symmetry in structural analysis

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70January, 1973

ST 1

By Peter G. Olocknerj! M. ASCE

9475 January, 1973 ST 1

y = utv, Ilie '" total deformation of concrete at extreme compress on

fiber of critical section, perpendicular to n~utral axis;

lis = total deformation of tension steel at critical section,

perpendicular to neutral axis;(e = concrete strain associated with compressive stress nor-

mal to neutral axis at crtucal fiber level, j c a» and de-

termined by Eq. 7;

e '" concrete strain at surface of slab above failure crackef 'I

attained at fal ure; ,(eu = ultimate concrete strain on extreme compression hber

for slab section in pure bending;

e s = f slE s = strain, at failure, of tension steel crossing column side

a'

I = st'rain, at failure, of tension steel placed perpendicular€sp=fspEs fll k

to free edge of slab at point of crossing a ure crac ;_.. (cll)2 + e (al1}2 steel strain, at failure, perpen-

€st - <-sp s,

dlcular to neutral axis ; ,1) es parameter in concrete stress-strain equation, dehned

by Eq. 6b j d f '(Je = rotation between two sections of slab compute rom

concrete strain, in plane perpendicular to neutral

of failure surface;(J rotation between twosections ofslab computed from s

S strain, in plane perpendicular to neutral axis of fa

surface; ( )j.t = roughness coefficientof steel surface, Wa,lther 22

posed values for J . L : for plain bars and wtres = 1.

deformed wtres > 1.25, and for efficiently deformed

and strands = 1.0 to 0.75;

II = bond coefficientjIJ i = inclination angle of failure surface to plane of slab

free edge of slab; and

w = frlf~.

Journal of the

STRUCTURAL DIVISION

Proceedings of the American Society of Civil Engineers

SYMMETRY IN STRUCTURAL MECHANICS

INTRODUCTION

This paper is a review of symmetry as it relates to applied and structural

mechanics; as such, its main purpose is to deflne symmetry, to presents exhibiting different kinds of symmetries and their response to

rtous loading conditions, and to demonstrate how the arguments of symme-

, may be systematically employed and exploited by engineers In their daily

in order to achieve Simplifications in analyses and corresponding

{or:.!)COI10tlllicenefits. To illustrate the full potential use and power of symme-

, desirable to Introduce the principle of superposition; however, for

sake ofbrevity, superposition is not reviewed in detail herein.

The motivation for this arttcle stems from conversations with students

r lectures in which symmetry princlples were briefly introduced, and

comments made by fellow engineers during the last few years, parUcu-

after a recent introductory presentation of these concepts (13). The

of this paper was further stimulated by the almost total nonexistence

rs dealing with these important concepts. Checking back through

of the Structural and Engineering Mechanics DIVisions of ASCE toand 1961, respectively, it was established that not a single article ap-

,'''Mllr,,1'I Inthese journals during this period dealing with symmetry prtnclple s,

this at a time of accelerated use of symmetry and group-theoretic meth-

inmany branches of science and some areas of engineering (4,8,12,20,24,

. . Apparently the last article to appear in an ASCE publication dealing ex-

~.(I;Iusl!velywith symmetry prtnctples was published some 32 yr ago (35). Thetotal absence of papers on thts subject in ASCE Journals Is surely

'IIlUH;a,~ve of the general scarcity of publications dealing with the fundamen-

of symmetry principles and the application of theoretic methods to

Discussion open June 1, onemust be flied with the Editor of Technical Publtcatlons, ASCE.

part of the copyrightedJournal of the Structural Division, Proceedings of theSociety of Civil Engineers, Vol. 99, No. STl, January, 1973, Manuscript was

Ittedfor review for possible publicationonSeptember 2, 1971.Prof., Dept.of Ctv, Engrg., Unlv. of Calgary, Calgary, Alberta, Canada.

71" •

j

!



'12 8T 1anuary, 19'13 8T 1

II

I"

,.



74 January, 1973 ST 1STRUCTURAL MECHANICS

formed as a result of whlch the body will be carried into an equivalent, o ridentical configuration. For structures, an equivalent configuration implies

self- coincidence with respect to both geometric and structural properties o~'

the corresponding parts. Thus, in structural mechanics geometric and struc-

tural symmetry are dealt with. For example, a Ir a me (see Fig. 1) may be.'

geometrically symmetric and yet structurally unsymmetric if the structural

properties of the columns, AB and CD, are not equal. From the point of

of the designer, only a structure exhlbittng geometrlc and structural sym

try of the same kind is truly symmetric; such a structure is sometimes

ignated as being completely symmetric (5).

The symmetry operations and elements primarily required in this analy-

sis are shown in Table 1. The type of symmetry most frequently encountered

flection takes the frame b k' t· "• showmg that a repetition : f cth~n:egs ~rlgl,nal configuration [F'tg; l(c)] thusE, l.e., a3 ,. E. In the ec. on 1~ equal to the identity operation,

•symmetry, in general Ct~~:eOf~~ee-tdlm~ntl0nal ,structures with a plane of

· plane which would p~rmit art t~O eX1S. a t axis of symmetry within thisplane. 0 a on equiva ent to reflection through that

. .Sh~n el:~~.i~: t:wo~~~a~~~nz~fotation, constdar !ne equllegged V-section

about this axis wi ll take the flg~:iS ~ftclearlt

yaIDaxis of symmetry: rotation

e n 0 equ va ent configurations. Rotatlon

.,.13)

TABLE 1.-SYMMETRY OPERATIONS AND ELEMENTS 2.

