A NOTE ON AUTOMATED DETECTION OF MOBILITY* OF SKELETAL STRUCTURES by Mehran Chirehdast and Panos Y. Papalambros Design Laboratory Department of Mechanical Engineering and Applied Mechanics The University of Michigan Ann Arbor, Michigan ABSTRACT A geometry-based rigorous approach to the detection of global and internal mobility of skeletal structures (trusses and frames) is presented. The method is useful for automated design of skeletal structures at the conceptual stage where the overall topology of the structure is known but no exact detail on geometry and size is available. Such topologies are, for example, those generated by interpreting homogenization images in topology optimization. The two-dimensional theory is illustrated utilizing two design examples, a classical two-bar truss design, and one generated in the framework of the integrated structural optimization system (ISOS), introduced in earlier publications. An extension to three-dimensional structures is also provided along with a space truss design example, solved in the literature using force-based techniques. May 1991 * Throughout this paper the phenomenon of mobility and immobility of structures is investigated. In order to maintain consistency same terms are used in reviewing the work of other researchers, even though they have used different terms. For instance, stability and instability are avoided because they are used in the context of buckling and equilibrium bifurcation, kinematic stability is an oxymoron, and rigidity is used for structures which are not deformable. 1

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1. INTRODUCTION The recent introduction of homogenization techniques for rigorous determination of proper structural topologies [1] has started to affect the research in the field of structural design. Indeed, it has worked counter to established intuition where conceptual design is performed by humans (or by some sort of "intelligent" computer programs), the subsequent details handled by computer-based analysis. Rather, in an integrated design environment using homogenization as part of the strategy [2, 3] the conceptual design phase is performed by a rigorous analytical optimization methodology, resulting in a gray scale image of the proposed topology. It is then that the human (or "intelligent" computer program) is called to interpret this image into the form of a realistic, manufacturable structure. In a three-phase process, described in the cited references, a design task is initiated by specifying only boundary conditions, type and amount of material, and generating the image by using a homogenization-based optimization procedure (Phase I); image processing and interpretation, possibly including alterations due to manufacturability requirements, yield a parameterized CAD-type representation (Phase II); detailed shape and size optimization that may include additional constraints puts the finishing touches on the design (Phase I). An early computer implementation of this strategy is the Integrated Structural Optimization System (ISOS) [2, 3]. One class of structures resulting from processing and interpreting homogenization images is skeletal structures, i.e., trusses and frames. As one may expect, mathematical optimization will tend to push the topological layout to its limits of efficiency, so skeletal interpretations of homogenization images may yield kinematically mobile structures. The homogenization process itself does not generally result in mobile structures, but the skeletal interpretation may do so. To illustrate this phenomenon two examples for skeletal structures developed through ISOS are provided here. The input to homgenization, consisting of an initial design domain, and boundary and loading conditions, is depicted in Fig. 1(a) and its output for a solid/void ratio of 1/3 is shown in Fig. l(b). Figure 1(c) shows an intermediate output of ISOS, that is a skeletal interpretation of the image shown in Fig. l(b). Is this skeletal structure mobile or not? (In structural mechanics and kinematics the former is called a "structure" and the latter a "mechanism".) For this particular design example, a truss (pin-joined) interpretation is mobile, whereas a frame (welded) interpretation is not. Another example is shown in Fig. 2 where both interpretations are immobile; this example has been extensively treated in [3]. 2

(a) (b) (c) Fig. 1: (a) Input to.homogenization, (b) optimal topology generated by homogenization, (c) mobile truss interpretation of image (b). A design engineer can use intuition and visual inspection to detect any mobility in the structure and subsequent detailed structural analysis would reveal any undetected mobility problems. Automating this task in the preliminary design stage using only geometric information requires operationally useful necessary and sufficient conditions for immobility. This article addresses the development of such conditions and the computational implementation in the ISOS environment. The ideas are akin to those in classical kinematics, but tailored to the present needs. Note that the developed theory is applicable outside the framework of ISOS as well, and can be used in any automated conceptual design environment. In the remainder of this introductory section we will make some links with previous work, before we proceed with developing the needed conditions. The well-known rule of static deternacy (indeterminacy), that twice the number of nodes minus the number of bars has to be equal to (greater than) the number of unknown external components, is mentioned in elementary texts as the necessary condition for immobility of trusses, see, e.g., [4, 5, 6]. Obviously necessary conditions do not guarantee immobility. Norris and Wilbur [4] introduce a set of heuristic rules based on engineering intuition as sufficient conditions for immobility. Weaver [5] suggests that the number and position of supports and members be adequate for immobility of the structure but without giving specific means of determining such adequacy. 3

