4.6 Mechanics of Solids

4.6 Mechanics of Solids

1. Deformable Body Analysis

1. Basic Assumptions:

   • Material is continuous, homogeneous, and isotropic.

   • Deformations are small compared to dimensions.

   • Stress-strain relationship follows Hooke's law (within elastic limit).

2. Analysis Steps:

   • Equilibrium: Apply $\sum F = 0$, $\sum M = 0$.

   • Compatibility: Deformations must be geometrically consistent.

   • Constitutive Relations: Use stress-strain laws ($\sigma = E\epsilon$).

3. Stress Types:

   • Normal Stress: $\sigma = \frac{F}{A}$ (tension/compression).

   • Shear Stress: $\tau = \frac{V}{A}$ (shearing force).

   • Bearing Stress: $\sigma_b = \frac{P}{A_{bearing}}$ (contact surfaces).

4. Strain Types:

   • Normal Strain: $\epsilon = \frac{\Delta L}{L}$.

   • Shear Strain: $\gamma = \text{angular distortion}$.

   • Volumetric Strain: $\epsilon_v = \frac{\Delta V}{V}$.

2. Determinate and Indeterminate Structures

1. Determinate Structures:

   • All support reactions and internal forces can be found using equilibrium equations alone.

   • Condition: $m + r = 2j$ (for trusses, where $m$ = members, $r$ = reactions, $j$ = joints).

   • Examples: Simply supported beams, cantilevers, simple trusses.

2. Indeterminate Structures:

   • Cannot be solved by equilibrium equations alone.

   • Condition: $m + r > 2j$ (for trusses).

   • Require additional equations from compatibility of deformations.

3. Degree of Indeterminacy:

   • External: $r - 3$ (for 2D structures).

   • Internal: For trusses: $(m + r) - 2j$.

   • Total: External + internal.

4. Solution Methods for Indeterminate Structures:

   • Compatibility Equations: Based on deformation constraints.

   • Method of Superposition.

   • Energy Methods: Castigliano's theorem, virtual work.

   • Slope-Deflection Method.

   • Moment Distribution Method.

3. Thin and Thick Cylinders

1. Thin Cylinders ($\frac{t}{D} < \frac{1}{20}$):

   • Hoop/Circumferential Stress: $\sigma_h = \frac{pD}{2t}$

   • Longitudinal Stress: $\sigma_l = \frac{pD}{4t}$

   • Assumptions:

     • Stress uniform across thickness.

     • Radial stress negligible.

   • Volumetric Strain: $\epsilon_v = \frac{pD}{4tE}(5 - 4\nu)$

2. Thick Cylinders ($\frac{t}{D} \geq \frac{1}{20}$):

   • Stress varies across thickness.

   • Lame's Equations (for internal pressure only):

     • Radial Stress: $\sigma_r = A - \frac{B}{r^2}$

     • Hoop Stress: $\sigma_h = A + \frac{B}{r^2}$

   • Boundary Conditions:

     • At $r = r_i$, $\sigma_r = -p_i$.

     • At $r = r_o$, $\sigma_r = -p_o$.

   • Maximum Stresses:

     • Hoop stress maximum at inner surface.

     • Important for pressure vessel design.

3. Compound Cylinders:

   • Shrink-fit cylinders to create compressive pre-stress.

   • Reduces tensile hoop stress during pressurization.

   • Increases pressure capacity.

4. Torsion of Non-Circular Sections

1. General Differences from Circular Sections:

   • Plane sections do NOT remain plane (warping occurs).

   • Shear stress distribution is NOT linear.

   • Maximum shear stress does NOT occur at farthest point from center.

2. Rectangular Sections:

   • Maximum shear stress at midpoint of longer side.

   • $\tau_{max} = \frac{T}{\alpha b h^2}$

   • $\theta = \frac{TL}{\beta G b h^3}$

   • $\alpha, \beta$ depend on $h/b$ ratio (from tables).

3. Thin-Walled Closed Sections:

   • Shear Flow (q): Constant around section: $q = \tau t = \text{constant}$.

   • Bredt-Batho Formula: $\tau = \frac{T}{2A_m t}$ Where $A_m$ = area enclosed by median line.

   • Angle of Twist: $\theta = \frac{TL}{4A_m^2 G} \oint \frac{ds}{t}$

4. Thin-Walled Open Sections:

   • Treat as series of rectangles: $J = \sum \frac{1}{3} b_i t_i^3$.

   • Maximum stress at thickest part.

   • Less efficient than closed sections for same weight.

5. Membrane Analogy (Prandtl):

   • Visual method to estimate stress distribution.

   • Shear stress ∝ slope of membrane.

   • Torsional constant ∝ volume under membrane.