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ME 323: Mechanics of Materials Homework Set 7

Fall 2019 Due: Wednesday, October 16

Note: Students are free to use Matlab /Maple /Mathematica to solve the final algebra and plot the deflection curves.

Problem 7.1 (10 points)

A rigid bar BEF is welded (fixed attachment) to an elastic cantilever beam ABCD (elastic modulus 𝐸 and second moment of area 𝐼) at the location B as shown in Fig. 7.1. A point load 𝑃 is applied at F. Using the second order integration method:

1) Calculate the deflection at the free end D 𝑣 in terms of 𝑃, 𝐿, 𝐸 and 𝐼. 2) Plot the deflection of the beam ABCD. (Non-dimensionalize the plot, i.e., 𝑥 becomes 𝑥/𝐿

and deflection 𝑣 becomes 𝑣/|𝑣 |, where |𝑣 | is the magnitude of deflection at free end D). Does the maximum deflection (absolute value) occur at free end D? (Answer Yes or No using the plot developed)

Fig. 7.1

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Problem 7.2 (10 points)

A cantilever beam ABC made from a material with elastic modulus 𝐸 and second moment of area 𝐼 is loaded as shown in Fig. 7.2. Using the second order integration method:

1) Determine the deflection at the free end 𝑣 in terms of 𝑃, 𝐿, 𝐸 and 𝐼. 2) Plot the deflection of the beam ABCD. (Non-dimensionalize the plot, i.e., 𝑥 becomes 𝑥/𝐿,

deflection 𝑣 becomes 𝑣/|𝑣 | where |𝑣 | is the magnitude of deflection at free end A). Does the maximum deflection (absolute value) occur at free end A? (Answer Yes or No using the plot developed)

For calculations, use 𝑞 =

Fig. 7.2

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Problem 7.3 (10 points)

A fixed beam AB (elastic modulus 𝐸 and second moment of area 𝐼) is subject to a trapezoidal load that varies from 𝑞 at 𝑥 = 0 to 2𝑞 at 𝑥 = 𝐿, as shown in the figure. Using second or fourth order integration method. Determine:

1) The maximum bending moment magnitude in terms of 𝑞, 𝐿, 𝐸 and 𝐼, and the corresponding 𝑥- coordinate location.

2) Calculate |𝑣 / | (the deflection magnitude at 𝑥 = 𝐿/2) in terms of 𝑃, 𝐿, 𝐸 and 𝐼.

3) Plot the deflection of the beam AB. (Non-dimensionalize the plot, i.e., 𝑥 becomes 𝑥/𝐿 and deflection 𝑣 becomes 𝑣/|𝑣 / |)

Fig. 7.3

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Problem 7.4 (10 points)

A steel (𝐸 = 30,000 𝑘𝑠𝑖) square beam ABCD with a side length 𝑎 = 6" is subject to loading as shown in Fig. 7.4. Using second order integration method:

1) Determine the reactions at the supports at ends A and D. 2) Plot the deflection of the beam ABCD.

Fig. 7.4