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A STUDY OF MATRIX STRUCTURAL ANALYSIS
by
MORRIS RAY SCALES, B.S, in C.E.
A THESIS
IN
CIVIL ENGINEERING
Submitted to the Graduate Faculty of Texas Technological College in Partial Fulfillment of the Requirements for
the Degree of
MASTER OF SCIENCE IN CIVIL ENGINEERING
Approved
Accepted
<9 Dfe' an of the Gracfuafiir Sc/iool
August, 1964
Ht Kl-Mt5.^{
yM L
5 19^
Acknowledgments
I am indebted to Professor Albert J, Sanger for his direction of
this report and to the other members of my committee, Professors
Keith R, Marmion and George A, Whetstone, for their helpful criticism.
This report is dedicated to my wife, Barbara, whose encouragement
and understanding made this work possible.
ii
, e s » s , o , o , o o a , , s 9 a o « » a ; ' 0
, 0 0 0 0 0 9 0 0 , o o e e a a a , 0 , 0
CONTENTS
Page
l i 1 ^ I yJr 1 / \ 0 1 J C 0 O O O O O O O O C I O O O O O O O O O O O C V C C i V
LIST OF FIGURES
I , INTRODUCTION,
Ila INFLUENCE COEFFICIENTS AND ENERGY THEOREMS, . , , = . 5
Influence Coefficients , , , , , * » , , , , o , 5
Ovram energy, o o o o a a o o o j a o o o i o , **'
V X r b U a J L TV o r i s . a a o o a o o a o a e a s a a e a
1 X X a rV /KV^C I M C i n v J L / a a a o a a o a a a o o o t a o a a a o e
I V , S T I F F N E S S M E T H O D . , 8 0 o a o o e o o e e « a o o e a * ^
I n t r o d u c t i o n a « a a o o a o a « o e a e o 9 g o 2 8
Pin-Ended Plane Frames , , , , , , , , a , , , , 29
Rigid-Jointed Plane Frames , , , o , , * , , . . 37
Three-Dimensional Analysis a , = , , . , , , , , 39
V 0 VJ vJINvrflilJO X vJINO 0 0 0 0 0 0 0 0 0 0 > c Q 3 0 f t o 5 o a o *^\)
LJJL^I \Jr i \ C 1 C t x C P I v i C d o o o o o o o o o o o o o o o o f t o o o o o ^ X
i\r tClHU l A o o o o o o o o Q o o o o o o o o o o Q o o o o o o o A ^
'V, Nomenclature , , , . o , a , , , . , , , , , , , , , 43
B, Fundamentals of Matrix Algebra, , a , , , , » , , , , 44
i i i
LIST OF TABLES
Table Page
I \7 * o o , , , a , o o o e , o o o o , o o o o o o c o v o o c * ' —
iv
LIST OF FIGURES
Figure Page
X, v sn Li jLever DO am , 0 0 , 0 0 9 0 0 0 0 0 0 0 0 0 0 , 0 0 ^
2 , Appl icat ion of Unit Loads, 0 0 0 0 0 0 0 , 0 0 0 0 0 0 7
3, Application of Unit Displacements, 0 , 0 , 0 0 = 0 0 , 9
4, Cantilever Beam Example of Force Method, , , , , , , 0 11
5, Propped Beam Example of Force Method , 0 0 0 , , , , , 20
60 Truss Example of Force Method, , , , , 0 , , , , , 0 , 23
7, Rigid Frame Example of Force Method, , , , , , , , , , 25
8a Typical Pin-Ended Truss Member , , , , , , , 0 , 0 , , 29
9o Truss Example of Stiffness Method, , , , , , , , , , , ^
10, Derivation of Transformation Matrix, , , , , , , , , , ^^
11, Typical Rigid Frame Member , , , , , , , , , , , , , , ^^
CHAPTER I
J INTRODUCTION
The development of new techniques of analysis to meet the in
creasing requirements of greater accuracy, and the revolutionary
development of the high speed electronic digital computer have
signaled a new era in structural analysis. The necessity for the
calculation of stresses and strains in complex elastic structures
has motivated the aeronautical industry to develop new and better
methods of analysis. Since the analysis of even a relatively
simple structure composed of linear elastic members undergoing
small deformations involves the solution of a number of simultaneous
linear equations in the static case and a corresponding number of
linear differential equations in the dynamic case, the advantages of
matrix formulation have been widely recognized. The purpose of this
paper is to present the basic principles governing the application of
matrix theory to structural analysis.
Matrix formulation provides a means of systematizing generalized
procedures for the machine computation of structural problems. This
is especially important in highly redundant structures where the use
of matrix language offers the following advantages in structural
analysis; (1) simplicity of presentation and calculation through
its concise mathematical language which is perfectly adaptable to the
electronic digital computer, and (2) division of labor into formula
tion and subsequent machine operation by someone who may be unfamiliar
with formulation procedures. This paper will deal only with the
formulation portion of matrix structural analysis.
Structural analysis consists of the computation of external
reactions, internal forces and accompanying stresses, strains, and
deflections of a structure. The two conditions which must be satisfied
in the analysis are: (1) the forces developed in the members must be
in equilibrium, and (2) the deformations of the members must be com
patible, or consistent with each other and with the boundary conditions
The forces and deflections must also be consistent with the stress-
strain relationship for the material used,
rThe analysis may be approached from two different points of view,
I-^ the^analysis of statically determinate structures, the external
reactions and internal forces and stresses are computed first. The
strains are then determined from the stresses, and subsequently the
deflections are computed. This same procedure may be followed in
analyzing indeterminate structures by a number of the classic methods
of analysis, such as the superposition-equation or dummy load method,
Castigliano*s second theorem, and the use of the three-moment equation.
In this approach, the forces acting on or in the members of the
structure are considered as unknown quantities. Since there are an
infinite number of such force systems satisfying the equations of
equilibrium, the correct one must also satisfy the conditions of
compatibility. Thus, the redundant forces and/or couples are first
computed by solving an equal number of simultaneous equations, each
of which expresses a known condition of the primary or statically
determinate cutback structure in terms of the redundants. This
approach is referred to as the force or flexibility method of
structural analysis. It is especially appropriate for structures
which are not highly redundant.
