O O CN - ^

CO

PLI

cy as O

>

JQ'^

DQd

"""

o

o

o

o

o

o

o

o

CO

o

o

CO

CN '

4PS

(U (U U JC CO 4J

(0 j c 0) 0) 6 0 r H X D 60 CO O C

U CO > ^ X "H

4-1 'V CO C (U (U CO X ^

CO u X X M (4^

CO O 0) r-l

J3 3 TJ •U 60 •H

C O M U

?; £• M M

11 II

pLi O *

•o c CO

X X

rH • CO >s

J3 rH O 0)

rH > 60 'H

4J (U U

X 0) 4-> a

(0 O 0) 4J Wl

iH •« 9) CO

r-) (U

CO cd Wl CO >s

X X M

II

Oi

c o B

r-i (U

CN 14-1

O

I

(U

i - H O

>

II

>

/ • N

CN

o 1

rH < ^ i i /

II

CO

>> 4-1 •H O

•H 4J CO CO

rH Cd

UH 0

3 rH 3

73 0 £ 1

U

rH CO

•H Wl 0) 4J

s (U X 4J

*u 0

0) 43 U

t4H O

0 •H 4J

s CO

» rH a «o 0 'H CO Wl CO 0 )

•H .U 0 CO

PL. a

1

CJ

APPENDIX B

The results of analysis of the 4-story, 6-story, 8-story and

10-story shear walls are given in Tables 7.2 through 7.10.

145

Table 7.2

146

Serial No.

1 2 3

4 5 6

7 8 9 10

11 12 13 14

15 16 17 18 19

20 21 22 23 24

25 26 27 28 29

Solid s Maximum Maximum

10-Story Shear Wall Analysis

Span of lintel beam (ft.)

s

48.0

40.0

30.0

20.0

10.0

6.0

1.0

hear wall of s strasi in wal dliplactmant

Width of each wall (ft.)

b

6.0

10.0

15.0

20.0

25.0

27.0

29.5

Depth of lintel beam (ft.)

d

4.0 6.0 8.0

4.0 6.0 8.0

2.0 4.0 6.0 8.0

2.0 4.0 6.0 8.0

1.0 2.0 4.0 6.0 8.0

1.0 2.0 4.0 6.0 8.0

1.0 2.0 4.0 6.0 8.0

ame overall dimensions: 1 - 12.00 kips/sq. ft. - 0.00288 ft.

Maximum stress in wall (kips/ sq. ft.)

211.0 116.0 83.0

102.0 67.4 51.7

87.9 55.5 39.6 32.2

47.0 34.4 26.9 23.7

32.7 27.6 20.3 17.6 17.0

24.9 20.8 17.6 16.6 15.9

16.4 16.2 16.1 16.3 17.1

Maximum displacement of wall (ft.)

0.11876 0.03912 0.02084

0.05528 0.02330 0.01390

0.06857 0.02618 0.01338 0.00879

0.02850 0.01229 0.00744 0.00568

0.01701 0.01067 0.00553 0.00437 0.00411

0.01024 0.00645 0.00438 0.00401 0.00378

0.00380 0.00370 0.00359 0.00355 0.00359

147

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iJ

a rJ

a

a I to

ki

tn

t: o u o

o • o

<>M 3

0.0

o • o -»

o 00 <r

§ 

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4J -H <^ a. —1 B a> o

• o O -O

rH C <V S 4J a o C (0 Wl 1-1 4) tw J .£5 •'

CM

00 NO rH

11

1.0

00

• o NO

ON

•r» iH

O • 00

00 r^

CO

r» CM ri

00

• cn r«.

NO

O r4

NO

• CM

r>. CM

«» • r^

O CM

m

NO CM 1—1

o

o fM -*

vO

>» in ro

r r>* 00 CM

-H

a (d V GO

o VO

*!> CM

17

5.9

• iH

o

NO

00 CM

• CM ON (N

cn

r>» O (M

o

• •n CM

ON

14?

