ABSTRACT

DYNAMIC ANALYSIS OF
NONLINEAR ELASTIC FRAMES

. \“
ax

by James K? Iverson

A method of solution for dynamically loaded elastic frame
structures is presented. The nonlinear effects of geometry changes
and axial thrust are considered in the solution. The method is
developed to analyze three dimensional structures subject to joint
loading or earthquake ground motion. A computer program written
in FORTRAN for use on the Michigan State University CDC 3600 computer
system was used to implement the solution.

The mass of the members is lumped at the joints. The
lumping procedure accounts for rotatory inertia as well as trans-
lational inertia. Mass-prOportional viscous damping is included in
the analysis. The nonlinear effects are taken into account in the
calculation of member forces. The analysis is formulated using the
joint displacements as variables, and the transient response of the
frame is obtained by a numerical integration of the equation of
motion of the joint masses.

Several studies of a simple cubic frame are presented.

The studies considered the order of magnitude of the nonlinear
effects, and the influences of dissymmetry in the three-dimensional

structure-load system. The results of the nonlinear studies show

James K. Iverson

that the consideration of geometry changes are significant only if
relatively large displacements occur. Axial force effects on flex-
ural stiffness may be significant for relatively common values of
axial load. The investigations also indicate that dissymmetry in
the distribution of mass, dissymmetry in stiffness, or a time lag in
the motion of the supports may cause considerable variations in the
response of the structure. Such variations in the response of the
structure can be predicted only with a truly three-dimensional
analysis.

The use of a damped dynamic solution to obtain a nonlinear

static analysis is also demonstrated for a plane frame.

DYNAMIC ANALYSIS OF

NONLINEAR ELASTIC FRAMES

by

a
James K. Iverson

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the'requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Civil Engineering

1968

ACKNOWLEDGMENTS

The author would like to express his gratitude to his advisor,
Dr. R. K. Wen. His advice, direction and editing were invaluable.

The gentlemen who served on the author's guidance committee, Dr. C. E.
Cutts, Dr. W. A. Bradley, Dr. J. L. Lubkin and Dr. C. C. Ganser have
contributed generously of their time and the author expresses his
sincere thanks for their assistance.

The author's graduate studies were made possible through a
Graduate Traineeship granted by the National Science Foundation and
administered by the Division of Engineering Research.

To the author's wife, Mary Lou, whose countless contributions
have made this work possible, and who "found" the time to type this

thesis, Special Thanks.

ii

TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . ii
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . v

LIST OF FIGIJRES o o o o o o o o o o o o o o o o o o o o o o 0 Vi

Chapter

I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . l
1.1 General . . . . . . . . . . . . . . . . . . . . . l
1.2 Literature Review . . . . . . . . . . . . . . . . 2
1.3 Scope and Outline of the Investigation . . . . . 4

1.4 Definitions of Member Incidence and
Coordinate Systems . . . . . . . . . . . . . . 6
1.5 Notation . . . . . . . . . . . . . . . . . . . . 8

II. FRAME JOINT REACTIONS . . . . . . . . . . . . . . . . . 11
2.1 General . . . . . . . . . . . . . . . . . . . . . 11
2.2 Linear Solution Used in Static Analysis . . . . . 11

2.2.1 Basic Member Stiffness Matrix . . . . 12
2.2.2 Member Rotation Matrices . . . . . . . 13
2.2.3 Force Equilibrium Matrix . . . . . . . 14
2.2.4 Relative End Diaplacements . . . . . . 15
2.2.5 Joint Stiffness Matrix . . . . . . . . 16
2.2.6 Solution of the Equilibrium

Equation . . . . . . . . . . . . . . 18
2.2.7 Member End Forces . . . . . . . . . . 18
2.3 Nonlinear Joint Reactions . . . . . . . . . . . . 19
2.3.1 Determination of Relative End
Displacements . . . . . . . . . . . 19
2.3.2 Determination of Member End
Forces . . . . . . . . . . . . . . . 22
3 Intermediate Rotation Matrix . . . . . 24
. .4 Rotation to Joint Global
Coordinates . . . . . . . . . . . . 25

III. DYNAMIC ANALYSIS 0 C C O O O I O O O O O O O O O O O O O 26

3.1 General . . . . . . . . . . . . . . . . . . . . . 26
3.2 Formation of Mass Matrix . . . . . . . . . . . . 26
3.3 Damping . . . . . . . . . . . . . . . . . . . . 32
3.4 Numerical Solution Procedure Used In

Dynamic Analysis . . . . . . . . . . . . . . . 32
3.4.1 Solution Procedure . . . . . . . . . . 32

iii

3.4.2 Time Increment Used in
Numerical Integration . . .
3.4.3 Change of Variables for Ground

Motion Problems

3.5 Computer Program

IV. APPLICATIONS . . . . . .

4.1
4.2
4.3

4.4

General . . . . .

Comparison with Modal Analysis Solution .
Static DiSplacements Using Damped

Dynamic Solution

Nonlinear Comparisons .
4.4.1 Effect of Geometry Changes . .
4.4.2 Effect of Compressive Axial

Forces .

Earthquake Loading

Effects of Dissymmetry in Distribution of

Mass, Stiffness and Loading .

Effect of Time Lag of Support Movements .

V. SUMMARY AND CONCLUSIONS

LIST OF REFERENCES . . . . . .

TABLES . .

FIGURES . .

APPENDIX I.

APPENDIX II.

BEAM COLUMN SOLUTION

COMPUTER PRwRAM

iv

Page

34

35
35

40
4O
41
41
43
43

44
45

46
49

51
54
57
59
78

83

Table

4.2

4.3

LIST OF TABLES

Page

Variation in Forces and Displacements in
Geometry Change Comparison . . . . . . . . . . 57
Effect of Dissymmetry in Stiffness . . . . . . . 58
Time Lag Studies . . . . . . . . . . . . . . . . 58

Figure
1.1
2.1

2.2

2.3

3.1
4.1

4.2

4.3
4.4
4.5
4.6
4.7

4.8

4.9
4.10
4.11

4.12

A.2

LIST OF FIGURES

Coordinate Systems . . . . . . . . . . . .
Member Relative End Displacements . . . . .

Relative Magnitude of Geometry Correction
Terms 0 O O O O O O O O O O O O O O O O O

Rotation of Member End Forces Due to
Chord Rotation . . . . . . . . . . . . .

Rigid Joint Mass . . . . . . . . . . . . .
cubic Frame 0 O O O O O O O I O O 0 O O O 0

Comparison of Solution Method with Modal
AnaIYSiS O O O O O O O O O O O O O O O O

Damped Response of Stiff Cubic Frame . . .
Static Solution of Plane Frame . . . . . .
Axial Load Effect on DiSplacements . . . .
Axial Load Effects on Column Forces . . . .
Axial Load Effects on Beam Forces . . . . .

Response of Cubic Frames to 1948 El Centro
Earthquake O O C O O I O I O O O O O O 0

Frame Used in Three-Dimensional Studies . .
Effects of Unsymmetric Mass Distribution .
Effect of Loading Pattern . . . . . . . . .
Normal Modes for Cubic Frame . . . . . . .

Coordinates for Member Solution . . . . . .

Coefficients in Expression for Axial Shortening

Due to BOWing O O I O O O O O O O O O I 0

vi

Page
59

6O

61

62
63

64

65
66
67
68
69

70

71
72
73
74
75

76

77

CHAPTER I

INTRODUCTION

1.1 General

The advent of high speed digital computers has made pos-
sible considerable advances in the study of nonlinear behavior of
structural frames. Generally speaking, there are two types of non-
linearities, material nonlinearities and geometrical nonlinearities.
Much work has been done in the former class of problems, both in
static analysis and dynamic analysis. However, practically all
published works in this area are limited to plane frames. The
influence of geometrical nonlinearities has been considered by a
number of investigators. Their studies are, in general, related
to the static behavior of structures.

The purpose of this thesis is to develop a method for the
dynamic analysis of elastic Space frames that takes into account the
nonlinear effects of geometry changes and axial thrust. Because of
the nonlinear nature of the problem, a linear static analysis is a
necessary prelude to this study. In order to demonstrate the prac-
ticality of the method, a computer program is prepared to implement
the analysis, and the results of certain numerical problems are

examined.

1.2 Literature Review

In the area of nonlinear dynamic response of frame struc-
tures Wen and Janssen (23)* studied plane frames using a lumped
mass and lumped flexibility model to investigate the effect of
elastic-inelastic material properties. The method is applicable to
frames with members having a general curvilinear moment-curvature
relation. Clough, Benuska and Wilson (1) presented a procedure for
computing the dynamic response of plane frames taking into account
the nonlinear effects of inelastic deformations in yield hinges at
the ends of the members. The procedure is more efficient in
handling frames of larger size but is limited to members with
bi-linear moment-curvature relation. It was used to study plane
frames subjected to earthquake loadings and to evaluate the
ductility requirements for various members. Damping was not in-
cluded. A continuation of this work including damping effects
was undertaken by Giberson (5).

Tezcan (19) presented a modal analysis solution for three
dimensional elastic frames which included damping. By use of the
acceleration response Spectrum derived for the 1940 El Centro earth-
quake, he obtained certain "equivalent static loads" which were
based on the natural periods of vibration of the structure. No
nonlinear effects were considered in this study. In this and all of
the above-mentioned papers masses were "lumped" at the joints or

node points with no rotatory inertia effects considered.

 

*Numbers refer to items listed in the List of References.

The area of static linear-elastic analysis of frames has
been widely investigated. In particular, the following works have
been referred to most frequently in the development of the linear
static analysis employed for the present study.‘ The network formu-
lation of linear frames has been presented by Fenves and Branin
(6). Their original work had the geometric properties of the mem-
bers combined with the stiffness properties. Subsequently, Dimaggio
and Spillers (3) and Livesley (13) independently presented versions
of a network formulation wherein the geometric pr0perties are com-
bined with the incidence properties. This results in a much im-
proved conditioning of the final simultaneous equations. The net-
work formulation uses a single 6 x 6 member stiffness matrix instead
of the more common 12 x 12 matrix of the direct stiffness formu?
lation.

Gere and Weaver's book (4) contains a comprehensive treat-
ment in the more common 12 X 12 stiffness matrix format. Weaver's
book (21) is primarily a presentation of programming techniques
and prepared programs for use in linear-elastic frame and truss
analysis.

The nonlinear effects of large displacements and axial
thrust in the stability of frames have recently been investigated
by a number of researchers. Saafan and Brotton (17) presented a
computer solution for plane frames which considers the effects of
member chord rotations and axial thrust. Saafan (16) continued the
study and included, in addition, the effects of member bowing on the

apparent axial deformation. Jennings (ll) derives stiffness

coefficients for plane members, considering geometry changes, and
studied a single bent plane frame dividing each member into several
"submembers" to increase accuracy. Zarghamee and Shah (24) devel-
oped stiffness coefficients in three dimensions for space frames,
that considered member chord rotations and bowing effects. They
used an iterative solution method to solve for the displacements.
Connor, Logcher and Chan (2) made a similar development in three
dimensions and studied the load-deformation relations of a small

three dimensional frame which had also been studied experimentally.

1.3 Scope and Outline of the Investigation

The investigation presented here will first develop a
method of solution for the nonlinear analysis of dynamically loaded
elastic space frames, and then incorporate it in a computer program
written in Fortran for use on the C.D.C. 3600 Computer System at
Michigan State University.

In the development of the analysis the following simpli-
fying assumptions were made:

1) Frame members remain elastic throughout the solution

and are of a uniform cross-section.

2) All joints are completely rigid.

3) All loads other than dead load are applied to the

joints.

4) Torsional-flexural coupling may be neglected in the

solution of the individual members.

5) The Bernoulli-Navier solution for a simple beam holds

for relatively large end displacements.

6) The rotation angle of the member chord relative to the
undisplaced position remains small enough that its
sine may be approximated by the angle, its cosine may
be approximated by unity, and the product of the sines
of two such rotation angles is zero.

The development of the analysis is based on straightforward
applications of principles of mechanics and, at places, a synthesis
of known results. In the preparation of the computer program,
emphasis was given to economy of computer time, ease of modification,
and generality. The frame is first analyzed statically for the
displacements and forces due to its dead loads, and these are used
as initial conditions for the dynamic solution.

The dynamic solution is accomplished using the displace-
ments of the joints as variables, and utilizing numerical inte-
gration to solve for the transient response of the joints to dy-
namic loading, either applied to the joints or through ground
motion. A mass lumping procedure (22) that accounts for rotatory
inertia is presented. Viscous damping is taken into account.

