# Uniaxial Tension: Stress State at Point A (Far from Discontinuities) ## 1. Reference Stress State **Description:** The plate is loaded in uniaxial tension (say, along the \(x\)-direction). Far from any holes or notches, the stress state is simple. - \(\sigma_x = \sigma\) (applied tension) - \(\sigma_y = \) - \(\tau_{xy} = \) ### Stress Element Sketch ``` ↑ | σ_x (tension) | --------------- | | | | → σ_x (tension) | | --------------- ``` *Alt text: A square element with arrows on left and right faces pointing outward, labeled σ_x, indicating tensile stress. No arrows on top or bottom.* ### Principal Stresses & Directions - **Principal Stresses:** - \(\sigma_1 = \sigma_x = \sigma\) - \(\sigma_2 = \sigma_y = \) - **Principal Directions:** - Along \(x\) and \(y\) axes (horizontal/vertical) - **Shear Stress:** - \(\tau_{xy} = \) --- # 2. Stress State Near a Circular Hole (Point B, Horizontal Centerline) ## 3. Which Stress Components are Nonzero? At point B (along the horizontal centerline near the hole): - **\(\sigma_x\):** Nonzero. It will be **greater than** the remote \(\sigma\) due to the stress concentration effect at the edge of the hole. - **\(\sigma_y\):** Zero, because on the horizontal centerline, symmetry and traction-free boundary conditions at the hole edge dictate \(\sigma_y = \) (for a thin, infinite plate). - **\(\tau_{xy}\):** Zero, by symmetry along the horizontal centerline. **Summary Table:** | Component | Nonzero? | Reason | |-------------|----------|------------------------------------------------------| | \(\sigma_x\) | Yes | Stress concentration at hole edge | | \(\sigma_y\) | No | Symmetry, traction-free boundary at the edge | | \(\tau_{xy}\) | No | Symmetry along horizontal centerline | ## 4. Maximum Normal Stress Comparison - **Maximum normal stress at point B:** **Greater than \(\sigma\)** - **Reason:** A circular hole causes a stress concentration. For an infinite plate in uniaxial tension, the maximum stress at the edge of the hole (horizontal axis) is: \[ \sigma_{max} = 3\sigma \] where \(\sigma\) is the remote applied stress. ## 5. Sketch: Stress Element at Point B ``` ↑ | (no σ_y) | --------------- | | | | → 3σ (tension, much larger than remote) | | --------------- ``` *Alt text: A square element with a very long arrow on the right face and none on the top/bottom, representing high tensile stress (σ_x) and zero σ_y.* --- # 6. Summary Table: Stress States | Location | \(\sigma_x\) | \(\sigma_y\) | \(\tau_{xy}\) | Max Normal Stress | |------------------|--------------|--------------|---------------|-------------------| | Far from hole (A)| \(\sigma\) | | | \(\sigma\) | | Near hole (B) | \(3\sigma\) | | | \(3\sigma\) | --- # Key Points - **At A (far):** Uniaxial stress, principal directions are axes, only \(\sigma_x\) nonzero. - **At B (edge of hole):** Only \(\sigma_x\) nonzero; it is much larger than remote stress (\(3\sigma\)); \(\sigma_y = \), \(\tau_{xy} = \). - **Stress element at B:** Long arrow in \(x\)-direction, none in \(y\). - **Maximum normal stress at B:** **Greater than** remote value (\(\sigma_{max} = 3\sigma\)).