00

• 00 ON cn

NO • r s rv

in

o CM

<J

ON CM m

00

<-i r^ <3

m

vO CN <J

<N

§ 0) OQ

«n « r>» (M

. rH O CM

»n

m CN|

CN|

00

cn

00

• 00 CM cn

r ON

»* (N

s • NO

«n

00

5 >» « VO

rH •*

m • r cn cn

m

CM <* CM

rH

CM O in

m rv NO >»

r«i

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m

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200.

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sr CM cn

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m • cn CM

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o

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rv

a\ cn -*

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(N -*

(N

vO «J >»

-*

a (0 0) PQ

cn

sr cn CM

183.

7

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rv CM

NO

ON «»

00

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rH

cn «* CM

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r

r^i VO

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cn

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rv

CM

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ON r-i

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m

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m

r^

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NO •* CM

m

o r4 CM

«n i-H i-H m

rv

iH O m

ON

>* <T m

vO

% 0) CO

.

m »» tn ri

13

1.6

vO

rv O

vO

rH

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NO

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fNl «»

00

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o • cn (N CM

cn • cn o CM

-* in Ov ri

(N

r-i •* CM

vO

fn m «N

vO

rv rv (N

rv

a (0 V (0

o o CM rH

10

7.6

cn rv ON

•n O m

vO

ON cn ri

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S • ON

(M

rv

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ON

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ON •

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cn

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cn •

\r\ o S cn

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rv <r f-(

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ON

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CM O i-H

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a ra 0) BQ

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1.0

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3 (U x>

«M r^

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c c O. rH .

to /

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Nt-i -H a / o 4» ra /

t-> V X x: ^ x/ U -ri / -—^ o- -i/ a >u / o

/ • C3 K O X )

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»n •

318

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r-i NO

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OV rH

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«n •

604

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153

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r") vO CM

CM H rv

m CM . tn

rH

•» cn

m vO

^ CM

m m ON iH

^ O CM (M

cn NO

in

r^

o\ o C-i

m vO

«» CM

in m ri cn

CM rv

NO m

m >?

m

O

n Q)

aa

149

Table 7.4(a)

Parameters in Shear Wall Analysis

Serial No.

1 2 3

4 5 6

7 8 9 10 11

12 13 14 15 16

17 18 19 20 21

22 23 24 25 26

27 28 29 30 31

Span of Lintel Beam

(ft.) s

48.0

40.0

30.0

20.0

10.0

6.0

1.0

Depth of Lintel Beam

(ft.) d

4.0 6.0 8.0

4.0 6.0 8.0

1.0 2.0 4.0 6.0 8.0

1.0 2.0 4.0 6.0 8.0

1.0 2.0 4.0 6.0 8.0

1.0 2.0 4.0 6.0 8.0

1.0 2.0 4.0 6.0 8.00

Ratio

Depth Span d/s

0.0833 0.125 0.1666

0.10 0.15 0.20

0.033 0.066 0.133 0.200 0.266

0.05 0.10 0.20 0.30 0.40

0.10 0.20 0.40 0.60 0.80

0.166 0.333 0.666 1.000 1.333

1.00 2.00 4.00 6.00 8.0

Ratio

Wall Stiffness Lintel Beam

Stiffness

13.5 4.0 1.60

52.10 15.30 6.50

8437.5 1054.7 131.8 39.0 15.7

13333.3 1666.6 208.3 61.8 26.0

13000.0 1630.0 204.0 60.0 25.5

9750.0 1219.0 152.4 45.2 19.0

2140.0 267.5 33.5 10.0 4.2

150

Table 7.4(b)

Other Parameters in Shear Mall Analysis

(i) Relating to span of lintel beam

Total Breadth of shear wall (ft.)

B

60.0

Span of lintel beam (ft.) s

48.0 40.0 30.0 20.0 10.0 6.0 1.0

s/B

0.80 0.667 0.50 0.333 0.167 0.10 0.0167

(ii) Relating to depth of lintel beam

Height of story (ft.)

h

12.0

,


d

1.0 2.0 4.0 6.0 8.0

Moment of Inertia of lintel beam (fti )

0.0833 0.6667 5.3333 18.00 42.6667

d/h

0.0833 0.1667 0.3333 0.50 0.6667

Y

0. 0. 0. 0. 0

I - d

h

9166 8333 6667 50 .3333

Table 7.5

151

Serial No.

