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ANALYSIS A}!D BEHAVIOUR
OF SKEW BOX BRIDGES
by
Stuart Gurevich, B.Eng. (Civil)
A thesis submitted ta the Faculty of Graduate Studies
and Research in partial fulfillment of the
requirements for the degree of
Master of Engineering
McGill University
Montreal, Canada
!4arch 1973
cv Stuart Gurevich 1973
"l\na.lvsis and J3ehaviour of SJ.:ep Bo:·: BrjJ \iE'~"
rages 18,19 - Notation
:i. l\" ."11
~. ·IIJ 5: ,"}i }r l "'l
= = = = =
node nUJn .. ~er, shape function corres~onain~ ta n~~c i, local eler'.cnt coor<'1inat.e~ ('1.:-; c:;'··('~··n tn "'icr. ?. 1 , value of ! an~, at nofe i, global coordinates 0:1' trÏJ1T1(7nlrlr '" 1.er 'r:>nt.
The ] ast 1ine of the page she.ul<1 rew:: "with re~pect te ~ and 1 are calcnlate0 hv finite ~t~~0r0ncPR"
Srction l!.::l
.... ictitious transverse ëliar.>hragms ~'ie:>:"e not 11.0:;0.,"( in thr: solution of the cantilever problem.
TO
Jl1Y
PARENTS
ANALYSIS AND BEHAVIOUR OF SKEW BOX BRIDGES
Stuart Gurevich
Department of Civil Engineering
and Applied Mechanics
McGill University
Montreal, Canada
ABSTRACT
f.i.Eng. Thesis
March 1973
A membrane finite element analysis is presented for
single and multi-cell skew box girder bridges with general
features including variable section and interior diaphragms.
Features of the program, which are designed to make its use as
economical as possible, include the simulation of transverse
bending stiffness by membrane elements, in-core solution for many
practical problems, and rapid preparation of data. The validity
of the method is discussed, and a study is presented of a one and
three cell box bridge, with reference to angle of skew and
diaphragm location.
ANALYSE ET COMPORTEMENT DE PONTS A POUTRE CAISSON
stuart Gurevich
Département de génie civil
et de m6canique appliquée
Universit~ McGill
Montréal, Canada
RESUHE
... t:-, These de ma1tr1se
Hars 1973
Une analyse par ~l~ents finis charg~s dans leur plan a
~té appliquée à l'étude de ponts ./... . .;
gauchês' ~ poutre caisson de
section variable, à cellule simple ou multiple, avec ou sans
diaphragmes internes. Le programme, conçu en vue d'une ~conomie
maximale, comporte entre autres particularités: la simulation,
par des éléments chargés dans leur plan, de la rigidité en
flexion transversale, la possibilité de ~
resoudre de nombreux
problèmes pratiques sans appel aux appareils périph~riques de
l'ordinateur, et l'entrée rapide et simple des données. Une
discussion est present~e quant au degré de , , ,
prec1s1on de la
m~thode. ", "'f' ... ~ Une etude a ete a1te d'un pont, ou l'on a varie le
nombre de cellules de la section, la position des diaphragmes, et
l'angle du pont par rapport a la rive.
ACKNOWLEDGID·IENTS
The work presented in this thesis was carried out under
the direction of Professor R.G. Redwood, Chairman of the
Department of Civil Engineering and Applied Mechanics. The
author wishes to express his deepest gratitude for the constant
guidance and encouragement received during the course of this
study.
Considerable thanks are also due to aIl my colleagues in
the Department, and in particular to Dr. U.J.U. Eka, Dr. A. Fam,
Professor R.G. Sisodiya, and Dr. R.A. Tinawi, who have always
been most willing to examine and offer constructive criticisms on
the progress of my work from its inception.
The National Research Council of Canada, which sponsored
the present work under grant Number A-3366 is also gratefully
acknowledged.
The typing of the manuscript of this thesis
was prepared by the author using the
IBM Administrative Terminal System (ATS)
TABLE OF CONTENTS
1 • CHlI_PTER 1 - INTRODUCTION
1.1 General
1.2 Review of previous Research
1.3 Scope of the Research Project
2 • CHAPTER 2 - THE FINITE ELm-mN'T PROGRM~
2.1 General
2.2 The Elements
2.2.1
2.2.2
2.2.3
2.2.4
Linear, Quadratic, Cubic, Triangle
WEB24
WEB20
WEB30
1
1
2
12
16
16
18
18
20
22
23
2.2.5 Concluding Rernarks 27
2.3 Formulation of the Element Stiffness Matrix 28
2.3.1 General
2.3.2 Numerical Integration
2.3.3 Verification of Element ~1atrj ces
2.3.4 Transformation Uatrices
2.4 Solutton Technique
2.4.1 Restraints
2.4.2 Loads
28
30
31
35
37
37
40
2.4.3 Solution of Equilibrium Equations 42
2.4.4 Stresses 43
2.5 Fictitious Diaphragms 44
2.6 Concluding Remarks 46
3. CHAPTER 3 - EXPERIMENTAL HODEL 57
3.1 Intronuction 57
3.2 Characteristics of Plexiglas 58
3.3 l~odel Fabrication 62
3.4 Load Cell 65
3.5 Testing Procedure 6R
3.5.1 Summary of Tests G8
3.5.2 Setup 69
3.5.3 Calibration of Load Cells 70
3.5.4 Loading 71
3.6 Analysis of Output 72
3.7 Model Idealisation for program SAPE 78
3.7.1 f.Iodel l 78
3.7.2 Models II and III 82
4. CHAPTER 4 - PROGRAM VERIFICATION
4.1 Cantilever problem
104
105
4.1.1 Tip Deflection, Reactions, Stresses 106
4.1.2 Condensed Element 110
4.2 Flat Plate problem
4.3 Cantilever Box problem
111
114
4.3.1 Torsional Deflection, Shear Stress 115
4.4 Five Cell Skew Box Problem
4.4.1 Deflection and Stresses
4.5 Concluding Rernarks
5. CHAPTER 5 - BEHAVIOURAL STUDY OF SKEW ---CELLULAR STRUCTURES
5.1 Introduction
5.2 Single Cell Box Bridge Investigation
5.2.1 General
5.2.2 Skew Angle
5.2.3 Diaphragms
5.2.4 Stress Contours
5.3 Three Cell Box Bridge Investigation
5.3.1 General
5.3.2 Ske'tol Angle
5.3.3 Diaphragms
6. CHAPTER 6 - CONCLUSIONS
6.1 Surnmary
6.2 Limitations
6.3 Recommendations for Future Work
7. APPENDIX
8. REFERENCES
116
117
119
128
128
129
129
130
132
134
136
136
137
138
150
150
152
154
156
158
-1-
1. CHAPTER 1 - INTRODUCTION
1.1 General
Recently there
percentage of cellular type
has been a vast
structures used
increase in the
in transportation
schemes. This preference can bè attributed to several factors;
improved structural efficiency, aesthetic appearance, ease in
providing electrical and other essential services, and economic
considerations, especially for medium length spans in the eighty
to ninety foot range. The main disadvantages were, first, the
inexperience of bridge contractors in dealing with this new type
of structure, and second, the need for new methods of analysis
applicable to cellular bridges. These drawbacks were especially
true when dealing with skew structures. The first problem has by
this time resolved itself, because much practical experience llas
been gained in this field. The research described in this thesis
will be applied towards the second problem: a method of analysis
of skew multi-cellular box girder bridges.
-2-
1.2 Review of Previous Research
~1any different techniques have been used by various
researchers for the analysis of cellular bridges. Among the most
prominent of these are the simple beam method, the finite
difference method, the folded plate method, the finite strip
method, and the finite element method. Each of these has a
particular type of cellular structure to which it is best suited,
and each is derived from its own set of basic assumptions and
limitations. For example, in the simple beam method, the
analysis is made as if the cellular box acted as a simple beam
and the flexure for.mula is used to predict the behaviour. Thus,
aIl points on a cross section are assumed to exhibit rigid body
displacements in their plane. This will only be true if
transverse distortion and warping of the box are negligible.
These are usually not the case in cellular structures, especially
when dealing with skew bridges. The method may have some
applicability in the central portion of a long narrow skew
bridge.
Perhaps the most suitable method as far as ease of
implementation and accuracy of results is the finite strip method
of Y.K. Cheung (1,2). Unfortunately, this technique is
restricted to either rectangular structures, or to curved
-3-
structures with rectangular ends, and is not applicable to the
skew box bridge. This restriction results from the method on
which the technique is based. A harmonic series is chosen to
represent the displacements of the structure and at the same time
must also satisfy the boundary conditions. For rectangular
bridges this requirement poses no problem, but a suitable
harmonic series has not yet been found to be able to satisfy the
boundary conditions for a skew bridge.
continuing toward this objective.
Further research is
Another widely used and highly efficient technique of
analysing cellular structures is tile folded plate method of
Scordelis (3). But this method also uses a harmonie series which
is chosen to satisfy the boundary conditions and therefore cannot
be used to analyse skew structures. Both techniques are able to
treat bridges with transverse diaphragms. A force method of
analysis is used in which the redundants are taken as the
interaction forces between the wall elements and the transverse
diaphragme These forces are determined as those required to
establish compatibility between these members. Once the
magnitude of the redundant forces is known the analysis can
proceed as for the case without transverse diaphragms.
A more general technique which is capable of analysing
bridges with skew is the finite difference technique as developed
by A. Ghali. In Reference (4) he has used the finite difference
-4-
technique to obtain influence coefficients for deformations,
reactions, and stress resultants of bridges with variaus angles
of skew. The initial step in this technique requires the writing
of the elasticity equations in differential forme The crux of
the finite difference technique lies in the rewriting of these
differential equations inta algebraic forme This is accomplished
by replacing the derivatives of the deformations by finite
difference approximations of the deformations at variaus points
separated by a distance ~.The smaller the interval ~ between
the points, the more accurate will be the results of the
analysis. Thus, a set of differential equations is converted to
a set of simple algebraic equations which may now be easily
solved by any one of a number of possible algorithms. The finite
difference technique ia reported to yield good results if the
bridge is relatively long as campared ta its width.
The last method to be discussed, the finite element
method, is perhaps the most versatile and is treated in greater
detail as it forms the basis of the work described in this
thesis. It consists of dividing the structure into a series of
finite elements interconnected at discrete nodal points. At each
nodal point there are a number of degrees of freedom to which
forces or restraints may be applied, or at which displacements
are to be calculated. Stiffness expressions for each individual
element are first generated, and then assernhled into an overall
structure stiffness matrix. A direct stiffness solution is then
-5-
used to find aIl of the unknown nodal displacements. Once these
are known, stresses at any point in the element can be
calculated. The method is quite general, allowing any type of
loading or support conditions1 can handle variations in cross
section and element properties1 can include any combination of
internaI or end stiffening diaphragms1 can include cutouts in
members; and is readily adaptable to the skew or curved
configuration. The main disadvantage of the technique lies in
the large number of equations to be processed for even moderately
sized structures even with a coarse idealisation. Each node
contributes several equilibrium equations to the overall
stiffness matrix and unless expensive and time consuming use is
made of auxiliary storage devices during the solution process a
very large capacity computer is required. Refined meshes, which
add additional elements and nodes are required in the vicinity of
concentrated loads and other locations of high stress variation
if accurate results are to be obtained in these regions.
The finite element technique has been used in several
ways to study the behaviour of the skew box bridge. These
differences in the application of the fini te element technique
concern themselves with1
1) the type of elements used for the structural idealisation,
2) the various parameters under study, and,
3)
-6-
the ways by which existing knowledge of
behaviour can be used to simplify, and hence
general finite element technique.
the structural
optimize, the
A major simplification to the finite element technique
was developed by Sawko and Cope. As described in Reference (5),
they realized that the major stress carrying modes for cellular
structures were direct stresses in the top and bottom flanges,
and flexural stresses in the transverse medium. The direct
stresses were readily obtained by using a p~cne stress finite
element approach, where each node was permitted only ~1ree
degrees of freedom corresponding to translations in the x, y, and
z directions, instead of a full six degree of freedorn analysis
involving three translations and three rotations. Thus, a
tremendous saving in storage requirements and solution time was
immediately obtained. The flexural stiffness of the transverse
medium was simulated by the use of equivalent diaphragms.
Section 2.5 of this report includes the mathematical derivation
of the characteristics of the equivalent diaphragms. This
simplification \'las weIl justified as it included the important
effects of cellular bridges and neglected those that were
unimportant. Thus, local longitudinal flange bending stresses
which were small cornpared to direct stresses were ignored. Local
flange torsional stresses were also ignored when compared to the
overall shear flow in the walls. Sal'7ko and Cope did carry out a
full six degree of freedom analysis for purposes of comparison,
-7-
and reported that while results of the approximate method were
not exact, they were accurate enough to be able to be used for
design purposes.
The work described in their paper had been applied to
rectangular structures and was being extended to the skew
condition. All finite element programs developed for this
project include the technique of equivalent diaphragms as
developed by Sawko and Cope. Specifie applications can be found
in Section 3.7 and throughout Chapter 4.
Crisfield has also applied several simplifications to
the finite element technique based on a knowledge of structural
behaviour of cellular bridges for the case when wall bending can
be neglected (e.g. if transverse diaphragms are included)~ The
actual procedure has been concisely stated in Reference (6) and
only a summary of his results are presented
assumptions are summarized as follows:
here. His
1) From simple beam theory the variation of vertical
displacement with depth has been omitted.
2) For linear variation of bending moment and constant shear
a long the depth~ the variation of vertical displacement must
() vary cubically in the longitudinal direction~ the axial
deformation must vary quadratically in the longitudinal
-8-
direction: and it must also vary linearly with the distance
from the neutral axis.
3) From elementary beam theory zero stress has been assumed
along the depth.
4) By examining the expression for the strain energy of the
element it was observed ~1at two additional nodal degrees of
freedom (in plane mid-depth rotations) need not be included
(7) •
5) The torsional stiffness can
displacement in the normal
distance from the neutral
longitudinal direction.
be included by having the
direction vary linearly with
axis, and quadratically in the
A diagram indicating the original degrees of freedam is
shown in Figure 1.1. After the simplifications just described
have been applied to the element the final nodal degrees of
freedom are as shawn in Figure 1.2. When the stiffness matrix of
the whole structure is formed, the stiffness of the flange
elements must be doubled to allow for the strain energy of the
top and bottom flanges. Similar reasoning will result in
simplifications for elements when wall bending is also included.
-9-
Although these results were developed for the case of
rectangular bridges, they were directly applicable to the skew
condition as shown in Figures 1.3 and 1.4. Figure 1.3 is an
example of the type of element used for multi-cellular skew
bridges including the presence of transverse diaphragms. Note r
tile use of plane stress triangular elements with midside nodes at
the ends of the flanges. If transverse diaphragms are not
present, the elements of Figure 1.4 can be used if suitable
transformations are made into skew coordinates.
