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Space Truss - Finite Element Analysis - Assignment, Exercises of Mathematical Methods for Numerical Analysis and Optimization

Baddi University of Emerging Sciences and Technologies Mathematical Methods for Numerical Analysis and Optimization

Main points are: Space Truss, Modulus of Elasticity, Bending Moment Curve, Deflection Curve, Beam Properties, Spring Stiffness, Four-Noded Bar Element, Shape Functions, Jacobian Determinant, Displacement Matrix

Typology: Exercises

2012/2013

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Finite Element Analysis

Question 1:

Analyze the space truss shown in the figure below. The truss is composed of four nodes,

whose coordinates (in meters) are shown in the figure, and three elements, whose cross-

sectional areas are all 42

10 10

m. The modulus of elasticity 210 EGPa for all the

elements. A load of 20 kN is applied at node 1 in the global xdirection. Nodes 2 to 4 are

pin supported and thus constrained from movement in the x, y, and z directions.

Question 2:

A beam is clamped at the left end and on a spring at the right end as shown in the figure

below. A force 3 000 F, N

acts downward at the right end as shown. The spring stiffness

3 000 k, Nm. The beam properties are 1 Lm, 2

1 000 EI, Nm. Determine the

deflection curve



vx and bending moment curve

 

M x .

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Finite Element Analysis

Question 1: Analyze the space truss shown in the figure below. The truss is composed of four nodes, whose coordinates (in meters) are shown in the figure, and three elements, whose cross- sectional areas are all 10  10 ^4 m^2. The modulus of elasticity E  210 GPa for all the elements. A load of 20 kN is applied at node 1 in the global x direction. Nodes 2 to 4 are pin supported and thus constrained from movement in the x , y , and z directions.

Question 2: A beam is clamped at the left end and on a spring at the right end as shown in the figure below. A force F 3 000 , N acts downward at the right end as shown. The spring stiffness

k 3 000 , N m. The beam properties are L  1 m , EI 1 000 , Nm^2. Determine the

deflection curve v x   and bending moment curve M  x .

Question 3: For the four-noded bar element shown in the figure below, show that the Jacobian determinant is J  L 2. Also determine the shape functions N 1 to N (^) 4 and the

strain/displacement matrix B.

Question 4:

Evaluate the integrals (a) I   ^11  x^2 cos  x 2  dx and (b) I   ^1 1  3 x  x  dx using three-

point Gaussian quadrature. x ,x 1 3 (^)  0 77459666924148_._ , x 2 (^)  0 , w ,w 1 3 (^) 5 9, w 2 (^) 8 9.

Question 5: Consider two triangular elements shown in figure below. The nodal displacements are given

Space Truss - Finite Element Analysis - Assignment, Exercises of Mathematical Methods for Numerical Analysis and Optimization

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deflection curve v x   and bending moment curve M  x .

Evaluate the integrals (a) I   ^11  x^2 cos  x 2  dx and (b) I   ^1 1  3 x  x  dx using three-

as  u ,v ,u ,v ,u ,v ,u ,v 1 1 2 2 3 3 4 4    0 1 0 0 1 0. , ,. , , 0 1 0 0 1 0. , ,. , . Calculate displacements  u,v 

and strains   u  x , v   y, u   y   v  x  in both elements.

L /

3 L /

L