Structural Analysis
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![### Calculating Stress Changes Below a Foundation
**Problem Statement:**
A square foundation with a width \( B = 5 \) feet is subjected to an applied load of 100 kips. Find the change in stress 8 feet below the bottom of the footing using the following methods:
e) 2:1 method
f) M and N method
g) Stress isobars
h) Newmark Method
#### Explanation:
1. **2:1 Method:**
- The 2:1 method is a simplistic approach to estimate the vertical stress distribution below a footing. The method assumes a uniform load distribution spreading out at a 2:1 slope from the base of the footing. The change in stress (\( \Delta \sigma \)) at a depth \( z \) below the footing can be calculated with the formula:
\[
\Delta \sigma = \frac{Q}{(B + z)^2}
\]
where \( Q \) is the applied load and \( B \) is the width of the footing.
2. **M and N Method:**
- The M and N method is a more refined approach based on elasticity theory, which accounts for the footing shape and depth more accurately than the 2:1 method. The method involves using tables or charts that provide influence values (M and N) based on the geometry and depth.
3. **Stress Isobars:**
- Stress isobars are contour lines representing constant stress values within a soil mass under a given load. They can be used to visualize how stress dissipates in the soil. These isobars are usually determined using theoretical formulas or numerical methods such as the Boussinesq equation.
4. **Newmark Method:**
- The Newmark method uses a graphical technique with influence charts to determine the stress distribution under a loaded area. Influence charts represent the change in stress at various depths caused by a uniformly loaded area.
Each of these methods provides a different level of accuracy and application complexity. For educational purposes, understanding the assumptions and limitations of each method is crucial before applying them to real-world scenarios.](https://content.bartleby.com/qna-images/question/24dbfb66-daf6-4b8e-a748-b5c870ef5592/761da8d7-66c6-4f87-8aa5-f16b812d9ee6/2ynrn1e_processed.jpeg)
Transcribed Image Text:### Calculating Stress Changes Below a Foundation
**Problem Statement:**
A square foundation with a width \( B = 5 \) feet is subjected to an applied load of 100 kips. Find the change in stress 8 feet below the bottom of the footing using the following methods:
e) 2:1 method
f) M and N method
g) Stress isobars
h) Newmark Method
#### Explanation:
1. **2:1 Method:**
- The 2:1 method is a simplistic approach to estimate the vertical stress distribution below a footing. The method assumes a uniform load distribution spreading out at a 2:1 slope from the base of the footing. The change in stress (\( \Delta \sigma \)) at a depth \( z \) below the footing can be calculated with the formula:
\[
\Delta \sigma = \frac{Q}{(B + z)^2}
\]
where \( Q \) is the applied load and \( B \) is the width of the footing.
2. **M and N Method:**
- The M and N method is a more refined approach based on elasticity theory, which accounts for the footing shape and depth more accurately than the 2:1 method. The method involves using tables or charts that provide influence values (M and N) based on the geometry and depth.
3. **Stress Isobars:**
- Stress isobars are contour lines representing constant stress values within a soil mass under a given load. They can be used to visualize how stress dissipates in the soil. These isobars are usually determined using theoretical formulas or numerical methods such as the Boussinesq equation.
4. **Newmark Method:**
- The Newmark method uses a graphical technique with influence charts to determine the stress distribution under a loaded area. Influence charts represent the change in stress at various depths caused by a uniformly loaded area.
Each of these methods provides a different level of accuracy and application complexity. For educational purposes, understanding the assumptions and limitations of each method is crucial before applying them to real-world scenarios.
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