Question

What is the failure theory to be considered for the loading of aluminum components under steady load conditions?

• A. Maximum principal stress theory
• B. Maximum principal strain theory
• C. Maximum shear stress theory
• D. Maximum strain energy theory

Hint:

The choice of failure theory depends on the material's ductility. For \textbf{ductile materials} (like aluminum, steel) under static loading, the \textbf{Maximum Shear Stress Theory (Tresca)} and \textbf{Distortion Energy Theory (Von Mises)} are commonly used. For \textbf{brittle materials} (like cast iron), the \textbf{Maximum Principal Stress Theory (Rankine)} is typically applied. Under cyclic loading, fatigue theories are used.

Answer 1

The Correct Option is C

Step 1: Understand the context of failure theories.
Failure theories are used in engineering to predict when a material will yield or fracture under complex stress states based on the results from simple uniaxial tensile tests. These theories are essential for design and safety.
The choice of failure theory depends on the material's behavior (ductile vs. brittle) and the loading conditions (steady vs. cyclic).
Step 2: Characterize aluminum and the loading condition.
Aluminum components: Aluminum alloys are generally considered ductile materials. Ductile materials typically fail by yielding (permanent deformation) before fracture under static loading.
Steady load conditions: This implies static loading, not fatigue (cyclic) loading.
Step 3: Review common failure theories for ductile materials.
For ductile materials under static loading, two prominent failure theories are typically considered: 1. Maximum Shear Stress Theory (Tresca Yield Criterion): This theory states that yielding begins when the maximum shear stress in the material reaches the maximum shear stress at the yield point in a simple tensile test. It is often a conservative (safe-side) prediction for ductile materials. \[ \tau_{max} \le \frac{S_y}{2} \] where \( \tau_{max} \) is the maximum shear stress in the component and \( S_y \) is the yield strength from a uniaxial tensile test. 2. Distortion Energy Theory (Von Mises Yield Criterion): This theory states that yielding occurs when the distortion energy per unit volume at any point in the material equals the distortion energy per unit volume at the yield point in a simple tensile test. This theory is generally more accurate for ductile materials than the Maximum Shear Stress Theory, especially for complex stress states. \[ \sigma_v \le S_y \] where \( \sigma_v \) is the Von Mises equivalent stress.
Step 4: Determine the appropriate theory for aluminum under steady load.
Aluminum is a ductile material. For ductile materials under steady (static) load conditions, both the Maximum Shear Stress Theory (Tresca) and the Distortion Energy Theory (Von Mises) are applied. Let's evaluate the given options in the context of typical engineering practice:
Maximum Principal Stress Theory (Rankine): This theory is generally suitable for brittle materials. Aluminum is ductile.
Maximum Principal Strain Theory (Saint-Venant): This theory is also less commonly used for ductile materials compared to Tresca or Von Mises. Maximum Shear Stress Theory (Tresca): This theory is widely used for ductile materials and provides a safe design.
Maximum Strain Energy Theory: This term often encompasses both total strain energy and distortion energy. The distortion energy component (Von Mises) is highly accurate for ductile materials. If "Maximum strain energy theory" implies only total strain energy, then it's generally not used for ductile materials because it predicts yielding under hydrostatic stress which is not observed. Given the options and the fact that aluminum is ductile, Maximum Shear Stress Theory is a standard and acceptable theory to consider. The final answer is \( \boxed{\text{Maximum shear stress theory}} \).