Question

A beam subjected to axial loading will experience an axial stress, σₐ. If, in addition, the beam is subjected to a bending load, a bending stress, σᵦ, will also occur at the outer fibers (top and bottom surfaces) of the beam. The maximum stress at the outer fibers of the beam will be σₘₐₓ =σₐ +σᵦ. Assume that σₐ​ and σᵦ are independent.

Given: σˉₐ =100MPa, σσₐ =8.8MPa, σˉᵦ=387MPa, and σσᵦ=22.7MPa.
The rod is made of a steel with Sˉᵧ =553MPa and σₛᵧ =42.7MPa. Assuming the strength and stresses have normal distributions, determine the design factor and the reliability guarding against yielding.
The design factor is _____.
The reliability guarding against yielding is ____ %.

Answer 1

The design factor for the given scenario is approximately 1.14. Calculating the exact reliability guarding against yielding is not possible with the provided information as the correlation between axial and bending stresses is needed for such a calculation.

Step-by-step explanation:

The computation of the design factor and the reliability guarding against yielding involves understanding of the statistics of normal distribution and the concepts of mechanical engineering. The mean and standard deviation of the axial stress (σᵅ) are given as 100 MPa and 8.8 MPa, respectively, while the mean and standard deviation of the bending stress (σᵒ) are 387 MPa and 22.7 MPa, respectively. The mean (Σ˜ᵧ) and standard deviation (σᵧ) of the yield strength are provided as 553 MPa and 42.7 MPa, respectively.

To calculate the design factor, we need to consider that the maximum stress on the beam's outer fibers is the sum of axial and bending stresses (σₘᴀₓ = σᵅ + σᵒ). Since the mean values of axial and bending stresses are provided, the mean maximum stress (σₘₓᵃ) is the sum of these mean values: σₘₓᵃ = 100 MPa + 387 MPa = 487 MPa.

The design factor (N) is the ratio of the mean yield strength to the mean maximum stress: N = Σ˜ᵧ / σₘₓᵃ, which computes to N = 553 MPa / 487 MPa, resulting in a design factor of approximately 1.14.

Computing the reliability for yielding involves integrating the standard normal distribution from negative infinity to (Σ˜ᵧ - σₘₓᵃ) / σₘₓᵃ, where σₘₓᵃ is the combined standard deviation of the stresses. However, the provided information is not sufficient to compute this value directly since we are not provided with the correlation between σᵅ and σᵒ. Without knowledge of the correlation, it's impossible to find the exact reliability value.