AASHTO-LRFD LIVE LOAD DISTRIBUTION SPECIFICATIONS

Transcription

1 AASHTO-LRFD LIVE LOAD DISTRIBUTION SPECIFICATIONS By Toorak Zokaie, 1 Member, ASCE ABSTRACT: The live load distribution factors contained in the AASHTO-LRFD Bridge Design Specification present a major change to the AASHTO-LFD specifications that have been in effect for more than 50 years. This change has generated some interest in the bridge engineering community and has raised some questions. The AASHTO-LFD formulas are based on the girder spacing only and are usually presented as S/D, where S is the spacing and D is a constant based on the bridge type. This method is applicable to straight and right (i.e., nonskewed) bridges only. The new formulas are more complex and consider more parameters, such as bridge length and slab thickness. It may not be obvious to the engineers what added accuracy and flexibility (e.g., skewed bridges) is gained by the increased complexity. This paper will present the background on the development of the formulas and compare their accuracy with the S/D method. A discussion on the extension of the single girder design (using formulas) to the skewed bridges is also presented. BACKGROUND The AASHTO-LRFD (AASHTO 1994) live load distribution formulas have resulted from the National Cooperative Highway Research Program (NCHRP) project, entitled Distribution of Live Loads on Highway Bridges (Zokaie et al. 1991). This project was initiated in 1985, long before the LRFD specifications were developed, to improve the accuracy of the S/D formulas contained in the AASHTO specifications (Standard 1996). Upon review of the S/D formulas, it was found that these formulas were generating valid results for bridges of typical geometry (i.e., girder spacing near 6 ft and span length about 60 ft), but would lose accuracy very soon when the bridge parameters were varied (e.g., when relatively short or long bridges were considered). It was therefore concluded that, in order to gain higher accuracy, additional parameters such as span length and stiffness properties must be considered. This study led to the development of a set of formulas that not only provided higher accuracy but also include a broader range of bridges. These formulas were adopted by AASHTO as the guide specifications for distribution of live loads on highway bridges (Guide 1994). The AASHTO-LRFD specifications presented differences in the live load model and the multiple presence factors. As a result, the original formulas were revised to retain their accuracy when applied to the LRFD live loads. These formulas were developed for several bridge types: beam-and-slab (reinforced concrete T-beam, prestressed concrete I-girder, and steel I-girder), multicell box girder, side-by-side and spread box beams, and slab bridges. However, the rest of this paper will concentrate on beam-and-slab bridges for simplicity. The methods used to develop the formulas, verification methods, and applicability discussions are applicable to all bridge types that were considered. In order to evaluate the existing formulas, it is necessary to compare their results with an accurate method. Finite-element or grillage analysis methods were used for this purpose, and bridge deck models were prepared based on generic geometric parameters and material properties. A database of several hundred actual bridge decks was also prepared, and analytical models were developed for all the bridges in the database. The positioning of the live load is one of the key items in the calculation of the distribution factor and is further described below. 1 Sr. Software Engr., Leap Software, Inc., 1144 Coloma Rd., Ste. 440, Gold River, CA toorak@leapsoft.com Note. Discussion open until October 1, To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on May 1, This paper is part of the Journal of Bridge Engineering, Vol. 5, No. 2, May, ASCE, ISSN /00/ /$8.00 $.50 per page. Paper No CALCULATION OF ACCURATE DISTRIBUTION FACTORS A grillage or finite-element analysis of the bridge is generally acceptable as an accurate analysis. However, two key points must be kept in mind to achieve accurate results: the computer program must be selected carefully so that the important parameters affecting the behavior of the bridge deck can be modeled; and the model must be prepared carefully to represent the true behavior of the bridge deck. After reviewing several computer programs and modeling details, the computer program GENDEK5A (Powell and Buckle 1970) was selected, and modeling details were finalized. This program was selected partly because it uses plate elements to model the deck slab, and it can model the eccentricity of the beams. The results of a number of field and prototype tests were compared with the analytical results, and it was found that GENDEK5A produces accurate results and compares well with test results. A typical bridge deck model is shown in Fig. 1. The distribution factors were calculated by loading the deck FIG. 1. Single-Span Beam-and-Slab Bridge Finite-Element Model JOURNAL OF BRIDGE ENGINEERING / MAY 2000 / 11

