1. The flexural stress σ of a beam with rectangular...

1. The flexural stress $\sigma$ of a beam with rectangular cross-section under pure bending is given by $\sigma = \frac{My}{I}$, where $M$ is the bending moment, $y$ is the distance from the neutral axis (N.A.) to the point of interest, and $I = bh^3/12$ is the area momentum of the cross-section. If $b$ and $h$ have the relative errors of 2% and 3%, respectively, calculate the relative error induced in the maximum flexural stress $\sigma_{max}$ at $y = \pm h/2$ [5 Pts].

Transcript

00:01 Hello student, in the given question we have to calculate in first part we have to calculate the expected value expected value of s max.

00:13 So, we have formula as expectation of s max which is equals to 3 into expectation of w into expectation of l upon 2 into expectation of b into expectation of h bracket square.

00:36 So, now substituting all this value in this formula we get 3 multiply by 200 multiply by 48 divided by 2 into 2 into 4 square.

00:47 So, which is equals to 144 psi.

00:53 So, expectation of s of max is equals to 144 psi.

01:02 Then in second part we have to calculate the variance variance of s of max using tyler's series method tyler's series method method.

01:25 So, the formula for variance is variance of s of max is equals to del of s of max upon del of w means partial derivative with respect to w bracket square into variance of w plus del of s max given del of l bracket square into variance of l plus del of s max divided by del of s max divided by del of b bracket square into variance of b plus del of s max given del of h bracket square in multiply by 44 variance of h.

02:20 So which is equals to 3 l upon 2 b h bracket square multiply by 0 .15 of w bracket square plus 3 w upon 2 b h bracket square of square into 0 .05 of l of l bracket square plus 0 square plus 0 square plus 3 w l upon 2 b h bracket square into 0 .15 of l of l bracket square plus 0 .15 h of cube.

03:08 So which is equals to 162.

03:14 Therefore variance of s of max which is equals to 162 then the third part sigma of s of max to find the sigma of s of max we have formula as sigma of s of max is equals to square root of square root of variance of s of max which is equals to variance of 162 means under root of 162 which is equals to 12 .7 of psi...

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