Crisfield has verified his analysis of rectangular
structures with several existing analytical and numerical
solutions, and with experimental perspex models. He has been
able to report excellent agreement in all cases. Unfortunately,
at the time he completed his paper there were no published
solutions involving the analysis of skew structures. However,
subsequent comparisons have been made with Crisfield's analysis,
and they are presented in Section 4.4.
Many applications of the finite element technique in
analysing skew and circular cellular bridges have been made by
Sisodiya, Ghali, and Cheung. Their work has been largely
concerned with the various types of elements that will provide
the best representation for the cellular structure. They have
also used the finite element technique to analyse the effects of
some of the more important parameters on these structures. Most
-10-
notable among these are the effects of internaI and/or end
transverse stiffening diaphragms.
Sisodiya et al used the finite element technique to
analyse box girder bridges by retaining aIl six nodal degrees of
freedom in .the analysis. Their research was divided into two
phases. First, they used existing parallelogram and triangular
elements to analyse skew and curved bridges (8,9). These
elements gave acceptable results, but the analysis took a very
long time and was therefore considered too costly for practical
use. Subsequently, in the second phase they developed new
elements (10) and were able to show good accuracy and much less
solution time for the structures analysed.
Theoretical studies were made investigating the effects
of end and intermediate diaphragms on cellular bridges with
various angles of skew (11). For a single cell skew box they
found that the addition of end diaphragms increased the reactions
at the obtuse corner point, while decreasing the reaction at the
acute corner point. In fact, uplift of the acute corner points
was reported under some loading conditions. The addition of a
central diaphragm was found to have no effect for uniformly
distributed loads, but it did aid in distributing tile affects of
an eccentric point load to the other web. The addition of two
more diaphragms at the quarter points did not result in any
futher changes. From their studies Sisodiya et al concluded
-11-
that the slight beneficial effect of transverse diaphragms were
not worth the time and cost involved in their design and
construction, and they therefore should be dispensed with.
-12-
1.3 Scope ~ ~ Research Project
The investigation reported in this work is comprised of
three sectionSi the theoretical, the program verification, and
the behavioural analysis. The theoretical sections deal with the
development of the fini te element program used for tile analysis
of cellular bridges. The program must satisfy the following
reguirements:
1) Be as efficient and economical as possible,
2) Reguire a minimum of input data,
3) Not reguire the use of auxiliary storage devices, and,
4) Be sufficiently general to allow for variations in member
properties, various support and loading conditions, the skew
configuration, and any probable combinations of internal
and/or external diaphragms.
The amount of input data can be minimised by using a few
large elements. Because of the nature of the problem, elongated
elements are appropria te, and their development will be described
in Chapter 2. The size of the computational problem is minimised
-13-
by treating elements in plane stress only, with three degrees of
freedom per node, and when necessary, using the method of Sawko
and Cope to account for the transverse bending stiffness of webs
and deck.
The verification of the program is the subject of
Chapt ers 3 and 4 of this report. The validity is established by
applying it to several problems. Included in these is a five
cell concrete skew box bridge for which Sawko and Cope have
published experimental results, and for which a corresponding
computer solution has been presented by Crisfield. In addition,
tests were carried out on rectangular and skew single cell
plexiglas models.
The agremaent obtained in these and O~l~~ examples gave
sufficient confidence that the program was capable of analysing
the skew cellular bridge to the required degree of accuracy. In
tile analysis phase, as desc~ibed in Chapter 5, the program was
used to investigate the effects of two major parameters
associated with skew multi-cellular box bridges, these being the
angle of skew, and the effects of diaphragms.
-14-
1 Y 8Y2 /
" " x
8X1 w2
Vc ~v2 ~
~ 8x2 u2 V, z
FIGURE 1.1 - Original Degrees of Freedom from Crisfie1d (6)
z
FIGURE 10 2 - Remaining Degrees of Freedom After Simplification From Crisfie1d (6)
-15-
FIGURE 1.3 - F1ange Representation with Diaphragms
FIGURE 1.4 - F1ange Representation Without Diaphragms
-16-
2. CHAPTER 2 - THE FINITE ELEMENT PROGRAM
2.1 General
As mentioned in the previous section, new, large, high
aspect ratio elements were to be developed for use in the
program. To ensure that a small number of elements could
accurately idealise the complex forro of skew box bridges with
diaphragms, simple rectangles and triangles would no longer
suffice. As the elements became larger and more elongated, more
nodes were added along tileir sides. With increasing number of
nodes, overall accuracy was greatly increased and the number of
such elements required to obtain an adequate solution decreased
rapidly.
It was decided to derive these new elements from the
family of isopararnetric elements, those elements in which the
sarne polynomial function is used to describe the assumed nodal
displacement expansion and the element geametry. This choice was
made for the following reasons. First, good results have been
reported by those researchers who have used them (12,32).
Second, a relatively simple programming effort was required for
-17-
automatic generation of the stiffness matrices of these types of
element. This will be further described in the following
sections. And third, the isoparametric elements were developed
as general quadrilaterals and therefore could be used for
idealising skew or curved structures.
-18-
2.2 The Elements
2.2.1 Linear, Quadratic, Cubic, Triangle
The various isoparametric quadrilaterals and the linear
strain triangle used in the idealisations are shown in Figure
2.1. Elements (a), (b), and (c) are the linear, quadratic and
cubic displacement quadrilaterals, respectively. The shape
functions as given by Ergatoudis et al (13) are as follows.
L inear element -
For nodes i = 1,2,3,4
Ni = (1 +!Si)(1 +"')T w here 'i and "/,. are the values of ~ and "l at node i.
Q uadratic element -
For nodes i = 1,2,3,4
Ni = ~ (~~L +f)(hlhf, +1)- !(t-12)(f+~1,)-!(1+~ei)(t-"l2) For nodes i = 5,6
J 2 Ni = 2. (f - ! ) ( 1 + 11i )
F or nodes i = 7,8
-19-
Cubic elernent -
For nodes i = 1,2,3,4
Ni= ;1{1+ig&>{I+'*/i) ~'O +.9{i~+1lil For nodes i = 5,6,7,8
Ni =n( t- t 2 )(1 +'«~J(t+"11;)
F or nodes i = 9, 1 0, 11 , 12
Ni;; tr(t+§'J(I- 'l~)(1 +'''l'J'li)
Elmaent (d) is the widely used linear strain triangle, for which
shape functions are, for example, given by Zienkiewicz (12) as
N 1 = N2 = N3 = N4 = N S =
N6 =
Wlere,
(2L1 - 1) L1
(2L2 - 1 ) L2
(2L3 - 1 ) L3
4 L1 L)
4 L 2 L 1
4 L3 L2
L1 = (a 1 + b 1x + c1y)/2A
L2 = (a 2 +b 2x+ c2y)/2~
L3 = (a 3 + b 3x + c 3y)/2 A
A = area of triangle
a 1 = x 2Y3 - x 3Y2
b 1 = Y2 - Y3
c1 a x 3 - x2 and the subscripts vary cyclically.
1
-20-
2.2.2 WEB24
Element (e) is used as widely as possible in the
idealisations considered, with its longer direction oriented
along the bridge span. It is used for bOtll web and flange
elements, and typically the shorter side of one such element will
represent the side of a celle It is the possibility of using
this elongated element in this way which limits the required
number of elements. The quadratic displacement variation on the
shorter side permits a linear stress variation, and the element
is tilUS considered suitable for the webs. This 24 degree of
freedom element in two dimensions is denoted WEB24.
Wi th curvilinear coordinates :s and '1 , the appropriate
displacement expansion for the displacement u i6 given by
U = ctl -+ Cl~! + ct3' + Ct., f1 + C(S S 2- + cc, ~t + ~7 g3 + Q(8 '$"ï.
+ oc, S "12. + 01'0 ~31 + 0(" g 4- -+ cl'l g 4-"1
and similarly for v. The terms in these expansions are indicated
by the use of Pascal's triangle, as shown in Figure 2.2, and
arise fram the nodal arrangement of Figure 2.1.
From these expansions, the shape functions can be
readily calculated, as per Reference (13). A brief summary of the
-21-
method follows. The vector G is defined as the terms in the -displacement expansion and is written as
3 ~ 2 1 ~'I e,,;l § = n s "1 ~ h[ S ~ 1 ~ g ~ '1 f"l S '7 ~ :>!J
The H matrix is defined as the value of the G vector at each node - -as shown in the following figure.
g=U s2. ,l 3 ~2., g ,t !1, !'i !.'f~ ! "l ~, !
NODE ~ '1
1 -1 -1 1 -1 ~1 1 1 1 -1 -1 -1 1 1 -1
2 1 -1 1 1 -1 -1 1 1 1 -1 1 -1 1 -1
3 1 1 1 1 1 1 1 1 1 1 1 1 1 1
4 -1 1 1 -1 1 -1 1 1 -1 1 -1 -1 1 1
5 -1/2 -1 1 -1/2 -1 1/2 1/4 1 -1/8 -1/4 -1/2 1/8 1/16 -1/16
6 0 -1 1 0 -1 0 0 1 0 0 0 0 0 0
7 1/2 -1 1 1/2 -1 -1/2 1/4 1 1/8 -1/4 1/2 -1/8 1/16 -1/16
9 0 1 1 0 1 0 0 1 0 0 0 0 0 0
10 1/2 1 1 1/2 1 1/2 1/4 1 1/8 1/4 1/2 1/8 1/16 1/16
11 -1 0 1 -1 0 0 1 0 -1 0 0 0 1 0
12 1 0 1 1 0 0 1 0 1 0 0 0 1 0
H -The shape functions are found from the following equation
-22-
with j and 1 taken as plus or minus 1 on the element
boundaries, tlle following are obtained for WEB24.
For nodes i = 1,2,3,4
Ni ,. tr ~Lf!2 - 'H' + '1 .. )( 12 + ~~i) + (3"Zt_ 3)(' +! !iil
For nodes i = 5,7,8,10
Ni = ; UI +"11j)(-g~~'f + ~~, (1 - ~2)~ For nodes i = 6,9
Ni = 1 (f +"1"ld(l -55 t + Ji §4-)
F or nodes i = 11, 1 2
Ni = ~ (1 +S5c)(1- "12. )
The general expressions for the partial derivatives
can be directly obtained from these relationships.
2.2.3 WEB20
d ;)Ni
an -~"l
The element (f) shown in Figure 2.1 is useful as a
transition between a parabolic element and the WEB24 element.
While several other eransition elements may be developed, this
one has been found most useful. With twenty degrees of freedom,
it is denoted WEB20, and the terms in its displacement expansion
are shown on Pascal's triangle in Figure 2.3.
shape functions are as follows.
The appropriate
-23-
, ( 2. 4 1!3 ~ ~ 2) N,=iï -3+8.§-5J +3:1"1+3"'l -B~ -3s"1+8s -35"1
N2.=;i-(-3 -S5 _5~2. -3'1 +3"lt+8s~ -3 ~~1 +8!" "'3 ~~Z)
~ Z 1
N..,=t(-I +~ --S'1+1 +~~
NS = -S (-~ + 2 S ~ + 53 + ~ 5 If )
N fi == t (1-, - ~ 5 ~ "\'" S ~~ + «8 ~ 't)
N 7 ~ t (5 + z st - S 3 - l s't)
Na -:: t (1 +'1- !.t_ g~h})
N, :- l (1 -g _'11 ~ ~~2)
N,,, :: t (1 + s _,.,2. _ ~., 2 )
2.2.4 ~ŒB30
Referring to Figure 2.2 of Pascal's triangle for the
WEB24 element, it is seen that there are three missing terms from
the complete displacement expansion of order sS" and "'l ~. These a13t. 'Il
terms are: ~ "1 , ~ "l,and ~ "l. It was thought that an improved
solution could have been obtained if these terms were included in
the displacement expansion. Initial attempts to group these
terms with others already included in the expansion, e.g.,
-24-
o(i[g2(I-hJ1.], o(,i+,[r;\'-"1a)], and Q(/-;.,J((I-,?%)], aIl met with failure,
as the resulting H matri~~ was always singular. To include these --terms, three interior nodes were added to the element and a new
set of shape functions was generated. This new element was
denoted 'NEB30 and is shown in Figure 2.4 with Pascal's triangle
and its displacement expansion. The shape functions for the
NI!B30 eler.lent are liste;i here for reference.
-25-
Some test runs were made with this element for purposes
of comparison, but it was always intended to eliminate the
additional six degrees of freedom provided by the three extra
interior nodes by the technique of static condensation. Thus,
the resulting element would make use of a full displacement
expansion while still having the same number of degrees of
freedom as the original WEB24 element. The method of static
condensation can be summarized as follows.
~I I(,t -~u.
"..,
\'lhere, A = nodal loading, -k = element stiffness matrix,
f = nodal displacements, -subscript 1 refers to those degrees of freedom to be
retained, and,
subscript 2 refers to those degrees of freedom to be
eliminated.
~1 can be set to zero because neither external loads nor other
elements are joined to these additional interior nodes:
or
-26-
S ubstituting for J~: .....
or A, = Il (' ... ~~,
w here k is the statically condensed element stiffness matrix of -size 24 X 24 for the case considered, and -,
~ ::. ~II - ~12. ~~" ~~, Stresses can be determined as follows:
€ - ~, ~~ [fJ -'"
where, é = strains at a point in an element, "'" B = the strain rnatrix for a point in an .... and the subscripts are as before.
Substituting for Jl:
é. ': B S, ..... """ ....
element,
w here, B is the statically condensed strain matrix equal to -
-27-
Several test problems were made with this condensed
element to de termine if improved results could be obtained.
Unfortunately, as shown in section 4.1.2 of this report, improved
results were not obtained, and for this reason it was decided not
to pur sue this possibility any further.
2.2.5 Concluding Remarks
While the quadrilateral elements in Figure 2.1 are shown
as rectangles, being isoparametric elements derived in terms of
the curvilinear coordinates ~ and '1 , they may take on more
general quadrilateral shapes, and also sides may be curved to the
degree permitted by the number of nodes on a side. It is this
feature which enables non-uniform bridge plans or elevations,
curvature or skew to be idealised in a straightforward manner.
-28-
2.3 Formulation 2! ~ Element Stiffness Matrices
2.3.1 General
Reference (13) describes in detail the technique used
to generate the element stiffness matrix of plane stress iso
parametric elements. The appropriate notations are as follows:
k = element stiffness matrix, -B = defines strains in terms of nodal displacements, -x,y = cartesian coordinate system,
u,v = nodal displacements in x,y coordinates,
e = strain matrix, ....,
l'l = element shape functions, ...,
~,1 = local element coordinates,
J .... = Jacobian matrix,
D ..... = plane stress elasticity matrix,
t denotes matrix transposition.
