Structural Rigidity

This article summarizes the presently available general theory of rigidity of 3dimensional structures. We explain how a structure, for instance a bar and joint structure, can fail to be rigid for two quite different types of reasons. First, it may not have enough bars connecting certain sets of nodes. That is, it may faij for topologlcrl reasons. Secondly,. although it may “count” correctly, it may still fail to be rigid if it is set up with some special relative positions of its nodes and bars; This second type of failure is a question not of topology but of projectbe geometry. What distinguishes structural engineering from mechanical engineering is the special attention paid to the question of rigidity.* Whereas mechanisms* (linkages) are useful primarily by virtue of the relative motion of their parts, buildings must be designed to stand rigid, and to continue to stand when subjected to a variety of external forces such -as gravity, loading and wind pressure. In this article we sketch the presently available general theory of rigidity for three types of structures: (I) bar and joint structures*, typified by trusses in wood, or the bolted ironwork introduced in the last century, (2) strut and cable structures*, such as the tensegrity structures popularized by Buckminster Fuller in the fifties, which maximize reliance on the available tensile strength of wire cables, and (3) hinged panel structures*, explored recently by Janos Baracs and his students, which expand the potential for construction with prefabricated concrete or moulded plastic panels. With regard to each of these types of structures, we shall point to a number of challenging unsolved problems. The rigidity we speak of is such that rigid structures will resist even Inflnltedmal motions. To be non-rigid in this sense, a structure need not be a true mechanism, with an easily observable motion. For mathematicians we can make the distinction clear by saying we will be using linear algebra and projective geometry rather than differential geometry. From the standpoint of geometry, we are looking at structures whose component parts are line segments joined to one another at nodes, or else polygonal pieces of planes in space, joined to one another along edges. These joints are articulated , so the bars at a node are at least locally free to change in angle relative to one another, and the panels are merely hinged, rather than welded, to one another. In Figure 1 we show (related) structures, each just rigid, of the three types. Bar and Joint Structures Rigidity theory for bar and joint structures in the plane was already well advanced in the latter half of the nineteenth century, thanks to the efforts of the English physicist James Clerk Maxwell and the Italian geometer Luigi Cremona (Maxwell 1864 and Cremona 1890). It was their theoretical advances which led to the development of graphical statics* as a practical discipline, at the hands of the German engineers Culmann, Henneberg and their followers (Culmann 1875 and Henneberg 1886). But during the last fifty years, the rigidity theory for bar and joint structures has been mucweglected, and the best work of the last century has, for the most part, been forgotten. It is one of the tasks of our research group 26 to correct, to extend (particularly from structures in the plane to structures in space), and to apply in new ways geometrical methods initially developed under this heading of graphical statics. Each bar and joint structure is given topologically as a graph consisting of nodea, certain pairs of which are joined by bars. Each subset of the set of bars of a given structure determines a au-uon the same set of nodes. We will discuss rigidity of structures primarily at the level of projective geometry, where the nodes of a structure are assigned positions in projective space (usually in a plane, or in 3-space) and each bar is represented as a line segment connecting its two nodes. (The degenerate case where the two nodes of a bar are in the same position we handle by assigning a direction to the bar at that point.) To say we are working in projective space means on one hand that we are making no use of angles or distance measurement or of the concept of being parallel. It also means that we have available all the points and lines on a “plane at infinity” which are missing from affine 3=space, so any line not lying entirely in a given plane must meet the plane in a single point. Two distinct coplanar lines always meet in a single point, and two distinct planes always meet in a line. The following introductory material we present for structures in three-dimensional space, and will make the obvious restriction to structures in the plane when it becomes necessary to do so. If a bar is 27 Note to Architects and Engineers This article is written by a mathematician, and is intended primarily for readers with a mathematical background. As such, an effort has been made to give mathematical definitions of terms commonly used in statics, to place the entire discussion in a context familiar to readers with some experience in linear algebra and projective geometry, and to lay the groundwork for future research. As a result, structures are treated here in a way which may seem unduly abstract, and unrelated to any direct and practical structural application. Even the examples given are those dictated by the theoretical development rather than those arising in architectural or engineering practice. Figure 1. Just-rigid -4 structures, of types. Part of this difficulty is in the nature of the subject, and must be patiently studied by practitioners and theoreticians alike. For instance, there is a dependence among 22 specified bars in the structure illustrated below, a dependence which causes the octahedraltetrahedral