A wooden beam is subjected to concentrated loads as shown below. Where along the span the maximum flexural stress (bending stress) will develop?

To determine where along the span of a wooden beam the maximum flexural stress (bending stress) will develop due to concentrated loads, we need to consider the loading conditions and the properties of the beam.

1. Break Down the Problem

• Identify the concentrated loads acting on the beam.
• Understand the support conditions (simply supported, cantilever, etc.).
• Determine the distance between the concentrated loads and the supports.

2. Relevant Concepts

• Bending Stress Formula: The bending stress ($\sigma$) at a distance $y$ from the neutral axis of the beam can be expressed as: $\sigma = \frac{M \cdot y}{I}$ where:
  • $M$ = moment at the section,
  • $y$ = distance from the neutral axis to the outermost fiber,
  • $I$ = moment of inertia of the beam's cross-section.
• Location of Maximum Moment: Maximum bending moments typically occur directly under the concentrated loads or between them, depending on the beam's supports and load arrangement.

3. Analysis and Detail

• Calculate the bending moment along the beam. This can be done using the method of sections or by constructing a shear and moment diagram.
• The maximum moment usually occurs at the point of loading or at mid-span for symmetrical loading.
• Evaluate the moment at each significant section of the beam to see where the maximum bending moment occurs.

4. Verify and Summarize

• By examining the shear and moment diagrams, we can establish the specific span locations that yield maximum bending moments.
• The maximum flexural stress will generally be found at the point of maximum moment, which could be directly under a load or at the support hinge, depending on the beam's configuration.

Final Answer

In summary, the maximum flexural stress will develop at the location along the beam where the maximum bending moment occurs. This is typically at the point of concentrated loads or at sections where loads create a high moment, which can be determined through moment calculations and shear force diagrams.