By David V. Hutton
This new textual content, meant for the senior undergraduate finite aspect direction in mechanical, civil and aerospace engineering departments, offers scholars a superb, useful knowing of the foundations of the finite point procedure inside of numerous engineering purposes. Hutton discusses easy concept of the finite aspect process whereas heading off variational calculus, as a substitute focusing upon the engineering mechanics and mathematical heritage which may be anticipated of senior engineering scholars. The textual content depends uncomplicated equilibrium ideas, advent of the primary of minimal power power, and the Galerkin finite aspect procedure, which with no trouble permits program of finite point research to nonstructural difficulties. The textual content is software-independent, making it versatile sufficient to be used in a large choice of courses, and gives a good choice of homework difficulties and examples.
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Extra info for Fundamentals of Finite Element Analysis
In the case of rotation, the partial derivative of strain energy with respect to a rotational displacement is equal to the moment/torque applied at the point of concern in the sense of the rotation. The following example illustrates the application in terms of a simple torsional member. 5 A solid circular shaft of radius R and length L is subjected to constant torque T. 9. Formulate the elastic strain energy in terms of the angle of twist at x = L and show that Castigliano’s ﬁrst theorem gives the correct expression for the applied torque.
Solution This example includes a nonhomogeneous boundary condition. In previous examples, the boundary conditions were represented by zero displacements. In this example, we have both a zero (homogeneous) and a speciﬁed nonzero (nonhomogeneous) displacement condition. The algebraic treatment must be different as follows. 3: Three-element system with speciﬁed nonzero displacement at node 3. Hutton: Fundamentals of Finite Element Analysis 30 2. Stiffness Matrices, Spring and Bar Elements CHAPTER 2 Text © The McGraw−Hill Companies, 2004 Stiffness Matrices, Spring and Bar Elements Since U 1 = 0 , we remove the ﬁrst row and column to obtain 4k −3k 0 −3k 5k −2k 0 U2 −F = F −2k ␦ 3 2F U4 2k as the system of equations governing displacements U2 and U4 and the unknown nodal force F3.
1a, which shows that no displacement constraint whatever has been imposed on motion of the spring element; that is, the spring is not connected to any physical object that would prevent or limit motion of either node. With no constraint, it is not possible to solve for the nodal displacements individually. Instead, only the difference in nodal displacements can be determined, as this difference represents the elongation or contraction of the spring element owing to elastic effects. As discussed in more detail in the general formulation of interpolation functions (Chapter 6) and structural dynamics (Chapter 10), a properly formulated ﬁnite element must allow for constant value of the ﬁeld variable.