0045-7949/8X s3.00 + 0.00
8 1988 Per~tttoo Press plc
compurers & stnicwas Vol. 29, No. 2, pp. 34>3JO, 1988
Printed ia Great Britain.
THE ADJOINT METHOD FOR DETERMINING
INFLUENCE LINES
A. D. BELEGUNDU
Mechanical Engineering Department, The Pennsylvania State University, University Park,
PA 16802, U.S.A.
(Received 25 August 1986)
Abstract-A
method for computing influence lines is presented. Influence lines are widely used in
structural design to determine worst-case locations of live loads. The method is a result of giving physical
interpretation to certain adjoint variables used in the field of design sensitivity analysis. The method is
general, and can be easily implemented in finite element programs. Computational aspects and example
problems are discussed.
I. INTRODUCIION
Recently, certain
design-specifically
adjoint
variables
where p] is an (n x n) structural stiffness matrix and
F is an (n x 1) vector of applied loads. From eqn (I),
we have z as a function of the load vector F. That is,
we have
used in optimal
in design sensitivity analysiswere given a physical interpretation [I]. The adjoint
variables were shown to be the influence coefficients
associated with a given function. In the present paper,
the above interpretation is shown to result in a new
and general approach for determining influence lines,
which can be easily integrated into existing finite
element codes.
Influence lines are used by structural analysts to
determine worst-case locations of live loads [2-4].
Influence Iines show the influence on a certain function of a unit load as it travels across the structure.
The function can be anything that varies as the load
moves across the span, such as moment or shear at
a given section in the girder or beam, or bar force in
a particular truss member, or deflection of a given
point on a structure. The ordinate of the influence
line, at a given point, is referred to as an influence
coefficient, and represents the value of the function
due to a unit load acting at that point.
The adjoint variable method for determining
influence lines is derived in the next section. Example
problems are given in Sec. 3. In the first of the
example problems, the adjoint method is compared
to the Muller-Breslau approach existing in the literature. A summary is given in the last section.
z=[i(l-‘F,
(2)
from which
dz/dF=[I(]-‘.
(3)
We can write
g s g(W)).
Now
dg/dF = dg(z)/dz dz/dF.
Using eqn (3) in (4), and noting
symmetric matrix, we have
fK1dgJdFT = dg(z)/dzT.
that
(4)
[K] is a
(5)
Letting
1= dg/dFT,
which is an (n x 1) vector, we can write eqn (5) as
2. DERIVATiON OF
THE ADJOINT METHOD
p]lz = dg(z)/dzT.
Consider a function g(z) where g can be the shear
force or moment at a beam section, or a bar force in
a particular truss member, or a deflection of a given
point on a structure. The (n x 1) vector z is a vector
of nodal ~spla~ments
obtained from the finite
element equations
Wlz= F,
(6)
Now consider the case when g is a linear function of
z. This is true in linear structural analysis when g is
a moment or shear, bar force, or a deflection. Since
z is a linear function of F, it also follows that g is a
linear function of F. C’onsequently, we have the
equation
(1)
345
g(F+u)=g(F)+dg/dF.u,
(7)
A.
346
D. BELEGUNDU
where g =g(z(F)) =g(F), and u is a change in F. If
the structure is initially unstressed, we have g = 0
when F=O. Letting F=O, u=(O ,..., l,..., O)T
represent a unit load along the ith degree of freedom
of the structure, and noting that li = dg/dF,, eqn (7)
yields
g(u) = li.
(8)
That is, li is the value of the function g due to a unit
load acting along the ith degree of freedom. By
definition [2], 1 is the influence coefficient vector
associated with the function g.
Therefore, it can be seen that solution of eqn (6)
yields the influence coefficients vector 1. Noting the
resemblance between eqn (6) and the equilibrium
equations in eqn (1), we can think of Iz as a displacement vector associated with the load vector dg/dz.
This important observation leads to the following
conclusion. In the finite element method, element
shape functions are used to obtain analytical expressions for displacements at interior points in the
structure from the nodal displacement vector z. Likewise, element shape functions can be used to obtain
analytical expressions for the influence coefficients
within the elements from the nodal values of A. These
expressions then can be used to generate the influence
lines (or curves).
