Retrieving the Flexural Rigidity of a Beam from Deflection Measurements

A rigorous investigation into the identi cation of the heterogeneous exural rigidity coe cient from de ection measurements recorded along a beam in the presence of a prescribed load is presented. Mathematically, the problem reduces to the need to solve the Euler-Bernoulli steady-state beam equation subject to appropriate boundary conditions. Conditions for the uniqueness and continuous dependence on the input data of the solution of the inverse problem for simply supported beams are established and, in particular, it is shown that the operator which maps an input de ection into an output exural rigidity is Holder continuous. Since the inverse problem can be recast in the form of a Fredholm integral equation of the rst kind, numerical results obtained using various methods, such as Tikhonov's regularization, singular value decomposition and molli cation are discussed. NOMENCLATURE E Modulus of elasticity I Moment of inertia L Length of the beam M Bending moment a Flexural rigidity f Transversely distributed load p Amount of noise u Transverse de ection Inverse of the exural rigidity Regularization parameter Standard deviation INTRODUCTION In the Euler-Bernoulli beam theory, it is assumed that the plane cross-sections perpendicular to the axis of the beam remain plane and perpendicular to the axis after deformation, resulting in the transverse de ection u of the beam being governed by the fourth-order ordinary di erential equation d dx2 a(x) du dx2 (x) = f(x); 0 < x < L (1) where L is the length of the beam, f is the transversely distributed load and the spacewise dependent conductivity a = EI (often called exural rigidity) is the product of the modulus of elasticity E and the moment of inertia I of the cross section of the beam about an axis through its centroid at right angles to the cross section. It should be noted that eqn.(1) can be split into a system of two secondorder ordinary di erential equations, namely, M (x) = a(x) du dx2 (x); 0 < x < L (2) dM dx2 = f(x); 0 < x < L (3) where M is the bending moment. Using a simple nite element methodology, it can be seen that the primary variables 1 Copyright c 1999 by ASME associated with eqn.(1) are the de ection u and the slope du=dx, whilst the bending moment M = a(du=dx) and the shear force (d=dx)(adu=dx) are the secondary variables, any four values of which are usually speci ed on the boundary, namely, x = 0 or x = L. The direct problem of the Euler-Bernoulli beam theory requires the determination of the de ection u which satis es eqn.(1) (orM and u satisfying eqns (2) and (3)), when a > 0 and f 0 are given and four boundary conditions (essential or natural) are prescribed. If at least one of these boundary conditions is on the de ection then this direct problem is well-posed. However, there are other problems (termed inverse) that may be associated with the Euler-Bernoulli eqns (1)-(3). For simply supported beams an inverse load identi cation problem, which requires the determination of the load f(x) which satis es eqn.(3) when a and M are given, has been investigated by Collins et al. (1994) in the practical context of recovering engineering loads from strain gauge data. This problem is ill-posed since it violates the stability of the solution, which relates to twice di erentiating numerically M , which is a noisy function. In this study, we investigate an inverse coe cient identi cation problem which requires the identi cation of the positive, heterogeneous exural rigidity coe cient a(x) of a beam which satis es eqn.(1), (or eqns (2) and (3)), when u and f are given. This problem is an extension to higherorder di erential equations of the inverse problem analysed by Marcellini (1982) for the one-dimensional Poisson equation. Prior to this study, a numerical algorithm has been recently proposed by Ismailov and Muravey (1996) for supported plates, when f > 0 and u 0. However, the theoretical investigation of the uniqueness of the solution is much more di cult since eqn.(1) does not, in general, have a unique solution, e.g. if a0 is a solution of eqn.(1) then for any linear function h, the function a1 = a0+ h(d u=dx) 1 may still be a solution of eqn.(1). Therefore, a necesary condition for the uniqueness of the solution of eqn.(1) is that the set S = x 2 [0; L] j du dx2 (x) = 0 (4) is not empty. However, at the other extreme, if S is a subset of [0; L] of non-zero measure then a(x) is not identi able on S. Further, based on the physical argument that the properties of the beam, namely E and I, should be positive, and considering only applications in which there is always a positive nite load acting on the beam, we can assume that there are 0 < f1 f2 < 1 such that f1 f(x) f2 and that there are 0 < 1 2 < 1 and Q > 0 such that a(x) belongs to the domain de nition set A = fa 2 L1(0; L) j 1 a 2; ka 0k Qg (5) where k:k denotes the L(0; L)-norm. For the present analysis we restrict ourselves to homogeneous boundary conditions to accompany the non-zero load f and the linear eqns (1)-(3). Also, since a > 0 then the set S given by eqn.(4) is S = fx 2 [0; L] j M (x) = 0g. Thus boundary conditions which ensures that S 6= ; include u(0) = u(L) or M (0)M (L) = 0. If only Dirichlet and/or Neumann type boundary conditions for u and M are considered then there are in total P4 k=0C k 4C 4 k 4 = 70 of these possibilities. However, half of these possibilities are equivalent, based on the invariance of eqns (1)-(3) under the translation x 7! (L x). The aim of this study is not to investigate all these possibilities but rather select realistic physical boundary conditions associated with beams that naturally occur in elasticity, such as beams supported at both ends, namely, u(0) = u(L) = M (0) = M (L) = 0 (6) Other types of boundary conditions have been investigated elsewhere, see Lesnic et al. (1999). The plan of the paper is as follows. In section 2 we prove that the operator u 7! a is Holder continuous and thus injective, i.e. the uniqueness and continuous dependence on the input data of the inverse problem. Further, when random noisy discrete input data is included, a regularization algorithm is developed in section 3 and the numerically obtained results are illustrated and discussed. MATHEMATICAL ANALYSIS We formally de ne the operator U : A! L(0; L) U (a) := ua = Z Z M a ; M = Z Z f; 8a 2 A (7) as a formal general solution of eqns (1)-(3), where the integral sign R is understood in the sense that the constants of integration are to be determined by imposing the boundary conditions (6). For convenience, and in order to simplify the algebraic manipulations, a uniform load f 1 was assumed which in turn gives 2 Copyright c 1999 by ASME M (x) = x(x L) 2 (8)