Solved Problems

Problem: A simply supported rectangular timber beam is 4 m long, 200 mm wide, and 300 mm deep. It carries a uniformly distributed load of 10 kN/m over its entire length. Determine the maximum flexural stress.

Problem: Using the same beam as Example 1, determine the maximum shearing stress.

Problem: A T-beam has a flange width of 100 mm, flange thickness of 20 mm, web thickness of 20 mm, and total depth of 120 mm. The web depth is $120 - 20 = 100$ mm. If the vertical shear force $V = 10$ kN, determine the shear stress at the junction of the flange and the web.

Problem: A W-shape steel beam (W410x60) is subjected to a maximum bending moment of $250 \text{ kN}\cdot\text{m}$. From the steel manual, the section properties are: depth $d = 407 \text{ mm}$, moment of inertia $I_x = 216 \times 10^6 \text{ mm}^4$. Calculate the maximum bending stress and state whether it exceeds an allowable stress of $165 \text{ MPa}$.

Problem: A solid circular steel shaft with a diameter of $D = 50 \text{ mm}$ is subjected to a vertical shear force of $V = 15 \text{ kN}$. Determine the maximum shear stress in the shaft.

Problem: A hollow rectangular aluminum beam has outer dimensions of $150 \text{ mm}$ (width) by $200 \text{ mm}$ (depth) and a uniform wall thickness of $10 \text{ mm}$. It is subjected to a bending moment of $M = 35 \text{ kN}\cdot\text{m}$. Calculate the maximum flexural stress.

Problem: A simplified steel I-beam is subjected to a shear force $V = 100 \text{ kN}$. The flanges are $200 \text{ mm}$ wide and $20 \text{ mm}$ thick. The web is $10 \text{ mm}$ thick, and the overall depth is $340 \text{ mm}$. The computed total moment of inertia is $I = 164.5 \times 10^6 \text{ mm}^4$. Calculate the shear stress just inside the flange and just inside the web at the junction.

Problem: An asymmetrical cast iron T-beam is subjected to a bending moment of $M = 15 \text{ kN}\cdot\text{m}$. The centroid of the cross-section has already been calculated to be $35 \text{ mm}$ from the top flange surface. The total depth of the beam is $120 \text{ mm}$, and the moment of inertia about the neutral axis is $I = 2.5 \times 10^6 \text{ mm}^4$. Calculate the maximum tensile and compressive bending stresses. Note that cast iron typically has different limits for tension and compression.

Problem: A $50 \text{ mm}$ by $200 \text{ mm}$ rectangular wooden joist can be oriented in two ways: standing upright (strong axis, $h = 200$, $b = 50$) or lying flat (weak axis, $h = 50$, $b = 200$). Demonstrate why orienting the beam along its strong axis is vastly superior for resisting bending.

Problem: A solid rectangular steel beam and a steel I-beam have the exact same cross-sectional area (meaning they weigh the same per meter). Explain conceptually why the I-beam can carry a significantly larger bending load before failing.

Problem: When a beam is subjected to a vertical shear force, the resulting internal shear stress is not uniform across the cross-section. Describe the shear stress profile for a solid rectangular section versus an I-beam section.

Problem: In a standard simply supported beam subjected to a uniformly distributed load, the structural engineer must check both flexural (bending) failure and shear failure. Where in the physical 3D space of the beam do the absolute maximum flexural stresses and absolute maximum shear stresses occur?