# 6.4 Static Analysis

## 6.4 Static Analysis

### 1. Design for Static Strength

#### Stress Analysis Fundamentals

1. **Stress Types in Static Loading**:
   * **Direct Normal Stress**: $$\sigma = \frac{F}{A}$$
     * Tensile: Positive (pulling)
     * Compressive: Negative (pushing)
   * **Direct Shear Stress**: $$\tau = \frac{F}{A}$$ (parallel to surface)
   * **Bearing Stress**: $$\sigma\_b = \frac{F}{A\_p}$$ (contact between surfaces)
2. **Bending Stress**:
   * In beams subjected to bending moments.
   * Formula: $$\sigma = \frac{M \cdot y}{I}$$
     * $$M$$ = bending moment
     * $$y$$ = distance from neutral axis
     * $$I$$ = area moment of inertia
   * Maximum at outermost fibers: $$\sigma\_{max} = \frac{M \cdot c}{I}$$
3. **Torsional Shear Stress**:
   * In shafts subjected to torque.
   * Formula: $$\tau = \frac{T \cdot r}{J}$$
     * $$T$$ = applied torque
     * $$r$$ = radial distance from center
     * $$J$$ = polar moment of inertia
   * Maximum at outer surface: $$\tau\_{max} = \frac{T \cdot R}{J}$$

#### Combined Stresses

1. **Superposition Principle**:
   * Total stress = sum of stresses from individual loads.
   * Valid for linear elastic materials.
2. **Principal Stresses**:
   * Maximum and minimum normal stresses at a point.
   * For plane stress: $$\sigma\_{1,2} = \frac{\sigma\_x + \sigma\_y}{2} \pm \sqrt{\left(\frac{\sigma\_x - \sigma\_y}{2}\right)^2 + \tau\_{xy}^2}$$
3. **Failure Theories**:
   * **Maximum Normal Stress Theory**: Failure when max principal stress reaches yield strength.
   * **Maximum Shear Stress Theory (Tresca)**: Failure when max shear stress reaches shear yield strength.
   * **Distortion Energy Theory (Von Mises)**: Failure when distortion energy reaches critical value.
     * Von Mises equivalent stress: $$\sigma' = \sqrt{\sigma\_1^2 + \sigma\_2^2 - \sigma\_1\sigma\_2}$$

#### Safety Factors

1. **Definition**: $$n = \frac{\text{Strength}}{\text{Stress}}$$
2. **Selection Factors**:
   * Material properties variability.
   * Load uncertainty.
   * Manufacturing tolerances.
   * Consequences of failure.
   * Cost/weight constraints.
3. **Typical Values**: 1.5-4 for ductile materials, 3-10 for brittle materials.

### 2. Transmission Components

#### Gear Design for Static Strength

1. **Bending Stress (Lewis Equation)**:
   * $$\sigma\_b = \frac{F\_t}{b \cdot m \cdot Y}$$
     * $$F\_t$$ = tangential tooth load
     * $$b$$ = face width
     * $$m$$ = module
     * $$Y$$ = Lewis form factor
   * Based on cantilever beam model of tooth.
2. **Contact Stress (Hertzian)**:
   * Surface pitting/failure.
   * $$\sigma\_H = \sqrt{\frac{F\_t}{b \cdot d\_1} \cdot \frac{E}{2\pi(1-\nu^2)} \cdot \frac{i+1}{i}}$$
     * $$E$$ = equivalent modulus of elasticity
     * $$\nu$$ = Poisson's ratio
     * $$i$$ = gear ratio
3. **Design Considerations**:
   * Material selection (steel, cast iron, bronze).
   * Heat treatment (case hardening for wear resistance).
   * Fillet radius to reduce stress concentration.

#### Shaft Design

1. **Strength-Based Design**:
   * Resist combined bending and torsion.
   * Equivalent torque: $$T\_e = \sqrt{M^2 + T^2}$$
   * Shaft diameter: $$d = \left\[\frac{16}{\pi\tau} \sqrt{M^2 + T^2}\right]^{1/3}$$
2. **Stress Concentrations**:
   * Keyways, grooves, holes, shoulders.
   * Use stress concentration factors: $$\sigma\_{max} = K\_t \cdot \sigma\_{nominal}$$
3. **Deflection Limits**:
   * Typically L/360 to L/1000 for beams.
   * Avoid excessive deflection affecting gear meshing.

#### Belt and Chain Drives

1. **Belt Tension Analysis**:
   * Tight side tension ($$T\_1$$) > Slack side tension ($$T\_2$$).
   * Power transmitted: $$P = (T\_1 - T\_2) \cdot v$$
   * Ratio: $$\frac{T\_1}{T\_2} = e^{\mu\theta}$$ (flat belts), $$e^{\mu\theta/\sin(\alpha/2)}$$ (V-belts)
2. **Chain Design**:
   * Based on breaking strength with safety factor.
   * Consider wear and fatigue life.

### 3. Fasteners and Connections

#### Bolt Joints

1. **Preload Importance**:
   * Maintains joint tightness under external loads.
   * Prevents separation and fretting.
   * Typical preload = 75% of proof load.
2. **External Load Distribution**:
   * Only portion of external load affects bolt.
   * Load factor: $$C = \frac{k\_b}{k\_b + k\_m}$$
     * $$k\_b$$ = bolt stiffness
     * $$k\_m$$ = member stiffness
   * Bolt load: $$F\_b = F\_{preload} + C \cdot F\_{external}$$
3. **Joint Stiffness**:
   * Bolt stiffness: $$k\_b = \frac{A\_b E}{L}$$
   * Member stiffness (cone model): $$k\_m = \frac{\pi E d \tan\alpha}{\ln\left\[\frac{(2L\tan\alpha + D\_h - d)(D\_h + d)}{(2L\tan\alpha + D\_h + d)(D\_h - d)}\right]}$$

#### Failure Modes

1. **Bolt Failures**:
   * Tensile failure (yielding/fracture).
   * Thread stripping (shear of threads).
   * Fatigue failure (cyclic loading).
2. **Joint Failures**:
   * Separation (loss of preload).
   * Slippage (shear failure).
   * Bearing failure (crushing of hole edge).

#### Design Calculations

1. **Required Bolt Size**:
   * Based on preload: $$A\_t = \frac{F\_{preload}}{0.75S\_p}$$
   * Based on external load: $$A\_t = \frac{F\_{max}}{S\_t}$$
     * $$A\_t$$ = tensile stress area
     * $$S\_p$$ = proof strength
     * $$S\_t$$ = tensile strength
2. **Number of Bolts**:
   * $$n = \frac{F\_{total}}{F\_{perbolt}} \cdot \text{Safety Factor}$$
3. **Bolt Pattern**:
   * Even spacing for uniform load distribution.
   * Pitch circle diameter for circular patterns.
   * Edge distances: 1.5d minimum, 2d preferred.

#### Other Connections

1. **Riveted Joints**:
   * Design for shear and bearing.
   * Efficiency = $$\frac{\text{Strength of joint}}{\text{Strength of solid plate}}$$
2. **Welded Joints**:
   * **Fillet Welds**: Designed for shear along throat.
     * Throat thickness = 0.707 × leg size
   * **Butt Welds**: Full penetration, strength = base metal.
   * **Allowable Stresses**: Based on electrode/weld type.
3. **Pinned Connections**:
   * Design for double shear.
   * Check bearing on connected members.
   * Consider pin bending for long pins.