Number Symmetry element Symmetry operation Symbol'x

(1) (2) (3) ( 4 ) _Q

2 . " ' /3

1 Plane of symmetry Reflection In plane (f

2 Axis of symmetry One or more rotations about axis C C311

3 Center of symmetry Inversion through center Ic

2 :3

FIG. 2.-CYCLOSYMMETRY ANDAXIS OF SYMMETRY

~:/3 W~l~h;;ng the section into its first equivalent configUration a

'ey:~~~o~:ta ti;nhth~~~ht~;e ;~~ ~l~~;ei:~~c~t~oS~t~o:~i:~::~:~:~~g~~:~~!~:

· or sue a rotation and axis of rotation is C in whtch n d t

. r of the axis of symmetry, i.e., the largest valuen~f n in a rota:t:~ ~~

Whi~hCWililake the figure Into an equivalent eonflg~ration. The axis

~Sl~:na'teaY

n s also called an n-fold axis Tntil

equilegged Y-section is a three-fold aXI~ ~~'ignaetadXlbsfcs:mmetry of, e y 3 C3 is also

j y

IIC2 I~l"CI) 1.. rn

. c . . . : l

~:n'2 x

KI "til-RC2 RC2

o a A A 0

U lI) [e II 1

2.

A

FIG. i.-MIRROR ORBILATERAL SYMMETRY

in nature as well as in structures, is referred to as bilateral or mirror

metry and is characte rtzed by a plane of symmetry. The symbol used

plane of symmetry and a reflection through the plane is a. The frame

in Fig. 1, has one plane of symmetry, orthogonal to the plane of the ira

reflection of every material point of the frame through this plane

an equlvalent configuratlon [Fig, 1(b)], which, were it not for the letter

buls used to designate the four corners, is indistinguishable from the

configuration, Clearly, in this two-dlmensional example, the same equtva

configuration can also be obtained by rotating the frame through 180· about

vertical, Y. axis of symmetry. Reflection through the y-plane of sym

or rotation through 1800 about the axis of symmetry is equivalent to

forming all coordinates x of the frame to - x. Repeating the operation of

2

2.

':1

75

I':;



ST 1 STRUCTURAL MECHANICS76

January, 1973 S T l

SYMMETRY IN STRUCTURAL MECHANICS

..(21 0 II 'I

..til<ttll it(2)

B C o B

+, = = - - ~c

!

A 0 Q

(bl

\!.

A

o

(42) . It is inter~sting to note, that aU three symmetry operations previously

analyzed in detatl, are equivalent to one or more reflection; thus reflection is

oftentreated as the fundamental operatlon of symmetry (45).Utilizing, symmetry arguments, it is clear that the center of gravity of

two-dtrnenstonat and three-dimensional symmetric h.omogeneous bodies will

H e along a line of symmetry and in a plane of symmetry, respectively. Two

planes of symmetry define uniquely the position of the centroid of a plane

, I lgure , while f~r a three-dimensional body a combination of symmetry ele-

ments may define the location of the center of gravity, e.g., a plane and an

axis; a center of inversion; or three planes of symmetry. Symmetry a rgu-

.ments are also useful in determining prtnclpal moments of inertia and prin-

ciple axes. From a brief study of the symmetry of such plane figures a~ the

: circle, the square ~ the equilateral triangle, the equilegged Y-section (16), it

, can be seen that, lf a plane area possesses more than two centroidal axes of

: symmetry in its own plane, all centroidal axes are principal axes. Although

: all axes of symmetry lying in the plane of the figure are principal axes not

, all principal axes are axes of symmetry. '

used to represent the operation of rotat130nthrough an angle 211/ . 3 . Similarly~ ~

C2 denotes a rotation through 411/3 and C3 through 61T/3 = 311. Whtle a plane o t ~s;mmetry has only one operation associated with. it, namely. reflectloD'

through the plane, an 1'1- fold axis generates n symmetry operations, namely"

C C2 CJ e1J = E . Note also that the Y-sectlon has three planes O r

n, 71 1 n··· " . .symmet ry. bisecting each leg (sec Fig. 2). The Ir ame shown 111 Flg. 1 bas a ,C axis of symmetry, The type of symmetry exhibited by the Y-section, whlch:

ls2associated with j-ot at inn through rational parts of a circle, is referred W

as cyclo-symmetry, " ' , 'Atwo- fold

axls ofsymnwt ry, c;.

til l'''UI~h theorigin

IS

exhibited byth e

frame shown in Fi~. 3(11) . Howf'V,'1', tnts h';UlH' may also be brought into an,

equivalentconfiguraUon hy lnvprslonthrouJ!:h the center of symmetry, Ie, also

located at the orIgin. The ope ra lion of inversion consists of changing the cn',~

B~ __ ~ __ ~C _

CEp. fT ER OF SYMMETRY. I e

ANO AXIS Of SYMMETRY. ClMost structural engineers have an intuitive understanding of bllateral

,symmetry and, as a rule, make use of symmetry arguments in simplifying

, t i le analysis of such structures as frames, folded plates, shells, etc. The'computational advantages resulting from the application of symmetry in stg-