1 2 3

4 5 6 7

8 9 10 11

12 13 14 15 16

17 18 19 20 21

Solid si Maximum Maximum

8-S


s

40.0

30.0

20.0

10.0

1.0

lear wall of s stress in wal displacement

Itory Shear Wall Analysj


b

10.0

15.0

20.0

25.0

29.5

ame overall 1 - 7.68 kip - 0.001184 f


d

4.0 6.0 8.0

2.0 4.0 6.0 8.0

2.0 4.0 6.0 8.0

1.0 2.0 4.0 6.0 8.0

1.0 2.0 4.0 6.0 8.0

dimensions a/sq. ft. t.

-S


68.3 48.4 38.5

55.4 38.1 30.2 25.3

34.3 23.9 19.8 17.8

22.8 19.3 15.4 13.9 13.5

11.5 11.2 11.1 11.3 11.8


0.02607 0.01188 0.00712

0.02789 0.01229 0.00677 0.00440

0.01302 0.00595 0.00364 0.00271

0.00745 0.00495 0.00271 0.00209 0.00187

0.00183 0.00172 0.00166 0.00163 0.00162

Table 7.6

152

Serial No.

1 2 3

4 5 6 7

8 9 10 11 12

13 14 15 16 17

18 19 20 21 22

6-Story Shear Wall Analysi


s

40.0

30.0

20.0

10.0

1.0


b

10.0

15.0

20.0

25.0

29.5


d

4.0 6.0 8.0

2.0 4.0 6.0 8.0

1.0 2.0 4.0 6.0 8.0

1.0 2.0 4.0 6.0 8.0

1.0 2.0 4.0 6.0 8.0

s


45.4 33.1 26.4

33.8 25.1 20.0 16.9

21.5 20.3 15.6 13.0 11.8

14.0 12.2 9.90 9.00 8.78

7.28 7.04 7.03 7.20 7.54


0.01208 0.00585 0.00347

0.01013 0.00545 0.00314 0.00204

0.00519 0.00460 0.00261 0.00165 0.00122

0.00269 0.00201 0.00119 0.00091 0.00079

0.00077 0.00071 0.00068 0.00067 0.00067

Solid shear wall of same overall dimensions: Maximum stress in wall • 4.32 kips/sq. ft. Maximum displacement - 0.000374 ft.

Table 7.7

153

4-Story Shear Wall Analysis

Serial No.

1

2 3

4 5 6 7 8

9 10 11 12 13

14 15 16 17 18

19 20 21 22 23

Solid si Maximum Maximum


s

40.0

30.0

20.0

10.0

1.0

lear wall of s stress in wal displacement


b

10.0

15.0

20.0

25.0

29.5

ame overall < 1-1.92 kipi - 0.0000733


d

4.0 6.0 8.0

1.0 2.0 4.0 6.0 8.0

1.0 2.0 4.0 6.0 8.0

1.0 2.0 4.0 6.0 8.0

1.0 2.0 4.0 6.0 8.0

dimensions 3/sq. ft. ft.


25.4 19.4 15.7

18.0 16.9 13.9 11.50 9.90

10.9 10.2 8.59 7.45 6.84

7.19 6.54 5.56 5.14 5.08

4.04 3.89 3.92 4.08 4.31

^ ~


0.00383 0.00210 0.00129

0.00266 0.00239 0.00164 0.00107 0.00074

0.00126 0.00113 0.00081 0.00057 0.00044

0.00070 0.00059 0.00041 0.00033 0.00028

0.00028 0.00025 0.00024 0.00023 0.00023

154

00 .

rv

« r^

•s

8

c rJ

C

(0 « u u o

«

.c (/}

>% k> o a I

00

20.0

30.0

o

~9

e (TJ

xn >M rH O «

.J

c c (TJ -w a. —« ,

tn /

8.0

6.0

4.0

2.0

1.0

8.0

6.0

4.0

2.0

8.0

6.0

o

>* --H 0 / O HI TJ /

»j at • x: c x>/ a. -*/ B OJ / 0 n / *-"

• * j / • o / O Xi

-1 c V 6 4-1 e o C (TJ l-i

^ a> Ni-1 J JJ vx

Moments in Lintel Beams (kip-ft.)

