THEORY OF ELASTIC WEIGHTS. J. T. Oden

Transcription

1 /0 THEORY OF ELASTIC WEIGHTS by J. T. Oden

2 ABSTRACT TITLE: THEORY OF ELASTIC WEIGHTS AUTHOR: J. T. Oden DESCRIPTORS: Structural deformations, secondary stresses, nonlinear structures, matri methods. SUMMARY: The general theory of elastic weights is derived from the geometry of closed curves and polygons. Application of the theory is demonstrated for a variety of structural problems.

3 , THEORY OF ELASTIC WEIGHTS 1 J. 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 structural analysis may be considered to be special applications of the general ideas of elastic weights. The etension of these ideas of include in the analysis thermal effects, initial imperfections, elasto-plastic and nonlinearly elastic behavior is presented as well as the relation bedveen elastic weights and the matri force method. 1 Associate Professor of Engineering Mechanics, University of Alabama in Huntsville.

4 INTRODUCTION The use of elastic or angle weights is nothing new in structural analysis. Virtually every tet on elementary structural 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 early 234 as 1868 by Mohr and later etended by Muller-Breslau '. Muller- Breslau presented the bar-chain method for the deflection analysis of simple trusses and used approimate epressions 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. Schwalbe 5 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 Patel 6 and Scordelis and Smith 7. The method was later generalized by Ramey8 for calculating deflections 2 Mohr, 0., "Behandlung der Elastischen Als Seillinie," Zeitschrift der Architekten-Ingenieur-Verein, Hannover, Muller-Breslau, H. F. B., "Bietrag Zur Theorie Des Fachwerkes," Zeitschrift der Architekten-Ingenieur-Verein, Hannover, Muller-Breslau, H. F. B., Die Graphische Statik Der Baukonstruktionen, Vol. II, Part 2, 2nd Ed., Leipzig, 1925, pp Schwalbe, W. L., "Conjugate Load Method in the Analysis of Thin Shells," Proceedings of the First Midwest Conference on Applied Mechanics, p Lee, S. L. and Patel, p. C., "Bar-Chain Method for Analyzing Truss De forma tion," Proceedings, ASCE, Vol. 86, Paper No. 2477, May, Scordelis, A. C. and Smith, C. M., "An Analytical Procedure for Calculating Truss Displacements," Proceedings, ASCE, Vol. 81, Paper No. 732, July, Ramey, J. D., "Elastic Weights for Trusses by the String Polygon Method," M.S. Thesis, Oklahoma State University, Stillwater, 1960.

5 - 2 - of trusses. Tuma and Oden 9 introduced the string polygon method for analyzing comple frames and the concept of elastic weights was used by Gillespie and Liaw lo 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 etension of a number of the earlier special methods to more comple structural systems and demonstrates of the relationships between various methods. In addition, the etension 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 eample, 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, therefore, 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 Frames with Straight Members," Proceedings, ASCE, Vol. 87, Paper No. 2956, October, Gillespie, J. W. and Liaw, B., "Frequency Analysis of Beams by Fleibility Method," Journal of the Engineering Mechanics Division, ASCE, Vol. 90, No. EM!, February 1964, p. 23.

6 ..... c I r- -- c / ' / / 70 / / ~I I E FIG. 1 CLOSED POLYGON '-, './ S I I J-S' / / / / / / / / FIG. 2 SIMPLE CLOSED CURVE

7 - 3 - by noting that if, for eample, 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 oj s q:ds=o.(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 and y components of the displacement of any joint i, ~., be denoted ~ by ~. and ~., respectively. Hence, if B displaces an amount ~ ~y ~ in deforming the polygon, ~B and ~BY are the projections of ~ on the and y aes. Assuming that joint A is stationary B and can be used as a reference, the closed deformed polygon is now traverse in a clockwise path and the algebraic sums of or y components of the joint displacements are recorded. A negative value is assigned to any component acting in the negative or y directions and a positive value to those in the direction of increasing 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 epressed mathematically for a polygon with joints as follows: \' ~. I ~ i~ o.(3)

8 ~ and n "'IJ. L iy i=l a..(4) Again, if 0 and 0 y are the and y components of displacement per unit length of elements of a deformed smooth curve S (that is, n approaches infinity), it follows from Eqs. (3) and (4) that and ()ds <P = 0 s \P () ds = O. s y (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(,y) that is analytic at all points within and on a closed curve S, g> f(,y) ds = 0, This implies that ~, 0, and 0 are analytic functions. y For a final geometric property, consider the typical side DE of a closed polygon shown in Fig. 3. Assuming A to be fied 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. (7) and ~y = tne sin e. - Ed., (8) Proceeding in this manner, similar relations are obtained between