(a) (b) (c) Fig. 2: (a) Input to homogenization, (b) optimal topology generated by homogenization, (c) immobile truss interpretation of image (b). In early work, Southwell [7] introduced tension coefficients to determine the forces in the bars of statically determinate trusses. Generalizing the set of loads applied at the nodes can reveal a unique relation of internal forces to external loads. This uniqueness, referred to as determinacy, is equivalent to immobility of the structure. Such a force-based method was applied as recently as 1980 to detect mobility of space trusses [8]. A similar approach was taken by Timoshenko [9] as a basis for deriving the zero-load test which is less attractive for automation than the original tension coefficient method. One may note that these procedures cannot be formally extended to statically indeterminate structures, and that no formal definition of immobility of skeletal structures is used for rigorous proofs. An additional disadvantage is that components of the structure have to be treated as external forces, resulting in a lack of differentiation between global and internal immobility, as will be discussed later in this article. Using classical mobility criteria based on potential energy of mechanical systems, such as Dirichlet's criterion (see, e.g., [10]), has the disadvantage of global kinematic complications. In a typical kinematics treatment, e.g., in [1 1], Griibler's formula is discussed as a method of determining the degree of freedom of a mechanism. A degree of freedom equal 4

to or less than zero, although necessary for immobility of the structure, does not guarantee that the structure is not mobile. Even more sophisticated formulae for the degree of freedom, such as the one derived by Kutzbach [12], represent only a necessary condition. Hunt [13] discusses how screw theory can be applied to determine the so-called actual degree of freedom of mechanisms. He also indicates (without elaborating) that the theory may be applied to cases where the degree of freedom of the mechanism is less than or equal to zero, i.e., where one is dealing with structures. As mentioned earlier, one way of detecting mobility is structural analysis, for example, using finite-element methods. For every skeletal structure a stiffness matrix is computed which, multiplied by the displacement vector, gives the loading. If the stiffness matrix of a structure is singular, without applying any loads there exist nonzero displacements. From our present viewpoint, the disadvantage of this approach is that detailed information on the structure is not available at the conceptual level of a design process. The approach taken here is purely geometric and ignores the loading conditions. The definition of immobility is in terms of geometric quantities (see Section 2), the loads contributing to mobility of the structure rather than causing it. The geometric configuration of the structure is considered as the primary cause of mobility. This approach is based on ideas introduced by GriUning in [14] for two-dimensional skeletal structures (both trusses and frames). The theory has been applied to two-dimensional statically-determinate topologies generated in the framework of ISOS with extensions to three-dimensional structures. As most of the homogenization results to date are two-dimensional, we will start with the two-dimensional case. The remainder of the article is as follows. In Section 2 some terms are formally defined and necessary and sufficient conditions for immobility are derived. In Section 3 some remarks useful for application purposes are provided. Two two-dimensional examples are given in Section 4 for illustration. A three-dimensional extension of the theory is provided in Section 5, followed by a space truss example in Section 6. 2. NECESSARY AND SUFFICIENT CONDITIONS FOR IMMOBILITY We begin with some definitions. A node or joint is a point in space used to connect one or more members of a skeletal structure. A linear member or element is the linear connection between two nodes. The linear elements are divided into bars and beams. Bars can carry only axial forces whereas beams can carry axial and shear forces and also bending moments. Bars and 5

beams can be connected at the nodes using hinges or welds. If a linear element is connected via a hinge, at its respective end it accepts only forces. In contrast, if a linear element is connected via a weld, at its respective end it accepts both forces and moments. A weld is considered stiff about those of its axes that can transmit moments. A clamp or a clampedend supports a structure by providing an external moment around a given axis. A (simple) support supports a structure by providing an external force component in a given direction. Definition 1. A skeletal structure is defined as a set of distinct nodes connected by a set of linear elements and externally supported by a set of clamps and supports. Bars, beams, nodes, simple supports, and clamps are collectively referred to as the members of the structure. A skeletal structure with only pin-joined bars is referred to as a truss, while one with only beams joined by welded joints is called a frame. A skeletal structure that is neither a truss nor a frame is referred to as a hybrid structure. If not explicitly specified, the term "structure" is used for hybrid structures for the remainder of this article. Definition 2. A skeletal structure is globally immobile, if and only if changes in the positions of its nodes (i.e., node displacements) with respect to fixed points are possible only by simultaneously deflecting the bars and beams, and/or deforming the stiff joints, and/or moving the supports, and/or deforming the clamps of the structure. Definition 3. A skeletal structure is internally immobile, if and only if changes in the positions of its nodes, relative to each other, are possible only by simultaneously deflecting the bars and beams and/or deforming the stiff joints of the structure. As it will be explained later in this article, the internal and global immobility of a structure differ only in terms of the kinematic boundary conditions used to support that structure. A skeletal structure is mobile (internally or globally), if and only if it is not immobile. An obvious simple theorem can be then stated as follows. Theorem 1. A structure is mobile, if and only if displacing any combinations of its nodes is possible without deflecting or deforming members of the structure. Proof: The proof follows trivially from the following logical equivalences and Definitions 1 through 3. (1) p X q -,p X -nq and (2) -,(p * q) - p A -iq. In case of Equivalence (2), 6