truss to be non-rigid. This type of phenomenon must be understood in itself, and cannot be eliminated by simple algorithms and practical rules of thumb. (This example will be the subject of a short article in Structural Topology (2). The other half of the difficulty in exposition can and should be corrected by the publication of further articles intended specifically for those with architectural or engineering training, perhaps with less mathematical background. In the second issue of the Bulletin, we will include a general exposition on structural rigidity, summing up the state of the art for those who wish to apply these methods in practice. The discussion and examples will be supported by intuitive and risual evidence, rather than by higher mathematics. Such “translations” will be a regular feature of the Bulletin, and will occur in both directions. Theoretical papers will be “translated” to draw out their intuitive content and practical consequences. Papers describing practical applications will be “translated” into scientifically precise language, to reveal some camouflaged but interesting and unsolved mathematical problem. Although we may take occasion, as we have here, to point out that a certain article is intended for readers with some specific training, we do not intend to discourage readers from attempting to digest articles posed in language and expressing ideas from fields other than their own. Also, though it is unavoidable at this stage that we write some articles differently for different audiences, it is in the nature of this publication project that “translations” will be less necessary as we go along. subjected to forces applied at its two ends, it will tend to move unless the two forces are equal in magnitude, opposite in direction, and directed down the line of the bar. In this single case the bar Is in equlllbrium under the apblied load; it is either in tension or in compression. Using this idea as a starting point, we apply one of two theorems from elementary linear algebra, to arrive at a very simple ‘explanation of the basic concepts of statics and mechanics of structures. With any structure S having V nodes and E bars in 3space, we associate a matrix M = M(S) called the coordfnrtlzlng matrix + of the structure, which has E rows and a total of 3V columns, arranged in V groups of 3 columns each. The entries in this matrix consist of six possibly non-zero entries in each row: if the row is that corresponding to a bar between nodes in positions a and b, then in the three columns for the node a we have the components of the vector a-b, and in the columns for the node b, the vector b-a. Multiplication by this matrix H is a linear transformation l M from RE to Rsv , and will convert an assignment s of scalars to the bars into an assignment sM of 3-vectors to the nodes. (A different matrix in which vectors of unit length replace the vectors a-b, gives a more useful interpretation of bars of length zero, but we shall not get into that here.) If we think of the scalars s as assigning a compression (if positive) or a tension (if negative) to each bar, measured in force per unit length, then the result sM of multiplication by M is the resultant force on each node due to the combined effect on that node from the tension and compression in the bars incident with that node. Flgure 2 shows one such resolution on a structure which is a skew quadrilateral in space. For any scalar assignment s to the bars, the negative of the resultant -sM is an oquilibrlum system of forces on the nodes, a system of forces which will have no tendency to move the structure in any way. If the forces -sM are applied externally to the nodes, the tension-compression assignment s is one possible static response of the structure to the applied load. Any two distinct possible static responses s, t to the same external load -sM = -tM differ by a scalar assignment s-t such that (s-t)M = 0, a scalar assignment which produces no resultant force on any vertex. It is thus an internal tension compression equilibrium, which we call a m* in the structure. (In engineering terminology, stress means force per unit cross-sectional area in a bar, so the scalar we assign to a bar must be multiplied by the length of the bar and divided by the cross-sectional area, to give the conventional measure of “stress”.) Let r(S) denote the rank of the matrix M, and call this the rank * of the structure. Let n(S), the nullity * of S, denote the dimension of the kernel of 9 M as a linear transformation from RE to R3V . Then r(S) = n(S)= E, where E is the number of bars, r= r(S) is the dimension of the space of resolvable external (equilibrium) loads, and n = n(S) is the dimension of the space of stresses (internal equilibria). For example, the skew polygonal structure in Figun 2 cannot normally be stressed. In order for\ a nonzero tension-compression assignment to resolve to zero at a node, the two bars at that node must be collinear. If the stress is non-zero in one bar, it must be non-zero in the two adjacent bars. Thus the structure normally’has rank 4, nullity 0, and has rank 3, nullity 1 if and only if the structure lies entirely along a straight line in space. This is the-simplest example of the phenomenon which is the main object of our study: under certain pr@ective geometrlc conditlona, a structure will have a lower rank than would be expected from purely topological consideration. An external equilibrium load, in general, is a system of vectors acting along lines in space, whose vector sum is zero and whose total moment is zero about any axis. An arrow from a point b to a point a can be viewed as a force acting at a point b, in the specified direction and with the magnitude la-bb( . We coordinatize each such force f = ab as a 6vector U bJ6) = (a,bl, a2-b2, a-b, aA-a3b2,