,nlflcantly reducing the number of independent redundants in highly indeter-

..mlnate structures, or the economies derived from the use of symmetry in

,modifying such methods of analysis as moment distribution or slope deflec-

' tion, are well known. Symmetry may also greatly reduce the amount of com-

:p.ltaUonal work in the analysis of frames, especially where sidesway is a

.problern (5). In studying the buckling of frames (17,41,42), symmetry may be

conveniently exploited to simplify the analysts. The symmetry of the stiffness

andflexibility matrices is taken for granted by designers in the linear analy-

,sis of perfectly elastic structures. The importance of symmetry in analyzing

t i l e elastic properties of homogeneous Hookean materials has long been rec-

,ognized, It is hard to believe that a structural analyst has not, at one time or

another, invoked symmetry and superposition principles (15) in replacing a

genea.l, unsymmetric loading condition on a mirror symmetric structure by

ordinates (x) of each materia I point in the structure to (- Xi) and, in the a b i ; ' ,anequivalsnt symmetric andantlsymmetric load system. Designers could, no

sence of a plane of symmetry, is characteristic of the type of symmetl '7i ;doubt,without too much difficulty, determine the vanishing and nonvanishing

referred to as skewsymmetry. Note that self-congruence of the skewsynt\, ~forceand displacement components on the plane of symmetry of the struc-

metric plane structure may also be acnJceved bl consecutive reflecUo~, aures shown in Figs. 4(a)-4(c) or arrive at the type and existing reaction

through the two planes of skewsymmetry, 1](1) and 0-(2), one along each of t l ! ' : ,~omponentsfor these structures when subjected to various loading conditions,

two coordinate axes [see Fig. 3 ( b ) ]. A , ,It may. however, prove more difficult to arrive at simUar conclusions purely

In addition to the three types of symmetry previously consjde red, mentl~ on the basis of intuItive arguments, for structures such as those ;nown in

should be made of translational symmetry and magnification symmetry, bOI!F~g8, 4(d)-4(h). Therefore, itmight be desirable to have a systematic proce-

of which are, strictly speaklng , associated only with infinite bodies (41,42)·t '~uraby which symmetry arguments may be used to determine existlng and

In all of the symmetry operations described, clearly some transformaU« yanlaning reaction components as well as force and displacement components

of coordinates of the material potnts is involved. Thus It is possible to thl~¢nplanes of structures of various symmetries.

of symmetry as invariance of the relationship between body and an overall ,t Symmetric and Antisymmetric Loads.-The concepts of symmetric and

coordinate system, while that system is subjected to some transformat1~ ,antlsymmetric loading conditions were used by Newell (35) in the analysis of

FIG. 3.-CENTER OF SYMMETRY AND INVERSION, SKEWSYMMETRY

7 7

. !.i



7B January, 1973 ST 1

frames with a plane of symmetry; structures with other types of symmetry

elements were not mentioned. Also, rather than defining symmetric and antie

symmetric loads, 12 conditions, labeled as self-evident by that author, were

quoted and their use recommended for the analysis of indet~rminate struc-tures with a plane of symmetry. Ref. 37 mentions symmetric and antrsym-

metric loads while brief desc rtptions, usually of bilateral symmetry and

corresponding definitions or examples of symmetric and antlsyrnmetrleloads are given in publications such as Refs. 10, 22, 34, and 50. However;

, to the best of this writer's knowledge, no rigorous definition for, or gen-

1.1 lb.

o

A (II

a

(el

r- ,

I A " '-

: ~ . . ~ . r t . ' . . ~" " Ia~" ~

Iv p

(t)

(h)

FIG. 4.-STRUCTURES OF VARIOUS SYMMETRIES SUBJECTED TO SYMMETRIC'AND ANTISYMMETRIC LOADINGS

ST1STRUCTURAL MECHANICS

79

Suchload systems are defined as symmetric loads.

Consider next the completely mirror symmetric frame shown in Fig. 6(a).

Again, applying the symmetry operation (T ( l ) or C2, the frame and load condi-

tionshown in Fig. 6( b) result. Although the structure itself is in an equivalent

posltion, the loads, Q, are in directions opposite to the corresponding vectors

! C zI

! Ol j C ZiT

p ° 1 1° -f s, ~~ lP_9- - F C o "l B

0'(1)

~

~A

OR CZ

°20 a H aA~

~tll I l t ~ o-

a M \ .. .. t v V t - 1 . A0 a A

. . V A - V oI H A I · I H o lI M A I - I M o l

( a l{b I

FIG. 5.-SYMMETRIC LOADING AND REACTIONS ON RECTANGULAR MIRRORSYMMETRIC FRAME

I c2I aUl

O f , 0a c

<T (I)

=--RC2

AI ? . . . . .