The other approach to the analysis of indeterminate structures
considers the displacements of the joints in the structure as unknown
quantities. Since an infinite number of systems of mutually compatible
deformations in the members are possible, the correct system is the
one for which the equations of eqilibrium are satisfied. Thus, the
key displacement components are expressed in terms of internal forces
and/or couples and substituted into the equations of equilibrium.
The solution of this simultaneous set of equations yields the values
of the displacements and the analysis is completed as if dealing with
a statically determinant structure. This approach is known as the
direct stiffness or displacement method of structural analysis and
has the advantage of being completely independent of the concept of
static determinateness or indeterminateness, It is the basis for
slope-deflection and the relaxation types of analysis, such as moment
distribution, (1)
The degree of redundancy determines the most appropriate method
of analysis for a particular structure. The basic principles in the
formulation of the flexibility and stiffness coefficient matrices
will be discussed in Chapter 11^ and example problems covering the
force and stiffness methods will be presented in Chapters III and IV,
respectively, A summary of matrix notation and algebra is presented
in the Appendix, The primary reference for this paper is Matrices
for Structural Analysis by Sydney John McMinn (2).
CHAPTER II
INFLUENCE COEFFICIENTS AND ENERGY THEOREMS
Influence Coefficients
If an elast ic spring deflects an amount A under the action of
an axial load W, and i f Hooke»s Law applies, the f l ex ib i l i ty f of
the spring is defined by the quotient,
f « ^ (2,1)
Thus,vthe f l ex ib i l i ty f is the displacement produced by a unit load
W»l, The deflection L may be writen as
A • f"W (2.2)
The reciprocal of the f l ex ib i l i ty is called the stiffness and is
defined by k « 1/f. The load is thus related to the displacement
by the equation
W . k»A (2,3)
where k i s the force required to produce a unit deflection.
The above relations can be extended to the two-dimensional body
by considering a load W. applied to a structure at point j and produc^
ing a directly proportional deflection Ai at point i . Thus Ai«fij*W.
where f i j i s a constant. Under the principle of superposition, the
action of a number of loads on the structure would result in the
equation Ai« f i i W • fi2'W2 • fi3''W3 • <""> * fin'Wn (2,4)
where the deflection Ai i s in the same direction as the load Wi,
This same procedure for displacements at other points or at i in
another direction will result in a set of simultaneous equations.
In matrix notation, these equations which replace equations (2.2)
and (2o3), respectively, may be written
A » F^W, and (2.5)
W « F"^»A « K-A (2.6)
The flexibility influence matrix F has as its general element f|^,
which is the displacement produced at point i, in the direction of
Wi, by a unit load at j, in the direction of Wj. The stiffness in
fluence matrix K has as its general element kij, which is the force
required at i, in the direction of Wi, to produce a unit displacement
at j in the direction of Wj. The elements must obey Maxwell's Law
of Reciprocity, resulting in symmetic influence coefficient matrices
as long as they are referred to orthogonal coordinate systems=
These influence coefficient matrices derive their names from
theii similarity to sets of influence lines. In order to obtain the
deflection Ai, the products of the applied loads times the correspond
ing ordinates of the influence line are summed. Here, the elements
of the ith row represent the ordinates of the influence line for Aj
at the individual load points
9* I
Figure 1, Cantilever beam (2,p,58)
The derivation of the influence coefficient matrices for the
simple cantilever of Figure 1 is presented as an example. The
canti lever has uniform cross-section and loads applied as shown To
obtain the f l ex ib i l i t y matrix, equation (2,5) becomes
P2 V2 -62
5ll ^21 ^31
! l 2 *22 '32
fl3 ^23
^33
:x2 •v2 (2,7) M; '2
where the subscripts of the loads and displacements denote the node
numbers. Unit loads are now applied to node 2 as shown in Figure 2,
and deformations due to shear are considered negligible.
Q)
(3)
1
Figure 2, Application of Unit Loads (2,po59)
The displacements are computed using the M/g^ diagrams of Figure 2
and the moment-area method.
For unit P2 applied alone.
U, * AaE ? V2 • 0 » ©2
8
For unit S2 applied alone.
u.
^ - 1»L 2 " E«I
62 «ilL '• E,I
For unit M2 applied alone.
k 2 L I
2L * L" 3 3EoI
. , L2
2E'I
u.
V2 •
® 2 - F T T
L
L
L 1
a „ L ZE«I
L
The signs are recorded according to pos i t ive and negative deflec-
t ion re la t ive to the coordinate system, and clockwise f») or counter
clockwise (••') rotation in order to conform with normal pos i t ive shear
and bending moment conventions. Tension on a member i s considered
p o s i t i v e . From these elements, the f l e x i b i l i t y influence matrix i s
P2 S2 M2
u-L3
SToT 2EcI -L2 J ^
IfoT E I_ The underlined terms are for guidance purposes only.
0.
L AoE 0
0 0
=L2 (2,8)
To obtain the stiffness matrix, unit displacements are applied
singly as indicated in Figure 3,
r ®miHi Rt
ra)
Figure 3 . Application of Unit Displacements (2,p,60)
Putting U2 « 1 with V2 « ©2 ' "
A'-E'l , Sj • 0 , and M2 « 0
For V2 • 1 with U2 • ©2 " ^
Fx
2E«I V2 Ml • M2 « i | - - (202 • ©1 -3 j ^ ) »
S2 « (M2+Mi)/L - 12E»I
6E'I
For ©2 « 1 with U2 • V2 • 0?
