Tutorial 1A: Load Rating of Three Span Bridge (U.S. Units)(LFD) Hand Calculation for Selected Items

Theoretical Background

This bridge will be load rated at two levels, Inventory and Operating Levels. Load ratings based on the Inventory Level results in a live load which can safely utilize an existing structure for an indefinite period of time. Load ratings based on the Operating Level describes the maximum permissible live load which the structure may be subjected.

The following equation will be used in determining load rating in structure:

R F = \frac{C - A_{1} D}{A_{2} L (1 + I)}

Art. 6-1a

where

RF	=	rating factor for the live load carrying capacity
C	=	the capacity of the member (allowable stress, Mn or Vn)
A₁	=	factor for dead loads
D	=	unfactored dead loads
A₂	=	factor for live loads
L	=	the live load effect on the member
I	=	the impact factor to be used with the live load effect

The above formula is applied to concrete tension and compression (Top and bottom fiber of concrete section), strand tension (in the most stressed strand), and moment and shear capacities. The factors A₁ and A₂ are specific to the quantity being checked and are different for Inventory and Operating ratings. The following table shows the factors used for various checks.

Rating Case	Item	Capacity/ Allowable	A1(DL.Fac)	A2(LL.Fac)
Inventory	Concrete Tension	6*sqrt(f'c)	1.0	1.0
	Conc. Comp.1	0.6*f'c	1.0	1.0
	Conc. Comp.2	0.4*f'c	0.5	1.0
	Moment Capacity Shear Capacity	PhiMn PhiVn	1.3 1.3	2.17 2.17
	Strand Tension	0.8*fy	1.0	1.0
Operating	Moment Capacity Shear Capacity	PhiMn PhiVn	1.3 1.3	1.3 1.3
Operating	Strand Tension	0.9*fy	1.0	1.0

Rating Calculations for Interior Beam (Span 1, Beam 2)

This section details the rating calculation for span 1, beam 2, an interior beam. These hand calculations are limited to checking at middle (0.5L) of span1. Inventory and operating ratings are normally done for the design load (e.g., HS20) as well as legal loads (e.g., 3S2, 3-3, and S). Overload trucks may include state specific trucks as well, such as California P13. Operating rating is also performed for permit trucks for permitting or routing purposes. The rating procedure is the same for all cases, and the only difference is live load response values. This section only illustrates the ratings due to the HS20 truck.

For easy verification of the values used in computing rating factors, we will add only a HS20 Truck under Design Live Loads in Precast/Prestressed Girder, and review the results from the beam level output for span 1, beam 2. This is the recommended method of getting details on values used by Precast/Prestressed Girder in computing the rating factors. Future versions of the program may include user options to print more information automatically.

A. Inventory Rating Level

Concrete Stress (Tension and Compression) Capacity

Inventory rating includes concrete stress check in tension and compression. Tension is checked at top and bottom of the section. When checking tension on bottom, the results from positive live load (causing tension on bottom) are used and when checking tension on top, negative live load results are used (causing tension on top).

When checking compression stresses, two cases are considered. In this case, when checking compression on top, positive live load results, and when checking compression on bottom, negative live load results are used.

Here are the rating equations for Inventory Level as on Art 6.6.3.3.:

Concrete Tension

{R F}_{I N V} = \frac{6 \sqrt{{f `}_{c}} - (F_{d} + F_{p})}{F_{1}}

Concrete Compression 1

{R F}_{I N V} = \frac{0.6 \sqrt{{f `}_{c}} - (F_{d} + F_{p})}{F_{1}}

Concrete Compression 2

{R F}_{I N V} = \frac{0.4 \sqrt{{f `}_{c}} - 0.5 (F_{d} + F_{p})}{F_{1}}

where:

- (28-day compressive strength of concrete), 6 ksi (6000 psi);

F_d - unfactored dead load stress;

F_p - unfactored stress due to prestress force after all losses;

F_l - unfactored live load stress including impact.

The allowable stresses used for Rating are as shown in the rating parameters dialog

Allowable Stresses:	Stresses (psi)
Allowable Tensile $6 \sqrt{{f `}_{c}}$	-464.758
Allowable Compression 1 $0.6 {f `}_{c}$	3600
Allowable Compression 2 $0.4 {f `}_{c}$	2400

Precast/Prestressed Girder already has computed the stresses: The calculations are at Midspan. Concrete stresses from analysis are as follows:

Table 1. At bottom (tensile stresses):
From:	Stresses (psi)
Self weight	-1062.6
Deck+Haunch	-1159.6
Prec DL	-239.4
Comp ADL	-158.6
TOTAL DL (Fd)	- 2620.2
Prestress (Fp)	3161.5 (compression)
Positive Live Load ( F_L⁺)	-794.7
Negative Live Load ( F_L^-)	-57.5 (tensile at top)