2 TABLE 2. Beam-and-Slab Bridge Wheel Load Distribution Factors for Sensitivity Study Parameters FIG. 2. Variation of Girder Spacing in Beam-and Slab Bridges FIG.. Relationship of Slab Thickness and Girder Spacing in Beam-and-Slab Bridges model with truck loads positioned at the longitudinal location that produces the maximum moment. The trucks were then moved transversely across the width of the bridge, and for each location the maximum girder moment was calculated. The largest girder moment for all locations was then selected as the maximum moment. This procedure was repeated for any number of trucks that fit on the bridge transversely, and the maximum moment was adjusted by the multiple presence reduction factor. The controlling moment was then selected. The ratio of this moment to the moment obtained from a simple beam loaded by one truck wheel line (one half of the axle loads) represents the wheel load distribution factor. Note that the LRFD specifications use a factor that is found by using the lane load (full axle load), rather than the wheel loads. BRIDGE SUPERSTRUCTURE DATABASE In order to get a representative sample of the bridges in the United States, several hundred bridges were selected randomly from the National Bridge Inventory File (NBIF). Bridge plans Variation from average properties Moment Values Multiple lane One lane Shear Values Multiple lane One lane (5) Average a S = 457 mm (1.5 ft) 0.2 S = 1,72 mm (4.5 ft) S = 1,840 mm (6.04 ft) S = 2,896 mm (9.5 ft) S =,992 mm (1.1 ft) L = 8.90 m (29.2 ft) L = m (9.0 ft) L = m (66.0 ft) L = m (84.0 ft) L = m (150.0 ft) t = mm (6 in.) t = (9.0 in.) I Ae 2 = m 4 (2.74 ft 4 ) I Ae 2 = m 4 (9.98 ft 2 ) I Ae 2 = m 4 (41.04 ft 4 ) I Ae 2 = m 4 (64.91 ft 4 ) J = m 4 (0.01 ft 4 ) J = m 4 (.19 ft 4 ) J = m 4 (.04 ft 4 ) J = m 4 (4.82 ft 4 ) Number of girders = Number of girders = de b = de b = m (0.5 ft) de b = 0.05 m (1.0 ft) de b = m (1.92 ft) de b = m (.0 ft) de b = 1.72 m (4.5 ft) de b = m (6.5 ft) Gaug m (4.0 ft) Gaug m (5.0 ft) Gaug 2.48 m (8.0 ft) Gaug.048 m (10.0 ft) Gaug.658 m (12.0 ft) a Average properties are: spacing (S) = 2.68 m (7.77 ft); span length (L) = 14.6 m (48.0 ft); slab thickness (t) = mm (6.95 in.); beam stiffness (I Ae 2 ) = m 4 (17.22 ft 4 ); torsional inertia (J) = m 4 (1.25 ft 4 ); number of girders = 5; d m (2.5 ft); gaug 1.8 m (6.0 ft). b Exterior girder distribution factors. were obtained from the state departments of transportation. Several parameters were extracted from the bridge plans and were stored in a database. This information was enough to carry out a finite-element or grillage analysis of the bridge deck. The information contained in the database included bridge type (i.e., T-beam, prestressed I-girder, or steel I-girder), span length, edge to edge width, skew angle, number of girders, girder depth, slab thickness, overhang, curb to curb width, year built, girder eccentricity (distance from centroid of the girder to the midheight of the slab), girder moment of inertia, and girder area. IDENTIFICATION OF KEY PARAMETERS The database was studied to identify the range and variation of each parameter. The minimum, maximum, mean, and stan- TABLE 1. Variation of Wheel Load Distribution with I, A, and e I Ae 2 I A/e m 4 (57,000 in. 4 ) m 4 (65,688 in. 4 ) m 2 (596.1 in. 2 ) m (22.11 in.) m 4 (57,000 in. 4 ) m 4 (47,445 in. 4 ) m 2 (614.5 in. 2 ) m (22.44 in.) m 4 (57,000 in. 4 ) 0.01 m 4 (75,255 in. 4 ) m 2 (586. in. 2 ) m (21.92 in.) m 4 (57,000 in. 4 ) m 4 (65,688 in. 4 ) m 2 (418.0 in. 2 ) m (26.40 in.) m 4 (57,000 in. 4 ) m 4 (65,688 in. 4 ) m 2 (825.6 in. 2 ) m (18.78 in.) 1.88 A e (5) g int (6) 12 / JOURNAL OF BRIDGE ENGINEERING / MAY 2000