M, = 2Ea (^2) . .^i-L 2 L L
M « 2E-I ,.,«. . 2E»I
Sj • (Ml •Mj) /L . ^
Pj = 0
10
Thus, the stiffness matrix is
K =
AoE L
0
0
12E°I
6E'I
11 0
6E'I ^ ^
4Ea L J
(2.9)
Obviously, the calculation of the flexibility matrix will become
prohibitive as the structure increases in size and complexity. It
will therefore be beneficial to offer an alternate procedure based
upon the principles of strain energy and its accompanying theorems.
This approach will also offer increased versatility through the easy
calculation of stresses and moments in addition to deflections. The
following sections contain derivations available in many references,
but offered here in matrix notation in order to facilitate an under
standing of the examples to follow, (2)
Strain Energy
Since we are dealing with elastic structures which obey Hooke's
Law, both strain energy and complementary energy are expressed in
terms of the loads and deflections by the same formula
In matrix notation
V » I E Wi Ai
V « i W* A, 2 .. -, (2.10)
where W and A are column matrices. Using equations (2c5) and (2.6),
V = i w* 1 F«W = J A*(F-^)*»F»K'A = * K'A, (2,11)
11
with F and K being symmetrical.
The strain energy for a member can also be expressed in terms of
the internal stresses o and strains e as
V • Z O'E'dxcdy^dz,
or for the whole structure,
V = i///ECToEcdx^dycdz
Preferring to work with large portions of the structure and
utilize superposition, we will refer to the cantilever shown in
Figure 4 and divide it into the lengths AB, BC and CD, The forces
acting on the elements of the stmcture are bending moment, shear
forces, and axial forces, which will be designated stress resultants.
The strain energy for the cantilever will be calculated as though
each length were a simple cantilever
I" I We iwc I
*1 ftM cU P i S4 B "^ -^c ^^Mo
/EOtns. fiOir^J^aOm >md
Figure 4, Cantilever Beam Example of Force Method (2, po61)
Let the column matrix S ^ represent the stress resultants
acting on member AB and let the column matrix © g represent the
corresponding displacements of the end B relative to end A. There
fore, the strain energy in AB is
V.„ « i S.o* AB ' J ^AB • ©AB (2.12)
Since 0^g - ^AB'^AB *"** ^AB'^AB'^AB* ^« ?«*
^AB = K B ' ^ A B ' S ^ B " 7 ®AB'^AB'©AB
The total internal strain energy is therefore
or in matrix notation
V « |s*o0.
where
' ^ A B '
^BC
^CD B 1
, and 0
e AB
01 BC
0 CD
in the form of partitioned matrices. Since
0 • f"S,
the diagonal partitioned matrix f is
0
0
^AB 0
BC
0 CD
Then the total strain energy for the structure is
V a i S*»f^S « l0*cko0.
where k is the inverse of f,
k » f"^ «
\i' ' 0
0 0
0
0
-1 'CD
AB
0
0
0
BC
0
0
CD
12
(2ol3)
(2J4)
(2,15)
(2,16)
(2,17)
(2,18)
These matrices f and k are the flexibility and stiffness matrices of
the unassembled structure,
13
We will neglect shear and axial deflections in this example,
with bending energy being the only strain energy considered. Then,
from (2.8) the f l ex ib i l i ty equation for each length is
0
3E^I
-L'
IE'I (2.19)
LM. 2E»I E°I
where S and M are the shear force and bending moment at the right=
hand end. It is fundamental to matrix analysis that structures
have loads applied at nodes only. The procedure to be followed in
the event of intermediately applied loads will be discussed subsequently
Using (2,19) to form the flexibility matrix f of (2,16) and
nuttinp in numerical values results in
5,760 -72 0
.10^ £oI
72
0
1.2 0
720 -18
0
0
0
0
0
0
0
0
18
0
0
0,6 0
0 720
0 -18
18
0,6
(2,20)
The strain energy due to external or internal conditions is the
same, or
W*. A = S*^ 0 (2.21)
The internal stress resultants are produced by the applied loads and
are related to them by a set of simultaneous equations
S = B«W (2,22)
14
For the example problem,
(2-23) Sg « Wg Wc Wijj
Mg » -60 Wc-120Wjj*Wg*M +V ;
Sc " Wc Wpj
Mc « -60Wp+lCf *Wo;
SD • %'^
MD - ^ ^
where M i s the bending moment at B, and F? i s the externally applied
moment at B, Placing equations (2,23) in matrix form (2.22) gives
hi MB
S^
M C
D
L"D.
1 -60 1 -120
o l
1
0
1
0
1
(2.24)
A W W
From equation (2^22) we have S * W "B . Substi tution of th is into
equation (2o21) results in
W*°A = W*''B*'0
Then
A = B ' 0
(2o2S)
(2.26)
The relationship between equations (2,22) and (2^26) is important,(2)
Virtual Work
If a structure which is in equilibrium under a system of external
loads is given a virtual deformation by the application of additional
15
loading, then the strain energy becomes
Expanding,
V+6V « i(W4.«W)* (A*6A ) (2.27) 2
V+5V • i[W*»A*6W*<.A-»'W*«6A*6W**6Al 2
Disregarding the higher order terms and subtracting V«iw*''A
results in
6V - i[6W*»A* W*'6A] (2,28)
Since 6W 'A* 6W •F-'W, and using the rule that the transpose of the
product equals the product of the transposes in reverse order, we have
6W*»F«W « [(FoW)*''6W]* « [W*-F*''6W]*
« [W*»F*6W]*, (2.29)
where F is symmetric. Prom A«F»W follows 6A« F-'fiW and
6W*»F''W • [W*''6A]* or
6W*»A « [W »6A] (2,30)
Since V is a scalar quantity, both 6W*»A and W*«6A are also
scalar quantities and equal to their own transposes. Therefore, from
equations (2,28) and (2,30),
6V « W*''6A (2.31)
5V « 6W -A (2.32)
Similarly, i t can be found that
6V « S*'60, and (2.33)
(5V « 6S* 0 (2.34)
Upon combining
6W*<'A - 6S*'0 (2.35)
16
The load term W includes the redundants as well as the applied
loads in the case of indeterminate stxuctures. However, in equation
(2.35), ^ is selected so that the redundants do not change. Thus,
the structure is reduced to a statically determinate structure by
disregarding the redundants. Under the principle of superposition,
the effects of the redundants are subsequently added. Therefore,
6W of equation (2,35) is applied to a statically determinate system
resulting in the relationship for 6S of •••(• T • ., ; ' L '
63 • B « 6W (2.36)
In the whole.structure, we have seen that
S - B«W
In matrix notation 6S includes as many elements as S, but those
corresponding to the redundant members are zero. Similarly, B^ is
the same size as B, but contains rows of zeros corresponding to the
zeros of 6S, But equation (2,35) can be written as
6W*»F"W - 6S*»f»S - 6W*'>Bj<.f<.B.W
from equations (2,36) and (2,22), Therefore, for an arbitrary 6W,
F » B*»f»B, or (2.37) < f •
F - B*-f-B (2.38) J ' - • :
where BQ applies to the statically determinate cutback structure of
an indeterminate system, and B applies to the determinate system.