HA

~H D

V A V D

IC 2..(n

"I , "Q

1Q

J a 0 a

~

A-V A Vo

, . I V AI ' I v D IHA'HD'O

C al (bl le Ie ral analysis of, symmetric and antisymmetric loads has been puhltshed

heretofore. , 'FIG. 6.-ANTISYMMETRIC LOADING AND REACTIONS ON RECTANGULAR MIRRORConsider the geometrically and structurally minor symmetric fram!.;'!SYMMETRICFRAME

shown in Fig. 5(a ) , subjected to load systems as shown. The frame clearlr; "

has a vertical plane of symmetry, (1(1>, as well as an axis of symmetry, C~ ; Inthe original configuration. The loading system can be brought into self-

in the plane of the frame; performing the operation of reflectlon through th ~t)lncldenceby reversing the signs of the load components Q 1nFig. 6(b) [see

plane, or rotation through 11 about C2, results in the structure and loadinti }. 6(c)]. The loading system assumed for this frame is thus brought into

condition shown in Fig. 5(b), which is equivalent to the original structure ~ ~ntlself-coincidence through the symmetry operation of the structure. An

load Thus this load system becomes self-COincident when the symmetry op:!gperation of reversal of signs must be applied to tho loading, in addition to

eratton of the structure is used to take it into an equivalent configuration! t b t l symmetry operation of the structure, to bring it into a configuration



80January, 1973

ST 1

equivalent to the original load configuration. Su~h.l?ad systems are referred

to as anUsymmetric. The following general defmltlons are offered for sym-

metric and antisymmetric loading conditlons:

1. A system of loads, when applied to a structure possessing certa~n geo-

metric and structural symmetry, is defined as being symmetric, If it Is

brought into an equivalent configuration by the symmetry operattontal of the

structure. '2. A system of loads, when applied to a structure posseesmg ce~taln geo-

metric and structural symmetry, is defined as belng antisyn:metr-Ic,ifit 18brought into an equivalent configuration only by the combmatton of the sym-

metry operation(s) of the structure and the operation of reversal of s igns

(t.e., directions) of all load vectors.

The symmetry or antisymmetry of a particular loading arrangement Is

thus relative and depends on the kind of symmetry of the structure to which

ST 1 STRUCTURAL MECHANICS

same skewsymmetric frame and loading condition as well as the two planes

of skewsymmetrv, ~(l) and i(a). Performing cons;cutively, the operations of

reflection through a(l) and a(a) [see Figs. 7(d), (e)] brings frame and loading

into selt-cotnctdence. Thus, the loading system shown is termed symmetric

for this par ttcula r structure. Similar reasoning quickly leads to the conclu-

sion that the load components shown in Fig. 8(a) are antisymmetrlc for that

structure.

The definitions and arguments used previously for forces are equally ap-

plicable to any polar (true) vector field, I.e., the symmetry and antisymmetry

of any force-field or displacement field is thus rigorously defined. In gen-

eralizing the definitions to vector fields, the case of axial vectors should be

V

hM-H0

° 1 1 °c~

p p . . . . -

-~

B CC2

I ! _ A

M ~ tV

(a)

r \ 1 - ' r j- -I H

l:)

o 8

d

o ~ t o+a

0H-~\...

1\

(b)

IIl

l'I,

Q

-'-t.• MA ' Mo

! H ,, !' IH 0 1

I""I-IVel(el

(dl

"I

i!2)

ld l

: , H,,' Ho· 0

VA· Vo

(9)e I (f )

FIG. 8.-ANTISYMMETRIC LOADING AND REACTIONS ON SKEWSYMMETRICFRAME

81

mentioned. As is well known, axial vectors, such as moments and rotations,

d o not behave as polar vectors under certain transformations. In particular,

FIG. 7.-SYMMETRIC LOADING AND REACTIONS ON SKEWSYMMETRIC FRAME, underan improper orthogonal transformation, such as the reflection, a minussign must be introduced into the transformation rule for components of first

the load system is being applied. One and the same load configuration may b@ order tensors. No such extra negation is required under proper orthogonal

termed symmetriC in one case and antlsymmetric when applied to anoth6r.~t'ansformations, such as rotations. Thus axial vectors are defined as being

structure. . ~8ymmetric and anttsymmetrtc by the previous deflnltions, as long as a syrn-

Two further examples, shown in Figs. 7 and 8, are analyzed herem. ThS',metry operaUon corresponding to a proper orthogonal transformation is

completely skewsymmetric or cycla-symmetric structure shown in FIg. 7 ( 0 ) ) : , u s e d ; under operations correspondlng to improper orthogonal transforma-

has an axis of symmetry, Ca, and is loaded as shown. Applying the aperat1o~i ':tlons,an extra negation is to be applied to all axial vectors. For example, if

Ca, brings the structure and loads Into self-coincidence. Fig. 7 (c ) shows thV:'mlrror symmetrlc couple loads, lylng in the plane of the frame of Fig. 5(a )

;'i

[ I :

'\

ti'II

' · ' 1. : ; : 1 1

fI

~

I )I~

\ '

1 . 1

l:

i·



STRUCTURAL MECHANICS

were applied, clockwise at point Band anticlockwise at point C, raspecttvely, eluded that for this frame when ' 83and if such loads were represented by the ir corresponding axial vectors, the· vertical reaction compone~ts are z z r : to the loading system shown, thedirection of the vectors being defined by the right-hand rule, then the opera" : tlon, Whereas the horizontal reaction~ va jlnhmagnltude and opposite in direc-tion, C2, would take such vectors into self-coincidence, confirming the mlr-]: . an 8 , i.e., I VAl = IVDI and HA = HDror symmetry of the couple loads. However, reflection through (1(1) W OU ld; .

