LIMITS OF REINFORCEMENT

Positive Moment Section

Check of Maximum Reinforcement [LRFD Art. 5.7.3.3.1]

[Provision deleted in 2005]

Check of Minimum Reinforcement [LRFD Art. 5.7.3.3.2]

At any section, the amount of prestressed and non-prestressed tensile reinforcement should be adequate to develop a factored flexural resistance, Mr, equal to the lesser of:

1.2 times the cracking strength determined on the basis of elastic stress distribution and the modulus of rupture, and
1.33 times the factored moment required by the applicable strength load combination.

Check at Midspan:

In 2005 Interims, Art. 5.7.3.3.2. was introduced the procedure to compute the cracking moment

M_{c r} = (f_{r} + f_{b p}) S_{b c} - M_{d / n c} (S_{b c} / S_{b} - 1) \geq S_{b c} f_{r}

LRFD, Eq.5.7.3.3.2-1

where:

f_r = modulus of rupture [LRFD Art. 5.4.2.6]

= 0.965 \sqrt{{f `}_{c}} = 0.965 \sqrt{45} = 6.473 M P a

f_pb = compressive strength in concrete due to effective prestress force only (after allowance for all prestress losses) at extreme fiber of section where tensile stress is caused by externally applied loads, MPa

= \frac{P_{p e}}{A} + \frac{P_{p e} e_{c}}{S_{b}} = \frac{4040443}{523107} + \frac{4040443 \times 647}{2.643 \times 10^{8}} = 7.723 + 9.890 = 17.613 M P a

M_d/nc = (moment due to non-composite dead loads)

(M_{g} + M_{s}) = (1619.6 + 2153.7 + 111.3) = 3884.6 k N - m

S_bc= composite section modulus for the extreme fiber of section where the tensile stress is caused by externally applied loads, mm³

S_b = non-composite section modulus for the extreme fiber of section where the tensile stress is caused by externally applied loads, mm₃.

M_{c r} = (6.473 + 17.613) (\frac{3.611 \times 10^{8}}{2.643 \times 10^{8}} - 1) = 7296.41 k N - m \geq - \frac{3.611 \times 10^{8}}{10^{6}} 6.473 = 2337.4 k N - m

1.2 M_{c r} = 7755.5 k N - m

At midspan, factored moment required by Strength I load combination is:

M_{u} = 9536.4 k N - m

Therefore, 1.33 M_u = 12692.46 kN-m.

Since 1.2 M_cr < 1.33 M_u, then 1.2 M_cr controls

M_{r} = 10899.1 k N - m > 1.2 M_{c r} = 8755.55 k N - m

Note: Contrary to the AASHTO Standard Specifications, LRFD requires that this criterion be met at all sections

Negative Moment Section

Check of Minimum Reinforcement [LRFD Art. 5.7.3.3.2]

For components containing no prestressing steel, the minimum reinforcement provision, herein, may be considered if:

ρ_{\min} \geq \frac{0.03 {f `}_{c}}{f_{y}}

where:

ρ_min = ratio of tension steel to gross area

f`_c = specified concrete strength, MPa

f_y = yield strength of tension steel, MPa

Therefore,

\frac{A_{s \min}}{b d} \geq \frac{0.03 {f `}_{c}}{f_{y}}

A_{s \min} \geq \frac{0.03 {f `}_{c}}{f_{y}} \times b \times d = \frac{0.03 \times 45}{413.7} \times 975 m m \times 1705 m m = 5424.7 {m m}^{2}