THEORY OF ELASTIC WEIGHTS

by

J. T. Oden

/0

ABSTRACT

TITLE: THEORY OF ELASTIC WEIGHTS

AUTHOR: J. T. Oden

DESCRIPTORS: Structural deformations, secondary stresses,nonlinear structures, matrix methods.

SUMMARY: The general theory of elastic weights is derivedfrom the geometry of closed curves and polygons.Application of the theory is demonstrated for avariety of structural problems.

,

THEORY OF ELASTIC WEIGHTS

1J. T. Oden • A. M.

SYNOPSIS

In this paper, the general theory of elastic weights is

derived from the geometry of closed polygons and simple closed

curves. It is shown that several well-known methods of struc-

tural analysis may be considered to be special applications of

the general ideas of elastic weights. The extension of these

ideas of include in the analysis thermal effects, initial imperfec-

tions, elasto-plastic and nonlinearly elastic behavior is presented

as well as the relation beDveen elastic weights and the matrix

force method.

1 Associate Professor of Engineering Mechanics, University ofAlabama in Huntsville.

INTRODUCTION

The use of elastic or angle weights is nothing new in

structural analysis. Virtually every text on elementary struc-

tural analysis makes some mention of the conjugate beam method

or the bar-chair method. The analogy between the geometry of a

closed curve and the equations of statics was pointed out as early234

as 1868 by Mohr and later extended by Muller-Breslau ' . Muller-

Breslau presented the bar-chain method for the deflection analysis

of simple trusses and used approximate expressions for elastic

weights to analyze straight and bent beams.

Since the work of these early investigators, the theory

of elastic weights has been applied by a number of authors to

many special structural problems. Schwalbe5 used the ideas of

conjugate loads to study certain problems in plates and shells,

and modifications of Muller-Breslau's bar-chain method were

presented by Lee and Patel6

and Scordelis and Smith7. The

method was later generalized by Ramey8 for calculating deflections

2 Mohr, 0., "Behandlung der Elastischen Als Seillinie," Zeitschriftder Architekten-Ingenieur-Verein, Hannover, 1868.

3 Muller-Breslau, H. F. B., "Bietrag Zur Theorie Des Fachwerkes,"Zeitschrift der Architekten-Ingenieur-Verein, Hannover, 1885.

4 Muller-Breslau, H. F. B., Die Graphische Statik Der Baukonstruktionen,Vol. II, Part 2, 2nd Ed., Leipzig, 1925, pp. 337-365.

5 Schwalbe, W. L., "Conjugate Load Method in the Analysis of ThinShells," Proceedings of the First Midwest Conference on AppliedMechanics, p. 74.

6 Lee, S. L. and Patel, p. C., "Bar-Chain Method for AnalyzingTruss De forma tion," Proceedings , ASCE, Vol. 86, Paper No. 2477,May, 1960.

7 Scordelis, A. C. and Smith, C. M., "An Analytical Procedure forCalculating Truss Displacements," Proceedings, ASCE, Vol. 81, PaperNo. 732, July, 1955.

8 Ramey, J. D., "Elastic Weights for Trusses by the String PolygonMethod," M.S. Thesis, Oklahoma State University, Stillwater, 1960.

of trusses.

- 2 -

Tuma and Oden9 introduced the string polygon method

for analyzing complex frames and the concept of elastic weights

was used by Gillespie and LiawlO for the frequency analysis of

continuous beams. A number of additional references on the subject

are given in the paper by Tuma and Oden 9

In this paper, a generalization of the elastic weight analogy

is presented which is based on geometric considerations of simple

closed curves. This permits the extension of a number of the

earlier special methods to more complex structural systems and

demonstrates of the relationships between various methods. In

addition, the extension of the elastic weight analogy to the

analysis of secondary stresses in trusses, effects of temperature

and initial imperfections, elasto-plastic and nonlinearly elastic

behavior is discussed.

GEOMETRY OF CLOSED CURVES AND POLYGONS

The concepts of elastic weights may be derived from certain

geometric properties of a simple closed polygon. Consider, for

example, the closed polygon ABCD shown in Fig. 1. Suppose that

the polygon is deformed so that a new polygon A'B'C'D' is formed.

