DEFLECTION AND CAMBER

Deflections due to self-weight and camber due to prestress are calculated using the initial modulus of elasticity of concrete. However, the deflections for remaining loads are computed using the final modulus of elasticity of concrete. Computation of deflection is based on the gross moment of inertia. Deflections due prestress and camber are computed for release stage and erection and final stage deflections are obtained using factors as suggested by the PCI Design Handbook.

Deflection Due to Prestressing Force at Transfer

Moment-area method is used to compute the uplift at the release stage.

Total prestress force before transfer is

P_{i} = 0.75 \times 1861.58 \times 98.7 \times 36 = 4960924.5 N

Total prestress force after transfer is

P = 4960924.5 \times (1 - 6.3 %) = 4648386.3 N

where:

P = total prestressing force at transfer, after losses of 6.3% at release, N

e_c = eccentricity of prestressing force at midspan = 722 - 75 = 647 mm

e_e = eccentricity of prestressing force at end = 722 - 236.11= 485.89 mm

e´ = difference between eccentricity of prestressing force at midspan and at end

e_c - e_e= 647 - 485.89=161.11mm

a = draped length= 9870 mm

L = beam length= 32700 mm

t = transfer length= 762 mm

Y_cg transfer = 236.11 - 762 = 223.7 mm

e_t = eccentricity of prestressing force at transfer length.= 722 - 223.6= 498.4 mm

M_a = P × e_t= (4648386.3)(498.4)(10^-6)= 2316.75 kN-m

M_b = P × e_c= (4648386.3)(647)(10^-6)= 3007.51 kN-m

where:

w = deck slab weight = 16.41 kN/m

L = span length from CL bearing to CL bearing = 32400 mm

E_c = modulus of elasticity of precast beam at 28 days = 33978.6 MPa

Table 1. TABLE A3-2.
1: (1/2)(762)(2316.755×106)	=8.8268 ×10¹¹ × (-15842)	=1.398 × 10¹⁷
2: (9048)(2300.54×1061)	=2.09583 ×10¹³ × (-11064)	=-2.31883 ×10¹⁷
3: (1/2)(9048) (3007.51×2316.35)(1061)	=3.12495 ×10¹²× (-9556)	= -2.9862 ×10¹⁶
4: (6540)(3007.51×1061)	= 1.96691 ×10¹³ × (-3270)	= -6.43179×10¹⁶
	=4.46309 ×10¹³ × (16350)	= 7.29716×10¹⁷
		= 3.8561 × 10¹⁷

Deflection due to prestress at transfer:

Δ_{p} = \frac{P_{i} e L^{2}}{8 E_{ci} I} = \frac{3.8651 \times 10^{7}}{E I} = \frac{3.8561 \times 10^{17}}{29966.30 \times 1.908 \times 10^{11}} = 67.443 m m ↑

Deflection due to prestress at erection:

Δ_{ρ} = 67.443 m m \times 1.8 = 121.398 m m ↑

Deflection due to prestress at final:

Δ_{ρ} = 67.443 m m \times 2.2 = 148.376 m m ↑

Deflection Due to Beam Self-Weight

Δ_{g} = \frac{5 w l^{4}}{384 E_{c i} I}

where:

w = beam self-weight = 12.325 kN/m

Deflection due to beam self-weight at transfer:

L = overall beam length (precast length)= 32700 mm

Δ_{g} = \frac{5 (12.325) {(32700)}^{4}}{(384) (29966.3) (1.908 \times 10^{11})} = 32.11 m m ↓

Deflection due to beam self-weight at erection:

Δ_{g} = 32.11 m m \times 1.85 = 59.40 m m ↓

Deflection due to prestress at final:

Δ_{g} = 32.11 m m \times 2.4 = 77.06 m m ↓

Deflection Due to Slab and Haunch Self-Weight

Δ_{S} = \frac{5 w L^{4}}{384 E_{ci} I}

where:

w = deck slab weight = 16.41 kN/m

L = span length from CL bearing to CL bearing = 32400 mm

E_c = modulus of elasticity of precast beam at 28 days = 33978.6 MPa

Therefore,

Δ_{S} = \frac{5 (16.41) {(32400)}^{4}}{(384) (33978.6) (1.908 \times 10^{11})} = 36.326 m m ↓