3 dard deviation for each parameter was determined. Fig. 2 shows the variation of the girder spacing values in the database as a sample. Furthermore, several parameters were plotted against each other to determine if they are correlated. For example, it was suggested that girder spacing and slab thickness may be correlated, or that larger span lengths result in larger moments of inertia and/or girder depths. This study revealed that, by and large, the parameters are not correlated. Fig. shows the relationship between the girder spacing and slab thickness as a sample. A hypothetical bridge deck model that is made of all mean values of the parameters was created and referred to as the Average Bridge. To identify which parameters are of considerable importance for live load distribution, a sensitivity study was performed. A bridge deck finite-element model was prepared for the average bridge and loaded with HS20 trucks, as described earlier, to calculate the live load distribution factors for shear and moment. A parameter was introduced for the longitudinal stiffness of the girder to cut down the number of variations. This parameter, (K g = I Ae 2 ), can replace the girder inertia (I), girder area (A), and girder eccentricity (e). A number of bridge decks with the same K g and different I, A, and e values were analyzed, and it was determined that the final distribution factors are not largely affected by this variation. The results are shown in Table 1. A similar analysis was performed for several models, keeping all parameters as mean value, except for one that was varied from its minimum to maximum. This process was repeated for all parameters. The values of the parameters used in this study and the resulting distribution factors are shown in Table 2. These results were also plotted to provide a visual examination of the importance of the parameter. FIG. 4. Sensitivity of Wheel Load Distribution Factors to Girder Spacing FIG. 6. Sensitivity of Wheel Load Distribution Factors to Composite Girder Bending Stiffness (K g ) FIG. 5. Length Sensitivity of Wheel Load Distribution Factors to Span FIG. 7. Sensitivity of Wheel Load Distribution Factor to Slab Thickness JOURNAL OF BRIDGE ENGINEERING / MAY 2000 / 1

4 After examining the results of Table 2 and graphs showing the variation of the distribution factor with each parameter, it was determined that the key parameters for each bridge type are girder spacing (S), span length (L), girder stiffness (K g ), and slab thickness (t). Graphs showing the effect of each parameter on the distribution factor are presented in Figs Since the design truck has a fixed gauge width, variation of truck axle width (gauge) was not considered. Most permit FIG. 8. Comparison of Predictions of Simple Formulas with More Accurate Analysis: (a) AASHTO S/D; (b) AASHTO Guide Specifications (before Revisions for LRFD) TABLE. Formulas for Moment/Shear Distribution (g) to Interior Girders Beam and slab Bridge type Concrete box girders Bridge designed for one traffic lane S S K g 4 f L Lt s 0.1 Slab 2f LW Multibox beam decks Beam and slab (slab on girder) Concrete box girder Multibox beams Note: f = 04.8 mm (1.0 ft) Bridge designed for two or more traffic lanes (a) Moment Range of applicability S S K g.5f S 16 f f L Lt s 0.15 S f f L N 1 L S 90f 0.25 c N N 800f 9f L b I k S 5f L L L J c L.5f 0.06 LW S 2f L L b b 1 I f L N J (b) Shear 2 S S S f 6f 25f f L 4.4f L b I 1.15 L J f L.1f L b b b I.2f L J 0.75f t s f 0.48f 4 K g 7f 7f S 1f 60f L 240f N c 8f L 70f 12f W 100f 20f L 15.5f 20f L 105f 1.2f 4 J 29.5f f 4 I 29.5f 4.5f S 16f 0.75f t s f 0.48f 4 K g 7f 7f S 1f 60f L 240f N c 20f L 15.5f 20f L 105f 1.2f 4 J 29.5f f 4 I 29.5f 4 14 / JOURNAL OF BRIDGE ENGINEERING / MAY 2000

5 trucks have a larger gauge width, which results in lower distribution factors. Therefore, using simplified formulas that are developed based on the design truck will yield conservative results for permit trucks. SIMPLIFIED FORMULAS In order to develop the formulas in a systematic manner, certain assumptions must be made. First, it is assumed that the effect of each parameter can be modeled by an exponential function of the form ax b, where x is the value of the given parameter and constants a and b are to be determined based on the variation of the distribution factor with x. Second, it is assumed that the effects of different parameters are independent of each other. This assumption allows each parameter to be considered separately. The final distribution factor will be modeled by an exponential formula of the form g = (a)(s b1 )(L b2 )(t b )( ), where g = wheel load distribution factor; S, L, and t = parameters included in the formula; a = scale factor; and b1, b2, and b are determined from the variation of g with S, L, and t, respectively. Assuming that for two cases all bridge parameters are the same except for S, then and therefore or b1 b2 b g1 =(a)(s1 )(L )(t )( ) b1 b2 b g2 =(a)(s2 )(L )(t )( ) b1 (g1/g2)=(s1/s2) b1 = ln(g1/g2)/ln(s1/s2) If n different values of S are examined and successive pairs are used to determine the value of b1, then (n 1) different values of b1 can be