IO rH »0 N f i O N ^ ^ o o i O i o r v

r H C M r v O » O C v » O C M C O O O O O N O - 4 ' O C V I ^ mi r-i ^ ,-( ^ ^

84.60

122.9

132.55

125.45

109.4

89.89

71.25

45.45

r H O O r v r v r H r v i / > ^ c M i o r v ' > 9 i o r v r v o • « . . . . . . cnrvrvoNiooOrHvo • * N o r v r v f v \ o \ o ^

C M g \ \ O t v f r ) s O f V v O C ^ O r H O ' J v O C ^ f O

. . . . . . . . r H O N ^ J r v O O O O O O l O r H r H C M ( N C M r N | C M C M

s O r r ) > « \ O v O O N C N | \ 0 > O 0 0 > O r H « 9 l / N v O ' 4

. . . . . . . . r H e M C O ^ » » > J ' > J ^

l O Q NO O O r H C r i v O r N r H O O a N

. . . . . . . . N O O N O I O C « 4 I O O N O • * i H r H O N « 0 ( S a » l O r-t (^ r* r-i w4 r-i

CM I O CO IO O N r H W O N O O C O ^ f M

• • • • • • • • O O C O O C M C O O O O C M O N ' ^ N O I O C N I H O O N O

r-i <-i t-i r-i r^

• * l / N r H « l O C r ) r H O N ^ < * N 0 0 C r ) O N O O N > O

. . . . . . . . O Q C n v O r H - » l O O i / t O O O N O N O N O O f v v O

C O O N C T I C M O O O O N N O v O i O > * 0 0 < * N r v O v O

P s l r - ( r v O ( N C M C M O N r H ( M C M C O c n ( O c r » p > |

O O O m o O r v c M r H r H C T N r H t v O - ^ C M <NioON-*>ONOrvrv O f v v o r r i o O O O O " ^ CNt C^ C\l CM rH rH

O O O O O O O O N O i - H r O f * N O N O O ^ * J ^ O

m i n c M - H O O r H v O v o - * O r H O N i O ( M Q o m rH (M CM i H rH rH

O O O O O i O O O O O v O O N t v O N O - t f O O v O

( M r v r O r H P v o s r H r M O O C M > 3 ' v T r > I O O N l ^

i-H rH rH rH i H

r H < N < * > » J l O v O r v O O

(T)<93<0'0<w'« 'TJ P Q C Q P O O Q C Q c a M t O

1 Axial Forces in Lintel Beams (kips)

l O r H N f i l O r H ^ l O f O l O n O O O i H H O O N O

. . . . . . . . C O I O I O N O N O V O I O C O

• ^ o o r v r M ( O i n H < e lOHOVCMOOOOCM . . . . . . . . ro io io iANONOioto

O N c o o t i o o ^ ^ ^ r v O O C M O C M C M O N O O N . . . . . . . . (Niio«eNO«ONeiocM

• 9 "tf 00 ON 00 ^ lO ^ « 0 0 < * C O O l O C M

. . . . . . . . C O I O V A N O N O N O I O C O

NO 00 lO IO IO ON (O » » r ^ « » r H C O H I O r H

. . . . . . . . c4<9' invo«ONOio^

O O O N O N C V i i ^ t H ^ I O

0<tfoorv\o«oirtrH . . . . . . . .

< « I O I O I O I O I O I O ^

0 0 r H O N N O r H < « N O r H r H N O O N ( S r v ^ ' ^ l / >

. . . . . . . . • t f io io io io io iocn

C O C O r H C M r H C O C O r v C O Q O c n C M O O O « » r H

. . . . . . . . • ^ I / N S O N O N O I O I O C O

mrvvoooivpHoocM ONvOiorvvoo>o«9 r r t i n v O N O N O N O I ' ^ ' O

r H C O O N O H O C M O O o o i o > » i O f o m o N r v « 3 - i r » i / N m i O l / ^ l O < r >

a N f > J i r > ^ c o s O ( M 0 0 ONOOrv^cMinNOO

• « . . . . . . >»irt ir>ir( ir im>/Nm

a O O v O C H C O ( M v O > 0 <r«3'rHrv>3-ir>m\o i / N v O v O i / N i O i n i / N C M

r H f M C O < t l O > O ^ v 0 0

(a (6<Q(T)<o iv io 'a tQOitOCQCQfOCOcCi

155

Tabic 7.8 (cont'd.)