In the example cantilever, matrix B is as indicated in equation
(2,24) and f in equation (2.20), Therefore, from equation (2,38),
the product f,B is
10 2 FT
5760
-72
0
0
0
0
72
1,2
0
0
0
080
144
720
-18
0
0
-72
1,2
-18
0,6
0
0
14,400
-216
1800
-54
720
= 18
.72
l c 2
18
0,6
18
0.6
17
(2,39)
and then F » B of .B or
ri2i E o i
576
-7,2
»7,2 1,008 -7.2 1,440
0,12 -14,4 Ool2 -21.6
1,008 -14,4 1,944 -16,2 2,916
-7,2 0,12 -16,2 0,18 -27
1,440 -21,6 2,916 -27 4,608
L -7,2 0,12 -16,2 0,18 -28,8
=7.2
0.12
-16,2
0.18
-28.8
0.24
(2.40)
The f l ex ib i l i ty matrix could have been calculated directly but
without the inherent simplicity and versat i l i ty afforded by the use
of strain energy and virtual work.
Since 6S « Bo«6W(2,36) and 6W*»A-6S*»0(2,35), then
6W*oA « 6W*«Bo*»0 , and (2.41)
A - Bo*-0 (2,42)
Obviously, the displacement Ai i s equal to the product of the ith
row of BQ and the column matrix 0, The ith row of B© , which is
the ith column of BQ, is the l i s t of stress resultants produced in
the s tat ica l ly determinate basic structure by a unit load at W.(2)
CHAPTER III
FORCE METHOD
The force method of analysis of statically indeterminate struc
tures is simply the adaptation of the dummy load-superposition method
of analysis familiar to all structural engineers to matrix notation
and subsequent computer solution. The flexibility matrix, F, of the
whole structure is obtained from the flexibility matrix, f, of the
unassembled structure through the formula F • B^ "f^B, equation (2,38),
The stress resultants are determined from S • B<>W, ectuation (2,22),
However, in the case of indeterminate structures, the matrix B can
not be calculated directly, so the structure is reduced to a
statically determinate structure by removing selected redundants.
Thus S • Bj,»W*BioR (3,1)
where S is the column matrix of the stress resultants, W is the
column matrix of the loads, and R is the column matrix of the re
dundants. The matrix B^ is derived from the statically determinate
structure, and B^ is derived from primary structures where only
dummy forces are acting at the point of application of the redundants.
If Si represents the stress-resultants resulting from the
redundants acting alone on the structure, then
Si • B^oR (3,2)
From equation (2,26), the displacements of the points of
application of the redundants are given by
Ar • Bi*»©,
18
19
where 0 is the column,of actual distortions. If we consider L^" 0,
where there i s no sett l ing of the supports, e t c , then
Ar - 0 « Bi*of,s (3,3)
Then, from equation (3 ,1 ) ,
0 « Bi*°f'[Bjj»W*BioR] or
Bj*«f<>BQ»W • - B *'f°Bi»R
Let
Thus
Dp - Bi*''foBo, and (3,4)
Dj • B2*»f«Bi
Djj'W - - Dj'R, or (3,5)
R • - Dj' oDooW (3,6)
Therefore, from equation (3 ,1 ) ,
S - BQOW - Bi»Di"^oDQ«W, and (3.7)
S - [Bo-BjoDf ^oDo] W, or
S « BoW
where
B « [Bo-Bi-Pi'l-Do] (3,8)
Uti l iz ing the equations above, any structure can be fully
analyzed as the following examples wil l indicate.
Example Problem 3 .1 .
The cantilever with the redundant at node 3 in Figure 5 is
analyzed. Shear deformation will be ignored.
20
^?\ m lO" " /30K" Ji5<"
"•4*°
Figure 5, Propped Beam Example of Force Method (2,p.77)
Let S - col(S2,M2,S3,M3,S4,M4) and W - col iyi2$^2»^S»^Af>^i^ ^
The applicable equations linking S and W are:
S2 .
M2 '
S3 '
Ms •
S4 .
M4 •
• " 2 * V 3 !
• -2loyl/^•^2•^i*^^* ^^o R3;
• V3: • . 60W4- f f3*R4j
. W 4 i
• " 4 .
Thus, the matrix B^ is derived by applying unit loads to the cutback
structure at the points of application of external loads in the
appropriate sense. The matrix Bj, is derived by applying unit forces
individually to the primary structure upon which only the dummy load
is acting.
(3.9)
B<
£1 M2
S3
! ^
£i M^
W2
" 1
0
0
0
0
0
A 0
-1
0
0
0
0
J5i 0
-1
0
-1
0
0
^
1
-210
1
-60
1
0
h 0
1
0
1
0
1
21
Bi - c o l ( - l , 150, - 1 , 0, 0, 0 )
The basic f l ex ib i l i ty matrix for each length, from equation (2 ,8 ) , is
0
' L3
L2 I FT M
Thus the flexibility matrix, f, of the unassembled structure is
576 -7,2 0 0 0 0
f • 10^ E«I
-7,2 0,12
1125 -11,25
-11,25 0,15
72
-1,8
-1,8
0,06 0 0 0 0 -J
Matrix multiplication leads to
Bi**f « 10"^ (-1656, 25,2, -1125, 11,25, 0, 0 )
Di « B,*»f«B2 • ,6561
D
1 - "1
D.