It should be noted that when g is a linear function
of z, the vector of dg/dz which forms the right-handside of eqn (6) is a constant vector, which does not
depend on z. The analytical expression of g(z) is
readily provided by the finite element method. Consequently, calculation of dg/dz is routinely handled in
computer programs. Details related to element shape
functions and calculation of dg/dz are given the next
section.
with the adjoint variable vector I determined from
the adjoint equations in eqn (6). This method of
computing derivatives has its origins in optimal control theory [S]. In [1], a physical interpretation was
given to the adjoint variable vector rZ.The algorithm
presented above and example problems in the next
section demonstrate that the adjoint method based
on eqn (6) is a powerful method for determining
influence lines.
3. EXAMPLE
Four example problems are presented. the first is a
four-bar truss, where calculations are done by hand
to illustrate the adjoint method. The problem is also
solved using the Muller-Breslau method. The remaining three problems are solved using a computer
program.
Four -bar truss
Consider the four-member statically indeterminate
truss shown in Fig. 1. It is necessary to obtain the
influence coefficients for the vertical reaction at the
roller support at node 2. In this problem, EA = 1 for
each member, where E is Young’s modulus and A is
cross-sectional area. Using the stiffness method of
analysis, the equilibrium equations Kz = F in (1) can
be written as
where Zi, Fi, i = 1, 2, 3, are indicated in Fig. 1. Using
the finite element method, the reaction at node 2 can
be expressed as
R = -A
Algorithm
The steps involved in the adjoint method
determining influence lines are as follows.
PROBLEMS
(0.96 z, + 2.50 z~).
(10)
for
Step 1. Compute dg(z)/dz.
Step 2. Solve for 1 from eqn (6).
Step 3. Obtain expressions for influence coefficients
within the elements using element shape functions.
Then construct the influence lines.
The present author uses the name ‘adjoint method
for the new method presented because eqn (6) is used
in optimal design to evaluate sensitivity coefficients or
derivatives of functions with respect to design parameters. More specifically, let b = (k x 1) be a vector of
design variables, and g(z, b) be a function with z
implicity dependent on b through the equilibrium
equations [K(b)]z = F(b). Then, the derivative dg/db
is calculated from
The influence coefficients for this problem are simply
the values of R due to unit loads along each degree
of freedom. The influence coefficients will be derived
using the adjoint equations of this paper, and by
another method taken from the literature.
F3rZ3
T
4
f
40
1
t-------i
Fig. 1. Four-member truss.
347
The adjoint method for determining influence lines
Adjoint method. From eqn (IO), we have
dR
- = -&
dz
which, when substituted
(0.96,0,2.50),
ment method, for any general function
example, consider
(11)
into eqn (6), yields
resulting in the influence coefficients being given by
A = -{0.237,0,
l.O)r.
-2,
(14)
where R, and R, are the reactions at two different
supports. In the adjoint method, dg/dz = dR, fdz +
dR,/dz is evaluated in the usual manner, and rZ is
obtained from eqn (6). Use of the Muller-Breslau
principle for eqn (14) is more involved and requires
special logic in the computer program. Alternatively,
use of superposition requires solution of two righthand-side vectors, as opposed to only one in eqn (6).
(12)
It should be noted, of course, that calculating 1
from eqn (6) for large problems is routinely handled
in computer programs.
Muller-Breslau
principle.
According
to this
principle [2,6], the roller support is removed and
replaced by a load P (Fig. 2). The influence
coefficients, say K,, are then given by
IC;=
g(z)= R,(4 + 4 (~1,
g(z). For
(13)
Pratt truss
Consider the Pratt truss in Fig. 3(a), which is a
problem obtained from [2]. It is necessary to obtain
the influence coefficients associated with the bar force
in member 3-9. This and subsequent problems are
solved using a computer program.
The calculation of dg/dz for this problem can be
illustrated by considering a typical two-dimensional
truss element as shown in Fig. 4. Using the finite
element approach, element forces in the local reference frame are given by
where Si are the displacements along z, , z,, zj due to
load P, and A is the displa~ment along P Fig. 2).
