CES 6116 FEM-Review Matrix Structural Analysis

Dr. F. Necati CatbasOffice:ENG-II 406Phone: 407-823-3743e-mail: catbas@mail.ucf.edu

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Coordinate Transformation-Truss Elements

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Global to Local Coordinate Transformation

In Global AxesIn Local Axes

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Transformation Matrix

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Member Angles

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Local to Global Transformation

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Transformation Matrix

Orthogonal Matrix

v=TTu

Local to GlobalGlobal to Local

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Truss Analysis - Example

Determine the joint displacements, member axial deformations, support reactions of the truss

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Member Stiffness Matrices

Truss Analysis - Example

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Truss Analysis - Example

{P}=[S]{d}

21

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Truss Analysis - Example

Q1=k1u1u1=T1v1

Member 1 Deformations in Global CS.

Stiffness and Member Forces in Local CS

MEMBER 1

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Truss Analysis -Example

F=TTQ

Member 1 Force Transformation from Local to Global

MEMBER 1

3412

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Truss Analysis-ExampleMEMBER 2

Member 2 Deform.s in Global CS. u2=T2v2

F=TTQ

Q2=k2u2

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Truss Analysis-Example

Member 3 Deformations in Global CS.

u3=T3v3

{Q}=[T]{F}

F3=K3v3

MEMBER 3

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Truss Analysis-Example

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Coordinate Transformation-Beam Elements

Global Coordinates

Local Coordinates

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Coordinate TransformationGlobal

CoordinatesTransformation MatrixLocal

Coordinates

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Coordinate Transformation-Derivations

Also we can write the following:

Finally, We can write {F} as follows:

[T] is orthogonal matrix Substituting, we obtain,

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Stiffness Matrix for a Plane Frame Element Oriented at an Angle φ

R=A/I

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Example Problem #1

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Example Problem #1 (cont.)

For Element 2-3

For Element 1-2

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Example Problem #1 (cont.)

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Example Problem #1 (cont.)

Combining Element 1-2 and Element 1-2 Stiffness Matrices

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Example Problem #1 (cont.)

Boundary Conditions

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Example Problem #1 (cont.)

Re-arranging based on BCs and known loadings, we get

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Example Problem #1 (cont.)Writing knowns and unknowns, we get 6 eqns for 6 unknowns:

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Example Problem #1 (cont.)

Using matrix inverse,solve for unknown disp.

Solve for unknown disp.

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Example Problem #1 (cont.)

Using disp found above,solve for unknown loads.

Solve for unknown disp.

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Example Problem #2

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Example Problem #2 (cont.)

Deformed shape of symmetric structure under anti-symmetric loading

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Example Problem #2 (cont.)

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Example Problem #2 (cont.)

Combining 2 element Stiffness Matrices,

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Example Problem #2 (cont.)

Deformations are obtained as follows,

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Example Problem #2 (cont.)

Forces and Moments at the Nodal Points (including supports)

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Example ProblemDetermine the joint displacements, member end forces and support reactions for the two member frame.

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Example Problem

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Fixed End Forces on Member 1

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Analysis of Member 2

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Member End Forces in Local Coordinates

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Example Problem

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Solution of the Assembled Structural System

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Computing End Forces for Member 1

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Computing End Forces for Member 1

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Equilibrium Check for Member 1

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Example Problem

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Example Problem