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 ρy = ρz = 1.2, and Az = Ax/ρy = Ax/1.2. This is the accurate interpretation, the approximation is Az = Az = 0.8*Ax.
In case of circular section ρy = ρz = 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 τxy(y,z), τxz (y,z), γxy (y,z) and γxz (y,z) with different values in each point. In case of a rib element, for example, there is only one τxy, τxz, γxy and γxz for a cross-section. This contradiction is being resolved by the shear cross-sectional factor (ρy, ρz), and the shear cross-sectional area (Ay, Az). 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 = τxy * γy *Ay+ τxz * γz *Az. 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, Ay and Az 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):
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 Vy and Vz shear force. Based on this, we perform the integration on the meshed cross-section. Finally, using the τxy = Vy/Ay and τxz = Vz/Az assumed to be average, constant shear stresses, we calculate the Ay and Az shear cross-sectional areas to fulfill the energy equilibrium.