The nonlinear effects of geometry changes and axial thrust
are accounted for in the determination of the frame reaction forces
used in the equation of motion for each joint. The geometry effect
enters into the analysis in two ways: a consideration of the rota-
tion of the "chord" of the member (the straight line joining its
two ends) and of the "bowing" of the member. The former affects
the end rotations of the member and the axial strain, whereas the

latter affects the axial strain. These considerations enter into

the calculations of the member distortions. The effect of axial
force (axial force is defined to be the and force acting in the
direction of the chord) simply modifies certain stiffness coef-
ficients of the member according to beam-column behavior.

The investigation also includes various studies made with
the computer program developed. A simple cubic frame is studied
under dynamic loading to investigate the nonlinear effects. The
capability to treat three-dimensional systems is utilized to study
the effects of dissymmetry in a similar cubic frame, and to inves-
tigate the response to three-dimensional ground motion from a

recorded earthquake.

1.4 Definitions of Member Incidence and Coordinate Systems

For the purposes of the analysis, the members will be as-
sumed to be directed (arrows on the individual members will be used
in sketches) between the two joints at their ends. The direction
or orientation may be arbitrarily assigned initially. However,
once assigned, it must be fixed throughout the analysis. The mem-
ber then is said to be directed from the joint at its "initial"
and to the joint at its "final" end. The member is "positively"
incident to the joint at its initial end and "negatively" incident
to the joint at its final end. Thus, in Figure 1.1 the member K is
positively incident to joint I and negatively incident to joint J.

Three coordinate systems are used in the analysis. (see
Figure 1.1):

1) Structure Global Coordinate System:

This system consists of a single set of cartesian

2)

3)

axes with its origin at any convenient fixed point.

The system is oriented with the X and Z axes horizontal
and the Y axis vertical. This system is used to spec-
ify the initial location (three dimensional vectors) of
the joints.

Joint Global Coordinate System:

This system consists of one set of cartesian axes for
each joint with the origin placed at the initial lo-
cation of the joint with its axes parallel to the
structure global axes. This system is used to des-
cribe the displacements of the joint and forces

acting on the joint, both of which are represented by
vectors with six elements. For displacements, the
first three elements in each vector represent the
translational displacements in the direction of the
three coordinate axes taken in order, and the second
three elements represent rotations about each of the
ordered axes, respectively. The force vectors are
made up of forces corresponding to these displacements.
Member Coordinate System:

This consists of two sets of rectangular cartesian
axes per member, located with the origins at the
location of each end of the unloaded member. Both
sets of axes will be oriented with the first axis
along the centroidal axis of the member directed from
the initial end to the final end of the member. The

second axis is directed along one of the principal

axes of inertia and the third axis is directed along

the other principal axis of inertia. These axes are

used to describe the displacements and forces acting

on the two ends of the member. As in the joint global

system, the diSplacements and forces are described by

six element vectors, with the first three elements

being translations and corresponding forces in the

directions of the three axes, and the second three

elements being rotations and corresponding moments

about the three axes.

1.5 Notation

The notation shown below has been used in this report:

A =
X

A
Y

{a}
[C]
{d}

{DP}

r-a

’11

L...)
II

C)
II

the cross-sectional area of a beam;

the equivalent shear area for shear stiffness for
shear force parallel to the member y axis;

the equivalent shear area for shear stiffness for
shear force parallel to the member 2 axis;

an arbitrary 6 element vector;

the viscous damping matrix;

member relative end displacements;

the vector of forces representing the damping
force on each joint;

vector of corrected relative member end dis-
tortions;

modulus of elasticity;

vector of end forces;

shear modulus;

{P}

{RE}

[3M0
{ST}

shear stiffness coefficient;

the member torsional constant;

the member moment of inertia about the member y
axis;

the member moment of inertia about the member z
axis;

intermediate member rotation matrix;

inertia tensor;

torsional constant;

joint stiffness matrix of the entire structure;
member stiffness matrix;

member stiffness coefficient;

member length;

bending moment;

bending moment on end i of member;

lumped mass matrix of the entire structure;
mass;

axial member load;

the vector of dynamic forces acting on the joints
or on a particular joint if subscripted;

the hyper vector of forces representing the frame
reaction on the frame joints;

member rotation matrix;

the vector of static load forces acting on the
joints;

shear force;

time;

At
[T]
{u}
{x}

F{}

(°)

('°)

10

time increment;

force equilibrium matrix;

the vector of displacements in member coordinates;
the vector of displacements in joint global
coordinates;

angle defining relative rotation of member chord
in the x-y plane (see Figure 2.3);

same as above in x-z plane;

damping constant;

vector of end forces at one end of a member;

mass unit density;

a finite, but small increment of the variable
preceded;

a vector quantity associated with the initial end
of a member;

a vector quantity associated with the final end

of a member;

AL)...

dt ’
£41.
2 .

dt

CHAPTER II

FRAME JOINT REACTIONS

2.1 General

The equation of motion for a space frame may be written as
[M] {3?} + {DP} + {RE} = {P} .....(291)

in which all quantities are in global coordinates: [M] is the mass
matrix; {a}, the acceleration vector; {DF}, the damping force
vector; {RE}, the "joint reactions"; and {P} the external loads

on the joints. The formulation of [M] and {DF}, and the solution of
the equation will be discussed in the next chapter. In this chap-
ter the joint reactions {RE} are derived.

In the linear static case {RE} = {P} = [K] {x}, where [K]
is the linear joint stiffness matrix. Since a linear static solu-
tion is a necessary prelude (to obtain the initial stresses and
displacements) to the dynamic analysis and also to serve as back-
ground material for the derivation of {RE}, the linear static

solution used in this analysis is outlined in the following section.

2.2 Linear Solution Used in Static Analysis

The linear static analysis is formulated using a stiffness
solution in a manner outlined in the paper by Dimaggio and Spillers

(3). Using this format each member's stiffness properties are

11

12

represented by a 6 x 6 stiffness matrix containing the direct stiff-

ness coefficients at the initial end of the member instead of the
more common 12 x 12 matrix including the direct stiffness coeffi-
cients at both ends as well as cross coefficients. The overall
joint stiffness matrix of the structure is assembled from these
member stiffness matrices using what Dimaggio and Spillers call a
"Modified Incidence Matrix" that includes the necessary geometry
and member incidence information. To save computer storage space
and computation time several modifications were made.

2.2.1 Bag 2 Member Stiffness Matrix,--The "Basic Member
Stiffness Matrix" [Km] is a 6 x 6 stiffness matrix containing the
direct stiffness coefficients for the initial end of a member ex-
pressed in member coordinates. This matrix is symmetric and has 8

independent non-zero elements

—k1 o o o o o]
0 k2 o o 0 k7
o 0 k3 0 k8 0
[Km] = 0 0 0 k4 o o .....(2-2)
0 0 k8 0 k5 o
0 k7 o o 0 k6

 

 

The non-zero elements are defined as:

 

 

 

EA 12EIz 1
k1: ; 1‘2: 3 ’
L L l+2gz
12EI 1 GJ
k - , k = -1
3 L3 1+2g 4 L

13

 

 

 

 

4EI 2+3 4E1 2+g
k5 = ——1 —Y—; k6 = —£ 2 ; ..... (2-3)
L 2+4gy L 2+4gz
6EI 1 6EI l
k = z . k = - ———l
7 L2 1+2 ’ 8 L2 1+2 ,
82 8y

where E and G are the modulus of elasticity and shear modulus, L is
the member length, J is the member torsional constant, and Iy and
I2 are the moments of inertia about the two principal axes of

bending. The shear stiffness coefficients gy and g2 are defined by

 

6E1
g = ___1_.
y GALZ
y
6E1z
g = ..... (2-4)
2 GAZLZ

where Az and Ay are the appropriate shear stiffness areas for shear
force along the z and y axes, respectively. For example, for I
beams Ay will be taken as equal to the area of the web. For the
purpose of this investigation the torsional constant J, of the I
sections which were used, was computed using the approximation

for thin rectangular sections

§bh3 .....(2-5)

1

Edi»

J:

i
where b and h are the long and short dimensions of the ith thin
rectangular section.

2.2.2 Member Rotation Matrices.--The member rotation
matrices are the transformation matrices that rotate a 6-e1ement
vector quantity (end force or end displacement) expressed in joint

global coordinates to the member coordinates of each member.

14

Since the joint global coordinates for all joints have a common ori-
entation, a single rotation matrix is sufficient for each member.
In vector form if {a'} is a vector expressed in member coordinates
and {a} is the same vector in global coordinates, {a'} = [RM] {a}.
Since this transformation is between two orthogonal coor-
dinate systems it follows that the inverse transformation from
global to member coordinates will be accomplished by the transpose
of the transformation matrix above. That is {a } = [RM]t {a'}.
The matrix [RM]is made up of the direction cosines between

the axes of the two coordinate systems

r__
811 812 813 I _T
I
821 S22 S23 : o
I
S31 S32 S33 .
[RM] = —-~-- ---»- "f -------------- (2-6)
: S11 S12 S13
0 I S21 S22 823
l
i 831 S32 S33
__ I _,

 

 

where sij denotes the cosine of the angle between the ith member
coordinate axis and the jth joint global coordinate axis.
2.2.3 Force Equilibrium Matrix.--The member force equilib-

rium matrix is denoted by [T] and is defined by the relationship:

{T}Final End = [T]{T}Initial End °°°° (2'7)

where {T} is the vector of member end forces. If the member and

forces at both ends are expressed in member coordinates

15

[T] = ..... (2-8)

 

 

 

 

This follows from the requirements of equilibrium and under the
assumption that no load is applied between the end points of the
member.

2.2.4 Relative End Displacements.--The displacement of an
end-loaded member may be described by its end displacements in mem-
ber coordinates, I{u} and F{u},where the presubscript I indicates
initial end and F indicates final end. These displacement vectors

are related to the joint displacements:

{u} = [RMJtM .....(2-9)

in which {x} is, in global coordinates, the diSplacement of the
joint corresponding to the member end.
Let I{d} be the diSplacement vector of the initial end of

the member relative to the final end; i.e.,
t
11d} = IlU} + [T] F{u} .....(2-10)
It follows that

{T}Initial End = [Km] I{d} "°'°(2'11)

16

2.2.5 Joint Stiffness Matrix.--By use of the previously
defined quantities the structural reactions at the joints may be
expressed in terms of the displacements of the joints. The equi-
librium equation for a statically loaded frame may therefore be

expressed in matrix form as
[K] {Kl = {ST} .....(2-12)

where {ST} is the vector of joint static loads in joint global co-
ordinates. The matrix [K] so defined will be called the "joint
stiffness matrix", or the ”overall joint stiffness matrix".

This joint stiffness matrix has certain advantageous
properties, namely, symmetry, Sparseness and bandedness.

In this solution method the joint stiffness matrix will be
considered as partitioned into 6 x 6 submatrices with each submatrix
corresponding to a given joint and the 6 dimensions of the submatrix
corresponding to the 6 coordinates of that joint. In expanded form

[K] {x} = {ST} then becomes:

 

 

 

 

 

 

 

 

___ __fl __ ___ __ __1
6 X 6 6 X 6 PK].- ”ST;
Sub- Sub- ..... x2 8T2
matrix matrix x3 8T3
X ST
4 4
X5 ST5
gulf-6 gm):-6 - ---- fd “-1 _ST§j Jt.1
matrix matrix -—_-:_..-- ';--_ -----
l I \ 2
I I \ X3 8T3
l | \ x ST
I I \ 4 ST4
L__ ' l \ x5 STS
' .xél Jt 2 g Jt 2
__HF ______ ' _____
: I
__ | __J __ : _}

 

 

 

 

17

where all quantities are expressed in joint global coordinates.
Each row of the partitioned matrix equation expresses an equilibrium
equation for a particular joint of the structure being considered.

Analyzing a row of the partitioned matrix [K] it can be seen
that the diagonal term must reflect the direct action of all members
incident to the joint corresponding to that row. Furthermore, there
must be non-zero submatrices in each column corresponding to the
"other" end for each member incident to that joint. All other sub-
matrices in that row will be null.

To conserve computer storage the computer program developed
for this study generates and stores only the non-zero 6 x 6 subma-
trices in the joint stiffness matrix, and only those that appear
in the upper triangular portion of [K]. It is now necessary to
compute the non-zero submatrices. Four possibilities exist:

1) A direct contribution from a member positively
incident to the joint being considered, which is
reflected in the 6 x 6 submatrix on the diagonal and
is equal to [RM]t [Km] [RM].