1.524158
.-4
Bj=Dj-^»Do«10-2
Bj*-f«BQ - 10"^ (-1656, -25,2, -36,45, -8748, 36,45)
DJ"^«DQ - 10"^ (-2524, -38,4, -55.56, -13,333.3, 55,56)
25.2401 0.3841 0,5556 133.3333 -0.5556
-3786.0090 -57.6135 -83.3340 -20,000.0010 83,3340
25,2401 0.3841 0,5556 133.3333 -0,5556
0 0 0 0 0
0
0
0
0
0
0
0
0
0
0
22
B-[Bo-Bi 'Dr l .D^]
0,7475 -0,0038 -0,0056 -0.3333 0.0056
37,8601 -0.4239 -0,1667 -10.0000 0,1667
i ' . . !• r,
foB • 10 -4
-0,2524
0
0
0
158.025
-0,840
283,951
2,840
0
L 0
-0.0038 -0,0056 -0.3333 0.0056
-60.0000 1.0000
1.0000 0
0,840
-0.023
-4,321
0,043
0
1.0000.
=2,000 -120,000 2,000
0,020 1,200 -0,020
5c000 299,999 -5,000
-0,087 -5.250 .0,087
72,000 - 1 . 8
-1,800 0,060.
158.025 0.840 -2,000 -120,000
F«BQ*'fB • 10"^
.
0,840 0.023 -0.020
-2,000 -0,020 0,067
120,000 -1,200 4,050
2,000 0,020 -0.067
The loads are expressed as
W - col ( 7 , 10, 30, 6 , 15 )
Therefore, from equation ( 3 , 6 ) ,
Rj • - D J - I ' D Q ' W • 9,89^
From equation (2,22)
S - B»W
-1,200
4,050
314.998
-5.850
2.000
-0.020
-0.067
-5.850
0.127
S - col ( 3 , 1 1 , 198,282, - 3 . 8 8 9 , -375 , 6 , 15 )
23
From equation (2,5),
F,W
A « col (,0365, ,0002, ,0011, .1072, .0021 )
Example Problem 3,2,
The truss, illustrated in Figure 6, is indeterminate to the
second degree. The members selected as redundants are 2-4 and 3-6
E'A*ccnsfenf
Figure 6. Truss Example of Force Method
The flexibility matrix for each member is
AoE so the flexibility matrix, f, for the unassembled stxucture is
1-2 1-4 1-3 2-4 2-3 4-3 4-6 4-5 3-5 5-6 3-6
A'E
144 0 0 0 0 0 0 0 0 0 0
0 192 0 0 0 0 0 0 0 0 0
0 0
240 0 0 0 0 0 0 0 0
0 0 0
240 0 0 0 0 0 0 0
0 0 0 0
192 0 0 0 0 0 0
0 0 0 0 0
144 0 0 0 0 0
0 0 0 0 0 0
192 0 0 0 0
0 0 0 0 0 0 0
240 0 0 0
0 0 0 0 0 0 0 0
192 0 0
0 0 0 0 0 0 0 0 0
144 0
0 0 0 0 0 0 0 0 0 0
240
24
Since there are only two external loads, B^ and Bj are given by
W w.
1-2 0 0
Bo «
1-4
1-3
2-4
2-3
4-3 -.1.
4-6
4-5
3-5
5-6
3-6
0,667
-0,833
0
0
0,500
0
0,833
-0,667
-0,500
0
0.500
0.625
0
- l o O O O
-0,375
0
0,625
-0,500
-0,375
0
B, «
2-4
-0 .600
-0c800
1,000
1.000
=0,800
-0 ,600
0
3-6
0
0
0
0
0
-0 ,600
-0 .800
0
J Lo
1.000
-0,800
-0,600
1.000J
Using matrix multiplication and the applicable equations,
* 1 r-86. ^1 'f- F T L 0
4 -153,6 240 240 -153,6 -86,^ 0 0 0 0 0 0 0 0 -86,4 -153.6 240 -153.6 -86.4 24 a
n * r n 1 r 829,44 51,84 Dj . Bj 'f .Bj • /[Tf L 51.84 829.44.
Dj"^ - A»E 0,001210 -0,0000761
-0,000076 0,001210J
The procedure from this point is identical with that of the
previous example problem, and the structure wil l be completely
analyzed.
25
Example Problem 3,3
The rigid frame indicated in Figure 7 is analyzed by the same
procedures. However, the computations will include the effect of
axial forces and deformations. The cutback structure is formed by
releasing the fixed reaction at node 5, and thus reducing the third
to H
1*< ®
30o1
T J @
A E - cor^sfartf
• — * r v s
Figure 7. Rigid Frame Example of Force Method
degree indeterminate structure to a statically determinate one. The
structure can now be treated as a cantilever fixed at node 1. Unit
loads are applied in place of the externally applied loads to obtain
BQ and unit M, V, and H at node 5 to obtain B^. The sign convention
is the same one mentioned in Chapter II, Therefore,
26
B
1-2
2-3
3-4
4-5
{ {
P 5"
f
M
0 1 0 0 0 0 0 0 0 0 0 0
w,
-1 0
240 0 1 0 0 0 0 0 0 0
ili h
B
L
0 1
-300 -1
0 -300
-1 0
-300 0 1 0
1 0
420 0
-1 280
0 -1
0 -1
0 0
Mt
0 0 1 0 0 1 0 0 1 0 0 1
The f matrix for each element or span of the frame, from equation
(2,8), is
f «
L A»E
0
0 0 L3 L2
3FT "21^1
The f matrix for the complete unassembled structure is
f «
^1-2 0
0
0
0
f
0
0
0 0
2-3 0 0
^3-4 0
0 f 4-5
The procedure from this point of the analysis is identical with that
of example problem 1. This solution will give results which differ
from those obtained by moment distribution due to consideration of
axial deformation. This variance can be quite significant
If a pinned reaction were present at node 5 in place of the
fixed one, BQ and B^ would be reduced by the row that corresponds to
27
the moment at node 5, Likewise, the M column of matrix Bj wculd
not ex i s t , and the f matrix of the element 4»5 wil l be reduced by
the row and column corresponding to the moment at node 5 which no
longer ex i s t s , (2)
CHAPTER IV
STIFFNESS METHOD
Introduction
In the direct stiffness method of analysis, the displacements of
the joints in a structure are considered to be the unknown quantities.
