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ME 475Computer Aided Design of Structures
Finite Element Analysis of Trusses – Part 1
Ron AverillMichigan State University
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Learning Objectives1. Describe the differences between truss and frame systems
2. Recall the 2D bar finite element equations and assumptions
3. Define the orientation angle for a 2D plane truss element
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Trusses
A truss is a structure made from slender members that are joined together at their ends.
The type of connection used to jointhe members is important indeciding how to represent themembers in a finite element model.
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Trusses versus FramesPin joints in trusses can transmit forces, but not moments.
So members do not bend.
Angle θ is free to change
during loading.
Rigid joints in frames can transmit forces and moments.
So there is bending in members.
Angle θ remains fixed
during loading.
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θ
θ
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Truss Assumptions1. Members are joined at their ends by frictionless pins
2. Loads are applied at the joints
These assumptions ensure that each F
truss member acts as a two-force member:
Tension
Compression
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Review of 1D Bar Finite ElementsA 2-noded linear bar element e is depicted as follows:
y
e
x
1 2
h Local coordinates x and y are associated with the element
Local nodes are always numbered “1” and “2” with x2 > x1
We use lower case letters for all local (element) quantities 6
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1D Bar Finite Element ApproximationsWithin a 2-noded linear bar element, we assume that the axial displacement u varies linearly between nodes 1 and 2:
where
x
1 2
h7
Element solution approximation
Interpolation functions
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1D Bar Finite Element EquationsFor a 2-noded linear bar element, the final form of the local finite element equations is:
where
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Stiffness matrix
Nodal displacementvector
Internal force vector
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2D Plane Truss ElementsThe members of a truss are really just bar elements that are oriented arbitrarily relative to the global XY coordinate system: Y x
y e 2
1 X** θ is measured counter clockwise (CCW) from X to x.** Local z and global Z coordinates are in the same direction.
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θ
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Element OrientationsThe orientation of an element is defined by the direction of the local x coordinate, which is from node 1 to node 2.
Note that
Y e 1
y x 2
X
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θ
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ExerciseDetermine the orientation angle for each of the truss elements:
1
1 Y 2 2 3
3 X
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45°
45°
The boolean array is:
The orientations are:
Element θ
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SolutionRecall: θ is measured counter clockwise (CCW) from X to x.
1
1 Y 2 2 3
3 X
12
The boolean array is:
The orientations are:
45°
45°
Element θ
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SolutionRecall: θ is measured counter clockwise (CCW) from X to x.
1
1 Y 2 2 3
3 X
13
The boolean array is:
The orientations are:
45°
45°
Element θ
1 135°
θ1
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SolutionRecall: θ is measured counter clockwise (CCW) from X to x.
1
1 Y 2 2 3
3 X
14
The boolean array is:
The orientations are:
45°
45°
Element θ
1 135°
2 270°
θ2
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SolutionRecall: θ is measured counter clockwise (CCW) from X to x.
1
1 Y 2 2 3
3 X
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The boolean array is:
The orientations are:
45°
45°
Element θ
1 135°
2 270°
3 225°
θ3