ANALYSIS OF FRAME STRUCTURES MADE OF
LiNEAR ViSCOELASTIC MATERIALS

Thesis for the Degree of Ph. D.
MICHJGAN STATE UMVERSSTY
MARK M. BERREO
3.970

THESW‘,

 

This is to certify that the

thesis entitled

ANALYSIS OF FRAME STRUCTURES MADE OF
LINEAR VISCOELASTIC MATERIALS

presented by

Mark M. Berrio

has been accepted towards fulfillment
of the requirements for

Ph. D. degree in Civil Engineering

 

 

fléflm’

Major professor

Date £120 (L /?70

0-7639

 

 

 

 

ABSTRACT

ANALYSIS OF FRAME STRUCTURES MADE OF
LINEAR VISCOELASTIC MATERIALS

BY

Mark M. Berrio

A matrix formulation method is presented for the
analysis of plane frame structures made of linear Visco-
elastic materials. The study will follow the isothermal
quasi-static linear theory of homogeneous viscoelastic
solids. Geometric non-linearity and effect of axial
forces on the flexural stiffness of the members is taken
into consideration.

A computer program written in FORTRAN is prepared
for the implementation of the analysis. Three numerical
examples are considered, and the results are compared
with those obtained from experimental work.

The method is based on the calculation of a
fictitious load vector, obtained from the structure's
deformation history and added to the actual external load
vector as time increases. The computation of the
fictitious load vector is carried out by a finite dif-

ference method scheme.

Mark M. Berrio

Creep tests are run to establish the viscoelastic
properties of the material experimentally, rather than
using spring dashpot models. A direct numerical inversion
is then carried out, yielding the relaxation modulus
values without resorting to the use of the Laplace

transform.

 

 

ANALYSIS OF FRAME STRUCTURES MADE OF

LINEAR VISCOELASTIC MATERIALS

BY

— .,— J. 1‘
Mark M: Berrio

A THESIS

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

DOCTOR OF PHILOSOPHY
Department of Civil Engineering

1970

PLEASE NOTE:

Some pages have indistinct
print. Filmed as received.

UNIVERSITY MICROPILMS.

ACKNOWLEDGMENTS

The writer would like to express his appreciation
to Michigan State University, and to his guidance committee
Dr. Charles E. Cutts, chairman, Dr. George E. LaPalm,

Dr. James L. Lubkin, Dr. George E. Mase and Dr. David H. Y.
Yen for their support and assistance.

The writer is especially grateful to Dr. George E.
LaPalm, who as major professor provided direction and
encouragement throughout the course of his study.

A special word of thanks is extended to the writer's
wife, Margaret, for her help in the laboratory work, for
her typing of the rough draft, and above all for her
patience and understanding during these three years at

Michigan State University.

ii

TABLE OF CONTENTS

Page
ACKNOWLEDGMENTS . . . . . . . . . . . . . ii
LIST OF TABLES . . . . . . . . . . . . . v
LIST OF CHARTS . . . . . . . . . . . . . vi

LIST OF FIGURES O O I O O O O O I O O I O Vii

Chapter

I. INTRODUCTION . . . . . . . . . . . l
1.1 Viscoelastic Materials . . . . . . l
1.2 Previous Work . . . . . . . . . l
1.3 Objectives and Outline of this Work . . 3
1.4 Assumptions and Limitations . . . . . 4
II. CREEP COMPLIANCE INVERSION . . . . . . . 6
2.1 Some Definitions . . . . . . . . 6

2 2 Relation Between Creep Compliance and
Relaxation Modulus . . . . . . . 8

2.3 Upper and Lower Bounds for Relaxation
Modulus . . . . . . . . . . 9
2.4 Inversion of Discrete Experimental Data . 22

III. FRAME ANALYSIS 0 O O O O O O O O O O 35

3.1 Elastic Analysis . . . . . . . . 35
3.2 Viscoelastic Analysis . . . . . . . 37
3.3 Transversal Deflections . . . . 40
3. 3.1 Deflections Due to Combined
Joint Loads . . . . . . . 47
3.3.2 Deflection Corrections Due to
Distributed Loads . . . . . 50
3.4 Equivalent Joint Loads . . . . . . 54
3.5 Numerical Computer Solution . . . . . 57
3.6 Quasi-Elastic Solution . . . . . . 62

iii

Chapter Page

IV. LABORATORY WORK . . . . . . . . . . 64

4.1 Materials . . . . . . . . . . . 64
4.2 Casting . . . . . . . . . . . 64
4.3 Creep Tests . . . . . . . . 70
4.4 Creep Compliance Values . . . . . . 78
4.5 Aging Tests . . . . . . . . . . 81
4.6 Frame Tests . . . . . . . . . . 81

V. RESULTS AND DISCUSSION . . . . . . . . 87

5.1 Introduction . . . . . . . . . . 87
5.2 Comparison of Relaxation Modulus Values . 87
5.3 Frame 1 Joint Displacements . . . . . 89
5.4 Frame 2 Joint Displacements . . . . . 89
5.5 Frame 3 Joint Displacements . . . . 92
5.6 Epoxy Resin Used for the Laboratory

Models . . . . . . . . . . . 97
5 7 Laboratory Equipment . . . . . . . 97
5.8 Relaxation Modulus . . . . . . . . 99
5.9 Matrix Analysis Formulation . . . . . 100
5 10 Number of Finite Parts per Member . . . 102

VI. SUMMARY AND CONCLUSIONS . . . . . . . . 105

LIST OF REFERENCES . . . . . . . . . . . . 108
APPENDIX A: NOTATION . . . . . . . . . . . 112

APPENDIX B: COMPUTER PROGRAMS . . . . . . . . 117

iv

LIST OF TABLES

Table Page

2.1 Upper and lower bounds for E(t). Time
intervals equal to one second. Total
period covered equal to three minutes . . . 28

2.2 Upper and lower bounds for E(t). Time
intervals equal to one minute. Total
period covered equal to three hours . . . 29

2.3 Upper and lower bounds for E(t). Time
intervals equal to one hour. Total period

covered equal to 30 days . . . . . . . 30
2.4 Upper and lower bounds for E(t). Time

intervals equal to 12 hours. Total period

covered equal to six months . . . . . . 31

2.5 Upper and lower bounds for E(t). Time
interval redefined as the inversion
proceeds. Intervals used: 1 second,
1 minute, 1 hour, 12 hours. Total period
covered equal to six months . . . . . . 32

2.6 Upper and lower bounds for E(t). Time
interval redefined as the inversion
proceeds. Intervals used: 6 seconds,
1 minute. Total period covered equal to
768 minutes . . . . . . . . . . . 33

2.7 Upper bound of E(t) obtained from discrete
values of D(t). Time interval redefined as
explained in Section 3.5. Total period
covered, 768 minutes . . . . . . . . 34

5.1 Axial force effect on joint displacements . . 103

5.2 Number of finite parts taken per member,
versus accuracy and computer time . . . . 104

LIST OF CHARTS

Chart Page

3.1 Concise general flow chart . . . . . . . 60

vi

Figure

2.1

LIST OF FIGURES

Time intervals . . . . . . . . . .
Member boundary values . . . . . .
Deflections u and boundary values . . . .

Deflections v . . . . . . . . . .
Member divided in n-l equal parts . . . .

Fixed end member with loading and end
reactions shown in the positive direction

Time interval redefinition . . . . . .

Arrangement of molds to cast three specimens
at a time . . . . . . . . . . .

End-piece . . . . . . . . . . . .
Holding fixture . . . . . . . . . .
Sample ready to be used . . . . . . .
Arrangement for welding of frame joints . .
Creep test set up . . . . . . . . .
Middle span deflections on creep tests . .
(Part I) Creep compliance curve . . . .

(Part II) Creep compliance curve . . . .

Aging test. Simply supported beam . . .
Aging test. Cantilever beam . . . . .
Frame 1 . . . . . . . . . . . .
Frame 3 . . . . . . . . . . . .

vii

Page
12
42
43
46

51

55

58

66
68
69
71
72
74
76
79
80
82
83
84

85

Comparison of relaxation modulus curves

Vertical deflections of midspan point A of

Frame 1 . . . . . . . .

Horizontal displacement of joint B of Frame

Vertical deflection of midspan point A of

Frame 2 . . . . . . . .

Horizontal diSplacement of joint B of

Frame 2 . . . . . . . .

Vertical deflections of midspan point A of

Frame 3 . . . . . . . .

Horizontal displacement of joint B of

Frame 3 . . . . . . . .

Relaxation modulus increase with age

viii

Page

. 88
. 9O
1. 91
. 93
. 94
. 95
. 96
. 98

CHAPTER I

INTRODUCTION

1.1 Viscoelastic Materials

 

Although the origins of the theory of viscoelasticity
can be traced to isolated cases during the second half of
the nineteenth century, it has not been until recent years
that the subject has received close attention.

Time dependent deformation or creep, as it is
commonly named, arises in a number of modern structural
situations. High temperatures induce material inelasticity,
loss of strength and stiffness, creep, etc. Asphalt pave-
ments and solid fuel in rockets offer two more instances of
engineering applications of viscoelastic materials.

However, it is due largely to the recent advances in
highpolymer technology that the theory of viscoelasticity

has attracted an ever-growing interest.

1.2 Previous Work

 

Recent investigations in linear viscoelasticity may
be divided into two general groups. The first group is
concerned with the basic constitutive equations, and deals
with the subject from the viewpoint of continuum mechanics.

The second group is concerned with the solution of

boundary value problems, with a number of solutions to
particular problems.

In 1954, Hoff (ll)* proved that, for a body subject
to stationary creep under the action of constant forces,
it is always possible to use an elastic analogue. Here
by state of stationary creep is meant one where the space
distribution of stress in the body remains constant. The
elastic analogy has been used by Hult (12), Patel and
Venkatraman (15), Odqvist (l4), and others, in the analysis
of trusses, beams and plates. Williams (19) carried out
a structural analysis using spring dashpot mechanical
models to represent the viscoelastic materials. Flfigge
(5) solves the beam problem by means of the correspondence
principle, which links the linear theories of visco-
elasticity and elasticity.

The correspondence principle can be stated in a
general form saying that, if the solution of an elastic
problem is known, the Laplace transform of the solution
of the corresponding viscoelastic problem may be found
by an appropriate replacement of the elastic constants by
viscoelastic Operator polynomials, and the actual loads
by their Laplace transforms (5).

The writer recognizes that this is hardly a com-

prehensive review of the recent work on viscoelasticity.

 

*
Numerals with no punctuation in parentheses refer

to entries in the list of references.

The list of papers could go on showing a number of
solutions to beam, bars, plates and columns, based on
spring dashpot models, differential Operators, Laplace
transform, etc. As it has been pointed out, most of the
investigation on linear viscoelasticity has been done
from a continuum mechanics or applied mathematics point
of View, and, therefore, somewhat different from the
framed structure analysis in which we are interested

in this study.

1.3 Objective and Outline of This Work

LaPalm (13) developed a numerical integration
scheme which allowed him to write lower and upper bounds
to the solution of the governing equation of a linear
viscoelastic beam-column. The objective of this study
will be the development of a practical computer matrix
method for the analysis of linear viscoelastic frames,
using LaPalm's results as a basis and point of departure.

Creep tests will be conducted to establish the
material's properties experimentally, rather than using
spring dashpot mechanical models. In Chapter II an
inversion of the creep compliance values will be performed,
yielding the relaxation modulus values. A computer pro-
gram will be written for this purpose, again making use
of results obtained by LaPalm (13).

Chapter III will provide the results needed to write

a computer program to carry out the analysis of framed

structures made of viscoelastic materials. This will
constitute the main goal of this study.

The laboratory investigation will be presented in
Chapter IV. Several creep tests will be run to establish
the viscoelastic properties of the material. Rigid two-
dimensional frames will be tested under constant joint loads.

The theoretical and laboratory results will be
presented and compared in Chapter V.

Finally, a brief summary with conclusions will be
given in Chapter VI.

Notation and listing of computer programs will be

included in Appendices A and B, respectively.

1.4 Assumptions and Limitations

 

Effects of changes in temperature, and temperature
gradient will be excluded from this study, i.e., the
structure will be considered at a uniform constant
temperature, free of thermal stresses.* The inertia
term is considered small, and therefore drOpped from the
equation of motion (quasi-static case).

The study will be restricted to the viscoelastic
linear range of the material. For the creep tests, the
space distribution of stress in the body will be considered
to remain constant, giving rise to a state of stationary

creep.

 

*
The test room was kept at all times at a virtually
constant temperature of 80.5°F.

The assumption of plane sections remaining plane
after bending will be used here. Equilibrium will be
enforced in accordance with the geometry of the deformed
structure. Finally, the effect of axial forces on the
flexural stiffness of the members will be taken into

consideration.

CHAPTER II

CREEP COMPLIANCE INVERSION

2.1 Some Definitions

 

The creep compliance, D(t), of a viscoelastic
material can be defined as the strain, varying with time,
due to a unit constant stress. If E(t) represents the
strain as a function of time, and Co the constant unit

stress, we can write

e(t) = D(t)OO (2.1)

The creep compliance is a property of the material
related to its internal constitution. In the case of
high polymers the orientation of crystals will have its
effect in the creep compliance of the material (1) (3).

The creep compliance is a monotonically increasing
function for all t>O (4). For high polymers, the rate
at which D(t) increases diminishes with time, approaching
zero as time approaches infinity (3).

Equation (2.1) shows that discrete values of D(t)
can be obtained readily by measuring the deformations,
A(t), that will occur when the material is kept under

constant load. See Chapter IV for a full description of

the way in which values of D(t) were obtained for the
models used in this work.

Some numerical formulations of problem solutions
are more easily handled using the so-called relaxation
modulus, E(t), which can be defined as the stress varying
with time due to a constant unit strain. Letting o(t)
be the stress as a function of time, and so be the

constant unit strain, we can write

0(t) = E(t) 60 (2.2)

The relaxation modulus is monotonically decreasing (4).
For details about the relaxation modulus and the internal
constitution of materials such as high polymers, the
reader is referred again to references (1) and (3).

It will be shown in Chapter III that correspondence
can be established between the role of the relaxation
modulus in the analysis of viscoelastic structures, and
that of the modulus of elasticity in the analysis of
elastic structures. Due to this circumstance, it will be
preferable to work with the relaxation modulus, rather than
with the creep compliance. This resolves, in some way,
the viscoelastic problem into a succession of elastic
solutions.

It can be seen from Equation (2.2), that by inducing a

constant unit strain, 60, and measuring the resulting stress

o(t), as a function of time, a direct numerical evaluation
of E(t) is possible. This is known as the relaxation test.
In a relaxation test, a displacement is induced and kept
constant. Then the force is measured as it varies with
time. The crux of the problem lies in the fact that forces
are read by measuring displacements. In essence, one is
supposed to measure the displacements which are occurring
while holding the displacement constant. This would
require more time and more sophisticated equipment than
were available for the present study.

