A 35 ft simply supported beam is loaded with concentrated loads 15 ft in from each support. On one end, the dead load is 8.0 kips and the live load is 18.0 kips. At the other end, the dead load is 4.0 kips and the live load is 9.0 kips. Include the self-weight of the beam in the design. Lateral supports are provided at the supports and the load points. Determine the least-weight W-shape to carry the load. Use A992 steel and Cb

From the load equation

F=stress*Area

Given stresses are 8 kips and 9 kips.

Hence the minimum weight supported=6.695 lbs.

ASD = 306 kips-feet

LRSD = 1387.5 kips-feet

Explanation:

a step by step process to solving this problem.

ΣM at A = 0

where;

RB * 35 - (8+18)15 - (4+9)20 = 0

RB = 18.57k

also E y = 0;

RA + RB = 18 + 8 + 9 +4 = 20.43 k

taking the maximum moment at mid point;

Mc = RA * 35/2 - (8 +18) (35/2 -15)

Mc = 292.525

therefore, MD = RA * 15 = 20.43 * 15 = 306.45 kips-feet

MD = 306.45 kip-feet

ME = 279 kip-feet .IE 18.57 * 15

considering the unsupported  length; 35 - (15*2 = 5ft )

now we have that;

L b = L p = 5ft

where L p = 1.76 r y(√e/f y)

L p = 1.76 r y √29000/50

r y = 1.4 inch

so we have that M r = M p  for L b = L p where

M p = 2 F y ≤ 1.5 s x   F y

Recall from the expression,

RA + RB = (8+4) * 1.2 + (18+9) * 1.6 = 57.6

RA * 35 = 4 * 1.2 * 15 + 9 *1.6 * 15 + 8 * 1.2 * 20 + 18 * 1.6 * 20

RA = 30.17 k

the maximum moment at D = 30.17 * 15 = 452.55 kips-feet

Z required = MD / F y = 452.55 * 12 / 50 = 108.61 inch³

so we have S x = 452.55 * 12 / 1.5 * 50 = 72.4 inch³

also r = 1.41 in

Taking LRFD solution:

where the design strength ∅ M n = 0.9 * Z x * F y

given r = 2.97

Z x = 370 and S x = 81.5, we have

∅ M n = 0.9 * 370 * 50 = 16650 k-inch = 1387.5 kips-feet

this tells us it is safe.

ASD solution:

for L b = L p, and where M n = M p = F c r    S x

we already have value for S x as 81.5 so

F c r = Z x times F y divide by  S x

F c r = 370 * 50 / 81.5 = 227 kips per sq.in

considering the strength;

Strength = M n / Ωb = (0.6 * 81.5 * 50) * (1.5) / 12 = 306 kips-feet  

This justifies that it is safe because is less than 306