The standard formulation of the plane stress stiff
ness matrix is given by
The strain matrix is given by
-29-
dU u, a;x.. t,
E= ()\1 v.t, a, B Vt
--- "'" DeJ +~V' dl:.. ()~
in which
B - [B, B~ ........ ] -...
with
dNi 0 d):.
Bi 0 JN,' - ~
~l'Ji JNi -d'1 aX,
As E is defined in terms of S and 7 it is necessary to change
the derivatives to ~)!and %,. Noting that
;}N· d)G. .!.L aN, aNi _1. -é)~ al a§ a)l- é>x.
- - J dN,; (J)C., dCJ -aN, atJi -a1 a"l d' a, ë}3
in "Thich J is the Jacobian matrix which can be easily evaluated -by a numerical process by noting that
-30-
dN. ONt 1-, '1. -~! ~J ~t 't J -- 8N. !!i.~ . a, a~
we can write
and thus calculate the expression for Bi. The only further
change whicll is required is to replace the element of area
as follows:
dx dy = det (J) d~ d1
and the limits of integration to -1 and +1 in the double
integration.
2 .3.2 Numerical Integration
The double integration over the elernent 3 s area 1s
simplified by applying Gaussian quadrature. This can be
described mathematically as
-31-
, , m "
h i ~ f(~, "1) d1 ds = h 6- f. Hi Hj f(ajl bi l
§"·I 1=-' where f(~,"1) = stiffness expression
h = elernent thickness,
H = Gauss weighting factor,
a,b = Gauss coordinates for numerical quadrature,
m,n = number of Gauss points in 5 , "1 directions.
After specifying the appropriate number of Gauss points,
the double summation proceeds automatically. The coordinate
value of the Gauss integration points, and of the Gauss weighting
factors, are stored as data in the prograrn. The elongated
elements will not require as many Gauss points in the short
direction ( , ) as in the long direction (J ) . Also, if the
elements are used in a non-rectangular (skew) configuration, it
will be necessary to increase the number of Gauss points in order
to obtain an accurate stiffness matrix.
2.3.3 Verification of Element l-iatrices
A program (entitled ESU'l'ES'l') was written t·o generate the
element stiffnoss matrix for each type of element in both
rectangular and skew configuration. Various combinations of
-32-
Gauss integration points were used in order to find the minimum
cornbination required to produce a stiffness matrix whose terms
remained constant. With tile generation of each stiffness matrix,
the following tests were made to ascertain its validity:
1) Check on shape functions: 'l'he value of Ni must be one at
node i, and zero at every other node. This follows directly
from the definition of the shape functions, i.e., the nodal
values of displacements.
2) The sum of the shape functions must be unit y: By decreasing
the size of the element to zero, ~,-x.~ _. ··.'lLi = constant.
Since ~ = N, l', + Nt Xl + ... + Ni. 'toi we now have
c = N,c + N~c + ... + NiC
t = N, + Nl + ... + Ni
3) The eum of each row (colurnn) of the elernent stiffness matrix
must be zero. This is so because each row (colurnn)
represents an equilibrium equation for a degree of freedom.
4) Direct stiffness AlI values on the principal diagonal of
the elernent stiffness matrix must be greater than zero.
5) Finite differences - The derivatives of the shape functions
with respect to and are calculated by finite differences
-33-
(second or fourth order) and compared with the analytical
expressions. The second order finite differences expression
for the first derivative is as follows (14):
dN, _ --~~i
Ni (J-c: +~ l "li) - Ni. (~, -)., "Ii) 2À
NiC;" "li+~) - NiC li) ~,-~)
ZÀ
\'lhere " is a small number. The fourth order expression is (14):
dN~ [Ni(~'+~J~i) -NiC~i-)/'li)J - t[Ni('i+2~,"li) -2Ni('ii.+~,t1fJ ~" 2- Il
+ 2 Ni (I.:-~) 1ë) - Ni (:Sl-2~"",,)J
~Nc and sil"lilarly for a1l •
6) Eigenvalues The elernent stiffness matrix must have three
eigenvalues of zero rnag:li tude ''1hich correspond to the three
rigid bod~' modes.
Table 2.1 SnOtolS each type of elernent analysed anù the
minimum required arrangenent of nauss integration points for the
rectangular and skew configuration.
-34-
Number of Gauss Points
ELEMENT
2 2
2 2
2 2
3 3
3 3
4
1.Q h
h 4 3 -10 h
5 3
3
3
TABLE 2.1 - Required Number of Gauss Integration Points for Various Elements
-35-
2.3.4 Transformation r.1atrices
The element stiffness rnatrix must be transformed into
the orientation of the three dimensional global axes system. The
direction cosine matrix of order two by three is generated as
folIOl-TS (15):
1:.2 -X, ~2. - ':f, la - z, cI~ <12.1 d21
ROT= - 'X!t - 't., '4., - 'j, Z .. -~,
cl 'II J'l' J'fI
are the global coordinates of node i, and dij is
the distance separating nodes i and j. This direction cosine
matrix is then expanded into a transfornation matrix
\"here i = the
rnatrix proceeds
k' = RMATt - -
RMAT -~
number of
as follows:
k RMAT - ~
QOT 0 0
o ROT
o ROT
, ~ 3
nodes. The rotation of the stiffness
whcre, Je = the stiffness rnatrix in the local system of
coordinates,
-36-
k' = the stiffness matrix in the global system of -coordinates,
superscript t denotes matrix transposition.
-37-
2.4 Solution Technique
2.4.1 Restraints
The element stiffness matrix is now ready to be assigned
to its proper location in the overall structure stiffness matrix.
This is a matrix in banded forro, the first column of which
corresponds to the principal diagonal. Before these equations
cau be solved for the unknown nodal displacements, the restraints
must be introduced into the stiffness array. Several methods are
available, as described below.
The first method requires that the rows and columns of
the overall structure stiffness matrix be re-ordered such that
those rows and columns corresponding to unrestrained degrees of
freedom be listed first. The solution is then based on the first
rO\'7s and columns, where nd refers to the number of
unrestrained degrees of freedom in the structure. Note that the
size of the computational problem has been reduced by n,
equations, ''Ihere n, is the number of restraints on the structure.
However, there are usually only a very few restraints acting on a
particular structure and the considerable programming effort
-38-
required for the row and column interchanges is hardly
worthwhile. For this reason this technique will not be given
further consideration.
The second method requires that each term on the
principal diagonal that corresponds to a joint restraint be
replaced by a very large number, L. Each corresponding term in
the loading vector must be replaced by the following terme
where
(Kil. ) X (L) X (P)
Rtl = stiffness coefficient for restraint i,
L = very large number, and,
P = prescribed displacement of restraint i.
In most cases the prescribed displacements will be zero (fixed
support), and then the appropriate terms in the loading vector
are simply set to zero. This procedure will yield a displacement
very close to the prescribed value depending on the size of the
large number chosen in relation to the other terms in the
stiffness rnatrix.
A third possibility is to replace every term in the row
and column corresponding to a joint restraint with a zero, except
for the term on the main diagonal which is assigned a value of
unity. Each row of the stiffness matrix corresponding to a joint
restraint is saved in an auxiliary vector prior to replacement by
zeros to be later used for the calculation of nodal reactions.
-39-
Care must be taken in manipulating the row and column subscripts
of banded matrices. Initially prescribed displacements can be
introduced by the following method. For each restraint the
entire loading vector is modified as follows.
for
where
For
FCi) = r(i) - !te j, i) X f (j) ~
F ( ') .- J = E (j)
i = 1,N but i not = j
j = 1,NR
i = equation counter,
N = total number of equations,
j = restraint counter,
NR = total number of restraints,
! = initial load vector,
! = modified load vector,
P = prescribed initial displacement.
the usual case of fixed supports, only the second
modification i9 necessary, i.e. aIl terms in the loading vector
corresponding to restraints are set equal to zero. While both of
the previous two techniques are considered acceptable, the second
was chosen for implementation into the program.
-40-
2.4.2 Loads
Concentrated loads applied to the joints are placed
directly into ~le loading vector. Distributed loads along an
element edge are converted to nodal loads according to the
technique of consistent loading. The method is based on the
principle of virtual work.
where
(0. )
J = nodal displacements, -P(x,y) = distributed loading on element, -u ,."
G -of.. -
= consistent nodal loading,
= nodal displacement function = g ~ , = variable terms in displac~nent function,
= coefficients of terms in displacement function
= H-1 S ,.. "", li = nodal values of terms in displacement function, ,...
and superscript t denotes matrix transposition.
From these definitions:
(b)
Substituting equation (b) into equation (a):
-41-
T he shape functions are given by:
and therefore,
Nt ::- H·,tG. t (cl) - -Substituting equation (d) into equation (c):
fe'f == ~A ~t P()(,~) dA (e) If the elernent is of constant thickness t, the integration can
be taken along a side, rS1 t'
fet{ :::t ls, ~ f(x,!j) ds and this can be further simplified by converting to curvilinear
coordinates:
f~~ = t C ~t f("~. "1) d.t (1) d a. where, i = ~ or "1 , depending on which side the loading occurs,
J = Jacobian matrix. -The loading will be considered only up to a parabolic variation,
and therefore (for the x-direction):
e (l<) == ~)( + Pa. /'" i + p.. - 2 ~x + Pa" i1. where, P,xJ Pax = the magnitude of the distributed load at the
corner points of the edge under consideration,
p~X = the magnitude of the distributed load at the
midpoint of the edge under consideration,
and similarly for the y-direction.
Equation (f) can now be integrated, and a five point
quadrature scheme is used to evaluate the integral. The
-42-
consistent loading vector is then rotated to the global frame of
reference and added to the joint loads in preparation for
calculation of nodal displacements. As a direct consequence of
the method used to introduce the restraints into the stiffness
matrix, and because aIl supports are completely fixed, any load
acting directly on a restraint is replaced by zero.
2.4.3 Solution of Equilibrium Equations
The set of sin1ulatneous equations representing the
degrees of freedom of the idealised structure are solved by
Cholesky's square root method (16). This has been found to be
the most efficient technique, as the banded rnatrix need be
decomposed only once, and then solved for each loading condition
by using the appropriate consistent loading vector. Degrees of
freedorn normal to ti!e flanges where no webs exist must be
suppressed to avoid a singular matrix. This is accomplished by
placing a value of unit y on the main diagonal at the appropriate
row. AlI the nodal displacements are calculated in the global
coordinate axes.
•
-43-
2.4.4 Stresses
Stresses can be calculated at any point in an element.
Their locations are specified in terms of the curvilinear
coordinates S and1. The stress matrix is obtained by generating
the strain matrix at each point and pre-multiplying by the plane
stress elasticity matrix. The nodal displacements are rotated to
the local element axes and then pre-multiplied by the stress
matrix to obtain the stresses a-~, a; , and 'L"KJ. These steps can
be summarised in matrix notation as follows.
where (f --12 B -RMAT --J ,..
= (D) X CB) X (RMAT) X (S ) - - -- "'" = stresses at the point,
= plane stress elasticity matrix,
= strain matrix at the point,
= rotation matrix,
= global displacements of nodes.
Finally, reactions are obtained by multiplying the
previously stored rows of the stiffness matrix by the vector of
nodal displacements.
-44-
2.5 Fictitious Diaphra~s
The representation of the distortional stiffness of a
cell by a fictitious diaphragm has already been rnentioned in the
introduction. Because of the indispensability of this technique,
further elaboration will be in arder, closely following the
discussion by Sa,.,ko and Cape fa und in nefernece (5).
Figure 2.5a shows the cross section of a single cell
Hith its fictitious diaphragrn of thickness t, and shear rnodulus,
G. The shear stiffness of the diaphragm will sirnulate the
distortional stiffness of the celle The simulation will be based
on the equality of deformations of the corners of the two systems
under equivalent loading conditions. Each configuration will
deforrn as shown in Figure 2.Sb. FOllowing the notation of the
figure, and denoting 9 as the rotation of joint 1, the stiffness
equations for the \Y'alls are given by
M = GElt Il h
e - SEI. 2.J hl "
(i. )
Mit, -6EI2. e 6EIt 2dh (il) V V 2..
F,2, - 12EIa hi e 12EI, 2.d
h3 "
(Lil)
-45-
The equilibriun condition M,a = -M,,, yields
and
e = t l, 6 dv - ~ 12. 2 d h l, v + I 2 h
cornhininl] equations (iv) and (iii) :
r, = -12. EI, 12 12 h(vI. + hla)
~hdv + 2;h] The elasticity eguation for the diaphragm
T = tG (2~'1 + 2;h) and comvaring t~lis t.o eguation (v) yields:
tG = -12. El,I2 T h(vI, +hla.) Fla.
S ince F,a. = - ~ equation (vii) reduces to
tG =
(iv)
{V}
is:
(vi)
(Vii)
(VLÜ)
and replacing G by E/2 (1 +~) the thickness of the fictitious
diaphragrn is given as:
(ix)
The fictitious diaphragrn will use the stiffness rnatrix
of the appropriate plane stress elernent, (depending on its nodal
arrangement), and have a ~lickness t, specified by equation (ix).
-46-
2.6 Concluding Remarks
Based on the the ory described in this section, ~ finite
element program in single precision has been written, entitled
SAFE, an acrony.m for ~tructural ~alysis by Finite Elements. A
generalised flow diagram of the program SAFE is shown in Figure
2.6. A brief sampling of execution times and costs can be found
in the Appendix.
-47-
3 6 3 -
74~ 8
-1 2 1 5 2
(a) Linear (b) Quadratic
4 7 8 3 -
10 4 4~12 6
94~ 4~11
-. -, 5 6 2 1 3 2 (c) cubic (d) Triangle
(e) WEB24
4 8 9 10 3 - .-...
L 1 '1 4~12
- - .--1 5 6 7 2
(f) WEB20
8 3 -
9 4• 10
-. - .-
1 5 6 7 2 FIGURE 2.1 - Isoparametric Elements
-48-
FIGURE 2.2 - WEB24 Displacement Function
-49-
FIGURE 2.3 - WEB20 Displacement Function
-50-
ft 8 9 10 3
n! 1~ 1~ 1~ !12 1 5 6 7 2
WEB30
Terms for Displacement Function
[ ~ ] = COC 1 + ... ~ , i- "'3"l + 0( .. §2 + "'5 ,,/:1. + "', ~ "l + '" 7 S 3 + 0(8 ~1 2 If 3 2. 2 '1 3 2.
