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The differential equations describing the pile and soil displacements are solved using the Ritz method and the finite difference method, respectively, following an iterative numerical scheme.
The approximate solutions of displacements are solved for the first three terms, and the corresponding internal stresses can also be obtained.
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Equations obtained by applying the principle of virtual displacements were solved in a closed form using double trigonometric series.
The implicit Euler time scheme is used, and the non-linear system giving the increment of the generalized nodal displacements is solved by the Newton-Raphson algorithm.
The coupled equations of fluid motion and pipe displacement are solved.
The model equations for small amplitude stress fiber displacement were solved in response to forces applied either orthogonal to or in the direction of the stress fiber axis as depicted in Figure 1.
Nonlinear governing equations are obtained based on the static version of virtual displacements and are solved via the multi-term Galerkin method.
The governing equilibrium equations are derived analytically from the Principle of Virtual Displacements (PVD), and are solved exactly referring to the Lévy-type procedure.
For this purpose, exact displacement shape functions are solved from the differential equilibrium equation, by using arbitrary boundary conditions.
An unstructured finite volume time domain method (UFVTDM) is proposed to simulate stress wave propagation, in which the original variables of displacement and stress are solved based on the dynamic equilibrium equations.
A wave solution is used to describe the displacements of the cylindrical shell, while the displacements of the conical sections are solved using a power series solution.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com