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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