A boundary shape function method for analyzing nonlinear composite beams, subjecting to nonlinear boundary moment conditions

2021 ◽  
pp. 113636
Author(s):  
Chein-Shan Liu
Keyword(s):  
Function Method ◽  
Nonlinear Boundary ◽  
Shape Function ◽  
Composite Beams ◽  
Moment Conditions ◽  
Boundary Shape ◽  
Nonlinear Composite

Solving nonlinear boundary value problems by a boundary shape function method and a splitting and linearizing method

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Chein-Shan Liu ◽  
Essam R. El-Zahar ◽  
Chih-Wen Chang
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Shape Function ◽  
Boundary Value ◽  
Linear Equations ◽  
Accurate Solution ◽  
Iteration Step ◽  
Nonlinear Boundary Value Problems ◽  
Boundary Shape ◽  
Nonlinear Boundary Value

Abstract In the paper, we develop two novel iterative methods to determine the solution of a second-order nonlinear boundary value problem (BVP), which precisely satisfies the specified non-separable boundary conditions by taking advantage of the property of the corresponding boundary shape function (BSF). The first method based on the BSF can exactly transform the BVP to an initial value problem for the new variable with two given initial values, while two unknown terminal values are determined iteratively. By using the BSF in the second method, we derive the fractional powers exponential functions as the bases, which automatically satisfy the boundary conditions. A new splitting and linearizing technique is used to transform the nonlinear BVP into linear equations at each iteration step, which are solved to determine the expansion coefficients and then the solution is available. Upon adopting those two novel methods very accurate solution for the nonlinear BVP with non-separable boundary conditions can be found quickly. Several numerical examples are solved to assess the efficiency and accuracy of the proposed iterative algorithms, which are compared to the shooting method.

scholarly journals Exact Static Analysis of In-Plane Curved Timoshenko Beams with Strong Nonlinear Boundary Conditions

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Sen-Yung Lee ◽  
Qian-Zhi Yan
Keyword(s):  
Boundary Conditions ◽  
Function Method ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Strong Nonlinearity ◽  
Timoshenko Beams ◽  
Static Deflection ◽  
Boundary Problems ◽  
Beam System ◽  
Differential Characteristic

Analytical solutions have been developed for nonlinear boundary problems. In this paper, the shifting function method is applied to develop the static deflection of in-plane curved Timoshenko beams with nonlinear boundary conditions. Three coupled governing differential equations are derived via the Hamilton’s principle. The mathematical modeling of the curved beam system can be decomposed into a complete sixth-order ordinary differential characteristic equation and the associated boundary conditions. It is shown that the proposed method is valid and performs well for problems with strong nonlinearity.

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