For the bar subjected to the linear varying axial load shown in Figure P3–54, determine the nodal displacements and axial stress distribution using (a) two equal-length elements and (b) four equal-length elements. Let A = 2 in.2 and E = 30 106 psi. “Compa

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Question

For the bar subjected to the linear varying axial load shown in Figure P3–54, determine the nodal displacements and axial stress distribution using (a) two equal-length elements and (b) four equal-length elements. Let A = 2 in.2 and E = 30 106 psi. “Compa

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Finite-Elements engineering Mechanical-Engineering

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Answer 1

Answer #1

(a)

Draw a 3-noded bar element with a linear varying axial load as follow:

Calculate the total force acting on the element 1.

The total force is from the right handed triangle,

Hence, the nodal forces is given by equation 3.10.33 in the text book,

Calculate the total force acting on the element 2.

Divide the load into uniform load and triangular load.

Consider the equation 3.10.33 in the text book for the triangular load, and adding the loads for the element 2,

Hence, the nodal forces are,

...... (1)

Substitute the known values in equation (1).

Calculate the stiffness matrix for element 1.

Here, the element stiffness matrix is, the cross sectional area of the element is A, modulus of elasticity is E and the length of the element is.

Substitute.

Calculate the stiffness matrix for element 2.

Here, the element stiffness matrix is, the cross sectional area of the element is A, modulus of elasticity is E and the length of the element is.

Substitute.

Calculate the global stiffness matrix,

...... (2)

Substitute the known values in equation (2).

Consider the equation,

...... (3)

Here, the global stiffness matrix is and the displacement matrix is.

Substitute the known values in equation (3).

Substitute,and.

Solve the matrix equation,

Solve the equations to get the nodal displacements,

Hence, the displacement at node 2 is and at node 3 is.

Calculate the stress in each element by using the equation,

Here,,and the angle made by the bar with x-axis is.

For element 1:

Here, the nodal displacement in x direction is u and the nodal displacement in y direction is v.

Solve the matrix equation,

Hence, the stress in element 1 is.

For element 2:

Solve the matrix equation,

Hence, the stress in element 2 is.

(b)

For a four element solution, the bar is divided into four elements of lengths 15 in each and there are 5 nodes

Calculate the total force acting on the element 1.

The total force is from the right handed triangle,

Hence the nodal forces is given by equation 3.10.33 in the text book,

Calculate the total force acting on the element 2.

Divide the load into uniform load and triangular load.

Consider the equation 3.10.33 in the text book for the triangular load, and adding the loads for the element 2,

Calculate the total force acting on the element 3.

Divide the load into uniform load and triangular load.

Consider the equation 3.10.33 in the text book for the triangular load, and adding the loads for the element 3,

Calculate the total force acting on the element 4.

Divide the load into uniform load and triangular load.

Consider the equation 3.10.33 in the text book for the triangular load, and adding the loads for the element 4,

Hence, the nodal forces are,

...... (4)

Substitute the known values in equation (4).

The global stiffness matrix for a 5-noded bar element is,

Here, the cross sectional area of the element is and modulus of elasticity is E.

Consider the equation,

...... (5)

Here, the global stiffness matrix is and the displacement matrix is.

Substitute the known values in equation (5).

Solve the matrix equation to get the nodal displacements,

Hence, the displacement at node 2 is, at node 3 is, at node 4 is and at node 5 is .

Calculate the stress in each element by using the equation,

Here,,and the angle made by the bar with x-axis is.

For element 1:

Here, the nodal displacement in x direction is u and the nodal displacement in y direction is v.

Solve the matrix equation,

Hence, the stress in element 1 is.

For element 2:

Solve the matrix equation,

Hence, the stress in element 2 is.

For element 3:

Solve the matrix equation,

Hence, the stress in element 3 is.

For element 4:

Solve the matrix equation,

Hence, the stress in element 4 is.

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Answer 2

For the bar subjected to the linear varying axial load shown in Figure P3–54, determine the nodal displacements and axial stress distribution using (a) two equal-length elements and (b) four equal-length elements. Let A = 2 in.2 and E = 30 106 psi. “Compa

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For the bar subjected to the linear varying axial load shown in Figure P3–54, determine the nodal displacements and axial stress distribution using (a) two equal-length elements and (b) four equal-length elements. Let A = 2 in.2 and E = 30 106 psi. “Compa

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Homework Answers

Add Answer to: For the bar subjected to the linear varying axial load shown in Figure P3–54, determine the nodal displacements and axial stress distribution using (a) two equal-length elements and (b) four equal-length elements. Let A = 2 in.2 and E = 30 106 psi. “Compa

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Earn Coins

For the bar subjected to axial load shown in Figure 1 to 2, determine the nodal...

Find all the unknown nodal displacements in the bar use 2 elements. A=2 in^2, E=30× 10^6...

3.24 Determine the nodal displacements and the element forces for the truss shown in Figure P3-24. Assume all elements have the same AE 4 15 m 4 2 20 m Figure P3-24 3.24 Determine the nodal...

Problem 2: a. For the plane truss shown in Figure 2, determine the nodal displacements, the element forces and stresses, and the support reactions. All elements have E-70 GPa and A-25 cm 100 k...

Question 2 (10 points) For the rod loaded axially as shown in the Figure, determine the axial displacement of the free end. Let E-30 x 10 psi, A 2 in2, and L-60 in. Use the finite element stiffne...

For the rod loaded axially as shown in the Figure, determine the axial displacement of the free end. Let E-30x 10s psi, A 2 in2, and L 60 in. Use the finite element stiffhess method. For...

Add Answer to:
For the bar subjected to the linear varying axial load shown in Figure P3–54, determine the nodal displacements and axial stress distribution using (a) two equal-length elements and (b) four equal-length elements. Let A = 2 in.2 and E = 30 106 psi. “Compa