For an A36 steel rod (yield stress of 36 ksi), a total of 0.011 strain occurs if a stress of 
55 ksi is applied as shown in the figure below. Determine the permanent strain for this 
stress level (55 ksi). The modulus of elasticity of steel is 29,000 ksi

To determine the permanent strain for the stress level of 55 ksi, we can approach this using the following steps:

### Step 1: Identify the Given Data
- **Yield stress** (\( \sigma_y \)): 36 ksi
- **Applied stress** (\( \sigma \)): 55 ksi
- **Total strain** (\( \epsilon_{total} \)): 0.011
- **Modulus of Elasticity** (\( E \)): 29,000 ksi

### Step 2: Calculate the Elastic Strain
Before the steel yields (up to 36 ksi), the strain is purely elastic. To find the elastic strain corresponding to the yield stress of 36 ksi, we use Hooke's law:

\[
\epsilon_{elastic} = \frac{\sigma_y}{E}
\]

Substituting the known values:

\[
\epsilon_{elastic} = \frac{36}{29,000} = 0.00124
\]

This is the elastic strain up to the yield point.

### Step 3: Find the Permanent (Plastic) Strain
The total strain at 55 ksi includes both elastic and plastic (permanent) components. Therefore, we can express the total strain as:

\[
\epsilon_{total} = \epsilon_{elastic} + \epsilon_{plastic}
\]

Rearranging to find the plastic (permanent) strain:

\[
\epsilon_{plastic} = \epsilon_{total} - \epsilon_{elastic}
\]

Substituting the values:

\[
\epsilon_{plastic} = 0.011 - 0.00124 = 0.00976
\]

Thus, the permanent strain for the stress level of 55 ksi is **0.00976** (dimensionless strain).

### Summary:
The permanent (plastic) strain for a stress level of 55 ksi is **0.00976**.

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Would you like further details or clarifications? Here are some related questions for deeper exploration:

1. How does stress beyond the yield point affect the strain of steel?
2. What is the significance of the modulus of elasticity in calculating strain?
3. What is the difference between elastic and plastic deformation in materials?
4. How does the yield stress relate to the design of steel structures?
5. What happens to a steel rod when subjected to stresses above its yield point?

**Tip:** Always distinguish between elastic and plastic deformation to understand how materials behave under various loads!

For an A36 steel rod (yield stress of 36 ksi), a total of 0.011 strain occurs if a stress of 55 ksi is applied as shown in the figure below. Determine the permanent strain for this stress level (55 ksi). The modulus of elasticity of steel is 29,000 ksi.

Calculate the permanent strain for an A36 steel rod subjected to a stress of 55 ksi, given a total strain of 0.011 and a modulus of elasticity of 29,000 ksi. This problem involves concepts like Hooke's Law and elastic-plastic deformation, suitable for undergraduate engineering students.

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