A cantilever beam of constant cross section is subjected to a positive moment $M$ and a positive axial force $F$ as shown. The normal stress due to the axial force $F$ is 200 MPa and the maximum compressive stress due to the moment $M$ is 100 MPa. If the point A is located on the top surface and the point B is located on the centroidal axis of the beam, the stress elements at these points are (with stresses shown in MPa)

Question

Prepp Team · Accepted Answer

To solve this problem, we need to calculate the stress at points A and B in the cantilever beam subjected to an axial force $F$ and a bending moment $M$.

Concepts and Calculations

Axial Stress: The axial stress $\sigma_a$ due to the force $F$ is given directly as 200 MPa (tensile).
Bending Stress: The stress caused by the bending moment $M$ can be calculated using the formula: $\sigma_b = \frac{M \cdot y}{I}$, where $y$ is the distance from the neutral axis and $I$ is the moment of inertia of the cross-section.
The maximum compressive stress at the top surface due to $M$ is given as 100 MPa (compressive).

Stress at Point A (Top Surface)

$\sigma_a = 200 \text{ MPa} \, (\text{tensile})$
$\sigma_b = -100 \text{ MPa} \, (\text{compressive})$.
Total Stress at A: $\sigma_A = \sigma_a + \sigma_b = 200 - 100 = 100 \text{ MPa (tensile)}$

Stress at Point B (Centroidal Axis)

$\sigma_a = 200 \text{ MPa} \, (\text{tensile})$
$\sigma_b = 0 \text{ MPa}$ (since $y = 0$ at the centroidal axis).
Total Stress at B: $\sigma_B = \sigma_a + \sigma_b = 200 \text{ MPa (tensile)}$