This is why creep tests were run, as will be seen in
Chapter IV. From the creep compliance values, the sought
relaxation modulus values were obtained, as will be shown

later in this chapter.

2.2 Relation Between Creep Compliance
and ReIaxation Modqus

 

 

The Laplace transforms of the creep compliance,
5(5), and relaxation modulus, E(s), are related by the
simple expression

D(s) E(s) = 4% (2.3)
s

A rigorous proof of Equation (2.3) is offered by Gurtin
and Sternberg (9) in the form of a theorem. The same

relation is derived in a simpler way by a number of

writers, e.g., Fung (7). Equation (2.3) is well known in

viscoelasticity, and what we propose to do is to carry
out an actual numerical inversion, i.e., given a set of
values of D(t), find the corresponding numerical values
of E(t).

The usefulness of a direct application of Equation
(2.3) will depend on the forms of E(s) and 5(5). For a
simple spring-dash model the inversion will be readily
performed. When a set of laboratory data for a real
viscoelastic material is the only information available,
a direct application of Equation (2.3) is impossible, and
a numerical scheme has been devised to make the inversion
practical.

2.3 Upper and Lower Bounds for the
RelaxatIBn Modqus

 

 

In reference (13) it is formally shown that an
approximation as to what might be considered as upper and
lower bounds to the true values of E(t) are given by the

following two expressions

 

 

k
1 - z E(tk - ti-l) [D(ti) — D(ti_l)]
i=2
E(tk) i D(tl) (2'4)
k
1 - ; E(tk - ti) [D(ti) - D(ti_l)]
E(tk) > 1:1 (2.5)

D(O)

10

where E(t) refers to the value of E(t) at time t The

k.

t

f1n1te 1ntervals of t1me t k_lr k

1' t2,..., ti'°'°’ t
need not be equal.

For the sake of completeness, and in order to
establish a more objective basis for the value to be
assigned to the expressions (2.4) and (2.5), a development
of such expressions follows. This deduction parallels a
similar one in reference (13), Appendix B.

Applying the convolution theorem to Equation (2.3),

the following expressions are formally obtained:

t

E(t) D(O) + [E(t-T) %? D(T) d1 = 1 (2.6)
O
t d

D(t) E(O) + [D(t-T) a? E(T) d1 = 1 (2.7)

O

The same results are stated and proved by Gurtin and
Sternberg (9).

Except at time equal to zero where the relaxation
modulus E(t) has a singularity (7), E(t) is a well

behaved smooth function, so that one can write:

t t.
k d k 1 d
I E(tk'T) a"? D(T) dT = i=1 J E(tk-T) d—{f D(T) (11' (2.8)
O t1-1
where tO < tl < t2 <...< ti-l < ti <...< tk-l < tk.

11

Also the following expression can be written

t.
l E(t -T) C—1—-D(T) dT - 1'm g E(t - ') d D( ') A
k d1 ‘ nix j=l k T3' at Tj Tj
ti-l (A1.) +0
3 max
(2.9)
where (see Figure 2.1):
ti-l = To < Tl < 11,..., Tn_l < Tn < In = ti
AT. = T. - T.
3 J 3-1
If E(t) decreases monotonically, then
I
E(tk-Tj-l) : E(tk-Tj) : E(tk-Tj) (2.10)
I
E(tk-ti_l) i E(tk-Tj) : E(tk-ti) (2.11)
. . . d . _
If, 1n add1t1on, d? D(T) : 0, letting A _ (ATj)maX,

one can write

n
. d u
E(tk-ti_l) 11m 2. d? D(Tj) Arj :
n+w j—l
A+0
< lim 2 E(t.T') (i I)(T') AT <
— n+oo j=l J a"? J —
A+O
n d .
< E(t -t.) lim 2 -—-D(T.) AT.
— k n+oo j=l dT J J
A+O

12

.mHm>HoucH oEHBII.H.N ousmflm

 

e
ce Hume he auflw me He 0e
w o n u o 1 x 9) v u i

n
1.... T 1;:
.filcP< A Alwe< A|MH< ._He< A

 

 

 

 

13

Combining Equations (2.9) and (2.12), one gets

t. t.

1 d 1 d
E(tk-ti_l) J 5? D(T) dT : J E(tk-T) a? D(T) dT
ti-l ti-l
t.
1 d
< E(tk-ti) J E? D(T) d1 (2.13)
ti-l
Integrating first and third integrals
t.
1 d
E(tk-ti_l)[D(ti) - D(ti_l)] : J E(tk-T) a; D(T) dT
ti-l

Taking a summation from 1 to k of each member

k k ti
2 E(t -t )[D(t )-D(t )] < Z J E(t -T) Q_ D(T) dT
._ k -1 i i-l — _ k dT
1—1 1—1
t
1-1
k
i i=lE(tk-ti)[D(ti)-D(ti_l)] (2.15)

Substituting Equation (2.8) in the middle member

14

t
k k d
E_ E(tk t _l)[D(ti)-D(ti_l)] J E(tk-T) a?-D(T) d1
1—1 0
k
i i=lE(tk-ti)[D(ti)-D(ti_l)] (2.16)

Adding E(t)D(o) to the three members of the above
inequality, and substituting the value of Equation (2.6),

we get the following two inequalities:

k
E(tk)D(O) + Z E(t

i=1 ‘ti-1)[D(ti)‘D(ti_1>1 1 1 (2.17)

k

k
E(tk)D(O) + Z E(t

-ti)[D(ti)-D(ti_l)] : 1 (2.18)
i=1

k

Taking the first term out of the summation, and transposing,

one gets from the Equation (2.17)

E(tk)D(O) + E(tk)[D(tl)-D(O)]

k
1 - i=2E(tk-ti_l)[D(ti)-D(ti_l)] (2.19)

| A

E(tk)D(o) + E(tk)D(tl) - E(tk)D(O)

—ti_l)[D(ti)—D(ti_l)] (2.20)

[A
H

I
P°M
m
‘3

15

Finally, cancelling and transposing, Equations
(2.4) and (2.5) are obtained. This development, although
somewhat detailed, should be considered heuristic, rather
than as a rigorous establishment of Equations (2.4) and
(2.5). Therefore, a test of the validity of these two
equations as upper and lower bounds, respectively,
follows.

At time t = t

k 1’ Equation (2.4) may be written as

1
EU(tl) — BTEIT‘ (2.21)

Here the subscript U has been introduced to identify
EU(tl) as the supposed upper bound to the exact value

E(tl) obtained from Equation (2.6) for t = t1, as follows

t
1
d
E(tl) D(O) = l -[ E(t1 - T) E1711”) dT . (2.22)
0

Let eU(tl) be defined so that
EU(tl) = E(tl) + eU(tl). (2.23)

Equation (2.21) becomes

eU(tl) D(tl) = l - E(tl) D(tl) (2.24)

Subtracting Equation (2.22) from Equation (2.24) yields

16

t1

t Dt = Et
eU(l) (1) J) <

O

l-T) 3% D(T) dT - E(tl)[D(tl) - D(0)1.

(2.25)

From Equation (2.16) one can write

t1 d
J E(tl-T) a?-D(T) dr 3 E(tl) [D(tl) - D(0)].

o
(2.26)
Thus
eU(tl) D(tl) 1 0 (2.27)
and, therefore
eU(tl) 1 0 (2.28)

which means that EU(tl), as given by Equation (2.21), is

the upper bound to E(tl).

At time tk = tn’ Equation (2.4) may be written as
n
EU(tn) D(tl) = 1 - i=2 EU(tn-ti_l) [D(ti) — D(ti_l)].
(2.29)

Let eU(tn) be defined so that

EU(tn) = E(tn) + eU(tn). (2.30)

17

From Equations (2.29) and (2.30) it follows that

n
eU(tn)D(tl) = 1 - E(tn)D(ti)-§:2EU(tn-ti_l)[D(ti)-D(ti_l)]
(2.31)
Equation (2.6) at time t = tn becomes
tn d
E(tn) D(O) = l — J E(tn -T) d? D(T) dT. (2.32)

O

Subtracting Equation (2.32) from Equation (2.31) yields

t
n
eU(tn) D(tl) = J E(tn -T) 5% D(T) dr - E(tn) [D(tl)—D(O)]
O
n
- i=2 EU(tn — ti-l) [D(ti) - D(ti_l)]. (2.33)

Since D(t) is monotonically increasing, it follows that

E(tn) [D(tl) - D(0)]> 0 (2.34)
and

n

1:2 EU(tn — ti-l) [D(ti) - D(ti_l)] > 0 (2.35)

Also from Equation (2.16) one has

18
t

o dT i=1

n
n
d
J E(tn-T) —— D(T) d1 _>_ z E(tn-ti_l) [D(ti) - D(ti_l)].
(2.36)
Now Equation (2.33) can be transformed into the following

inequality

n
eU(tn) D(tl) 11:1 E(tn - ti-l) [D(ti) - D(ti_l)]

- E(tn) [D(tl) — D(0)J

n
— I EU(tn - 11-1) [D(ti) - D(ti-l)]

(2.37)

which can be written in the following form

n
eU(tn) D(tl) 3 I: [E(tn‘ti—l) - EU(tn—ti_1)][D(ti)-D(ti_l)].

2

(2.38)

If EU(t) is greater than E(t) in the above eXpression,
the right-hand side of the inequality will be the product
of negative and positive values, and, hence, negative.
Therefore no conclusion can be drawn with respect to the
sign of eU(tn). Hence, the test fails to prove whether or

not EU(tn) provides the upper bound to E(tn).

19

Similarly, at time t = t

k 1' Equation (2.5) can be

written as

EL(tl) D(O) = 1 - E(O) [D(tl) - 0(0)] (2.39)

where the subscript L has been introduced to identify
EL(tl) as the supposed lower bound to the exact value
of E(tl).

Let eL(tl) be defined so that
EL(tl) + eL(tl) = E(tl) (2.40)

Now Equation (2.39) can be rewritten as follows
eL(tl) D(O) = -1 + E(O) [D(tl) - 0(0)] + E(tl) D(O).
(2.41)

Adding Equation (2.22) to Equation (2.40) yields

t
1
d
eL(t1) D(O) = E(O) [D(tl) — D(O)] — J E(tl-T) a?D(T) dT.
o

(2.42)

From Equation (2.16) one has
t
1 d

J E(tl-T) a—FDH) dT :E(0) [D(tl) - D(O)]. (2.43)
0

Thus

20

eL(tl) 0(0) 3 0 (2.44)

and, therefore,

eL(tl) Z 0 (2.45)

which means that EL(tl) is the lower bound to E(tl).

At time tk = tn, Equation (2.5) can be written as
n
EL(tn) D(O) = l - i=1 EL(tn-ti) [D(ti) - D(ti_l)].
(2.46)

Let eL(tn) be defined so that
EL(tn) + eL(tn) = E(tn) . (2.47)

From Equations (2.46) and (2.47) it follows that

eL(tn) D(O) = - l + E(tn) D(O)
n
+ 1:1 EL(tn - ti) [D(ti) - D(ti_l)]. (2.48)

Adding Equation (2.32) to Equation (2.48) yields

n
eL(tn) 0(0) = I=1 EL(tn-ti) [D(ti) - D(ti_l)]

tn d
- J E(tn - T) a? D(T) dr. (2.49)

O

21

Since D(t) is monotonically increasing, it follows that

P-M'J

=1 EL(tn-ti) [D(ti) - D(ti_l)] > 0 (2.50)

Also from Equation (2.16) one has

tn d n
J E(tn’T) a? D(t) dr 3 ;_l E(tn—ti) [D(ti) - D(ti_l)].
o l-
(2.51)
Now Equation (2.49) can be transformed into the
following inequality
n
eL(tn) 0(0) 3 £31 EL(tn-ti) [D(ti) - D(ti_l)1
n
- i=1 E(tn-ti) [D(ti) - D(ti_l)] (2.52)

which can be rewritten in the following form

n
eL(tn) D(O) 1 i=1 [EL(tn—ti) — E(tn-ti)][D(ti) - D(ti_l)1.

(2.53)

If EL(t) is smaller than E(t) in the above
expression, the right hand side of the inequality will

be the product of negative and positive values, and hence,

22

negative. Therefore here again the test fails, and
nothing can be concluded about the conditions under
which eL(tn) Z 0.

Although no rigorous conclusions have been drawn
about the validity of Equations (2.4) and (2.5), these
equations provide a valid numerical approximation, as it
is shown later in this chapter. Furthermore, the agreement
of experimental and analytical results, shown in Chapter V,
provides a strong verification of the good approximation
obtained as an upper bound for the values of E(t) using
Equation (2.4). Also it has been shown that for one
interval of time, Equations (2.4) and (2.5) provide upper

and lower bounds in the form

 

EU(tl) D(t (2.54)

and

1 — E(O) [D(tl) - 0(0)]
D(O)

(2.55)

 

EL(tl)

2.4 Inversion of Discrete
ExperImental Data

 

 

The writer has tested both Equations (2.4) and (2.5),
assuming for the values of D(t) those given by the

eXpression prOposed by Williams (19)

23

D(t) = Dg + —e——.—-9- , (2.56)

T n
O
(14'E—)

where D9 is the compliance at short time, or glassy com-
pliance, D(O). De is the compliance at long time, or
equilibrium compliance D(m). And To and n are parameters
chosen to best fit the particular set of experimental data.

The numerical values adOpted for D9, De’ I and n

0!

were found for the material studied in reference (13).
With these values, the actual expression for the values of

D(t) was

1 8.0 -6
D(t) + 2.0 + (1 + 13850000 0.2 x 10 (2.57)

t )

 

 

The results obtained for (2.4) and (2.5) converged
smoothly toward each other when the intervals of time were
small (see Tables 2.1 and 2.2).* For time intervals of one
hour or longer, the supposed lower bound solution showed
some oscillations that would cross the upper bound for the
first few steps, as can be seen in Tables 2.3 and 2.4.

A method for obtaining smooth solutions without

oscillations for the lower bound will now be shown.

 

*
The discrepancy shown in Tables 2.1 through 2.6
was calculated according to the formula:

upper bound - lower bound

lower’bound X 100'

 

24

In order to establish the time interval, two
aspects of the problem must be taken into consideration.
Use of large intervals results in undesirable oscillations
of the lower bound. On the other hand, use of very small
intervals would be impractical to span a reasonably long
period of time.

