Problem 54 - An Introduction to the Mechanics of Solids in SI Units

Question

A threaded steel rod is subject to a mean tensile stress of \(210 \mathrm{MPa}\). The ultimate strength for the material is \(900 \mathrm{MPa}\), and \(K_{f}\) for the threaded section is \(2.6\). Estimate the maximum alternating stress that can be applied.

Step-by-Step Solution

Answer

The maximum alternating stress that can be applied to the rod is approximately 77.88MPa.

Step 1: Goodman Equation

The Goodman relation is given by: \(\frac{\sigma_m}{S_u} + K_f \frac{\sigma_a}{S'} = 1\), where \(\sigma_m\) is the mean stress, \(S_u\) is the ultimate strength, \(\sigma_a\) is alternating stress, \(K_f\) is the fatigue stress concentration factor and \(S'\) is the endurance strength. We are looking for \(\sigma_a\), the maximum alternating stress.

Step 2: Substitute Known Values and Solve for the Unknown

Since \(S'\) can be approximated as 0.5*\(S_u\) for steel, we have \(S' = 0.5 * 900MPa = 450 MPa\). Re-writing the Goodman equation for \(\sigma_a\) gives \(\sigma_a = \frac{(1 - \frac{\sigma_m}{S_u}) * S'}{K_f}\). Substituting known quantities into the expression, \(\sigma_a = \frac{(1 - \frac{210MPa}{900MPa}) * 450MPa}{2.6} = 77.88MPa\).

Step 3: Final Answer

The maximum alternating stress that can be applied to the rod is approximately 77.88MPa.

Key Concepts

Understanding Mean Tensile Stress

Mean tensile stress is a crucial parameter in the mechanics of materials, particularly when assessing the performance of components under load. It refers to the average tensile stress that a material experiences when subjected to a constant or static load over time. This type of stress is vital in scenarios where components are subject to direct pulling forces that could lead to elongation or stretching. In the context of the original exercise, the threaded steel rod is subjected to a mean tensile stress of 210 MPa. This value indicates the continuous stress level imposed on the rod, which can affect its performance and potential for failure. Understanding the mean tensile stress helps engineers design materials that can withstand constant loads, ensuring structural integrity and reliability over long periods. When combined with other factors like alternating stress and stress concentration, it is integral for predicting a component's lifespan.

Exploring Ultimate Strength

Ultimate strength, often referred to as ultimate tensile strength (UTS), is the maximum stress that a material can endure before failure. It represents a key property of materials and is essential for determining how much load a material can handle. For the threaded rod in the exercise, the ultimate strength of 900 MPa signifies the peak stress level the steel can withstand from tension before it fractures. Engineers use this parameter to ascertain the safety margin and to choose appropriate materials for applications involving high loads. The ultimate strength is a benchmark for evaluating different materials and understanding how materials will perform under extreme conditions. In our equation, it helps calculate the alternating stress that can be safely applied, ensuring that the material does not exceed its capacity to bear stress and thereby avoiding structural failure.

Fatigue Stress Concentration Factor Explained

The fatigue stress concentration factor, denoted as \(K_f\), is a modifying factor used in the fatigue assessment of materials. It accounts for the increased stress and reduced fatigue life due to geometrical features like notches or threads. In the context of the threaded steel rod, the fatigue stress concentration factor is given as 2.6. This indicates that the presence of threading increases the localized stress by a factor of 2.6 times compared to what would be expected in a smooth, uninterrupted section. This rise in localized stress can accelerate the fatigue process, leading to premature failure of the material.Understanding \(K_f\) helps engineers design structures that minimize the risk of fatigue failure by considering geometric discontinuities. By incorporating \(K_f\) in the Goodman equation, designers can more accurately predict the maximum alternating stress that can be safely applied, ensuring the component's durability and longevity in service.