University:
McMaster University
Course:
MATLS 2H04 | Introduction to Materials Engineering
Academic year:

2023
Views:

185

Pages:

2
Author:

Sabrina S.

State of Plane Stress one area element is stress free all stress components related to this area element are 0 now only have 3 components σx, σy, τxy plane stress is important for 2 reasons when we analyse stress states close to a free surface of a structure, which is one of the most common practical cases if no forces stresses are acting on the surface, it can be analysed this way analysis of thin-walled structures (eg. thin-walled pressure vessels, composite laminates, plate & shell structures where the thickness is much less than the in-plane dimensions) 4.3.2 Stress Transformations directional dependence of stress P σ = μ/A = ρ/A V σ = ?? N A₀ = A/cosθ σ = N₀/A₀ = Pcosθ/A T = Vθ/A₀ = Psinθcosθ/A N₀ = Ncosθ V₀ = Nsinθ cannot treat stress as vectors because both the force component acting on the area element & the size of the area element the force is acting on, are both functions of θ ΔS = area of the triangle aligned w x ΔSx = ΔSsinθ ΔSy = ΔScosθ x' ΔSy τx'y' σx σx' ΔS ΣFx' = 0 ⇒ σx' = σxΔSycosθ + σyΔSxsinθ + τxyΔSxcosθ + τxyΔSycosθ τxy ΔSx σx = cosθsinθ * 2τxysinθcosθ ΣFy' = 0 ⇒ τx'y' = (σx - σy)sinθcosθ - τxy(sin²θ - cos²θ) σy' = σxcos²(θ+90°) + σysin²(θ+90°) + 2τxysin(θ+90°)cos(θ+90°) σy Image Analysis Visual Elements: The image contains handwritten equations and diagrams related to stress transformation. The text is organized into two main sections. Mathematical equations are used to express stress transformation formulas. Diagrams of stress components on an element and a loaded beam are included. Arrows indicate directions of stresses and coordinate axes. Textual Content: diff form of stress transformation eq's [\sigma_{x}^{\prime}=\frac{\sigma_{x}+\sigma_{y}}{2}+\frac{\sigma_{x}-\sigma_{y}}{2}cos~2\theta+\tau_{xy}sin~2\theta] [\tau_{x,y}^{\prime}=-\frac{\sigma_{x}-\sigma_{y}}{2}sin~2\theta+\tau_{xy}cos~2~\theta] [\sigma_{y}^{\prime}=\frac{\sigma_{x}+\sigma_{y}}{2}-\frac{\sigma_{x}-\sigma_{y}}{2}cos~2~\theta-\tau_{xy}sin~2\theta] e.g// Stress transformation - axially-loaded beam find the stress components in area element rotated from x-axis by [30^{\circ}] [\sigma_{x}=\frac{P}{A}] [\sigma_{y}=0] [\tau_{xy}=0] [\theta=30^{\circ}] [\sigma_{x}^{\prime}=\frac{\sigma_{x}+0}{2}+\frac{\sigma_{x}-0}{2}cos~60+0~sin~60] [=\frac{\sigma_{x}}{2}+\frac{\sigma_{x}}{2}(\frac{1}{2})] [=\frac{3}{4}\sigma_{x}] [\tau_{x,y}^{\prime}=-\frac{\sigma_{x}-0}{2}sin~60+0~cos~60] [=-\frac{\sigma_{x}}{2}(\frac{\sqrt{3}}{2})] [=-\frac{\sqrt{3}}{4}\sigma_{x}] [\sigma_{y}^{\prime}=\frac{\sigma_{x}+0}{2}-\frac{\sigma_{x}-0}{2}cos~60-0~sin~60] [=\frac{1}{4}\sigma_{x}] (Tay) shear stress convention if +ve ↑ if -ve ↓