result in antiself-colncidence of the axial vectors; an operation of negation

would be required, as stated in the aforementioned.Symmetric and Antisymmetric Reactions, - Before considering the sym-

metry and antisymmetry of the reaction components for the structures pre-viously mentioned, a moment of reflection, regarding a fundamental axiombased on age-old experience gained from nature (22), may be appropriate,'The axiom may be stated as: symmetric causes, when acting on symmetrIcentities, result in symmetric effects. The writer is not aware of any counter-

example to this axiom, at leastas long as effects are interpreted as the directresults of the causes. Such interpretation, together with a proper under-

standing of the fundamental concepts of stabillty, rules out as a counter- I

example the antisymmetric buckling mode of the mirror symmetric frame o(Fig. 5(a) subjected to equal vertical axial loads at paints Band C. Admitting:this axiom in structural mechanics demands that the operations which bringstructure and loads Into self-coincidence must also take all force and dg,·

formation components into an equivalent configuration. Symmetric loading'systems can thus result only in symmetric reaction components and anti ; :

symmetric loads produce onlyantisymmetric reactions; tne deformation r e · !sponse of a structure must also exhibit the same kind of symmetry as that o nthe loading precipitating such response. Similar conclusions could be rea cheliby postulating a unique solution to a given load system; however, as unique'.ness is primarily associated with linear systems, and is not admissible In '

many classes or problems, hypothesizing unique effects to certain causes;may be more restrictl.ve than admitting the axiom on symmetry q u o t e d ;

previously. :

Now, assume reaction components at A and D for the completely mirror)

symmetric frame shown in Fig. 5(a). Because the symmetry operation of t i l e 'structure brings both structure and load into self-COincidence, the reacUoa "components in Fig. 5(b) must also be indistinguishable from those of Fi~i.

5(a), thereby indicating HA '" HD '" H, VA '" VD :c V, and MA = MD " M : ' "

Se]f-congruence of structure and loads determine, to a certain extent, th e i

reaction components. The sense of the horizontal reactions remains i n d e ~ iterminate, as the H components at the base of the frame could equally w eq . , 'be acting outward on the basis of symmetry arguments. However, the numb$I,of possible unknown external reaction components for this frame is reducedfrom six to three by means of symmetry arguments. :;Consider the frame shown in Fig. 6( a) with assumed reaction componsm

as indicated. Although the symmetry operation a(l) or C2 takes the frame.l .an equivalent configuration, the loading system and reactions can be broug

into self-COincidence only by an additional operation of sign reversal, Ift'

operation of sign reversal, when applied to the forces acting on the equival .

structure, is to bring these vectors into self-co!.ncidence wlth ·tne origl~loads and originally assumed reactions, certain reaction components have

vanish. Clearly, a reversal of sign brings the V reaction components I nself-COincidence, but results in H components ill the equivalent conftguratl

which are of opposite sign to those initially assumed. It Is therefore c

82 January, 1973 ST 1 STl

I

[0

[bl

FlG. 9.-DETERMINATION OF EXJSTINTlON COMPONENTS, MIRROR SYMME~:rD VANlSHING FORCE AND DEFORMA-LOADING C FRAME SUBJECTED TO SYMMETRIC

[b)

e Mf MFS

.... MF-O

Pf ·0

'"·0

~j I ,7v M

[e)

JG . lO. -DETERMINATION OF EXIStlQNCOMPONENTS, MIRROR SYMM~~~~CA~~}:~NlSHING FORCE AND DEFORMA_.Ie LOADING ...runE SUBJECTED TO ANTISYMMET_

T I P ' Symmetry arguments have been used t~oblem to a s tatically determlnat 0 reduce a statically indeterminatej Ne t e one.. • x, consider the Skewsymmetric structures and assumed reaction com-

!



January, 197385TRUCTURAL MECHANICS

:'.

I'

1

'. 'J

84

onents shown in Figs. '7 and B. Applying symmetry operation C2 [Fig. ' 7 ( a ) ; \ \ L symmetry may be used to determine reactton components and deformationrb)] or ~(l) and 0(2) [Fig. '7(c), (d). (e) ] brings the structure, loads, and reac.!: fpatterns for the structures shown in Fig. 4, when subjected to various sym-

tion components into self-congruence, thus tndtcatlng tha~ the reactlon com'~i ; metric or anUsymmetric loading conditions. ,ponents assumed are symmetric. The sense of the reachon components are:, ,With such significant, simplifications in the analysis of structures whenindeterminate from symmetry arguments, as H, V, and M components of 0P": subjected to symmetric and antisymmetric loading conditions, it is desirableposite sign would equally well satisfy the arguments of, symmetr_y. The re-., ' ,to reduce any loading condition to an equivalent symmetric and antisymmetricaction components shown in Fig. B(a) are clearly antlsymmetrlc for this, loading system, and use superposition to obtain the total effect in all thosestructure, provided H vanishes. Again, a statically indeterminate problem i$, ,problems where superposition is applicable (15). Decomposition of a generalreduced to a determinate one by means of symmetry arguments. Analogously," J road system into symmetric and anUsymmetric components was introduced

, byAndree (1) and remains one of the most useful tools of the designer in re-ducing the complexity of linear analysis of structures. This procedure wasalso used recently in the analysis of shells of double curvature subjected to, unsymmetric loading condit ions (1,10).