obtained. If these b1 values are close to each other, an exponential curve may be used to accurately model the variation of the distribution factor with S. In that case, the average of (n 1) values of b1 is used to achieve the best match. Once all exponents (i.e., b1, b2, etc.) are determined, the value of a can be obtained from the average bridge, i.e. b1 b2 b a = g0/[(s0) (L0) (t0) ( )] This procedure was followed during the entire course of the study to develop new formulas as needed. In certain cases where an exponential function was not suitable to model the effect of a parameter, a slight variation from this procedure was used to achieve the required accuracy. However, this procedure worked quite well in most cases, and the developed formulas demonstrate high accuracy. VERIFICATION AND EVALUATION Since certain assumptions were made in the derivation of the formulas and some bridge parameters were ignored altogether, it is important to verify the accuracy of these formulas when applied to real bridges. The database of actual bridges was used for this purpose. Bridge in the database were analyzed by an accurate method. The distribution factors obtained from the accurate method were compared with the results of the formulas. The ratio of the formula results to accurate distribution factors was calculated and examined to assess the accuracy of the formula. Average, standard deviation, and minimum and maximum values of the ratios were obtained for each formula. The formula that has the smallest standard deviation is considered to be the most accurate. The minimum TABLE 4. Formulas for Moment/Shear Distribution (g) in Exterior Girders Bridge type Bridge designed for one traffic lane Bridge designed for two or more traffic lanes (a) Moment Beam and slab (slab on girder) Use simple beam distribution e g interior a Concrete box girders W e 7f 7f d e f Use simple beam distribution e g interior 27.7f d e 28.5f Multibox beam Use simple beam distribution e g interior 26f d e 25f (b) Shear Beam and slab (slab on girder) Use simple beam distribution e g interior 6f d e 10f Concrete box girders Use simple beam distribution e g interior 8f d e 128.5f Use simple beam distribution e g interior 8f d e 10f Multibox beams Use simple beam distribution e g interior 51f d e 50f a g interior = distribution factor for interior beams; f = 04.8 mm (1.0 ft). W e 7f Range of applicability f d e 5.5f W e S 0 d e 4.5f f d e 2f f d e 5.5f 2f d e 5f 0 d e 4.5f f d e 2f JOURNAL OF BRIDGE ENGINEERING / MAY 2000 / 15

6 and maximum values present the extreme predictions that each formula produces based on the database of actual bridges. Although these values may change if a different database is used, the values allow us to identify the shortcomings of a formula so that the formula can be fine-tuned to be more accurate. These shortcomings are not readily identified by the average and standard deviation values. FINE-TUNING FORMULA A scattergram and a histogram (bar graph) were developed for visual inspection of the results of a formula. The results obtained from a formula are compared with the results obtained from a more accurate analysis (e.g., finite-element deck analysis), and the values that were inspected were the ratios of the distribution factors obtained from the formula to the one obtained from the more accurate analysis. This value was referred to as the g-ratio. This visual inspection allows us to identify the revisions and fine-tuning that can improve the results. The following goals were set for the g-ratios in fine tuning each formula: 1. The standard deviation for the g-ratios obtained using the new formula must be less than the one obtained from the existing AASHTO formulas. This is a measure of the accuracy of the formula as applied to a large number of bridges. 2. The average value of the g-ratios must be greater than unity by one standard deviation. This builds slight conservatism in the formula, allowing the formula to be conservative in most cases, but not overly conservative.. Minimize the standard deviation. Slight variations of the formulas were examined to get the most accurate results. The goal was to limit the standard deviation to 0.05, but in some cases where this was not possible, values of up to 0.1 were accepted. 