2) A direct contribution from a member negatively inci-
dent to the joint being considered which is reflected
in the 6 x 6 submatrix on the diagonal and is equal
to [RMJ‘ [T1 [Km] mt [RM].

3) An indirect contribution due to the displacements of
the joint at the other end of a member positively
incident to the joint being considered. This corre-

Sponds to an off-diagonal submatrix in the column

18

corresponding to the other joint, but in the row cor-
responding to the joint being considered, and is equal
to [RMJ‘ [Km] mt [RM].

4) The same contribution as in 3) from a member negatively

incident is equal to [RM]t [T] [Km] [RM].

2.2.6 Solution of the Equilibrium Equation.--The equilib-
rium equation [K] {x} - {ST} is solved for the unknown displacements
using the Gauss Elimination Method (18). The method was used in this
investigation with the 6 x 6 submatrices as outlined in the previous
section as elements, and considering the symmetry of the joint stiff-
ness matrix. The basic method is well documented in most numerical

analysis texts. The reduction equation is

_ -l _
[AJij reduced - [Ajij present - [AJikEA1kkEA1kj °°°° (2 14)

where the subscript k corresponds to the diagonal pivot element
(actually a 6 x 6 matrix) currently being used.

Because of symmetry [A]ik = [A]:i. The submatrix [A]k1 is
in the pivot row and is in the stored portion of [K]. Also, any of
the reduced matrices will be symmetric since the joint stiffness
matrix is symmetric. These two facts allow treating only the sub-
matrices in the upper triagular portion of [K], which coincides
with the treatment in the generation of [K]. Back suhxitution is
accomplished in the usual fashion.

2.2.7 Member End Forces.--Member end forces are obtained

from the computed displacements using the basic equations:

19
M = [Km] IN} = [Km] 1M + £ij mt Fin} =
[Km] [RM] I{x} + [Km] [T]t [RM] F{x} ..... (2-15)

and

F{T} = [T] I{¢}. .....(2-16)

where I{x} and F{x} are the displacements in joint coordinates at

the initial end of the member and the final end, respectively.

2.3 Nonlinear Joint Reactions

The determination of the frame joint reactions {RE}, in the
nonlinear dynamic analysis is accomplished by solving each member for
its and forces corresponding to the current Specified joint displace-
ments and then, for each joint, summing the end forces for all the
members incident to that joint. The basic problem then becomes the
solution of a member with specified end diSplacementS for its end
forces. As in the linear solution the relative end displacements
are first obtained and then the appropriate Stiffness coefficients
are applied to obtain the end forces. The nonlinear effects of
geometry changes are accounted for in the determination of the rela-
tive end displacements,while the effects of axial force on flexural
stiffness are reflected in the stiffness coefficients.

2.3.1 Determination of Relative End Displacements.--Figure
2.1 shows a member initially undistorted with the member axis co-
incident with the x axis. After distortion,the initial end will
have undergone a displacement I{u} and the final end F{u}. As in

the linear case,these member end displacements are obtained from

20

the appropriate joint diSplacements by a linear rotation as shown in
Equation ,(2-9). The translations of the initial end of the distorted

member relative to the final end are:

2 I 2 F112 ooooo(2'17)

The primary'nonlinear effect of geometry change is caused by
the rotation of the "chord” of the distorted member relative to its
initially undistorted configuration. The "chord" refers to the
straight line jointing the end points of the member. Referring to

Figure 2.1, chord rotations in the x-y and x-z planes are approx-

imated by
~d2
A1 = -'
L
~d3
A2=_ 0.000(2-18)
L

The relative diSplacements in the nonlinear solution are more con-
veniently expressed relative to the chord of the deformed member.
The relative axial displacement, the axial twist, and the rotations

at each end of the members, completely define the member distortions,

e1:
(12 d2 L L
2 3 2 2
e1 = d1 ' ' ' (293 ' e385 ' 265) '
2L 2L 30 30
(2e2 - e e - 2e2) = (axial shortening)
4 4 6 6

21

= u - u = (relative twist about x axis)

e3 = Iu5 - A2 = (rotation about y axis at initial end)

e4 = Iu6 +-A1 = (rotation about 2 axis at initial end)
.....(2-19)

e5 = FuS - A2 = (rotation about y axis at final end)

e6 = Fu6 +'A1 = (rotation about 2 axis at final end)

The second and third terms in the expression for axial
shortening, e1, represent the effects of chord rotation and the last
two terms reflect the effect of apparent shortening due to "bowing".
The terms due to chord rotation may be develOped as follows. The
true shortening of the chord distance after displacement can be

represented by

2 2 %

= L - [(L-d1)2 + d + d3 ] .....(2-20)

echord rot. 2

Expanding the second term in a binomial expansion and dropping terms

over second order

dlz d22 d32
S d =d1- - - +"" 0.000(2-21)
chor rot. 2L 2L 2L 2
2 2 2 2 dl
For most cases d <<d and d <<d , the term.--‘is consequently
1 2 l 3 2L
ignored, thus
d22 d32
3 =d - - 00-00(2-22)
chord rot. 1 2L 2L

The correction terms for apparent member shortening due to
bowing are shown in Appendix 1, for the case where the effects of

axial force are considered and also where they are not considered.

22

The terms used in this investigation did not include the effects of
axial force in the correction for axial Shortening due to bowing.
The special case shown in Figure 2.2 was computed to determine the
magnitudes of the correction terms. The axial load used was equal
to the Euler load for the member considered. The neglect of axial
force in the bowing correction causes an error of approximately 12%
in the total correction terms. The axial forces found in practical
situations are normally considerably less than the Euler load to
preclude buckling of the member. Consequently, the neglect of axial
force effects in the bowing correction was considered justified.

2.3.2 Determination of Member End Forces.--In terms of the

relative diSplacements e given above the member end forces at the

1

initial end I{F} are computed using the following member stiffness

relations:

EA
IF1 =';- e1 = (ax1al force)
EIz
IF2 = ;§-'C3z (e4+e6) = (shear force)
EI
F = + =
I 3 £51 C3y (e3 e5) (shear force)
..... (2-23)
GJ
IF4 =‘;- e2 = (twisting moment)
EI EI
F = C + = ‘
I 5 'Ex' 1ye3 “:1 Czye5 (bending moment)
EI EIz
_._JE +.___ = .
IF6 - L Clze4 L sze6 (bending moment)

23

The coefficients used in F1 and F4 are the usual axial and twisting

stiffnesses. The derivation of the coefficients used in F2, F3

F5 and F6 is outlined in Appendix 1. Shear stiffness effects are
not included.
In terms of the stiffness coefficients defined previously,

these forces may be written as:

1F1 = k191

c

.22:
IF2 k7 (e4+e6)

6

fix
IF3 _ k8 (83+es)

6

..... (2-24)

IF4 ' k4e2

 

I 5 5 3 9 5
C C
12 22
F = k -- e + k --'e
I 6 6 4 4 10 2 6
where k1, k4, k5, k6, k7, k8 are as prev1ously defined in Section
2.1.1 and
2E1 l-g
k9=——1—1_
L 1-2
gy
2E1 1-g
klo = -—5- z ..... (2-25)
L 1-2gz

The coefficients Cly’ CZy’ C3y and Clz’ C22, C3z represent the

”beam-column effects", and their derivation is outlined in Appendix 1.

24

The first subscript corresponds to the subscript used in the Appen-
dix, the second indicates the appropriate plane.

Since these coefficients are functions of the axial force
on the member, the member end forces cannot be computed until the
axial force is known. Although an iterative procedure may be fea-
sible, in the present investigation the axial force from the forces
obtained in the previous time-step solution is used to obtain the C
coefficients. That is, the axial force used is one time increment
behind its true value. The type of numerical integration used re-
quires extremely small time increments and the variation in the
axial force was found to be quite small. A maximum variation of
approximately 3% of the Euler Load was noted for one time increment
in a member experiencing axial loads from O to 0.7 of the Euler load.

The forces at the final end of the member are now deter-

mined by equilibrium:
F{F} = [T] I{F} .....(2-26)

2.3.3 Intermediate Rotatigngatrix.--Considering Figure
2.3, it can be seen that due to the chord rotation the orientation
of the member end forces {F}, solved for as outlined in Section 2.3.2,
is no longer the same as the original member coordinate axes which
correspond to the undisplaced member orientation. For ease in the
programmed computation, all member forces are given in the original
member coordinate system. Therefore, the member forces {F} must be
transferred to that coordinate system. This is accomplished by the

introduction of an intermediate rotation matrix [IRM] such that:

25

{T} = [IRM] {F} .....(2-27)

To develop [IRM] the effect of the twisting of the member will be
neglected, the rotation angles will be assumed to be small enough

so that cos A'; 1, sin A': A and sin Al x sin A2 0. With these

assumptions [IRM] appears as

F 1 A1 A2:
-A1 1 O : 0
-A2 0 1 :
: 1 A1 A2
0 '5 -A1 1 o
_ : -A2 0 1 J .

 

 

2.3.4 Rotation to Joint Global CoordinateS.--The solution
for the member forces is completed by rotating the member end
forces in member coordinates, {T}, to joint global coordinates using
the transpose of the appropriate rotation matrix as outlined in
Section 2.2.2. After that, as mentioned previously, at a given
joint, the forces of the incident members are summed to yield the

corresponding joint reaction.

CHAPTER III
DYNAMIC ANALYSIS

3.1 General

The equation of motion (Equation (2-1)) was introduced in
Section 2.1 Chapter II was devoted to the solution for the joint
reactions {RE} in that equation. This chapter will discuss the for-
mation of the mass matrix [M] and the solution for the damping
force {DF}, and will present the procedure used to solve the equa-

tion of motion to obtain the response of the structure.

3.2 Formation of Mass Matrix

The procedure used to form the mass matrix was developed
to permit a "lumped" mass system that accounts for rotatory inertia.
The variables used in the dynamic solution are the displacements of
the joints, hence the procedure lumps the mass of the frame at the
joints. It is assumed that associated with each joint is a rigid
body made up of one-half of the length of all members incident to
the joint as shown in Figure 3.1. This typical joint mass has a
"mass center" that is generally not located at the joint. The
problem then is the determination of the properties of the rigid
body, or joint mass. It is convenient to determine the properties
about the joint mass center and to transform these to the coordi-

nates of the joint, in which the equation of motion will be treated.

26

27

Initially, the mass matrix for each % member is computed.
For descriptive purposes each % member will be called a "branch".
The branch mass mB for branch i is determined by

L

mBi = (p Ax+DM)-; .....(3-1)
where p is the density and DM is the superimposed dead load mass per
unit length. The branch inertia tensor about the branch mass cen-
ter, in member coordinates, [ITB], is determined next. Since pris-

matic members are assumed, the member axes will also be the princi-

pal axes of inertia.

 

 

"A. H N“
[1TB]i = H B F .....(3-2)
1 N F C_
and
2 2 9L
A - vol. p(z +y )dV = ;—'(Iy+Iz)
2 2 2
B — Ivol. p(z +x )dV + flan. DM x dL
L3 pL
=--(pA+DM)+-I .....(3-3)
96 x 2 Y
2 2 2
c — fvol. p(y +x )dV + f1en. DM.x dL
L3 pL
=— (pA +DM) +--I
96 x 2 z

28

where it has been assumed that the superimposed dead load is concen-
trated as a line along the longitudinal member axis, and all cross-
sectional properties are for member i. The products of inertia F,

N and H are zero Since the member axes are principal axes.

The joint mass m, is computed next

k
m=2 .....(3-4)
1:1 mBi

where k is the number of branches incident to the joint being
considered.
The global coordinates of the joint mass center are computed

by taking moments about the joint global axes

k

2
_ i=1 m31x1

Y = .....(3-5)

where (Xi, Yi’ Z1) =Vi is the position vector of the ith branch

mass center in joint global coordinates. Vi is determined by

‘ vi = 2 (Pi-Ii) .....(3-6)

where F1 and I1 are the position vectors of the final and intial

ends of member i in structure global coordinates.

The elements of the individual branch inertia tensors are

29

transferred to the joint mass center using the parallel axis theorem

([ITB]i)J.M.C. = h b f ..... (3-7)

 

 

and

A + m(y2+22)

9)
II

b = B + m(x2+zz)

c = c + m<y2+x2)
o 0000(3-8)
f = -myz
n = 'mXZ
h = -mxy

where (x,y,z) = U is the position vector of the joint mass center

i
relative to the ith branch mass center in member coordinates and

is determined by
Ui = [RMT]i (Vi-W) .....(3-9)

where [RMT]i is the three dimensional rotation matrix for member 1
and W = (X,Y,Z) is the position vector of the joint mass center in
joint global coordinates.