1.0

. .—_

.

——»

10.0

>v

Span of

X. li

ntel beam

8.0

6.0

4.0

2.0

1.0

8.0

6.0

4.0

2.0

o

O V ^

J: C ^

ii / a /

/ 6 ' rH C

c B •rt V rJ Xi (f

rom bottom; N.

Moments in Lintel Beams (klp-ft.)

IO I O I O I O O O N O H I O

• • . . . . • • r ^ N O N c o r v r H C O Q C M r v e o r ^ r o o N N o f f 0 0 O N ^ « • « • r H r v H r H

_ lO IO NO r H O r H r H O O C M C O ^ . . . . . . . . 0 0 « « I O ^ O N I O I O N O 0 N r v > 0 0 N O C H < « \ 0 ON o c^ rv ^o -i^ CM

ri 405.00

436.00

392.00

322.3

246.85

171.8

99.73

25.66

CMlOCMiO00<jNlOC0 c o v o r v c M r v f o O N r H O O N r H v O O O r H ^ r v ON ON 0> rv m . * CM

r H O N N O - a - i r i » » - . * c M . ^ r v r v ^ i n a o ^ ^ c o

( M v O l O C M r v C M O O f O CM CM CM CM rH rH

CO 00 O O N ' ^ r H 0 0 l O « » 0 0

o o c o r v O N C M < * r v o o N O r H O r v « » O N O C M r-i t^ t^ r-i r^ ri

lO 00 rH rH N O O N O O r H I T I r H O O l O

r H C O - * O O O i r > C M l O O C 0 C 0 C M C T > f v i O < M rH rH rH rH

r H r H \ O 0 0 r H C O « * N O i r » > O r H r H r H r H l O v O

r H C M O O - ^ l O l O C O l O i / - ) r v r v r v v o i n « » c M

< M ^ f v O ( M i O r H O N ^ O O i O i A m s t O ' v

^ c M r v C T N O N O O r v — t . - H C M t N C M C M f M C M t N

r O \ O i / ^ N O Q 0 r o c O r H 0 0 r v O 0 D ( ^ ^ ' * O

C M ^ v O v C ^ c v r v t v

r H f M ( O « » » ' > N O r v ' ) 0

( g r a r s B t d i o ' o n 4 l ( U 0 ) ( U ( U O I < U V p Q C Q C O G a P Q C Q e O f A

Axial Forces in Lintel Beams (kips)

O N ^ W f v f O r v f v c O • t f C O 0 0 . f N O ^ C O 0 0

( O N O ^ « o o o o o o a o • o

a o o N i o r v c o N O < « N O a O v O r H t O O O O O l O C O . . . . . . . . C O N O O O O O O O O O O O I O

r H P v O N ^ C O l O l O l O O O C M ^ O i O O O O C O

. . . . . . . . c o v o f ^ r v o e o o r v ^

O O O N C O O O C O i - 4 0 0 0 0 C M C M C O N O f v O O r v C M

. . . . . . . . C O I O N O N O N O N O N O C O

N O O C O i H > 9 r H O O r v f l O ( 3 0 " * < ' l C O . J O

. . . . . . . . C M ^ ^ m i O v O v O N O C O

CO ON ^ rH rH 00 O r v C O r H C O C O f > i O O O

C O I O N O V O N O V O N O C O

O C M O 0 0 - * r v , H O e M r H « » C 0 C 0 C 0 O l O

. a . . . . . .

( M I O V O N O V O N O N O C O

r o v o Q r v c o o o r v c M O N O N O O r H P O i H O N - * . . . . . . . . C M « * I O V O N O V O I O ( 0

f v . r v r v r o ( M r H c O N O o o r v r v t M P O ( M ( O c o

c M - * u ^ N O \ O v o i n c o

V O O O ' > * C O O N « * 0 ^ J C M ^ O C M O ' v O O

t M ^ i T l N O v O N O i O r O

, H « N r O > » l O N O f v O O

e Q m c O A C Q B Q e Q M

156

o CO

i i • > «J

c rJ

C

O CM

O >o

O

O .