Straightforward calculations result in
A=95P
6, = 22.5 P
6, = 0
p’ = F’]z’,
(15)
where k’ is the element stiffness matrix and z’ is the
element nodal displacements in the local (x’ - y’)
reference. Equation (15) can be written in expanded
form as
r
1 0
-1
01
f-q
6, = 95 P,
which, in view of eqn (13) yields
IC = - f0.237,0,1 .O)r,
which is the same as that obtained earlier.
From this example, one can see that implementing
the Muller-Breslau
principle
requires
special
modifications of the computer program, since removal of the roller support requires modifications of
the boundary conditions. Furthermore, the principle
is not as efficient and as easily applicable as the
adjoint method within the context of the finite ele-
P
Fig. 2. Muller-Breslau method.
Evidently, p ; = pi = 0 and p ; = -p ; . The bar force is
simply the component pi in the element force vector,
which follows the convention that positive means
tension. We thus have
P; = k;UW.
(17)
where k; is the third row of [k’]. The vector ze = (z,,
22, Z3r zJT is the element displacement in global
reference as shown in Fig. 4, and [fl] is a transformation matrix, with z’ = [/?]z’. From eqn (17), we
obtain
where [asjaz] is a (4 x n) matrix consisting of 0s and
1s. The vector ap;/az obtained from eqn (18) now
forms the right-hand-side of eqn (6), which yields the
influence coefficient vector 1. The vector I is given in
Table 1. It should be noted that the calculations in
eqn (18) are routinely handled in computer programs.
A. D. BELEGUNJXJ
348
(b)
0.239
Fig. 3. (a) Pratt truss. (b) Influence line for bar force in member 3-9 in Pratt truss. (c) Influence line for
shear in Pratt frame--values to be multiplied by lo-‘.
In Fig. 3(b), an influence line is constructured by
considering only the influence coefficients associated
with the y degrees of freedom at nodes 1-7 along the
bottom chord. The ordinates of Fig. 3(b) represent
values of the force in member 3-9 due to downward
unit loads at nodes 1-7.
example is converted into a frame by treating all
joints as rigid. All members have a rectangular
cross-section with width = 10 in. and depth = 100 in.
Young’s modulus E = 30 x 106psi. The function g ,
for which influence coefficients are required is defined
as
Pratt frame
g =f,lF,,
The application of the adjoint method to frames is
illustrated here. The Pratt truss of the previous
(19)
where f, is the maximum flexural shear stress in the
cross-section of element 3-9 and F3 = 14,400 psi is the
Fig. 4. A typical two~mensional
truss element.
The adjoint method for determining influence lines
349
Table 1. Influence coefficient vectors for example problems
Pratt frame
Pratt truss
Node/
direction
Value
Node/
direction
lx
2x
0.0
0.0
2x
ITx
4y
0.167
0.0
0.333
0.0
-0.5
0.0
-0.333
0.0
-0.167
-0.222
0.167
-0.222
- 0.667
-0.222
-0.5
- 0.222
-0.333
-0.222
-0.167
IX
3yx
&
Syx
1::
1iX
lT9
20
30
48
58
68
7%
88
98
100
118
128
allowable shear stress. The function g in eqn (19)
represents the ratio of actual to allowable shear stress
in element 3-9.
Since f, = 1.5 V/A, with V = shear force and
A = cross-sectional area, we have
dg/dz = 1.S(d V/dz)/(A . F,).
(20)
In eqn (20), dV/dz is computed in a manner similar
to that described in the Pratt truss example earlier.
Substitution of dgjdz into eqn (6) yields the influence
coefficient vector I, which is given in Table 1. Also,
the influence line constructed from the influence
coefficients corresponding to the y degree of freedom
of nodes l-7 is shown in Fig. 3(c). The influence
curves are constructed using element shape functions,
as explained in the next example.