The branch inertia tensors expressed at the mass center are
then rotated to joint global coordinates and summed to obtain the

joint inertia tensor at the mass center [IT].

30

k
[IT] =i§1 [RMT]: ([ITB]i)J M C. [RMT]i .....(3-10)

The joint mass matrix at the mass center, [M]J M C can

then be assembled

 

 

rm 0 o l T
o m o l 0
I
o o m I
[M]J.M.C. = ------ T —————— .....(3-11)
0 1 [IT]
L i _

and is then inverted.

The inverted joint mass matrix at the joint mass center is

now transferred to the joint,

[M131 = [H]t [MJEM'C' [H] .....(3-12)

where [H] is a force transformation which is explained as follows.
The transformation matrix [H] is defined as a matrix such

that [H] transforms a force vector acting at the joint to an equiva-

lent vector acting at the joint mass center in similarly oriented

coordinates, that is,

{F}J[M.C. = [H] {F]J .....(3-13)

If the space coordinates of the joint mass center in joint global

coordinates are X, Y, Z, then

31

 

 

1 0 0| 1
|
O 1 O I O
I
o o 1:
[H]='- ------- I' .......... (3-14)
0 Z-Y:l o o
-zo xlo 1 o
I
Y -X 0 I 0 0 l J
_ I

The relationship between displacements corresponding to

these force sets then is

{x}J = [H]t {x}J.M.C. ..... (3-15)

It follows that

{5% = [Hlt {§}J.M.C. .....(3-16)

The equation of motion of the joint about the joint mass

center may be rearranged to form

[MJJ.M.C.{§}J.M.C. = {P}J.M.C. ' {RE}J.M.C. " {DF}J.M.C.

= {R}J.M.C. ..... (3-17)
where {R} is a resultant force vector. Then
{i}. - = [M]'1 {R} <3-18)
J.M.C. J.M.C. J.M.C. "°'°

Substituting the relationships in Equations (3-13) and (3-16)

{5:}J = [H]t [mime [H] {R}J ......(3-19)

32

From which it follows that

pa]? = [Hlt [MJETMfi' [H] .....(3-20)

The mass matrix of the structure [M] is defined as a diago-
nal matrix containing the individual 6 x 6 joint mass matrices, com-

puted as outlined above, as the diagonal elements.

3.3 Damping

The exact form of damping in most structures is practically
unknown. However, damping forces are generally relatively small so
that it is not essential that they be represented precisely in a
mathematical analysis. For these reasons a simplified form of
damping is assumed. A viscous (resistive force proportional to
velocity) type of damping is assumed in this investigation, and the
damping coefficient matrix, [C] is assumed to be proportional to the

mass matrix. Writing [C] - 1[M], the equation of motion becomes

[“1 {35} + MM] {5!} + {RE} = {1’} .....(3-21)

3.4 Numerical Solution Procedure Used In Dynamic Analysis

3.4.1 Solution Procedure.-—The procedure used in the dy-
namic solution will be described in this section. Assume that the
state of the frame is known at t = t1. This includes the displace-
ments, velocities and accelerations of the joints, and the internal
forces of the members corresponding to the displacements. It is
desired to determine the state of the frame for t = t1+At where At

is one time increment. Knowing this,the same procedure can be used

to solve for the state of the frame at time t = t1+2At, and so on.

33

The solution process for the typical step from t to t1+At is out-

1
lined as follows:

1) Levy's numerical integration formula,

2
X = 2X - X + mt) 3‘. 000.0(3-22)
tl‘i’flt t1 t1 At 1:1

is used to obtain the displacements at t = t1+At where
the subscripts refer to the time at which the vari-
ables are to be evaluated.

2) The frame joint reactions {RE} are determined for
t = t1+At from the displacements computed in step 1,
using the procedure outlined in Section 2.3.

3) The damping force is computed using the damping matrix
given in Section 3.3. In order to save computation
time, by avoiding iteration, which would otherwise be
necessary, the velocities at t = t are used to compute

1

the damping force at t +-At; i.e.,

1
{1)F}t1mt = )([M] {5:}t1 ..... (3-23)

This was thought justifiable because of the approxi-
mate nature of the damping representation and the fact
that velocities do not change a great deal in one time
increment.

4) The equation of motion is solved for the accelerations

at t1+At:

{fltlflt =[M1-1 [{P}t1+At ' {RE}t1+At ' {DF}t1-IAI:]

.;...(3-24)

34

5) The velocities are computed using the Euler-Fox

numerical integration formula

At
.2 = a. + — [as + a: ]
t1+At t1 2 t1 t1+At

.....(3-25)
These 5 Steps then complete the solution for all the variables at
time = t1+At, and a similar solution process may be repeated for
the next advance into the time domain.

3.4.2 Time Increment Used in Numerical Integration.--It is
well known that in the solution of problems of structural dynamics
by numerical integration the time increment must be less than a
certain fraction of the smallest period of vibration in order to
assure stability of the integration procedure. For Levy's formula,
this fraction is l/n (20). Strictly speaking, this applies only to
linear problems.

In order to obtain the smallest period, a solution of the
eigenvalue problem would be necessary. This would be time consuming
and a quick approximate estimate is desirable. To estimate the
smallest period of a frame the following approximation is suggested.
The translational mass is estimated for all free joints and the
greatest axial stiffness for any member incident to each joint is

used in the expression for the "period", Ta:

 

 

= mass _
Ta Zn’V/axial stiffness "°°°(3 26)

The smallest T8 for all members is used as an estimation of the
smallest natural period, Ts' This approximation was used for the
stiffer cubic frame Shown in Figure 4.1 to yield TS = 0.0087 seconds

versus 0.0052 seconds obtained from a modal analysis program. The

35

approximation yields 0.0093 seconds for the flexible frame shown in
Figure 4.1 versus 0.0041 seconds from the model analysis. While in
error by a factor of approximately 2, these approximations would
still give a usable first estimate of the integration interval, if
a generous factor of 0.1 or less is used in lieu of the coefficient
l/n mentioned earlier.

3.4.3 Change of Variables for Ground Motion Problems.--If
the input of the problem is ground or support motion (assumed trans-
lational motion only), it is convenient to define the joint displace-
ment {x} as relative to that of the ground {x0}. The absolute joint
displacement is then {x} + {x0}. Therefore, the equation of motion

becomes

[M] {52} + [c] {)2} + {RE} = {P} - [M] {520} .....(3-27)

in which {20} is the ground acceleration. It is seen that -[M] {*0}
plays the same role as an external loading. In this equation the
damping force is assumed to be proportional to the relative velocity

{x} and [C] = 1[M] as before.

3.5 Computer Program

An outline of the program developed for this study is pre-
sented in this section, the program itself is given in Appendix 2.
The major steps in the program are described in the order in which
they are executed:
l) The basic physical information concerning the frame is
input. This information includes the locations of all

joints (in structure global coordinates), the incidence

2)

3)

36

relations and the geometrical and mechanical properties
of each member. The input of member properties is
presently limited to prismatic members with an I cross-
section, although it is a very simple matter to modify
the program to include any other cross-section so long
as the members are prismatic. Since only I sections
are considered, the depth, flange width, flange thick-
ness, and web thickness are input for each member.
Also, the program is presently developed to use a sin-
gle modulus of elasticity, shear modulus and density
for all members; however, the computational portion of
the program is set up to use separate properties for
each member if this is desired. Any superimposed dead
loads in addition to the weight of the members and any
static loads on the joints are also input at this time.
Only concentrated joint static loads and uniform super-
imposed dead loads may be considered.

From the information input in l, the member stiffness
matrices and the member rotation matrices are computed.
Then the joint stiffness matrix is assembled. As out-
lined in Section 2.2.2 only the nonzero upper-diagonal]
6 x 6 submatrices of the joint stiffness matrix are
generated. A "locator" matrix is used to indicate
which of the 6 x 6 submatrices are nonzero and where

it is stored.

The frame is next analyzed statically for displacements

4)

5)

6)

37

and forces due to dead loads and any other static joint
loads. The dead loads are treated by computing the end
forces on each member due to its own weight and any
additional superimposed uniform vertical dead load
which may be acting on the beams. The negative of
these end actions for all members incident to a given
joint are then summed and taken as a fictitious joint
load to be used in the Static solution. The diSplace-
ments and forces so determined are taken as the initial
conditions for the dynamic solution.

The mass matrix is computed using the procedure out-
lined in Section 3.2.

The necessary data to perform the dynamic analysis is
input. This includes the integration interval, how
long the integration should continue, how often the
"running" information should be output and what infor-
mation should be monitored for maximum and minimum.

The dynamic loading information is input. Two types

of dynamic loading are included in the system devel-
oped. One type is base motion (earthquake loading),
the second is a dynamic forcing function applied to

all specified free joints. Both types utilize a dis-
crete input method to allow storage of a loading ma-
trix in the core memory of the computer to decrease
computation time. The loading is input for trans-

lational coordinates only and is input at every 0.01

38
second in three dimensions. If intermediate values are
needed, a straight line interpolation procedure be-
tween the discrete points is used.

The base motion input is developed from the ground
acceleration records of the El Centro, California, May
18, 1940, earthquake. The North-South, East-West and
Up-Down directions were used. These records were
converted to input data in a format such that the
accelerations in the order of directions to be loaded
for the particular investigation were assembled in
sets of three with one set for each 0.01 of a second
of elapsed time. If needed, straight line interpol-
ation of the random accelograms is used.

To increase the capabilities of the program it is
also possible to introduce a time lag in the base motion
application to any of the support joints desired. The
displacement of these lagging support joints must be
computed relative to the displacements of the "base"
support joints. If {x0} is taken as the displacements
of the "base" support joints and {xL} is taken as the
displacements of the lagging support joints the rel-

ative support joint displacement is the difference,

{le = {XL} - {x0} .....(3-28)

The diSplacements of the support joints are computed

by integrating the earthquake accelogram (using Levy's

7)

8)

39

numerical integration method).

The forcing function input is Specified so that,
as in the base motion case, the three coordinates of
the forcing function may be input at 0.01 second inter-
vals. This allows a wide variety of forcing functions
in all three dimensions to be used.

The dynamic analysis is accomplished using the proce-
dure outlined in Section 3.4. Information is output
at designated intervals during the solution to permit
a running record of the displacements of one joint and
the initial end forces of one member. The displace-
ments of one coordinate is also plotted at this same
interval. Designated displacements and forces are
monitored for maximum and minimum.

The maximum and minimum values determined for the

specified forces and displacements are output.

CHAPTER IV

APPLICATIONS

4.1 General

Several studies of structural behavior were made using the
computer program which embodies the solution method developed in the
previous chapter. Initially a comparison with a linear solution by
a separate computer program using the normal modes method was made
to verify the reliability of the method. The system is then used
to demonstrate its applicability to the solution of nonlinear static
problems by use of sufficiently large damping coefficients. Such a
solution is compared with some published data on a nonlinear static
problem.

The program is then used to solve a dynamically loaded
cubic frame, first experiencing large geometry changes and then with
large axial loads in the columns. Linear and nonlinear analyses are
compared for both cases. Next an earthquake analysis is presented.
To highlight the three-dimensional features of the behavior of a
space structure, a cubic frame is studied for the effects of unsym-
metrical mass distribution, and for variation of frame stiffness,
both under earthquake loading. An analysis using a torsional type
of loading is also presented. Finally, the variation of the re-
sponse of a structure due to a time lag in the application of the

earthquake shock wave is considered.

40

41

4.2 Comparison with Modal Analysis Solution

Although each part of the program was checked individually,
and certain sections, for example, the Statics part, were verified
completely, it was deemed desirable to obtain a check on the dynamic
part of the program. For this purpose another computer program was
developed for the solution of dynamically loaded linearly elastic
frames using the method of normal modes (see, for example, Hurty
and Rubinstein (10)). The stiff cubic frame shown in Figure 4.1 was
analyzed with both solution methods for a step function loading with
a magnitude of 10 kips applied on joints 1, 2, 3 and 4 in the posi-
tive X-direction. As this loading caused sufficiently small dis-
placements and internal forces, the nonlinear solution should agree
with the linear normal modes solution.

The translation of joint 1 in the positive X-direction is
shown in Figure 4.2 for approximately four fundamental periods of
vibration. The maximum difference between the displacements from
the two solutions was less than 1% of the maximum displacement, and

the two solutions essentially coincide when plotted.

