PRESTRESS LOSSES [LRFD INTERIMS 2005 - APPROXIMATE METHOD, ART.5.9.5]

A. Elastic shortening losses (LRFD Interims 2005, Art. 5.9.5.2.3a)

Due to Initial Prestress and due to Beam self weight

Δ f_{p E S} = \frac{E_{p}}{E_{c i}} f_{c g p}

Where:

E_p - modulus of elasticity of prestressing steel;

E_ci - modulus of elasticity of concrete at transfer or time of load application

Δ f_{p E S} = \frac{E_{p}}{E_{c i}} f_{c g p}

P_{i} = N o s t r a n d s \times A r e a s t r a n d s \times 0.75 \times f_{p u}

P_{i} = 27 \times 0.153 \times 0.75 \times 270 = 836.5

e_c = 18.48 in

M_g= 1097.7k.ft

A_beam = 843 in2

I_beam= 203086 in4

f_{c g p} = \frac{P_{i}}{A} + \frac{P_{i} e_{c}^{2}}{I} - \frac{M_{g} e_{c}}{I} = \frac{0.961 \times 836.5275}{843} + \frac{0.961 \times 836.5275 \times {18.48}^{2}}{203086} - \frac{1097.7 \times 12 \times 18.48}{203086} = 1.107 k s i

Therefore,

f_{p E S} = \frac{E_{p}}{E_{c i}} f_{c g p} = \frac{28500}{4066.84} 1.107 = 7.76 k s i

The Elastic Gains

The elastic gains are calculated as $\frac{E_{p}}{E_{c i}} (\frac{M \times e_{c c}}{I})$ for all loads other than prestress and self weight.

Due Precast Loads:

M_{p r e c a s t} = 40.5 k f t

{E G}_{p r e c a s t} = \frac{E_{p}}{E_{c t}} (\frac{M \times e_{c c}}{I}) = \frac{28500}{4496.06} (\frac{40.5 \times 12 \times 18.48}{203086}) = - 0.28 k s i

Due Composite Loads:

M_{c o m p o s i t e} = 93.8 + 158.6 = 252.4 k f t

{E G}_{c o m p o s i t e} = \frac{28500}{4496.06} (\frac{252.4 \times 12 \times 18.48}{203086}) = - 1.75 k s i

Due Live Loads:

M_{l i v e} = 792.6 k f t

{E G}_{l i v e} = \frac{28500 \times 0.8}{4496.06} (\frac{792.6 \times 12 \times 18.48}{203086}) = - 4.39 k s i

Adjustment

K_{r}^{r e l e a s e} = \frac{\frac{\frac{E_{p}}{E_{c i}} \times (1 + \frac{A_{g r o s s} \times e_{c c}^{2}}{I_{g r o s s}}) \times A_{p s}}{A_{g r o s s}}}{(1 (\frac{\frac{E_{p}}{E_{c i}} \times (1 + \frac{A_{g r o s s} \times e_{c c}^{2}}{I_{g r o s s}}) \times A_{p s}}{A_{g r o s s}}))} = \frac{\frac{\frac{28500}{4496.06} \times (1 + \frac{843 \times {(18.48)}^{2}}{203086}) \times 4.131}{843}}{(1 + (\frac{\frac{28500}{4496.06} \times (1 + \frac{843 \times {18.48}^{2}}{203086}) \times 4.131}{843}))} = 0.0699

Due Precast Loads:

Δ_{{E G}_{p r e c a s t}}^{a d j u s t} = K_{r}^{r e l e a s e} \times {E G}_{p r e c a s t} = 0.0699 \times 0.28 = 0.02 k s i

Due Composite Loads:

Δ_{{E G}_{c o m p o s i t e}}^{a d j u s t} = K_{r}^{r e l e a s e} \times {E G}_{c o m p o s i t e} = 0.0699 \times 1.75 = 0.12 k s i

Due Live Loads:

Δ_{{E G}_{l i v e}}^{a d j u s t} = K_{r}^{r e l e a s e} \times {E G}_{l i v e} = 0.0699 \times \frac{4.39}{0.8} = 0.38 k s i

Art. 5.9.5.3 Approximate Estimate of Time Dependent Losses

The time dependent losses are given on Equation 5.9.5.3-1. The first term corresponds to creep losses, the second term to shrinkage losses, and the third to relaxation losses.

B. Creep of Girder Concrete (Art 5.9.5.4.2a)

Δ f_{s r} = 10.0 \frac{f_{p i} A_{p s}}{A_{g}} y_{h} y_{s t}

In which:

y_h (correction factor for relative humidity of the ambient air)

y_h = 1.7 - 0.01H = 1.7 - 0.01 x 70 = 1.00 [Eq. 5.9.5.3-2]

y_st (correction factor for specified concrete strength at time of prestress transfer to the concrete member)

y_{s t} = \frac{5}{(1 + f_{c i})} = \frac{5}{(1 + 4.5)} = 0.9091

f_pi - prestressing steel stress immediately prior to transfer

- jacking x f_pu = 0.75 x 270 = 202.5 ksi

Therefore:

{Δ f}_{C R} = 10.0 \frac{f_{p s} A_{p s}}{A_{g}} γ_{h} γ_{s t} = 10.0 \times \frac{202.5 \times 4.131}{843} \times 1.00 \times 0.9091 = 9.02 k s i

C. Shrinkage of Girder Concrete (Art 5.9.5.4.2b)

{Δ f}_{S H} = 12.0 γ_{h} γ_{s t} = 12 \times 1.00 \times 0.9091 = 10.9092 k s i

D. Relaxation of Prestressing Strands (Art. 5.9.5.4.2c)

An estimate of relaxation loss taken as 2.5ksi for low relaxation strand, 10.0 ksi for stress relieved strand, and in accordance with manufacturers recommendation for other type of strand.

Δ f = 2.5 k s i

Total losses at transfer:

Δ_{f p i} = Δ_{E S} = 7.76 k s i

Stress in tendon after transfer:

f_{p t} = (f_{p i} - Δ_{f p i}) = 202.5 - 7.76 = 194.74

Force per strand

= (Stress in tendons after transfer) (area of strand)

= 194.74 \times 0.153 = 29.768 k i p s

Therefore total prestressing force after transfer is:

P_{i} = 29.795 \times 27 = 804.465 k i p s

\frac{Δ_{f p i}}{f_{p i}} = \frac{7.76}{202.5} = 3.83 %