The following procedure was adopted and used with
satisfactory results on both counts. The method consisted
essentially of redefining periodically the length of the
intervals to be used. In this way, through the use of
small intervals at the beginning of the inversion,
oscillations were avoided. The use of larger and larger
intervals as the experiment progressed made the method
suitable to cover any practical period of time. There is
no limit to the ways of redefining the time intervals to
be used, and only practical considerations of the case
involved will serve as a guide in this respect. Although
a logarithmic increment of time is feasible, this would
require some modifications in the computer program, which
is written for equal intervals of time.

Tables 2.5 and 2.6 show the solutions obtained for
two different redefinitions of the time intervals. In
the case of Table 2.5, the inversion was stated with

intervals equal to one second, and this was carried out

25

for a period of 180 seconds. Then the first redefinition
was made in the following way:

E ( 60 seconds) becomes E (1 minute)

E (120 seconds) becomes E (2 minutes)

E (180 seconds) becomes E (3 minutes)
In this way, the values for one, two, and three minutes
were obtained without any oscillations, and the inversion
carried out again up to one hundred and eighty minutes.
A similar redefinition was then made:

E ( 60 minutes) becomes E (1 hour)

E (120 minutes) becomes E (2 hours)

B (180 minutes) becomes E (3 hours).
After rerunning the inversion for thirty-six hours, a
third redefinition took place:

E (12 hours) becomes E (1 half day)

E (24 hours) becomes E (2 half days)

E (36 hours) becomes E (3 half days).

The sequence of half-days was run for 360 steps,
covering a period of time equal to six months.

Neither the upper nor the lower bound solutions
showed any kind of oscillations. The greatest discrepancy
between upper and lower bounds was of the order of 4% of
the magnitude of the upper bound, and occurred in the
fourth step of the inversion with intervals equal to
half day. This figure dropped to 0.97% in the 13th step,

and it was 0.06% at the end of the sixth-month period

tested.

26

The redefinitions used in the solutions shown in
Table 2.6 were similar to the ones just described, but
the intervals were chosen so as to be compatible with
those that can be obtained in the laboratory, using simple
equipment: a dial gauge and a timer. This means that
the closest that the readings can be spaced would be from
.06 to .10 minutes apart, when the help of a second
operator is used.

In this second example, the first 30 steps of the
inversion were run with intervals equal to six seconds.
The following redefinition was used

E (10th interval) becomes E (1 minute)

E (20th interval) becomes E (2 minutes)

E (30th interval) becomes E (3 minutes).
This was the only redefinition used, and the inversion
was then carried out for 768 minutes.

Although the last example took the values of D(t) to
be inverted from Equation (2.57), it is obvious that those
same values could have been taken from a set of discrete
values obtained from lab observations. The only condition
imposed upon the set of D(t) values is that an entry
should exist for every time interval considered in the
inversion. This condition does not present any practical
difficulty, since the time intervals considered in the
inversion not only are compatible with those readily
obtainable in the laboratory, but also happened to provide a

set of points well-spaced to plot a time-deflection curve

27

during the course of the laboratory test, as will be
shown in Chapter IV.

Finally, Table 2.7 shows the results of an inversion
when actual discrete laboratory data are used for the creep
compliance values. The time interval redefinition used in
the frame analysis, explained in Section 3.5 and Figure
3.6, was the one used in this example. Furthermore, the
values of E(t) shown in Table 2.7, which are the upper
bounds given by Equation (2.4) were the ones adOpted as
true values of E(t) in the theoretical analysis of framed

structures of Chapter III.

TABLE 2.l.--Upper and lower bounds for E(t).
intervals equal to one second.

28

to three minutes.

Time

Total period covered equal

 

 

Time Upper Bound Lower Bound Discrepancy
(seconds) (psi) (psi) (%)
0 500000 500000 .00
1 469140 467110 .43
2 464830 464380 .10
3 462060 461690 .08
4 459980 459680 .07
5 458290 458030 .06
6 456860 456630 .05
7 455620 455420 .04
8 454520 454330 .04
9 453530 453360 .04
10 452620 452460 .04
15 449000 448880 .03
20 446290 446190 .02
40 439220 439150 .01
60 434720 434670 .01
80 431360 431320 .01
100 428660 428620 .01
120 426380 426350 .01
140 424410 424380 .01
160 422660 422640 .01
180 421100 421080 .01

 

TABLE 2.2.--Upper and lower bounds for E(t).

equal to one minute. Total period covered equal to three

29

hours.

Time intervals

 

 

Time Upper Bound Lower Bound Discrepancy
(minutes) (psi) (pSi) (%)
0 500000 500000 .00
1 435090 425400 2.28
2 426690 425440 .29
3 421370 419850 .36
4 417410 416260 .28
5 414230 413250 .24
6 411560 410700 .21
7 409260 408490 .19
8 407230 406530 .17
9 405410 404760 .16
10 403750 403160 .15
15 397200 396750 .11
20 392350 391990 .09
40 380010 379790 .06
60 372350 372190 .04
80 366740 366610 .04
100 362270 362160 .03
120 358550 358460 .03
140 355360 355280 .02
160 352570 352490 .02
180 358070 350000 .02

 

TABLE 2.3.--Upper and lower bounds for E(t).

equal to one hour. Total period covered equal to 30 days.

30

Time intervals

 

 

Time Upper Bound Lower Bound Discrepancy

(hours) (psi) (psi) (%)
0 500000 500000 .00
1 373590 330810 12.93
2 359540 362900 -.93
3 350910 344140 1.97
4 344610 341950 .78
5 339630 336620 .90
6 335510 333120 .72
7 331980 329810 .66
8 328900 326970 .59
9 326170 324410 .54
10 323700 322090 .50
11 321460 319970 .47
12 319410 318010 .44
13 317510 316200 .41
14 315750 314510 .39
16 312550 311450 .35
18 309710 308710 .32
20 307170 306250 .30
24 302730 301940 .26
28 298950 298260 .23
32 295660 295040 .21
36 292750 292190 .19

 

31

TABLE 2.4.--Upper and lower bounds for E(t). Time intervals
equal to twelve hours. Total period covered equal to six

months.

 

 

Time Upper Bound Lower Bound Discrepancy
(half days) (psi) (psi) (%)
0 500000 500000 .00
1 321300 221900 44.79
2 304220 335230 -9.25
3 293990 268210 9.61
4 286650 290610 -1.36
5 280930 272250 3.19
6 276250 274640 .58
7 272280 268040 1.58
8 268840 266430 .90
9 265800 262960 1.08
10 364080 260810 .87
15 252630 250960 .66
20 245230 243940 .53
30 234840 233950 .38
40 227530 226850 .30
60 217320 216850 .22
120 200200 199960 .12
180 190460 190300 .09
240 183690 183570 .07
300 178530 178430 .06
360 174380 174300 .05

 

TABLE 2.5.-«Upper and lower bounds for E(t).
redefined as the inversion proceeds.
1 minute,

1 second,

1 hour,

32

12 hours.

equal to six months.

Time interval

Intervals used:
Total period covered

 

 

Time Upper Bound Lower Bound Discrepancy
(hours) (psi) (psi) (%)
1 372360 372190 .05
2 358550 358450 .03
3 350070 350000 .02
4 344680 338830 1.73
5 339680 336490 .95
6 335550 332400 .95
7 332020 239390 .80
8 328930 326560 .73
36 292760 292110 .22

(half days)

1 319430 317760 .42
2 302740 391820 .31
3 292760 292110 .22
4 286820 274910 4.33
5 281050 276870 1.51
6 276330 270240 2.26
7 272350 268360 1.49
8 268900 264770 1.56
50 221900 221200 .31
100 204660 204300 .17
220 185730 185560 .09
360 174390 174290 .06

 

TABLE 2.6.--Upper and lower bounds for E(t).
redefined as the inversion proceeds.

6 seconds,

1 minute.

33

Time interval

Intervals used:
Total period covered equal to 768
minutes.

 

 

Time Upper Bounds Lower Bounds Discrepancy
(minutes) (psi) (psi) (%)
1 434781 434489 .07
2 426422 426241 .04
3 421134 420998 .03
4 417418 415936 .36
5 414238 413156 .26
6 411569 410616 .23
8 407231 406468 .19
10 403757 403112 .16
12 400847 400285 .14
16 396124 395670 .11
20 392350 391965 .10
24 389194 388858 .09
32 384083 383812 .07
40 380007 379778 .06
48 376605 376405 .05
64 371108 370947 .04
96 363094 362976 .03
160 352566 352486 .02
256 342473 342418 .02
384 333471 333431 .01
768 317509 317486 .01

 

34

TABLE 2.7.--Upper bound of E(t) obtained from discrete
values of D(t). Time interval redefined as explained in
Section 3.5. Total period covered, 768 minutes.

 

 

Time D(t) E(t)
(minutes) (sq. in./lb.) (lbs./sq. in.)

1 2.275 x 10"6 4.394 x 105
2 2.312 x 10'6 4.323 x 105
3 2.337 x 10'6 4.276 x 105
4 2.356 x 10‘6 4.242 x 105
5 2.372 x 10'6 4.213 x 105
6 2.386 x 10'6 4.187 x 105
8 2.408 x 10'6 4.149 x 105
10 2.427 x 10"6 4.116 x 105
12 2.444 x 10'6 4.087 x 105
16 2.472 x 10'6 4.040 x 105
20 2.495 x 10'6 4.002 x 105
24 2.516 x 10"6 3.968 x 105
32 2.550 x 10’6 3.915 x 105
40 2.581 x 10"6 3.866 x 105
48 2.612 x 10'6 3.817 x 105
64 2.662 x 10'6 3.745 x 105
96 2.739 x 10‘6 3.634 x 105
160 2.847 x 10'6 3.494 x 105
256 2.954 x 10‘6 3.368 x 105
384 3.048 x 10'6 3.260 x 105
768 3.208 x 10"6 3.099 x 105

 

CHAPTER III

FRAME ANALYSIS

3.1 Elastic Analysis

 

The governing equation for elastic beam-columns is (17)

d4 d2
EI —-1- y(x) + N——2- y(x) = q(x) (3.1)
dx dx

where EI represents the flexural rigidity of the beam in
the plane of bending, N is the axial load, and q is the
load distributed along the member length.

In the theory of matrix structural analysis, the

expression
-1
{x} = [K] {P} (3.2)

is well known, where the column matrix {X} represents the
structure joint displacement vector, [K] is the structure
stiffness matrix, and {P} is the load vector. Brackets {}
will be used throughout this work to indicate a column
matrix.
The elements of the load vector {P} act only at the

joints of the structure. In general, however, a structure
will also have loads acting on the members. When this

is the case, in order to use Equation (3.2), the loads

35

36

acting on the members must be transformed to equivalent
joint loads. Then the vector {P} may be considered as

made up of two parts
{P} = {w} + {Q} (3.3)

where {W} represents the actual loads acting at the joints,
and {Q} is the equivalent joint loads. {P} is then called
the combined joint load vector.

The vector {Q} is evaluated in such a manner that
the resulting joint displacements of the structure cal-
culated from Equation (3.2), when using the combined joint
load vector {P}, are the same as the displacements pro-
duced by the actual loads. Furthermore, the support
reactions for the structure subjected to the combined
loads are the same as the support reactions caused by the
actual loads. The evaluation of {Q} and its properties
can be found in any standard matrix structural analysis
textbook, for example, reference (8).

Unfortunately the substitution of loads acting on
the members by the equivalent joint load vector (0) will
not preserve the actual transversal deflections, y(x),
along those members which originally were loaded. When
the actual force acting on the member is replaced by the
equivalent joint load, and included in the combined joint
load vector {P}, although the end displacements remain

unchanged, the transversal displacements along the member

37

are no longer equal to y(x). We call these displacements,
caused by the combined joint load vector {P}, u(x). To
find the actual deflections y(x), a correction v(x) has to
be added to u(x). The deflections v(x) are obtained by
loading the member with its original load system, while

keeping the ends clamped (10). It can then be written

y(x) = u(x) + v(x). (3.4)

3.2 Viscoelastic Analysis

 

For viscoelastic beam-columns LaPalm (13) shows that

Equation (3.1) takes the form

4 t

4
IE(t) 3‘4 y(x,0) + J IE(t-T) %; [a—z-y(X.T)lar
3x 8x
0
82
+ N(t) ——7 y(x,t) = q(x,t) (3.5)
EX

with an upper bound given by the solution of

4 2

3 a
IE(t -t _ ) ———-y(x t ) + N(t ) —-— y(x,t )
k k 1 8x4 ' k k ax2 k
k-l a4
= q(x,tk) - I 1:1 [E(tk-ti_l) - E(tk-ti)] ;;4'Y(x'ti)

(3.6)

38

and a lower bound given by the solution of

4 2
8 3

k-l 4
_ _ _ a
_ q(x,tk) I 1:0 [E(tk ti) - E(tk-ti+l)] 3;; (x,ti)

(3.7)

subject to boundary conditions given in terms of trans-
versal displacements and rotations at the ends of the
member, as shown in section 3.3.

The left-hand sides of both Equation (3.6) and (3.7)
have the same form as Equation (3.1), while the right-
hand sides, representing the forcing function, are made
up of the actual load q(x,tk) at time tk’ plus a fictitious
load derived from the history of the solution.

Now the writer proposes to parallel the relation
between Equation (3.1) and (3.2), with that between

Equation (3.6) or (3.7) and

1

{xm} = [K(t)1' {P(t)} (3.8)

Equation (3.8) produces the joint displacement
history of framed structures made of linear viscoelastic
materials, as the writer intends to show. At a given time,

t Equation (3.8) becomes

kl

39

-1
{X(tk)} = [K(tk)] {P(tk)} (3.9)

which could be identified with Equation (3.2), except for
the vector {P(tk)} that has a peculiarity of its own.

The vector {P} in Equation (3.2) has two parts, given
by Equation (3.3), where {Q} represents the equivalent
joint load, taking the place of the actual distributed
loading, q(x), as appears in the governing Equation (3.1).
Likewise, it follows that the vector (P(tk)} in

Equation (3.9) is given by

{P(tk)} = {W(tk)} + {Q(tk)} + {F(tk)} (3.10)

where the new term {F(tk)} is a fictitious equivalent

joint load vector taking the place of the forcing term

-1 4
a

-1 4
8
[E(tk-ti) - E(tk-ti+1)] ;;[ Y(Xrti)

of the governing Equation (3.7). This means that the
vector (P) is a function of all previous deformed con-
figurations of the structure. This could be rephrased,

saying that the deflections to be obtained at a given

40

time, tk, will depend not only on the actual loading at
that moment, but also on the whole previous deflection
history.

Before an evaluation of {F(tk)} can be attempted,
the set of previous solutions y(x,tl), y(x,t2), ... ,

y(x,tk_l) must be available.

