Answer:
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