In general, each side of the polygon changes in length and there

occurs at each joint a change in the original angle between the

sides of the polygon. Let ~AB' ~BC' ~CD' ~DE' and ~EA denote

the changes in length of each side and ¢A' ¢B' ¢C' ¢D' and ¢E

denote the angle changes that occur at joints A, B, C, D, and E,

respectively. From elementary plane geometry it is recalled

that the sum of the deflection angles of any closed polygon is

2n radians. Since the deformed polygon is also closed, the sum

of its deflection angles must, again, be 2n. It follows, there-

fore, that the algebraic sum of all of the angle changes that

occurred at each joint of the polygon is zero. This is also seen

9 Tuma, J. J. and Oden, J. T., "String Polygon Analysis of Frameswith Straight Members," Proceedings , ASCE, Vol. 87, Paper No. 2956,October, 1961.

10Gillespie, J. W. and Liaw, B., "Frequency Analysis of Beams byFlexibility Method," Journal of the Engineering Mechanics Division,ASCE, Vol. 90, No. EM!, February 1964, p. 23.

I..

cr- --__-- -- 0'

....

c //

/ 70//

~I......----......----

......----I

E

FIG. 1 CLOSED POLYGON

SII

J-S'//

//

//

//

'- --------- ./,-- ---'

FIG. 2 SIMPLE CLOSED CURVE

x

x

- 3 -

by noting that if, for example, joint A undergoes an angle change

¢A' E remaining stationary, part of the angle change at B is

necessarily -¢A' Thus, the angle changes manage to cancel in

such a way that their algebraic sum is always zero, Mathematically,

this means that for a polygon with n joints,

, ,(1)

Furthermore, Eq. (1) is valid for any number of sides, and it is

equally valid for an infinite number; that is, a smooth closed

curve. Hence, if ~ is defined as the angle change per unit

length of a deformed curve S in Fig. 2,

cD q:ds=OoJ s

.(2)

Note that no restrictions were imposed on the magnitude of the

angle-changes; and, hence, Eqs. (1) and (2) apply to large

deformations as well as small.

Referring, again, to the polygon in Fig, 1, let the x and

y components of the displacement of any joint i, ~., be denoted~by ~. and ~. , respectively. Hence, if B displaces an amount~x ~y~ in deforming the polygon, ~Bx and ~BY are the projections of

~ on the x and y axes. Assuming that joint A is stationaryB

and can be used as a reference, the closed deformed polygon is now

traverse in a clockwise path and the algebraic sums of x or y

components of the joint displacements are recorded. A negative

value is assigned to any component acting in the negative x or y

directions and a positive value to those in the direction of

increasing x and y. Obviously, each sum is zero since, on

closing the traverse, we arrive at precisely the same point from

which we started. This fact is expressed mathematically for a

polygon with joints as follows:

\' ~.I ~xi~

o .(3)

~.

- 4 -

andn

"'IJ.L iyi=l

a . .(4)

Again, if 0 and 0 are the x and y components of displacementx yper unit length of elements of a deformed smooth curve S (that

is, n approaches infinity), it follows from Eqs. (3) and

that

<P ()ds = 0x

s

and

\P () ds = O.s

y

(4)

·(5)

.(6)

Note that no restrictioI1Shave been placed on the magnitude of

these displacements.

Equations (2), (5), and (6) also follow from Cauchy's Integral

Theorem which states that for any function f(x,y) that is analytic

at all points within and on a closed curve S, g> f(x,y) ds = 0,

This implies that ~, 0 , and 0 are analytic functions.x yFor a final geometric property, consider the typical side DE

of a closed polygon shown in Fig. 3. Assuming A to be fixed for

clarity, suppose that a small angle change ¢E occurs at E which

causes D to move to a positions D". Then let DE undergo a small

change in length, ~E' so that D acquires its final position, D'.

Since ¢E and ~E are small in comparison with the dimensions,

it is easily verified from the geometry of Fig. 3 that

and

~y = tnE sin ex. - ¢Ed .