3.3 Transversal Deflections

 

At a given time, tk' y(x,tk) can be written as yk(x),
and considered as a function of x alone. The Equation (3.6)

may be written as

d4 2

d
IE(t -t._ ) ——-y (x) + N y (x)
k 1 1 dx4 k k dx2 k

k-l d4
= qk(x) - 11-1 [E(tk-ti_l) - E(tk-ti)] 3;; yi(x) (3.11)

The handling of the distributed load q(x) is a well-
established procedure in the analysis of elastic structures.
The reader is referred to reference (8). Therefore, since
its consideration at this point would only make the study
more cumbersome, without producing new results, we will
drop it now, and consider loading applied at structure
joints alone. Equation (3.11) becomes, finally

2 II
ykIV + kk yk = fk (3.12)

where

41

d2
“‘2 Yk(x)
dx

Nk

IE(tk-ti_

 

l)

-1

H453‘

=1

4

d
[E(tk-ti_1) - E(tk-ti)] E;z-yk(x)

 

The solution of Equation (3.12) is subject to the

boundary conditions

{y} = {Y*}

where

{10:4

 

 

Y
ll.
91'
dx x=o
Yl
x=L
g1|
dx x=L

 

y(O)

y'(0)

y(L)

y'(L)

 

(3.13)

(3.14)

42

 

 

N

.mosam> humccson nonfiozuu.a.m enamem

15.»

 

 

 

N

% AOV ..h

E»

43

.mmsam> mumccson cam 5 mCOAuoonmonn.m.m mnsmflm

 

 

 

 

 

44

and {Y*} represents the member end displacements
obtained from the structure joint displacements, {X}, in
Equation (3.9), see Figure 3.1.

The value of fk in Equation (3.12) is a function
of the solutions yl(x), y2(x), ... , yk_1(x), obtained at
times t1, t2, ... , tk-l' Therefore, at time t1, f1 = 0,
and the initial solution yl(x) is obtained from the
homogeneous case of Equation (3.12). Instead of solving
equation (3.12) directly, the value of y(x) will be
obtained in a manner similar to the procedure followed
for the elastic case with equation (3.4).

Let uk(x) be the transversal displacements obtained
for a member at a given time tk' when the actual loading
acting on this member has been substituted by equivalent
joint loads and incorporated in the combined joint load
vector {P(tk)} in Equation (3.9). The displacements uk(x)
are not equal to the displacements yk(x) obtained from the
actual load system. A correction vk(x), defined later in
this section, will be needed. Let u E uk(x). Then the
deflections u, shown in Figure 3.2, are given by the

equation

uIV + kZuII = o (3.15)

under the boundary conditions

45

{U} = {Y*} (3.16a)

where {U} is defined in a manner similar to the way {Y}
was defined for boundary condition (3.13), i.e., by the

column matrix
{U} = {u(o) u'(o) u(L) u'(L)} (3.16b)

and {Y*} is obtained from the structure joint deflections.

In order to obtain the member deflections y for the
actual loads, a correction, v, must be added to the
deflections, n. In order to obtain this correction, we
clamp the ends A' and B', in Figure 3.2 and apply the
distributed load, f, as shown in Figure 3.3.

The deflections, v, are given by the equation

v1V + k2v11 = f (3.17)

subjected to the homogeneous boundary conditions

{v} = {O} (3.18)
where, as before,

{V} = {v(O) v'(o) V(L) V'(L)} (3.19)

and {Y*} is obtained from the structure joint displace-
ments.

Finally, because of linearity of the governing
equation, the member deflections due to the actual loads

are given by

46

 

.> mcofluomamooll.m.m ousmflm

47

Likewise, the fourth derivations of the deflections will

be given by

yIV = uIV + VIV (3.21)

3.3.1 Deflection Due to
Combined JoInt Loads

 

 

Three different cases will be considered for
Equation (3.15), corresponding to a member under tension,
compression or no axial load.

For no axial load, the deflections u will be given

by equation

uIV = 0 (3.22)
with boundary conditions

{0} = {Y*} (3.23)
yielding the solution

x3 + A x2 + A x + A4 (3.24)

where

48

_ 3 * 2 * 3 * 1 *
12‘ {2‘6 5‘6 ”'st LY4
*
A3 ‘ Y2
*
A4"Y1 (3.25)

and the fourth derivative is obviously given by Equation
(3.22).
For members under compression, the governing

equation will be

uIV + kZuII = 0 (3.26)

with boundary conditions (3.23), yielding the solution

u = Bl cos kx + B2 s1n kx + B3x + B4 (3.27)

with the fourth derivative given by

uIV = B k4 cos kx + B k4

1 2 Sln kx (3.28)

where

49

'k

*1: *94‘
(-Yl LY2+Y3)(k cos kL-k)-(-Y2+Y4)(sin kL-kL)
l I (cos kL-1)(k cos kL-k) - (-k sInIkL)(sin kL-kL)

 

* * ' Y* Y* 1(*
(cos kL-l)(-Y2+Y4) - (—k s1n kL)(- l-L 2+ 3)

2 _ (cos kL-l)(k cos kL-k)5- (-k sIn kL)(sin kL-kL)

 

(3.29)

Finally, for members under tension the governing

equation becomes

0 .
- k“u" = 0 (3.30)

with boundary conditions (3.23), yielding the solution

u = Cl cosh kx + C2 Slnh kx + C3x + C4 (3.31)
with the fourth derivative given by
uIV = Clk4 cosh kx + C2k4 sinh kx (3.32)

where

50

 

 

* * 'k * *
(-Y -Y L+Y )(k cosh kL-k) - (-Y +y )(sinh kL-kL)
C = 1 2 3 2 4
1, (cosh kL-l)(k cosh kL-k) - (k sinh kL)(sinh kL-kL)
* * _ * * *
C = (cosh kL-1)(-Y2+Y4) - (k Slnh kL)(-yl-Y2L+Y3)
2 (cosh kL-l)(k cosh kL-k) - (k sinh kL)(sinh kL-kL)
= *—
c3 Y2 Czk
*
C4 — Yl - Cl (3.33)

3.3.2 Deflection Corrections
Due to Distributed Loads

The value of k in Equation (3.17) changes with time,
and so will the form of the solution, whiCh would become
complex and awkward to handle in a general algebraic
formulation. More desirable and practical, when
digital computers are to be used, would be a numerical
solution.

A finite difference method will be used to approxi-
mate the second and fourth derivatives, with an error of
the order of hz, as follows, see reference (16):

v. - 2v. + v.
.II 1-1 1 1+1 (3.34)

Iv _ Vi-z ‘ 4Vi-1 + 6V1 ' 4V1+1 + v1+2
vi - 4 (3.35)
h

51

 

 

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HI: oucfl pocfl>flc HoQEmzll.v.m ousmflm

 

 

 

 

 

 

52

Dividing the member under analysis in n-l equal
parts with n nodes (see Figure 3.4), by substitution of
relations (3.34) and (3.35) into Equation (3.17), we can
write

2 2

2 2
vi_2 + (h k -4)vi_l + (-2h k +6)vi

+ v. = h4f. (3.37)

2 2
+ (h k -4)vi+ 1+2 1

1

From Equation (3.37) and boundary conditions (3.18), the

following system of equations is obtained:

v1 = 0

(-2h2k2+7)v2 + (h2k2-4)v3 + v4 = h4f2

(h2k2-4)v2 + (-2h2k2+6)v3 + (h2k2-4)v4 + v5 = h4f3

v2 + (h2k2-4)v3 + (—2h2k2+6)v4 + (h2k2-4)v5 + v6 = h4f4
v3 + (h2k2-4)v4 + (-2h2k2+6)v5 + (h2k2-4)v6 + v7 = h4f5

2

2 2 2 2 2
v + (h k -4)vn + (-2h k +6)vn + (h k 4)vn

n-5 -4 -3 -2

53

vn_4 + (h2k2-4)vn_3 + (-2h2k2+6)vn_2 + (h2k2-4)vn_l
= h4fn-2
vn_3 + (h2k2-4)vn_2 + (-2h2k2+7)vn_l = h4fn_l
vn = 0 (3.38)

This system of linear algebraic simultaneous
equations is readily solved for v. In the present case,
the Gauss elimination method (6) was used. The listing of
the computer program appears in Appendix B.

Knowing the values of v, the fourth derivative, vIV,
is obtained by numerical differentiation. The derivatives
corresponding to the nodes 1 and n were obtained by
forward and backward differences, respectively, using the
following expressions (16), with an error of the order of

h2:

- 24vi+3 + llvi+4 - 2v

 

 

V IV = 3V1 ' 14V1+1 + 26v1+2 1+5
1 h47
(3.39)
v IV = -2vi_5 + llvi_4 - 24Vi-3 + 26Vi-2 - 14Vi-l + 3vi
i h4

(3.40)

54

while for the nodes 2, 3, ... , n—l, Equation (3.35)
was used.

Applying Equations (3.20) and (3.21), one is now in
a position to evaluate the deflections and their fourth
derivatives for each member of the structure. The deflec-
tions y obtained at times tk, k = 1, 2, ... , will give
the deformation history of the structure, while the
fourth derivatives, yIV, at times t1, t2, ... tk' will
provide the forcing function, f, for time tk+l' according

to Equation (3.12).

3.4 Equivalent Joint Loads

 

As it was pointed out earlier in this chapter, the
forcing function, f, in Equation (3.12) acts as a dis-
tributed load, and in order to take into consideration
its effect on the structural analysis, an equivalent joint
load, F, must be formed, to be added to the structural
loading, as shown in Equation (3.10).

The evaluation of F will parallel that of Q, with f
playing the role of q. The distributed load q can, in
many practical cases, be approximated by a simple function
of x, for which the reactions are often found tabulated,
or easily calculated. In the case of f, all one has is a
set of discrete values, as many as thought necessary, and
a general algebraic solution is not readily available.

Therefore a numerical procedure will be used.

55

 

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56

Loading the member with the function f, as shown in
Figure 3.4, Equation (3.37) will give the values of the
deflections v. Then, neglecting the effects of shearing
deformations and shortening of the beam axis, one has, for

an elastic material (17)

Q.)
N

EI = -M (3.41)

D.
X

Applying the correspondence principle (5), Equation (3.41)

becomes, for a linear viscoelastic material, at time tk'

_ _ II

From Equation (3.42), one has

I

3
ll

- E(tk) I yI (0) (3.43)

I

M = - E(tk) I yI (L) (3.44)

where ML and MR represent the end moment reactions for

x = 0 and x = L, respectively. The subscripts L and R
stand for left and right, not to be confused with L which
represents the length of the member (see Figure 3.5).

Taking moments about point n and 1, one obtains

-1

 

50
II
I
1"th
P‘P1U

l h .
E(fi+fi+1)h[L - 5 - (1-l)h] (3.45)

57

 

%(fi+fi+l)h[% + (i-l)hl (3.46)

where h represents the length of the n-l equal segments,
in which each member of the structure is divided.

R’ RL

and RR for each member of the structure, are applied as

loads at the structure joints, and the set of these loads

The negative of the values obtained for ML' M

forms the vector F.

3.5 Numerical Computer Solution

 

The numerical solution of Equation (3.12) by means
of digital computers faces the problem of storage, since
the forcing function at a given time, tk, will be a
function of all previous deformed configurations. If the
structure has m members, each divided in n-l parts, and
the period of time to be analyzed has k time units, the
required storage space would be m(n-l) k. In this way,
the length of the period of time for which the structure
can be analyzed is limited considerably, even if
auxiliary storage space, such as tapes, drums, etc., is
used.

In order to avoid this difficulty, a redefinition
of the time interval is used in this work. Figure 3.6
represents schematically this redefinition. The t-axis
serves as basic time reference, and it will give the

structure's age in minutes. The r-axis represents the

58

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59

cycle or iteration, i.e., it tells how many times the
time intervals have been redefined. The numbers on the
tr-plane are the time interval labels, which will be
represented by the letter 1.

Every cycle goes from 1 to 6, and is defined in
such a manner that the age of the structure during

iteration r at interval 1 will be given by

t = i - 2r minutes (3.47)

In this way, the storage required of past deformed
configurations to produce the forcing function f, will
reduce to 6m(n-l). The choice of 6 intervals in each
cycle was arbitrary to a great extent, and a different
number might be shown to produce better results. Six
was chosen because it is a small number, offers three
points to overlap with the past known data, and advances
an equal amount toward new data.

At this point, one has all the necessary means
to produce a numerical solution by programming a high-
speed digital computer. A concise general flow chart of
such a program appears on Chart 3.1. Double-lined boxes
indicate that the step is obtained by means of a sub-
routine subprogram.

The Chart 3.1 starts with the label DATA, under
which are included the general information about the

structure properties, loads, member subdivision, time

 

 

READ DATA

 

 

 

 

 

COUNTERS

t=l, i=1, 'r=o

 

60

 

 

 

 

 

 

LOAD VECTOR
P=W

 

FORM FICTITIOUS
EQUIVALENT
JOINT LOADS
F

 

 

 

 

 

 

 

 

 

STRUCT, JOINT
DISPLACEMENTS

 

 

 

 

 

LOAD VECTOR
P=W+F

 

 

 

 

 

 

 

 

l

 

 

 

PRINT X

 

 

 

1 t=i-2r

 

 

 

 

 

 

 

 

 

 

MEMBER
BOUNDARY VALUES
X+Y*

 

 

REDEFINE
TIME
INTERVALS

 

 

 

 

 

 

DEFLECTIONS
U

 

 

 

r=r+1

 

 

 

 

 

 

 

 

V

 

 

 

 

DEFLECTIONS
V

 

MEMBER
TOTAL DEFLECTIONS

 

Y=U+V

 

 

 

 

 

 

 

 

 

 

 

 

i=i+1

 

 

 

NO

YES

 

IV

-—+l Y + Y

 

 

 

Chart 3.l.--Concise general flow chart.

61

limit, etc. If the relaxation modulus, E(t), is not given
as an algebraic function, its tabulated set of values will
be also read at this moment.

To define time, three counters will be used. Basic
time, in minutes is represented by t, the time interval by
i, and the cycle number, or iteration, by r. So the program
starts by setting t=l, i=1, and r=0.

The actual loads applied at the joints are given by
the vector W. So, at time t=l, since the structure was
previously undeformed, the load vector P will equal W.

Now keeping the time fixed, an elastic analysis is
run, by means of a subroutine. The analysis follows the
procedure outlined by Wang (18) in a second order
structural analysis, taking into consideration the effect
of axial forces on the flexural stiffness of the members,
and enforcing equilibrium in accordance with the deformed
geometry of the structure. The listing of this sub-
routine is not given in Appendix B, and the reader is
referred to reference (18).

The deflections u and v were defined previously, and
are calculated by two separate subroutines, whose listing
appears in Appendix B.

Finally, the fictitious equivalent joint loads are
obtained by the finite difference method described in
paragraph 3.4, and the program listing can also be found

in Appendix B.