Force and Displacement Components in Svmmetric Structures.-The sym-metry arguments used in the previous section are valid for any vector, andare thus equally applicable to internal force and displacement components at'rarUcular sections or points in the structure. Consider, e.g., the structureshown in Fig. 5(a ) ; expose the internal force components by taking a vertical, section through tile structure at point F, as shown in Fig. 9(a). Destgnate the; Internal bending moment, axial force, shear, vertical and horizontal dis-

'" " placement, and rotation at section F by MF, PF• S, w, u, and B, respectively,";iand assume their directions as shown in Fig. 9(a). Applying the symmetry

:; t operation of reflection through a(l) or rotation around C2 brings the struc-

, I ture, the loads, and the reaction components into self-coincidence [Fig. 9(b )J .• ',For self- coincidence of the internal force and displacement components at F,

H is clear that S, u, and e must vanish; the sense of PF, MF, and w is againIndeterminate from purely symmetry arguments. Similar arguments are ap-

. plied to the structures subjected to the loading conditions shown in Figs.,.lO(a), l1(a), and 12(a), and the vanishing andnonvanishlng force and displace-

,ment components established., Because none of the symmetry arguments used herein depend on the rna-'!erial properties of the structure, the results obtained from such arguments',are applicable to structures made of any material. Also, no reference was:made to the magnitude of deformations or the linearity of the kinematic r-ela-: tlons and thus, symmetry arguments are equally applicable to the analysis of'tile deformation of geometrically linear or nonlinear structures, even though,as a result of educat ion, training, or experience, symmetry is usually thought,ofa s being useful in connect ion with l inearly elast ic and geometrically linear'~tructures. Finally, it should be noted that in all of the examples presentediIn d etail, a single symmetry operation was sufflcient to bring the structure: Into a n equivalent configuration. There are, of course, cases where morethan one of these symmetry operations are required to take the structure: Into self-coincidence. Symmetric and antisymmetric loads, reactions, and"deformations are again defined as being those which are brought into self-coincidence and antiself-coincidence, respectively, by the symmetry opera-

, t ions of the structure.

!.IF' 0

u '0

w • 0

lal lbl

FIG 11 _DETERMINATION OF EXISTINGANDVANISHINGFORCEANDDEFOR-MATIONCOMPONENTS,SKEWSYMMETRICFRAMESUBJECTED TO SYMMETRIC

LOADING

101

e-leI

CONCLUSIONS

FIG. 12.-DETERMINATION OF EXISTINGANDVANISHINGFORCEANDDEFORMA·TION COMPONENTS,SKEWSYMMETRICFRAME SUBJECTED TOANTlSYMMETRIC After the definitions of symmetry elements and operations are introduced,LOADING :tbese concepts are used in rigorously deflning symmetric and skewsymmetric

, ,; !

, ,,I,

~ .-~I,

!. I

~I

H iIiI

"I:



86January, 1973 STRUCTURAL MECHANICS

structures, symmetric and antisymmetric loads, reactions and displa .'. IQ.Csonk~: ~.• "Sym~etrically a~d Antisymmetrically Loaded Symmetric and Antisymmetric

ments, including polar and axial vector fields. The decomposition of any gen- . Shells, (in Hungarian) HllIlgar.all Academy of Sciences, Vol. 33, 1%4, pp. 262-269.

eral load system into symmetric and antisymmetric components and the II .. Drucker, D. C., Illtroduction 10 Mechal li cs of Deformable Sol ids, McGraw-Hili Book Co.•

power and advantage of such decomposition is mentioned. Symmetry argu-. ' Inc., New York. N.Y., 1967. pp. 51-52; 74; 7 8; 102; 1 12; 1 67-172; 2 07; 3 45-347.ments are used to establish existing and vanishing reaction components In 12.~;!~~ov.L. M., Group Theory mId Irs Physical Applicatiolls. University of Chicago Press.

statically indeterminate structures. Finally, a systematic procedure is de » 13GIockner, P. G., "The Principles of Superposition and Symmetry in Structural Mechanics ..

scribed and illustrated by several examples, whe reby symmetry Ann.ualMeeting of The Engineering Institute of Canada and The Association of Professional

are exploited in determining vanishing and existing force and .dlsplace Engineers of Alberta. Calgary, Canada, May 28. 197Lcomponents at certain sections in stru~tures po~se SSin.g varlO~s kinds 14.Glockner, P. G., "Symmetry and Superposition in Structural Mechanics," Research Report

symmetry, subjected to symmetric or antLsymmetnc loadlO~ condLtions. No. CE 71-lJ, The University of Calgary, Calgary, Alberta, Canada, June, 197LThis analyals hopefully will encourage structural daslgner's to 15.Glockner,P.G., and Vishwanath, T.• "Superposition in Structural Mechanics" Research Report

these prinCiples' and use them more regularly in their dally activities. . No. CE 71-16, The University of Calgary, Calgary, Alberta. Canada, July, 1'971.

though symmetry arguments are used in some speci~lized areas, a c· 16,.GIockner,P. G., "Symmetry and Superposition: Where, When, and How?" Research Report

trated research effort is required to utilize and exploit symmetry ULlll1~~ ..g. ,_.' No. CE 71-17, The University of Calgary. Calgary, Alberta. Canada. July, 1971.