. (7)

, (8)

Proceeding in this manner, similar relations are obtained between

..

h

A

xFIG. 3 DEFORMATION OF POLYGON SEGMENT

FIG. 4 CONJUGATE MOMENTS

- 5 -

the components of the joint displacements and the changes in

side length and joint angle.

THE ANALOGY

On examining the relations developed thus far, it is seen

that they suggest that an analogy exists between the well-known

equations of statics and the geometry of a closed polygon. In

fact, by replacing the words "angle-change" by "force," "change

in length" by "moment," and "closed polygon" by "a system in

equilibrium" or by the words "structural member," the preceding

discussion acquires an amazing similarity to a discussion of the

laws of statics. Moreover, if, instead of ¢ and ~ the symbols

P and M are used for the angle changes and joint displacements,

since P usually denotes a force and M a moment, Eqs, (1), (3),

and (4) becomen

Ll\ == O.

i=l

n

IM. == 0~x

i=l

andn

LMiY = O.

i=l

(9)

. (lO)

,(11)

It follows that the geometry of deformation of a closed polygon

(or curve) may be evaluated by employing statics to an imaginary

structure of the same dimensions as the polygon, loaded by forces

which are equal to the angle-changes which occur at joints of the

polygon and by moments which are equal to the displacements of the

joints of the polygon, Since, in the discussions to follow, it is

intended to relate these concepts to structural problems, it is

correct to assume that angle-changes in elastic systems are to be

considered. For this reason, the angle changes, P. , corresponding~

•

- 6 -

to elastic systems are called elastic weights. The imaginary

structure on which they act is called the conjugate structure

and the equations of statics (Eqs. (9), (10), and (11)) associated

with the conjugate structure are called elasto-static equations

Equations (9), (10), and (11), then, state that the conjugate

structure of any closed polygon is in the elasto-static equilibrium,

Furthermore, if a portion of the conjugate structure is isolated

as a free-body (for example, side DE as shown in Fig. 4) and

if the x and y components the conjugate bending moment at D- -

are denoted by M Dx and~y'

it is seen that DE is in

elasto-static equilibrium provided

and

~x == r~E cos ex. + PEh.

~y = ~E sin ex. - PEh,

. . (12)

.(13)

where ~E is the applied conjugate moment equivalent to ~E

the displacement of D relative to E on member DE. Comparing

Eqs, (l2) and (13) with (7) and (8), it is seen that displacements

of points on the polygon become bending moments of the conjugate

structure. Similar considerations show, in addition, that shears

of the conjugate structure are equal to changes in slope of sides

of the polygon,

It is also clear from Fig. 4 that the directions of the conju-

gate loading and the stress resultants may be adjusted so that they

are consistent with the directions of the displacements of the

polygon. Angle changes are replaced by force vectors normal to the

plane in which the change occurs; displacements in a prescribed

direction represented by conjugate moments in that same direction.

For the present purposes, it suffices to apply elastic weights in

the positive z direction and conjugate moments in a clockwise

direction around the polygon. Hence, the conjugate of the polygon

in Fig. (1) is the structure shown in Fig, 5, and that of the

deformed curve in Fig. 2 is shown in Fig. 6.

z

MEA E

o

xFIG. 5 CONJUGATE OF CLOSED POLYGON

FIG. 6 CONJUGATE OF CLOSED CURVE

x

o·

- 7 -

Some simple examples of closed curves in elastic systems

are shown in Fig, 7, The closed traverse 122'1'1 formed by

a portion of the elastic curve of the beam shown in Fig. 7a

IIIUst,according to the above theory, be in elasto-static

equilibrium. Hence,

, (14)q:ds = 0

122'111

cp 0 ds = rh 0 ds = rh. x 'J' y ';t'122'1'1 122'1'1

A displacement of a support of a beam such as that shown in

Fig. 7b, "opens" a polygon 012340, unless elasto-static equilibrium

is restored by applying a conjugate moment on the conjugate

structure of a magnitude and direction equal to this displacement,

The mechanical hinge at 4 is accounted for by applying an elastic

weight at the corresponding point on the conjugate structure which

represents the rigid-body rotation of the structure about the

hinge. Similarly, the sum of the displacements and angle changes

around any closed path in the truss in Fig. 7c must vanish.