Table 2. At top (compression stresses):
From:	Stresses (psi)
Self weight	1027.8
Deck+Haunch	1121.6
Prec DL	231.6
Comp ADL	50.2
TOTAL DL (Fd)	2431
Prestress (Fp)	-687.4 (tensile)
Positive Live Load ( F_L⁺)	251.6
Negative Live Load ( F_L^-)	181.6 (compression at bottom)

Concrete Tension Rating Factors

For positive moment we have tension in concrete at bottom:
${R F}_{I N V}^{C T e r m (+)} = \frac{6 \sqrt{{f `}_{c}} - (F_{d}^{b o t} + F_{p}^{b o t})}{F_{L L + I}^{b o t (+)}} = \frac{- 464.758 - (- 2620.2 + 3161.5)}{- 794.7} = 1.27$
For negative moment we have tension in concrete at bottom:
${R F}_{I N V}^{C T e r m (-)} = \frac{6 \sqrt{{f `}_{c}} - (F_{d}^{b o t} + F_{p}^{b o t})}{F_{L L + I}^{b o t (-)}} = \frac{- 464.758 - (- 2620.2 + 3161.5)}{- 181.6} = 5.54$

Concrete Compression

For positive moment we have compression in concrete at top:
${R F}_{I N V}^{C C o m p 1 (+)} = \frac{0.6 \sqrt{{f `}_{c}} - (F_{d}^{t o p} + F_{p}^{t o p})}{F_{L L + I}^{t o p (+)}} = \frac{3600 - (2431 - 687.4)}{251.6} = 7.38$

${R F}_{I N V}^{C c o m p 2 (+)} = \frac{0.4 \sqrt{{f `}_{c}} - 0.5 (F_{d}^{t o p} + F_{p}^{t o p})}{F_{L L + I}^{t o p (+)}} = \frac{2400 - 0.5 (2431 - 687.4)}{251.6} = 6.07$
For negative moment we have compression in concrete at top:
${R F}_{I N V}^{C C o m p 1 (-)} = \frac{0.6 \sqrt{{f `}_{c}} - (F_{d}^{t o p} + F_{p}^{t o p})}{F_{L L + I}^{t o p (-)}} = \frac{3600 - (2431 - 687.4)}{57.5} = 32.9$

${R F}_{I N V}^{C c o m p 2 (-)} = \frac{0.4 \sqrt{{f `}_{c}} - 0.5 (F_{d}^{t o p} + F_{p}^{t o p})}{F_{L L + I}^{t o p (-)}} = \frac{2400 - 0.5 (2431 - 687.4)}{57.5} = 26.58$

Prestressing Steel Tension Capacit

The rating equation for Inventory Level as on Art 6.6.3.3 for prestressing steel tension.y

{R F}_{I N V}^{P S S t T} = \frac{0.8 {f *}_{y} - (F_{d} + F_{P})}{F_{l}}

The stress in strands shall be checked for tension. In this case, for inventory level,

the allowable stress is 0.8f*_y .

0.8f*_y in turn is estimated to be as in the following table:

Type of tendon:	f*_y
Low Relaxation Strand	0.9 f_pu
Stress Relieved Strand	0.8 f_pu

The prestressing steel type used in tutor1.csl is 1/2" - 270 K (low relaxation steel) (f_pu = 270 ksi)

Therefore,

f*_y= 0.9 = 0.9*270 = 243 ksi = 243000 psi

and the allowable stresses for prestressing steel for inventory level as in Article 6.6.3.3 is:

0.8f*_y=0.8(243000)=194400psi

The dead load stress in strand is the final prestress stress after all losses. This can be calculated as fpj- fs. 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*f_pu), and total loss of 43532.81 psi, the final strand stress is 158967.19 psi, which used for the dead load term "D",i.e. (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_{L L + I}^{b o t t o m s t r a n d} = \frac{E_{s}}{E_{c}} \times (f_{b o t} + \frac{(f_{t o p} - f_{b o t}) \times e_{b o t}}{H})

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 (72 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 = 28000 ksi;

E_c - modulus of elasticity of 28-day strength concrete = 4695.98 ksi;

f_{L L + I}^{b o t t o m s t r a n d} = \frac{28000}{4695.98} \times (- 794.7 + \frac{(251.6 + 794.7) \times 2.0}{72}) = 4565.604 p s i

Therefore the Rating Factor for Inventory Level for prestressing steel tension.