p corresponds to existence of nonzero nodal displacements and q corresponds to existence of nonvanishing deformations in the structure. Theorem 1 means that a nodal displacement in a rigid structure (where no deformations take place), exists if and only if the structure is mobile. This statement can be easily verified for any mechanism. Based on this theorem, analytical expressions for necessary and sufficient immobility conditions of two-dimensional structures can be derived. The following symbols are used. All coordinates of the nodes and their deflections are expressed with respect to a Cartesian coordinate system as shown in Fig. 3. The position of node i is expressed in terms of its coordinates xi and yi. The differentials Axi and Ayi denote the infinitesimal displacement components of that node along the x and y axes respectively. The length of linear element (ik) connecting nodes i and k is denoted by sik, its change by Asik. The welded joint r connects linear elements fri) and (rk), Or denotes the angle between those members and A0r the infinitesimal change in that angle. A simple support r constrains the movement of node r along a prescribed direction for a given distance cr, (see also Fig. 4). A clamped-end r constrains the angle of rotation of beam (rk) about an infinitesimal angle of A. Note that a clamped-end r does not necessarily impose any constraints on the displacement of the corresponding node r. With the above notation, the following corollary to Theorem 1 can be stated. Corollary 1. A structure is mobile, if and only if the deformation quantities (As, AO, c, and t) can vanish while the displacement quantities (Ax and Ay) do not. Note that this corollary and other theorems discussed here are not limited to twodimensional structures. We now proceed to derive operationally useful expressions for necessary and sufficient conditions for immobility (or mobility) of skeletal structures. As it can be readily seen from Fig. 3: Sik2 = (Xk-Xi)2+(yk-y)2 (1) (sik+Asik)2 = [(Xk-Xi)+(AXk-Axi)]2+ [(Yk-Yi)+(Ayk-Ayi)]2 (2) 7

y k X Fig. 3: Positions of nodes and dimension and orientation of their connections. The deflection Asik of member (ik) is presumably small compared to sik so that the Asik Axk-Axi quantity si is much smaller than one and can be neglected. The same is true for~~~~~~~~~~~~~~~~Sik ~~~Sik and yk y Subtracting the two equations above, dividing by Sik, and taking this approximation into account, we obtain Ask =(Axk-Axi)(xk-xi) + (Ayk-Ayi)(Yk-yi) ~sik = (3) Sik As shown in Fig. 3, let ark be the angle between the x axis and the direction of element (rk), and let Ork be the same angle after the nodal positions have changed. The change in that angle is denoted by AccOk; i.e., Aark = rk-ark (4) For small AWOrk we have A(rk tan(AOCrk) = tan(%k-cark) = tan(G&-t(5) 1+tan(q,k)tan(ark) The following equation can be easily derived from geometry 8

tan(ok) = Yk-Yr (6) Xk-Xr where Xk ~ Xr. Similarly for crk, we have tan(grak) = YkYr+ r (7) Xk-Xr+AXk.AXr Axk-AXr AYk-yr Again assuming that the absolute values of quantities A and are much Srk Srk smaller than unity the following equation is obtained from Eqs. (5), (6), and (7). (Xk-Xr) (AYk-AYr)-(Yk-Yr)(AXkk-AXr) Srk2 The case where Xk = xr can be treated by the following coordinate transformation: x* = y and y* = -x, where the new coordinates have * as —a superscript. A back transform leads to the same final result (Eq. (8)) as for the case where xk ~ xr. Noting from Fig. 3 that AOr = aOri-Aark (9) and using Eq. (8) we obtain that Adr = (xi-xr)(Ayi-Ayr)-(yi-Yr)(Axi-Axr) (Xk-xr)(AYk-Ayr)- (Yk-Yr) (Xk-AXr) A~e, Sri2 Srk2, (10) As illustrated in Fig. 4, for every simple support r which constrains the deflection of node r for a distance cr in a direction along a line at an angle 5 with the x-axis, the following relationship is obtained Cr = Axrcos(4)+Ayrsin(4) (11) A relation can be derived from Eq. (10) for every clamped-end r where the angle of deflection of the beam (rk) is restricted to an angle Tr. Note that if AXri = 0, then tr=Aoc1k. Therefore 9

(Xk-Xr) (AYk-AYr)- (Yk Yr) (AXk-Axr) /1,\ tr=- - Srk2 _r yC Ayr, 1t AXr X Fig. 4: Support r whose movement is constrained along a given axis. Equations (3), (10), (11), and (12) provide the sought immobility criteria in the form of linear equations relating the displacement quantities (Ax and Ay) to the deformation quantities (As, AO, c, and t). The following symbols are used for the numbers of various members: k = number of nodes, r = number of linear elements, w = number of welds, a = number of support components, ande = number of clamped-ends. Now, Eq. (3), (10), (11), and (12) can be uniformly represented in the matrix notation Ax = b, where x = (Axl, Ay,,..., Axk, Ayk)T, an n-vector, b = (Asl,..., Asr, A01,..., AOw, c1,..., c., t1,..., t)T, an m-vector, and A is an [m x n] matrix. Note that n = 2k and m = r+w+a+e. Using Corollary 1 we conclude that a structure is globally mobile, if and only if the linear system Ax = o has a nontrivial solution. Linear algebra theory leads to the following results (see, e.g., [15]). Theorem 2. A structure is globally immobile, if and only if rank(A) = n. Corollary 2. A structure is globally mobile, if and only if rank(A): n. Comparing Definitions 2 and 3, one can conclude that internal immobility can be studied the same way as global immobility by removing the external constraints on the structure and fixing the Cartesian coordinate system to the structure. Removing the external constraints corresponds to neglecting equations for supports and clamped-ends, i.e., Eqs. (11) and (12). Fixing the Cartesian coordinate system to the structure can be done by fixing 10