It is important to note that the term "elastic weight" is

actually a misnomer. Nowhere in the preceding discussions have

any elastic properties been mentioned; the relationships apply

to any closed curve and are purely geometric,

THE CONJUGATE BEAM METHOD

The well-known conjugate beam method is a special application

of the theory of elastic weights, The angle change per unit length

is a straight elastic beam is

.(15)

where v is the transverse deflection, From the Bernoulli-Euler

beam theory,

d2 M--Y. xdx2 = EI

, .(16)

(a)

(b)

4

(c)

FIG. 7 CLOSED CURVES IN STRUCTURES

- 8 -

in which M is the bending moment and EI is the flexural rigidity.xThus, the angle change in a unit length is

, (17)

P is called an elemental elastic weight, According to Eq. (2),

the sum of the elemental elastic weights developed around any

closed curve must vanish if deformations are to be compatible, If

Eq, (17) is integrated from point i to point j on the elastic

curve, an equation is obtained for the segmental elastic weight:

P -nl rjji - l"j - ¢. =~ .i

M dxxEI

. (18)

represents the angle change between tangents to the elasticP ..J~

curve at i and j, Thus, Eq, (18) is a statement of the first

area-moment proposition,

The deflection of j relative to i (the tangential deviation)

is equal to the conjugate bending moment at j due to the elastic

weights between i and j :

0" = m .. =J~ J~(19)

where d .. is the distance between points i and j and S is~J

the coordinate measured from j to the point of application of pThus, Eq. (19) is a statement of the second area-moment proposition.

According to Eqs, (5) and (6), the sum of the conjugate moments

developed on any closed curve in the structure must vanish if

deformations are to be compatible,

THE STRING POLYGON METHOD

Since the analogy between the geometry of deformation and

statics is complete, it is permissible to replace the distributed

elemental elastic weights with a statically equivalent force system

- 9 -

for purposes of writing the equations of elasto-static equilibrium.

Such a process is the basis of the string-polygon method (11).

Consider a segment ij of a straight beam under general loading.

The distributed elemental elastic weights (Eq. (17)) between i and

j are statically equivalent to a segmental elastic weight at i

given by

P .. =M. GoO +M. F .. + 'T ..~J J J ~ ~ ~J ~J ,(20)

and a segmental elastic weight at j given by

p .. = M. G.. + M. F .. + 'T ..J ~ ~ 1J J J ~ J ~

.(21)

j

G .. , GooJ ~ ~J,are

G.. is the end slope at~J

i , T .. is the end slope at j due toJ1

are known constants for a given member.These

due to a unit moment at

applied loads, etc,

M. and M. denote the bending moments at i and j~ J

F .., and F .. are angular flexibilities and 'T.. and T ..~J J ~ ~J J ~

angular load functions. F .. is the end slope of the simple beamJ~

ij at j due to a unit moment at j ,

Equations (20) and (21) are quite general, With the proper

choice of the flexibilities and load functions, special effects

can be accounted for such as beam-column action, elastic foundations,

shear deformation, unsymmetry of the cross section, temperature

changes, and prestressing forces. In the case of a curved beam

segment, an additional term representing the angle-change due to

a thrust must be added to Eqs, (20) and (21). It is also necessary

to add a conjugate moment vector directed from i to j to account

for the fact that i displaces relative to j

When the entire loading on a conjugate structure is replaced

by statically equivalent loads acting at the ends of arbitrarily

selected segments, it forms a closed polygon called a string poly-

gon of the structure, The elasto-static equations now give the

geometry of deformation of the polygon rather than the structure;

but the vertices of the polygon may be chosen so that they correspond

to the joints of the structure. The total angle change occurring

in joint j of the polygon is called the joint elastic weight at j.

- 10 -

The joint elastic weight is the sum of the segmental elastic

\veights at j :

P. = P .. + P'kJ J ~ J

Thus, from Eqs, (20) and (21),

P. = M. G.. + M. (F.. + F.1) + M. Gk. + ('r.. + T. k)J ~ ~J J J ~ J <: -1< J J ~ J

.(22)

.(23)

In analyzing any complex frame, segmental elastic weights are

calculated by means of Eqs. (20) and (21) and are applied on the

conjugate of the string polygon, Three independent elasto-static

equations are written for each closed polygon. These equations,

plus three equations of statics, form a system of consistent

independent equations from which the end moments are obtained.

