To determine the behavior of the steel rod under the applied force, we need to analyze the situation using the concepts of stress and yield strength. The yield strength of the structural steel is given as 2.5 x 10^8 N/m², and we can calculate the stress experienced by the rod when a force of 90 kN is applied. Let's break this down step by step.
Calculating Stress
Stress is defined as the force applied per unit area. The formula for stress (σ) is:
σ = F / A
Where:
- F = applied force (in Newtons)
- A = cross-sectional area (in m²)
Finding the Cross-Sectional Area
The cross-sectional area (A) of the rod can be calculated using the formula for the area of a circle:
A = πR²
First, we need to convert the radius from millimeters to meters:
R = 9.5 mm = 0.0095 m
Now, substituting R into the area formula:
A = π(0.0095 m)² ≈ 2.827 x 10^-4 m²
Calculating the Stress
Now we can calculate the stress:
F = 90 kN = 90,000 N
Substituting the values into the stress formula:
σ = 90,000 N / 2.827 x 10^-4 m² ≈ 3.18 x 10^5 N/m²
Comparing Stress to Yield Strength
Now that we have the stress, we can compare it to the yield strength of the steel:
Yield Strength = 2.5 x 10^8 N/m²
Since the calculated stress (3.18 x 10^5 N/m²) is significantly less than the yield strength (2.5 x 10^8 N/m²), we can conclude that the rod will not undergo permanent deformation.
Understanding the Options
Now, let's analyze the provided options:
- (a) Permanent deformation: This is incorrect because the stress is below the yield strength.
- (b) Elongates in the elastic limit: This is the correct answer. The rod will experience elastic deformation, meaning it will return to its original length once the force is removed.
- (c) No change: This is not accurate; there will be some elongation within the elastic limit.
- (d) Diameter reduces to half: This is also incorrect as the stress is not sufficient to cause such a change.
Final Thoughts
In summary, when a force of 90 kN is applied to the steel rod, it will elongate within the elastic limit but will not experience permanent deformation. This understanding of stress and yield strength is crucial in structural engineering, as it helps ensure that materials are used safely and effectively in construction and design.
