Beams and Frames

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Beams and Frames

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• beam theory can be used to solve simple beams
• complex beams with many cross section changes are
  solvable but lengthy
• many 2-d and 3-d frame structures are better
  modeled by beam theory

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One Dimensional Problems
The geometry of the problem is three dimensional,
but the variation of variable is one dimensional
Variable can be scalar field like temperature, or
vector field like displacement. For dynamic
loading, the variable can be time dependent
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Element Formulation

• assume the displacement w is a cubic polynomial
  in

a1, a2, a3, a4 are the undetermined coefficients

• L Length
• I Moment of Inertia of the cross sectional
  area
• E Modulus of Elsaticity
• v v(x) deflection of the neutral axis
• dv/dx slope of the elastic curve (rotation of
  the section
• F F(x) shear force
• M M(x) Bending moment about Z-axis

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• Applying these boundary conditions, we get

• Substituting coefficients ai back into the
  original equation for v(x) and rearranging terms
  gives

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• The interpolation function or shape function is
  given by

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• strain for a beam in bending is defined by the
  curvature, so
• Hence

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• the stiffness matrix k is defined

To compute equivalent nodal force vector for the
loading shown
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Equivalent nodal force due to Uniformly
distributed load w