The procedure to be followed in the analysis is similar to that of the
Force Method with the complex structure replaced by an equivalent
idealized structure consisting of basic structural parts that are
connected to each other at selected node points. Stiffness matrices
are needed for each basic structural unit appearing in the idealized
structure. While all other nodes are held fixed, a given node is
disnlaced in a particular coordinate direction, The forces required
for this displacement and the resulting reactions generated at connected
nodes are determined by the stiffness matrix of each individual member.
These forces and reactions thus form one column of the composite stiff
ness matrix. Repeating this process until all possible freedoms at a
node have been considered will result in the complete development of
the stiffness matrix. In the general case, this matrix will be of
order 3n by 3n, where n equals the number of nodes. This singular
matrix so developed will be rendered nonsingular by imposing the
desired support conditions and striking out columns and corresponding
rows for which zero displacements have been specified. With given
external forces at the nodes, matrix calculations applied to the
stiffness matrix will yield all components of node displacement plus
28
29
external reactions. Internal forces can then be found by applying
the appropriate force^deflection relations. The sign convention used
in the Stiffness Method will disagree somewhat with those of the Force
Method of the previous chapter, since the sign convention of the
Stiffness Method agrees with the usual positive directions of the
coordinate system and clockwise rotation will be taken as posit ive,
Pin-Ended Plane Truss
Figure 8 shows a typical pin-ended truss member. In a pin-ended
truss, the only forces that can be applied to a member are equal and
opposite axial forces acting at the nodes. Each node has two possible
degrees of freedoms, translations in the x and y directions. There
fore, prior to introduction of boundary or support conditions, the
st i f fness matrix of this member will be of order 4 by 4,
£>„^iu->a'
cos Ousya
I L^'^^ (b)
Figure 8, Typical Pin-Ended Truss Member
30
To develop one column of the member K, subject the member to
"2 ^ 0,u,«v,sv2«0, as indicated in Figure 8 (a) . Then
AL = U2eosajj « U2X ,
where the change in the angle is neglected since i t is a virtual dis
placement. The axial force necessary to produce AL is
Therefore, the components of P at node 2 are
Px2 - Pc°s ©x - ( ^ ) U 2 ^ ^
V2 p cos e y -{^|U2^^ Under the conditions of equilibrium, the forces at node 1 are
^xi • -Px2
Pyl " -Py2
This gives, from equation (2,6),
F, xl
>1
x2
_Py2_
AE r
L -Xy
X2
Xy
"1
^1
u^
JL^^J To develop another column, subject the member to V2 f* 0,Ui«U2«vi"0,
as indicated in Figure 8 (b), Then
AL • V2 cos © • V2P
P . AE ^ IT
Fx2» P cos ©x« _
Fy2" P COS ©y" —-
Therefore ^^ AL - V2y
A^ V2yX
P cos ©v" i^ V2p2
Pxl" -Px2» and Fy^ - -F y2
31
Repeating this process gives
K = AE L
"1
X2
Xy
-X^
= Xy
l l Xy
2 y
-Xy
. y 2
^
-x2
-Xy
X2
Xy
IL -Xy
2
- y
Xy 2 y
(4,1)
This matrix is singular, since its determinant vanishes and its in
verse does not exist, but imposing boundary conditions will render it
nonsingular. Thus, boundary conditions will prevent the member from
moving as a rigid body. The stiffness matrix of the truss indicated
in Figure 9 will be developed. The length of the members, since
variable, will remain inside the brackets, giving X«X ,i/«y , etc.
The data for the truss is as indicated
AE conshan+
Figure 9. Truss Example of Stiffness Method
in Table 1
32
Member
1-2
1-3
2-3
2-4
3-4
X
0
L
L
L
0
y
L
L
0
=L
-L
Length
L
Vzl
L
|2L
L
TABLE 1
X
0
l / f 2
1
1/P 0
y
1
i / R
0
X2
0
1/2
1
- l / f 2 1/2
-1 0
2 y
1
1/2
0
/ l / 2
1
Xy
0
1/2
0
-1 /2
0
P 0
1/2 KJL
1/L
1/2f2L
0
y
l/L
1/2(2L
0
fl /2f2L
1/L
XIT
0
1/2y2L
0
- l /2 ) f2L
0
AE
1 The composite singular matrix for the structure is (4,2)
l /2 f iL l/2ldL 0 0 -l/2|f2L -l/2)f5L 0 0
l/2lf2L l/L*l/2lf2L 0 -1/L ^l/2flL -l/2!f2L 0 0
0 0 l/L+l/2lf2L -l/2y2L -1/L 0 .1/2K2L l/2f2L
0 -1/L .1/2/21 l/L.l/2f2L 0 0 l/2lf2L l/2|flL
0 l/L*l/2f2L 1/2 RL 0 0
0 1/2RL l/2+l/2(2L 0 -1/L
l/2lf2L 0 0 l/2f2L -l/2ir2L
l/2f2L 0 -1/L -l/2)f2L 1/L-1/2(2L
Imposing the boundary conditions u^-vj « u -v ^O by striking out their
rows and corresponding columns gives the nonsingular composite s t i f f -
-l/2ir5L -1/212L -1/L
-l/2f2L -l/2r2L 0
0 0 -l/2y2L
0 0 1/2|(2L
ness matrix (4,3^
Px2
Fy2
''x3
L'y3j
AE IT
1 1+IT7
1 -7T7
- 1
0
1 -Tf7
1 l-TTT
0
0
- 1
0 1
i*ir? 1
Tf?
