mee342_hw1_2026.md

layout	default
title	MEE342 Homework 1 2026

Disclaimer

Images for Problems 2 and 3 are taken from Shigley's.

Problem 1

For the plane stress $\sigma_x = -8MPa$, $\sigma_y = 7MPa$, $\tau_{xy} = 6MPa$ cw, draw a Mohr's circle diagram properly labeled, find the principal normal and shear stresses, and determine the angle from the $x$ axis to $\sigma_1$. Draw stress elements as in Fig. 3-11c and d (Shigley's) and label all details.

Problem 2

A countershaft carrying two V-belt pulleys is shown in the figure. Pulley A receives power from a motor through a belt with the belt tensions shown. The power is transmitted through the shaft and delivered to the belt on pulley B. Assume the belt tension on the loose side at B is 15 percent of the tension on the tight side.

• Determine the tensions in the belt on pulley B, assuming the shaft is running at a constant speed.

• Find the magnitudes of the bearing reaction forces, assuming the bearings act as simple supports.

• Draw shear-force and bending-moment diagrams for the shaft. If needed, make one set for the horizontal plane and another set for the vertical plane.

• At the point of maximum bending moment, determine the bending stress and the torsional shear stress.

• At the point of maximum bending moment, determine the principal stresses and the maximum shear stress.

Problem 3

The cantilevered bar in the figure is made from a ductile material and is statically loaded with $F_y = 200lbf$ and $F_x = F_z = 0$. Analyze the stress situation in rod AB by obtaining the following information.

• Determine the precise location of the critical stress element.

• Sketch the critical stress element and determine magnitudes and directions for all stresses acting on it. (Transverse shear may only be neglected if you can justify this decision.)

• For the critical stress element, determine the principal stresses and the maximum shear stress.

Problem 4

Consider a cantilever beam as shown in the figure with length $l$ and the left end fixed to a wall.

1. Derive the deflection $y$ of the beam along $x$ under a single downward force $F$ on the right, and assume that the moment of inertia $I$ is constant along $x$. (Show complete derivation instead of the final function $y(x)$)

2. Now consider that the beam has two connected parts, with the part on the left (of length $l/2$) having a moment of inertia $I_1$, and the part on the right $I_2$. Derive the deflection $y$ again. Ignore stress concentration.

3. Further, consider that the cross-sections for the two parts are both circular, and the total volume of the beam is constant. What should $I_1$ and $I_2$ be for the beam to have minimal maximum deflection? (Optional)

Problem 5 (From bending moment to deflection)

Consider beam analysis where the beam is positioned along the $x$-axis and deflection along $y$-axis. Let $M(x)$, $E(x)$, $I(x)$, $\rho(x)$ be the bending moment, Young's modulus, moment of inertia along $y$-axis, and the curvature radius, respectively. Show step-by-step the following relationship:

$$ \frac{1}{\rho(x)} = \frac{M(x)}{E(x)I(x)}. $$

Then explain that for small beam defection, we have

$$ \frac{1}{\rho(x)} \approx \frac{d^2y}{dx^2}. $$

Problem 6 (Stress due to bending)

Consider beam analysis where the beam is positioned along the $x$-axis and deflection along $y$-axis. Focus on a cross-section at some location $x$ along the beam. Let $M$, $V$ and $I$ be the bending moment, shear force, and moment of inertia along $y$-axis, respectively, at this cross-section. Let $\sigma(y)$, $\tau(y)$, and $b(y)$ be the normal stress, shear stress, and cross-section width, respectively, along $y$-axis, considering the neutral axis of the cross-section as the origin. Let $c$ be the distance from the neutral axis of the cross-section to its top (we assume that the cross-section is symmetric along its neutral axis). Show step-by-step the following relationship:

$$ \sigma(y) = -\frac{M}{I}y, $$

and

$$ \tau(y) = \frac{V}{Ib(y)}\int_y^c y'b(y')dy'. $$

Problem 6 (Shigley's 11th Exercise 4-11 Superposition)

For the wire form of diameter d shown in figure below, determine the deflection of point B in the direction of the applied force F (neglect the effect of transverse shear).

Problem 7 (Shigley's 11th Exercise 4-122)

The steel beam ABCD shown is simply supported at C as shown and supported at B and D by shoulder steel bolts, each having a diameter of 8 mm. The lengths of BE and DF are 50 mm and 65 mm, respectively. The beam has a second area moment of 21 $\times 10^3$ $mm^4$. Prior to loading, the members are stress-free. A force of 2 kN is then applied at point A. Determine the stresses in the bolts and the deflections of points A, B, and D.

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