PSI - Issue 70

Aamir Anwar Nezami et al. / Procedia Structural Integrity 70 (2025) 105–112

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( , , ) = ( ( ) −1 ) 0 ( , ) + 2 1 ( , ) + 3 2 2 ( , ) + ∑ ( , ) − ( , ) −1 =1 where, ( , , ) = [ , ] ; 0 ( , ) = [ 0 , y0 ] ; 1 ( , ) = [ 1 , y1 ] ; 2 ( , ) = [ 2 , y2 ] ; = ℎ 1 ∑ ∫ [ 4 ( 4 ) 4 ( 5 ) 4 ( 5 ) 5 ( 5 ) ] +1 = 1 ; ( ) =[ 4 ( 4 ) 4 ( 5 ) 4 ( 5 ) 5 ( 5 ) ] ; Here k, m, h, and N, resembles ℎ layer, the mid-layer index, the total thickness of the laminate, and the total number of layers, respectively . The functions − ( , ) = (− + +1 ), + ( , ) = ( − +1 ) are Heaviside-Unit-Step functions representing regions below and above the mid-layer, respectively. The term γ ( , ) accounts for the discontinuity (jump) in transverse shear strain ( ) across the layer interfaces. By enforcing shear stress continuity conditions at the interfaces and applying shear stress-free boundary conditions at the top and bottom surfaces of the laminate, and after appropriate algebraic manipulation, the resulting displacement fields are obtained as expressed in Equation (2) with five unknowns only. +∑ ( , ) + ( , ) −1 = (1) 1 ( ) ( , , ) = ( , ) − ( , ) + 1 1 ( ) 0 ( , ) + 1 2 ( ) 0 ( , ) 1 ( ) ( , , ) = ( , ) − ( , ) + 2 1 ( ) 0 ( , ) + 1 2 ( ) 0 ( , ) z ( ) ( , , ) = ( , ) (2) Where 1 1 ( ) , 1 2 ( ) , 2 1 ( ) , and 2 2 ( ) he zigzag functions, depending on the geometric and material properties of the laminae. To appreciate the accuracy of present zigzag model, we will see some numerical examples in the following section. 3. Numerical Results Numerical studies have been carried out in order to assess the existing model's accuracy and reliability. The calculations were carried out in accordance with the methods Pagano (1970a) and Pagano (1970b) devised for an exact three-dimensional elasticity solution. Pagano acquires the plot in the figure referred to as Exact-3D. Additionally, numerical outcomes from the Enhanced-RZT developed by Sorrenti and Di Scuiva (2021) and MZT developed by Murakami (1986) are also shown. No in-plane tractions are applied to the top and bottom surfaces, instead, a sinusoidal distributed transverse load is applied to these surfaces. The total transverse load is equally divided into two parts. One half is applied on top and the other half on the bottom external surfaces in a three-dimensional solution. For buckling problems, only axial compressive load is applied. 3.1 Field variables for symmetric cross-ply laminate having simply supported boundary conditions on all four sides Navier's method is employed to derive solutions for symmetric cross-ply orthotropic rectangular plates with simply supported boundary conditions (SS-1) subjected to a transverse bi-sinusoidal distributed load ( , ) = 0 ( x ) ( y ) , the equilibrium equations and boundary conditions are fulfilled using the following trigonometric function sets: ( , ) = mn ( x ) ( y ) , ( , ) = ( x ) ( y ) , ( , ) = ( x ) ( y ), 0 ( , ) = ( ) sin( ) , 0 ( , ) = ( ) cos( ) (3) 3.2 Field variables for symmetric angle-ply laminate having an infinite length in the y-directions and simply supported along the two opposite edges Under transverse sinusoidal distributed load ( , ) = 0 ( x ) ( y ) , the equilibrium equations and boundary conditions can be addressed using the following trigonometric function sets. [ , , 0 , 0 ]= [U mn , V mn , , ]cos( ) , and = W mn sin( ) (4)