4.3 Static Displacements Using Damped Dypamic Solution

One of the interesting uses of the solution method presented
here is the application to nonlinear static problems. A static solu-
tion may be accomplished by using a monotone-increasing dynamic load
(a step function was used here) which attains a maximum magnitude
equal to the corresponding static loading. The damping coefficient
is made sufficiently large to cause the oscillation of the structure

to decrease rapidly to a displacement configuration equal to the

42

desired static one.

To illustrate the effect of the magnitude of the damping
coefficients, the stiff cubic frame was Studied with 10 kip loads on
joints 1, 2, 3 and 4 for damping coefficients of I = 10, 40 and 60
kip-in. / sec. (see Section 3.3). The results for the displacement
of joint 1 in the X-direction are shown in Figure 4.3. Since the
displacements are small, little nonlinear effect should be present.
It is noted that a rapid approach to the linear static solution with
little oscillation was obtained with the damping coefficient of 60
kip-in. / sec. which is approximately twice the eigenvalue (31.5)
corresponding to sway displacements in the X-direction. This is
essentially the critical viscous damping of the structure in the
first mode.

To further illustrate the method, the nonlinear static be-
havior of the plane frame shown in Figure 4.4 was considered. This
problem had been studied by Saafan (17). The results from the anal-
yses with the method presented here are compared with Saafan's re-
sults in the same figure. The agreement was good, the maximum dif-
ference within the range of loads considered being approximately 10%.
The difference in the two solutions is thought to be due to the fact
that the method presented considers the intermediate rotations in the
force orientation, which Saafan's analysis does not, and that Saafan
considers the axial force effects in his bowing corrections, which
this solution does not. This comparison also provides further veri-

fication of the reliability of the solution procedure developed.

43

4.4 Nonlinear Comparisons

The difference in the frame response as computed by the
nonlinear analysis presented and by a linear analysis was studied to
determine the areas where Significant variations would appear and to
determine the relative magnitudes of such variations. The linear
analysis was obtained by deleting the nonlinear considerations used
in the solution for frame joint reactions.

In the following two sections the nonlinear effects of
geometry changes and (compressive) axial loads are considered. It
is recognized that these effects cannot be really separated. How-
ever, the loading for each study was chosen so as to accentuate the
effects under study.

4.4.1 Effect of Geometry Changes.--The study was made to
consider the effects of geometry changes on the response of the
structure. The flexible cubic frame was subjected to a step-function
loading applied on joints 1, 2, 3 and 4 in the positive X-direction.
All displacement components of joint 1 were monitored for maximum
and minimum. The significant results for 15 kip and 20 kip loads
are shown in Table 4.1. The greatest variation is seen to be in the
vertical displacement of joint 1. This is attributed to the more
accurate consideration in the nonlinear analysis of the axial dis-
placement of column 6 (which is directly under joint 1).

Various internal forces throughout the frame were monitored
and the results from the linear and nonlinear solutions are also
shown in Table 4.1 for some of the significant forces (forces with

relatively large magnitude) which showed differences. The variation

44

in the more significant forces was less than 5% even for the 20 kip
loading which produced a sway ih the X-direction of approximately

0.3 of the column height. The moment about the Z-axis at the initial
end of column 6 and the moment about the Y-axis at the initial end of
beam 4 are shown as representative. The axial forces in most members
showed considerable variation and again this is attributed to the
more accurate solution for axial displacement that is used in the
nonlinear solution. The bending moment about the X-axis at the
initial end of member 6 exhibited substantial variation (19% maxi-
mum), but is of minor importance because of its relatively low
magnitude.

It was found during this investigation that the nonlinear
solution for axial force is quite sensitive to accuracy in the dis-
placement solution. A shorter integration interval had to be used
in order to obtain accurate axial forces.

4.4.2 Effect of Compressive Axial Forces.--The investi-
gation of axial force effects was made using the flexible frame, Sub-
jected to a step function loading in the X-direction on all free
joints. The magnitude of the step function was held at 10 kips. In
addition, identical vertical static loads were applied to all four
free joints and the magnitude of these static loads was increased to
approximately the Euler load of the columns (42 kips). The same
displacements and forces were monitored for absolute maximum as in
the previous section. Representative results are shown in Figures
4.5, 4.6 and 4.7.

Linear dynamic solutions for all magnitudes of vertical

45

static loads Showed no Significant variation from the solution with
no static loads, for the response variables monitored (with the
obvious exception of the maximum Y-displacement of joint 1 which
increased as a result of the direct compression of the column by the
statically applied load).

In general, the nonlinear analyses showed larger response
than that of the linear analysis. The difference increases with an
increase in the axial loads. Figure 4.5 shows the variation in dis-
placements of joint 1 in the X and Y-directions. Considerable vari-
ation between the nonlinear and linear solutions was found with in-
creasing axial loads as may be noted in the figure. Figure 4.6
shows two of the and forces of column 6. Again considerable increase

(70%t) occurs. Figure 4.7 shows two initial end forces in beam 1.

4.5 Earthquake Loading

To illustrate the application of the method developed to an
earthquake response problem, both the flexible and stiff cubic
frames were subjected to the well known 1948 El Centro earthquake
for a 5 second duration. The ground motions in all three dimensions
were taken into account, with the X-axis corresponding to the North-
South direction of the earthquake input.

To more closely simulate realistic conditions, a superim-
posed beam dead load of 0.03125 kips per inch was applied to members
1, 2, 3 and 4 to approximate a roof or floor dead load, and a damping
coefficient of 1.8 kip-sec. / in., approximately 3% of the "critical"
damping factor defined in Section 4.3, was used. The results are

shown in Figure 4.8.

46

The maximum displacement and forces for certain joints and
members which are considered significant are shown in the chart, at
the bottom of the figure. It is seen that the same earthquake pro-
duces quite different responses in the two different structures
which have similar mass distribution but considerably different stiff-

ness properties.

4.6 Effects of Dissymmetry in Distribution of Mass, Stiffness.
and Loading

The effects of dissymmetry in the frame were investigated
for both mass distribution, stiffness variation, and loading pattern.
The frame shown in Figure 4.9 was used with the flexible columns and
beams shown in Figure 4.1. The superimposed dead load was set at
10 lbs. per inch. The loading used was one second of the El Centro
earthquake from 1.5 seconds to 2.5 seconds. This portion was used
since it produced the most violent motion during the long term re-
sponse studies reported in Section 4.5.

Mass dissymmetry was introduced by varying the superimposed
beam dead load on members 1 and 3, but keeping the sum of the dead
loads on both beams at 20 lbs. per inch. The dead load is included
in the mass lumping procedure, and consequently this produced unsym-
metric mass distribution. The results are shown in Figure 4.10. The
greatest variations in the forces studied occurred in the column
bending moments. Columns 6 and 7 are shown as representative. AS
more mass is shifted from beam 3 (connected to column 7) to beam 1
(connected to column 6), considerable increase in the maximum bending

moment in column 6 was experienced with a similar decrease in the

47

maximum bending moment on column 7.

The maximum twisting moments on the columns also increased
with the dissymmetry in mass distribution. The graph for column 5 is
shown as a representative example in Figure 4.10. The maximum hori-
zontal bending moment in the beams increased approximately three
times, and the results for member 4 are shown in the same figure.

The results illustrate that mass distribution is a very significant
consideration in determining the stresses experienced by the members
of a structural system under an earthquake loading.

Dissymmetry was introduced in the stiffness of the frame by
replacing members 3 and 7 (in Figure 4.2) with members 3 and 8 in the
stiff set of members shown in Figure 4.1. Again the maximum twisting
and bending moments in the columns varied considerably. The values
for column 5 are shown in Table 4.2 along with the maximum horizontal
bending moments in beam 4 and the maximum displacement of joint 1 in
the X-direction.

A point to be observed here is that if the bent composed of
members 5, l and 6 is considered alone considerable difference in the
forces it experiences results depending on which of the three dimem-
sional structures it is actually a part. For purposes of comparison
this bent was analyzed as a plane problem subjected to the X-com-
ponent of loading used above. The results are shown in the last
column of Table 4.2. It is of interest to note that the results are
closer to those of the unsymmetrical frame.

As a possible means of estimating the variation in struc-

tural dynamic response caused by unsymmetrical stiffness, static

48

analyses of both the modified and original three dimensional frames
were made with 10 kip loads on joints 1, 2, 3 and 4 in the positive
X-direction. The ratio of the displacement of joint 1 in the X-direc-
tion for the unsymmetric frame versus the symmetric frame was found

to be 1.10. The ratio of the same maximum displacements in the dy-
namic analysis was 1.34, an increase of approximately 20% over the
static case.

To consider the effects of loading pattern, the same frame
was loaded with 10 kip step loads on joints 1 and 4 in opposite direc-
tions in one case, and in the same direction in another as Shown in
Figure 4.11. The maxima of the X-displacement of joint 1, the twisting
moment and bending moment about the Z-axis for column 6 and the
bending moment about Y-axis for beam 4 for the two solutions are
shown in the table in the figure.

As would be expected, the twisting moment in column 6 and
Y-bending moment in beam 4 are considerably higher for the reversed
loading case. However, the relatively large values of the displace-
ment and column bending moment which occurred in the reversed loading
case as compared with the symmetrical case were unexpected as an
essentially torsional mode response was anticipated. To explain this
result the normal modes were obtained for the frame investigated.

The first four modes and their periods are Shown in Figure 4.12.
These modes all primarily involve sway displacements of the columns.

It is of interest to note that the first torsional mode has
a larger period than that of the translational mode in the X-direction.
The large X-diSplacement of joint 1 and bending moment in column 6

would indicate that the response is mainly in the fourth mode.

49

A further examination of the time-history plot of the response (for
its "periodicity") confirms this observation.

The foregoing would indicate that in the presence of marked
dissymmetry in a three-dimensional structural-load system, the be-
havior of the system can be accurately predicted only through a

three dimensional analysis.

4.7 Effect of Time Lag of Support Movements

In this section the method of solution was used to consider
the effect of the finite Speed of travel of an earthquake shockwave
through the foundation material. The method of including this time
lag in the analysis is discussed in Section 3.5. It is recognized
that this is an approximation of the true situation since foundation
damping, secondary wave fronts and similar effects are ignored, but
the purpose here is to show a method of considering this phenomenom
and to obtain a quantitative assessment regarding its effect on a
simple three dimensional structure.

This study again used the cubic frame with the flexible
members and with the cross member in the top level deleted as shown
in Figure 4.9. Loading was one second of El Centro earthquake
starting at 1.5 seconds and using all three dimensions. A super-
imposed dead load of 10 lbs. per inch on beams l, 2, 3 and 4 was
used to simulate a roof or floor loading. The time lag was applied
to support joints 6 and 7 which would correspond to a shock wave
moving in the positive X-direction. A speed of seismic wave of
1000 ft. per second is used assuming a sandy soil (8). The 20 foot

span of the frame then yields a time lag of 0.01 seconds.

50

The maximum absolute values of the displacements of joint 1,
the initial-end forces of members 1, 2 and 4 and moments at the ends
of all columns were monitored. The variation of representative
forces and displacements are shown in Table 4.3. Variation in the
column end moments as illustrated by the moment about the X-axis at
the initial end of column 8 was not large with a maximum of approxi-
mately 10%. Other variations of somewhat larger percentage, but for
forces of lesser magnitude may be noted in the figure. Differences
in displacements from those corresponding to zero lag time were
generally small except for the case of 0.01 second lag time. In the
latter case, a difference of 26% and 130% are noted for joint 1 in

the X-direction and joint 2 in the Z-direction, respectively.

CHAPTER V

SUMMARY AND CONCLUSIONS

The report has presented a method of analysis for dymani-
cally loaded elastic Space frames considering the nonlinear effects
of geometry changes and axial thrust. The solution method has been
embodied in a computer program written in FORTRAN. The program has
been used to study the behavior of a simple cubic frame under various
conditions.

The analysis uses essentially a lumped parameter model.
Numerical integration is utilized to obtain the transient response
of the frame. The nonlinear geometry and axial thrust effects are
accounted for in the determination of the frame reactions corre~
sponding to the joint diSplacements. Consideration of geometry
changes leads to corrections in the relative member displacements
for rotations of the member chord and for apparent axial shortening
due to bowing. The flexural stiffness coefficients are continuously
revised to reflect the beam-colhmn effects of the axial thrust.

The solution utilizes a mass lumping procedure which accounts
for rotatory and translational inertia.. Mass proportional damping
may be included in the solution. The intial conditions for the dy-
namic analysis are determined by a linear static analysis of the
frame under dead loads.