one node to be the origin of the coordinate system (without loss of generality, we can assume that node to be Node 1) and constraining the motion of an additional node along one axis, such that the other axis passes through that node (again without loss of generality, we can assume that node to be Node 2). This procedure corresponds to translating and rotating the coordinate system and also setting to zero the displacement components Axl, Ay1, and Ay2. Note that the Cartesian coordinate system needs to be translated such that Node 1 is its origin and the x axis passes through Node 2 (see [15] for details on coordinate transformations). Now the linear system of equations represented by Eqs. (3) and (10) remains to be studied, that is Aix = b, where x = (Ax2, Ax3, Ay3, Axk, Ayk)T, an ni-vector, b = (Asl,..., Asn, A01..., Aow)T, an mi-vector, and Al is an [mi x ni] coefficient matrix, with ni = 2k-3, and mi = r+w. Note that Eqs. (3) and (10) need to be represented in terms of the transformed nodal coordinates. The following result is then easily shown. Theorem 3. A structure is internally immobile, if.nd only if rank(Ai) = ni. This concludes the basic two-dimensional immobility results. 3. REMARKS ON APPLYING THE THEORY Some remarks are now in order regarding the proper application of the above results in practice. For trusses no welded joints and clamped-ends exist, therefore e = w = O. No other difference exists in the study of their immobility compared to that of hybrid structures. However, note that according to our earlier definitions, trusses have certain modeling restrictions: (a) no moments can be applied; (b) forces can be applied only at the joints. As stated in [8], if a truss is mobile its equivalent frame model may be immobile. In most of such cases the nodal deflections of the frame are still considerably large, since the bending effects are dominant in order to sustain the structure which would be mobile if only axial forces were used. For a joint with k beams welded on it, there exist only k- 1 linearly independent angular equations (Eq. (8)), although the number of possible combinations of enclosed angles (pairs of linear elements) equals k(k-l). The quantity a denotes the sum of the number of linearly independent constraints for each support. For example, for a pinned support that number is equal to 2, the corresponding equations being Ax = Ay = 0. 11

Half-hinges are used to model some hybrid structures. Figure 5 depicts a halfhinge. The coordinate changes of half-hinges are not treated as independent unknowns of x, the vector of the displacement quantities. This is due to the fact that the coordinates of half-hinges are linear functions of the end nodes of the beams on which they appear. However, the length of the bars or beams whose end nodes are half-hinges provide us with a constraint. Alternatively, one can introduce two independent displacement components (Axh and Ayh) for each half-hinge and add two deflection equations (constraints) for every beam on which the half-hinge appears. These equations are Axi+t(Axj-Axi)-Axh = 0 (13) Ayi+t(Ayj-Ayi)-Ayh = 0 (14) where i and j refer to the two end nodes of the beam on which the half-hinge appears. Parameter t = x-xi if xi ~ xj, otherwise t = yh-Yi. Note that according to the definition of a Xi-Xj~i ~~Yj-yi skeletal structure (Def. 1), xi ~ xj or yi * yj. Equations (13) and (14) can be derived using the fact that a half-hinge h belongs to the line passing though nodes i and j, both before and after nodal displacements take place. A comment needs to be made on the completeness of the set of joints used to connect linear elements for both two- and three-dimensional structures. Although the joints commonly used in structural design are treated in this article, the set of joints is by no means complete. Equations for other joints, mainly used in mechanism design, can be derived similarly. I k Fig. 5: A half-hinge h in the middle of linear element (iy) and at the end of (kh). Finally, we recall that the rank of a matrix is the number of linearly independent rows or columns and the most commonly used algorithm to determine the rank of a matrix is some sort of Gaussian elimination. After performing Gaussian elimination, the matrix consists of some rows containing only zeros and some with non-zero elements, the rank of 12