9 .. h A FIG. 3 DEFORMATION OF POLYGON SEGMENT FIG. 4 CONJUGATE MOMENTS

10 - 5 - the components of the joint displacements and the changes in side length and joint angle. THE ANALOGY On eamining the relations developed thus far, it is seen that they suggest that an analogy eists 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) become n Ll\ == O. i=l n IM. == 0 ~ i=l (9). (lo) and n LMiY = O. i=l,(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 ~

11 - 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 eample, side DE as shown in Fig. 4) and if the and y components the conjugate bending moment at D - - are denoted by M D and it is seen that DE is in ~y' elasto-static equilibrium provided ~ == r~e cos e. + PEh... (12) and ~y = ~E sin e. - PEh,.(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 conjugate 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.

12 z o MEA E FIG. 5 CONJUGATE OF CLOSED POLYGON FIG. 6 CONJUGATE OF CLOSED CURVE

13 o Some simple eamples 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, cp 0 ds = rh 0 ds = rh. 'J' y ';t' 122'1'1 122'1'1 122'1 1 1 q:ds = 0, (14) A displacement of a support of a beam such as that shown in Fig. 7b, "opens" a polygon , 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. d2 = EI,.(16)

14 (a) (b) 4 (c) FIG. 7 CLOSED CURVES IN STRUCTURES

15 - 8 - in which M is the bending moment and EI is the fleural rigidity. Thus, 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 r j ji - l"j -. ~ =. i M d EI. (18) P.. represents the angle change between tangents to the elastic 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 p Thus, 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

16 - 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 ~ M. and M. denote the bending moments at i and j ~ J F.., and F.. are angular fleibilities and 'T.. and T.. ~J J ~ ~J J ~ angular load functions. F.. is the end slope of the simple beam J~ ij at j due to a unit moment at j, due to a unit moment at applied loads, etc, These.(21) G.., Goo J ~ ~J, are G.. is the end slope at ~J i, T.. is the end slope at j due to J1 are known constants for a given member. Equations (20) and (21) are quite general, With the proper choice of the fleibilities 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 polygon 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. j

17 The joint elastic weight is the sum of the segmental elastic \veights at j : P. = P.. + P' k J J ~ J.(22) Thus, from Eqs, (20) and (21), P. = M. G.. + M. (F.. + F. 1 ) + M. G k. + ('r.. + T. k) J ~ ~J J J ~ J <: -1< J J ~ J.(23) In analyzing any comple 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 the J 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, P l, P 2,.., 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 aial force is developed in each member and the member

18 t k k A 8 A B z Copla.nar Frame Conjugate Frame P;A' 0 ZI Z _ is. JI jk... P. ~~- JI P. jk P I} ia ~p AI n" [Iasta -stat j cs 21 p. tg I P. AI 1 5egmenta I Elastic. Weights is K ~ kj\t~b X Joint E-lastic Weights FIG. 8 STRING POLYGON PRINCIPLE

19 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, P ih, P jh ' and P kh ' equal to the angle-changes at i, j, and k, and to conjugate moments, M.., and M., and K " equal to the change in ~J J k -K~ length of the corresponding members, Assuming that the aial forces, N ij, N jk, and N ki, 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 M jk = ~jk = N jk ).,jk, (24) and ~i = ~i = Nki ).,jk where A.., = d../ea.,, ).,'k= d.k/ea' k ' and ~. = dk./e~., ~J ~J 1J J J J ~ ~ ~

20 and d ij and A ij are the length and the area of member ij, etc. The quantities A.., A' ' and \. are the aial fleibilities ~J J k 1 of members ij, jk, and ki. Physically, A.. is the displacement ~J of point j relative to i of member ij (the change in length) due to a unit aial force, The elastic weights may now be determined from the equations of elasto-statics equilibrium: M.. + M' l + ~. + P' H y.. + P kh Yk' = 0 ~J J {X ~ J J M.. + M' k +~. +P H.. - PkH k. =0. (25) ~JY J Y ~y J J 1 ~ P ih + PjH + PkH = 0 where y,. = y. - y., Yk' = Y k - y.,.. =. -., k = k -. J~ J ~ 1 ~ J~ J ~ ~ ~ and the and Y subscripts indicate 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 ' k P' H =~ ~ D H (26) where L6 H and ~HY are the algebraic sums of the 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