0
0 1
IT? 1
i*ifr
"2
^2
u. 3
L^3 J
33
Inverting this 4 by 4 matrix, and using equation (2.5) will give the
values of the displacements at nodes 1 and 2, Considering the
partitioned matrix (4,2), it can be represented as
2^2. D
B 2x4 2x2
4x2 4x4 H
4x2
2x2 I 2x4 2x2
Expanding equation (4,3) gives the following additional sets of
equations?
(4,4) "F^ "" XI
L^ij • [B]
"u," V "3 V3
F X4
L^J - [H]
Uo 2
^2 "3
(4.5)
Equations (4,4) and (4,5) will give the reactions after the joint
deflections at nodes 2 and 3 have been solved. In the matrix (4.2),
the 2 by 2 submatrices along the diagonal represent self terms, or
deflections at a node due to two forces. The elements along the
diagonal are known as self-self terms and are displacements at a node
due to forces at the same node in the same direction. The submatrices
off the diagonal are connecting terms between two nodes. The self
terms in matrix (4,2) consisted of the sum of the matrix self terms
for all members entering the particular node. The connecting terms
are simply the connecting terms for the particular member between the
two nodes in question. The terms all come fro» the basic member
34
matrix (4,1) with the appropriate sign. (1)
Another approach offered is a consideration of the basic stiff
ness matrix (2.9) derived for the cantilever in Chapter II. This
matrix was derived based on each load being applied as an axial force
and a shear or transverse force. In order to bring all joint displace
ments into the same reference system, a rotation or transformation
matrix T is utilized. If we consider the member 1-2 in
p. /\—4 (a) (b)
Figure 10, Derivation of Transformation Matrix (2,p.112)
Figure 10 (a ) , and displace a joint 6^ only, with 6y«0, then the
displacement in the P direction is 6x°cosa and in the S direction
= 6x'sin a. Repeating with 6yf«0, and 6jj"0» *"^ adding the two
sets gives
6p « 6«ocos o ••• 6v''Sin a
fie ~ -6v»sin a ••• fiv'cosa
35
In matrix notation this is
cos a sin a
-sina cos a
Calling the coefficient matrix T, where 0j»T»Aj, and replacing its
elements with their equivalent from Figure 10 (a) gives
6 . y .
X r Y
T
Y r X r
(4.6)
When dealing with a pin-ended truss member, the shear forces are zero
and the axial forces are in equilibrium. The stress resultants are
12 • • 1
Si '21 • . e i
fip 1
>Si , and ©2
With node 2 fixed, Sj2*l^i2*®i» where
12
E«A
0
(4.7)
Repeating with node 1 fixed gives S22"ki2'©2'' '^^ reaction produced
at node 1 with this translation is Si2«-S2i"-J^i2°®2» Superposing
these gives
Si2 "1^12 (®i-62) • "^21-
Utilizing T, the components of the applied forces Sj2 in any direction
are related to the components Sj2* in the x and y directions by
Si2 « TcSj2'
The transformation matrix T is orthogonal, or T'^ * T'
36
Therefore
» - 1
^12 -"T 12
= T-i°k.(02-02)
= T-l-kj2 T(Ai-A2)
The joints are in equilibrium, so
Wi-J Sj2' • 0
Therefore
W = Z T-^°kj2°'^(^r^2^ ^^"^^
= (E T-lckoT)Ai-J:(T-i»k°T«A2)
where the summations are taken over all joints 2 which are directly
connected to node 1.
Wi);h equations (3,6) and (3.7)
(T-lck«T)j2 = H ^
V Y Since ~ • cos a and — » sin a,
L L (T-l»k»T) E o A
L
X^
L 2
XY
J7
'7 Xy
XY 1 7
Y2
Xy
"2 y
12
with the notation previously used. This will be recognized as the
individual self-self term contained in matrix (4.1), Thus the member
stiffness matrix can be derived as shown in Chapter II and rotated
to its proper orientation relative to the x and y axes by
M - [ "'] hp] b] C2)
37
Rigid-Jointed Plane Frames
The member 1-2 indicated in Figure 11 has three possible
degrees of freedom at each end.
Figure 11, Typical Rigid Frame Member
With node 2 fixed, slope deflection gives the following relationships:
M 12
M
2EI L
2EI 21
. . ,6Si 4EI01 6El5s (201-3-^) - -=-1; 1 ^
' - 6Elgsi L
(Mi2*M2l) . 6Eiei 12EI«sl
F'A'fipl L
In matrix form, this is
p.
Ml
E'A L
0
0
0 0
12E'I -6E.I
•^i? [7 -6E«I 4E'I
°P1
«si
©1
or Si"ki2*©i« The reactions produced by these displacements at
node 2 are related to St2 by
38
S2 - -Sj
M2 s -Mj-Sj^L
or in matrix form.
• P 2 "
^2
. " 2 .
B
.1 0
0 -1
0 =L
0
0
• p l "
Si
_ " l . If A is f ixed.
^2
^2
_ " 2 .
S
E,A L
0
0
0 0
12E-I 6E »I
U ^ L2 6E»I 4Ea
T? L
«P2
«S2
3. The reactions at node 1 are
a 0
0 -1
0 L
"Pl"
h
_"l_
s
0
0
1
" P 2 "
^2
3. As with the truss members, the displacements in the x and y
directions are related to the displacements in the P and S directions
by the transformation matrix
cos ot
- s i r a
sm a
cos a
0 0
39
L
Y L
Y
r
L
0
0
Using the transformation matrix and the proper member K matrix, the
absolute s t i f fness matrix can be obtained, and then the structure
may be analyzed as shown in the pin-ended truss example, (2)
Three-Dimensional Analysis
The procedure followed in the case of three-dimensional s t ructures
i s but an expansion in scope and size of the appropriate two-dimensional
case by the addition of the th i rd dimension. I t i s obvious that the
composite s t i f fness matrix wil l become quite large and the analysis
more complex
CHAPTER V
CONCLUSIONS
The complexity of modern structures is increasing the use of
computers for analysis which was formerly prdiibitive without the
use of simplifying assumptions which were self-defeating and
uneconomical.