the original matrix being equal to the smaller of the two numbers of nonzero rows and columns of the transformed matrix. In practice, zero will be a number which is small relative to the others. Thus, prior to performing the Gaussian elimination, we should normalize (scale) the row vectors of the original matrix which represent comparable physical quantities. Any number in the resulting matrix whose absolute value is smaller than a certain amount, say 10-2, should be regarded as zero. 4. TWO-DIMENSIONAL EXAMPLES In this section two truss examples are discussed to illustrate the applicability of the above results. Figure 6 shows a statically determinate two-bar truss with two pinned supports. This configuration is immobile except for the case where the angle a becomes zero or i radiants. This fact will be verified using the theory. According to the first comment in the previous section e = w = 0, r = 2, a = 4, k = 3, and therefore the matrix A is 6 x 6. The four equations for the supports (Eq. (11)) are easily written by setting the x and y displacement components of Nodes 2 and 3 to zero, i.e., Ax2 = Ay2 = Ax3 = Ay3 = 0. Two equations are set up for the deflection of the two bars according to Eq. (3): for Bar 1 (connecting Nodes 1 and 2): Asl = Ax2 - Axl, and for Bar 2 (connecting Nodes 1 and 3): As2- (Ax3 -Axl)cos(a)+(Ay3-Ay1)sin(a). Thus, the homogeneous system of linear equations has the following matrix representation. -1 0 1 0 0 0 Ax -cos(c) -sin(a) 0 0 cos(a) sin(a) AY1 0 0 1 0 0 0 AX 2=0 0 0 0 1 0 0 AY2 0 0 0 0 1 0 Ax 0 0 0 0 0 1 AY3 L~Y3~ ~(15) Now, rank(A) = 6, if and only if sin(a) ~ O. Therefore, according to Theorem 2, the necessary and sufficient condition for the mobility of the structure is that a be equal to zero or in. One may also verify that regardless of the value of angle a, the structure is internally mobile. 13

3 a x Fig. 6: Two bar truss design example. The second example is a skeletal structure generated by homogenization in ISOS. Figure 7(b) shows a computer-generated skeletal image of the stiffest possible topology for a given amount of material for the design problem illustrated in Fig. 7(a). Using computer vision techniques the image of Fig. 7(b), i.e., the output of homogenization, is transformed into the truss structure shown in Fig. 7(c). The nodal coordinates and their connectivities are extracted, and support conditions are retrieved automatically and stored in data structures, serving as the input to the mobility detection algorithm. For this particular problem this information has the form shown in Table 1. (a) (b) (c) Fig. 7: (a) Initial design domain and specifications as input to ISOS, (b) computer-generated optimal topology, (c) final design interpretation of the structure as a truss. 14

Table 1. Input to the mobility detection algorithm. Node Position H y 1 40 1 2 8 19 3 72 19 4 19 31 5 62 31 6 35 37 7 47 36 8 1 1 9 80 1 connected nodes 2934675 148 1 59 126 1 37 147 1 56 12 13 Boundary Conditions Force Applied * roller along H-axis pinned Joint * This fact is not needed for the purposes of mobility detection. The size of matrix A for this problem is 18 x 18. The output of the algorithm shows the rank of the matrix to be equal to 18, and hence the structure is globally immobile. Note that because of the given support conditions global and internal stabilities are equivalent. 5. EXTENSION TO THREE DIMENSIONS Definitions and theorems introduced in Section 2 are also valid for three-dimensional structures. The main difference is in the equations derived in Section 2, even though some of them can be easily extended. Some types of joints do not exist in two-dimensional structures. Equations for them will also be derived in this section. Derivations of those equations trivially extendable from the two-dimensional to the three-dimensional case are not discussed here. The nomenclature introduced in Section 2 is still valid unless stated otherwise. Equation (3) can be extended to Eq. (16) where zi denotes the z-coordinate of node i and Azi the infinitesimal displacement of that node along the z-axis. ASik = (AXk-AXi) (Xk-Xi)+(yk-AYi) (Yk-Yi)+(Azk-zi)(Zk-Zi) Sik (16) Equations for supports where the movement is restricted along a given direction are derived as follows. Two angles characterize the restricting direction of support r. These two angles are shown in Fig. 8 and are denoted by 0 and p, c, being the amount by which 15

the support is forced to move. The following equation can be derived for c, as a function of the displacement components of the node and the two introduced angles. Cr = AXrCOS((p)cos(o)+Ayrsin((p)cos(0)+Azrsin(0) (17) z r Cr p~ Az r y AX X Ay Fig. 8: Spatial position of a constrained support r. For the two-dimensional case equations for angles of clamped-ends and welds were derived similarly, whereas in the three-dimensional extension different approaches need to be taken to derive those equations. We will start with equations for clamped-ends. Figure 9 shows a clamped-end r in three-dimensional space supporting member (ri). The displaced positions of the nodes are underlined. An orthogonal coordinate system (el, e2, e3) is introduced where el is a unit vector along the initial position of member (ri), and e2 and e3 3re perpendicular to el and to each other, i.e., el = (Xixr Yi-Yr ZiZr) (Xirir, Zir) Sir.Sir (18) r(1, o, o) e2 = j(Yir -Xir, 0) Y X ir2+Xir2 e3 =el x e2 if Xir= Yir= 0 otherwise (19) (20) 16

