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38649 results in Cambridge Textbooks

Page 739 of 1933

  • 3 - Beams and Frames

  • Sung W. Lee, University of Maryland, College Park, Peter W. Chung, University of Maryland, College Park
  • Book:
    Finite Element Method for Solids and Structures
    Published online:
    08 July 2021
    Print publication:
    17 June 2021, pp 69-110

  • Summary

    In this chapter we first consider finite element modeling of slender bodies undergoing bending deformation. This will be followed by a discussion on frame structures which can be modeled as an assemblage of slender bodies rigidly connected. First, we will introduce the Bernoulli--Euler theory of beam bending as a review and extension of what is typically covered in an undergraduate sophomore-level course on mechanics of materials. We will then introduce the frame element which can be used to model frame structures deforming in the 2D plane and 3D space.

  • 6 - Virtual Displacement and Virtual Work

  • Sung W. Lee, University of Maryland, College Park, Peter W. Chung, University of Maryland, College Park
  • Book:
    Finite Element Method for Solids and Structures
    Published online:
    08 July 2021
    Print publication:
    17 June 2021, pp 165-183

  • Summary

    In this chapter we introduce the concept of an arbitrary virtual displacement which may also be considered as an arbitrary weight function. This will be used to express the equilibrium equation for 3D solids and structures in a scalar integral form. Subsequently, the divergence theorem is applied to transform the scalar integral into another form to which the force boundary condition can be introduced. This results in the statement for the principle of virtual work involving internal virtual work and external virtual work. Internal virtual work and external virtual work will then be expressed in matrix form so that they can be used for the finite element formulation in later chapters. We then consider plane stress and plane strain problems in which the principle of virtual work can be expressed in 2D domains in accordance with simplifying conditions. In the last section, the Lagrange equation is derived within the context of deformable solid bodies, starting from the principle of virtual work.

  • 11 - Heat Transfer

  • Sung W. Lee, University of Maryland, College Park, Peter W. Chung, University of Maryland, College Park
  • Book:
    Finite Element Method for Solids and Structures
    Published online:
    08 July 2021
    Print publication:
    17 June 2021, pp 283-316

  • Summary

    In this chapter we present the finite element formulation of heat transfer problems which can be used to determine temperature distributions in solid bodies, starting with heat conduction in the 1D domain. Similar to the notion of virtual displacement in earlier chapters, a virtual temperature or an arbitrary weight function is introduced to derive an integral equivalent of the governing equation to which the finite element formulation is applied. Methods for heat conduction and convection, in 1D, 2D, and 3D domains, including time-dependent effects, will be covered. Mathematical equivalence with other scalar field problems is also discussed.

  • 1 - Introduction to the Finite Element Method

  • Sung W. Lee, University of Maryland, College Park, Peter W. Chung, University of Maryland, College Park
  • Book:
    Finite Element Method for Solids and Structures
    Published online:
    08 July 2021
    Print publication:
    17 June 2021, pp 1-41

  • Summary

    The finite element method is a powerful technique that can be used to transform any continuous body into a set of governing equations with a finite number of unknowns called degrees of freedom (DOF). In this chapter, we will introduce the fundamentals of the finite element method using a system of linear springs and a slender linear elastic body undergoing axial deformation as examples. These simple problems are chosen to describe the essential features of the finite element method which are common to analysis of more complicated structural systems such as 3D bodies.

Page 739 of 1933