Shear cross-section area Ay/Az #

In AxisVM, the calculation of shear cross-section areas is based on the following article: Pilkey, W. D., Analysis and Design of Elastic Beams – Computational methods, John Wiley & sons, Inc., 2002.

Rectangular and circular cross-sections #

In case of rectangular cross sections ρ_y = ρ_z = 1.2, and A_z = A_x/ρ_y = A_x/1.2. This is the accurate interpretation, the approximation is A_z = A_z = 0.8*A_x.

In case of circular section ρ_y = ρ_z = 7/6.

Rectangular and circular sections can be calculated analytically, and in these cases, AxisVM will use this approach. In the case of general cross-sections, the theory is briefly discussed below.

General cross-sections #

Shear stresses and shear deformations due to shear forces are functions of τ_xy(y,z), τ_xz (y,z), γ_xy (y,z) and γ_xz (y,z) with different values in each point. In case of a rib element, for example, there is only one τ_xy, τ_xz, γ_xy and γ_xz for a cross-section. This contradiction is being resolved by the shear cross-sectional factor (ρ_y, ρ_z), and the shear cross-sectional area (Ay, A_z). In the first case, the energy of the shear deformation has to be integrated for the whole cross-section, while in the second we can only calculate with W = τ_xy * γ_y *A_y+ τ_xz * γ_z *A_z. This formula means, that the τ and γ values are constant for all of the points of the cross-section, and the integration is simplified into multiplication by area. However, A_y and A_z shear cross-sectional area is set in a way, that the constant τ and γ values due to a given shear force results in the same shear energy, as it was integrated based on changing (in cross-section) τ and γ values for the same shear force.

This theory is discussed on pages 219-222 in the following book, (which is in Hungarian):

Cholnoky__Rugalmassagtan.pdf (fw.hu)

In the case of general sections, the calculation is performed based on this theory, but numerically. The first step on the meshed cross-section, we calculate – with finite element method – the distribution of the shear deformation in the cross-section for V_y and V_z shear force. Based on this, we perform the integration on the meshed cross-section. Finally, using the τ_xy  = V_y/A_y and τ_xz = V_z/A_z assumed to be average, constant shear stresses, we calculate the A_y and A_z shear cross-sectional areas to fulfill the energy equilibrium.