CM

f*" _ I O ON 0 0 r H NO ON r v vo eo >9

• • • . . . rH O IO lO rH rH 0> CM rH ON rv .*

op CO in rH 00 CM O ON CO CM \0 ^

00 ON rH CM ON 00 •o rv 00 rv m CO

rH vo rv o O 00 <o CM rv rv CM •« 00 CM «0 lO CM rH CM ^ ^ •* .* CO

CO 00 00 rH O rH •O CO •.» CO ON

vO

o • CO

NO

o • IO

(M

o< • IO

NO ON O 00 • «o

• IO

CM (M • •*

^s

. 4J IM 1

a •H M >mf

I V BQ

r i

«

int

rJ

e f i

•en

ts

x

NO

NO ON

• O

in sO

CM

o r i

CM CM . rv NO

00 CO • CO

ro

O CM CO CO rH r-i

IO IO

• rH

IO «» 00 CO r^

O «» • IO

ON

CO rv • O lO

00 •« •» . rv

IO

• CM ^ r^

CO CM

. ON ON

CO

<o -» m

-H

e (0 (U BQ

rH

. CM r i

•9 •

r^

ON

r^ ^ r-i

IO ON rH CO o o o • r-i

r-i

CO CO r-i

. . . CM CM CM

IO CM CO rv FH rv 0> CM rH O 00 IO r-i

CM 00 ON IO rv . rv

ON

r^ NO •

NO IO

00 NO

• <» rH

NO

•

00 00 rv . . . 5 rH O 00 rv in

rH O rH NO <0 CO . . .

IO rH rH IO IO >*

tn CO •< rv 00 IO

• • • IO IO ^ f^ ri ri

00 IO 00 rH CM

. . . CO >0 00 r« rH 00 r-i

m ON

• • *

CO rH

f-H fv

o CO

<M

a (fl V

(O

NO r^

lO (M

• o CO rH

00

o vO CO

CO

a (0 » O 00 IO r-i

rv 00 ON o\ in m

ON ON m rv vo in

>» in >o

B a a CO (T) (Tl 9) 0> 0) (4 CO OQ

•« lO CM 00 CO O O ON rH 00 00

(O in in NO in CO

rH CO >0 <«'«' (M rH O ON rH 00 NO • « . . . .

CO m m NO IO CO

CO

a

i so

V

s ri

a

0) u u o

rH fv fO CM rv O 00 NO rv o rv 00 . . . . . .

CM in NO IO CO

O «o rv CNi i^ CNJ rH 00 O vO NO •4' . . . . . . CM CO in in in "*

^ ON ON ON 00 (M m (M 00 00 NO CM

• • . . . . CO in m m in ^

•9 O ri ri T^ mi NO «^ O O NO rv . . . . . .

CO m «o NO IO CO

ON Ol rH rH CO 00 NO CM (M rv .«

. . . . . . CO IO »0 NO IO CO

CM CO lO g ON NO ^ r«. rv

IO NO NO to CO

CM CO 00 NO NO O CM IO NO IO •« (O . . . . . .

.tf in in in in ^

ON m vO >0 NO ON CO rv 00 NO »* vO . . . . . .

«» m m m in CO

rv ri (M 00 o m m o CM ON m rH ^ vo »o in in (o

I-H CM <o -J m vo a a a a n) (Q (Q <w

a a (0 (T) (u at 9) V CJ t) CQ DQ 03 CO CO CQ

157

e o u

JO (0 H

O

O

O

00

o «o

o

o CM

o 00

o >o

o

o

/-\ . 4J

1 a •H .li \^ n § CQ

>-i 9> M

e •ri rJ

e 09 4J

c n o X

in ON

r-i tn ri

in CO

* 00 tH rv

m

ON 00 CM

rv • CO

NO

CM NO

m r^

CM

ON rH rH

r-i ri . -^ rv

r-i

o in CO

in --i

<7<

in >o r ^

I-H

Bea

m

m CO

S fH

IO rv

rv CM rv

•n r-i

• CO ON (M

NO CO

• NO NO

ON

m rv .-i

rH

CO

•» t-i

m m • 00

00

NO rv

>o «»

on rv

r-i

<- vO

(VI

fM

a (Tl 01 BQ

in CO . . in «

:3s ri

in vo . . 9 ri

Q fO NO *7

m NO CM

• • r-i CO >» rv CM rH

^0 00 \0 • • m o

in >»