62

3.6 Quasi-Elastic Solution

If the effect of axial force on the flexural stiff-
ness of the structure is neglected, the solution can be
obtained by application of the super-position principle
(7), i.e.,

t
X(t) = S(t) P(O) + J S(t-T) %?-P(T) d1 (3.48)
o

where S(t) is the time—dependent solution for a unit step
loading, and P(t) is the time-dependent load.

If the structure is subjected to a constant load,
P = P(O), as in the case being considered, the integral
of equation (3.48) will vanish, and for any given time tk,

one obtains

X(tk) = S(tk) P (3.49)

The difference between Equation (3.40) and Equation
(3.2), is that while for the elastic case the flexibility
‘ matrix is kept fixed, for the quasi-elastic case at every
time, tk’ the modulus of elasticity, E, is replaced by
the corresponding value of the relaxation modulus. Of
course, as it has been pointed out above, the flexibility
matrix S(tk), unlike K-1(tk) in Equation (3.9), does not
include the effect of the axial force in the members'

stiffness. The solution of Equation (3.49) provides an

63

alternative approximation to the viscoelastic deflection

history of the structure, called the quasi-elastic solution.

CHAPTER IV

LABORATORY WORK

4.1 Materials

The materials, equipment, and laboratory tests are
similar to those used by LaPalm (13). At risk of
being repetitious, a brief description of the laboratory
work will be given in this chapter.

The material used was epoxy resin ARALDITE 502 in
combination with ARALDITE Hardener 951, both manufactured
by CIBA Products Company, Summit, New Jersey. This
mixture cures in a short time at room temperature, has a
wide linear range, and can be reproduced. Also,
individual pieces can be welded together, making possible
models of different shapes. These properties are claimed
by the manufacturer, and confirmed in reference (13),

and by tests conducted in this work.

4.2 Casting

Several prOportions of resin and hardener were tried,

and a final mixture of ten parts of hardener per hundred
parts of resin (in weight) was adopted for all lab tests.
The casting procedure and environmental conditions were

always duplicated as closely as possible to ensure maximum

consistency of results.

64

65

All mixtures were prepared in a room whose tempera-
ture was held at 75°F 1 1°F. The amount of ARALDITE and
hardener mixed was always the same, and followed the same
sequence: 160 grams of ARALDITE were placed in a con-
tainer whose total capacity was twice as much, while 16
grams of hardener were measured in a different smaller
container. The reason for placing the resin and the
hardener in two different containers, rather than adding
the hardener directly to the resin (or vice versa), was
to avoid partial curing of the mixture during the process,
which actually happened the first time the mixture was
made. Two different containers allowed enough time for
a careful and accurate measurement of both resin and
hardener. The disadvantage of this procedure is that
part of the hardener will remain on the walls of the
container, producing a proportion of ingredients other
than expected. This remainder was measured repeatedly,
giving an average of 0.63 grams. An extra amount of
hardener equal to this average was added to the theoretical
16 grams, and a check of the actual waste was subsequently
carried out, giving the real proportion used for every
particular case. These values fluctuated from 9.97 to
10.03%, when a proportion of 10% was desired.

Once the hardener was poured into the resin, the
mixture was stirred vigorously for one minute, and then

centrifuged for two minutes at about 1500 rpm.

66

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67

Both the casting and the curing of the samples were
done in the same room where all the tests were run. This
room was kept at constant temperature of 80.5°F, with a
fluctuation of i 0.5°F. The relative humidity fluctuated
from 40 to 60 per cent.

The molds were made of aluminum, and consisted of
machined, prismatic bars resting on a heavier base, and
arranged in such a way as to produce specimens 3/4" x l"
x 22" in size (Figure 4.1).

As mold release, three different products were used
with no apparent difference: Silicone Release Agent,
manufactured by Dow Corning Corporation, Midland, Michigan;
FreKote 33 Release Agent, FreKote, Incorporated, Boca
Raton, Florida; and Johnson Paste Wax, Johnson and Son,
Incorporated, Racine, Wisconsin.

Before pouring the mixture, the mold was levelled
to obtain a bar with uniform height, and clamped to
avoid any leakage. Once the mixture was poured, the
mold was covered to protect the sample from dust. The
sample was kept in the mold for seven days, at the end
of which the different pieces of the mold were released.
An effort was made not to cause any disturbance of the
sample.

If the sample was to be used in a creep test, end-
pieces (Figure 4.3) were attached to its ends to provide

supports, and to enable the application of end moments.

68

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70

In order to prOperly center the and pieces, and place

them in a plane normal to the sample axis, a holding

fixture (Figure 4.3) was used. The final beam model

ready to be used in a creep test can be seen in Figure 4.4.
For frame models, the triple mold of Figure 4.1 was

used. After one week of curing, the ends of the three

bars were mitered and welded together. For this purpose,

the pieces of Figure 4.1 and Figure 4.4 were combined

to support the frame, while the joints were welded, as

shown in Figure 4.5.

4.3 Creep Tests

The creep tests consisted of measuring the middle
span deflections of a simple supported beam under equal
constant bending moments, applied at the supports.

The following equipment was used:

1. Beam mounted on end-piece (Figure 4.5), with two
inch diameter pulleys, from which a given weight could be
hanged, producing the desired end bending moment.

2. Containers with lead pellets were used as loads.
In this way, any arbitrary weight could be chosen,
including the effect of hangers.

3. A Starret dial indicator was used to measure the
middle span deflections. Some of the dial characteristics
are as follows: range 0.4"; graduation 0.0001"; minimum
load to overcome internal resistance 0.062 pounds; spring

constant 0.0063 pounds per inch.

 

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73

4. To measure time, use was made of an ordinary
electric clock (Westclox), and a digital electric timer
(model T-lOl), manufactured by Nuclear Instrument and
Chemical Corporation, Chicago, Illinois. This timer
measured hundredths of a minute, and had a range of
999.99 minutes. After this number was reached, the
counting would start from zero again.

5. Finally, a hydraulic jack was used to apply the
loads. A 2" x 6" x 30" piece of lumber was attached to
the head of the jack. Before starting the test, the head
of the jack was raised, and two weights, resting on the
piece of lumber, were centered under the points of appli-
cation and hung from the pulleys. A slow uniform downward
motion of the jack was obtained by means of a control
valve. A ballast of 56 pounds (two lead bricks) helped
to further stabilize the motion and reduce the change of
velocity while the load was transferred from the jack to
the test model. In this way the load was applied at
both ends uniformly and without significant impact effect.

Figure 4.6 shows the general set-up of a specimen
under a creep test.

The help of a second laboratory operator was used to
better synchronize the zero time with the application of
the load, and it proved to be especially useful in the
reading of the deflections at intervals of time as short

as 0.06 minutes. It was pointed out in Chapter III that

 

74

 

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to facilitate the evaluation of the relaxation modulus,
very small intervals of time were to be taken for the
first few minutes of the creep test.

Tests were run at different specimen ages. Finally,
35 days from the day of casting was adopted as the
specimen age for all tests. This age was adOpted
partially arbitrarily, and partially influenced by the
aging tests (see section 4.5).

To ensure that the values for the creep compliance
were obtained within the linear range of the material,
several tests were run under different loads. In fact,
the maximum stress reached was 305 psi, a figure well
below the 2500 psi claimed by the manufacturer* as linear
range.

A better understanding of the results plotted in
Figure 4.7 will follow from an estimate of the uncertainty
involved in the tests. The effective beam length, from
center to center of supports, was 19.25 i 0.001", and the
depth 0.75" i 0.01". The effective length (pulley's
radius plus hanging wire's radius) was 0.953" 1 0.001".
The weights were measured in a scale with sensitivity of
0.01 lb. The prOportion of hardener to resin varied

from 9.97% to 10.03%, which, according to the

 

*

Technical Service Note, A/24UX-l, Structural Resins
Department, CIBA Products Corporation, Kimberton,
Pennsylvania, 1969.

 

(inches)

Deflections

76

 

 

 

 

  
 

 

 

 

 

 

 

o = 305
M = 28.593
.120 —
P = 30
.110 5 P = load applied (lbs.)
M = moment (in lbs.)
0 = stress (psi)
.100 '
.090
o = 203
-°30 F M 19.062 '
.070
.060 =
M = 14.296 ° 153___
.050
o = 102
.040
.030
3
.020' 1 l l l J_‘ 1
0 100 200 300 400 500 600

Time (minutes)

Figure 4.7.--Middle span deflections on creep tests.

 

77

manufacturer's literature, would mean a fluctuation of
i 1% of the mixture's stiffness.

Now an estimate of the uncertainty is readily
available. The middle span deflection of a simply

supported beam under equal concentrated end moments is

 

given by
2
A = 551 (4'1)
or else
2
A = 3WRL3 (4.2)
2Ebh

where W is the hanging weight; R, the lever length; E, the

Young's modulus; b, the beam width, and h, the beam depth.
Taking logarithms of both sides of Equation (4.2),

changing the negative signs to positive (2), and dif-

ferentiating, one obtains

dA _ aw dR dL 68 db dh

where each quotient gives the relative uncertainty.
Assigning numerical values to the right-hand side,

Equation (4.3) becomes

—— = 0.061 (4.4)

78

This means that the uncertainty of the values for the
deflections is i 6%, which more or less fits the results

shown in Figure 4.8.

4.4 Creep Compliance Values

The deflections plotted on Figure 4.7 gave rise to
different sets of values for the creep compliance. These
values were averaged, and plotted on a semilogarithmic
paper. A curve was drawn to best fit the tendency of the
points plotted, and the values read from the curve were
adopted as the "true" values of the creep compliance
(see Figure 4.8).

The justification of this procedure lies, first of
all, on the fact that in any measurement, the average of
the different observations is the best estimate of the
true value (2). Then, in this case, the scale selected
in the semi-logarithmic paper to plot the mean values
obtained for the creep compliance, had the same sensitivity
as the dial gauge used to read the deflections, i.e.,
four significant figures.

The values read from the curve of Figure 4.8 appear
tabulated on Table 2.7 together with the corresponding
values of the relaxation modulus used in the numerical

examples of frame analysis presented in Chapter V.

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4.5 Aging Tests

 

Two samples were tested a number of times at dif-
ferent ages to determine the curing process, the aging or
the strengthening of the samples as they grow older.
Although different results were obtained in the different
tests, as can be seen in Figures 4.9 and 4.10, the tests
cannot be considered conclusive, since there could be a
number of factors, other than age, affecting the stiffness
of the material.

An aging test, isolating age from all other possible
causes, may prove useful, but in the present case would be

beyond the sc0pe of this study.

4.6 Frame Tests

Three different plane portal frames were made and
tested. All were made up of rectangular bars with cross
section of l" x 3/4". They will be referred to as Frame 1,
Frame 2 and Frame 3. Frame 1 was a rigid frame fully
clamped at both supports, and tested under loads of 2 and
10 pounds. A sketch of the general set-up is shown in
Figure 4.11. Frame 2 was essentially like Frame 1, with
the beam two inches longer, and loads of 5 and 15 pounds,
instead of 2 and 10, respectively. Frame 3 had one support
clamped, and the other hinged, and was tested for 2 and

10 pounds as sketched in Figure 4.12.

6

Creep Compliance (10 /psi)

82

4.76

 

4.76
”A L=l9.25" ‘9

9 day old specimen

     
  

LA)
0
U1

LA)
0
O

 

 

 

 

 

T L l l l l l

0 20 40 60 80 100 120

Time (minutes)

Figure 4.9.--Aging test. Simply supported beam.

Creep Compliance (106/psi)

83

P=l lbs

“I“:

L=18" 1
8 day old specimen

 

 

 

 

 

Shear deformation
has been neglected.

 

l l l l l l

0 20 40 60 80 100 120

Time (minutes)

Figure 4.lO.--Aging test. Cantilever beam.

84

 

 

 

 

 

92" 92"
L 8 L 8 4L gDial indicator
i 1 i Pulleys
4:9
\\Dial indicator
a?)
In
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Supporting frame

 

 

 

10
lbs

 

 

 

 

ballast

lbs

 

 

 

 

 

 

Hydraulic jack\\N

 

 

 

 

 

 

 

Figure 4.11.--Frame l.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Ball bearings
2
10
lbs lbs
ballast
J 111

 

 

 

 

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Figure 4.12.--Frame 3.

r“ r'n“ ‘

711

If,

K// Hydraulic jack

86

The vertical load was hung directly from the top
middle point of the beam, while for the horizontal load
two pulleys mounted on ball bearings were used. The
loads, as in all other tests, were applied with the help
of a hydraulic jack to avoid impact, and reduce to a
minimum the effects of acceleration.

The tests were conducted when the frames were 35
days old, and the readings were spread to follow the
pattern shown in the t-axis of Figure 3.6. As before, the
help Of a second Operator was used to synchronize the

zero time with the application of the loads.

 

CHAPTER V

RESULTS AND DISCUSSION

5.1 Introduction

 

In the first part of this chapter, results from the
various methods used in this study will be presented and
compared. The second part will consist of a general dis-
cussion, together with some conclusions.

5.2 Comparison of Relaxation
Modulus vaIues

 

 

Figure 5.1 shows two curves Obtained for the
relaxation modulus values for the epoxy resin used in the
lab tests of this work.

Curve 1 was Obtained by inverting the creep compliance
values given by the Equation (2.56), using for the
parameters the values obtained by LaPalm (13). Curve 2
was obtained by direct inversion of the laboratory results
obtained in this work. More exactly, curve 2 is the
inverse Of the creep compliance curve of Figure 4.8, which
is the average of the different laboratory results.

The discrepancy of both curves is of the order of

2%. This confirms the reproducibility of the material used

87

5)

Relaxation Modulus (psi/10

 

88

 

Curve 1

3.3” (LaPalm)

_ urve 2
.b test)

 

l L l l J l

100 200 300 400 500 600
Time (minutes)

«i.
l
0

Figure 5.l.--Comparison of relaxation modulus curves.

 

89

for the tests, and adds support to the values obtained

thus far for the relaxation modulus Of such material.

5.3 Frame 1 Joint Displacements

 

Frame 1 is sketched in Figure 4.11. The vertical
deflection of the middle span point, A, and the horizontal
displacement of the right-hand corner, B, are plotted in
Figures 5.2a and 5.2b, respectively.

Figure 5.2 shows the vertical deflections of midspan
point A, and horizontal displacements of joint B of Frame 1.
Each one of them is represented by three curves. Curve 1
represents results Obtained from Equation (3.8) by means
of the program VIELANAL, developed in Chapter III and
listed in Appendix B. Curve 2 represents the quasielastic
solution, i.e., at every time, t, an1 elastic solution is
carried out by simply substituting the modulus of
elasticity by the relaxation modulus E(t). Finally,

curve 3 represents the displacements measured in the

laboratory using models, as described in Chapter IV.

