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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
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