(X ir AY ir o r' Ir ir I -3 Fig. 9: A clamped-end and its characteristic dimensions and angles. where x indicates a vector product. To simplify notation, we set xir = xi-Xr. Similar notation is introduced for the other coordinates and nodes. The displaced position of node i is denoted by i and its coordinates can be expressed in the initial (ex, ey, ez) and the new (el, e2, e3) coordinate systems as follows: i = (Xir+Axir)ex+(Yir+Ayir)ey+(Zir+Azir)ex = il el+J2 e2+i3 e3 (21) Without loss of generality we can assume that the origin of both coordinate systems is L, the displaced position of node r. Taking the inner product of both sides of Eq. (21) with e2 and e3 gives the equations for iL and j3, respectively. i2 = (Axir, Ayir, Azir)e2 = Axire2x+Ayire2y+Azire2z (22) i3 = (AXir, Ayir, Azir)e3 = AXire3x+Ayire3y+Azire3z (23) Components of e2 and e3 can be found from Eqs. (19) and (20), respectively. As mentioned earlier (for the derivation of Eq. (17)), two angles are needed to prescribe the position of a line with respect to a given reference frame, assuming that one point of that line is fixed. In the clamped-end case these two angles are Yr and 6r, shown in Fig. 10. For small angles the following approximations are valid ir 6r 2 (25) Sir 17

so that, approximately (Axire3x+Ayire3y+Azire3z),yr = sir (26) -r = (Axire2x+Ayire2y+Azire2z) Sir (27) Figuire 10 depicts the cross-section of a type of clamp for which the angular deflection of the linear element (ri) in a given plane, with the given normal vector e2 = (e2x, e2y, e2z), is not restricted. The out-of-plane angular deflection about e3, denoted by r, is specified and can be obtained in a manner similar to i in Eq. (27). Fig. 10: Cross-sectional view of a clamped-end without in-plane angular constraints. A slightly more complicated situation arises when one is dealing with welded joints. Figure 1 1 shows a welded joint r connecting two elements (ri) and (rk). The displaced positions of nodes are underlined. As mentioned earlier, two angles are needed to determine the relative position of two elements with respect to each other. The only way to obtain two meaningful angles is to project the nodes onto two different planes. There are infinite possibilities for two normal planes in the three-dimensional space. However, as it will be explained later, there is only one alternative pair of projections which provides us 18

with reasonable results. First we discuss the unit vectors introduced for the purpose of the projections. The origin of all unit vectors is assumed to be in r. Vectors el and ek are parallel to elements (ri) and (rk) respectively and e3 is their cross-product. Vector el is along the vector sum of ei and ek, and (el, e2, e3) represents a right-handed coordinate system. Thus, the following equations can be derived for these vectors: e =(Xi-Xr, Yi-yr, Zi-Zr) (Xir, yir, Zir) Sir Sir (Xk-Xr, Yk-Yr,Zk-Zr) = (Xkr, ykr, Zkr) Skr Skr (el x ek) e3 =Ilei x ekil = (el + ek) el = ilei + ekll e2 = e3 x el (28) (29) (30) (31) (32) where 11.11 denotes the Euclidean norm of a vector. k Fig. 11: A welded connection and its associated angles and dimensions. The first projection is along the vector e3 (thus, onto the plane spanned by ei and ek). The infinitesimal in-plane angle of deflection oar equals the difference of the two angles (i and ok. The final result for this angle is given in Eq. (33) below. Replacing the components of the given vectors in that equation will lead to unnecessarily long expressions and is therefore avoided. 19

or = (Axir, Ayir, Azir)(e3 x ei)-(Axkr AYkr, AZkr)(e3 x ek) (33) The second projection can be carried out along any vector in the plane spanned by et and ek. However, in order to avoid singularities and also for the linearization to be valid, the projection of neither of the elements may vanish or even become small compared to the original lengths of the members. The following procedure provides a general projection fulfilling the above requirements. The projection is carried out along e2 (thus, onto the plane spanned by el and e3) as long as the angle between ei and ek is less than or equal to 2; otherwise the projection is carried out along el (thus, onto the plane spanned by e2 and e3). The following equation gives a general formula for the out-of-plane angular deflection denoted by fr. (AXir, Ayir, AZir)e3 (AXkr, Aykr, AZkr)e3 (Xir Yir, Zir)el (Xkr, Ykr, Zkr)el if ee2 Wr (34) (Ai y, zir)ei (AXkr, Aykr, AZkr)e (4 (Xir, Yir Zir)e2 (Xkr, Ykr, Zkr)e2 Note that Yr is a smooth function of ele2, since at ele2 = O projection of the vector (xir, yfir, Zir) onto el and e2 are equal and the same is true for the vector (xkr, ykr, zkr). Eqs. (33) and (34) are derived for the case where nodes r, i, and k span a unique plane. They can be easily extended to include the singular case where the three nodes are aligned. The vector el is then a unit vector parallel to the line (rik), and vectors e2 and e3 are any pair of orthogonal unit vectors perpendicular to el. Equations (19) and (20) give one way of computing these vectors. An equation relating displacement and deformation of revolute joints can be derived similarly to Eq. (34). Revolute joints act as a hinge in the plane spanned by the three nodes and as an out-of-plane weld. A door hinge is an example of a revolute joint. It is obvious that although Eq. (33) is still valid, angle y is no longer prescribed. The prescribed out-ofplane angle of deformation is denoted by X and computed exactly as Wx in Eq. (34) for welds. Similar immobility arguments are valid as for the two-dimensional case and will be discussed here. Equations (16), (17), (26), (27), (33), and (34) provide the sought immobility criteria in the form of linear equations relating the displacement quantities (Ax, Ay, and Az) to the deformation quantities (As, c, y, 6, A, a, yr, and A). The following symbols are used for the numbers of various members: k = number of nodes, r = number 20

