The Asymptotically-Correct Stiffness Matrix

Hodges and Volovoi (2000, 2002) applied the VAM to thin-wall open/closed shells and strips depic­ted in Figure 1, and obtained the asymptotically-correct stiffness matrix for each case respectively. The intersection of the middle surface of the shell with the plane of the beam cross-section defines the contour ^, which does not change along the span of the beam.

The energy functional of the composite beam in this case is the elastic shell energy per unit area written in terms of the six generalized shell strain measures:

2 U shell = fTQf + 2фт Sf + фтРф fT = [ Y11 hpn hp12 ]

фТ = [ 2Y12 Y22 hp22 ], (10)

where Q, S, and P are material property matrices that depend on the 2D elastic modulus tensor of the material, which is obtained from the reduced stiffness coefficients Qmn of Classical Laminate Theory (Badir, 1992).

The six generalized shell strain measures are given in terms of the curvilinear displacements, which in turn can be expressed in terms of the cartesian displacements m1, u2, and u3:

Y11 = 0,1 P11 = П3Д1,

2yi2 = Під + П2Д P12 = ПЗД2 + 4^(ni,2 – ЗП2д), (11)

K22 = «2,2 + P22 = 0,22 – (^),2,

The subscripted comma indicates a differentiation with respect to the curvilinear coordinate sub­sequent to it, and R is the radius of curvature defined using the cartesian coordinates x2 and x3 along b2 and b3 respectively as R = X2/X3 = —X3/X2. The over-dot implies a differentiation with respect to the curvilinear coordinate s. The generalized strain measures in Equation (11) are substituted in the shell energy in Equation (10) and integrated over the contour to give the functional:

І 2 U shell ds = F (f, ф). (12)

It is reemphasized in this problem, that two small dimensionless parameters are present: the slender­ness ratio n1 = a/l, and the thinness ratio n2 = h/a. All deformation modes are assumed to have the same order of magnitude O(e), and therefore they are all included in the development to make it general: extension, torsion, bending about the b2 axis, and bending about the b3 axis.

An asymptotic expansion of the shell energy functional in terms of the small parameters is sought in the form:

S 2Ushell = O(c2 • n? • 4) + O(c2 • n1 • n0) +

+ &(e2 • n? • n1) + O(e2 • n1 • n1) + ••• + O(e2 • n1 • nJ2).

Only the first term in Equation (13) is retained, since it is the dominant one, to give the elastic energy per unit length:

8 = £TSe, (14)

where eT = [u1 в’ — u’3 u"], which is the classical linear strain measures vector.

The matrix S, which has a closed form solution given in Hodges and Volovoi (2000, 2002), is the 4×4 asymptotically-correct stiffness matrix, which is beyond the Euler-Bernoulli Theory of bending and St. Venant’s Theory of torsion in terms of its rigor. It takes into account all in-plane warping deformations since y12 is never assumed to be zero throughout the development, in addition to the shell bending strain measure p22. Given the natural boundary conditions on the tip of the anisotropic thin-wall beam, the correct elastic response can be obtained from the flexibility matrix, which is the inverse of the stiffness matrix:


Me / M2 M3

Table 1. The elements of the stiffness matrix obtained using the thin-walled anisotropic beam theory and VABS for Figure 2.

Stiffness Element


VABS (Popescu and Hodges, 2000)



1.2185 x 106 lb

1.2500 x 106 lb



0.0522 x 106 lb – in

0.0521 x 106 lb – in



0 lb – in

0 lb – in



0 lb – in

0 lb – in



0.1730 x 105 lb – in2

0.1770 x 105 lb – in2



0 lb – in2

0 lb – in2



0 lb – in2

0 lb – in2



0.0508 x 106 lb – in2

0.0543 x 106 lb – in2



0 lb – in2

0 lb – in2



0.1283 x 106 lb – in2

0.1340 x 106 lb – in2