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Mathematical Preliminaries
Published in Michael R. Gosz, Finite Element Method, 2017
A method for solving systems of linear algebraic equations that is amenable to numerical computation employs matrix methods. To illustrate how the procedure works, let us write the system of equations (2.20) in the form Ax = b as follows: [1−10−12−10−11]{x1x2x3}={001}
Introduction
Published in Praveen Nagarajan, Matrix Methods of Structural Analysis, 2019
There exists two ways by which a structure is analyzed using matrix methods, namely the flexibility method or the force method and the stiffness method or the displacement method. The flexibility method of analysis is discussed in Chapter 3, whereas Chapters 4 and 5 deal with the stiffness method of analysis. The methods to develop the force–displacement relation of elements are covered in Chapter 2. In this chapter, the important terms and their definitions used in matrix methods of analysis are discussed.
Discrete Physical Systems
Published in Gianni Comini, Stefano Del Giudice, Carlo Nonino, Finite Element Analysis in Heat Transfer, 2018
Gianni Comini, Stefano Del Giudice, Carlo Nonino
In structural engineering, the computational procedure outlined in this section is known as the matrix method of structural analysis.11,12 Accordingly, the matrix K is known as the stiffness matrix, φ is the vector of nodal displacements, s is the vector of internal nodal forces and r is the vector of external forces which correspond to reactions pertaining, typically, to support points.
Form Finding and Structural Optimisation of Tensile Cable Dome Using Parametric Modelling Tools
Published in Structural Engineering International, 2021
Different numerical methods have been introduced over the years. In 1975, Argyris developed a form-finding method based on the finite element method.7 Barnes introduced a dynamic relaxation method (DRM), which is defined by a system of springs and particles.8 The applied load causes the particles to move until they eventually stabilise in a state of equilibrium. In the 1970s, Schek introduced the force density method, which remains the most used method.9 The stiffness matrix method comes from structural analysis and uses the elastic and geometric stiffness matrices. This method includes unnecessary material properties that increase cost and have difficulty in controlling a stable convergence.10 The force density method can solve the equilibrium equation of a tensile structure that is transformed into a discrete cable network, nodes and lines.11 In a cable network, pin-jointed network structures reach the state of equilibrium when the internal and external forces are balanced.12 The force density method is material independent. It produces results that may serve only as preliminary, and additional iterations may be necessary.
A method to estimate the hydroelastic behaviour of VLFS based on multi-rigid-body dynamics and beam bending
Published in Ships and Offshore Structures, 2019
Da Lu, Shixiao Fu, Xiantao Zhang, Fei Guo, Yun Gao
In order to facilitate the calculation, the matrix method is used in structural mechanics. The coordinate system of a beam element, which satisfies the right-handed principle, is set as follows (see Figure 4): the neutral axis of the beam element is taken as the X axis and the two principal axes of inertia on the cross section of the beam element are, respectively, Y axis and Z axis.