The procedure is indicated in Fig. 8.

It is interesting to note that, in the case of a continuous

beam, P. is zero if the supports are chosen as joints of theJ

polygon. In this case, Eq. (23) reduces to the well-known

three-moment equations.

DEFOR}~TIONS OF COPLANAR TRUSSES

Application of the general theory of elastic weights to

determine deformations of coplanar trusses is referred to as the

general bar-chain method, The adjective "general" is used to

distinguish the theory from the more specialized form found in

the literature; and the meaning of the term "bar-chain" will

become apparent in the developments which follows.

Consider the simple coplanar trusses of general shape shown

in Fig. 9. The truss is subjected to a general set of joint forces,

Pl, P2, . ., as is indicated, and the geometry is typified by

a number of triangular cells, A, B, . . " F, formed by the truss

bars, The most significant feature of the geometry of this

structure is that the truss members form several closed polygons,

The manner in which a truss deforms is conceptually very

simple; an axial force is developed in each member and the member

t

x

k

BA

Conjugate Frame

8

k

A

xCopla.nar Framez

~

ZI Z _ is. JI jK .

P;A' 0 kJ\t~B~~- JI P.P. jk

P I}AI

iA 1..~pP.

AI n"

X

[Iasta -stat j cs 5egmenta I Elastic.Weights

21 p. tgI isK

x

Joint E-lastic WeightsFIG. 8 STRING POLYGON PRINCIPLE

- 11 -

undergoes a change in length, There occur small changes in the

angles between various members to accommodate these changes in

length, the members rotate in their hinged joints, and the total

structure reaches a deformed configuration. The deformation of

a given cell of the truss, therefore, is defined by changes in

the length of the members forming the cell plus angle-changes at

each joint of the cell.

According to the theory of elastic weights, angle-changes

may be represented by force vectors and relative displacements

(or changes in length) may be represented by conjugate moments

acting on a conjugate structure. Thus, there must be associated

with a typical cell, H, of the truss a conjugate cell which is

in elasto-static equilibrium, The angle changes occurring at joints

of the truss cell are elastic weights acting at the corresponding

joints of the conjugate cell; changes in the length of sides of

the cell are conjugate moments acting along the corresponding

sides of the conjugate cell,

Consider a typical cell, H, of a coplanar truss defined by

members connecting to joints i, j, and k. The conjugate of cell

is s~own in Fig. 10, subjected to joint elastic weights, PiH, PjH '

and PkH ' equal to the angle-changes at i, j, and k, and to- - -

conjugate moments, M .., and M.k

, and K " equal to the change in~J J -K~length of the corresponding members, Assuming that the axial forces,

Nij

, Njk

, and Nki

, developed in members ij, jk, and ki, respectively,

are known, the conjugate moments are easily determined by the

formulas

M.,=~ .. =N.,)., ..~J ~J ~J ~J

Mjk = ~jk = Njk ).,jk

and

~i = ~i = Nki ).,jk

where A.., = d ../EA., , ).,'k= d.k/EA'k ' and ~. = dk./E~. ,~J ~J 1J J J J ~ ~ ~

, (24)

..- 12 -

ijof memberirelative tojof point

and dij and Aij are the length and the area of member ij, etc.

The quantities A .. , A'k

' and \. are the axial flexibilities~J J 1

of members ij, jk, and ki. Physically, A.. is the displacement~J

(the change in length)

due to a unit axial force,

The elastic weights may now be determined from the equations

of elasto-statics equilibrium:

M .. + M'l + ~. + P'H y .. + PkH Yk' = 0~Jx J {X ~x J J1 1

- - - -M .. + M'k +~. +P·Hx .. - PkHxk. =0 . (25)

~JY J Y ~y J J1 ~

- - -P iH + PjH + PkH = 0

where y,. = y. - y. , Yk' = Yk - y. , x.. = x. - x. , xk· = xk - x.J~ J ~ 1 ~ J~ J ~ ~ ~

and the x and Y subscripts indicate x and y components,

respectively, of the conjugate moments.

