Let's break down the problem step-by-step: --- ### **Problem Restatement** - A **circular bar** is under **pure torsion**. - We are asked to: 1. Draw the **Mohr's circle** for a small surface element at the bar's surface. 2. Find the **orientation** (`x', y'`) where the **principal stresses** reside. --- ## **Step 1: Stress State at the Surface** For a bar under pure torsion: - On the surface, the only nonzero stress is the **shear stress** (τ). - **No normal stresses** (σx = σy = 0). Let τ = Torsional shear stress at the surface. The **stress element** at the surface: \[ \begin{bmatrix} 0 & \tau \\ \tau & 0 \\ \end{bmatrix} \] Where τ = \( \frac{T r}{J} \) (T = torque, r = radius, J = polar moment of inertia). --- ## **Step 2: Draw the Mohr's Circle** **Given:** - σx = 0 - σy = 0 - τxy = τ ### **Mohr's Circle Construction:** - **Center:** At (σ_avg, 0) = (0, 0) - **Radius:** R = sqrt[(σx - σy)²/4 + τ²] = sqrt[τ²] = τ The circle is **centered at the origin** with **radius τ**. #### **Mohr's Circle Diagram:** ``` σ ↑ | * | * | ---------+--------→ τ | * | * | ``` - The top and bottom points are at (0, +τ) and (0, -τ). --- ## **Step 3: Principal Stresses** Principal stresses are the maximum and minimum normal stresses (σ₁ and σ₂). - On Mohr's circle, principal stresses are at the intersection with the σ axis. \[ σ_{1,2} = σ_{avg} \pm R = 0 \pm τ = +τ, -τ \] --- ## **Step 4: Orientation of Principal Planes** The angle θ_p (from the x-axis to the principal plane) is given by: \[ \tan 2θ_p = \frac{2τ_{xy}}{σ_x - σ_y} = \frac{2τ}{0} = \infty \] So, \[ 2θ_p = 90^\circ \implies θ_p = 45^\circ \] **Therefore, the principal planes are at 45° to the x and y axes.** --- ## **Summary Table** | Quantity | Value | |-------------------------|---------------------------| | **Principal Stresses** | \( +τ \), \( -τ \) | | **Orientation (θ_p)** | \( 45^\circ \) | | **Mohr's Circle Center**| (0, 0) | | **Mohr's Circle Radius**| \( τ \) | --- ## **Final Answer** ### **1. Mohr's Circle** - Center at (0, 0) - Radius = \( τ \) (the maximum shear stress at the surface) - Principal stresses: \( +τ \) and \( -τ \) ### **2. Principal Planes** - **Orientation:** \( 45^\circ \) to the axes (x, y) - The axes x', y' for principal stresses are rotated \( 45^\circ \) from the original x, y axes. --- **Diagram (for your notes):** ``` (Mohr's Circle) σ ↑ | *----* (at +τ and -τ on τ axis) | -----+----→ τ | ``` **Principal axes are at 45° to x and y.** --- ### **Boxed Final Statement:** **The Mohr's Circle has center (0, 0) and radius τ. The principal stresses at the surface are +τ and -τ, acting on planes oriented 45° to the x and y axes.**