On the foundations of anisotropic interior beam theories

Abstract

This study has two main objectives. First, we use the Airy stress function to derive an exact general interior solution for an anisotropic two-dimensional (2D) plane beam. Second, we cast the solution into the conventional form of 1D beam theories to clarify some basic concepts related to anisotropic interior beams. The derived general solution provides the exact third-order interior kinematic description for the plane beam and includes the Levinson/Reddy-kinematics as a special case. By applying the Clapeyron's theorem, we show that the stresses acting as surface tractions on the lateral end surfaces of the interior beam need to be taken into account in all energy-based considerations related to the interior beam in order to avoid artificial end effects. Exact 1D interior beam equations are formed from the general 2D solution. Finally, we develop an exact interior beam finite element based on the general solution. With full anisotropic coupling, the stiffness matrix of the element becomes initially asymmetric due to the interior nature of the plane beam. By redefining the generalized nodal axial forces of the element, the stiffness matrix takes a symmetric form.

Modeling the non-trivial behavior of anisotropic beams: A simple Timoshenko beam with enhanced stress recovery and constitutive relations

“…In particular, Figure 4(b) highlights that shear does not vanish in the beam mid-span τ (l/2, y) = 0, despite the vertical internal force vanishes V (l/2) = 0. Further comments about this peculiarity of simply supported beams can be found in [22]. More interestingly, Figure 4(b) highlights that axial stress does not vanish at the bearing σ x (l, y) = 0, despite both bending moment and axial internal force vanish M (l) = N (l) = 0.…”

“…Similarly, also the shear stress τ explicitly depends on the transversal load q. To the authors' knowledge, only Karttunen and Von Hertzen [22] and Hashin [17] obtained similar dependencies, but their analysis was limited to homogeneous beams. Furthermore, d V σx and d q σx depend on E xx /G x and E 2 xx /G 2 x , respectively (see Equation (17c) and (17d)), and similar coefficients was reported also by Karttunen and Von Hertzen [22].…”

“…This section compares the solution of the beam model discussed in Section 2 with the analytical solution derived in [22] for a simply supported homogeneous beam. Numerical results are obtained assuming the following parameters h = 0.2 mm; l = 2 mm; q = 1 N/mm; α = 0 E 11 = 10 4 MPa; E 22 = 5 · 10 2 MPa; G = 10 3 MPa; ν = 0.25 (26) It is worth mentioning that the material is highly anisotropic: E 11 /E 22 = 20 and E 11 /G = 10.…”

“…Assuming θ = 45 deg the rotation of the constitutive relation (6) leads to set E xx = E yy = 1.904 GPa; E xy = 40.000 GPa; G x = G y = −1.052 GPa; G = 0.322 GPa (27) Figure 4 compares the cross-section distribution of stresses evaluated according to reference [22] ψ ref , the proposed beam model ψ mod , and standard Timoshenko beam ψ T . Numerical results demonstrate that the proposed beam model provides results substantially identical to the reference solution.…”

Two-scale constitutive modeling of a lattice core sandwich beam

“…In the homogeneous solution, the integration constants c 1 , c 3 and c 4 correspond to rigid body motions and the five remaining constants are related to the stress resultants (cf. [43]).…”