Although each elastic weight can be easily determined from

the statics of each conjugate cell, it is also possible to obtain

a general formula for the joint elastic weights by solving Eqs. (25)

for One finds

x'kP'H =~~ DH (26)

where L6Hx and ~HY are the algebraic sums of the x and y

components, respectively, of the conjugate moments and

, (27)

Equations (25) provide a means for determining each elastic

weight corresponding to cell H. The total angle-change occurring

at a joint k of the truss is simply the sum of elastic weights

at k of each cell having joint k in common. Hence,

. (28)

The conjugates of any number of adjacent cells can be arbitrarily

added together to form a variety of conjugate structures as is

illustrated in Fig, 11. Regardless of the number of cells, as long

FIG. 9 SIMPLE COPLANAR TRUSS

x

~H I_ - -r--""- -, / "--- / ""- "-

/ J "-,,-/ "-

I -- -~m

~Hz

FIG. 10 CONJUGATE CELL H

- 13 -

as the appropriate changes in length and angle are used, the

resulting conjugate structures are elasto-static equilibrium.

If the conjugate structure is "cut" so that no closed

polygon is formed, conjugate moments and shears must be added

to keep the "structure" in elasto-static equilibrium, This

is illustrated in Fig. 11 c and d, In fig, 11 d, for example,

a cut is made from joints 1 to S of the truss and a conjugate

moment MIS is applied along the line 15, as shown, to provide

equilibrium of moments, Conjugate forces, Rlz and RSz are

necessary for the elasto-static equilibrium of forces. Accordingly,

MIS is the displacement of joint 1 relative to 5 and R5z is

the change in the angle between the lines S6 and 51, etc.

It is from free bodies such as those shmvn that truss

deformations can be evaluated. For example, to evaluate the

displacement of a top chord joint of the truss in any direction,

we traverse a chain of bars such as l234S, preferably starting

from a fixed joint such as joint 1, and close the polygon by

applying the proper conjugate moments and forces for equilibrium

(MSl' Rlz, and RSz in this case), The displacement of point 3,

for example, in some direction n is simply the component of

the conjugate bending moment at 3 in the direction n ,

SECONDARY STRESSES IN TRUSSES

By combining the ideas of the string polygon method and the

bar-chain method, an approximate procedure for analyzing secondary

stresses in trusses is obtained.

Following the standard procedure for analyzing secondary

stresses, the structure is first analyzed as a pin-connected truss;

the changes in length of each member are computed and applied

as conjugate moments along the appropriate sides of each conjugate

cell, These are accompanied by elastic weights at each joint due

to rotations of the members about the hinged connections. These

elastic weights are calculated from simple elasto-statics or from

zisIA

z I ~j51

z

(a)

== ......

bar-chain

~F

5

-x

x

x

FIG. 11 COMBINING CONJUGATE CELLS AND FORMING BAR-CHAINS

- l4 -

Eq. (26). If the joints are now assumed to be rigid, secondary

moments are developed which create additional elastic weights at

each joint. The magnitudes of these secondary elastic weights are

given by Eqs. (20) and (21). Thus, the total elastic weight at

joint i of cell H is

x'k y'k~ 'fJj. - ~ "£f\. + M. Gk. + M. (F .. + F. k) + MJ' G. iD Hx D Iffy -K ~ ~ ~J ~ JH H

+ Too + 'f'k~J ~,(29)

The remainder of the procedure is identical to the string

polygon analysis of frames: three elasto-static equations are

written for each conjugate cell and the resulting set is solved

for the secondary moments. The effects of large guss~t plates can

be included by accounting for a variation in the flexual stiffness

of each member when computing the angular flexibilities and load

functions.

If a better approximation is desired, a new set of axial

forces and conjugate moments can now be computed, this time taking

into account the effects of secondary moments, These are applied

on the appropriate sides of the conjugate cells and the entire

procedure is repeated, If desired, this cyclic procedure can be

continued until the results of a given cycle are not significantly

different than those of the preceding one.