Deflection due to deck slab at final:

Δ_{s} = 36.326 m m \times 2.3 = 83.551 m m ↓

Deflection Due to Precast DC

For uniform load along the entire span

Δ_{p} = \frac{5 w L^{4}}{384 E_{ci} I}

where:

w = 0.848 kN/m

L = span length from CL bearing to CL bearing = 32400 mm

E_c = modulus of elasticity of precast beam at 28 days = 33978.6 MPa

Therefore,

Δ_{P} = \frac{5 (0.848) {(32400)}^{4}}{(384) (33978.6) (1.908 \times 10^{11})} = 1.877 m m ↓

Deflection due to Precast - DC at final:

Δ_{P} = 1.877 m m \times 3 = 5.631 m m ↓

Deflection Due to Barrier (Composite DC)

w = barrier weight = 8.756 kN/m

L = 33000 mm

E_c = modulus of elasticity of precast beam at 28 days = 33978.6 MPa

I = composite moment of inertia of bridge = 1.7595 × 10¹² mm⁴

Δ_{b} = - 0.288 ↓

Note: This value was calculated using a continuous model.

Deflection due to Barrier (Composite DC) at final:

Δ_{b} = 3 (0.288) = 0.864 m m ↓

Deflection Due to Future Wearing Surface (Composite DW)

w = wearing surface weight = 14.8428 kN/m

L = 33000 mm

E_c = modulus of elasticity of precast beam at 28 days = 33978.4 MPa

I = composite moment of inertia of bridge = 1.7595 × 10¹² mm⁴

Δ_{w s} = 0.488 ↓

Note: This value was calculated using a continuous model.

Deflection due to Barrier (Composite DC) at final:

Δ_{w s} = 3 (0.488) = 1.464 ↓

Deflection and Camber Summary for Center-Span

$(Δ_{p} + Δ_{g}) = 68.156 - 32.11 = 36.1 ↑$

Total deflection at erection, using PCI multipliers (see PCI Design Handbook):

121.398 - 59.40 - 36.326 - 1.877 - 0.288 - 0.488= 23. 019 mm↑

Total deflection at final:

= 148.375 - 77.062 - 83.551 - 5.631 - 0.864 - 1.464 = - 20.197 mm↓

Deflection Due to Live Load and Impact

Live load deflection is not a required check per the provisions of LRFD and is usually not a problem for prestressed concrete I shapes, especially when they are constructed to act as a continuous structure under superimposed loads. If the designer chooses to check deflection, the following is recommended, per LRFD:

$\frac{S p a n}{800} = \frac{33000}{800} = 41.3 m m$ [LRFD Art. 2.5.2.6.2]

If the owner invokes the optional live load deflection criteria specified in Art. 2.5.2.6.2, the deflection is the greater of [LRFD Art. 3.6.1.3.2]:

that resulting from the design truck alone, or
that resulting from 25% of the design truck taken together with the design lane load.

LRFD states that all the beams may be assumed to be deflecting equally under the applied live load and impact [LRFD Art. 2.5.2.6.2].

Therefore, the distribution factor for deflection is calculated as follows:

= \frac{n u m b e r o f l a n e s}{n u m b e r o f b e a m s} = \frac{3}{4} = 0.75

LRFD Art. C2.5.2.6.2

However, it is more conservative to use the distribution factor for moment and we will use that in our calculations.

The live load deflections from continuous model are:

Δ_lane = -1.206 mm

Δ_truck = -2.056 mm

Δ_truck+IM = -2.7345 mm

Δ_lane +0.25 Δ_truck+IM = -1.206 + 0.25(-2.7345) = -1.8896 mm

Δ_{LL bridge} = -2.7345 mm

D.FM+ = 0.9125

Δ_{L L b e a m} = - 2.7345 (\frac{1.7595 \times 10^{12}}{4.23 \times 10^{11}}) 0.9125 = 13.379 m m ↓ < 41.3 m m