of linear elements, a = number of support components, e = number of clamped-ends, p = number of clamped-ends where the in-plane angle is not restricted, w = number of welds, and h = number of welded(stiff) joints where the in-plane angle is not restricted. Now the equations numbered above can be uniformly represented in the matrix notation Ax = b, where x = (Axl, Ayl,..., Axk, Ayk)T, an n-vector, b = (As1,..., Asr, cl,..., Ca, Yi1,. /Ye' 61,..., 6e U17... o w'Up. ) *, 1 *...) 1v X) khJ)T, an m-vector, and A is an [m x n] matrix. Note that n = 3k and m = r+a+2e+p+2w+h. Theorem 2 and its corollary derived in Section 2 are valid for three-dimensional structures as well. In a way very similar to the two-dimensional case, internal immobility is studied by ignoring the external constraints and fixing the Cartesian coordinate system to the structure. Ignoring the external constraints corresponds to discarding Eqs. (17), (26), and (27). Fixing the coordinate system to the structure is analogous to the two-dimensional case and requires two steps. First a secondary coordinate system attached to the structure must be introduced. Second, the coordinates of nodes must be transformed from the initial coordinate system to the newly introduced one. Since the procedures to perform the second step can be found in references on elementary analytical geometry, such as [15], we will discuss only the first step in more detail. We assume that the structure has at least three non-aligned nodes, otherwise it can be treated by means discussed in Section 2. The origin of the new coordinate system is Node 1, its x axis passes through Node 2, and Node 3 is on the xy plane. In order to uniquely determine the new coordinate system we assume the y coordinate of Node 3, y3, is positive. Therefore, the following displacement components have to vanish: Axl = AY1 = AZ1 = AY2 = Az2 = Az3 = O. The new coordinate system is now uniquely fixed to the structure; its unit vectors (ex, ey, e) can be found as follows. (x2-xl, Y2-yl, z2-z1) (35) S12 = (x3-xl Y3Y1, Z3-zl) (36) S13 ez = ex x v (37) ey = ez x ex (38) where v is an auxiliary vector. As in the two-dimensional case, the system of Eqs. (16), (33), and (34) represented in the form Aix = b, determines the internal immobility of the structure, where x = (Ax2, Ax3, Ay3, Ax4, Ay4, AZ4,.., Axk, AYk, Azk)T is an ni-vector, b = (Asl,..., Asr, Y1,.. Cw 1, *., yw, Xi,..., kh)T an mi-vector, Al is an [mi x nil coefficient matrix, ni = 3k-6, and mi = r+2w+h. Theorem 3 and its corollary can be applied here to determine the internal immobility of the structure. 21

6. A THREE-DIMENSIONAL TRUSS EXAMPLE As mentioned earlier, a kinematic immobility problem of a three-dimensional structure using a force-based approach is solved in [8]. We will solve the same problem using the theory developed in this article and obtain the same results. Figure 12 shows the front and right-side view of a truss structure which consists of two n-gons (in this case pentagons) lying in parallel planes and connected via triangles. Structures of this type have applications in aerospace engineering. The corners of the n-gon lying in the xy plane are pin-supported. The structure has 2n nodes and 3n bars. Since the structure is assumed to be a truss, only equations like Eqs. (16) and (17) need be considered. Equation (17) for the pinned-supported nodes (on the xy plane) can be expressed as follows: 0 = Axi = Ayi = Azi for i = n+l,..., 2n (39) y h Fig. 12: A three-dimensional truss structure (n = 5). Replacing the displacement components of the pinned nodes in equations for length deformation similar to Eq. (16) provides us with the required immobility matrix A, which in this case reduces to a [3n x 3n] matrix. One can verify that matrix A has the format of Eq. (40) by writing Eq. (16) for each of the bars in the n connecting triangles; we only state the result, namely A is the matrix 22

