Service Limit States (Inventory and Operating Level)

These calculations are for Midspan location. Because our total moment is greater than 0, we have compression at top and tension at bottom. For both cases we will compute the Rating Factors.

Compression at top (Compression Stress RFs are for Service 1 Limit State)

{R F}_{I N V} = \frac{f_{R}^{t o p} - γ_{D} \times f_{D}^{t o p}}{γ_{L} (f_{L L + I}^{t o p})}

We obtain thee values from File > Print / Positive Envelope Stresses option. In the following table we list the stresses due to dead load at midspan location at the top of the precast beam.

Stresses due to dead loads (ksi) at midspan top
DC	Self-Weight	0.886
	Deck Haunch	0.996
	Diaphragm	0.162
	Comp DC	0.039
DW	Comp DW	0.032
	Total (f_D)	2.115

And for Live Load:

Stresses due to live loads (ksi) at midspan top
LL+IM	0.292

f_R - flexural resistance at top

f_R = f_pb + allowable compression stress at top

f_pb - stress due to effective prestress: -0.856 ksi

Allowable compression stress - 0.6 x 5 = 3.00 ksi

Therefore, f_r = 3.00 ksi - (-0.856 ksi) = 3.856 ksi

γ_L=1.00; γ_D=1.00;

{R F}_{I N V} = \frac{f_{R}^{t o p} - γ_{D} \times f_{D}^{t o p}}{γ_{L} (f_{L L + I}^{t o p})} = \frac{3.856 - 1.00 \times 2.115}{1.00 \times 0.292} = 5.96

Tension at bottom (Tension Stress RFs are for Service III Limit State)

{R F}_{I N V} = \frac{f_{R}^{b o t} - γ_{D} \times f_{D}^{b o t}}{γ_{L} (f_{L L + I}^{b o t})}

We obtain these values from File > Print / Positive Envelope Stresses option. In the following table we list the stresses due to dead load at midspan location at the bottom of the precast beam.

Stresses due to dead loads (ksi) at midspan bottom
DC	Self-Weight	-0.748
	Deck Haunch	-0.841
	Diaphragm	-0.137
	Comp DC	-0.137
DW	Comp DW	-0.111
	Total (f_D)	-1.974

And for Live Load:

Stresses due to live loads (ksi) at midspan top
LL+IM	f _LL+I	1.020

f_R - flexural resistance at top

f_R = f_pb + allowable compression stress at top

f_pb - stress due to effective prestress: 2.562 ksi

- 0.19 \sqrt{f_{c}} = - 0.19 \sqrt{5} = - 0.425 k s i

Therefore, f_R = -0.425 ksi - 2.562 ksi = -2.987 ksi

γ_L= 0.80; γ_D=1.00;

{R F}_{I N V} = \frac{f_{R} γ_{D} f_{D}}{γ_{L} (f_{L L + I}^{t o p})} = \frac{- 2.987 - 1.00 \times (- 1.974)}{0.800 \times (- 1.020)} = 1.24

Prestressing Steel Tension Capacity

The rating equation for prestressing steel tension is:
${R F}_{I N V}^{P S S I T} = \frac{0.9 f_{y} - {(F}_{d} + F_{y})}{F_{I}}$

The stress in strands shall be checked for tension. In this case, the limiting steel stress is 0.9F_y.

f_py in turn is estimated to be as in the Table 6-10 from the LRFR Manual:

Type of tendon fpy

Low Relaxation Strand 0.9f_PU

Stress Relieved Strand 0.85f_PU

The prestressing steel type used in this example is 1/2 (low relaxation steel) (fpu = 270 ksi).

Therefore
$f_{p y} = 0.9 f_{p u} = 0..9 \times 270 = 243 k s i$

and the limiting stresses for prestressing steel as per Art. 6.5.4.2.2.2 is:
$0.9 \times f_{p y} = 0.9 \times 243 = 218.7 k s i$

The dead load stress in strand is the final prestress stress after all losses. This can be calculated as $f_{p j} - Δ f_{s}$ . Note that these values are reported in Precast/Prestressed Girder, but are based on the location of centroid of all prestressing steel. The stress in the bottom row is usually slightly higher, however, in lieu of better estimates, this value is used.

Based on initial stress of 202.5 ksi (0.75*fps), and total loss of 41.84 psi, the final strand stress is 160.66 ksi, which used for the dead load term D (F_d + F_p) from the formula of the Rating Factor.

The live load increment of stress is calculated by interpolating the girder top and bottom live load stresses, and finding the concrete stress at location of bottom strand. Then using modular ratio to find the steel stress,

$f_{p e r m i t}^{b o t t o m s t r a n d} = \frac{E_{s}}{E_{c}} \times {f_{P E R M I T}^{B O T} + [\frac{f_{P E R M I T}^{T O P} + f_{P E R M I T}^{B O T}}{H}] \times e_{b o t}}$

where:
f_bot - bottom fiber concrete stresses due to total live load;

f_top - top fiber concrete stresses due to total live load;

h - total section height (54 in);

e_bot - the eccentricity of the bottom row of strands which is assumed in Precast/Prestressed Girder to be located at 2 inches from the bottom flange of the beam;

E_s - modulus of elasticity of the prestressing tendon: 28500 ksi;

E_c - modulus of elasticity of 28-day strength concrete: 4030 ksi;

h - total section height (54 in);

We should compute the live load stresses at top and at bottom.

We have the Live Load Moment for Design Truck Load = 1486.0 k.ft, previously computer, and the section modulus at top (S_tc = 61012.22 in³) and at bottom (S_tc = 17471 in³).

Therefore, the formula to compute the stresses from these live loads at the bottom of the section is:
$f_{D E S I G N}^{T O P} = \frac{M_{L L + I M}^{D E S I G N}}{S_{t c}} = \frac{1486.0 \times 12}{61012.22} = 0.292 k s i$

$f_{P E R M I T}^{B O T} = - \frac{M_{L L + I M}^{P E R M I T}}{S_{b c}} = - 1.020 k s i$

$f_{P E R M I T}^{b o t t o m s t r a n d} = \frac{28500}{4030} \times {- 1.020 + (\frac{0.292 + 1.020}{54}) \times 2} = - 6.87 k s i$

Therefore, the Rating Factor for Design Load for prestressing steel tension:
${R F}_{D E S I G N}^{P S S i T} = \frac{0.9 f_{p y} - (F_{d} + F_{p})}{{- f}_{L L + I}^{b o t t o m s t r a n d}} = \frac{218.7 - 160.66}{6.87} = 8.45$

Type of tendon	fpy
Low Relaxation Strand	0.9f_PU
Stress Relieved Strand	0.85f_PU