+ O(CJ ~ "1 + O(,o! + 0<." ! 1 + 1),(,2. .i '7 ;. ex'3 S '? + al,'4- S '? '1 t + 0('5 ~ "l .
FIGURE 2,4 - WEB30 Element
I~ h , I,
12
Horizontal Loading
F = -Th/2 41
Vertical Loading
F12 = -Tv/2
-51-
h
v G
(a)
Shear Flow T
FIGURE 2.5 - Notation for Derivation of Equivalent Diaphragm Stiffness from (5)
-52-
FIGURE 2.6 - General Flow Diagram: Program SAFE
START
, U ID&O~1~T~O~~~R~OB~~----~6
RE AD , PRINT:
Structure Title and Properties
, ~C~AL~L;"';S~D~E.~T~AJIr--___ ", READ, PRINT:
RETURN
CALL ESM 1
Coords of Corner Nodes, Nodal Arrangement of Elements
, CALCULATE, PRINT:
Midside Node Coords
READ, PRINT:
Restraint Data
• _1 DO 1 TO NELEM 1
GENERATE:
Element Stiffness Matrix of Element l
J-----~·l
-53-
1 r-~--U;C~AL~Lk1R~O!!T~.N.fTE~r-I-~ ROTATE:
t STORE:
Rows of Structure Stiffness Matrix Corresponding to Restraints
• ENTER:
Constraints into Structure Stiffness Matrix
SUFRESS:
RETURN
Unsupported Degrees of Freedom if Necessary
RETURN y
ASSIGN:
Rotated Element Stiffness Matrix to Proper Location in Structure Stiffness Matrix
1
LC~AL~L~D~BAND~~j-I----~ DECOMPOSE:
Structure Stiffness Matrix bv Choleski
@I-I'4IE---RETURN------I1
Element Stiffness to Global System
1
-54-
3~----__ ~ss<orrLYMEE:::-------1
RETURN , PRINT:
Nodal Displacements
CALL ESTREM 1
For Nodal Displacements by Choliski
f
IDO 1 TO NELEM 1
GENERATE:
Stress Matrix for each Stress Point in Element l
RECALL:
Nodal Displacements for Element l
l ~C~AL~L..!R~O!.!T!iAT~ED----~ROTATE :
Jlli UK1'II , GENERATE, PRINT:
Stresses at Each Point
&-- RETURN ___ ---'1
Nodal Displacements of Element l to Local Axes of Element l
1
cp IDD~OUl:]TO~Ni!LfCcr ...... -~ 5
1 CALL LDATA
RETURN , Il..!C~AL~L.1S~B~AND~r-"'--1 3
-55-
READ,PRINT:
Number of Loaded Nodes, Number of Loaded Elements Loading Title
, READ,PRINT,ASSIGN:
Concentrated Nodal Loads into Loading Vector
GENERATE, PRINT:
Consistent Loading Vector
CALL ROTATE ~ ROTATE:
RETURN " 1
SUM:
Concentrated and Consistent Loads
ZERO:
Actions Applied at Restraints
Consistent Loading Vector to Global Reference System
1
4
GENERATE,PRINT:
Reactions
More Loading Conditions
NO
NO
-56-
YES---il~"'0
-57-
3. CHAPTER 3 - EXPERIMENTAL MODEL
3.1 Introduction
A simple cellular bridge model was constructed for the
purpose of verifying the analysis embodied in the program SAFE.
The description of material properties, details of construction
of the model, experimental technique, and discussion of results
forro the contents of this chapter.
Among the most popular materials for model analyses are
plastics and metals. Many studies have been made investigating
the characteristics of each type of modeling material, and from
these reports it was decided that an acrylic plastic (plexiglas)
would be the most suitable for the present application in terms
of cost, ease of manipulation, and accuracy of experirnental
results (17,18,19,20,21,22,23,24,25).
-58-
3.2 Characteristics of Plexiglas
The advantages of using plexiglas for model tests can be
listed as follows:
1) Readily available and inexpensive,
2) Its transparency is an aid to fabrication and gaging,
3) Ease of manipulation with ordinary hand tools,
4) Isotropie, hamogenous, and non-brittle,
5) Relatively small value of Young's modulus permits small loads
to produce measurable strains and deflections,
6) Depth-thickness ratios of the model will be similar to those
of actual structures.
The main disadvantages are:
1) Temperature and humidity effects,
2) Creep effects,
3) Non-linearity,
4) Dependence upon loading history,
5) Local stiffening due to strain gages, and,
6) Low thermal conductivity of plexiglas.
Techniques which reduce or eliminate these adverse effects will
now be considered.
-59-
While precise termperature and humidity controls were
not available in the testing laboratory, aIl tests were performed
after hours in a closed room which minimised the variation of
these quantities.
Creep effects and non-linearity were perhaps the most
serious drawbacks of plastic materials, and hence, they were
investigated in greater detail. A series of flexure tests were
performed on the simply supported beam shown in Figure 3.1. The
model material and beam sample were taken from the sarne plexiglas
sheet. Central deflections and quarter point strains were
measured at regular time intervals for up to eighty minutes at
several stress levels. The variation of strains at each constant
stress level is plotted in Figure 3.2. From this graph it was
seen that almost aIl of the creep occurred during the first few
minutes of load application. In fact, after ten minutes of load
application, the error in assuming there was no further creep was
5~, 7~, 6~, and 6~ for stress levels of 100, 250, 500, and 1000
psi, respectively. A stress strain curve was then drawn for t =
.10 minutes of load application, shawn in Figure 3.3. From this
curve a slight non-linearity was apparent. The graph can be
divided into two regions, the second beginning at 460 psi.
Young's modulus, as found from the slope of the first region was
415,000 psi. The implications of these tests may be summarized
as follows:
1) the stress level should remain under 460 psi at aIl times,
-60-
2) Young's modulus can be taken as 415000 psi,
3) AlI readings should be delayed ten minutes after application
of load in order to be able to use this value of Young's
modulus. An analogous technique was recommended by
Carpenter, Roll, and Zelman (18),
4) Poisson's ratio was found to be 0.35 (obtained from tension
tests of plexiglas coupons).
The manufacturer's specification lists the material properties of
plexiglas as 400,000 - 450,000 psi for Young's modulus, and 0.35
for Poisson's ratio (26).
A direct consequence of the creep phenomenon in
plexiglas is that this material has "memoryR (17). That is, the
behaviour of a plexiglas model will be dependent on its previous
loading history. The most practical method of eliminating this
problem is to allow sufficient time to pass between tests. For
the tests described in this report, the model was permitted to
recover for a period of time at least twice as long as it had
been loaded. In addition, if size and symmetry conditions
permit, re-orienting the model in an upside down position was
found to eliminate almost completely the effects of previous
loading history.
-61-
Strain gages applied to the surface of plexiglas sheet
can greatly increase local stiffness due to the larger modulus of
elasticity of the gage wire. Litle (22) has found errors of up
to eighty percent, especially when dealing with sheet thicknesses
of 1/32 inch. Corrective measures include the use of thicker
material (greater than 1/8 inch), and the use of foil type gages
which have a greater surface area, and hence, lesser stiffening
effect.
The low thermal conductivity of plexiglas will not
permit the dissipation of heat energy supplied by the gage. Local
heating will then occur which will effect a local change in
material properties. Since the .amount of heat generated is
proportional to the square of the applied voltage, modern strain
gage recording devices use a system whereby voltage is applied in
short pulses lasting only a few milliseconds. The use of foil
gages is also suggested since their greater surface area is an
aid to heat dissipation.
-62-
3.3 Mode1 Fabrication
A single sheet of acry1ic plastic (plexiglas) was
obtained from Rohm and Haas Company of Canada, the usua1
producers of this materia1. The nominal thickness of the sheet
was 1/4 inch, but when precise1y measured with ca1ipers the
thickness was found to be uniform1y constant at 0.230 inch. A11
the required component parts for the bridge (inc1uding webs,
f1anges, and diaphragms) and severa1 standard testing coupons
were eut from this sheet.
The first mode1 to be fabricated consisted of a
rectangu1ar (non-skew) single ce11 box with a central transverse
diaphragm, but without end diaphragms. This was denoted Mode1 l
(Figure 3.4). The span was 38 inches, the overal1 width was five
inches, and the overal1 depth was three inches. Prior to
assemb1y strain gages were applied to the webs and flanges in the
longitudinal direction at a distance of five inches fram the
central cross section. This is a1so shawn in the figure. The
gages were type TML FLG-6 (longitudinal), and TML PR-S (rosette),
of the Tokyo Sokki Kenkyujo Company of Japan. CN adhesive was
used for bonding the gages to the plexiglas surface according ta
the manufacturer's specifications.
-63-
The model was now ready for final assemblY. Three
possible techniques of assembly were available. The first method
(mechanical) consisted of using machine screws. The second and
third techniques (chemical) utilised epoxy type glues, or acrylic
solvents, respectively. The first technique was rejected because
of the uncertainties involved in determining the required minimum
fastener spacing, and in the unknown local effects of the screws
on the gage readings. The choice between an epoxy glue or an
acrylic solvent was not critical, but the solvent method was
preferred for the following reason: upon application, the epoxy
resin would undergo a complex chemical polymerisation which
would result in the addition of considerable strength and
rigidity to the joint. On the other hand, the application of an
acrylic solvent could only result in a partial dissolution of the
plastic material at the joint, which would simply allow the
members to fuse into each other. Consequently, the solvent
technique for the bonding of plexiglas was thought to provide a
more accurate representation of th~ conditions in a monolithic
structure. Methylene chloride was the actual solvent used for
the bonding.
Two methods of application were available. For the case
of an uncomplicated geometry and moderately sized elements, a
trough was made from aluminum foil (about 1/2 inch deep). The
trough followed the outline of the edge to be bonded. It was
then filled to about half with the methylene chloride. The edge
-64-
of the plexiglas member was held submerged in the solvent for
about fifteen minutes, or until it became visibly soft. It was
then brought into contact with its mating surface, correctly
positioned, and lightly clamped for 8-12 hours, by which time the
joint had fully hardened. If the mernbers became too large, or
the geometry too complicated, Methylene chloride could be applied
directly to the bonding surfaces by means of a hypodermic
syringe. Care had to be taken to use a syringe made of glass,
and not one of plastic.
Structural tests were then performed on Model I. Details
are presented in sections 3.5 and 3.6. Following the completion
of the tests on Model l, a sixt Y degree skew model was obtained
by cutting into the configuration shown also in Figure 3.4. This
was denoted Model II. The large value of the angle of skew was
chosen in order to magnify discrepancies between experimental
results and output from prograrn SAFE. A third model, denoted
~1odel III was obtained by adding end diaphragms to Model II.
Because the joining of the end diaphragms was not cri tic al to the
behaviour of the bridge, a sufficiently rigid joint was made by
using screws. Models II and III were then tested, and results
are presented in sections 3.5 and 3.6.
-65-
3.4 Load Cell
When dealing with skew bridges, uplift of the acute
angle corner points is to be expected. In order to be able to
measure this uplift, two load cells were designed and built. The
load cells were to satisfy the following requirements:
1) Young's modulus should be sufficiently low to allow small
loads to pro duce measureable strains.
2) The load cells should exhibit a negligible amount of
settlement under maximum load.
3) The dimensions of the load cells should facilitate the
placing of at least three strain gages.
4) The load cells should possess enough stability to be able to
resist small lateral forces.
The use of 5/8 inch diameter plexiglas dowel cut into
lengths of 1.5 inches satisfied these criteria. The sensitivity
of this load cell can be estimated from the following formulaz
p e=At
-66-
where e = longitudinal strain in load cell,
P = load on cell,
A = cross sectional area of load cell,
E = Young's modulus = 415,000 psi.
Thus, for a load of one pound, the load cell will give a
reading of about eight micro-inches per inch. The settlement can
be estimated from the following expression:
where dL
dl = PL AE
= settlement of load cell,
L = original length of load celle
At a maximum load of 100 pounds, the settlement is only
0.001 inch, a quantity tao small to affect the behaviour of the
model.
Figure 3.5 gives detail drawings of the load cell, and
also the experimental setup of the cells. In order to be sure
that no actual uplift of the acute corner points would occur, a
pre-Ioad was applied to these points by means of the weight
hanger system shown in the same figure. The maximum uplift force
was estimated to be about five pounds, and hence, a preload of
ten pounds was applied for the duration of the experiment. A load
cell reading of twelve pounds would indicate that two pounds of
applied loading were taken up at that reaction point, while a
-67-
reading of seven pounds would indicate an uplift force of three
pounds. The counterweight was necessary to ensure that the axis
of the weight pan remained vertical.
-68-
3.5 Testing Procedure
3.5.1 Summary of Tests
Six tests were performed on the plexiglas models as
follows:
1) Model l (rectangular bridge), symmetric midspan load.
A. Deflections
B. Strains
2) Model l, torsional midspan load.
A. Deflections
B. Strains
3) Model II (skew bridge - no end diaphragms), symmetric mid
span load.
A. Deflections
B. Reactions
4) Model II, torsional midspan load.
A. Deflections
B. Reactions
5) Model III (skew bridge - with end diaphragma), symmetric
midapan load.
A. Strains
-69-
6) Model III, torsional midspan load.
A. Strains
3.5.2 Setup
The testing setup consisted of a heavy framework table
to which two paraI leI I-beams were securely clamped. This system
formed a stable base on which the model could be mounted. Four
plexiglas dowels, 1.5 inches long by 5/8 inch in diameter were
used for reaction points. Circular depressions were hollowed at
the top of each dowel, into which were placed quarter inch nylon
balls. These provided point supports for the model. The balls
were weIl greased to ensure free rotation about any axis. Two of
these dowels (those located at the acute corner points) were
affixed with strain gages and acted as load cells. The model was
mounted on these supports, and the preload weight hangers were
put into position. It was important to ensure that the nylon
ballon which each hanger rested was indeed placed exactly above
the lower nylon ball of the load celle If not, an additional
bending moment would be induced throughout the model. The strain
gages were inputed into a fifty channel recording instrument
manufactured by B & F Instruments Inc., USA, of which twenty five
channels were used. Each active channel was brought to a zero
-70-
reading, and then calibrated according to its gage factor, as per
the operating instructions of the apparatus. And finally, three
deflection dial gages, each reading to 0.0001 inch were placed
along the central cross section1 one at each of the load
positions, and one at the center. The entire setup is shown in
Figure 3.6. The model was then ready for testing.
3.5.3 Calibration of Load Cells
The first step in the testing procedure was the
calibration of the load cells. Because of small, but significant
differences among tests, such as temperature and humidity
variations, cell seating, and even previous history, it was
thought best to do a new calibration for each test. Load was
applied to the preload weight hangers in two pound increments up
to twenty pounds, weIl beyond the expected maximum value of load
at that point. The average of the three gage readings were
plotted against the applied load to forrn the calibration curve.
