Finite deformations of full sine-wave St.-Venant beam due to tangential and normal distributed loads using nonlinear TSNDT

We analyze deformations of a beam initially in the form of a full sine wave and loaded by distributed tangential and normal surface tractions with the goal of finding how the beam curvature and the consideration of all nonlinearities affect the maximum values of stresses and deflections. The curvature of the beam in the left half is opposite of that in the right half of the beam, and the radius of curvature varies from point to point. The problem has been analyzed by using the third order shear and normal deformable beam theory (TSNDT) that accounts for all geometric nonlinearities. The beam material is St. Venant-Kirchhoff for which the second Piola-Kirchhoff stress tensor is a linear function of the Green-Lagrange strain tensor. It is found that for quasistatic deformation with non-dimensional pressure, (q) over bar (0) = q(0)L(4)/100E(L)H(4) = 2.6, the axial stress at the point (3C/4 , H/2) from the nonlinear theory equals nearly 4 times that from the linear theory. The lateral deflection at the point (3C/4 , 0) from the nonlinear theory is about 1.5 times that from the linear theory. Here q(0) is the uniform pressure applied on the bottom surface of the beam, E-L Young's modulus in the longitudinal direction for infinitesimal deformations, H the beam height, L the horizontal distance between the two end faces, and L the arc length. Significant features of the work include using the TSNDT, accounting for all geometric nonlinearities, using a materially objective constitutive relation, considering curvature varying from positive to negative, and applying both tangential and normal surface tractions.