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    3d Stiffness Matrix

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    Tanmoy Mukherjee
    This document introduces the derivation of element stiffness matrices for one-dimensional finite elements using classical structural analysis methods. It discusses two approaches: the first uses concepts like moment-area analysis, while the second is based on energy considerations and shape functions. It then derives the stiffness matrices for various elements like trusses, beams, frames, and grids by relating forces and displacements based on axial, flexural, torsional and shear deformation. The document aims to assemble individual element stiffness matrices which can then be used to form a global structure stiffness matrix.

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    3d Stiffness Matrix

    Uploaded by

    Tanmoy Mukherjee
    1K views38 pages
    This document introduces the derivation of element stiffness matrices for one-dimensional finite elements using classical structural analysis methods. It discusses two approaches: the first uses concepts like moment-area analysis, while the second is based on energy considerations and shape functions. It then derives the stiffness matrices for various elements like trusses, beams, frames, and grids by relating forces and displacements based on axial, flexural, torsional and shear deformation. The document aims to assemble individual element stiffness matrices which can then be used to form a global structure stiffness matrix.

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    3 d stiffness matrix

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    This document introduces the derivation of element stiffness matrices for one-dimensional finite elements using classical structural analysis methods. It discusses two approaches: the first uses concepts like moment-area analysis, while the second is based on energy considerations and shape functions. It then derives the stiffness matrices for various elements like trusses, beams, frames, and grids by relating forces and displacements based on axial, flexural, torsional and shear deformation. The document aims to assemble individual element stiffness matrices which can then be used to form a global structure stiffness matrix.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    1K views38 pages

    3d Stiffness Matrix

    Uploaded by

    Tanmoy Mukherjee
    This document introduces the derivation of element stiffness matrices for one-dimensional finite elements using classical structural analysis methods. It discusses two approaches: the first uses concepts like moment-area analysis, while the second is based on energy considerations and shape functions. It then derives the stiffness matrices for various elements like trusses, beams, frames, and grids by relating forces and displacements based on axial, flexural, torsional and shear deformation. The document aims to assemble individual element stiffness matrices which can then be used to form a global structure stiffness matrix.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 38
    At a glance
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    The key takeaways are that different approaches can be used to derive stiffness matrices of structural elements, including classical methods and energy methods using shape functions, and various types of elements will be discussed.

    Influence coefficients are defined as the effect on a degree of freedom due to a unit action at another degree of freedom, for an individual element or whole structure. Examples of influence coefficients are shown in Table 2.1.

    The two major beam theories discussed are Euler-Bernoulli beam theory, which is the classical formulation, and Timoshenko beam theory, which accounts for transverse shear deformation effects.

    Introduction to Finite Elements

    Lecture 2
    Matrix Structural Analysis of Framed
    Structures

    Introduction
    In this chapter, we shall derive the element
    stiness matrix [k] of various one dimensional
    elements. Only after this important step is well
    understood, we could expand the theory and
    introduce the structure stiness matrix [K] in its
    global coordinate system.

    Introduction
    As will be seen later, there are two fundamentally
    dierent approaches to derive the stiness
    matrix of one dimensional element. The rst one,
    which will be used in this chapter, is based on
    classical methods of structural analysis (such as
    moment area or virtual force method). Thus,
    in deriving the element stiness matrix, we will be
    reviewing concepts earlier seen.

    Introduction
    The other approach, based on energy
    consideration through the use of assumed shape
    functions, will be examined later. This second
    approach, exclusively used in the nite element
    method, will also be extended to two
    dimensional continuum elements.

    Influence Coefficients
    In structural analysis an inuence coecient Cij
    can be dened as the eect on d.o.f. i due to a unit
    action at d.o.f. j for an individual element or a hole
    structure. Examples of Inuence Coecients are
    shown in Table 2.1.
    It should be recalled that inuence lines are
    associated with the analysis of structures subjected
    to moving loads (such as bridges), and that the
    exibility and stiness coecients are components
    of matrices used in structural analysis.

    Influence Coefficients

    Flexibility Matrices

    Remember the Virtual Force Method!

    Flexibility Method

    Stiffness Coefficients

    Force Displacement: Axial Def.

    Force Displacement: Flexural Def.

    Force Displacement: Flexural Def.

    Force Displacement: Torsional Def.

    Force Displacement: Torsional Def.

    Force Displacement: Shear Def.

    Force Displacement: Shear Def.

    Force Displacement: Shear Def.

    Effect of Translation on Shear Def.

    Effect of Rotation on Shear Def.

    Putting it All Together


    Using basic structural analysis methods we have derived
    various force displacement relations for axial, exural,
    torsional and shear imposed displacements. At this point,
    and keeping in mind the denition of degrees of freedom,
    we seek to assemble the individual element stiness
    matrices [k]. We shall start with the simplest one, the
    truss element, then consider the beam, 2D frame, grid,
    and nally the 3D frame element.
    In each case, a table will cross-reference the force
    displacement relations, and then the element
    stiness matrix will be accordingly dened.

    Truss Element

    Beam Elements
    There are two major beam theories:
    Euler-Bernoulli which is the classical
    formulation for beams.
    Timoshenko which accounts for transverse
    shear deformation eects.

    Euler-Bernoulli Beam Theory

    Euler-Bernoulli Beam Element (Flexure)

    Timoshenko Beam Element

    Timoshenko Beam Element

    Timoshenko Beam Element

    Timoshenko Beam Element

    2D Frame Element (Beam+Truss)

    Grid Element

    3D Frame Element

    3D Frame Element

    Remarks on Element Stiffness


    Matrices

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