PSI - Issue 64

Niels Pichler et al. / Procedia Structural Integrity 64 (2024) 409–417 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

413

− = 0

5

( 3 ) where is the transverse force, is the longitudinal force, and is the bending moment acting on the infinitesimal beam element of height ℎ , represented in Figure 4b). The distributed transverse loading is denoted and the distributed shear loading is denoted .

Assuming a generic stress-strain relationship = 1 ( ) and = 2 ( ) the following beam constitutive behavior relating internal load and displacement can be obtained through integration of the stress fields over the beam height. = ∫ − ℎ 2 − ℎ 2 = ∫ − 1 (− , + , ) ℎ 2 − ℎ 2 = ( , , , ) ( 4 ) = ∫ − ℎ/2 −ℎ/2 = ∫ − 2 (0.5(− + , )) ℎ/2 −ℎ/2 = (− + , ) ( 5 ) = ∫ − ℎ/2 −ℎ/2 = ∫ − 1 (− , + , ) ℎ/2 −ℎ/2 = ( , , , ) ( 6 ) Therefore, the equilibrium equations ( 1 )( 2 )( 3 ) are given by: , , + , , − , − − ℎ2 = 0 ( 7 ) (− , + , )+ =0 ( 8 ) , , + , , − =0 ( 9 ) The functions and are assessed across a grid of , , , , and over a range of = 0.5(− + , ) , employing trapezoidal integration of the corresponding stress profiles. Bivariate splines are used to interpolate the assessed values of and , while a cubic spline is employed for . Figure 4: a) Schematic of the model, boundary and continuity conditions; b) Loaded infinitesimal beam segment.