«» -» NO o

in CM r-i r-i

m .» rv ON

00 ON (M ON r^

ri -9 CM 00 . . CO rv

00 NO

00 r-i m m

rv (N «» «»

rv in 00 (M

in vo --I -i

vO o rH Vj

CO (*N

ro -J

B (0 CQ OQ

IO

o •» • * CO CO ON IO •* r-i

IO ri NO ON . . ON CO

IO rv CM

in m •* . . CO rv

O CM r-i

NO ON ON tH

• m

•<t rv CM

ON O CM

00 CO

CO CO ro CO

CO o vO CO

(M r^ ON •»

• • ON •* .* CM

CO sO (O 00

m o CO CM

CO 00 rv vo in CM - i rH

in in ^ (N

m fO

in vo

(T) n 0) 9) BQ OQ

^v 10

a •H ja

M

Q (0 0) BO

r-i 9>

C •H _> e •H

CB

U 0 Ch

r i

(« •H

iS

ON rH in «0 ON m PNl CO 00 CO CO rH

CO rv CO 00 vo

NO CM O ON CO CO NO vO rH lO NO <0 . . . . . . CO \0 OO 00 00 ^

CO ON rv iH O 00 m rH ON O IO . . . . . . CO \o rv rv 00 .^

00 CM 00 ON CO lO O CM CM \0 ON <4' . . . . . . CO m ^ vo lO CO

00 NO CO rv in NO sO rv 00 CM m rH

. . . . . . (M •* m NO NO CO

rH CM CM CO (M rv rH O ON rH O (M . . . . . .

CO in in vo NO ^

<7 00 00 in <« <tf CO ON ON (M rH ON . . . . . . (N ^ m vO NO CO

CM rH vO ON ON rv rv CO 00 rH O NO . . . . . . CM •«» in vO vO (O

rH CO CM (o o m vo m NO o o rv (M -J in sO vO CO

\0 »J ( 00 o in rH 0\ ^ >0 ot. —1

rM CO in u-i in vj

rH (M ro >j in vo

B B B B (0 (T3 0) V <U Q) 'U V OQ BQ OQ BQ cQ OQ

158

oo

4) u C

"H . J

C

u

«a •H

I

3

(0

« cn >% u o «i OS I

o

o CM

O • s

o o

a 4) XI

U_l rH O Oi

4J

c c (0 -H arH tn >

o NO

o * «»

o CM

o • rH

o • 00

o • NO

o • •*

o • CNl

o 1-^

o 00

o NO

o <3-

V4-I rH B

o a> (Q «-) 9> .

XI c x>/ 4J -r / a.—i/ 9> / a /

Lintel

beam no.

^^ a o 4-1 4-1

(from bo

^-\ 4J MH

k •H

i « BQ

rH V W a •H iJ

C •H

nts

S S X

CM ON

(M CO

CM • in

rH

rv ON

• CM

CO >* • o

CO rv • ON

m

o rv • rv

CO

NO • rv

ri

ri «* • CO

00 -» •

o

CM rv • CO

00

in

m m

00 <N

(TN CM

rH

Beam

»» 00

40.

r-i CM

• o (M

NO r-i

• -*

o NO • o

CM ON • ri

rv

NO ri

• 00

tH CM •

•» CM

CM 0\ •

«»

o rv •

O

00 tn • in

ON

>» 00

68.

m CO

(JN

ro

CM

Beam

CM «»

37.

;^ • o CM

ON •» •

<»

m NO • o

r^ rv .

O NO

«» NO • «»

«»

ON NO .

«* CM

NO tn • m

rv rv •

O

o CO • (M

rv

CM CO

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higher order finite element analysis of shear a

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