The Variational Asymptotic Method
A brief symbolic outline of the VAM and its general features is presented to complement the subsequent discussion. Let the 3D elastic energy of the beam be symbolically defined by the energy functional F with the small parameter that is for now called n such that (Cesnik, 1994):
F(V, n) = Ex(Y, z) + 8n0(T, z1,n),
Z1 = [ W11 W12 W13 ] , (1)
where Г is a 6x 1 column matrix that represents the 3×3 symmetric Biot-Jaumann strain field, and Y is a function of the axial coordinate along the span of the beam only (in this case it corresponds to Г) (Danielson and Hodges, 1987).
The energy functional F is decomposed into two parts: Ei(Y, zi) which contains all terms of order n0 = 1 andEn0(Y, z1, n) that contain all terms of order n1 and higher with respect to this small parameter. The vector z1 represents a perturbation in the classical 3D displacement field, which in reality is the in/out-of-plane warping functions to a first correction, that gives rise to low and high
order terms as is apparent from its appearance in both parts of the energy functional. In order to find the first correction to the displacement field, z1, the high-order component of the functional is discarded and then the functional is minimized with respect to z1. The solution of the Euler minimization problem is not unique and the displacement field is four times redundant. The four rigid body modes have to be eliminated from z1 (the warping field) over the surface area of the cross-section S, therefore, the following four constraints are imposed:
minzj F = minZ1 g1(T, z1),
f G1(z1) ds = 0, f C2(z1) ds = 0, f C3(z1) ds = 0, f 64^1) ds = 0. (2)
J S J S J S J S
When Equations (2) are solved over the cross-section they yield what is called the “zeroth- approximation" or the building block of the solution, z1. This must not be confused with the order of the components of z1 itself, which could be of some order of n, but rather it refers to it being obtained by minimizing the part of the energy that has zeroth order of n (i. e., n0). In most cases, there is no closed form solution for z1, and the problem is discretized over the cross-section with the constraints leading to a Sturm-Liouville problem followed by finite element calculations, which is the methodology of VABS (Cesnik et al., 1993, Cesnik and Hodges, 1997). The order of the components of z1 is not known a priori but determined throughout the minimization procedure. The solution of Euler equations of the functional is symbolically written as:
zi = fi (Т, Z2, Z3), (3)
where Z2 and Z3 are the perpendicular axes defining the 2D Cartesian plane of the cross-section.
The displacement field is then perturbed again. Let the new perturbation be called z2 such that:
z2 = z1 – f1(T, Z2,Z3). (4)
The new perturbation is substituted back into the energy functional of Equation (1) to obtain:
F (V, n) = F1(T) + S2(T, z2) + SV1(T, z2,n),
z2 = [ W21 W22 W23 ] . (5)
The function F1 (T) represents all the terms that do not contain the new unknown, z2. It is subscripted with 1 to indicate that it contains contributions from the first correction to the displacement field z1. The function E2 contains the lowest-order terms involving z2, while En1 contains all high-order terms.
Following the same procedure as before, the high order terms (i. e., En1) are discarded and the functional is minimized with respect to z2 subject to the same constraints:
minz2 F = F1(T) + minz2 82(T, z2)
f 61 (z2) ds = 0, S
f 62(z2) ds = 0, 1 C3(z2) ds = 0, 1 C4(z2) ds = 0. J S J S J S
z2 = f2(T, Z2, Z3).
The process is repeated until the new perturbation yields no terms in the energy functional of order that is of the highest yielded by the previous perturbation, and at this point the displacement field is said to have converged. For example, assume that the perturbation zk produced terms in the energy
Fig. 1. Shell/thin-walled bar coordinate systems (Hodges and Volovoi, 2000).
functional with one being the highest order having an order of O(n). A further perturbation, zk+1, produces terms that are of order O(n + 1) and higher, then at this stage the iteration is terminated. In fact, one may not have to go as far as this in order to obtain the correct elastic behavior of the beam since n is a small parameter to start with. Terms of order O(n0 = 1) are the only ones needed to obtain the asymptotically-correct global elastic behavior of the beam.
The energy functional can then be written after the kth perturbation as:
F(Г, n) = Fx(T) + F2(T) + ••• + Fk(T) + Sk+1(T, zk+1, n) + Enk(T, zk+1, n). (8)
Alternatively, the energy functional can be expanded implicitly as an asymptotic series in the small parameter n:
F(Г, n) = O(n°) + 0(nh) + O(n2) + O(n3) + ••• + O(nk). (9)
It must be evident by now that no ad hoc assumptions are made in order to arrive at the asymptotically-correct stiffness matrix that can be extracted from terms of order O(n°). The process of dropping high order terms during the minimization or truncating them from the asymptotic expansion of the functional is equivalent to a correct and systematic neglect of insignificant terms in more conventional solution methods that rely on the Theory of Elasticity, equilibrium equations, and boundary conditions. But these conventional methods usually involve a multitude of partial differential equations, rendering the identification of these insignificant terms extremely difficult if not outright impossible.