INITIAL IMPERFECTIONS AND THERMAL EFFECTS

The influences of initial imperfections and temperature can

easily be accounted for using the theory of elastic weights. Consider,

for example, a beam segment ij which is initially an amount €

too long and which is warped in such a way that initial end slopes

w. and w. exist at points i and j . If it is assumed that~ J

these imperfections are small in comparison with the length of the

beam, their influence on the behavior of a complex structure

containing such a member is determined by simply applying elastic

weights W, and W. at points i and j of the conjugate structure.~ J

- lS -

The lack-of-fit € is applied as a conjugate moment acting about

member ij of the conjugate structure. Elastic weights calculated

by means of Eq. (29) are then superimposed on those due to initial

imperfections. Statics of the conjugate structure then gives the

desired equations of the end moments and axial loads in terms of

€ , W. , W. .~ JThermal effects are accounted for in a similar manner. In the

case of a nonuniformly heated truss, for example, the extension of

member ij due to a temperature increase T is

"'- .. = ex. LT.~~J.(30)

ex. being the coefficient of thermal expansion, The displacement

~T" is appplied as a conjugate moment acting about member ij~J

of the conjugate structure. In the case of beam members, end slopes

wTi

and WTj

due to temperature are computed assuming that nlember

ij is unrestrained, These become thermal elastic weights acting

at i and j in the conjugate structure. The analysis then

proceeds as discussed previously.

ELASTO-PLASTIC AND NONLINEARLY ELASTIC STRUCTURES

Since the concept of elastic weights is based on purely kinematic

considerations, the method can also be applied to elasto-plastic

and nonlinearly elastic structures, In these cases the elastic

weights become nonlinear functions of the end moments and forces

and the elasto-static equations must be solved numerically.

Consider, for example, a straight beam ij which is constructed

of a material obeying a Ramberg-Osgood12 type stress-strain law

y = aE + can .(31)

where Y is strain, a is stress, and c and n are parameters

defining the shape of the stress-strain curve, Conceivably, a

relation of similar form applies to the moment-curvature relation.

Thus, if p is the elemental elastic weight, for a nonlinearly

12 Ramberg, W. and Osgood, W. R., "Description of Stress-Strain Curvesby Three Parameters," NACA, TN 902, 1943.

- 16 -

elastic or elasto-plastic beam,

- 1\ np = (EI + kMx ) dx . . (32)

where k and n are appropriately selected parameters, The moment

M in this equation is expressed, by statics, as a function of thex

end moments M, and M. The segmental "elastic II weight at j is~ Jthen

. (33)

in which L is the length of member ij and g is the distance

from j to the point of application of p. If no intermediate

loads are present, substituting Eq.(32) into Eq, (33) and expressing

M as a function of M. , M. and S givesx ~ J

elastic weight due to linearly elastic

The total elastic weights

n+l n+2(M.L) lL(n+l) (M.-M.)-2LM, + (M.L) ]J J 2~ ~ ~ . ,(34)

(M. - M.) (n + 1) (n + 2)J ~

k+

(EI)o Ln+l

_(e)P.. is theJ~

and is given by Eq. (20).

_(e)= P ..

J~

in which

behavior

P. ,J~

become nonlinear functions of the end moments. The remaining

steps in the analysis are similar to those in linear structures:

elastic weights are applied on the conjugate structure and, using

statics, a system of equations involving the end moments is obtained,

For nonlinear structures these elasto-static relations represent

a system of nonlinear simultaneous equations in the redundant moments

which must, in general, be solved by iteration or a related

numerical technique,

THE MATRIX FORCE METHOD

The matrix force method13 can be interpreted as an application of

the theory of elastic weights. In this method a complex structural

13 Argyris, H. J., "Energy Theorems and Structural Analysis,"Butterworth Scientific Publications, London, 1960.

..

