Answer the question in paper Suppose we know little about the strength of materi

Question

Answer the question in paper

Suppose we know little about the strength of materials but are told that the bending stress sigma in a beam is proportional to the beam half-thickness y and also depends on the bending moment M and the beam area moment of inertia I. We also learn that, for the particular case M = 2900 in · lbf, y= 1.5 in, and I = 0.4 in4, the predicted stress is 75 MPa. Using this information and dimensional reasoning only, find, to three significant figures, the only possible dimensionally homogeneous formula sigma = y f(M, I).

Explanation / Answer

Writing dimensions,

Stress sigma = MPa = Pa = N/m^2 = (kg-m/s^2)/m^2 = kg/(m-s^2) = [M L^-1 T^-2]

Bending moment M = in.lbf = in.(lb-in/s^2) = lb-in^2/s^2 = [M L^2 T^-2]

Half thickness y = in = [L]

Area MOI I = in^4 = [L^4]

Thus there are three fundamental dimensions, M,L and T and four independent variables (sigma, M, y and I). So only 4 - 3 = 1 dimensionless parameter can be formed.

Dimensionless parameter P = [M L^-1 T^-2]^a * [M L^2 T^-2]^b * [L]^c * [L^4]^d

Thus, P = [M]^(a+b) * [L]^(-a + 2b + c + 4d) * [T]^(-2a - 2b)

Thus, we get,

a+b = 0...........(1)

-a + 2b + c + 4d = 0............(2)

-2a - 2b = 0............(3)

Eqns 1 and 3 are identical. Both yield a = -b

Putting it in eqn 2, b + 2b + c + 4d = 0

3b + c + 4d = 0

Taking a = 1 and d = 1, we get, b = -1 and c = -1.

Thus, P = Sigma*(M^-1)*(y^-1)*I

or, P = Sigma*I/(My)

Sigma = P*(My/I)

Taking P = 1

Sigma = My/I

Sigma = (2900*1.5)/0.4 psi

Sigma = 10875 psi

Sigma = 74.980 MPa

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