Beam
Consider the beam shown in Fig. 5(a). It is necessary to determine the influence line associated with
the moment at support b of this beam. A finite
element model of this beam is shown in Fig. 5(b). It
may be noted that a lesser number of elements will
Beam
Value ( x 10m5)
0.136
0.129
0.239
0.107
-0.084
0.002
-0.477
-0.004
-0.287
-0.0
-0.147
-0.055
0.152
-0.160
-0.223
-0.185
-0.565
-0.195
-0.297
-0.194
-0.146
0.181 E-03
1.167 E-03
-7.661 E-03
1.363 E-03
0.310 E-03
0.413 E-03
0.401 E-03
1.637 E-03
- 7.689 E-03
1.205 E-03
0.377 E-03
0.425 E-03
Node/
direction
Value
2x
0.0
-0.550
-0.165
0.0
-1.110
0.0
0.0
0.495
-0.002
- 1,015
-0.044
-0.002
-0.683
0.168
-0.002
0.132
-0.002
0.332
0.012
-0.002
0.226
-0.044
-0.002
-0.036
-0.002
-0.120
-0.006
-0.002
-0.096
0.012
-0.002
0.018
;
3x
i;
4x
e
5x
;
6x
;
7x
e
8x
Y
e
9x
;
10x
0
11x
Y
9
12x
;
13x
8
also yield the same solution, since exact shape
functions are used in the finite element analysis of
beams (as per elementary beam theory). Solution of
eqn (6) yields the 1 vector, as given in Table 1. In
order to draw the irdluence lines, however, it is
necessary to know the values of 1 within the elements.
This is obtained by using element shape functions as
described below.
A typical two-dimensional beam element has six
degrees of freedom as indicated in Fig. 6. The value
of 1 at an inte~ediate
point x (Fig. 6), is given by
4x) = i N,(x)&
(21)
i-l
where Ni are Hermite cubic shape functions as given
in, for example, [7]. For element 1-2, we have
1, = 2, = L, = A4= 0 and, from Table 1, A5= -0.550,
1, = -0.165. In eqn (20), it should be noted that 1 (x)
represents the value of the moment at support b due
to a vertical load at point x. Consequently, for
element l-2, we have
n(~)=(3<~-2<~)1,+f-<*+,~)L&,
(22)
A. D. BELEGLJNDU
b I=900
12
3
4567
8
&‘
d I= 1600
~I=1200
9
IO
II
12
13
tbl
7%
‘z&h’
(c)
-0.12
-1.1 I
-0.096
-1.015
Fig. 5. (a} Beam problem. (b) Finite element model of beam. (c) Influence line for moment at support
b of beam.
4. SUMMARY
Fig. 6. A typical two-dimensional beam element.
where < =x/L,
written as
and L = 5’. Equation
;3&-LA,)+
f
(22) can be
;-2L,+L&),
(23)
0
which, upon substituting
&, yields
n(x)=
--
REFERENCES
numericai values for 1, and
2
A new method for computing influence lines based
on eqn (6) is presented. Element shape functions can
then be used to construct influence lines. The method
is applicable to statically determinate or indeterminate structures alike, and is applicable to any
problem which can be modeled using finite elements.
In addition to structural engineering problems, mechanical design problems (such as determining
influence lines for bearing reactions in cams modeled
by finite elements) can also be handled. The adjoint
method, as it is referred to herein, is especially
suitable for implementation into finite element programs. The method can be implemented in a general
manner, and its logic is not problem dependent.
x3
30.86 + 480.76 ’
(24)
where 0 G x G 5’. In general, a(x) for any element
can be obtained from eqn (21), and is routinely
handled in finite eIement codes. The complete
influence line for the beam associated with the moment at support b is constructed in Fig. 5(c). It
should be noted that use of eqn (21) can include axial
and shear effects, depending on the choice of Ni.
Inclusion of such effects by the moment ~st~bution
method used in [2, p. 4961 is very difficult.
1. A. D. Belegundu, Interpreting adjoint equations in
structural optimization. AXE, J. Struct. Engng 112,
1971-1976 (1986).
2. C. H. Norris and J. B. Wilbur, Elementary Structural
Analysis. McGraw-Hill, New York (1960).
3. W. J. Lamach, Inruence Lines: for Statically Indeterminate Plane Structures. Macmillan, London (1964).
4. J. F. Fleming and A. P. Shah, Truss influence lines on
a time sharing system. Comput. Struct. 4, 933-950
(1974).
5. A. D. Belegundu, Lagrangian approach to design sensitivity analysis. ASCE, J. Engng Mech. 111, 680-695
(1980).
6. T. M. Carlton, Principle of virtual work in relation to
Muller-Breslau’s principle. Inc. J. Mech. Sci. 22,
523-525 (1980).
7. J. S. Przemieniecki, Theory of Matrix Structural Analysis. McGraw-Hill, New York (1968).