The computer program is verified by comparing a nonlinear

51

52

solution under practically linear conditions with a solution obtained
by the method of modal analysis. In addition, good agreement was
found between a published nonlinear static analysis of a plane frame
and the results obtained by the present program using a large
damping coefficient in the dynamic analysis. This serves not only
as a verification of the nonlinear aspects of the solution, but also
demonstrates a very useful application of the solution method.

The applications of the computer program have also included
studies of a simple cubic frame. These were made to demonstrate the
magnitude of the nonlinear effects considered in the analysis; to
investigate the effects of dissymmetry of mass, stiffness and loading
and to study the effect of time lag in the motion of the supports.
The last two effects can be studied only with a three-dimensional
analysis.

The numerical results Show that the geometric nonlinearities
have relatively small effect on the major internal forces (those
having larger magnitudes). For example, under a step loading which
produced a maximum sway of 30% of the column length, the maximum
column moments change only 5%. However, the axial thrust effect is
quite significant. A variation of 30% in column moments was found
for the case with static column loads equal to 50% of the Euler load.

The studies considering the effects of unsymmetrical mass
and stiffness distributions demonstrated significant variations in
maximum forces experienced throughout the frame as the frame pro-
perties were changed. Variations of 50% to 100% were noted in

various maximum forces and displacements.

53

In conclusion, it has been demonstrated that a three-dimensional,
dynamic, nonlinear analysis of structural frames is practicable with
the use of a computer. Furthermore, the studies made with the com-
puter program have Shown that nonlinear geometry effects are not
critical until quite large displacements occur. However, axial force
seems quite significant. The studies of the various unsymmetrical
frames indicate that three-dimensional analysis can provide signifi-
cant increases in the accuracy of analysis of Space structures.

Future extensions of this investigation could include: (i) a
further study of the application of dynamic analysis using large
damping coefficients to obtain nonlinear static solutions; (ii) the
marriage of this solution with one that includes the effects of non-
linear material properties which is in progress at Michigan State

University.

10.

LIST OF REFERENCES

Clough, R.W., Benuska , K.L. and Wilson, E.L., "Inelastic Earth-
quake ReSponse of Tall Buildings," Proceedings of the Thi£g_
World Conference on Earthquake Engineering, Volume II, New
Zealand, 1965, pp. 11-68-84.

Connor, J.J., Logcher, R.D., and Chen, S.C., "Nonlinear Analysis
of Elastic Framed Structures," Journgl of the Structpral
Division, ASCE, Vol. 94, No. 8T6, Proc. Paper 6011, June
1968, pp. 1525-1547.

DiMaggio, F.L. and Spillers, W.R., "Network Analysis of Struc-

tures," Journal of the Engineering Mechanics Dipision,
ASCE, Vol. 91, No. EM3, Proc. Paper 4376, June 1965.

Gere, J.M. and Weaver, W., Analysis of Framed Structures, D.
VanNostrand Company Inc., Princeton, New Jersey, 1964.

Giberson, M.F., The Response of Nonlinear Multi-Story Structures
Subjected to Earthquake Excitation, A Report on Research
Conducted Under a Grant from the National Science
Foundation, California Institute of Technology, Earthquake
Engineering Research Laboratory, Pasadena, California, 1967.

Fenves, S.J., and Branin, F.H., "Network-Topological Formulation

of Structural Analysis," Journal of the Structural Divisiqn,
ASCE, Vol. 89, No. ST4, Proc. Paper 3611, August 1963,

pp. 483-514.

Goldstein, H., Classical Mechanics, Addison-Wésley Publishing
Co. Inc., Reading, Mass., 1950.

Hardin, B.O. and Richart, Jr., F.E., "Elastic Wave Velocities in

Granular Soils," Journal of the Soil Mechanics and.
Foundations Division, ASCE, Vol. 89, No. SMl, Proc. Paper

3407, February, 1963, pp. 33-65.

Housner, GQW. and Jennings, P.C., "Generation of Artifical Earth-
quakes," Journql of the Engineering Mechanics Division,
ASCE, Vol. 90, No. EMl, February 1964.

Hurty, W.C. and Rubinstein, M.F., Expamics of Structures,
Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1964.

54

11.

12.

l3.

14.

15.

16.

17.

18.

19.

20.

21.

22.

55

Jennings Alan, "Frame Analysis Including Change of Geometry,"
Journal of the Structural Division, ASCE, Vol. 94, No.
8T3, Proc. Paper 5839, March 1968, pp. 627-644.

Lee, S., Manvel, F.S. and Rossow, E.C., "Large Deflections and
Stability of Elastic Frames," Journal of the Engineering
Mechanics Division, ASCE, Vol. 94, No. EM2, Proc. Paper
5895, April 1968, pp. 521-547.

Livesley, R.K., MatrixMethods of Strpctural Analysis, Pergamon
Press, London, England, 1964.

Massachusetts Institute of Technology, Dept. of Civil Engir
neering, STRESS: A Users Manual and STRESS: A Reference
Manual, The M.I.T. Press, Cambridge, Mass., 1965.

Renton, J.D., "Stability of Spaceframes by Computer Analysis,"
Journal of the Structural Division, ASCE, Vol. 88, No.
ST4, Proc. Paper 3237, August 1962, pp. 81-103.

Saafan, S.A., "Nonlinear Behavior of Structural Plane Frames,"
W. ASCE: Vol- 89. N0-
ST4, August 1963, p. 557.

Saafan, S.A. and Brotton, D.M., "Elastic Finite Deflection
Analysis of Rigid Frameworks by Digital Computer,"
Proceedings of the Symposium op the Use of Computers in
Civil Engineering, Vol. I, Lisbon, Portugal, 1962,
pp. 6.1-6.22.

Salvadori, M.G. agd Baron, M.L., Numerical Methods in Engi-
neering, 2n Edition, Prentte-Hall Inc., Englewood Cliffs,

New Jersey, 1961.

Tezcan, S.S., "Earthquake Analysis of Space Structures by

Digital Computers," Proceedings of the Third WOrld Con-

ference on Earthquake Engineering, Volume II, New
Zealand, 1965, pp. II-631-647.

Tung, T.P. and Newmark, N.M., A Review of Numerical Integra-
pipn Methods for Dynamic Response of Structures; Techni-
cal Report to Office of Naval Research, Contract N6ori-
O71(06), Task Order VI, Project NR-064-183, Civil Engi-
neering Studies, Structural Research Series No. 69,
University of Illinois, Urbana, Illinois, March 1954.

Weaver, W., Computer Programs for Structural Analysis, D.

VanNostrand Co., Inc., Princeton, New Jersey, 1967.

Wen, R.K., "Unpublished Class Notes on Structural Dynamics,"
Department of Civil Engineering, Michigan State University,
Summer Term, 1968.

56

23. wen, R.K. and Janssen, J.G., "Dynamic Analysis of Elasto-
Inelastic Frames," Proceedings of the Third WOrld Con-
ference on Earthquake Engineering, Volume II, New
Zealand, 1965, pp. 11-713, 11-729.

24. Zarghamee, M.S. and Shah, J.M., ”Stability of Spaceframes,"
Journal of the Engineerinngechanics Division, ASCE,
Vol. 94, No. EM2, Proc. Paper, 5878, April 1968,
pp. 371-384.

57

Table 4.1.--Variation in Forces and Displacements in Geometry Change
Comparision

All values maximum absolute experienced during 1 second of vibration.
Displacements in inches and radians and forces in kips and kip-inches.

 

 

 

 

 

 

 

 

 

 

 

 

 

15 kip loading 20 kip loading
Linear Non-linear Linear Non-linear

vertical dis?‘ 0 042 7 28 0568 12 53
of joint 1 ' ' ° '
DiSp. of joint 1
in X-direction 56.72 55.32 75.63 72.48
Rotation about
X-axis of joint 1 0.0162 .0225 0.0218 0.0335
Rotation about
Y-axis of joint 1 0.0775 0.0861 0.1034 0.1169
Rotation about
Z-axis of joint 1 0.0852 0.0894 0.1132 .1186
Axial force in
col. 6 (tension) 25.50 29.55 34.20 53.91
Axial force in
beam 1 (tension) 5.51 9.71 7.33 28.97
Mbm' ab°“t Z'aXiS 3457 3540 4610 4825
init. end col. 6
¥°T' ab°ut Y'aXis 565.0 583.9 754.6 766.9
lnlt. end beam 4
Mom. about X-axis
init. end col. 6 125.42 147.65 166.8 206.7

 

 

 

 

 

 

58

Table 4.2.--Effect of Dissymmetry in Stiffness

(Maximum Absolute Values)

 

 

 

 

 

 

Item Sym. Frame Unsy. Frame Plane Frame
Twisting mom. col. 5

(kip-in.) 17.85 38.84

Bending mom. Z-axis

col. 5 (kip-in.) 131'7 180' 188'
Bending mom. Y-axis

Displ. joint 1 in

X-dir. (inches) 2'41 2'23 2-51

 

 

 

 

 

Table 4.3.--Time Lag Studies

(Maximum Absolute Values)

 

 

 

 

 

 

 

 

 

 

 

 

 

Time X mom. Z mom. Y mom. X displ. Z diSpl.

Lag col. 8 col. 8 beam 2 of of

(sec.) joint 1 joint 2
0.0 18.73 122.2 16.73 2.41 0.487
0.01 18.62 143.0 17.92 3.04 1.15
0.02 18.57 134.1 17.52 2.48 0.505
0.03 18.47 139.3 14.90 2.49 .513

 

Time lag is the delay in

6 and 7.

application of earthquake load to joints

 

59

 

r/,Structure

, 1’ ix

 

 

 

 

 

 

 

Structure Global Coordinate Axes

Joint I

   
  

Member K

 

 

Member Coordinate Axes

Figure 1.1 Coordinate Systems

60

 

 

 

 

Intermediate Deflections in x-y Plane

Deflected Axis of Member

   
  

 

 

u
OF 4 Undistorted
K: Position
j 1 x
/ I /

—.—’

x-z Projection

Deflected Member in Three Dimensions

Figure 2.1 Member Relative End Displacements

61

 

7

E '
Unloaded- I
Position \\\\\4

rad.

33 X 103 psi
423 in.

 

 

LI_L = 200”

L_

Cantilever Column Investigated

 

 

u _ u )2 Bowing Bowing Total Total 7
I 2 F 2 Ax. For. Not Axial Force No With Diff
2L Considered Considered Ax. For. Ax. For. '
1.000 0.333 0.501 1.333 1.501 12.5

 

 

 

 

 

 

 

 

Values in inches

Figure 2.2 Relative Magnitude of Geometry Correction Terms

62

.L‘
—‘

T
1.2 IF2

IFl {/
T l
I 1 "\‘é \ \ x-y Projection
F <’
I 6 ‘\
A1 1

\\ I
\.-_- ,5'- _,

 

 

Undistorted Member
Location

 

4%”:
A __ // I\
21”, " " l
ITIX/ E V
- x-z Projection
F F I
I 1 I 5 :\‘ F
I 3
I 3

Note: Only the initial end
forces are labeled.

 

Figure 2.3 Rotation of Member End Forces Due to Chord Rotation

63

 

.4
WII
hie-J

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4+
llll
I|.|
I
H l
.4
Joint
F‘ V
.4
4“ / 1r __
r r \A-_E
' 7
\\Joint Mass Center
N \\
.4
FINN \
4:?
Typical Joint
Joint J

 

 

Typical Branch

Figure 3.1 Rigid Joint Mass

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.1 240"
\ \ n...
1 ,4C) 2 Step Function
"r’F” Loading
4 '0)
'3 1%) I
~¢
N
4 """"i
N\“rTypical Beam
Cross.section
Orientation
AL— ;5 jL6———X
(3
K :‘V‘A
Typica1.001Umn
Cross section
Orientation
8
A» J,
\2
Flexible Members Stiff Members
Memb. Flange Flange Total web Flange Flange Total web
No. Width Thick. Depth Thick. Width Thick. Depth Thick.
Q) 5.25 0.313 8.00 0.250 10.00 0.62514.00 0.375
® 1! II n u u n n n
® I! n u u n u n 11
C4) 11 n n H II n n 11
© H n n n n n n 11
(Q 5.00 0.375 5.00 0.250 '-' 0.50010.00 "
® II II II II II H H II
n n n n n n n n
H H H H H H H H

 

 

 

 

 

 

 

 

 

 

 

Figure 4.1 Cubic Frame

65

.mcowusfiom
osu mnu aw mocmummmwv
manmofiuo: 0: ma mumcu

xomusoom waauuoHd cwnuwz

0.0 5.0 0.0

mfimzfimc< Hmvoz saws vonumz.:owuaaom mo comwumaaoo N.q madman

. . .mumnams mmaum nuws cacao "mamum
¢.m.~.a muCHOH co cowuo::m amum M 0H ”wood

“muoz mvcoomm ca mafia

m.0 «.0 m.0 «.0 H.0

 

fil¢ _ _

_ _ _ _ _

.umm m0m.0 um :mm0.m .QmHQ .xmz

 

(saqour) uornoalrp-x u; I Juror go nuamaoatdsrq

of Joint l (in.)

