https://file.aiquickdraw.com/mathbot/file/172573433055023k847y4.jpg

The problem statement provides information about a beam subjected to a completely reversed bending moment, and we are asked to find the factor of safety (FOS). Let's break this down step by step to calculate the factor of safety.

### Given:
- Bending moment amplitude, \( M = 5 \, \text{kNm} = 5000 \, \text{Nm} \)
- Ultimate strength, \( \sigma_u = 500 \, \text{MPa} \)
- Endurance limit, \( \sigma_e = 250 \, \text{MPa} \)
- Section modulus of the cross-section, \( Z = 50,000 \, \text{mm}^3 = 50 \times 10^6 \, \text{mm}^3 = 50 \times 10^{-6} \, \text{m}^3 \)

### Formula:
To calculate the factor of safety (FOS) under fatigue loading conditions, the typical approach is to use the Goodman criterion for completely reversed bending:

\[
\text{Factor of Safety (FOS)} = \frac{\sigma_e}{\sigma_{\text{bending}}}
\]

Where \( \sigma_{\text{bending}} \) is the bending stress induced by the applied moment:

\[
\sigma_{\text{bending}} = \frac{M}{Z}
\]

Now, let's compute the bending stress and subsequently the factor of safety.The calculated factor of safety (FOS) is \( 0.0025 \), which is extremely low and indicates that the beam would fail under the given loading conditions. This result seems unusual given the typical safety factors expected in engineering.

It might be a good idea to revisit the data units or assumptions. One possible reason for this discrepancy could be the incorrect interpretation of units, such as the bending moment or section modulus. Let me try adjusting these units to check the validity of the solution by converting section modulus to proper dimensions of \( mm^3 \).The corrected factor of safety (FOS) is **2.50**. This is a more realistic value for a beam under fatigue loading conditions, ensuring that the beam can safely withstand the applied loads with an appropriate safety margin.

Would you like any further details or clarifications?

Here are five related questions to consider:
1. How is the endurance limit determined for different materials?
2. What other safety factors are used in engineering besides the Goodman criterion?
3. How do changes in section modulus affect the safety of a beam?
4. How does fatigue loading differ from static loading in materials analysis?
5. Why is it important to consider the completely reversed nature of bending in fatigue analysis?

**Tip:** Always ensure consistent unit usage when working with mechanical and material properties to avoid miscalculations.

A beam is subjected to completely reverse bending moment of amplitude 5 kNm. Ultimate strength is 500 MPa and endurance limit is 250 MPa. Section modulus of the cross section of beam is 50000 mm³. Factor of safety is [Round off to the two decimal places].

Learn how to calculate the factor of safety for a beam subjected to a completely reversed bending moment of amplitude 5 kNm. This solution involves using the Goodman criterion, taking into account the ultimate strength, endurance limit, and section modulus. The final factor of safety is calculated as 2.50.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation