1. Types of Variable Loads

Variable loads can be classified based on how stress varies with time:

(a) Completely Reversed Load

• Stress alternates equally between tension and compression
• Example: rotating shaft under bending

$$\sigma_{max} = -\sigma_{min}$$

(b) Repeated (Pulsating) Load

• Stress varies between zero and a maximum value

$$\sigma_{min} = 0, \quad \sigma_{max} \neq 0$$σmin​=0,σmax​=0

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(c) Fluctuating Load

• Stress varies between two unequal values

$$\sigma_{min} \neq \sigma_{max}$$σmin​=σmax​

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🔷 2. Important Stress Parameters

For fluctuating loads, we define:

Mean Stress

$$\sigma_m = \frac{\sigma_{max} + \sigma_{min}}{2}$$σm​=2σmax​+σmin​​

Alternating Stress

$$\sigma_a = \frac{\sigma_{max} – \sigma_{min}}{2}$$σa​=2σmax​−σmin​​

Stress Ratio

$$R = \frac{\sigma_{min}}{\sigma_{max}}$$R=σmax​σmin​​

These parameters are essential for fatigue analysis.

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🔷 3. Fatigue Failure and Endurance Limit

• Fatigue failure occurs due to repeated stress cycles.
• Materials fail at stress levels much lower than yield strength.
• Endurance limit (Se): Maximum stress a material can withstand for infinite cycles.

Key Points:

• For steels: endurance limit exists
• For non-ferrous materials: fatigue strength is specified for a finite number of cycles

4. S-N Curve (Wöhler Curve)

• Shows relationship between stress amplitude and number of cycles to failure
• Logarithmic scale is used for cycles

Regions:

• High stress → low cycles (fatigue failure quickly)
• Low stress → high cycles (infinite life possible for steel)

5. Factors Affecting Endurance Limit

Actual endurance limit is modified as:$$S_e = S’_e \cdot k_a \cdot k_b \cdot k_c \cdot k_d \cdot k_e \cdot k_f$$

Where:

• $k_a$ka​: Surface finish factor
• $k_b$kb​: Size factor
• $k_c$kc​: Load factor
• $k_d$kd​: Temperature factor
• $k_e$ke​: Reliability factor
• $k_f$kf​: Miscellaneous factors

6. Stress Concentration and Notch Sensitivity

• Geometric discontinuities (holes, fillets, keyways) cause stress concentration

$$K_t = \frac{\text{Maximum stress}}{\text{Nominal stress}}$$

• Fatigue stress concentration factor:

$$K_f = 1 + q (K_t – 1)$$

Where:

• $q$: notch sensitivity

7. Failure Theories for Fluctuating Loads

To design safely, we use fatigue failure criteria:

(a) Goodman Theory (Linear Relation)

$\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_u} = \frac{1}{n}$

• Conservative and widely used
• $S_u$Su​: Ultimate strength
• $n$n: Factor of safety

(b) Soderberg Theory (Most Conservative)

$$\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_y} = \frac{1}{n}$$

• Uses yield strength $S_y$Sy​
• Safest but may lead to overdesign

(c) Gerber Theory (Parabolic)

$$\frac{\sigma_a}{S_e} + \left(\frac{\sigma_m}{S_u}\right)^2 = \frac{1}{n}$$

• More accurate for ductile materials
• Less conservative

8. Design Procedure

Step 1: Determine Loads

• Identify maximum and minimum stresses

Step 2: Calculate:

• Mean stress $\sigma_m$σm​
• Alternating stress $\sigma_a$σa​

Step 3: Find Material Properties

• Ultimate strength $S_u$Su​
• Yield strength $S_y$Sy​
• Endurance limit $S_e$Se​

Step 4: Apply Modifying Factors

• Calculate corrected endurance limit

Step 5: Choose Failure Theory

• Goodman / Soderberg / Gerber

Step 6: Calculate Factor of Safety

9. Design for Infinite Life vs Finite Life

Infinite Life Design

• Stress must be below endurance limit
• Used in shafts, structural parts

Finite Life Design

• Based on S-N curve
• Used in components like aircraft parts

10. Practical Design Considerations

• Improve surface finish
• Avoid sharp corners → use fillets
• Use shot peening (induces compressive stress)
• Avoid stress raisers
• Proper lubrication and alignment
• Select materials with high fatigue strength

11. Applications

• Rotating shafts
• Springs
• Connecting rods
• Gear teeth
• Turbine blades

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