X-Displ.

 

66

Loading: 10 Kip Step Load
on Jts. 1,2,3,4
in X-Direction.

 

 

A in kip-in.
sec.
r =
X 40 A = 10 Static Soln.
1.86"
A
K = 60
1 l l 441
O 0.1 0.2 0.3 0.4

Time (in seconds) —>

Figure 4.3 Damped ReSponse of Stiff Cubic Frame

67

 

All members of uniform

 

 

 

 

g section with I = 252 in.3,
11—— A = 13.38 in.2
x
:o
O
N
_!L__ ‘L it
200" v
Plane Frame
0.6_
_ Linear-Elastic
,‘ o;3_ Method presented
a /
CO A ,
'3 Saafan (16), Figure 7
0.40
.2
a /
U... 003'— ,/
o
H
.E.’ 0.2-
a
#1
E:
E; (J 1_
O
l L L l I

 

 

0 20 4O 60 80 100

Vertical Deflection of
Loaded Joint (in.)

Figure 4.4 Static Solution of Plane Frame

of

Max. Absol. Displ.

Max. Absol. Vert.

68

 

 

 

 

 

   

  

 

 

 

6O _

.5}
~1 50 _ Nonlinear (IV/’I”’//,,.——v-
.2. 707.
“U Increase
x 40.. '4;
.5 1LLinear 37.8" n:
H 30 -- .61
u 0
a
'8
'1 20 1 l l I J

0 10K 20K 30K 40K

Vertical loads on Jts. l, 2, 3, 4

10_

51
~/ Nonlinear
H
u .
-§ 5 q:
.3 a.
{+4 0
0 Linear yields negligible :0
a. .
-3 vertical displacements
Q - I I 1 ) J

0 10K 20K 30K 40K

Vertical load on Jts. 1, 2, 3, 4

Figure 4.5 Axial Load Effect on Displacements

69

 
  
 
 
 

 

 

 

 

 

4000 ,
c; l
«4
.35 Nonlinear Ki
8 \D
E a.”
. .
N .3 3000 717. increase —~ .
P: 0 r3
0 'o p
m 1:
gm {Linear 2300 K-in.
0 «H
g 'E' #1
H L I I
2000 0 10K 20K 30K 40K

Vertical loads on Jts. l, 2, 3, 4

 
   

 

 

"? 4000 _

a

w-I

I
. E4,
:§~o
:4: .4
. 8 3000 __ Nonlinear (4
p1 n1
ova m
.215 .
<1 H :3
- m c).
x a
i9 E Linear 2025 K-in.

2000 WK 0K

Vertical loads on Joints 1, 2, 3, 4

Figure 4.6 Axial Load Effects on Column Forces

10F

70

Nonlinear

C01 PEA

 

 

Max. Absolute Axial
Force Beam 1 (KlpS)

00

1
10K

\Linear = 3 . 68K

l
20K

Vertical load on Joints 1, 2, 3, 4

 

   

Nonlinear

1
30K

  
  

[Linear = 2022K"

 

40K

C01. PE\

 

U-

 

 

3500-

5 3000-
'r-I
I
D.
001-!
1554
ov
a
[:11-4
9

353 2500
am
I—l
0'0
cot:
.031
<0
0“
X'El

£.« 2000

0K:

10K

Vertical load on Joints 1, 2, 3, 4

Figure 4.7 Axial Load Effects on Beam Forces

30K

40K

71

 

r-I
.5
'1
41
o
‘A
Fan
a.m
:04:
0H0
Qt:
I-H
p<\u
1 2 3
Time (Sec.) —————

Stiff Cubic Frame Response

X-Displ. of Jt 1
(inches)
I I
hand
C}
<
<4

I
w

l 2 3 4 5

Time (Sec.) -——-—

Flexible Cubic Frame Response

 

 

 

 

 

Stiff Flex.
Transl. Jt l in X-dir. 1.55" 1.96"
Transl. Jt 1 in Z-dir. 0.41" 1.34"
Z-Mom. Init. End Col. 7 1054K” "165K”
Z-Mom. Init. End Beam 1 852K" 261K"

 

 

 

 

 

Comparison of Max. Absolute

Values of Forces and Displacements

Figure 4.8 Response of Cubic Frames to 1948 El Centro Earthquake

72

1?

 

 

Superimposed
“$1 I I I 14?; 3:33
1 . +7 2

 

G)’

 

 

 

 

 

 

 

 

 

 

 

 

 

L
m

‘L.

‘Kl

’\

Note: Column and beam cross-sections used in the

Flexible Members in Figure 4.1 are used here.

Figure 4.9 Frame Used in Three-Dimensional Studies

200 '-

150
:5
-.-4
0
Ga
-.—4
a
4.)
I:
<1)
E
g? 100

30

20

73

   
 
   

Max. Absol. Z-Bend Mom.
Initial End Column 6

Max. Absol. Z-Bend Mom.
Initial End Column 7

  
 
 
  

Max. Abso .
Twist. Mom.
Col. 5

    
 

Initial Beam 4

l l 1 Q

 

10

Figure 4.10

1/2 3/8 1/4 1/8 0

Portion of 20 Ih/in. S.D.L. which is applied to
Beam 3 - Remainder on Beam 1

Effects of Unsymmetric Mass Distribution

74

 

 

 

 

 

 

 

Anti-symmetric Loading

 

 

 

 

__._-1. +x

 

 

 

Symmetric Loading

Selected Maximum Displacements and Forces in 1

Second of Motion

 

 

 

 

 

 

 

Symmetric Reversed
Loading Loading__
DiSpl. Joint 1
in X-dir. (in.) 20'59 17-16
Twisting Mom. _ _
Col. 6 (Kip-in.) 14'32 284'6
Bend. Mom. Z-axia
Col. 6 (Kip-in.) 1256 1021
Bend Mom. Y-axi
' . 416.4
Beam 4 (Kip-in.)1 55 3

 

 

 

Figure 4.11 Effect of Loading Pattern

75

Axis for all modes

 

 

 

 

 

 

 

 

 

 

<=X
:- """ 1 I\
l \
I ‘7
I I
I I
______ I L, I
\
\ \ v
‘ \l
T = 0.523 sec. T = 0.378 sec.

Dashed lines show beam configuration
in the mode shown.

 

 

 

 

 

 

 

 

”7
r '1 " /
I I /” /
I I /
I | / /
I : / /
I / J
L __J 4"
L”
T = 0.329 sec. T = 0.251 sec.

Top view of frame is shown as modes primarily

involve only column sway.

Figure 4.12 Normal Modes for Cubic Frame

76

M20 S O
y L Chord

 

Deflected Member

I
“yo/r I

P {\‘Cmfi /‘>“‘""

S
20

Figure A.l Coordinates for Member Solution

77

Coefficient

   
     
   
 

 

 

 

 

 
  

  
 

 

0.10 - 2 2
of (91 +62 )
with Axial Force
Considered
\ 1 .
15 (No Ax1al Force)
0.05 -
.4
'I'
E 0 A J 2-
m . 1 p
'3 0.5 1.0 Euler
a: .
t 1
8 / - 35- (No Axial Force)
-0.05 -' Coefficient
of 9102 with
Axial Force
Considered
-O.10 L

Figure A.2 Coefficients in Expression for Axial Shortening
Due to Bowing

APPENDIX I

BEAM COLUMN SOLUTION

A.l.1 Stiffness Coefficients
In this section the stiffness coefficients of a member
under the influence of an axial force are derived. These coefficients
are available elsewherg for example, in Reference 4. However, they
are derived here for completeness and to outline the assumptions
necessary to use these relationships in a three dimensional solution.
The assumptions that torsional-flexural coupling may be
neglected and that the Bernoulh:Euler equations apply permit the dis-
placement-force relationships for the individual member to be ex-

pressed by the three equations.

dzy
EIz‘d—E = -Mzo + Spo - Py 0.000(A91)
x
dzz
EI -—— I -M + S x - Pz .....(A-2)
de2 yo 20
= £435 (A-3)
twist GJ "’°°

where as shown in Figure A.l; P is the axial force, M20 and Syo are

the moment and shear force components in the x-y plane at the initial
end of the member, and Myo and S20 are similar components in the x-z
plane. The differential Equations QI-D and (A-2) are identical in form

and consequently a single solution will be given and subscripts will

78

79

be drOpped. Rearranging and letting MI be the end moment at the

initial end and S the shear, yields the general differential equation:

dzy P 1
-—— +'-— y =-—- (S x - MI) .....(A-4)

dx2 E1 E1

which has the solution;

MI sin p x
y = -— cos p x +-——-—-——— [MI (1 - COS p L) - SL]
P P sin p L
S MI
+—X"_ 0.009(A'5)
P P
P
where p2 =--.
EI

Differentiating for the lepe yields:

dy MI p cos p x
'- ='- p sin p x + ----—- [MI (1 - cos p L) - SL]
dx P P sin p L
S
+‘— .....(A-6)
P
dy .0 dy
At x = 0, the slepe-- = 0 ; and at x = L, the slope - = 0 .
1 2
dx dx
Substituting these relationships yields two equations for MI and S in

terms of 91 and 02, the coefficients of which are the stiffness coef-

 

ficients:
El E1
MI = (C1 -—) 01 + (C2--) 02 .....(A-7)
L L
EI
S = (C3 -§) (01 + 02) .....(A-8)
L
where
pL (sin pL - pL cos pL)
C1 =

Q ..... (A-9).

80

pL (pL - sin pL)

 

 

C = 0.000(A-10)
2 0
2
(pL) (l-cos pL)
C3 = 0.000(A-11)
Q
and
9 = 2 - 2 cos pL - pL sin pL .....(A-12)

The C coefficients then are the parameters reflecting the effects of
the axial thrust. Identical coefficients apply for both Equations

(A-1) and (A-2) with the apprOpriate moment of inertia used in computa-
P

tion of p = —..
EI

Similar coefficients result for the case of negative axial
force (axial tension):
pL (pL cosh pL - sinh pL)

C = .....(A-13)
Q

 

pL (sinh pL - pL)
C = .....(A-14)
Q

 

(pL)2 (cosh pL - 1)

 

.....(A-15)
0

where now

Q'Z‘ZCOSh pL+pL Sinh pL 000.0(A-16)

81

A.l.2 Solution for BowingiEffect on True Length

The apparent axial deformation, 6, caused by bowing may be

approximated by the expression

1 L 232;
6 .2‘f0 (d dX 000.0(A'17)
Substituting the expression for'gi in Equation (A-6), integrating over

the beam length and substituting for M and S in terms of 61 and 92

I
from the stiffness expressions in Equations (A-7) and (A48) yields

1

e = ---E [03 - 02 (2 sina + sina cosa)
4p§

+ a(2 + 2 cosa - 4 cosza) - 2 sina + 2 sina cosa]

.....(A-18)

1
(6 2 + 6 2) + -——— [- 03 cosa + 3&2 sina + 6a
1 2 2p02

(cosa - l) + 28ina (l - cosa)] 0102

where a s pL.

The above formula is quite lengthy and some simplification is
possible if the effect of the axial force is neglected. The general

differential Equation (A-4) is simplified to

dzy 1

_.—‘-(Sx-M) 0.000(A'19)
dxz EI I

This equation may be solved by direct integration to yield,

dy M S 2 2
__....l. __I: __£__§_ -
(x 2) + (6 2 ) ......(A 20)

dx EI EI

The end moments with neglect of axial force become linear functions

of the end displacements;

82

4EI 2E1

MI = —;— 01 +'-;— 02 .....(A-21)
4E1 2E1

MF=T 02+“: 91 .....(A-22)

These relationships are substituted into Equation (A-20). The
resulting expression for-gi in terms of the end rotations is substi-
tuted into Equation (A-17) to yield

L 2 2 L

e = -—-(91 +02 ) -'- 9102 .....(A-23)

15 30
The coefficients of (012 + 022) and 0102 from the more exact expres-
sion (A-18) and the simplified expression (A-23) were evaluated for
various values of axial load, and the variation is shown in Figure

A.2. It is seen that the differences of these coefficients are

rather small for axial force less than one-half of the Euler load.