5.4 Frame 2 Joint Displacements

 

It was pointed out in Chapter IV that Frame 2 was
like Frame 1, except for the horizontal member, which was
two inches longer. Thus, Figure 4.11 can also serve as a
sketch for Frame 2 if we read 10 1/2" instead of the 9 1/2"
shown as the length for half Of the horizontal member

referred to.

 

90

.H oamum mo d chom cmmmcHE mo cOHDOOHwoO HMOHpHo>n:.MN.m oHDmHm

AmouchEV mEHB

 

com com com com oov com com OOH o
q _ H H H 4 — -
1
11. H oEmnm 11 .
m a .
All

 

 

 

mmu muoumuoan
m o>usu

".2
:oHumsUm
H o>udo

OHgmmHmuHmmsv
m w>uso

 

meo.

omo.

mmo.

ooo.

mmo.

(sauna?) HornoaIJGG

91

.H oEmum wo m ucHon mo ucoEOOMHQmHO HmuCONHHomII.nN.m ousmHm

AmouscHEv 0849

com com com com oov

com

com

00H

 

 

— 1 M H H

H ofimum

 

 

 

+ >
and» NHODMHOQMH I m o>uso

 

 

Hw.mv soHumsvm I H m>uso

 

(loHumMHouHmmsv I N m>uso

H

 

omo.

mmo.

ovo.

mvo.

omo.

(seqour) quemeoetdsrg

92

Figure 5.3 shows the joint displacements for Frame 2.
The comments made in section 5.3 about curves 1, 2 and 3

can be applied here to Figure 5.3.

5.5 Frame 3 Joint Displacements

 

Frame 3 is sketched in Figure 4.12. Unlike Frames 1
and 2, it is unsymmetric. Also, one of the supports is
hinged, while the other remains clamped.

Figure 5.4 represents the vertical deflection of the
midspan point A, and the horizontal displacement of joint B.
The comments about curves 1, 2 and 3 can also apply here.

At this point, a remark about frame stresses seems
to be in order, since so far, nothing has been said about
the range within which the frame tests have been conducted.
For Frames 1 and 2 the maximum normal stresses occurred
in the midspan point, A (see Figure 4.11), in both cases,
and they were 335 psi and 364 psi, respectively. For
Frame 3, the maximum normal stress occurred at the joint
C (see Figure 4.12), and it was equal to 406 psi.

Although these figures exceed the maximum stress value
for which the material was tested in this study (305 psi,
see Figure 4.7), the agreement of laboratory results with
the numerical linear analysis causes the writer to feel
that it is plausible to assume that the frame tests were

run within the linear range of the material.

93

 

 

 

__, l
A B
.090—
4"” Frame 2 ”L”
Curve 1
quasi-elast'o
.085r
Curve 2
Equation (3.:
3;.080 -
3 Curve 3
o laboratory test ,
é 0
g .
H .075 f
.IJ o
o
m .
H
‘H
w
o
.070 - ‘
.065

 

l l I I l l

d 100 200 300 400 500 600
Time (minutes)

Figure 5.3a.--Vertical deflection of midspan point A
of Frame 2.

 

94

.m wEwum mo m ucflon mo ucmamomammwo HmuconuomII.nm.m musmflm

Ammuscflev mafia
com con com com oov com com ooa o

 

_ - _ 1 q - d d t

lomo.

 

 

 

 

 

 

 

.s .
m ¢ .\ mac
1 T .
. . -omo.
o -
ilvll
umwu muoumuoan I m w>nsu
Immo.
,m.mv coflumsvm I N m>nsu
oflummHmIHmmso I H m>uso
Iovo.

 

(SEQOUT) UOInoaIJBQ

95

.m mfimum mo m ucflom commoflfi mo coapomammo Honduuw>II.Mv.m musmflm

Ammuscwfiv mafia

 

 

 

com com com com oov com com ooa o
q . .q . q q _ a _
“1
m wEmum
4% one.
o.
<
0 II? m . owe.
a o
o I
. iomo.
p o lOOHo
ummu unoumuonma I m m>uso
“ m coflumsvm I H m>us
roaa.

 

.ummHmIHmmsw I N 0>HDU

(saqouI) uorqoetgefl

96

 

.m mamum mo m ucH0w mo ucofimomHmmHU HmucoNHuomII.Qw.m musmflm

Ammuscflfiv mafia
com com com com cow com com OOH o

— _ q _ . _

m mfimum Aw

 

 

>

ummu muoomuoan I m m>nsu

H m>ndu

Am.mv couumsw
oabmmHoIHmmsw I m m>uso

 

ono.

omo.

omo.

ooa.

CHH.

(seqout) quemeoetdqu

97

5.6 Epoxy Resin Used for the
Laboratory Models

 

 

The epoxy resin used for laboratory models has a
wide range for which the strains are prOportional to the
stresses. Also, Figure 5.1 shows that, within this linear
range, the behavior of the resin is viscoelastic.

The material does not have a fixed stiffness in its
early age. This means that, when comparing test results
from different samples, attention should be given to the
age of such samples. Figure 5.5 shows the tendency with
which the resin stiffness increases with age. The curve
was obtained from the values corresponding to 120 minutes
shown in Figures 4.9 and 4.10. This is 120 minutes after
application of load. The inverse of these deflections was
multiplied by a reducing factor. This curve, based only on
two different tests, does not pretend to be conclusive,
but, as stated above, it shows the general tendency of the
material stiffness increasing with age.

In the present work, the aging of the material during
the test period was disregarded, since all loads were

applied to the material at the same material age.

5.7 Laboratory Equipment

 

The laboratory equipment was simple but adequate.
Still, a redesign of molds and end-pieces is recommended.
The end-pieces should be designed as an integral part of the

molds. This would avoid disturbances of the Specimen

98

.mmm nuHB ommmuocfl mDHSGOE coaummemmII.m.m oudmflm

“mumps made
on om om. ow om om OH o

 

_ _ q _ _ _ d

mumme Hm>maflucmu 4

mpmmB Emmm mHmEHm o

 

 

(SOT/18d) snInpom uotqexetea

99

while attaching the end-pieces, and, in addition, would
provide more accuracy in the attachment of such pieces.
Furthermore, the molds should be designed in such a way
to allow the test to be set up before the molds are
released. Finally, a set of screws could be used to
allow the molds to be removed smoothly, leaving the
specimen undisturbed and ready for the desired test

without requiring any further handling.

5.8 Relaxation Modulus

 

The presence of oscillations of the lower bound for
the values of the relaxation modulus (see Tables 2.3 and
2.4) still remains open to further study.

However, it has been shown that, by means of an
adequate redefinition of time intervals, this difficulty
can be overcome. In this way, we can invert creep com-
pliance values to obtain the corresponding relaxation
modulus without recourse to the Laplace transform.
Furthermore, this method is applicable to creep values
obtained from an algebraic formulation, as well as those
obtained from discrete laboratory data.

A relaxation test is still desirable and recommended
to prove the accuracy of both Equation (2.4) and Equation

(2.5).

100

5.9 Matrix Analysis Formulation

 

In Chapter III a matrix analysis for framed
structures was developed, and a digital computer program,
VIELANAL (see Appendix B), was written. The results of
this numerical solution differed from those obtained by
models tested in the laboratory by discrepancies of the
order of 10%.

In section 4.3 the uncertainty of the measurement
of midspan deflections of a simple beam model was esti-
mated to be of the order of 6%. In the case of a frame
with three members, it seems reasonable to assume that this
uncertainty will increase. Additional sources of error
can be stated, such as imprOper set up, friction in the
pulleys used for transmission of horizontal load, and,
for Frame 3, friction in the hinged support. Finally,
part of the 10% of discrepancy between the theoretical
and experimental results has to be accounted for by the
fact that the numerical analysis was an upper bound of
the solution.

This program was based on Equation (3.3), which
provides an upper bound to the governing Equation (3.2).
By a simple change in indices, using E(O) instead of E(l)
and (1+1) instead of (i), the program can be run for the
lower bound given by Equation (3.4). However, this
requires knowing the value of E(O), which is not readily

available.

101

E(O) can be approximated from Equation (2.5),

yielding

E(O) = $7, (5.1)

where the value of D(O) would have to be obtained from
extrapolation in the semilogarithmic curve of Figure 4.8.

In the present work, however, the extrapolation from
the curve of Figure 4.8 is not well determined, due to
lack of enough data near time equal to zero, producing
little more than meaningless results.

The difficulty of obtaining the glassy properties
of the material increases when we keep in mind that, while
on the one hand, a step load at time zero is desired, on
the other hand, the application of such load has to be
smooth and gradual to avoid impact effect.

When the nonlinear geometry and the axial force
effects were omitted (see Table 5.1), the joint displace-
ments dropped about one per cent. It must be kept in
mind, however, that the samples run in this study were
kept under practically linear geometry conditions, to
ensure linear behavior in the material. Under a heavier
loading system, the discrepancy between linear and non-
linear geometry and thrust effects may be of relevant
importance, and, therefore, further research in this area

is recommended.

102

Figures 5.2 through 5.7 show that the matrix formula-
tion developed in Chapter III, based on LaPalm's Equation

(3.3), produces satisfactory results.

5.10 Number of Finite Parts per Member

 

To calculate the fictitious joint loading, a finite
difference method was used. Table 5.2 shows the accuracy
obtained and the computer time needed for a given number
of equal parts into which a member is divided.

From Table 5.2 we can conclude that a choice of 10
or 15 parts per member may be considered economical and

still accurate enough for most practical considerations.

 

103

TABLE 5.l.-—Axia1 force effect on joint displacements.

 

I. Vertical deflection of midspan point A (see Figures

4.11 and 4.12) at time t

Axial Force Effect

Axial Force Effect

= 1.

 

Frame taken into Neglected Discrepancy
ConSideration
in in %
1 0.0461 0.0459 0.5
2 0.0600 0.0597 0.5
3 0.0762 0.0755 1.0

 

II. Horizontal displacement of joint B (see Figures 4.11

and 4.12) at time t = 1.

Axial Force Effect

Axial Force Effect

 

Frame taken into Neglected Discrepancy
Con81deration
in in %
1 0.0318 0.0315 1.0
2 0.0327 0.0324 1.0
3 0.0717 0.0710 1.0

 

104

TABLE 5.2.--Number of finite parts taken per member, versus
accuracy and computer time.

 

 

Number of Equal Computer Time* Accuracy**

Parts (seconds) (%)

6 0.14 85.4

10 0.16 92.7

15 0.19 95.4

20 0.25 95.9

40 0.78 97.2

50 1.33 97.7

 

*
CDC 3600 Digital Computer

**
A fixed-end beam under a load equal to cos x was
considered. The values obtained from the numerical
analysis were compared with the exact solution.

CHAPTER VI

SUMMARY AND CONCLUSIONS

A matrix method for numerical analysis of framed
structures made of linear viscoelastic materials has been
presented. The solution method has been embodied in a
computer program written in Fortran. This program has
been tested for plane frames, with different supports
conditions, in a CDC 3600 and in a CDC 6500 digital
computer.

The analysis uses constant loads concentrated at
the joints, as the actual external loading system. Then,
as time increases, fictitious joint loads are computed
and added to the actual external joint loads. The
fictitious joint loads were derived from the deformation
history of the structure. This was carried out by means
of a finite difference method scheme.

In order to store the information needed from the
past deformed configurations of the structure, a redefini-
tion of time interval was used as many times as needed
to cover any desired time Span. This time interval
redefinition is flexible, and can be modified and adapted
to particular cases, according to the structure size, and

computer store capacity.

105

106

In the examples worked in this study, small loads
were used. The purpose of using small loads was two-fold.
First, this assured working within the linear range of
the material. Second, the joint displacements were kept
small, reducing to a minimum the reaction of the internal
spring of the dial indicator. This resulted in a case
with small axial thrust and practically linear geometry.
As a result, the nonlinear geometry and axial forces
included in this study had negligible effects on the
joint displacements and stresses of the frames. When the
nonlinear geometry and axial forces effects were omitted,
the lateral sway dropped only by 2%. Of course, for
heavier loading, this discrepancy may increase considerably.

Laboratory tests were run with symmetric and
asymmetric frames made of a linear viscoelastic material.
The laboratory results were compared with those obtained
from the numerical analysis with satisfactory agreement.

A computer program to evaluate the creep-relaxation
inversion has also been presented. In this program, a
redefinition of time interval was also needed. However,
this redefinition differs in nature and purpose from the
one used in the frame analysis.

The relaxation modulus values are obtained from the
creep-compliance values without recourse to the Laplace

transform. The creep compliance can be given either as a

"I

107

general algebraic expression or as a set of discrete
values, obtained from laboratory tests.

In conclusion, a matrix method of analysis for
frames made of linear viscoelastic materials has been
developed. This analysis is carried out without expressing
the viscoelastic material prOperties in terms of spring
and dashpot models. Furthermore, this method is general,
and parallels those existing for elastic structures under
static constant loadings.

Future extensions of this study could include:

1. Effect of nonlinear geometry and axial thrust

for structures under large deflections.

2. Nonlinear viscoelastic materials.

3. Loading that changes with time.

4. Dynamic analysis.

L IST OF REFERENCES

108

10.

11.

1.2.

13.

LIST OF REFERENCES

Alfrey, T., "Mechanical Behavior of High Polymers,"
High Polymers, Vol. VI, Interscience Publishers,
Inc., New York, 1965.

 

Baird, D. C., Experimentation: An Introduction to
Measurement Theory and Experiment DeSign, Prentice-
HaIl, Inc., Englewood Cliffs, New Jersey, 1962.

 

Battista, O. A., Fundamentals of High Polymers,
Reinhold Publishing Corporation, New York, 1958.

 

Bland, D. R., The Theory of Linear Viscoelasticity,
Pergamon Press, Oxford, 1960.

Flfigge, W., Viscoelasticity, Blaisdell Publishing
Company, Waltham, Massachusetts, 1967.

 

Froberg, C. E., Introduction to Numerical Analysis,
2nd edition, Addison-Wesley Publishing Company,
Reading, Massachusetts, 1969.

Fung, Y. C., Foundations of Solid Mechanics, Prentice-
Hall, Inc., Englewood Cliffs, New Jersey, 1965.

 

Gere, J. M., and W. Weaver, Analysis of Framed
Structures, D. Van NostrandCCompany, Inc., Princeton,
New Jersey, 1968.

 

Gurtin, M. E., and E. Sternberg, "On the Linear Theory
of Viscoelasticity," Archive for Rational Mechanics
and Analysis, Vol. 11, 1962, pp. 291-356.

 

Hall, A. S., Woodhead, R. W., Frame Analysis, 2nd
edition, John Wiley and Sons, Inc., New York, 1965.