In aIl cases a linear relation between load and axial strain was
exhibited as can be seen from a typical curve in Figure 3.7.
After the calibration, ten pounds were left on each hanger to
prevent uplift.
-71-
3.5.4 Loading
Loads were hung from the two lateral extreme points of
the central cross section for symmetrical loading, and at only
one lateral extremity for torsional loading. The hangers did not
rest directly on the model. In order to alleviate the severity
of point loading, a small ruhber pad (3/4 inch diameter) was
placed on the model at the load position, on top of which was
placed a one cent piece, (also 3/4 inch diameter). The load
hangers rested on these buffers.
Loading was applied in increments, from 7 pounds up to
57 pounds for the torsional loading case, and from 14 pounds to
104 pounds for the symmetrical loading cases. AlI readings were
delayed by the prescribed length of time. The automatic strain
gage recorder read the 25 gages within a time interval of 14
seconds.
-72-
3.6 Analysis of Output
3.6.1 Reactions and Deflections
From the calibration curve the reactions at each load
increment were obtained, and were then plotted against the
applied load. The points were approximately colinear, and
therefore a straight line through them was fitted by eye. The
slope of this line was the reaction per unit load, and thus, the
reaction at any other load could be directly obtained. As an
example, Figure 3.8 is a typical graph for detez~ining the
reaction per unit load for the skew Model II under both loading
conditions. Table 3.1 summarizes the results for the tests on
~!odel II and gives the comparison with the values obtained from
the program SAFE at an applied load of 27 pounds. A discussion
of these results will be deferred until the various possible
idealisations of the model are described.
The method for deflections was analogous to that
described in the previous paragraphe Deflections were plotted
against the incre~ental loading, a straight line was fitted
through the points, the slope thereby representing the deflection
per unit load. Figures 3.9 and 3.10 show the deflection-load
-73-
relationship for the torsional and symmetric loading cases of
Model l and Model II, respectively. Tables 3.2 and 3.3 summarize
the results and list the comparisons with program SAPE for Model
land Model II, respectively. The idealisation of the models for
program SAPE will be discussed in a subsequent section.
3.6.2 Strains
Longitudinal strains were measured at eleven points
around cross section A-A (Figure 3.4). Since program SAFE
calculated membrane stresses only, (from which membrane strains
were directly ohtained), the hending component at each gaged
position had to he eliminated. This was accomplished by gaging
both the inner and outer surfaces at each position and using the
average of the two readings. For each pair of gages, these
average values were plotted against the incremental loading, a
straight line was fitted through the points, and the slope gave
the longitudinal strain per unit load. This procedure was
repeated for each gaged location. Figure 3.11 shows the strain-
'load relationship of each gaged location on Model III for both
symmetric and torsional loading conditions. Figures 3.12 and
3.13 show the comparison of longitudinal strains with the program
SAPE for Model land Model III, respectively. AlI tests were
-74-
redone in order to as certain that repeated measurements would
yield the same results.
Finally, the maximum stress in each model under both
loading conditions had to be checked to see that it fell beneath
the level stipulated in section 3.2. Because there were no gages
at the cross section of maximum stress (for the skew model the
location of this section was not even known) , approximate
techniques or the finite element program were used. For the
straight through box with a symmetric load of 154 pounds, the
simple beam flexure formula gave a maximum stress of 400 psi.
program SAFE calculated the same value and was used to estimate
the maximum stresses for the other test cases: Model l,
torsional load 250 p~i; Model II, symmetric load 125 psi;
Model II, torsional load - 100 psi. Thus for evexy test case the
maximum stress was weIl below the level at which creep or non
linearity could have become significant.
Applied Measure-
Load ment
lbs. at
A = 27
C
B = 27
A = 0
C
D
B = 27
A 0
cL 81 7 Model II
Experi- prograrn, Coarse Mesh prograrn, Fine Mesh
mentally with No No with
Measured Fiet. Fiet. Fiet. Fiet.
Reaction Dia. Error Dia. Error Dia. Error Dia. Error
lbs. " " " "
5.52 5.59 1 8.77 59 8.85 60 5.61 2
7.88 7.15 9 8.71 11 8.75 11 7.12 10
-1.35 -1.5 11 .04 -- .02 -- -1.56 15
TABLE 3.1 - Reactions from MODEL II
1 -...1 Ut 1
Loading Experi-
Case mentally
lbs. Measured
Defleet-
ions, in.
A = 27
.0319
B = 27
A = 0
at A=.0130
at B=.0200
B = 27
L-.
A
1 lB -1 Madel l
program, CoaI.'se Mesh program, Fine Mesh
lNo With No With
Fiet. Fiet. Fiet. Fiet.
Dia. Error Dia. Error Dia. Error Dia.
~ % %
.0324 2 .0324 2 .0324 2 .0324
.0124 4 .0125 4 .0124 4 .0125
.0201 0 .0201 0 .0201 0 .0201
TABLE 3.2 - Defleetions from nODEL l
,
Error: 1
% 1
2
4
0
1 -.J 0\ 1
Loading Experi-
Case mentally
lbs. Measured
Defleet-
ions, in.
A = 27
103.1
B = 27
A = 27
at A=89.1
at B=15.3
D = 0
A / 1 :7 Madel II
8 program, Coarse Mesh program, Fine Mesh
No With No With
Fiet. Fiet. Fiet. Fiet.
Dia. Error Dia. Error Dia. Error Dia. Error
~ % % %
94.3 9 78.8 24 95.5 7 79.4 23
122.7 38 95.0 7 124.1 39 95.8 7
28.4 85 15.9 4 28.3 85 16.2 6
TABLE 3.3 - Defleetions frorn r10DEL II
1
1
1 ...J ...,J 1
-78-
3.7 Model Idealisation for program SAFE
3.7.1 Model l
In keeping
elernents were
idealisations.
used
As
,OTi th the aim of rnaximized efficiency, WEB 2 4
as often as possible for the model
a result of the nodal arrangement of this
element, high aspect ratios were permitted, and hence, the coarse
mesh had only two elements spanning the entire length. To test
for further improvement with a finer mesh, the model was also
idealised with four ~mB24 elements spanning the length. In both
cases only one element was used to represent the entire depth and
width of the cell. The elernents were numerically integrated
using four Gauss integration points in the long direction and two
in the short direction. Also, analyses were made with and
without the use of equivalent diaphragms to simulate the
distortional stiffness of the cell. For the fully symmetric case
distortion was at a minimum and little improvement waa expected.
Under torsional loading distortion could have become aignificant
but equivaJ.ent diaphragms \'Tere found not· to be necessary to give
satisfactory results. Figure 3.14 shows the coarse idealisation
with and without equivalent diaphragma, and Figure 3.15 shows the
fine mesh idealisation with and without equivalent diaphragma.
-79-
With each idealisation is included the half bandwidth and number
of degrees of freedom. The following is a sample calculation for
determining the thickness of an equivalent diaphragm for the
coarse mesh idealisation.
b = t<element Span) = +(19) ='t.75 in. 3 l = bd =
1 12.
12, =
h = "".75
V - '1.75
~ = 0.35
in.
in.
3 *-75' .230 12.
'T.75 .230 Il
t = '+8 I! Ii {I +-.) hv(vI, + hI;t) -
- .00'#-816 .n.'+
3 - .001t816 in.'"
.003173 m.
Note that the thickness of the equivalent diaphragm is
several orders of magnitude smaller than that of the wall
elements (0.230 inch). Even so, their effect can he qui te
significant, especially in sorne cases when dealing with non
symmetrical structures or loading. Table 3.4 compares reaction,
deflection, and stress output for the four cases analysed
including both loading conditions. Because aIl idealisations
produced the same results, the coarse mesh (two elements per
side) without equivalent diaphragms has heen used for comparison
with the experimental results. From the table it is seen that
J
-80-
deflections have been calculated with an error of about 1%.
Reactions were not measured for the rectangular model, and the
stresses listed in the table are there to observe the variation
in stress results for several idealisations. For comparison with
strain gage readings, longitudinal strains were calculated frorn
the fOllowing relation:
Strains are compared with the experimental model in
Figure 3.12. For the symmetric loading case the maximum
difference was about 17%, although at several locations exact
agreement has been obtained. This was also true when the model
was acted upon by a torsional load.
-81-
Madel l
Symmetric Load Torsional Load
At C = 27 lbs. At C = 27 lbs.
At D = 27 lbs. At D = 0
Idealisation React j)efl stress React DefI Stress
at A at B at B at A at B at B
lbs. in. psi lbs. in. psi
2 Elem/side
No Fict Dia 13.5 .0320 -135 13.2 .0164 -67
2 Elem/side
With Fict Dia 13.3 .0320 -135 13.3 .0164 -67
4 Elem/side
No Fict Dia 13.5 .0320 -135 13.3 .0164 -67
4 Elem/side
With Fict Dia 13.4 .0320 -135 13.5 .0164 -68
TABLE 3.4 - Comparison of Idealisations for MODEL !
-82-
3.7.2 r.~odels II and III
Models II and III were each idealised by the Saffie
scheme, but with the difference that Model III included two end
diaphragms whereas Model II did note Several possibilities
existed for the representation of the structure, but because of
the type of high aspect ratio element used for the idealisation,
(i.e. WEB24), it was not possible to refine the mesh by simply
adding more elements. Three elements along its length were quite
sufficient to completely simulate the structurels behaviour.
Two idealisations were considered, and each was analysed
with, and without, fictitious diaphragms. The first was the
simplest idealisation possible, where the triangular flanges at
the span ends were represented by a single triangular element.
The rest of the idealisation followed fram this, as shown in
Figure 3.16. Note that the protruding walls and end diaphragms
(Model III only) were each idealised by one quadratic element.
Because it was desired to have the end diaphragms (Model
III only) and aIl wall elements represented by WEB24 elements, a
second idealisation was considered. In ûrder to accomplish this
the triangular flanges at the span ends were br ok en up
triangular element and a transition WEB20 element.
into a
This was
necessary since a triangular element with five nodes along one
-83-
side had not been developed. Because of this nodal arrangement
at the end flanges, it was necessary to divide the width of the
central portion of the cell into two WEB24 elements. It is hoped
that the preceding description ''lill be better understood if
reference is made to Figure 3.17.
For Mode 1 II, deflections and reactions have already
been compared with experirnental results in Tables 3.3 and 3.1.
For the case without fictitious diaphragma identical results were
obtained with the coarse and fine idealisations. For the cases
with fictitious diaphragms, identical results (although not the
same as obtained without fictitious diaphragms) were also
obtained with the coarse and. fine idealisations. Thus, the
degree of subdivision did not seem to affect displacement
results. There was, however, a marked improvement in comparison
with experimental values with the inclusion of fictitious
diaphragms. The same effect was observed when comparing
reactions. With the inclusion of fictitious diaphragms the
differences with experimental results were kept below 15% ,
whereas without their use errors increased to about 60%, and
uplift of the acute corner reaction points could not be obtained.
The idealisation used for Model III was that shown in Figure
3.17. Note that the idealisation did include the use of
equivalent diaphragms. Excellent agreement with experimental
results for longitudinal strains at section A-A was obtained as
ia shown in Figure 3.13. For both loading conditions most gaged
J
-84-
locations gave discrepancies of appreximately 10% while many
exhibited exact agreement. A few gages gave errers of up to 40~,
but these eccurred at regiens of small strain.
-85-
8 in. p
Dial Gage ----i ....
8 in.
LStrain Gages
FIGURE 3.1 - Plexiglas Bearn For Material Properties
.518 in.
1 li
!-If 1. 540 in.
CIl al .IJ ;::J ~
"M ::0:
Q)
e "M H
0 00
0 r--.
o '"
0 ll"\
0 ..j"
0 cq
0 N
(/) 0.
0 0 -,
(/) 0. a lJ') N
o o ll"\
,
-86-
o o o ~
(/) 0.
0 a lJ') ,
o o ll"\ ~
(/) a. 0 a a -
o o o N
o o ll"\ N
e CIl Q)
pq
CIl CIl ~
co "M ~ Q) ~
p..
c:: "M
0-Q) Q) ,..
t:.)
N
M
tLI
~ c.!l I-t ~
1000
800 t = 10 minutes
600
/ 1 00
" 1
'o-i rn p..
b 400
200
300 600 900 1200 1500 1800 2100 2400 2700
E micro in! in
FIGURE 3.3 - Non-Linearity of Plexiglas
II)
al ..c:: 0 Cl .....
C""I
-88-
Load Positions ---__. -+----.1._ A
19 inches 5 14 inches
-+----.1._ A
----0-~~ Il "~
Set> _..ll._ ---
~----i'-A
8.5 in. 8 in. 5 11.5 in.
r 5 inches l 1 1 1 1
1 1 t = 0.230 in.
MODEL l
MODEL II (no end diaphragms)
MODEL III ( with end diaphragms)
SECTION A - A
1 1 ~ Longitudinal Strain Gage
1 1
FIGURE 3.4 - PLEXIGLAS MODELS
LOAD CELL
5/8 in. dia.) l 0.7 in.
-89-
D
"'----'
~ in. dia. nylon ball
Acute Corner of Skew Mode 1 ____ .1
Counterweight
\ inch diameter nylon ball
1.5 inch
Vertical Strain Gages
t-II ....... -- Hanger
L.JI----- Load Ce 11
---Preload
,----Weight Pan
FIGURE 3.5 - Load Cell Detail and Installation
Preload Rangers (2) Rangers (2)
FIGURE 3.6 - Setup of Experimental Model
Supports (4)
l - Bearn (2)
Stable Base
(2)
1 \0 o 1
20
18
16
14
CIl 12 "0 t:: ;:::1 0 0..
