> [!info] Order of Studying > 1. Indeterminate structures + compatibility > 2. Axial + thermal deformation > 3. Torsion: stress, twist, composite/indeterminate > 4. Shear-force & bending-moment diagrams > 5. Flexural + transverse shear stress > 6. Generalized Hooke's law / Poisson effects > 7. Stress cubes, inclined-plane stress, gears, factor of safety | | \# | Focus | Why start here | | --- | ---------- | ----------------------------- | ----------------------------------------------------------------- | | [x] | **3.5-1** | Indeterminate axial members | Clean practice for **equilibrium → deformation → compatibility** | | [x] | **3.6-5** | Thermal + indeterminate axial | Combines $E,A,L,\alpha$, rigid supports, and thermal effects | | [x] | **4.5-1** | Basic torsion | Internal torque, $\tau=\frac{T\rho}{J}$, and $\phi=\frac{TL}{GJ}$ | | [x] | **4.6-1** | Indeterminate torsion | Fixed at both ends; solve torques, stresses, and rotation | | [x] | **4.6-9** | Composite shaft | Common twist and torque sharing between materials | | [ ] | **5.4-11** | Shear/moment diagrams | Full $V$-$M$ diagram practice | | [ ] | **6.3-5** | Flexural stress | Practice $\sigma=-\frac{My}{I}$ and $M=\frac{EI}{\rho}$ | | [ ] | **6.8-13** | Combined beam review | $V$-$M$ diagrams + max shear stress + max flexural stress | # General Problem-Solving 1. **FBD + equilibrium** - $\sum F_x=0,\quad \sum F_y=0,\quad \sum M=0$ 2. **Load–deformation / stress–strain relation** 3. **Compatibility** if indeterminate 4. Solve for forces/torques → stresses/deformations - **Determinate:** equilibrium is enough - **Indeterminate:** equilibrium + deformation + compatibility > **Workflow:** Equilibrium → Deformation → Compatibility → Solve # Stress, Strain & Material Relations $ \sigma=\frac{F}{A},\qquad \varepsilon=\frac{\Delta L}{L},\qquad \sigma=E\varepsilon $ $ \tau_{avg}=\frac{V}{A},\qquad \gamma=\frac{\tau}{G},\qquad G=\frac{E}{2(1+\nu)} $ For pins: $ \tau=\frac{V}{nA} $ - $n=1$: single shear - $n=2$: double shear - Tension: $\sigma>0$ - Compression: $\sigma<0$ ## Inclined Plane $ \sigma=\frac{P}{A}\cos^2\theta $ $ \tau=\frac{P}{2A}\sin2\theta $ - $|\tau|_{max}$ at $\theta=45^\circ$ ## Factor of Safety $ FS=\frac{\text{strength}}{\text{applied stress}} $ # Generalized Hooke's Law / Poisson Effects $ \nu=-\frac{\varepsilon_{lat}}{\varepsilon_{axial}} $ $ \varepsilon_x= \frac{1}{E}[\sigma_x-\nu(\sigma_y+\sigma_z)] +\alpha\Delta T $ Cyclic for $y,z$. $ \gamma_{xy}=\frac{\tau_{xy}}{G} $ - **Plane stress:** $\sigma_z=0$, usually $\varepsilon_z\neq0$ - **Plane strain:** $\varepsilon_z=0$, usually $\sigma_z\neq0$ - Thermal effects produce **normal strain**, not shear strain # Axial & Thermal Deformation General: $ \Delta L=\int\frac{F(x)}{A(x)E(x)}dx $ Constant properties: $ \boxed{\Delta L=\frac{FL}{EA}} $ Thermal: $ \varepsilon_T=\alpha\Delta T $ $ \boxed{\Delta L=\frac{FL}{EA}+\alpha\Delta TL} $ Fully restrained: $ \Delta L=0 \Rightarrow \boxed{\sigma=-E\alpha\Delta T} $ - Heating + restraint → compression - Cooling + restraint → tension ### Stiffness $ k=\frac{EA}{L} $ Higher $E$ or $A$ → stiffer Higher $L$ → less stiff ## Rigid-Bar Compatibility For small rotation: $ v=x\theta $ $ \frac{v_1}{x_1}=\frac{v_2}{x_2} $ For angled members: $ e=u\cos\theta+v\sin\theta $ # Stress Cube $\tau_{ij}$: - $i$ = face normal - $j$ = stress direction $ \tau_{xy}=\tau_{yx},\quad \tau_{xz}=\tau_{zx},\quad \tau_{yz}=\tau_{zy} $ Positive normal stress = tension. # Torsion ## Circular Shafts $ \gamma=\rho\frac{d\phi}{dx} $ $ \boxed{\tau=\frac{T\rho}{J}} $ $ \boxed{\tau_{max}=\frac{Tc}{J}} $ $ J_{solid}=\frac{\pi r^4}{2} $ $ J_{hollow}=\frac{\pi}{2}(r_o^4-r_i^4) $ Angle of twist: $ \boxed{\Delta\phi=\int\frac{T(x)}{GJ}dx} $ Constant $T,G,J$: $ \boxed{\Delta\phi=\frac{TL}{GJ}} $ Torsional stiffness: $ k_t=\frac{GJ}{L} $ ### Key Relationships - $\tau\propto\rho$ → zero at center, max at outside - $J\propto r^4$ → diameter strongly affects stress/twist - Larger $G,J$ → less twist ## Indeterminate Torsion Use: 1. Torque equilibrium 2. $\Delta\phi=TL/GJ$ 3. Compatibility Fixed at both ends: $ \sum\Delta\phi_i=0 $ ## Composite / Bonded Shafts Same rotation: $ \phi_1=\phi_2 $ Same twist rate: $ \frac{d\phi_1}{dx}=\frac{d\phi_2}{dx} $ $ T_i=G_iJ_i\frac{d\phi}{dx} $ Therefore: $ \boxed{T_i\propto G_iJ_i} $ - $\phi$ is the same across the cross section - $\gamma$ varies linearly with radius - $\tau=G\gamma$ can **jump** when $G$ changes ## Gears $ r_1\phi_1=-r_2\phi_2 $ $ \frac{T_1}{r_1}=\frac{T_2}{r_2} $ Meshing gears rotate in opposite directions. # Beams: Load, Shear & Moment $ \boxed{\frac{dV}{dx}=w(x)} $ $ \boxed{\frac{dM}{dx}=V(x)} $ Therefore: $ \Delta V=\int w\,dx,\qquad \Delta M=\int V\,dx $ - Slope of $V$ = $w$ - Slope of $M$ = $V$ - Area under $w$ = $\Delta V$ - Area under $V$ = $\Delta M$ - Point load → jump in $V$ - Point moment → jump in $M$ # Beam Stress ## Flexural Stress $ \boxed{\sigma_x=-\frac{My}{I}} $ $ \boxed{\sigma_{max}=\frac{|M|c}{I}} $ - Linear with $y$ - Zero at neutral axis - Maximum at extreme fibers Moment-curvature: $ M=\frac{EI}{\rho} $ ## Transverse Shear $ \boxed{\tau=\frac{VQ}{It}} $ $ Q=A^*\bar y^* $ - $I$ = entire cross section - $A^*$ = area above/below point - $\bar y^*$ = distance from neutral axis to centroid of $A^*$ - Usually maximum near neutral axis - Zero at outer surfaces ## Area Moment of Inertia $ I=I_c+Ad^2 $ Rectangle: $ I=\frac{bh^3}{12} $ Circle: $ I=\frac{\pi r^4}{4} $ Hollow circle: $ I=\frac{\pi}{4}(r_o^4-r_i^4) $ > $I$ → bending, $J$ → torsion # High-Priority Relationships | Change | Effect | | ---------------------- | ---------------------------------------------------- | | $E\uparrow$ | axial deformation $\downarrow$ | | $A\uparrow$ | axial deformation $\downarrow$ | | $L\uparrow$ | axial deformation $\uparrow$ | | $G\uparrow$ | torsional twist $\downarrow$ | | $J\uparrow$ | torsional stress and twist $\downarrow$ | | $\|y\| \uparrow$ | bending stress $\uparrow$ | | $I\uparrow$ | bending stress $\downarrow$ | | $M\uparrow$ | bending stress $\uparrow$ | | $V\uparrow$ | beam shear stress $\uparrow$ | | $\frac{EA}{L}\uparrow$ | member attracts more load in an indeterminate system | | $\frac{GJ}{L}\uparrow$ | shaft attracts more torque | # Exam Traps / Must-Know Concepts - **Indeterminate:** equilibrium alone is not enough - Include **thermal strain** in compatibility when $\Delta T\neq0$ - $\Delta L=0$ does **not** imply $\sigma=0$ - Plane stress $\neq$ plane strain - Use $J$ for torsion; $I$ for bending - Composite shafts share **twist**, not necessarily torque/stress - If $T(x)$ varies, integrate $\frac{T(x)}{GJ}$ - Bending stress depends on $M$; shear stress depends on $V$ - Use the full-section $I$ in $\tau=\frac{VQ}{It}$ - Keep force/torque/stress signs consistent