ME 323: Mechanics of Materials Homework 9 Fall 2021 Due Wednesday, November 10 Problem 9.1 (10 points)

A stepped shaft ABC is made of members 1, 2 and 3. Member 1 is a hollow shaft and members 2 and 3 are solid shafts. Members 1 and 2 are fit tightly together. Connector B is a rigid connector whereas A and C are fixed ends. The shear modulus of members 1, 2 and 3 are 1.5G, 5G and 7G, respectively.

A) Determine whether the structure is statically determinate or indeterminate. B) Use Castigliano’s theorem to find the torque on each member. C) Use Castigliano’s theorem to find the angle of rotation at B.

Use: d1 = 4D; d2 = 1.5D; d3 = 2D

ME 323: Mechanics of Materials Homework 9 Fall 2021 Due Wednesday, November 10

ME 323: Mechanics of Materials Homework 9 Fall 2021 Due Wednesday, November 10

ME 323: Mechanics of Materials Homework 9 Fall 2021 Due Wednesday, November 10

ME 323: Mechanics of Materials Homework 9 Fall 2021 Due Wednesday, November 10 Problem 9.2 (10 points)

The propped cantilever ABC of constant EI is subjected to a distributed load w0 as shown in the figure. The beam is also supported by an elastic rod BD. Use the Castigliano’s theorem to determine the deflection of point B on the beam. Ignore the shear energy due to bending in your analysis.

ME 323: Mechanics of Materials Homework 9 Fall 2021 Due Wednesday, November 10

ME 323: Mechanics of Materials Homework 9 Fall 2021 Due Wednesday, November 10

ME 323: Mechanics of Materials Homework 9 Fall 2021 Due Wednesday, November 10

ME 323: Mechanics of Materials Homework 9 Fall 2021 Due Wednesday, November 10 Problem 9.3 (10 points)

A rod is made up of 3 members. There is a rigid connector at B and C. The areas of the members are A1=A3=A

A2=2A. The Length of each member is L1=L2=L and L3=2L. The modulus of the members are E1=E3=E a0nd E2=3E. The rod has two forces acting on it at the connectors B and C, equal to 3P and 2P respectively.

A) Determine the Stiffness matrix [K] B) Determine the Forcing vector {F} C) Reduce the system equations using boundary conditions D) Determine the displacement at point B and C.

Figure 2

ME 323: Mechanics of Materials Homework 9 Fall 2021 Due Wednesday, November 10 Saturday, November 6, 2021 15:41

ME 323: Mechanics of Materials Homework 9 Fall 2021 Due Wednesday, November 10

ME 323: Mechanics of Materials Homework 9 Fall 2021 Due Wednesday, November 10

ME 323: Mechanics of Materials Homework 9 Fall 2021 Due Wednesday, November 10 Problem 9.4 (2.5 +2.5 points)

1) A shaft has a stiffness matrix [𝐾] =𝐾1 −𝐾1 0

−𝐾1 𝐾1 + 𝐾2 −𝐾20 −𝐾2 𝐾2

. The right end of the shaft is free.

Simplify the matrix.

a) 𝐾1 −𝐾1−𝐾1 𝐾1 + 𝐾2

b) 𝐾1 + 𝐾2 −𝐾2

−𝐾2 𝐾2 row 1 and column1 can be removed as the left side only is fixed

c) 𝐾1 −𝐾1

−𝐾1 𝐾1 + 𝐾20 −𝐾2

2) What is the order of the reduced stiffness matrix for a shaft composed of 4 elements and with both ends fixed. a) 4 b) 3 order of a stiffness matrix is (N+1) where N is the number of elements. Then each fixed end will

reduce the order by one. (4+1)-2= 3 c) 5 d) 6