APPENDIX 2

C MPUTER PRmRAM

A.2.l General

The computer program utilizes a main program called
CONTROL and ten subroutines. Four of the subroutines STATIC,
KSTORE, MAELIM and MAINVERT deal only with the initial static solu-
tion of the frame. The subroutine MEMPRO computes the member
stiffness matrices, rotation matrices, the incidence relations, and
the joint loads due to the dead load on the members. This data:
necessary for the static solution, is also used in the dynamic
solution.

A single subroutine called RMATRX computes the inverted
mass matrix. The subroutine SETUP utilizes two M.S.U. library
plotter subroutines PLOT and CHAR and is used to draw and title
apprOpriate coordinate axes for use with the computer plotter,
CALCOMP 563, which is used as one method of data output. The
remaining three subroutines, INTEGN, MEMREA and ROTATE are used for
the dynamic solution.

The program CONTROL is the controlling section for the
entire program. It is also used to read in control data for the
dymanic solution and to plot and print output data from this solu-
tion. The subroutine STATIC accomplishes a similar function in

the static solution. All basic information concerning the frame is

83

84
read by the subroutine MEMPRO. All other data is read either by
STATIC or CONTROL and all data output occurs in these two routines.
The processes of the static and dynamic solutions are un-
related except as noted above, in using certain basic frame infof-
mation from MEMPRO. Consequently, the basic outlines of the two
solutions are given separately in the following sections along with

the primary Operations that are performed in the various subroutines.

A.2.2 Dynamic Solution

The main program CONTROL using the parameters DTIME and
LOOP (see definitions of variables in Section A.2.4) calls the inte-
gration subroutine INTEGN which then integrates for LOOP cycles and
returns to CONTROL which prints and plots the designated displace-
ments and the current time. This process is repeated JS times.
Subroutine INTEGN performs the numerical integration. It also calls
the subroutine MEMREA which computes the frame reaction on each joint
for the current displacement configuration, and monitors for maximum
and minimum forces and displacements. MEMREA in turn calls RIEATE
which is used simply to rotate forces or displacements from member

coordinates to global joint coordinates, or vice-verse.

A.2.3 Static Solutigg

The main subroutine STATIC is called from CONTROL. STATIC
performs no signicant computation but calls MEMPRO, KSTORE and MAELIM,
in this order, and prints the required output data. MEMPRO, as out-
lined previously, reads the basic frame data, computes member stiff-

ness and rotation matrices and.incidence data, and computes dead

85

load-joint loads. KSTORE forms and stores the joint stiffness ma-
trices. MAELIM accomplishes the solution of the linear equations for
the static displacements, and computes the end forces. MAINVERT
simply inverts 6 x 6 matrices as needed in the solution of the linear

equations.

A.2.4 Variables used in Computer Program
The variable names used in the program are listed below in the
order encountered in the program:

Program CONTROL

MEMBS = number of frame members;

JINTS = number of frame joints;

NJFREE = number of free (non-support) joints;
U(i,j) a displacement of joint 1 corresponding to j

coordinate;
MSTIFF(i,j) - jth stiffness coefficient for member 1;
MSTIFO(i,j) = jth stiffness coefficient for member 1

with no axial effect;

SEC = total number of seconds of integration;

JS = number of times subroutine INTEGN will be
called;

LOOP = number of numerical integration steps per-

formed each time INTEGN is called;
DTIME = At for numerical integration;
JOPL, JCOPL a joint number and coordinate whose dis-
placement is to be plotted at each

return to CONTROL;

86

JOPR, MFORPR = joint number and member number of the

JOINT

ICOOR

NDIS, NFOR

JDIS(i) , JCORD(i)

JFOR(i) , lJCORF(i)

DAMP

NOLIN

ILOD(i)

IBE

IE(i)

IQAKE

FAC

joint-displacements and member-forces
to be printed at each return to CONTROL.
JOPC in R format;
JCOPL in R format;
= number of displacements and forces to

be monitored for maximum and minimum;

ith joint number and its coordi-
nate of the displacement which

is to be monitored;

ith member number (negative if
negative end forces are desired)
and the coordinate of the force
which is to be monitored;

damping coefficient A;

parameter which directs nonlinear analysis
if it is l;

the dynamic loading incidence parameter; if
it is 1 loading is applied as input, if it
is -1 loading negative of that input and if
0 no loading is applied to joint 1;

100th of a second time lag in loading;

value is 1 if the ith joint has a time lag
in loading;

parameter to direct earthquake;

scaling factor that multiplies the magnitude

of loading input;

87

RLD(i,j) = the jth load value (of 3) at ix0.01 seconds;
MM = parameter used to control dynamic loading;
MM 2 1 when time is 0 and increases by 1
each 0.01 second;
TIME = time;
DISM(i), TMAXD(i) = minimum value and time of occurrence
for the ith diSplacement monitored;
DISL(i), TMIND(i) = the minimum value and time of
occurence for the ith displace-

ment monitored;

FORM(i), TMAXF(i) the maximum value and time
of occurrence for the ith force

monitored;

FORL(i), TMINF(i) the minimum value and the time

of occurrence for the ith force

monitored;
Subroutine MEMPRO
COORD(i,j) = jth structure global coordinate of
joint 1;

JP(i), JN(i) = positive and negative joint of
member 1;

MODUL, SMODUL = modulus of elasticity and shear
modulus of the material used in the
frame;

WEIGHT a weight of the material in the frame in kips

per cu. ft.;

88

DEADL(i) = uniform Superimposed dead load on member 1
in kips per inch;
LE = number of members with superimposed dead load;
DENSTY(i) a density in kip-in. units;
E(i), G(i) = modulus of elasticity and shear modulus
for member 1;
AREAX(i), AREAY(i), AREAZ(i) = the cross-sectional
area and the shear areas for shear load

along the y and z axes respectively;

IXX(i), IYY(i), IZZ(i) = J, Iy and I2 respectively for
member 1;
COMP(i,j) = projection of member i on the jth struc-

ture global axis;

LENGTH(i) = length (in inches) of member 1;

COSX(i), COSY(i), COSZ(i) - X, Y, and Z projections of

member 1 each divided by the member length;

L1 = a logical variable used to protect against
dividing by zero in the computation of the
rotation matrices;

ALPHA(i) = rotation of the y axis of member 1 about

the x axis measured in decimal degrees
from the vertical plane containing the
x-axis;

RM(i,j,k) = 6 x 6 (j and k) rotation matrix for member i;

AIFA = ALPHA (i) in radians;

SHGl, SHGZ s gz and gy;

89

NMEM(i) = number of members incident to joint i;

MEMBER(i,j) member number of the jth member incident

to joint 1; it is negative, if negatively

incident;

STLOAD(i,j) = static load on joint i corresponding to
the jth global coordinate;

DLOAD = total member dead load per inch due to member

weight and any superimposed dead load.

FF(i)

ith component of the member end force at the
member end acting on the joint being considered.
Subroutine INTEGN
LOOP - number of steps of numerical integration
accomplished on one call to this subroutine;
MMC, MMl = parameters used for loading control;
MMB, MMBl = lagging loading control parameters;
BLOD(i) = lagging loading load increment for one inte-
gration interval for the ith coordinate;
RLOD(i) = ordinary load increment for one integration
interval for the ith coordinate;
U(i,j) = diSplacement of joint i corresponding to the jth
joint global coordinate;
UM(i,j) = joint diSplacement as above at the previous
integration step;
UMM(i,j) = joint displacement as above at two steps
preceding the current time;
UDDT(i,j) = acceleration of the ith joint mass corre-

sponding to the jth joint global coordinate.

90

UDTM(i,j) = acceleration as above at the previous

integration step;

REA(i,j) = {RE} for the ith joint mass and the jth
coordinate;
RLOAD(i,j) = current dynamic load on joint i corre-

sponding to joint coordinate j;

SJA(i) = acceleration in the ith coordinate direction
of ordinary support joints in earthquake
loading;

SJAL(i) = acceleration in the ith coordinate direction
of the lagging support joints in earth-
quake loading;

SJD(i), STDL(i) = ordinary and lagging support joint
displacements for ith coordinate at most
recent integration step;

SJDM(i), SJDLM(i) = ordinary and lagging support joint
displacements for ith coordinate
at previous integration step;

SJDMM(i), SJDLMM(i) = ordinary and lagging support

joint displacements for ith coordinate at
two previous integration steps;

RSJD(i) = relative displacement of the lagging sup-
port joints to the ordinary support joints
for the ith coordinate;

RESPON(i,j,k) = inverse of the 6 x 6 joint mass

matrix ([RM]) for joint 1;

91

STLOAD(i,j) = static load on joint 1 corresponding
to coordinate j;
UDT(i,j) - velocity of joint 1 corresponding to
coordinate j;

Subroutine MEMREA

JP(i) = positive joint of member 1;

JN(i) = negative joint of member 1;

DELY - A2 for the member being considered;
DELZ = A1 for the member being considered;
DD(i) = ei;

PP(i) = ith component of I{F};

PPN(i) = ith component of F{F};

PRM(i,j) = [IRM];

RFOR(i) = ith component of I{T};
RFORN(i) = ith component of F{T};
FOR(i) = ith component of the initial-end member end
forces in joint global coordinates;
FORN(i) a ith component of the final-end member end
forces in joint global coordinates;
P(i) = the axial force on member 1, positive for

member compression;

SKLIl = .JP/EIy x L;
SKLIZ = «JP/BIz x L;
COSHl, SINHl, COSHZ, SINHZ = Cosh (SKLIl), etc.;

PHIl, PHIZ = Q for Iy and I2 respectively, (see
Appendix 1);

3111, $211, --- = c , 0

1y --- etc. (see Appendix 1);

2y’

92

Subroutine RMATRX
MMASS(i) = mass of %'of member i;
JMASS a joint mass;
XBAR, YBAR, ZBAR - coordinates relative to the joint
of the joint mass center;
X1, Y1, Zl = coordinates relative to the joint of the
mass center of the member being considered.
RINERM(i,j) = the mass inertia tensor for the member
being considered;
RINER(i,j) = the mass inertia tensor for the total
joint expressed at the mass center;
RINERI(i,j) = the inverse of RINER;
RESPON(i,j,k) = inverse of the 6 x 6 lumped mass for
joint 1 in joint global coordinates;
Subroutine ROTATE
L = a control variable indicating whether the
rotation is.f0r forces 6r displacements;
II = a variable indicating the member number of

the apprOpriate rotation matrix;

B(i,j) - 6 element vector of displacements or forces
in joint global coordinates, for joint 1, if
displacements, for member 1, if forces;

JJ = joint number being considered in a displace-
ment rotation;

C(i) = 6 element vector of displacements or forces

in member coordinates;

93

Subroutine SETUP
PLOT a an M.S.U. Computer Lab. Library subroutine

used to control the CALCOMP 563 plotter;

CHAR = a similar library subroutine used to print
letters on the plotter;

KP = the coordinate number of the displacement
to be plotted;

JOINT = the joint number to be plotted in R format;

ICOOR = the coordinate number to be plotted in R
format;

SY = 100 times the inverse of the scale on the

Y (vertical plot) axis;
Subroutine KSTORE
IAA(i,j) = the "layer" (first index number) of the
location in KSTOR of the 6 x 6 submatrices;
if the submatrix is null, the value is zero
if the submatrix is below the diagonal,
the value is the negative of the locatibn
of the transposed matrix in the upper
diagonal portion of [K];
KSTOR(i,j,k) a the ith 6 X 6 (j and k) nonzero sub-
matrix in the upper triangular portion
of [K].
IND = the number of nonzero 6 x 6 submatrices

stored in KSTOR:

94

Subroutine MAELIM
RLD(i,j) = the static load on the ith joint in the

jth coordinate direction;.

TEMP(i) = a 6 element temporary storage matrix used
in the reduction of the loading vectors;

EDIS(i) = the 6 element vector of linear relative end
deformations in global coordinates;

GK(i,j) = the 6 x 6 member stiffness matrix in

global coordinates;
Subroutine MAINVERT
B(i,j) = the 6 X 6 submatrix of [K] which is to be
inverted;
A(i,j,k) = KSTOR(i,j,k);
II = the ith index of KSTOR designating the 6 x 6

submatrix to be inverted.

95

A.2.5 Computer Program

 

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