 

Hoff, N. J., "Approximate Analysis of Structures in
the Presence of Moderately Large Creep Deformations,’
Quarterly of Applied Mathematics, Vol. XII, 1954,
pp. 49-55.

 

Hult, J. A. H., Creep in Engineering Structures,
Blaisdell Publishing Company, Waltham, Massachusetts,
1966.

LaPalm, G. E., "The Creep Buckling of Viscoelastic
Beam-Columns," A Thesis Submitted to the Faculty of
Purdue University, January, 1968.

109

14.

15.

16.

17.

18.

19.

110

Odqvist, K. G., "Applicability of the Elastic Analogue
to Creep Problems of Plates, Membranes and Beams,"
Creepiin Structures, Colloquium, Stanford University,
Academic Press Inc., Publishers, New York, 1962.

 

Patel, S. A., and Venkatraman, B., "On the Creep-
stress Analysis of Some Structures," Cree in
Structures, Colloquium, Stanford UniverSity,
Academic Press Inc., Publishers, New York, 1962.

 

Salvadori, M. G., and M. L. Baron, Numerical Methods
in Engineering. 2nd edition, Prentice-Hall) Inc.,
Englewood Cliffs, New Jersey, 1962.

 

Timoshenko, S. P., and J. M. Gere, Theoryyof Elastic
Stability. 2nd edition, McGraw-Hill Book Company,
New York, 1961.

 

Wang, Chu-Kia, Matrix Method of Structural Analysis,
2nd edition, International Textbook Company,
Scranton, 1970.

 

Williams, M. L., "Structural Analysis of Viscoelastic
Materials," AIAA Journal, Vol. 2, No. 5, May, 1964,
pp. 785-808.

 

APPENDICES

111

APPENDIX A

NOTATION

112

A1,...’ A4

B1,..., B4

b
C1,..., C4

D(t)

e

[—0.

W W 71

constants

APPENDIX A

NOTATION

of integration

constants of integration

member width

constants of integration

creep compliance

compliance
compliance
modulus of
relaxation
fictitious

fictitious

at long time

at short time

elasticity

modulus

equivalent joint load vector

distributed load at time tk

length of each element when the member is

divided in

n-l equal parts

member depth

moment of inertia

index

stiffness matrix

constant

index

member length

bending moment

113

114

bending moment at time tk

bending moment at left support

bending moment at right support

number of members in a structure

axial force

axial force at a time tk

number of nodes when a member is divided in
n-l equal parts

combined joint load vector

combined joint load vector at time tk
equivalent joint load vector

equivalent joint load vector at time tk
distributed load

distributed load at time tk

pulley's radius

transversal reaction at left support
transversal reaction at right support
index

time dependent solution

time

a given time

member end displacements caused by the
combined joint load vector

member transversal deflection caused by

the combined joint load vector

II
v

III
u

III
v

III

IV
u

IV
V

W(tk)

X(tk)

Y*

IV
Y

115

u at time tk

first, second, third and fourth
derivatives, respectively, of u with
respect to the spacial variable x
transversal deflection of a member with
clamped supports.

v at time tk

first, second, third and fourth
derivatives, respectively, of v with
respect to the spacial variable x
actual joint load vector

actual joint load vector at time tk
joint displacement vector

joint displacement vector at time tk
coordinate along the member axis
member end displacements

boundary values

member transversal deflections under

actual loading

y at time tk

first, second, third and fourth
derivatives respectively of y with

respect to the spacial variable x

116

middle Span deflection
strain as a function of time
constant strain

stress as a function of time
constant stress

time

APPENDIX B

COMPUTER PROGRAMS

117

APPENDIX B

COMPUTER PROGRAMS

Two computer programs have been written and used
for the present study. Program CREEPINV computes Equations
(2.21) and (2.22) yielding the values of the relaxation
modulus, when the values of the creep compliance are given.
Program VIELANAL solves Equation (3.8), giving the
deformation history of a plane structure made of linear
viscoelastic material. This program uses the subroutine
ELASTAN to make an elastic analysis at every fixed time tk.
The listing of this subroutine is omitted, and the reader
is referred to reference (18) for a complete description
and listing of it.

The important FORTRAN names used in these programs
are defined below in alphabetical order. Any name which

appears in more than one routine but maintains the same

meaning is defined only once.

A(I,J) = coefficients of systems of simultaneous
equations
AREA(M) = area of cross section of member M

AXF(M) axial force in member M

A1(M),..., A4(M)

constants of integration corresponding

to member M

118

Bl(M),..., B4(M)

C1(M),..., C4(M)

D(KE)

DEL(M)

E(KE)

EAOL(M)

EIOL(M)

F(M)

FICT(IX,M,KY)

FML(M)

FMR(M)

FSL(M)

FSR(M)

HO(M)

119

constants of integration corresponding
to member M

constants of integration corresponding
to member M

value of creep compliance at time KE
length of each element into which the
member M is divided

value of relaxation modulus at time KE
modulus of elasticity times cross
section area of member M

modulus of elasticity times moment of
inertia of cross section of member M
axial force in member M

fictitious load at point IX of member M
at cycle KY.

fictitious moment at left end of member
M

fictitious moment at right end of
member M

fictitious Shear at left end of

member M

fictitious shear at right end of
member M

initial horizontal coordinate of end

of member M.

INERT(M)
K(M)
K2(M)
K4(M)

KE
KELIMIT
KI

KY

L(M)

NM
NP

NPE(M,J)

NPR

NPS

P(I)
PFICT(I)
PP(I)

u(x)

U4(X)

UU(X)

UU4(X)

120

moment of inertia of member M

square root of K2(M)

axial force/fluxural rigidity of member M
square of K2(M)

time

time limit

iteration number

cycle number

length of member M

number of nodes on a member divided in N—l
equal parts.

number of members in the structure

total degree of freedom

global degree of freedom of coordinate J of
member M

degree of freedom in rotation

degree of freedom in linear displacement

load at coordinate I of the structure
fictitious load at coordinate I of the structure
load at coordinate I of the structure
transversal deflection at point x of a member
under compression

fourth derivative of U(X)

transversal deflection at point x of a member
under tension

fourth derivative of UU(X)

UUU(X)

V(IX)

V4(IX)

VO(M)

W(I)

X(I)

XJ(I)

XLO(M)

Y(IX)

Y4(IX)

Y14(IX,M)

Y24(IX,M)

Y34(IX,M)

121

transversal deflection at point x of a member
under no axial load

transversal deflection at a point IX of a
member with clamped ends

fourth derivative of V(IX)

initial vertical coordinate of end of

member M

load applied at coordinate I of the structure
joint displacement along coordinate I of the
structure

joint displacement along coordinate I of the
structure

initial length of member M

total transversal displacement at point IX
of a member

fourth derivative of Y(IX)

storage of Y4(IX) for use in member M during
cycle 1 of new iteration

storage of Y4(IX) for use in member M during
cycle 2 of new iteration

storage of Y4(IX) for use in member M during

cycle 3 of new iteration.

LISTING 0: COMPUTcp PROGRAMS

PQOGPAM VIELANAL
C:=::::::===:==:::=::===================C
C ANALYSIS OF VISCOELASTIC STRUCTURES C
C==::==:::==:::=:=::::==:::::=::=:=::==:C
QFAL K9K20K4vLoINFQT
DIMFNSION F(76PI
DIMFNQION PD(pn)
DIMENSION NPEI706)9HOI7)oVOI7)oXLO(7)oFAOLI7)oEIOL(7)
DIMENSION AQFAI7IQINFDTI7)
DIMFNSION AXF(7I
DIMFNSION XJI’I)
DIMFNSION X(éq?)
DIMFNSION KI7)qK?I7)oK4(7)9LI7)
DIMENSION W(POI
DIMENSION Y(IO)
DIMENSION Y4(10)
DIMENSION I:ICT(109706)
DIMENSION DEL(7IOFI7)
DIMENSION A1(7)IA?(7)0A3(7)0A4(7I
DIMFNQION RII7IQQ?(7)OR?I7IORQI7I
DIMFNSION C1(7)oC?(7)oC1I7)oC4(7)
DyMFNQION PFIPT(?1)IADUM(?00?“)'V°“M(7")
DIMFNSIDN FMLI7IQFMDI7)quLI7IOF§QI7I
DIMFNSION V(?O)0V4(Ph)
DIMFNSION Y14I?Do7)IY?4(?Oo7)oY34IPOo7)
COMMON/STRUCT/NPQNPPQNPSqNM
COMMON/LOAD/Pp
COMMON/MEMBER/MEMN01NPEOHOOVOOEAOLCEIOLQXLOQL
COMMON/AXIAL/AXF
COMMON/JDISPL/XJ
COMMON/QFLAX/FQKFLIMIT
COMMON/TIME/KF
SINHIX) = 009*(FXDIX) - FXPI’XII
COSHIX) 2 0.G*(¢XD(X) + pXPI-XII
U(X): A1(M)*COSIK(M)*X) + A?IM)*SIN(K(M)*X) +A3IM)*X +
AQIM)
UAIX) = AIIM)*K4(M)*POSIK(M)*X) + A?(M)*K4(M)*SIN(KIM)*X)
UU(X) = PIIM)*CO<H(KIM)*X) + R?(M)*QINHIK(M)*XI +
R3IM)*X + 94(M)
UU4(X) = BIIM)*K4(M)*COSHIKIM)*X) + B?IM)*K4(M)*SINH(K(M)*X)
UUU(X) = CIIMI*X**3/6.o + C2IM)*X*x*o.= + C3<M)*X+C4(M)
DFAD IDoKELIMIT
10 FOQMATIIF)
C
CALL QFLAX
C

122

123

C
C D‘AD AND pQINT STPUCTUQF pQOpEQTIES AND LOADING
C
QFAD POQNPQNPDQNDQONM
?O FORMATI4Iq)
PFAD ?IOIWII)QI=IONP)
PI FOPMATIRFIOoO)
POINT ??.ND.NDD9ND<9NM
?? FODMATI///IOX.*§TDHFTUPF PPOPFQTIF9*//
IOX0*TOTAL OFGRFE OF FPFFDOM =*9I1/
1OX0*DGQ. O: 59. IN POTATION =*QI3/
IOX9*DGPQ 0F F9. LIN. TDANSO =*0I1/
IOX0*NUMRER OF MEMBERS =*0I3//)
PRINT 23
P3 FOQMATIIOXO*FXTFRNAL LOADS AT THE JOINTS*/)
DO 240I=19Np
?4 pQINT ?RQI9W(I)
7R FOQMATIIQXQ*W(*OI30*) =*QF707)

DJV"

POINT ?6
95 poQMAT(//Inx,*MFMn=R: PROP=QTIFq*//1nx.
I *MFMRFP N91 NP? ND? N94 NPS N96*09Xo
p *HO v0 ADEA MOM. OF INEDTIA*/)

DO ?9 I=19NM
RFAD P7. MEMNO.(NPF(IoJ)oJ=106)9HO(I)0VO(IIOAREA(I)9
I INFPTII)

27 FORMATI7I590FIDoO)
pRINngOMEMNOO(NpE(IOJ’QJ3l06)OHO(I’0VO(IIOAREA(I,Q
I INFPTII)

?R FORMATIII304XI6150?X9?FI1.29F13040F14o6)

7° CONTINUF
DEAD 100N

10 FOPMATIIS)
PARTS = N-I
pDINT RIQDAPTQ

31 FORMATI//IOX0*=ACH MFMRFR I9 DIVIDPD IN*9F5.193X0
I *FOUAL DAQTS*)
KP =1
KY = 1
K! = n
SE = 5(1)

DO 32 I=10NM
XLO(I) = SORTIIHOII)*HO(I) + VO(I)*VOIIIII
LII) = XLOII)
FAOLII) = FF*ADFA(II/XLOII)
3? FIOLII) = FF*INEDTII)/XLOII)
DO 33 I=IQNP
11 POI!) = W(I)

CALL ELASTAN

 

124

pPINT 340KF

14 FORMATIIHIQQRX\* ------------------ */56X0*-*016X9*-*/
I 56Xo*- TIMF = *0139* -*/56Xo*-*016X0*-*/
P 56Xo* ------- -----------*//)
C
C pDINT STRUCTUQF JOINT DISPLACFMFNTS
C
DOINT an

40 FORMATI/IIOX0*THF STQUCTUQE JOINT DISPLACEMFNTS APF*/I
DO 41 I=10NP

41 PPINT 4?QIOXJ(I)
42 FOQMATIIQX9*X(*0120*) =*9E13.5)
pDINT 6D

6D FORMATI///lOX¢*TRANSV¢QSF DFFLECTIONS OF MFMRERS*)
DO 100 M=10NM
DEL(M) = L(M)/DADTS

F(M) = AXF(M)
K?(M) = -F(M)/(FF*IN¢QTIM) )
K4(M) = K?(M)*K?(M)
K(M) = QODTIAPCIV?IM)))
C
CALL UDFFL‘XIIOM)OF(M)OK(M)OL(M)QA1(M)OA2(M)QA3‘M’O
I A4IM)IRIIM)082(M)083(M)OB4(M)QCIIMIOCZIMIOC3(M)O
C4(M))
C

IFIFIM) )62066064
6? DO 63 IX=1oN
SS = Ix-l
xx = ss*DFL(M)
Y(IX) = U(XX)
61 Y4(IX) = U4IXX)
GO TO 69
64 DO 6‘5 IX=IoN
S9 = IX-l
XX = SS*DEL(M)
Y(IX) = UU(XX)
6G Y4(IX) = UUQIXX)

GO TO 68
66 DO 67 IX=IQN
Sq = IX-I
XX 3 §¢*DFLIM)
Y(IX) = UUU(XY)
67 Y4(IX) = O.“
69 POINT 6IOM
61 FOPMAT(//IOX9*MFMRFQ*OIP/I

POINT 69o((IXoY(IX))oIX=IoN)
69 FOQMATISI5X9*Y(*0I29*) =*0FI305))
DO IDO J=?o6
J? = J
J1 = J-I

mm

I!”

14?
Ian

,7an
7m

90"!

on?