{R F}_{I N V}^{P S S t T} = \frac{0.8 {f *}_{y} - (F_{d} + F_{P})}{F_{l}} = \frac{194400 - (158967.19)}{4565.604} = 7.76

Flexural and Shear Strength Capacity

{R F}_{I N V} = \frac{ϕ R_{n} - 1.3 D}{2.17 L}

Art.6-1a

ϕR_n = nominal strength of section satisfying the ductility limitations of Art. 9.18 and Art. 9.20 of AASHTO. Both moment ϕM_n and ϕV_n should be evaluated;

D = unfactored dead load moment or shear;

L = unfactored live load moment or shear (this take in consideration impact factor).

Flexural Strength

The calculation will be made at Midspan:

{R F}_{I N V}^{F l e x u r a l} = \frac{ϕ M_{n} - 1.3 M_{D L}}{2.17 L}

Art.6-1a

a. Dead Load

The Dead Load will remain the same and we can have these values from File->Print Shear/Moment Envelope. In the following table we have the Dead Load values - Midspan location.

	Dead Loads (MDL) (k.ft) at midspan
Self-Weight	1320.8
Deck+Haunch	1441.3
Prec DL	297.6
Comp ADL	266.2
TOTAL	3325.9

b. Live Load

The same situation is for live load. The only difference is that for this type of load we have the value which already includes impact factor.

	Live Loads (MLL+I) (k.ft) at midspan
Positive Live Load	1334.1
Negative Live Load	-304.8

$ϕ M_{n}$ for positive moment capacity is already computed in Precast/Prestressed Girder® in Ultimate Moment Part (M_u-prvd) = 9351.6 k.ft and Φ = 1.0 for flexural members. In checking negative moment capacity, value of negative live load moment is used. Since the negative moment steel was provided only towards the end of span 1 and no steel is provided at the midspan location, the capacity of 0.0 kip-ft. is used.

Prestressed concrete members should meet the requirements of AASHTO Design Art. 9.18.2.1:

ϕ M_{n} \geq 1.2 \times M_{c r}

While there is no reduction in the flexural strength of the member when this provision is not satisfied, the owner may choose to limit live loads to those that preserve the relationship between $ϕ M_{n}$ and M_cr that is presented for a new design. For this option, an adjustment to the value of the nominal moment capacity $ϕ M_{n}$ used in the flexural strength rating equations is necessary.

Thus when $ϕ M_{n} \geq 1.2 \times M_{c r}$ , the nominal moment capacity becomes kΦM_n

Let's also check this requirement in our case - Tutor 1, span 1, beam 2 at midspan:

As already mentioned:

ϕ M_{n}

Also M_cr can be taken from the Ultimate Moment Output of Precast/Prestressed Girder

M_cr = 5209.6 k.ft => 1.2M_cr= 6251.52 k.ft

K = \frac{ϕ M_{n}}{1.2 \times M_{c r}} = \frac{9351.6}{6251.52} = 1.5

Because the equation $ϕ M_{n} \geq 1.2 \times M_{c r}$ is satisfied then no adjustment is needed.

a) for Positive Moment capacity

Therefore for Flexural Strength at Midspan we will have the following rating factors:

{R F}_{I N V}^{M +} = \frac{ϕ M_{n} - 1.3 M_{D L}}{2.17 M_{L L + I}^{+}} = \frac{9351.6 - 1.3 \times 3325.6}{2.17 \times 1334.1} = 1.74

b) for Negative Moment capacity

$ϕ M_{n}$ = 0 k.ft at midspan)

{R F}_{I N V}^{M -} = \frac{ϕ M_{n} - 1.3 M_{D L}}{2.17 M_{L L + I}^{-}} = \frac{0 - 1.3 \times 3325.6}{2.17 \times (- 304.8)} = 6.54

Shear Strength

Shear rating is only done for one shear value, i.e., there are no positive and negative shear ratings, and this may be a source of conservatism. These results are presented for critical location H/2 (3.35ft), however all locations are checked by the program.

Based on the shear reinforcement input, at the H/2 location, we have 2 legged US#3(M10) stirrup at 3" spacing.

{R F}_{I N V}^{S h e a r} = \frac{ϕ V_{n} - 1.3 V_{D L}}{2.17 V_{L L}}

a. Dead Load

The Dead Load will remain the same and we can have these values from File->Print Shear/Moment Envelope. In the following table we have the Dead Load values - H/2 location.

	Dead Loads (kips) at H/2
Self-Weight	43.3
Deck+Haunch	47.2
Prec DL	9.7
Comp ADL	11.0
TOTAL	111.2

b. Live Load

The same situation is for live load. The only difference is that for this type of load we have the value which include impact factor.

