FINAL STRESSES - POSITIVE MOMENT ENVELOPE AT MIDSPAN

Prest (Prestress)

Precast-top = $P (\frac{1}{A} - \frac{e}{S_{t}})$

P = Net prestress force at final

= (1178.1 k)(1-0.2151) = 924.65 k

Precast-top = $924.65 k (\frac{1}{767.0 {i n}^{2}} - \frac{31.57 i n}{15420.7 {i n}^{3}}) = - 0.6877 k s i = - 687.7 p s i$

Bottom = $924.65 k (\frac{1}{767.0 {i n}^{2}} - \frac{31.57 i n}{14915.1 {i n}^{3}}) = 3.1629 k s i = 3162.9 p s i$

Self-Wt

Precast-top = $\frac{(1320.8 k - f t) (12000 \frac{l b - i n}{k - f t})}{15420.7 {i n}^{3}} = 1027.8 p s i$

Bottom= $\frac{(1320.8 k - f t) (12000 \frac{l b - i n}{k - f t})}{14915.1 {i n}^{3}} = 1062.6 p s i$

DL-Prec.

Precast-top = $\frac{(297.6 k - f t) (12000 \frac{l b - i n}{k - f t})}{15420.7 {i n}^{3}} = 231.6 p s i$

Bottom= $\frac{(297.6 k - f t) (12000 \frac{l b - i n}{k - f t})}{14915.1 {i n}^{3}} = 239.4 p s i$

Topping

Precast-top = $\frac{(1441.2 k - f t) (12000 \frac{l b - i n}{k - f t})}{15420.7 {i n}^{3}} = 1121.6 p s i$

Bottom= $\frac{(297.6 k - f t) (12000 \frac{l b - i n}{k - f t})}{14915.1 {i n}^{3}} = 239.4 p s i$

DL-Comp(+)

S = Section modulus at top of topping

S_tt= $\frac{I_{c o m p}}{n y_{t t}}$

Note: The topping width was reduced earlier in order to compute the effective composite moment of inertia. However, in reality, the modulus of elasticity of the deck is less than that of the precast. Consequently, for a given strain, the stress in the concrete of the deck will be n times less than that of the beam.

S_tt= $\frac{1101617 {i n}^{4}}{(80.50 i n - 54.69 i n) (0.7638)} = 55882.7 {i n}^{3}$

S_t= $\frac{1101617 {i n}^{4}}{(72.00 i n - 54.69 i n)} = 63638.3 {i n}^{3}$

S_b= $\frac{1101617 {i n}^{4}}{(54.69 i n)} 20143.2 {i n}^{3}$

Topping-top = $\frac{(266.7 k - f t) (12000 \frac{l b - i n}{k - f t})}{55882.7 {i n}^{3}} = 57.3 p s i$

Precast-top = $\frac{(266.7 k - f t) (12000 \frac{l b - i n}{k - f t})}{63638.3 {i n}^{3}} = 50.3 p s i$

Bottom= $\frac{- (266.7 k - f t) (12000 \frac{l b - i n}{k - f t})}{20143.2 {i n}^{3}} = - 158.9 p s i$

LL+I(+)

Topping-top = $\frac{(1667.6 k - f t) (12000 \frac{l b - i n}{k - f t})}{55882.7 {i n}^{3}} = 358.1 p s i$

Precast-top = $\frac{(1667.6 k - f t) (12000 \frac{l b - i n}{k - f t})}{63638.3 {i n}^{3}} = 314.4 p s i$

Bottom= $\frac{- (1667.6 k - f t) (12000 \frac{l b - i n}{k - f t})}{20143.2 {i n}^{3}} = - 993.4 p s i$

Total, Final 1 (P/S + DL + LL)

Topping-top = 57.3 psi + 358.1 psi = 415.4 psi

Precast-top = -687.7 psi + 1027.8 psi + 231.6 psi + 1121.6 psi + 50.3 psi + 314.4 psi = 2058.0 psi

Bottom = 3162.9 psi -1062.6 psi -239.4 psi- 159.6 psi -158.9 psi -993.4 psi = -451.0 psi

Total, Final 2 (P/S + DL)

Topping-top = 57.3 psi

Precast-top = -687.7 psi + 1027.8 psi + 231.6 psi + 1121.6 psi + 50.3 psi = 1743.6 psi

Bottom = 3162.9 psi - 1062.6 psi-239.4 psi - 1159.6 psi- 158.9 psi = 542.4 psi

Total, Final 1 (50%P/S + 50%DL + LL)

Topping-top= $\frac{57.3 p s i}{2} + 358.1 p s i = 386.7 p s i$

Precast-top= $\frac{- 687.7 p s i + 1027.8 p s i + 231.6 p s i + 1121.6 p s i + 50.3 p s i}{2} + 314.4 p s i = 1186.2 p s i$

Bottom= $\frac{3162.9 p s i - 1062.6 p s i - 239.4 p s i - 1159.6 p s i - 158.9 p s i}{2} - 993.4 p s i = - 722.9 p s i$