10 "0 t1l 0
....:1 8
6
4
2
-91-
20 40 60 80 Axial Strain -
100 inlin
120 140
FIGURE 3.7 - Load Ce11 Calibration Curve
-92-
0 / ~ ;'J X Load Position
0 / Ij< ;fo
;fo 0 Load Cell
0 / * 8
6
4 CIl "0 ~ :;J 0 p..
~ 2 0 • .-1 .u CJ ~
~
10
30 40 50 60
Load - pounds
-2,
FIGURE 3.8 - Reaction-Load Relationship, Model II
100 1
6
80..L 0
C""I 0 1 0 r-I
:>< rn 60 Cl)
..c: CJ é
"" é 0
"" 40 ~ CJ Cl)
r-I ~
8
20
l
1 • 1 ~ Load Position
~ 1
~-- 1
o Dia1 Gage
o o 100
Load - pounds
FIGURE 3.9 - Def1ection-Load Re1ationship Of Mode1 l
120 140 160
1 \0 W 1
-94-
0 / ;; 7 x Load Position
[] / : 7 0 Dial Gage
6 / * 7 100
-.::t 80 1 0 .-l
>:: tIl <lJ ..c: 0 ~
"M 60
~ 0
"M l..)
c.J <lJ .-l 4-1 <lJ 40 c:l
20
10 20 30 40 50 60
Load - pounds
FIGURE 3.10 - Def1ection-Load Re1ationship of Mode1 II
12 11 10 9
o N
o o r-f
o N
1 2 3
8 7 6 Gage Positions at
Section A-A
5
o Symmetric Load
o Torsiona1 Load
Position 10
o Load
2 Load
-95-
1 0
L 0
o N
1
~ 1
o \0
1
o lI"I
20 Load
60
Position 12
100
100
Position 1
o Load
1 0
FIGURE 3.11 - Strain-Load Relation6 for Mode1 III at Section A-A
o lf"I .-1
1
o .-1 1
o N
10
20
-96-
Position 3
o 90 Load
Position 5
o 1 0 Load
FIGURE '3.11, Continued
o N .-1 1
1
20
Position 4
60 Load
Positicn 2
o Load
100
o
o
o \D
1
-97-
Load
20 40 60
Position 7
Gage Positions 6 & Il were not functioning
Figure 3.11, Continued
-98-
-240 -240
/ " I:J ~ -240 ~- - - - Q... - - .A. \ -240
P " ;1 ~
240 1 ;) 240
L-O ___ ~ __ ~_J 240 240
Symmetric Load = 54 pounds
-111 -156
Torsional Load = 27 pounds
S.A.F.E.
o Strain Gage
FIGURE 3.12 - Longitudinal Strains (micro in/in) from Madel 1 at Section A-A
-99-
-60 -112
-53 l " ~~-Q- ---0.__ Q 1 - -0-"",_ -106 ~ ,
~ , ,6 '0,
59 L- \ 106
L __ C2.. ___ o... l 53 -- --- .....
97
Symmetric Load = 54 pounds
-61
--- ..... --. 0.. "0--_ , --o 61 ,
P 'b L __________ .. __ ~\
16 L_ J -,:,--------O ----0-
61
Torsiona1 Load = 27 pounds
S.A.F.E.
o Strain Gage
FIGURE 3.13 - Longitudinal Strains (micro in/in) from Madel III at Section A-A
-100-
Without Fictitious Diaphragms
Fictitious Diaphragms (6)
With Fictitious Diaphragms
FIGURE 3.14 - Coarse Mesh Idealisation of Model l
Degrees of Freedom = 144
Bandwidth = 78
E = 415000 psi
\) = 0.35
.-101-
Without Fictitious Diaphragms
__ -- Fictitious Diaphragms (14)
With Fictitious Diaphragms
FIGURE 3.15 - Fine Mesh Idealisation of Model 1
Degrees of Freedom = 264
Bandwidth = 78
E = 415000 psi
-.) = 0.35
-102-
Without Fictitious Diaphragms
With Fictitious Diaphragms
FIGURE 3.16 - Coarse Mesh Idealisation of Models II & III
Degrees of Freedom = 186
Bandwidth = 78
E = 415000psi
~ = 0.35
-103-
Without Fictitious Diaphragms
With Fictitious Diaphragms
FlGUR$ 3.17 - Fine Mesh Idealisation of Models II & III
Degrees of freedom = 318
Bandwidth = 114
E = 415000 psi
~ = 0.35
-104-
4. CHAPTER 4 - PROGRAM VERIFICATION
The program has heen used to analyse a number of
structures for which exact analytical solutions, precise
experimental, or computer generated results have been previously
published by various authors. Four of these analyses are
presented in this thesis:
1) Cantilever problem (27),
2) Flat Plate prohlem (28),
3) Cantilevered Box problem (29), and,
4) Five Cell Skew Box problem (30,31).
A description of each problem and comparison with the finite
element program follows.
-105-
4.1 Cantilever problern
The cantilever problem is shown in Figure 4.1. It has
been idealised hy various combinations of linear, quadratic,
cubic, WEB24, and tVEB20 elements as shown in Figures 4.2a-f,
respectively. The cantilever was 48 inches long, twelve inches
deep, and had a thickness of one inch. Young's modulus and
Poissons ratio were taken as 30000 ksi and 0.25, respectively.
The cantilever was completely built in at the root, and was
loaded by a parabolically varying shear force per unit thickness
at the free end of zero magnitude at the upper and lower
extremities, and a maximum of five ksi at mid-depth. Thus, the
total load applied to the structure was fort y kips. It was
interesting to see how the program used the method of consistent
loading to allocate the distributed load to the nodes along the
loaded edge. Figure L~. 3 shows this on a diagram of the loaded
element(s) for each idealisation. Note that the sum of all the
consi stent loads was ah7ays fort y kips.
-106-
4.1.1 Tip Deflection, Reactions, and Stresses
The exact value of the tip deflection was given by the
follm-ling relation (27, page 167):
where S = p = E = l = 'J = A =
+ Lt+5--J PL 2.. rr
tip deflection,
applied load,
Young's modulus,
cross section moment
poisson's ratio, and,
cross section area.
of inertia,
This value was exact if the root section was free to warp
but the three nod.es at the root ''1ere fixed.
By substituting the appropriate values one obtains:
S = .34133 + .01400
S = .35533 in.
This value was compared to that obtained with each
idealisation in Table 4.1. Results were excellent in aIl cases,
except for the linear element. This was to be expected as two
-107-
nodes along each side were not sufficient to accommodate a cubic
variation in displacement.
-108-
A B Cantilever problem
Idealisation Tip Reactions Stresses
(see Figures Deflection kips ksi
4.2a-f) in. X y A B
a .1544 0.00 39.96 43.50 14.50
b .3485 0.00 39.80 59.68 20.23
c .3504 0.04 39.80 60.35 20.10
cl .3470 0.01 40.20 59.87 19.91
e .3367 0.03 39.76 59.96 19.98
f .3562 0.03 39.00 60.12 20.09
Exact .3553 0.00 40.00 60.00 20.00
TABLE lJ.1 - Reaul ta from Cantilever problem
-109-
Tahle 4.1 also lists the reactions calculated for each
case. The exact total values were zero and fort y kips in the x
and y-directions, respectively. Because the actual cantilever
was not supported at discrete points as in the idealisation, the
reaction at each nodal point by itself was rneaningless. However,
the surn of aIl the reactions in any particular direction should
be equal and opposite to the applied loads. The prograrn cannot
give any indication to the distribution of reactions for the case
of a continuous support. Again, results obtained were in
excellent agreement, and in aIl cases fell within 3% of the exact
values.
Longitudinal stresses, r~ , were compared to the exact
values given by the simple beam flexure folmula:
My l
This expression gave a value of eighty ksi at the root,
decreasing linearly to zero at the free end. Results for aIl
idealisations at selected points are shown in the sarne table.
Agreement was excellent in aIl cases except for that of the
linear elernent which was only capable of accommodating a constant
stress variation.
AlI idealisations using elements superior to the linear
element gave excellent results for deflections, reactions, and
-110-
stresses. Small discrepancies did exist between solutions,
however the se were most likely caused by round off errors in
single precision.
4.1.2 Condensed Element
The cantilever problem was solved one more time to test
the effectiveness of the WEB30 element, and that of the WEB30
element condensed to the configuration of the WEB24 element. The
results of three runs are shown in Figure 4.4. They were the
standard WEB24 element, the WEB30 element, and the WEB30 element
statically condensed to the configuration of the WEB24 element.
The WEB30 element gave identical results whether it was used as
is, or if it underwent condensation. There was a significant
improvement in the calculation of the tip deflection as
campared to the ordinary WEB24 idealisation. However, stresses
throughout the element gave less accurate results, and for this
reason it was decided not to pursue the possibility of using this
element any further.
-111-
4.2 Flat Plate problem
The Flat Plate problem is shown in Figure 4.5a. The
plate was 32 inches square and one-tenth inch in thickness. The
load was applied parabolically to opposite ends as shown in the
figure. Because of the symmetry of structure and loading, it was
only necessary to analyse one quarter of the plate. Two
idealisations were used to analyse the quarter plate: four
quadratic elements~ and two WEB24 elements. The Flat Plate
problem demonstrates the greater accuracy obtained when using
consistent loading to deal with distributed loads. To illustrate
this, the problem was also solved using "lumped" loads, i.e.,
loads obtained by just averaging the distributed load in the
vicinity of each node. Figure 4.5d shows the nodal loads as
generated by the consistent loading technique, and Figure 4.5e
shows the loads nssigned to each node as input data by "lumping".
Deflections for the two idealisations are tabulated in
Table 4.2. For each case the errors using ·lumped" loads were
usually not greater than seven percent, while the errors using
consistent loading were usually within one or two percent.
-112-
Stresses at selected points in the plate are tabulated
in Table 4.3. Greatly irnproved accuracy is also evident when
using consistent Ioads rather than "lumped" loads.
F -113-
G
H
f ~ Flat plate
Exact 4 Quad Elements 2 WEB24 Elements
tlode Defl. Lumped Error Consis- Error Lumped Error Consis-
X .001 Loads % te nt Lds % Loads % tent Lds
A 1.476 1.566 6 1.476 0 1.518 3 1.477
B 1.390 1.318 5 1. 390 0 1.344 3 1.392
C 1.143 1. 214 6 1.142 0 1.183 3 1.147
D .7786 .7620 2 .7754 0 .7690 1 .7664
E .3674 .4830 31 .3786 3 .4623 26 .3956
TABLE 4.2 - Deflection, Flat Plate problem
Exact 4 Quad Elements 2 WEB24 Elements
Node Stress Lumped Error Consis- Error Lumped Error Consis-
psi Loads % tent Lds " Loads % tent Lds
F 975.0 867. 11 969. 1 965. 1 974.
G 852.6 807. 5 851. 0 844. 1 860.
Il 609.4 602. 1 601. 1 590. 3 587.
l 249.4 260. 4 258. 4 261. 5 254.
TABLE 4.3 - Stresses, Flat Plate Problem
A
B
C
o E
Error
%
0
0
0
1
7
Error
%
0
1
4
2
-114-
4.3 Cantilever Box Problem
The bl0 problems just described dealt with structures
lying in a single plane. The Cantilever Box problem was the
first non-planar problern to be tested, and results were very
encouraging as shown below.
The structure consisted of a four-"Talled rectangular box
without end or interior diaphragms. The box was 48 inches long
and was completely built in at one end. The cross section of the
box "Tas t"Telve inches by twelve inches, and a torsional moment of
1l~40 in-kip acted on the free end. The thickness of each wall
was taken as unity. A diagram of the probleM including the
method of application of the torsional load is shown in Figure
4.6a. Only one mesh idealisation was considered which involved
the use of one WEB24 element representing each wall. This
idealisation is also shown in the same figure.
-115-
4.3.1 Torsional Deflection and Shear Stress
From Shanley (29, page 486) the exact expression for the
torsional rotation is given by:
ML l 't4-o !:ta 1 .,8'= d3t G 12.3 - 300 1 12.000
",here, ..9' = angular rotation of box,
M = torsional moment,
L = length of box,
d = depth (width) of box,
t = thickness of walls,
G = shear modulus.
From program SAPE, the Midside nodes displace 0.01998
inches (see Figure 4.6c) which corresponds to an angular rotation
of O.O/~c) 8 1 ---6 300
the exact solution. Similarly, the corner nodes displace 0.2005
inch in the y- and z-directions which correspond to an angular
rotation of
1 - 300
The exact value of the shear stress is given by
M t.)(~= ld 1 t = 5.0 Ksi.
The progrrum yields the identical result.
-116-
4.4 Five Cell Skew Box problem
The final test problem to be analysed \'Tas a concrete
model of a five cell ske"T box bridge for \'Thich Sawko and Cope
have published experinental results (30), and for which the
corresponding output from a computer analysis has been presented
by Crisfield (31). This problem was particularly appropriate as
it tested the ability of skew (non-rectangular) WEB24 elements to
solve the structure. As well as this, it was a very large
problem for an in-core solution and a fine mesh idealisation was
not possible for lack of additional computer memory. However,
resul ts gi ven by the program SAFE \'Tere comparable in most cases
with those given by the more complete finite element solutions of
Reference (31), and in general gave results which exhibited
better agreement with the experimental values.
The geometry of the model was as shown in Figure 4.7.
It consisted of the flanges, six webs and end diaphragms, and was
constructed of concrete with longitudinal prestressing to prevent
cracking. The elastic modulus was 27.7 kM/sq mm and Poisson's
ratio was taken as 0.18. Two concentrated load positions were
considered, hoth on the span centreline and located on webs as
shown in Figure 4.7.
-117-
The idealisation used for prograrn SAPE is also shown in
the sarne figure. It is relevant to note the large number of
fictitious diaphragms used to represent the transverse flexural
rigidity. The fact that these could not be place exactly normal
to the l'7ebs "Tas shown, by a number of trial runs wi th varying
degrees of skew for such diaphragIl1.s, not to cause significant
variations in stress or deflection results. The solid lines on
the plan view in Figure 4.7 represent the boundaries of the
flange elements, indicating that only two elements were used
across the bridge span.
Among the results presented by Crisfield were included
several solutions based on two different prograrns, each with a
fine and coarse idealisation. Only one of these results is given
in the following comparisons, this corresponding to a fine mesh.
Where both fine mesh solutions are given in Reference (31), the
one judged as exhibiting the better agreement with experiment
will be shol'm in the following figures.
4.4.1 Deflections and Stresses
The bridge l'1as analysed for deflections along the
centreline for each of the two loading conditions, and for
1
-118-
deflections along the loaded web, for each of the two loading
conditions. The results are shown in Figures 4.8a-d including
output from program SAFE, computer output from Crisfield, and
experimental results from Sawko and Cope. Excellent agreement
has been ohtained in aIl cases.
The bridge was also analysed for longitudinal stresses
across the centreline for each of the two loading conditions.
These results are shown in Figure U.9. From the first loading
condition, the results from both the Crisfield analysis and the
program SAFE poorly approximated the experimental value of the
stress at the position of the applied load. It should be noted,
however, that from this type of analysis, results at the location
of concentrated loads were not expected to be very accurate.
This problern did not occur at other points along the centreline.
-119-
4.5 Concluding Remarks
The agreement ohtained in these examples, including the
plexiglas models gave sufficient confidp,nce that the program was
working properly and that the simplifications on which it was
based were sufficient to analyse this type of structure. The next
step in the project was to apply the program to a study of the
main parameters affecting the behaviour of skew rnulti-cellular
bridges. This is the suhject of the next chapter.