21 FIG. 9 SIMPLE COPLANAR TRUSS z ~H --, ~H I _ - -r--"" / "- --- / "" "- / J "-,,- / "- I -- -~m FIG. 10 CONJUGATE CELL H

22 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 eample, 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, R lz and R Sz are necessary for the elasto-static equilibrium of forces. Accordingly, MIS is the displacement of joint 1 relative to 5 and R 5z 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 eample, 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 fied joint such as joint 1, and close the polygon by applying the proper conjugate moments and forces for equilibrium (M Sl ' R lz, and R Sz in this case), The displacement of point 3, for eample, 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 approimate 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

23 z is IA ==... ~F 5 (a) - z I ~ j5 1 z bar-chain FIG. 11 COMBINING CONJUGATE CELLS AND FORMING BAR-CHAINS

24 - 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 'k y'k ~ 'fjj. - ~ " f\. + M. G. k + M. (F.. + F. k) + MJ' G. i D H D Iffy -K ~ ~ ~J ~ J H H + Too + 'f',(29) ~J ~ k 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 fleual stiffness of each member when computing the angular fleibilities and load functions. If a better approimation is desired, a new set of aial 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 eample, 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. eist 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 comple 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

25 - 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 aial loads in terms of, W., W.. ~ J Thermal effects are accounted for in a similar manner. In the case of a nonuniformly heated truss, for eample, the etension of member ij due to a temperature increase T is "'-.. = e. LT. ~~J.(30) e. being the coefficient of thermal epansion, 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 w Ti and W Tj 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 eample, a straight beam ij which is constructed of a material obeying a Ramberg-Osgood1 2 type stress-strain law y = a E + 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 Curves by Three Parameters," NACA, TN 902, 1943.

26 elastic or elasto-plastic beam, - 1\ n p = (EI + km ) d.. (32) where k and n are appropriately selected parameters, The moment M in this equation is epressed, by statics, as a function of the end moments M, and M. ~ J then The segmental "elastic II weight at j is. (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 epressing M as a function of M., M. and S gives ~ J P., J~ _(e) = P.. J~ in which behavior k + (EI)o L n + l _(e) P.. J~ is the and is given by Eq. (20). 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 ~ elastic weight due to linearly elastic The total elastic weights 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 matri force method 13 can be interpreted as an application of the theory of elastic weights. In this method a comple structural 13 Argyris, H. J., "Energy Theorems and Structural Analysis," Butterworth Scientific Publications, London, 1960.

27 system is assumed to be composed of a number of component parts called elements, The elements can be any type of solid, fleible 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 threedimensional 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 eample, 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 displacements. These so-called node forces may be forces, moments, becouples, 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 eternal 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 matri (), The remaining known eternal forces are given by a column matri (P). Node forces o (p.) on element i of the structure are related to their corresponding ~ node displacements (6.) according to the formula ~ (p.) = [r, ] (P ) + Cr. ] (), ~ ~o 0 ~, (31) The matrices of the form [r ij ] are rectangular matrices, not necessarily 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 eternal loads ~o 0 about the coordinate aes at each unsupported node of element i and

28 adds a statically equivalent force vector to each node. The matri [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 eist, This means that the total angle-change (sum of the elastic weights) and the relative displacements (sum of the conjugate moments) must vanish, Thus, for a structure with n elements n \' [ T L r i ) f- 'Pi} = (0 J. j=l.(33) Introducing Eq. (31) into (30) and substituting the results into Eq. (33) gives n (' L [r. \. i=l ) T [f.] 1X ~ n [rio]) (p) + CI [rix]t[f i ] [r ix ]} (}= [O}. i=l. (34) from which the redundants can be solved. Equation (34) is precisely the same result obtained in using the matri force method.

29 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 conjugate 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.

30 NOTATION n A parameter - p Elemental elastic weight v C Transverse Coordinate Constant displacement E Young's modulus F.., G.. ~J 1J I L Angular fleibilities Moment of inertia Length of beam segment M Bending moment M., H,, ~ ~ J M., M,, ~ ~ J N., ~J - P.. ~J P. J End moments Conjugate moments Aial force Segmental elastic weight Joint elastic weight T Temperature distribution e. Coefficient of thermal epansion y Etensional strain {) 0 ' y E: A.. ~J ; a,.., ~J Displacements per unit length Initial imperfection Aial fleibility Beam coordinate Normal stress Angular load function

31 i ' ij Angle-changes Angle-change per unit length W. 1 Initial end slope