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tanmoy-mukherjee
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3 d stiffness matrix
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