-120-
48 inches
-~~~--------------------------~ 1 E= 30000 ksi
Parabolic end shear
.ci / ... X ~ 1,/ v=0.25
1 t = 1.0 L 5 k/in.
FIGURE 4.1 - Cantilever Problem
FIGURE 4.2 - Various Idealisations
1 1 III l : 1 : 1 : 1 : 1 (a) L(neâr Elements (b) Quadratic Elements
1 . : . 1 . : . 1 (c) Cubic Elements (d) WEB20 Elements
! : : : 1 1 : : : 1 : : : 1 (e) WEB24 Element (f) WEB24 Elements
-121-
(
~ 20.0
Linear Element
~ 20.0
1 ... ... - - t 4.0
4' ~ WEB20,
32.0 Quadratic) or
WEB24 Element
... ... \ ~ 4.0
... ... t 2.0
4 t 18.0 Cubic Element
4 4 ~ + 18.0
- ... - -- - -\ t 2.0
... ... -- + 0.5
4 , t 14.0
.- ~ ... ... ~ 11.0 Two WEB24 Elements
4 + 14.0
- ... ... , . +
0.5
FIGURE 4.3 - Allocation of Distributed Load by Consistent Loading
Exact
Original WEB24
WEB30 (No Conden-sation)
WEB30 (Condensed)
-122-
S T RES SES
80.0 60.0 20,0 0.0
79.1 59.3 39.6 19.8 0.0
1 0.0: : : 0.0 1 -79.1 -59.3 -39.6 -19.8 0.0
84.2 58.4 41.6 18.6 4.6
1°·0 : : : o.oj -84.2 -58.4 -41.6 -18.6 -4.6
84.2 58.4 41.6 18.6 4.6
Io.o ~ ~ : 0.0 j -84.2 -58.4 -41.6 -18 0 6 -4.6
FIGURE 4.4 - Cantilever Prob1em Using WEB30 Element
Tip
Def1ection
0.3553
0.3367
0.3522
0.3517
(b)
Total Load
1066.6 lbs ..
(d)
-123-
1__ 32 inches "1 100 pli
t = 0.1 in. E =10~OOO,000 psi G = 4,,000,000 psi ~ = 0.25
(a)
(c)
136.7 197.9
493.3 Total -- 370.8
Load 206.6 295.8
1066.6 lbs.
226.7 170.8
3.3
FIGURE 4.5
(a) Flat Plate Problem
31.3
(e)
(b) t Plate Idealisation Using 4 Quadratic Elements
(c) t Plate Idealisation Using 2 WEB24 Elements
(d) Load Allocation by Consistent Loading
(e) Load Allocation by Lumping
48 inches
each arrow represents 20 pounds of force
(c)
-124-
~I
(a)
1440 in.k • .... x
0.2005 in.
FIGURE 4.6
(a) Cantilever Box Prob1em
(b) Idealisation and Leading
(c) Torsional Disp1acements
t lO.2005 in.
in.
-125-
Oross Section of Model
38.1
25.4 -+--
101.5 Dimensions in mm.
152.3
25.4
38.1
50.9 50.9 203 50.9 50.9
Load Case 1 (10 KM) + ~ Load Case 2 (10 KN)
190.41 1: l : 1: l :I : 1 t Cross Section
Plan of Idea1ised Structure
Line of Support
of Support
2150 2150 o Load Position
FIGURE 4.7 - Sawko and Cape Madel (Refs. 30,31)
l
-126-
mm. ~Load Case 1 .2
.1-
.3
.2
.1
-.1
.2
.1
.3
.2
.1
(a)
tLoad Case 2 o Model Ref. 31
~ Analysis Ref. 31
-0--0-. S.A.F.E.
o
(c) Load Case 1
(d) Load Case 2
FlGURE4.8(a),(b) - DEFLECTION ACROSS CENTRELINE
(c),(d) - DEFLECTION ALONG LOADED RIB
.9
.8
.7
N~ .6 -z .... 5
CIl CIl Qi
t .4 CI)
.3
.1
-127-
Load Case 1
FIGURE 4.9 - Stresses Along Centre1ine
-128-
5. . CHAPTER 5 - BEHAVIOURAT...I STUDY OF SlŒW CELLULAR STRUCTURES
5.1 Introduction
The agreement obtained with the experimental model and
with the analytical solutions of Sa"Tko and Cope, Crisfield,
Shanley, Felippa, and Timoshenko gave sufficient confidence that
the prograM could be used to investigdte the effect of a number
of pararneters associated with skew rnulti-cellular structures.
T"70 separate investigations "Tere made and are presented in this
chapter. Both studies concerned themselves with the behaviour of
cellular structures with variations in the angle of skew, and
with various combinations of end and internaI stiffening
diaphragms. One study \-Tas a general analys.:i.s of a three cell
skew bridge for which the deflections, reactions, and stresses
were plotted under various conditions of skew angle and diaphragm
configuration. The other study "Tas a more detailed analysis of a
single cell skew structure. Stress contours on the upper flange
were also incluàed in this study.
-129-
5.2 Single Cel1 Box Bridge Investigation
5.2.1 General
A single ce11 hox bridge ,'ras ana1ysed to determine the
effects of variation in the angle of skew and in the effects of
various diaphragm configurations. The ana1ysis concered itse1f
\'lith a study of longitudj.na1 stresses a10ng the top of the webs,
and the distribution of longitudinal stresses acting on the top
f1ange. The bridge mode1s are shown in Figure 5.1. Their
properties are sumrnarized as fo11ows:
1 ) The span of the bridge 't'las thirty feet,
2) The single ce11 was seven feet wide, ~
3) The depth of the bridge was 3.5 feet,
4) Top and hottom f1anges \'lere five inches thick,
5) Webs and diaphragms were six inches thick.
The bridge was supported at four locations, these being
at the obtuse and acute corner points of the span. Line loads of
1400 pounds/foot were app1ied to each web. The total load
app1ied to the hridge was then about 85000 pounds, which was then
a110cated to each node according to the technique of consistent
loading.
-130-
5.2.2 Skew Angle
Ta study the effects of variation in angle of skew, the
three one cell bridges of 30, 45, and 60 degrees skew were
analysed l'li th end cUaph:r.agrns only. Longi tudinal stresses along
the top of the '\J'eb are shm'ffi in Figure 5.2. The marks on the
upper side of the abscissa represent the element boundaries along
the span length for the thirty degree case, those on the lower
side, for the sixt Y degree case, and those crossing the abscissa,
for the 45 degree case. The bridge of thirty degree skew
sustained the largest longitudinal compressive stress. As the
angle of Ske\'T increased, the maximum stress decreased. This was
expected since the ohtuse corner reaction points were closer to
each other "Ti th increasing angle of ske\'l. Thus, the bridge
tended to span diagonally a shorter distance between these
reaction points. The thirty degree bridge with its small angle
of skew behaved similarly .to a rectangular bridge, as the maximum
stress occurred at almost midspan. As the angle of skew
increased, the location of maximum stress shifted towards the
acute corner. For the 45 and 60 degree case, the peaks in
longi tuclinal stress \'lere located at a point on the web
perpendicularly across from the obtuse corner.
-131-
Each of the three bridges suffered tensile stresses in
the vicinity of the ohtuse corner.
of a support reaction located
This '<las due to the presence
there. Since there were end
diaphr.agms on these bridges, there existed a certain arnount of
continuity between the webs and the end diaphragms. As these two
members becarne more continuous (by increasing the angle of skew),
the tensile stress at t~is reaction point increased. The
analagous condition of tensile stresses in the upper fibers
exists at intermediate supports of a continuous bearn. As a
result, the tensile stress for the sixt Y degree skew case was
approximately 2.5 times greater than for the thirty degree skew
case.
Simple bearn theory predicted a state of zero stress at
the longitudinal extremities of the bridge. This condition was
satisfied for rectangular structures, but from Figure 5.2, at the
acute corner point, it was seen that this was not the case for
ske'toT bridges. A state of zero stress did not exist at this
location because of the end diaphragm and flange that join the
web non-orthogonally at this point. The departur.e from
orthogonali ty increased 't<1i th the angle of skew, and thus the
thirty degree structure showed the smallest non-zero stress at
the end point, the sixt Y degree structure W~2 approximately
doubly stressed, and the 45 degree structure was stressed between
the other t'toTO.
-132-
5.2.3 Diaphragms
For the discussion of the effect of diaphragm location,
refer ta Figure 5.3. The sixt Y degree bridge was analysed with
three diaphragrn configurations: no diaphragms, end diaphragms
only, and end plus transverse diaphragms. In this case, the
transverse diaphragms were perpendicular ta the webs, and passed
through the obtuse corner points.
The maximum longitudinal compressive stress occurred in
the structure that made no use whatsoever of diaphragms. The
location of the maxj~um stress was shifted towards the acute
corner and occurred within the triangular region of the bridge
extrernity. A slight reduction in compressive stress, arnounting
ta approximately ten percent, was obtainen by the addition of end
diaphragms. The location of maximum stress was not altered by
their inclusion. For the case of end and transverse diaphragms,
the compressive stress in this region was sharply reduced to
about twenty percent of the value of the case with no diaphragms.
Thi~ effect was due to the additional support provided by the
transverse diaphragm on the webs. The peak stress in this case
was only ahout half as large as the no-diaphragm case, and
occurred at the location of the perpendicular centreline of the
bridge.
-133-
The intensity of tensile stresses in the vicinity of the
ohtuse corner ~Tas largely dependent on the degree of continuity
linking the two reaction points at each of the span ends. As
previously mentioned there was a considerable amount of
continui ty in this hridge (sixt Y degree ske't-'l) ~Then end diaphragms
were included, and hence, the large tensile stress at the obtuse
corner. For the case "Ti thout end Liaphragms, continui ty was
almost non-existent (except for small contributions by the
flanges) and the tensile stress in this region was therefore only
about one-tenth of the value of the case including end
diaphragms.
Longitudinal stresses were not equal to zero at the
acute end of the structure for the same reason descrihed in the
section of 8kew angle. It was interesting to note the very large
value of the compressive stress for the structure without
diaphragms. The addition of end diaphragms caused this stress to
decrease by about fort y percent. A further reduction could be
obtained if transverse diaphragms were also used, although it
\-TOuld not arnount to more than an additional ten percent.
-134-
5.2.4 Stress Contours
Stress contours Here dra,·J'n for longitudinal stresses in
the upper flange for the various casses studied. The contour
interval '-Tas taken as thirty psi, negative numbers were used to
denote compressive stresses, and pea]~ values of stress were as
noted in the figure. Figure 5.4 shows the upper flange of the
bridge with end diaphragma and angle of skew of thirty, fort y
five, and sixt Y degrees. The most important aspect revealed in
these figures was the increasing region of tensile stress in the
vicinity of the obtuse corner as the angle of skew increased.
This information is of great interest to the designer of skew
cellular structures since the vast rnajority of these structures
are fabricated from concrete. Also related to this effect was
the fact that as the angle of skew increased the maximum tensile
stress increased while the maximum compressive stress decreased.
In aIl cases there \'7as a high stress gradient in the vicini ty of
the obtuse corner.
Figure 5.5 shm'Ts the contours of the upper flange for
the bridge of sixt Y degree skew with the diaphragm configurations
of end plus transverse, end only, and none. From these figures
it ~Tas ohserved that the removal of the end diaphragms did not
greatly effect the stress condition of the upper flange. There
-135-
was a beneficial effect of a smaller tensile region near the
ohtuse cor.ner and a smaller peak tensile stress. The maximum
compressive stress, which occured perpendicularly opposite from
the obtuse corner point, was slightly increased. However, the
stress at the exact center of the flange was almost doubled. In
both cases there was also a high stress gradient in the vicinity
of the acute corner. End and transversediaphragms had a very
pronounced effect on the upper flange. Only a small tensile
region existed, and the peak stress was only about one-tenth that
of the other cases. The location of the maximum compressive
stress was shifted to the middle of the structure and was
significantly reduced to less than one-half of the other two
cases. The stress gradient over the entire area was very small
in comparison with the other cases.
-136-
5.3 Three Cell Box Bridge Investigation
5.3.1 General
Bridges ",i th three cells as shown in Figure 5.6 were
analysed using the program SAFE. The investigation concerned the
effects of variation in angle of skew and in the location of
diaphragms. In aIl the following, the web and flange thicknesses
rema.in unchanged, ~li th the magnitudes gi ven in Figure 5.6. These
dimensions are typical of existing box girder bridges surveyed by
Scordelis (3). Diaphragms were assigned a thickness equal to
that of the \-Tehs.
Loading due to self-weight was considered, distributed
as line loads uniformly along the top of each web, with the
transverse distribution assigned as one-third to both of the two
interior "7ebs and one-sixth to the outside webs. In handling
distributed loads with high-order elements of the kind used in
this program, it was most important to use a consistent loading
vector.
5.3.2
degrees.
of the
-137-
Ske"T Ang le
Four angles of skew were considered: 0, 30, 45, and 60
In each case end diaphragms were used, and the length
"Tebs ~.ras kept constant at eighty feet. Thus, the total
load was the same in each case. The idealisations are shown in
Figure 5.6.
Deflections and stresses across the centreline (in this
case perpendicular to the "Tebs) are shown in Figures 5. 7 and 5. 8,
respectively. In both cases considerable reductions occurred as
the angle of ske,,; increased from zero to sixt Y degrees.
Deflections on the centreline \'Tere reduced to about 30%.
The reactions were provided only at the ends of the
webs, and the variation of their magnitudes is shown in Figure
5.9. There was a beneficial effect introduced by the continuity
of the "Tebs "dth the end diaphra.gms, as the angle of skew
increased. This was why the reactions at support points were
reduced and in sorne cases tended to uplift.
-138-
5.3.3 Diaphragms
The three cell sixt Y degree skew box was analysed with
three diaphragm configurations; no diaphragms, end diaphragms
only, and both end and transverse diaphragms, the latter spanning
from the obtuse corners in a direction normal to the \-lebs.
Deflections and stresses across the centreline, which as
before was considered normal to the webs, are plotted in Figures
5.10 and 5.11 , respectively. Both l'Tere reduced by the
introduction of the end diaphragms, this resulting from the
continuity provided, as described earlier. The transverse
diaphragm additionally reduced the stresses, particularly in the
outer \'Tebs, because of the support provided ,dthin their length.
The reactions with the three diaphragm arrangements are
shown in Figure 5.12. For a given angle of skew, the normal
stresses corresponding to positive moments within the span were
greatly nependent upon the reaction at the obtuse corner. The
larger this was, the lower the stresses. It can be seen from
Figure 5.7 that this reaction was increased very significantly by
the use of end diaphragms, and increased further still hy the
additional use of the transverRe diaphragms.