r-Rcos(cc) r(1 -cos(ax)) -Rsin(a) h 0 -rsin(2a) 0 r(cos(c)-1 ) 0 0 rcos(ax)-Rcos(a) 0 rsin(2x) rsin(2ox)-Rsin(a) O O.... h O * 0.* rcos(2(k-1 )a)-Rcos((2k-1 xa) rsin(2(k-1 )c)-Rsin((2k-1 *.... r(cos(2(k-1 )a)-cos(2ka)) r(sin(2(k-1 )a)-sin(2ka).... 0 0 )a) h 0 0 ) 0 r(cos(2ka)-cos(2(k-1 ) a)) r(sin(2kx)-sin(2(k-1 ))) o rcos(2kax)-Rcos((2k-1 )a) rsin(2kx)-Rsin((2k-1 )o) rcos(2(n-1 )c)-Rcos((2n-1 )a) rsin(2(n-1 )a)-Rsin((2n-l)a) r(cos(2n-l)a)-l) ' r(sin(2(n-1 )a) 0 h h 0 0 0 0 r(1 -cos(2n-1 )x)) -r(sin(2(n-1 )a) 0 0 r-Rcos((2n-1)a) -Rsin((2n-l)cx) h 0 - 0 (40) According to [8], the structure is immobile if n is odd, and mobile if n is even. The Appendix provides an outline of a proof that the determinant of A vanishes only for even n's. Using known facts from linear algebra and Theorem 2 of this article the above result can be confirmed. 7. CONCLUSION A rigorous procedure has been introduced to detect internal and global mobility of skeletal structures without performing any stress or force analysis. The computations require information of the nodal coordinates, their connectivity, and the support conditions of the structure. The two-dimensional analysis has been computationally integrated into the structural synthesis program ISOS. ACKNOWLEDGEMENTS This research has been partially supported by NSF Grant No. DDM 89-17430 at the University of Michigan. The first author has also been supported by a Horace H. Rackham Pre-doctoral Fellowship at the University of Michigan. This support is gratefully 23

acknowledged. The authors also wish to acknowledge the useful comments by Professors Alejandro Diaz and Sridhar Kota. REFERENCES [1] Bends0e, M., Kikuchi, N., Generating Optimal Topologies in Structural Design Using a Homogenization Method. Computer Methods in Applied Mechanics and Engineering, 71, 197-224, (1988). [2] Papalambros, P.Y., Chirehdast, M., An Integrated Environment for Structural Configuration Design. Journal of Engineering Design, 1(1), 73-96, (1990). [3] Bremicker, B., Chirehdast, M., Kikuchi, N., Papalambros, P.Y., Integrated Topology and Shape Optimization in Structural Design. Journal of Mechanics of Machines and Structures, to be published, (1991). [4] Norris, C.H., Wilbur, J.B., Elementary Structural Analysis. 3rd Edn., McGrawHill, New York, (1976). [5] Weaver, W. Jr., Matrix Analysis of Framed Structures. 3rd Edn., Van Nostrand Reinhold, New York, (1990). [6] White, N.R., Gergely, P., Sexsmith, R.G., Structural Engineering. combined Edn., John Wiley & Sons, New York, (1976). [7] Southwell, R.V., An Introduction to Theory of Elasticity. 2nd Edn., Oxford University Press, Oxford, (1941). [8] Hoff, N.J., Fernandez-Sintes, J., Kinematically Unstable Space Frameworks. In: Ed. Nemat-Nasser, S., Mechanics Today, 5, 95-111, Pergamon Press, Oxford, (1980). [9] Timoshenko, S.P., Young, D.H., Theory of Structures. 2nd Edn., McGraw-Hill, New York, (1965). [10] McCuskey, S.W., An Introduction to Advanced Dynamics. Addison-Wesley, Reading, MA, (1962). [11] Sandor, G.N., Erdman, A.G., Mechanism Design: Analysis and Synthesis. Prentice-Hall, Englewood Cliffs, NJ, (1984). [12] Bottema, O., On GrUbler's formulas for mechanisms. In: Applied Scientific Research, vol. A.2, 162-164, (1950). [13] Hunt, K.H., Kinematic Geometry of Mechanisms. Clarendon Press, Oxford, (1978). [14] Grining, M., Die Statik des Ebenen Tragwerkes. Verlag von Julius Springer, Berlin, (1925). 24

[15] Friedberg, S.H., Insel, A.J., Spence, L.E., Linear Algebra, 2nd Edn., PrenticeHall, Englewood Cliffs, NJ, (1989). APPENDIX To obtain the determinant of matrix A in Eq. (40) the following procedure should be performed. The first row is multiplied with -1 and added to the last row. The only nonvanishing member of the last row is 2Rsin(2(c), standing in the second column. Thus, the determinant is expanded with respect to that row. A [3n-1 x 3n-1] matrix remains whose only non-vanishing member (h) is in the second column and the first row. Expanding this determinant with respect to its second column gives us a [3n-2 x 3n-2] matrix of the form r(l -cos(2a)) r(cos(ax)-1) rsin(2a) o 0 B1 B2 B3k-2 Bk- Bk 0 r(1-cos(2a)) 0.. o r(cos(32)-1) -r(sin(2a)) 0 where Bi, i = 1,.., 3(n-1), are (3n-4)-dimensional vectors. The following equation can be verified by induction on even and odd n's C = (cos(2a)-l) det[B2, B3,..., Bk] - sin(2a) det[B1, B3,..., Bk] = (cos(2a)-l) det[B1,..., Bk-3, Bk-l, Bk] + sin(2a) det[B1,..., Bk-2, Bk] where C, a nonzero number, is used for later reference, and k equals to 3n-4. The determinant of A (the immobility matrix) can be written in the following form det(A) = (-1)(3n+l)2RhrCsin(2a)(1+(-1)(3n+l)) It can be easily verified that det(A) vanishes if n is even, and is nonzero otherwise. 25

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