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.