- 17 -

system is assumed to be composed of a number of component parts

called elements, The elements can be any type of solid, flexible

body capable of transmitting loads; they are usually taken to be

the most geometrically simple structural member that can adequately

represent the response of the structure to a given stimulus,

Springs, torque tubes, beams, plates, shells, and even three-

dimensional bodies can be used as elements. The geometry of an

element is defined by the location of a number of points on the

element called nodes. For example, if a beam is chosen as the

structural element, its end points are taken as nodes, The complete

behavior of an element is defined in terms of the forces of moments

which act at the node points and their corresponding node displace-

ments. These so-called node forces may be forces, moments, be-

couples, torques, etc,; in other words, there is related to each

node a system of generalized forces and displacements called

node forces and node displacements. All external loading is

represented by forces and moments acting at certain nodes on the

boundary of the structure, The boundary lines of the element

form a system of closed three-dimensional "space" polygons which

must be in elasto-static equilibrium,

Following the classical concepts of structural analysis, the

structure is reduced to a statically determinate system by

releasing a certain number of redundant generalized forces which

are denoted by the column matrix (x), The remaining known

external forces are given by a column matrix (P). Node forceso(p.) on element i of the structure are related to their corresponding

~node displacements (6.) according to the formula~

(p.) = [r, ] (P ) + Cr. ] (x),~ ~o 0 ~x, (31)

The matrices of the form [rij

] are rectangular matrices, not neces-

sarily square, which automatically replace a generalized force

system at j by a statically equivalent one at i. Thus, the

operation Cr. ] (P ) sums the moments of the known external loads~o 0

about the coordinate axes at each unsupported node of element i and

- 18 -

adds a statically equivalent force vector to each node. The matrix

[r ..] is the transpose of [r,. ) ([r., ) = [r.,]T )J ~ ~J J ~ 1J

Returning now to the concept of elastic weights, it is

evident that the displacement vector for element i can be considered

as a vector of conjugate moments and elastic weights developed by

deforming the space polygon which forms the boundary of the element.

Using the previous notation,

[p,}= [o.}.J ~

.(32)

From the principle of least work, the relative displacements due to

the redundants at the hypothetical cuts in the structure must be

zero for compatibility to exist, This means that the total

angle-change (sum of the elastic weights) and the relative dis-

placements (sum of the conjugate moments) must vanish, Thus, for

a structure with n elements

n\' [ T f-L r ix ) 'Pi} = (0 J .

j=l

.(33)

Introducing Eq. (31) into (30) and substituting the results into

Eq. (33) gives

n(' L T\. [r. ) [f.]

1X ~

i=l

n

[riO]) (p) + CI [riX]T[fi] [riX]} (x}= [O}.i=l

. (34)

from which the redundants can be solved. Equation (34) is precisely

the same result obtained in using the matrix force method.

CONCLUSIONS

If it is assumed that the angle-changes which occur in a

structure are small in comparison with characteristic dimensions

of the structure, these angle-changes may be treated as vector

quantities, Kinematic relations which depict the geometry of

the deformed structure then become a complete analog of the

equations of statics of a hypothetical structure called the con-

jugate of the real structure, The conjugate structure is

loaded by angle-changes analogous to forces and by displacements

analogous to moments,

This analogy is readily derived from the geometry of simple

closed curves and polygons. Since the analogy is based upon

purely kinematic considerations, the resulting equations are

independent of the material properties of the structure.

Because of this, a variety of special effects can be easily

accounted for including secondary stresses, initial imperfections,

nonuniform temperatures, and elasto-plastic and nonlinearly

elastic behavior.

•

n

-pv

x

C

E

F .. , G ..~J 1J

I

L

Mx

M. , H, , ~~ J- - -

M. , M, , ~~ J

N. ,~J-P ..

~J

P.J

T

ex.

y

{) 0x' y

E:

A ..~J

;

a

,. . ,

~J

NOTATION

A parameter

Elemental elastic weight

Transverse displacement

Coordinate

Constant

Young's modulus

Angular flexibilities

Moment of inertia

Length of beam segment

Bending moment

End moments

Conjugate moments

Axial force

Segmental elastic weight

Joint elastic weight

Temperature distribution

Coefficient of thermal expansion

Extensional strain

Displacements per unit length

Initial imperfection

Axial flexibility

Beam coordinate

Normal stress

Angular load function

•

¢i ' ¢ij

W.1

Angle-changes

Angle-change per unit length

Initial end slope