42
-139-
14 360 inches ~
(a)
360 inches ·1 r
(b)
FIGURE 5.I(a) - Idealisation for Single Cell Box Study (b) - Variation in Angle of Skew
30°
.~
CI)
Q.
CI) CI)
Q) S-I .u tf.)
.... t1I ~ .~
'0 ~ .u .~
00 ~ 0
...:1
110
70
30
-10
-50
-90
-130
-170
-140-
300
obtuse
FIGURE 5.2 - Longitudinal Stress Along Top of Web
for Bridge with End Diaphragms Only
at Various Angles of Skew
110
70
30
-.-1 -10 CIl
p.
CIl CIl <Il l-< ~ UJ
,..-4 -50 Cd I:l
-,-1 '0 ::l ~ -,-1 bD I:l 0
t-1 -90
-130
-170
-141-
end
none
acute obtuse
FIGURE 5.3 - Longitudinal Stress Along Top of Web
for Bridge of 60 degree Skew and Various
Diaphragm Configurations
o
-14
-36
-142-
-115 -168 o 80
122 0 -14
~77Y-~====~~7n
-144 o 122
-150 o 140
FIGURE 5.4 - Contours of Longitudinal Stress in Upper F1ange,
SingJ.'~ Ce Il Box Study
Contour Interval = 30 psi
-36
-75
-30
-36
-143-
-160 o 90
-67
-57
-67 o 12
140 o -150
o
-150 o 140
FIGURE 5.5 - Contours of Longitudinal Stress in Upper Flange,
Single Cell Box Study
Contour Interval ~ 30 psi
36
o
4
-144-
T T T i 1
1 1 1 l ... T T i 1 1 l 1 l J. T J l 1 1 1 l
1 l l
Top and Bottom Flange Thickness = 6
Web and Diaphragm Thickness = 8
End Supports Under Webs Only
E ::1 3000 ks i, ~::; O. 18
Dimensions in Inches
T 1 ... T
1 4 ... T
l 4
FIGURE 5.6 - Bridge Idealisations for Behavioural Analysis
-145-
1.0
~ ____ o----<_~--~----o----~-~ ___ ~~ 3~ 0.8 ... ~ - U
U)
~ CJ t: 0.6 - ~ H
... ou 45
0 U) -t: a -'..1 .j.J
CJ <li
.-l ~ 0.4 - f-<li ~
- -a 600
• il'"'"
0.2 1 1 1 1 1 ~
FIGURE 5.7 - Deflections Across Centreline - Three Cell Bridges
Effect of Angle of Skew (End Diaphragms Only)
-146-
700 r----c~-___o_---"'-' cf Skew
600
3cf -.-1 500 (1)
a.. ...
tI.I <LI tI.I tI.I <LI 1-< .w
450 CIl
400
300
200
FIGURE 5.8 - Longitudinal Stresses Across Centreline - Three Cell Bridges
Effect of Angle of Skew (End Diaphragms Only)
-147-
160
120
CIl ~
-.-/ ~
... 80 CIl t:: 0
-.-/ .u 0 tU <11 ~
40
-20 Acute Obtuse
FIGURE 5.9 - Vertical Support Reactions - Three Cell Bridges
Effect of Angle of Skew (End Diaphragms Only)
-..-1 CIl Cl.
" CIl CIl Q) 1-1 ~ CI)
-148-
0.4 CIl None Q)
..c CJ 0.3 c
H End " CIl End and Transverse c 0.2 0
-..-1 ~ CJ Q)
r-l 0.1 4-1 Q)
,::)
FIGURE 5.10 - Def1ections Across Centre1ine - Three Cell Bridges
Effect of Diaphragms (60 Degrees Skew)
350
250
150
None
End
End and Transverse
FIGURE 5.11 - Longitudinal Stresses Across Centreline
- Three Ce11 Bridges
Effect of Diaphragms (Sixt Y Degrees Skew)
-149-
160
End and Transverse
End
120
tIl 0- None .,-1
~
tIl 80 p 0
.,-1 .w C) CIl al p::;
40
Acute Obtuse
FIGURE 5.12 - Vertical Support Reactions - Three.Ce11 Bridges
Effect of Diaphragms (60 Degrees Skew)
-150-
6. CIIAPTER (5 - CONCLUSIONS
6.1 Surmnary
The objectives of the research project were the
development and verification of a finite element program which
was used to analyse the behaviour of multi-cellular skew box
bridges. Various simplifications were introduced into the
analysis in order to yield a program as efficient and economical
as possihle \'Thile still providing an acceptable degree of
accuracy. These simplifications includeà limiting the nodal
degrees of freedom of the inealised structure to three orthogonal
translations, and the development of high aspect ratio elongated
isoparametric elements. Because of the restriction of
translational degrees of freedom, plate bending of elements could
not be included and other methods of taking into account the
transverse stiffness had to be used. The technique chosen was
that of equivalent diaphragms as developed by Sawko and Cope. As
a result of the use of elongated elements with many midside
nodes, it was demonst~ated that the simple allocation of
distributed forces (lumped loads) would not result in accurate
output. The automatic allocation of distributed loads according
-151-
to the technique of consistent loading was then incorporated into
the program. Thi~ caused greatly improved results to be obtained
for aIl examples considered.
The verification of the program was accomplished in two
parts. A series of experimental tests wer~ carried out on a
single cell plexiglas mooel of a rectangular bridge and a skew
bridge, with and without end diaphragms, but always including a
central diaphragme The second part involved cornparisons with
analyses previously puhlished by various authors. Results in aIl
cases were favorable, and gave sufficient confidence that the
program could be used for the final phase of the research project
- the behavioural analysis of skew cellular structures.
Two separate investigations were carried out, the first
dealing with a single cell skew structure, and the second with a
three cell skew bridge. In both cases the nnalysis dealt with
the effects of variation in angle of skew and in various
diaphragm configurations on the cellular structure.
-152-
6.2 Limitations
The limitations of the analysis described in this report
are enurnerated as follo,,'s.
1) The rnaterial is elastic and isotr.opic throughout,
2) Material non-linearities due to plasticity effects are not
considered,
3) Elements within themselves have constant properties (although
separ.ate elements rnay have different properties),
4) Size of structure - A complete in-core solution was necessary
as a result of the requirement that auxiliary storage devices
"lere not to be used. Thus, a large capacity computer "las
required. Even so, only problems of limited size could be
held in the computer 1 s memory. ~1ore information of sample
computer times and storage requirements are presented in the
Appendix.
1
-153-
5) Skew supports At present only supports acting in one or
more of the orthogonal global directions are permitted to act
on the structure.
6) Overhangs The top deck of most cellular bridges extends
heyond the web to form overhangs. Because of the type of
analysis emhodied in the finite element program SAPE, only
in-plane stiffness can be directly analysed. Bending
stiffness of a cellular cross section has been simulated by
the use of fictitious diaphragms. It is not possible to use
this technique for the overhangs.
the overhangs cannot he taken
approximate techniques.
Thus, bending stiffness of
into account, even by
7) Loading - Another consequence of using elements with only in
plane stiffness is that they cannot accept loads normal to
their surface. For example, vertical loads cannot be
arbitrarily applied to the upper deck, but must be placed
over webs or diaphragms. A similar restriction would hold
true for horizontal loading on web elements.
8) niscrete support points Continuous supports can not be
accounted for in the finite element technique.
-154-
6.3 Recommendations for Future Work
1) Curved structures The elements de~cribed in this thesis
(except for the triangular element) are general
quadrilaterals, and can therefore take on a curved shape
suhject to the number of nodes along a side. Some
preliminary studies have been made in this direction with
excellent results, although they are not presented in this
work. These elements qhould readily lend themselves to the
idealisation of curved bridges with little additional
prograrnming effort.
2) Parameters - The two major parameters involving skew cellular
structure~ have been discussed in sorne detail. Other factors
do exist that also affect the behaviour of these structures.
The most notable among these are:
A) Number of cells,
B) Width of cells,
C) Depth/span ratio of the bridge, and,
D) Width/span ratio of the bridge.
3) Automation of input data This can be done for several
standard types of structures. It was not attempted during
-155-
the development phase of program SAFE due to the variety of
different geometries studied.
4) Automatic calculation of fictitious diaphragm thickness -
Each diaphragm requires a tedious calculation to determine
its equivalent thickness. Since this is a function of only
the previously defined geometry the calculation should
proceed automatically after input data has been entered.
5) Non-uniform elements The elasticity matrix could be
modified to deal "7ith elernents whose thickness varies along
their length or width.
-156-
7. APPENDIX - PROGRAM SAFE
AIl computer programming described in this thesis was
done on an IBM 360/75 installed at McGill University. Core
storage requirements and execution tirnes reported in this section
should be generally true for any installation of the sarne
machine, but program costs will be dependent on the charging
algorithm used at McGill. A simple listing of the rates will not
be of any help as programs executed at different hours of the day
are not charged at the sarne rate. program execution costs are
presented with this factor in mind. All times and costs apply
only to the ·~O STEP" as object decks from FORT~~~ IV G Level
were used as often as possible. Data in the following table
refer to the overall joint structure stiffness matrix. An
additional 100 k (approx.) is required for the remaining parts of
the program. At present, 400 k bytes of main memory are
available on the McGill system.
Degrees of
Freedom
99
144
186
264
318
46B
Bandwidth
50
78
78
78
114
131
-157-
Core
20 k
44 k
58 k
82 k
144 k
245 k
Solution
Time
2 sec
10 sec
12 sec
19 sec
43 sec
94 sec
Cost
$0.35
$2.25
$2.13
$3.70
$10.00
$18.60
8. REFERENCES
1. Cheung, Y.K.
2. Cheung, Y.K.
3. Scordelis, A.C.
4. Ghali, A.
5. Sawko, F.
Cope, R.J.
6. Crisfield, M.A.
-158-
"Orthotropic Right Bridges by Finite
Strip Method", Second Int. Sym. of
Concrete Bridge Design, 1969.
"Folded Plate Structures by Finite
Strip Method", Proc ASCE, Dec 1969.
"Analysis of Simply supported Box
Girder Bridges", Report n. SESM-66-17,
Dept. of Civil Engrg.,University of Cal.
BerkelYi Calif., October 1966.
"Analysis of Continuous Skew Concrete
Girder Bridges·, First Int. Sym on Conc.
Bridge Design, 1967.
"Analysis of Multi-Cell Bridges without
Transverse Diaphragms - a Finite Element
Approach", The Struct. Eng., Nov. 1969.
"Finite Element Methods of Analysis of
Multicellular Structures", Proc. IeE
7. DeVeubeke, F.B.
8. Sisodiya, R.G.
Ghali, A.
Cheung, Y.K.
9. Sisodiya, R.G.
Cheung, Y.K.
Ghali, A.
10. Sisodiya, R.G.
Cheung, Y.K.
Ghali, A.
11. Sisodiya, R.G.
Ghali, A.
Cheung, Y.K.
12. Zienkiewicz, o.c.
-159-
March 1971, Vol. 48.
"Displacement and Equilibrium Models in
the Finite Element Method", STRESS
ANALYSIS, John Wiley, London, 1965,
Ch 9, pp 145-197.
"Finite Element Analysis of Skew Box
Girder Bridges", Trans. EIC, March 1972,
Vol 15.
"Finite Element Analysis of Skew,
Curved Box-Girder Bridges", Int. Assoc.
for Bridge and Struct. Eng., Zurich, 1970.
"Ne\'T Fini te Element wi th Application to
Box Girder Bridges", Paper no. 7479,
Journal ICE, London 1972.
"Diaphragms in Single and Double Cell Box
Girder Bridges with Varying ~lgles of Skew"
Journal ACI, Vol 69, no 7, July 1972.
The Finite Element Method in Engineering
Science, McGraw Hill, 1971.
13. Ergatoudis, A.
Irons, B.'1.
Zienkiewicz, O.C.
14. Timoshenko, S.
Goodier, J.
15. Przemieniecki, J.
1 6. Weaver, i'7.
17. Roll, F.
18. Carpenter, J.E.
Roll, F.
Zeman, l-1.
19. Litle, W.A.
Cohen, E.
Somerville, G.
-160-
"Curved, Isopararnetric, Quadrilateral
Elements for Finite Element Analysis",
Int. J. Solids structures, 1968,
Vol 4, pp 31-42.
Theory of Elasticity, McGraw Hill, 1951.
Theory of Matrix Structural Analysis,
Z,!cGraw Hill, 1968.
Computer Prograrns for Structural Analysis,
Van Nostrand, 1971.
"Materials for Structural Models",
Proc. st Div ASCE, June 1968, Vol 94.
"Technique and Materials for Structural
Models", ACI Publication No. 24, Paper
SP24-3.
"Accuracy of Structural Models", ACI
Publication No 24, Paper SP24-4.
20. Carpenter, J.E.
21. Fialho, J.F.
22. Litle, W.A.
23. Preece, J.
Davies, T.
24. Myers, D.
Cooper, P.B.
25. Breen, J.
26.
27. Filippa, C.
-161-
"Structural Model Testing - Compensation
for Time Effects in Plastics", Journal
peA, VolS, No 7, Jan 1963.
"The Use of Plastics for Making Struct
ural Models", Bulletin RILEM,' No. 8,
Sept. 1960, pp. 65-74.
"Reliability of Shell Buckling Predictions
Based Upon Experimental Analysis of Plastic
Models", Report No. T63-7, MIT, Cambridge,
Mass. August 1963.
Models for Structural Concrete, Adelphi,
r.ondon, 1964.
"Box Girder Model Studies", Proc ASCE
ST Vol 95, 1969.
"Fabrication and Tests of Structural
Models", Proc ASCE ST Vol 94, June 1968.
Canadian Plastics Catalogue, Johnston
Industrial Plastics, Montreal, 1970.
Refined Finite Element Analysis of Linear
28. Timoshenko, S.
Goodier, J.
29. Shanley, F.R.,
30. Sawko, F.
Cope, R.J.
31. Crisfield, M.A.
32. Gurevich, S.
Redwood, R.G.
-162-
and Non-linear ~ Dimensional Structures,
PhD Thesis - University of California,
Berkely, 1966.
~heory of Elasticity, McGraw Hill,
Second Edition.
Strength of Materials, McGraw Hill, 1953,
p. 485.
"Experimental Stress Analysis of Beam
and-Slab, and Cellular Skew Bridges",
Paper 35, University of Liverpool, Eng.
Reply to discussion on Reference 6,
Proc ICE Vol 51, pp 150-164, March 1972.
"Approximate Analysis of Multicell Skew
Box Bridges", Proceedings of the Spec
ialty Conference on Finite Element Method
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