6: Allowable Strength

Introduction to Allowable Strength

• Definition: Allowable strength is the stress that can be applied to a material without expecting failure during service.

• Critical Importance: Using the mean strength for design fails because it suggests a 50% chance of failure at service loads.

Calculation of Allowable Strength

• Allowable or design strength is calculated using Weibull statistics.

• Weibull modulus: A material characteristic influenced by the type of material and its processing.

Inputs for Allowable Strength Calculation

• Mean Strength: The average strength of the material when subjected to testing.

• Probability of Survival: The design requirement indicating how much failure is tolerable in service.

Factor of Safety

• Definition: Factor of Safety (FoS) is the ratio of mean strength to maximum service stress (allowable stress).

• Can be formulated using Weibull statistics and is key to determining allowable stress.

Equation for Factor of Safety

• Derived from:
  ext{FoS} = \frac{\text{Mean Strength}}{\text{Allowable Stress}}

• Involves the Weibull modulus and the probability of survival.

Weibull Statistics

Probability of Survival Equation

• The probability of survival can be determined from applied stress:
  Ps = 1 - Pf
  where P_f is the probability of failure.

• Exponents and logarithmic manipulations help to express the factor of safety accurately.

Relationship to Allowable Stress

• Rearranging yields:
  \text{Allowable Stress} = \frac{\text{Mean Strength}}{\text{FoS}}

• Allows the calculation of maximum service stress based on both material properties and design safety requirements.

Examples of Calculating Allowable Stress

• Example 1: Little consequence of failure

  • Probability of failure: 10^{-3} (1 in 1000)

  • Probability of survival: $99.9$%

  • For Weibull modulus of $10$:

  • FoS = $1.9$

  • Required allowable stress reduces the mean strength by nearly a factor of $2$.

• Example 2: Severe consequences of failure

  • Probability of failure: 10^{-7} (1 in 10,000,000)

  • Probability of survival: $99.9999$%

  • For Weibull modulus of $10$:

  • Required mean strength reduction = $4.8$.

  • For Weibull modulus of $15$: only $2.8$ reduction needed.

Importance of Weibull Modulus

• Materials with a higher Weibull modulus allow for smaller factors of safety, facilitating less material usage while maintaining safety under high-stress scenarios.

• Demonstrates the critical relationship between material properties and the design process.

Conclusion

• Understanding and applying Weibull statistics for calculating allowable strengths is crucial for safe and efficient design in engineering applications.

• This approach allows engineers to design components that can withstand service loads with a calculated probability of survival, ensuring reliability and safety in practical applications.