The choice of which method of analysis to use is complex. The
force method requires the inversion of a matrix of the order of n by
n, where n is the number of redundants. The stiffness method requires
the inversion of an n by n matrix, where n is the number of possible
freedoms of the joints. The complexity of aircraft structures has
resulted in the aeronautical industry advocating the use of the direct
stiffness method, (1) Therefore, it appears that the redundancy of
the structure will determine the method of analysis to be used.
Because higher orders of redundancy should prove to be most economi
cally feasible for computer use, and matrix notation is compatible
to computer use only, the direct stiffness method of analysis should
offer the most promising approach to future development and study.
40
LIST OF REFERENCES
1, Turner, M. J., et al. "Stiffness and Deflection Analysis of Complex Structures", Journal of the Aeronautical Sciences, Vol. 25, No. 9, September, 1956, p. 805.
2, McMinn, Sidney John, Matrices for Structural Analysis, London, E, § F. No Spon, Ltd., 1962. — = = ===«,.==
3, Pipes, Louis A. Matrix Methods for Engineering, Prentice-Hall, Inc., Englewood ciitts, N. J., 1963. '^
4, Morice, P, B, Linear Structural Analysis, The Ronald Press Company, New YorkT*^
41
APPENDIX
A, Nomenclature
B, Fundamentals of Matrix Algebra
42
43
APPENDIX A: NOMENCLATURE
A = cross sectional area of member
B - coefficient matrix for complete structure
BQ - coefficient matrix for primary structure
Bj - coefficient matrix for reactions
E - modulus of elasticity
f - flexibility matrix for unassembled structure
F - flexibility influence matrix
Fy. - force at node 1 in the y direction
I - moment of inertia of member
k - stiffness matrix of a member
K - stiffness influence matrix
M - moment
P - axially applied load
R - reaction
S - transverse load
S,2 - internal stress resultants
T - transformation matrix
u - translation in the x direction
V - translation in the y direction
V - strain energy
W - load column matrix
A - deflection column matrix
0 - rotation angle
X - angle between x - axis and member
p - angle between y - axis and member
44
APPENDIX B: FUNDAMENTALS OF MATRIX ALGEBRA
I, Definition of a Matrix
A matrix is a rectangular array of elements defined by
certain rules of operation that prescribe the manner in which
these arrays are to be manipulated, A matrix of order n by s has
n rows and s columns and is denoted in the following manner:
A « [a] «
a n ^12
*21 ^22
a
a
13
23
^15
^25
/ n l *n2 an3 • ^ ^5
The elements a.^^ of the matrix [a] may be real or complex
numbers. The general element a^^ is that element in the ith row
and jth column where i » 1, 2,°"'», n and j e l , 2, 3,'°°°s, A
square matrix is one where n • s, or the number of rows equals the
number of columns, A matrix must be square in order to form a
determinant from its elements. The matrix with an asterisk, [aj*,
is obtained by a complete interchange of the rows and columns, so
that a^j - a^^, and is called the transposed matrix of [a], A
symmetric matrix is a square matrix whose elements are symmetric
about its principal diagonal so that a j « a... It is obvious
that this implies [a] * [a] . A matrix may be a column matrix,
designated by A * col.(a2j[, a22» ^is) » or a row matrix, designated
as A s (an, ai2» ^i$) >
45
II, Matrix Algebra
Matrices are equal if and only if their corresponding
elements are identical. If two matrices are of the same order,
then their sum is c^j « a^j -»• b^^ and their difference is
Cij » a.j -b...
The product of a scalar term and a matrix is effected by
multiplying each element of [a] by the scalar and obtaining a
new matrix whose elements are ka^j. The product of a matrix [a]
of order (n,s) by a matrix [b] of order (s,m) is defined to be
the matrix [c] given by
^ij • ^.1 ^ij * ^Jk
where the product is [a] [b] • [c], The number of columns of [a]
must equal the number of rows of [b] for multiplication to be
possible, and the matrices are then said to be conformable*
For example, if
i-s Z &ii • b,
Then C «
2
3
2
3
1
4
r 4
, and B «
~ 6 "
3
6
3
"2x6+
3x 6*
Obviously, the order of [cj is 2 by 1, or the number of rows of
[ a ] , the pre-mul t ip l ie r , by the number of columns of [b] , the
pos t -mul t ip l ie r . Matrix mult ipl icat ion is not normally commuta
t i v e , since [a] [bji^fb] [ a ] , except in special cases.
46
The determinant of a square matrix is equal to the determinant
of its elements. The cofactor matrix of any square matrix [a] with
n rows and columns, denoted by cof A, is the matrix obtained by
replacing each element of A by its cofactor. The cof actor of [a]
is the product of the determinant of the matrix with n-1 rows and
columns of [a], by (-l)i^J.
The inverse of the matrix A, denoted by A"^ is a matrix
such that
AA"^ « I
where I is the identity matrix. A"^ is given by
1. (cof A)* '^ det A
where (cof A)* is called the adjoint matrix. The identity matrix
contains only unit elements along the leading diagonal, with the
other elements all zero. This is but one means of determining
the inverse of a matrix. The determinant of a matrix must not
vanish for the inverse to exist. If the inverse of a matrix
exists, the matrix is said to be a nonsingular. (3)
An important rule is that * * *
(AB) = B - A
or the transpose of the product equals the product of the transposes
in reverse order.
Many normal rules of algebra are applicable. For example,
W s K'A
K-i«W = loA A = K-1»W ' F»W
• t X A S TFCHNnLDGICAi: CniJlXGfS LIBRARY