# Step-by-Step Solution Let's break down the problem as requested, using **Tresca (Maximum Shear Stress)** theory and **Maximum Normal Stress** theory, which are both relevant for brittle materials. --- ## **Given Data** - \( \sigma_x = -80\, \text{MPa} \) - \( \sigma_y = 90\, \text{MPa} \) - \( \tau_{xy} = 60\, \text{MPa} \) - Ultimate strength (\( \sigma_{ult} \)): **350 MPa** - Poisson's ratio (\( \nu \)): **.20** --- ## **Step 1: Find Principal Stresses** The principal stresses (\( \sigma_1, \sigma_2 \)) for a 2D stress state are given by: \[ \sigma_{1,2} = \frac{(\sigma_x + \sigma_y)}{2} \pm \sqrt{ \left( \frac{\sigma_x - \sigma_y}{2} \right)^2 + \tau_{xy}^2 } \] Plug in the values: \[ \frac{(\sigma_x + \sigma_y)}{2} = \frac{-80 + 90}{2} = 5\, \text{MPa} \] \[ \frac{(\sigma_x - \sigma_y)}{2} = \frac{-80 - 90}{2} = -85\, \text{MPa} \] \[ \sqrt{(-85)^2 + 60^2} = \sqrt{7225 + 360} = \sqrt{10825} \approx 104\, \text{MPa} \] So, \[ \sigma_1 = 5 + 104 = \textbf{109 MPa} \] \[ \sigma_2 = 5 - 104 = \textbf{-99 MPa} \] --- ## **Step 2: Tresca (Maximum Shear Stress) Theory** Tresca theory states that yielding begins when the maximum shear stress in the material reaches the maximum shear stress in a simple tension test (half the ultimate strength for brittle materials): \[ \tau_{max} = \frac{|\sigma_1 - \sigma_2|}{2} \] \[ \tau_{max} = \frac{|109 - (-99)|}{2} = \frac{208}{2} = 104\, \text{MPa} \] **Factor of Safety (FOS) using Tresca:** \[ \text{FOS}_{\text{Tresca}} = \frac{\text{Ultimate Strength}/2}{\tau_{max}} = \frac{350/2}{104} = \frac{175}{104} \approx \boxed{1.68} \] --- ## **Step 3: Maximum Normal Stress Theory** For brittle materials, the maximum normal stress theory (Rankine theory) is often used, which states failure occurs when the maximum principal stress exceeds the ultimate strength. \[ \text{FOS}_{\text{Max Normal}} = \frac{\text{Ultimate Strength}}{|\sigma_{max}|} \] Here, \( |\sigma_{max}| = \max(|\sigma_1|, |\sigma_2|) = \max(109, 99) = 109\, \text{MPa} \): \[ \text{FOS}_{\text{Max Normal}} = \frac{350}{109} \approx \boxed{3.21} \] --- ## **Step 4: Brief Discussion** - **Tresca Theory:** Chosen because it is widely used for ductile materials but can offer a conservative estimate for brittle materials when shear failure is possible. It considers the maximum shear stress, which is important in many engineering applications. - **Maximum Normal Stress Theory:** Directly applicable to brittle materials, as they usually fail in tension rather than shear. It compares the maximum principal (normal) stress to the ultimate strength. --- ## **Final Answers** **a) Tresca Factor of Safety:** \[ \boxed{1.68} \] **b) Maximum Normal Stress Factor of Safety:** \[ \boxed{3.21} \] --- **Summary Table:** | Failure Theory | Factor of Safety | |-------------------------|:----------------:| | Tresca (Shear) | 1.68 | | Max Normal Stress | 3.21 | --- **Why these criteria?** - **Tresca** gives a conservative estimate for safety, especially if there is any possibility of shear-driven failure. - **Max Normal Stress** is most appropriate for brittle materials, as they tend to fracture when the maximum normal stress exceeds their strength. --- If you want to see the calculation steps for other criteria (such as von Mises), let me know!