11 t. ' 17.Gregory, M., Elastic Instability, E. and F. N. Spon, London, England, 1961, pp. 196-216'

and associated group-theoretic methods to their fu est exten m 286-291; 320. .

structuralmechanlca. Increased research activity in this field will, "1°H all G G A I' d G "'II 'A ., ,.. , P P Ie roup L eory, mencan Elsevie r Publishing Co Inc New York N Y

edly bring about numerous new applications from which unforeseen helneIUs:.i,' 1967. ...., • . .,

may'result. If this article helps to stimulate and bring about such Hall, M., Jr., The Theory of Groups, The Macmillan Co.• New York, N.Y., 1959.

activity, it wlll have served one of its main p~rpo.ses. It should als.o help Ham.er~esh, M., Group Theory and Its Application to Physical Problems, Addison-Wesley

establishing a common terminology for the appltcatLOn of symmetry 10 Publishing Co., Inc.• New York. N.Y.. 1962

tural mechaniCS. Higman. B., Applied G"oup-Theoretic and Matrix Methods . Oxford University Press Oxford

England, 1955. ' ,

Hoff, N. J ., The Analysis of Structures. John Wiley and Sons, Inc., New York, N.Y. , 1956,pp. 49-51; 115-116. 350-351.

,.. Jaeger, F. M., Le Principe de Symetrie et ses Applications. Gauthier-Villars Paris France, 1945. ' • .

Jansen, L.. and Boon, M., Theory of Finite Groups, Applicatioll in Physics, North-Holland,The Netherlands, 1967.Knox. R. S .• and Gold, A., Symmetry ill the Sol id S ta te Benjamin Co. New York NY1964. • , , ..•

Koepcke, W.• "Application of the Method of Moment Dis tr ibution to Symmetr ic FramedStructures," (in German), BlIuplallung ulld Bautechnik, Vol. 3. July, 1949.pp. 214-216..Leech, J. W., and Newman, D. J., How to Use Groups . Methuen, 1969.

Lightfoot, E., Momellt Distribution. John Wiley and Sons . Inc., New York. N.Y., 196122-32. ' pp.

29.Loebl, E. M.• ed., Group Theory and lis Applications. Vol. I, II. Academic Press. New York,

N.Y.. 1969, 1971.

I.Andree, W. L.. 211r Bereclwullg Statisch Ullbestimmter Systeme• Das B ~ U Verfahren, ' .Lomont, 1. S . • Appl ica tions o f Fini le Groups , Academic Press, New York. N.Y., 1959.R. Oldenbourg. 1919. 31.Lyubarskii, G. Y., The Application of Group Theory in Physics, Pergamon Press, Inc., New

2.Barnhart, C. 1.., Thorndike-Barnhart Comprehensive Desk Dict iOl lary, Doubleday, New York, N.Y., 1960.N.Y., 1955. 32.McWeeny, R., Symmetry-An Ill trodl4ction to Group Theory and Its Applications, Pergamon

3. Bhagavantam, S., and Venkatarayudu. T., Theory of Groups alld Its Application to Press,Inc.• New York. N.Y., 1963.Problems, Bangalore Press, 1951. 33.Melvin. M. A., and Edwards, S. J r., "Groups Theory of Vibrations of Symmetric Molecules,

4.Birss, R. R., Symmetry alld Maglletism, John Wiley and Sons,.lnc .• ~ e.w York. N}"., 'Membranes and Plates," ]ol4rnal o f the Acoust ica l Soc ie ty o f America , Vol. 28, No.2, 1956,

5.Celis, A. J .• and Glockner, P. G .• "The Zero Sidesway Ax,ISof RIgId Frames, . pp.201-216.the Structural Division, ASCE. Vol . 89 ,No. ST5, Proc. Paper 3662, Oct . 1963,pp. 65-89 . 3~.Neal, B . G., Structural Theorems alld Their Applicatiolls , Pergamon Press. Inc.• New York,

6.Charlton. T. M., Prillciples of Structural Analysis. Longmans, 1969, p p, 37,50,71,82. N.Y., 1964, pp. 29-35.90, 120, 130, 131. . . . . 35.Newell. J. S., "Symmetric and Anti-Symmetric Loadings," Civil Engineering, ASeE. Vol.

7. Clirnov, B., "Zur Berechnung von Doppe lt Gekriimmlen Schalen bel Antimetrischer 9,No.4, Apr., 1939,pp. 249-25LEinseitiger Belastung," Bauplallung-BCllltechllik. Vol. 16, No. 11, pp. 556-558: Vol. 16, 36.Nicolle. J. , Die Symrnetrie ulld Ihre Anwendullgen. Deutscher Verlagder Wissenschaften. Berlin,

12. pp. 607-609. 1962. . WestGermany, 1954.8.Cotton, F. A., Chemical Appl ica tions o f Group Theory , 2nd ed., John WIley and Sons, 37.Norris,C . H., and Wilbur, J. B., Elemelltary Structural Analysis, 2nd ed., McGraw-Hill Book

New York, N.Y., 1971, Co., Inc., 1960,pp. 475-476. .9. Crandall, S. H., and Dahl, N. C., eds., A ll Ill troduction to the Mechanics of Solids, McGrikw-liilltu Nussbaum, A., Appl ied Group Theory for Chemist s, Phys ici st s a lld Engi 'l eer s, Prentice-Hall,

Book Co., Inc. , NewYork, N.Y., 1959.pp. 61; 184-187; 238-241; 264-265; 274-275; Inc. ,Englewood Cliffs, N.J., 1971.

ACKNOWLEDGMENTS

The writer acknowledges with sincere thanks the stimulating

made by Mansa C. Stngh, during several conversations, which ~ere pa

larly helpful in clarifying his own thinking on questions concermng the

mental axiom on symmetry.

