CHECK OF CONCRETE STRESSES AT SERVICE LOADS

The total prestressing force after all losses, Ppe = 4040443 N

Stress Limits for Concrete [LRFD Art. 5.9.4.2]

Compression

Due to permanent load, (i.e., beam self-weight, weight of slab and haunch, weight of future wearing

surface, and weight of barriers), for load combination Service I

For precast beam = 0.45f`_c = 0.45 (45) = +20.250 MPa

For the slab = 0.45 (28) = +12.600 MPa

Due to permanent and transient loads, i.e., all dead loads and live loads, for load combination Service I

For precast beam = 0.60 f`_c = 0.60 (45) = +27.000 MPa

For slab = 0.60 f`_c = 0.60 (28) = +16.800 MPa

Due to live loads plus one-half of dead loads, for load combination Service I

For precast beam = 0.40 f`_c= 0.40 (45) = +18.000 MPa

For slab = 0.40 f`_c = 0.40 (28) = +11.200 MPa
Tension

For components with bonded prestressing tendons:

For load combination Service III= $- 0.50 \sqrt{{f `}_{c}}$

For precast beam= $- 0.50 \sqrt{45} = - 3.354 M P a$

For topping= $- 0.50 \sqrt{28} = - 2.65 M P a$

Check of Stresses at Midspan

Concrete Stresses at the Top Fiber of the Beam

Under permanent and transient loads, Service I (Final 1):
$f_{t g} = 13.813 + \frac{M_{L L + I}}{S_{t g}} = 13.813 + \frac{2485.2 \times 10^{6}}{9.872 \times 10^{8}} = 13.813 + 2.517 = 16.330 M P a$

Comp
$f_{t g} = \frac{P_{p e}}{A} - \frac{P_{p e} e_{c}}{S_{t}} + \frac{(M_{g} + M_{s})}{S_{t}} + \frac{(M_{b} + M_{w s})}{S_{t g}} = \frac{4040443}{523107} - \frac{(4040443) (647)}{2.173 \times 10^{8}} + \frac{(1619.6 + 2153.7 + 111.3) \times 10^{6}}{2.173 \times 10^{8}} + \frac{(87.4 + 148.2) \times 10^{6}}{9.872 \times 10^{8}} = 7.723 - 12.03 + 17.877 + 0.2386 = 13.81 M P a$
ressive stress limit for concrete = +27.000 MPa OK
Under permanent load, Service I (Final 2):

Using bending moment values given in Service Limit State I: Moments and Shears Printout, concrete stress at top fiber of the beam is:

Compressive stress limit for concrete = +20.250 MPa OK
Under live loads plus one-half of dead loads (Final 3):
$f_{t g} = 2.517 + 0.5 \times 13.81 = 9.421 M P a$

Compressive stress limit for concrete = +18.000 MPa OK

Concrete Stress at the Top Fiber of the Slab, Service I

Note: Compression stress in the slab at service loads never controls the design for typical applications. Thecalculations shown below are for illustration purposes only and may not be necessary in most practical applications.

Under permanent and transient loads (Final 1):
$f_{t c} = \frac{(M_{w s} + M_{b} + M_{L L + I})}{S_{t c}} = \frac{(148.2 + 87.4 + 2485.2) (10^{6})}{8.532 \times 10^{8}} = 3.19 M P a$

Compressive stress limit for concrete = +16.800 MPa OK
Under permanent loads (Final 2):
$f_{t c} = \frac{(M_{w s} + M_{b})}{S_{t c}} = \frac{(148.2 + 87.4) (10^{6})}{8.532 \times 10^{8}} = 0.276 M P a$

Compressive stress limit for concrete = +12.600 MPa OK
Under live loads plus one-half of dead loads (Final 3):
$f_{t c} = \frac{0.5 M_{w s} + 0.5 M_{b} + M_{L L} + I}{S_{t c}} = \frac{(0.5 \times 148.2 + 0.5 \times 87.4 + 2485.2) \times 10^{6}}{8.532 \times 10^{8}} = 3.05 M P a$

Compressive stress limit for concrete = +11.200 MPa OK

Check of Tension Stress at the Bottom Fiber of the Beam, Service III:

f_{b} = \frac{P_{p e}}{A} + \frac{P_{p e} e_{c}}{S_{b}} - \frac{(M_{g} + M_{s})}{S_{b}} - \frac{(M_{b} + M_{w s}) + 0.8 M_{L L + I}}{S_{b c}} = \frac{4040443}{523107} + \frac{(4040443) (647)}{2.643 \times 10^{8}} + \frac{(1619.6 + 2153.7 + 111.3) \times 10^{6}}{2.643 \times 10^{8}} + \frac{(87.4 + 148.2 + 0.8 \times 2484.2) \times 10^{6}}{3.611 \times 10^{8}} = 7.723 + 9.890 - 14.703 - 6.158 = - 3.248 M P a

f_ps = average stress in prestressing steel, MPa

Tensile stress limit for concrete = -3.354 MPa OK