??D

O

?30

P41

pop
743

I

125

DO 100 IX=10N

FICT(IXQMQJ) = (F(J?*?**KII’FIJI*2**KII)*Y4(IX)
/ F(I)

IFIKYOLTOGI GO TO POO

KI = KI + I

DO IQO M=IQNM

DO 14? IX=10N

Y14IIX9M) = YPAIIYQM)
CONTINUV

KY = 4

GO TO POI

KY = KY + I

KF KY*?**KI

EF = E(KE)

DO 203 I=19NM

FAOLII) = EF*APEA(II/XLOIII
FIOLII) = FP*INFPTIII/XLO(I)
POINT anKF

NDOW : N-?
NCOL = N~I
ITPP = O

'30 770 M‘-=I QNM

CALL EOJTLD (FICT(1¢M0KY)vNoNROWcNCOLoDEL(M)oFIM)oK2IM)o
FMLIM)OFMQ(M)OFSLIM)QFSPIM)QEFQINERT‘M)OL(M)Q

ADUMOYDUV)
pFICTII) = +FMP(1) -FML(?)
PFICTI?) = FSPII)
PFICTI3I = -FQL(?)
DFICTI4) = +FMPI?) +FMLISI
DFICTIQI = no“
pFICTIOI = +FQDI?) +F§L(1)

PFICTI7) = +FMPI3) ~FMLI4)
PPICTIR) = FqL(4)

PFICT(Q) = —FQP(1)

DO 230 I=10NP

PP(I) = W(I) + PFICTII)

CALL ELASTAN

DO POO M210NM

IFI APSI ARQIFIM)) - APSIAXF(M)I I 0LT. ARSIOoOl*AXF(M))
GO TO PQO

on P4! I=IONM

F(I) = AXFIII

ITFR = [TFO + l

IFIITEQ - 50)?4?07610?41

GO TO PO?

PPINT ?440M

I

 

W

O

?44

126

FOQMATIIHIIQXq*AXIAL FOQCF
*DOFS NOT

GO TO IOOO

CONTINHF

POINT ??°

FOQMAT(///IDXQ*JOINT DISPLACFMFNTS*/)

DO ??7 I:IOND

POINT ??80IOXJIII

IN MFMRFQ*0120
CONVFDGE WITHIN SO ITFDATIONS*)

POQMATIIOX9*X(*0I?Q*) =*.FI$.3)
XIAII) = XIPo?) = XJI3)

xIs.I) = ~XJI?) s XII07I = XJ(?)
X(écl) ‘ X(39?) = -XJ(I)

X(Ao?) XIIQEI XJI‘)

X(qo?) Y(Eq?) XJI6)

X(éo?) X(303) = ~XJI4)

XIQQS) X(?94) = XJIR)

X(Oc?) = XJ(Q) $ X(loa) = -XJ(°I
X(éofi) = X(Soa) = ~XJI7I

TDANSVFDSF OFFLSPTIoNc P

32h

up,

179

191

‘3?!)

POINT 6O

DD 300 M=IQNM

F(M) 2 AXF(M)

KPIM) = “F(M)/IPF*INPQTIMI I

K4(M) = KPIM)*K2(MI

K(M) = QQQTIARSIKPIM))I

CALL HOFFL (XIIOM)QF(M)QK(M)OLIMIOAIIMICAPIMIQA1IMIO
AQIMIQQI(MIOR?IM)093(MIOBAIM)OCI(M)OC2IM)OC3IM)O
C4IM) )

CALL VDFFL (FICTIIOMOKY)ONONPOWONCOLQDELIM)9K2(M)O

V9V49A9YOHM)

IFIF(M)
DO 321
S9 = lx-l

xx = sq*DFL(M)

Y(IX) = U(XX) + V(IX)
Y4(IX) : U4(XX) + va(IX)
an Tn 1?6

DO RP? IX=IQN

SQ = IX-I

XX SC!DFL(M)

Y(IX) = UU(XX) + V(IY)
vaIIX) = HuatvyI + VAIIX)
GO TO 1?6

00 325 IX=IQN

I320o??403?2
Ix=IoN

 

127

sq = lX-l
xx SS*DFL(M)
Y'IX) = UUU(XX) + V(IX)
1?: Y4(IX) = V4IIX)
BPA DDINT 3IOoM
11m FOQMAT(//IOX.*MFMRFD*oIP/)
POINT 3?70I(IY0YIIYI)OIY=IONI
177 FOPMAT(R(SX0*YI*II7I*) =*0F13o53)
GO TO I340033?034Oo3?4034003361QKY
33? DO 333 Ix=IoN
111 Y14(IXIM) = Ya(IX)
GO TO 34m
114 DO 33% lx=l.N
11: Y?A(IX9M) = YAIIX)
GO TO 14h
116 no 337 IX=IQN
317 Y34(IX0M) = YAIIYI
r
c panFlNF FICTITIOUS OISTDIQHTFO LOADING C
C

14fi CONTINUF
JJ = ?
JV 2 KV + I
DO 299 J=JY06
DO PQR IX=ION

PPR FICTIIXOMQJ’ = FICTIIYQMQJI +
I IFIJJ*?**KI) - P((JJ—])*?**KI))*Y4IIX)
? / FIII

900 JJ = JJ+I

1mm CONTINHF
I: (KFoCFoKFLIMIT) GO TO 1000
GO TO I!“

IODD CONTINUF
FNO

SURQOUTINE PELAX

C
c THIS SURROUTINE DFFINFS on means RELAXATION MODULUS
r

COMMON/DFLAX/FIKCLIMIT

DIMFNSInN F(768)
r
C DSAD QFLAXATION vunuLuc VALUF:
c

DO IO I=1927

PFAD 50K9DUMMY
R FOQMATIIRQFIOoO)

E(K) = DUMMY

 

.I

O

C
C
C

C
C
C

n

In

1“

91

p?

91

?4

7O

128

CONTINUF
OF TUDN
FNn

qURQOUTINE FOJTLD (FICTQNONQOWQNCOLODFLQFQKZ.

I FMLoFMQQFQLOFQPvFQINFPTQLQAQY)
OPAL K90 INFPTQL
DIMENSION FICTIN) QAINPOWQNCOL)QY(NPOW)

YMITIALIZ‘r

OO 1 O I=I o NOON
DO I O J: I 0 NC 0L
A(IQJI = 0.0

C = OFL*DFL*K9

QOUAQE PAQT OF MATRIX A

Np Nan — 9
NI NQOW—I

DO ?n I=IQNP

AII0I+PI = IOA

nn 1] [=1.Nonm

A(IQI'pI = Ian

on P? I=IONI

AIIOI+II = C-a.0

D0 P1 I:PQNROW

AIIOI°II = C-aon

DO 24 I=20NI

AIIQI) = GOO—?0O*C

A(IOI) A(NPOMQNQOW) = 7.O-?9O*C

II

FODM LAQT COLUMN OF AUGMFNTFO MATPIX A

DELO : ”FL*DFL*QFL*nFL

DO SO I=IQNPOW

AIIONCOL) = OFL4*FICT(I+I)
LL 3 NPOW - I

CALL QIMFOSIAqNQOWoNCOLoLLoY)

va = I—S.o*vIl)+a.0*v(P)-Y(3) )/(OFL*OFL)
YD? = (~S.O*Y(NQOW)+a.O*Y(NPOW-l)-Y(NQOw-2)
FML = -F*INFDT*YL?

FMD = —F*INFQT*YD?

QHML = QUMD = n.n

on an I:?.NDOW
DI = I‘I

I/IOFL*OFL)

C

(‘
C

C
C
(‘

C
C
C

1")

9O

1O

SHML = SUML+FIFTII)*DFL*(L-DI*DFL)
<2! IMD :: CHMD+F1PT( I I*OCL*IDI*OFLI
cQL = I-SUML - O.S*FICT(I)*OFL*(L~nPL*n.2q)
1 —FIrT(N)*nFL*nFL*o.1?S+FML-FMD)/L
ch : {—QHMQ—O.:*CICT(N)*DFL*(L—HFL*n.pR)
1 -FICT(1)*o=L*ncL*n.I?G - FML + FMQ) / L
DCTHDN
FMD
QI IRDOUTINF UOF'FLIXQF oKoLo AI 0A?9A10A40RI QPZQRTBORAO
1 C1.C?.C3.Cd)
OPAL ’(oL
DYMFNSION X(fi)
QINH(X) = O.S*(F¥D(X) - FXPI—X))
COCH(X) = OoR*IFXD(X) + Pxp(-X))
IFIFIIOIFOoPO
MCMRCD UNDFD COMDDVQQION
Q = SIN(V*L)
C : FOS(K*L)
n 2 ((6-1.f‘)*(u"¥-r‘-I()+V*S*Ic-'(*LI
A1 : II—XI?)-L*XI1)+X(=))*(K*C-K)-(-XI3)+X(6))*(S-K*L)I
I /O
A? : (IF-1.O)*(-X(3)+X(6))+K*S*(-X(P)-L*X(3)+XIS)))/D
A1 = xI1I-K*Ap
A4 = X(P) — A1
Dr'TUDN
MFMRFD UNDFD TFMSIOM
Q : QINH(K*L)
C = COCHIK4L)
n = r-1.n)*(v*P-w)—K*Q*(C-V*L)
9! = ((-X(?)-L*X(3)+X(S))*IK*C-K)-(-X(3)+XI6))*IS*K*L))
I /O
R“ : ((C-I.“)*(-Y(1)+Y(6))-K*S*(-X(?)-L*X(3)+X(=))I/D
n1 : x(1) - r*n?
Ga : XI?) - q1
DCTUDN
MFMRCD UNocn NO AxIAL FODFF
r1 = IIa.n*x¢p)+5.o*L*XI1I-12.0*X(=>+6.O*L*XI6)I
1 /(L*L*L)
Co : (~A.O*YIfi)-A.O*L*X(1)+A.O*XI:)-90O*L*XI6II/(L*LI
C"! = XI”)
Pa = XI?)

130

QFTUQN
END

SURQOUTINF VDFFL (FICTQNQNPOW9NCOLODELQK29VOV40Avy)
QF’AL KP. INF-QT .L

OIMFNSION FICTIN)OAINQOWQNCOLIOYINPOW)

DIMFNQION VINIQVQIN)

C
C INITIALI7F
C
DO IO I=1oNROW
on In J=IQNCOL
IO AIIQJ) = 0.0
C = OFL*DFL*K?
r
C COUAPF DAPT OF MATRIX A
C
N? = Nnnw - 2
N1 = NQOw-l
DO 7” I=I ON?
on A(I.I+¢) = I.o
DO PI I=3ONQOW
91 A(IQI‘?I = I.“

DO ?9 I=IQN‘
P? A(IOI+I) = C-40O
DO 23 I=ZQNPOW
?1 A(IGI'I) = C-4oO
DO ?4 I=29NI

Pa AIIOI) = 6.0-?.O*C
A(1.1) = A(NQOWQNDOW) = 7.0-?.O*C
r
C FODM LACY POLUMN OF AUGMFNTFD MATRIX A
C

DFLa : OFL*DFL*DFL*OFL
DO 30 I=IQNQOW

1o A(IoNCOL) = DFL4*FICT(I+I)
LL = NPOW - 1

C

CALL QIMFOS. (AQNDOWQNCOLQLLQY)
C

VII) = VIM) = 0.0

NM : N—1

DO 4O I=?oNN
4O VII) = Y(I‘I)
VHII) = I-IOQO*V(?)+?60O*V(1)-?40O*V(¢)+IIoO*VIR) -
I 20O*V(6) I/OFLA
VQIN) =I~1490*V(N-1)+P6.O*V(N-?)-24.O*VIN-3) +
I IIoO*V(N-4)-?.O*V(N-5) )/DEL4

 

an

11
14

131

V4(?) = I70O*V(?)-aoO*V(3)+V(4) I/DFLQ

VQIN-I) = (7.O*V(N-1)-4.O*VIN-2)+VIN~3) I/DFLA

NN = N-?

DO GO I:10NN

VAII) = (\III-9I‘AOA*\I(I-I )+6."*V(H-4.O*V( I+II+VII+?II
/ DPLQ

DVTUDN

END

QURQOUTINE SIMFQQIAONQMQLOXI
DIMFNSION A(N.M)oX(N)

DO I? K=IoL

JJ=K

RycthQIAIK.K))

VD]:I(+1

SFAQCH POD THF LAQCFCT DIVOTAL ELFVFNT
DO 7 I=VPIIN

AR:AR<(A(I.K))

IF (RIG-AR) 6.7.7

RIG=AR

JJ=I

CONTINUF

IF (JJ-V) RQIOQR

INTFDCHANGF DOWQ 1F NFCFQQAQY
DO O J:I(.M

TFMp:-AIJJOJI

A(JJOJ)=AIKOJ)

A(woJ)=T¢MD

DO 11 IzkplgN
OUOT=AIIOK)/AIKQK)

DO 11 J=KDIIM
AII9J):AII9J)-OUDT*A(KOJ)

DO I? I=KDIIN

AIIQK)=O.

xINyzAIN.M)/AIN.NI

FvALUATF x ev RAPK SUPSTITUTION
DO I4 NN=IOL

CHM=n.

IzN-NN

IDI=I+I

DO I‘3 JzIploN
SUM=SHM+AIIIJ)*X(J)
X(I)=(A(IIM)-SUM)/A(III)
DFTUPN

FND

132

PQOGQAM CPEEPINV
C
C IMVFPSION OF DISCQFTF VALUFQ OF D(T)
C
COMMON nI30).FI1n)oN
DO IO I=103O
IO PPAD IIoDII)
II FODMATISXIFIS.AI
E1!) = 1.0/D(t)
E(?): (I.O-F(I)*IOI?)-D(I))I / 0(1)
5(3): (1.0 - FI2)*(nI2)-D(I)I - E(I)*(n(3)-o(2))) / D(I)

N = 30
c
CALL SOLUTN
c
DDINT so
on FORMATIIHI.OX.*TIMF D(T) . EIT)*/
I sx.*TFNTHc or MIN. co.IN./Le. Les./qo.IN.*//I

DO 1") IzloN
RD PPINT BIoIoDII)oFII)
3I FOPMATIII3QF2O050FI605)
DO 40 I=103
K = IO*I
D(I) = OIK)
an F(I) = E(K)
DO 41 I=496
AI PFAD IIODII)

N = 6
C
CALL SOLUTN
C
POINT so
an FOQMATIIHIoQXo*TIMF D(T) E(T)*/
I 9X0*MINUTES SOOINO/LBO LRSO/SOOINO*//I

DO 51 I=106
PPINT 310IoD(I)9FII)
RI PUNCH SPOIoFII)
I‘5? FORMATIISQFIOol)
DO 100 I=Io7
00 6O K=103
DIK) = DI?*V)
6O FIN) = FI?*K)
DO 6! K=406
61 PFAD IIoDIK)
N = 6

CALL SOLUTN

DO 69 J=406
KF = J*?**I

6?
IO"

Innn
In“!

133

pPINT 1)oKFoDIJIQFIJ)
PUNCH qPQKEQFIJI
CONTINHF

FND

SURQCUTINF qOLUTN
COMMON DI30)0FIEO)QN
DO IOOI K=49N

<3th 2 O.(‘

DO IOOO 3:?9K

SUM = SUM+FIK~I+1)*IO(II—DII-1))
F(K) = (1.0-SUMI/OII)
DFTUQN

FND

IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII

LLL||LLLLLLLLLILLLLLLLLLLLLLLL LLLLI LLLL LLLI ILLLL LLLLLLL