	Live Loads (kips) at H/2 (Vmx)
TOTAL	56.5

c.ΦVn

For H/2 location we have 2 legs US#3(M10) @ 3 in (stirrup spacing)

V_{s - p r o v d} = \frac{A_{v} \times f_{y} \times d_{v}}{s} = \frac{0.220 \times 60 \times 64.4}{3} = 283.36 k i p s

V_c = 223.2 kips

ϕ V_{n} = ϕ (V_{s - p r o v e d} + V_{c}) = 0.9 \times (283.36 + 223.2) = 455.9 k i p s

Therefore for Shear Strength at H/2 we will have the following rating factors:

{R F}_{I N V} = \frac{ϕ V_{n} - 1.3 V_{D L}}{2.17 V_{L L}} = \frac{455.9 - 1.3 \times 111.2}{2.17 \times 56.5} = 2.53

B. OPERATING RATING LEVEL

The Operating rating is calculated in similar fashion to the Inventory. The main differences are that Operating stress rating. Furthermore, the load factors are different, and allowable strand tension is 0.9.

Flexural and Shear Strength Capacity

{R F}_{O P} = \frac{ϕ R_{n} - 1.3 D}{1.3 L}

Art. 6-1a

Flexural Strength

The calculation will be made at Midspan:

{R F}_{O P}^{F l e x u r e} = \frac{ϕ M_{n} - 1.3 M_{D L}}{1.3 M_{L L + I}}

Art. 6-1a

ΦM_n, M_DL and M_LL were already computed for Inventory Rating Level. Therefore,

for Positive Moment capacity
${R F}_{O P}^{M +} = \frac{ϕ M_{n} - 1.3 M_{D L}}{1.3 M_{L L + I}^{+}} = \frac{9351.6 - 1.3 \times 3325.6}{1.3 \times 1334.1} = 2.90$
for Negative Moment capacity
${R F}_{O P}^{M -} = \frac{ϕ M_{n} - 1.3 M_{D L}}{1.3 M_{L L + I}^{-}} = \frac{0 - 1.3 \times 3325.6}{1.3 \times (- 304.8)} = 10.91$

Shear Strength

Shear rating is only done for one shear value, i.e., there are no positive and negative shear ratings, and this may be a source of conservatism.

These results, as for Inventory Rating Level, are presented for critical location H/2 (3.35ft).

Also ϕV_n, V_DL and V_LLwere already computed for Inventory Rating Level

Therefore,

{R F}_{O P}^{F l e x u r e} = \frac{ϕ V_{n} - 1.3 V_{D L}}{1.3 V_{L L}} = \frac{455.9 - 1.3 \times 111.2}{1.3 \times 56.5} = 4.23

The program automatically computes the gross vehicle weight (GVW) based on the individual axle loads. This value is used in Rating Summary to compute the controlling rating value for each truck (for both Inventory and Operating Rating Levels) in tons as shown below.

GVW = First Axle Intensity + Σ(Remaining Axle Parameters Intensity)

GVW for HS20 Truck = 8 + 32 + 32 = 72 kips = 36 tons

Therefore the Rating in tons is equal with:

{R F}_{O P}^{t o n s} = {R F}_{O P}^{S h e a r} \times G V W = 2.60 \times 36 = 93.6 t o n

The critical (lowest) rating factor for shear, 2.60 comes from H/2 distance from the right end of the beam.

Prestressing Steel Tension Capacity

Here is the rating equation for Inventory Level as on Art 6.6.3.3 for prestressing steel tension.

{R F}_{I N T}^{P S S t T} = \frac{0.9 {f *}_{y} - (F_{d} + F_{p})}{F_{t}}

the allowable stress is $0.9 {f *}_{y}$

Therefore the allowable stresses for prestressing steel for operating level as in Article 6.6.3.3 is:

0.9* = 0.9*243000 = 218700 psi

The rest of data is as for Inventory Rating Level therefore,

{R F}_{I N V}^{P s s t T} = \frac{0.9 {f *}_{y} - (F_{d} + F_{p})}{f_{L L + I}^{b o t t o m s t r a n d}} = \frac{218700 - 158967.19}{4565.604} = 13.08

Summary of Rating Factors

Please notice that this is a part of the Precast/Prestressed Girder Rating output which shows the results for the hand calculations made for tutor1.csl, span1, beam 2. The values which are in blue are the ones computed in hand calculations.

Load Rating of Prestressed Concrete Girder Bridges

(Manual for Condition Evaluation of Bridges, 2nd Edition 1994, with interims up to & incl. 2003)

Span:1, Beam:2

Inventory Rating Level (Art. 6.6.3.3)

Operating Rating Level (Art. 6.6.3.3)

Notation:

CC1-T/ CC1-B - Concrete Compression 1 at Top/Bottom;

CC2-T/ CC2-B - Concrete Compression 2 at Top/Bottom;

CTens-T/ CTens-B - Concrete Tension at Top/Bottom;

PSStT - Prestress Steel Tension.