Question

If two loads are applied to a beam as shown in the a The weight If the weight of the beam is random, the of the beam itself contributes to the bending moment. Assume that the beam is of unitorm thickness and density so that the resulting load is uniformly distributed on the beam. load from the weight is aso random; denote this load by W (kip-t). the fixed loads X, and X2 are independent rv's with means 4 and s kips, what are the expected value and variance of the bending moment? (Hint: If the lood due to the beam were w kip-ft, tf the beam is 14ftlong with ai-Sfeet and a2-i0 feet, w has mean isand standard deviaten Ols, and respectively, and stenderd deviations 1 and o.5 kip, respectively, the contribution to the bending moment wouild be wxdx.) expected value ) kip-ft ee variables (x, x, and w) are normally distributed, what is the probabity that the bending moment wilt be at most 300 uip-tr (Round your answer to four decimal places.)

Answer 1

a) The random variable representing the bending moment is

r14 0

The expected value is

E (M) 98E(W) + 5E (X1) + 10E (X2) E (M) 98 × 1.5 + 5 × 4 + 10 × 8 E (M) = 247

The variance of the moment is

Var (M) = 982Var (W) + 52Var (X) + 10-Var (X2) Var (M) = 982 × 0.254 52 × 12 + 102 × 0.52 Var(M) 56.125

b) We have . The probability,

MN(247,56.125)

M47,5 M-247 300-247 MN(247,56.125) V56.125 V56.125 P (M < 300)- P (Z < 7.0745316) P (M , 300) Φ (7.0745316)