Design for Variable loads and Fluctuating Loads free study notes

1. What is Variable loads and fluctuating loads

2. 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

$$\sigma_{min} = 0, \quad \sigma_{max} \neq 0$$

(c) Fluctuating Load

• Stress varies between two unequal values

$$\sigma_{min} \neq \sigma_{max}$$

3. Important Stress Parameters

For fluctuating loads, we define:

Mean Stress

$$\sigma_m = \frac{\sigma_{max} + \sigma_{min}}{2}$$

Alternating Stress

$$\sigma_a = \frac{\sigma_{max} – \sigma_{min}}{2}$$

Stress Ratio

$$R = \frac{\sigma_{min}}{\sigma_{max}}$$

These parameters are essential for fatigue analysis.

4. 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

5. 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)

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. Applications

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

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