APPENDIX I.-REFERENCES

87



88 January, 1973 ST1' BTl

i'!I

I

t

39. Or chin, M., and Jaffe, H. H., Symmetry, O rb it al s, a nd S pe ct ra , 1 97 1.40. Pat rashen, M. I., and Trifonov, E. D., ApplicCil io tlS of Grollp Tlleory ill QuallfUm Mechanics, .

MIT Pre ss , Cambr idge, Mass. , 1969.

41. Renton, J. D., "Buckling of Frames Composed of Thin-Walled Members," T h il l-Wa rr e d ' ;

Structures, A . H . C hilv er, ed . • J ohn Wiley and Sons. Inc. , New Yor k, N.Y., 1967. pp. 1 6 -2 4 ;' , .

4~a. .

42. Renton, J. D,. "On the S tab il it y Analys is o f Symmet ri cal F ramework s, " Quarterly Journal

o f M e ch an ic s a nd A pp li ed M nl he ma li cs , Vol. 17, 1964, pp. 175-197,

43. Schonland, D., Molecular Symmetry, D, Van Nostrand Co., Inc . . Princeton, N.J ., 1965.

44. Schouten, J. A., Der Ricci-KarkW, Springer-Verlag , Berlin, Germany, 1924,

45. Shubnikov. A. V., and Belov, N. V., C o lo re d S y mme rr y (Tran s. f rom 1951 Russi an edi ti on ),Pergamon Press , Inc .. New York, N.Y., 1964.

46. Shubnikov, A. V., Symmetry-Tire Laws of S ym me try arid Their Application in Science,

E llg in ee ri ng a nd A pp lie d A rt (in Rus sian) , Mos cow, Union of the Soviet Socialist Republk,

1940.

47. Singh. M. C., and Mishra, A. K .. "Analysis of Modes of vibrauons of Symmetric Networks

by the Group Representation Theory." Proceedings, Third Canadian Congres s of Applied

Mechanics, P. G. Glockner, ed., The University of Calgary, Calgary, Alber ta, Canada, M a y ,

1971, pp, 127~128.

48. Steinman, D. B., "Moment Dis tribution by Symmetr y and Anrisymmetry," Engineering News-:

Record. Vol. 134 , Jan. 25, 1945, pp. 105 -108 .

49. Toupin. R. A .. "Elastic Materials with Couple Stres ses." A rc lt iv es f or Raliollal Mechanics

a n d An ar y si s, Vol. II, 1962, pp. 385~414.

50. Van den Brock, J. A., Elastic Ellergy Theory, 2nd ed., John Wiley and Sons, Inc., New

York, N .Y. , 1942 , pp. 95-100.

51 . V i shwanath , T., and Glockner, P. G., "Effects of Axisymmetric Load on Inflated Noniinear

Membranes" (to be published).

52. Webs ter , A . M., Webster 's Collegiate DiCliOtlClry, 2nd ed., Thomas Allen LId., Toronto , Ontario ,'

Canada, 1941.

53. Wey l, H. , Symmetry. Princ eton Univer si ty P ress, P ri nce ton. N. J. , 1952.

54 . Weyl, H. , The T he or y o f G ro up s ,HId Quantum Mecllanics, Dover Publications. I nc ., N ew

York, N .Y. , 1932 .

55, Weyl, H., T he C la ss ic al G r ou p s, T he ir IIIva r ia l l l s a l ld Rep rese l lta t i o ll s , Princeton University P r e s s ; ,

Princeton, N.J., 1946.

56.Wigner, E. P. , Group Tileory a l ld i t s Ap p li c at i on s 10 tile Q ua nt um M e cl lC wi cs o f A to mi c S pe ct ta ,

Academic Press, New York, N.Y., 1959, .i

57 . Wilson, E. B. J r . • D e ci ns , J. C., and Cross. P. A . Molecular Vibrations, McGraw-Hili Boot '

Co .. Inc., New York, N.Y., 1955. '

58. Yang, T-L, "Symmetry Properties and Normal Modes of Vibration," Joftr llal of NonlinlllT'

Mechanics, Vol. 3 , 1968, pp. 367-381.

APPENDIX ll,-N<YI'ATION

The following symbols are used in this paper:

A,B,C,D

e n

c : rE

F

H,HA,HD

= paints in structure;

= n -Iold axis of symmetry, rotation through 211/n;

= m successive rotations of 2rr/n;

identity operation;

point lying on plane or axis of symmetry of structure;

horizontal force reaction components;

Ie

K,K., K2

L

M, MA, Mn

MF

P

PFQ, Q •• Q2 ' Q 3

S

U

V, VA, Vn

W

wx, Y

oa

a(i)

a(il

an

STRUCTURAL MECHANICS

inversion, center of inversion;

bending stiffness of members in rigid frame structure;=: length of member;

moment reaction components;

internal bending moment at point F;

horizontal concentrated load on rigid frame'

internal axial force at point F; ,

lateral concentrated loads On beam or frame'internal shear force at point F; ,

horizontal displacement component of point F;vertlcal force reaction components;

distributed load;

vertical displacement component of point F;

rectangular cartesian coordinates;

rotation of the tangent to the elastic curve at point F;

~lane of symmetry, reflection through plane of symmetry;

,th plane of symmetry;

ith plane of skewsymmetry; and

= n consecutive repreUtions of the operation of reflection.

, ~.

89

I',. 'j,

!!'

"

symmetry in structural analysis

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