4. Simplify the formulas as much as possible. In order for the formulas to be practical for design, a slight loss of accuracy may be acceptable for greater simplicity. Visual inspection and judgment were the key to this finetuning process. The trends were examined, and the formulas were fine-tuned by trial and error. When the overall accuracy was acceptable but the mean value needed to be adjusted, a constant was added to the formulas. As a result, some formulas take the following form: g = c a(s b1 b2 )(L )( ) In some cases, an exponential form does not produce accurate results, and other forms must be pursued. For instance, it was found that, for a spacing greater than 16 feet, the formula does not produce accurate results. The formula could be revised to be more accurate for these bridges, but the overall accuracy would suffer. Therefore, it was proposed that the lever rule be used for bridges with spacing greater than 16 feet. This would preserve the overall accuracy and produce acceptable results for the higher spacings as well. Fig. 8 shows the histogram of the g-ratios for the AASHTO S/D formula and the proposed formula. These graphs demonstrate the better predictions obtained from the new formulas, allowing us to have more confidence in our analysis. FINAL RESULTS The above procedure was repeated for each case of moment and shear, and for single-lane and multilane loading. After several trials, and after reaching the desired accuracy, the formula was finalized. The final results for these four cases are shown in Table. Please note that these formulas are independent of the units of measure and are based on the values of parameters. For example, in S/L or b/l, units of S and L or b and L are the same. In some cases, a constant distance of one foot ( f ) is used to create a unitless ratio, such as the ratio S/4f in the beam-and-slab formula for single-lane loading. Since the formulas were calibrated against this database, the range of variation of each parameter in the database was presented as the range of applicability. This is not to imply that the formula would give erroneous results when a parameter is outside its range, but that it could be less accurate. In most cases, the formulas are far more accurate than the S/D methods, even outside their range. The main purpose of presenting these ranges is that if the engineer wishes to obtain results that are generally within 5% of a detailed analysis and if the bridge TABLE 5. Formulas/Correction Factors for Calculation of Interior Moment and Obtuse Corner Girder Shear for Skewed Supports Bridge type Beam and slab (slab on girder) Concrete box girders, slabs, multibox beams, and spread box beams Beam and slab (slab on girder) Concrete box girder Multiple Presence Factors in AASHTO Specifica- TABLE 6. tions Number of lanes Bridge designed for any number of traffic lanes (a) Moment 1 c 1 (tan ) 1.5 Kg S c = 0.25 Lt L If q is less than 0, c 1 = 0.0 If q is larger than 60, use as tan( ) 1.0 If q is larger than 60, use as 60 Multiple Presence Factor 16th edition Range of applicability LRFD or more s (b) Shear 1.0 c 1 tan 1 c 1 = 0. K g 5 Lt s 1.0 c 1 tan( ) L c 1 = d 1.0 c 1 tan( ) c = 1 Ld 6S Multibox beams a 1.0 c1 tan( ) c = 1 L 90d Note: f = 04.8 mm (1.0 ft) a Applies to all beam (interior and exterior)..5f S 16f.75f t s f.48f 4 K g 7f 4.5f S 18f.75f t s f.48f 4 K g 7f 4 6f S 1f 20f L 240f f d 9f 20f L 15f 1.5f d 5.5f 20f L 105f 1.5f d 5f 16 / JOURNAL OF BRIDGE ENGINEERING / MAY 2000

7 falls outside the range of applicability, then a more detailed analysis may be warranted. Another issue of concern for design is that some formulas have an inertia term. Since the member size is not known, an iterative procedure is needed. Additionally, in most cases the beams are not prismatic, and the inertia varies along the length of the beam. Which inertia value should be used for design? All the inertia effects are included in one term (K g /Lt ) 0.1, and this term has the least effect on the final value. Therefore, the entire inertia term can be taken as unity for preliminary design. For nonprismatic beams, the inertia value at the location of highest moment (e.g., midspan) can be used. The value of K g can be taken as (I Ae 2 ) for beams designed for composite action with the slab, and as the girder inertia (I) for noncomposite beams. EXTENSIONS After establishing the base formulas (i.e., formulas for flexure and shear, for single and multiple lane loading, in the interior girder), several extensions were investigated. These include continuity, edge girder, and skew effects. For most of these cases, the results of the base formulas can be adjusted by a correction factor. Therefore, the final distribution factor would be g s = f s g 0, in which g s is the final factor, f s is the correction factor, and g 0 is the result of the base formula. A similar procedure was used to study the sensitivity of the factor to various bridge parameters and to develop its formula. Continuity Effect: It was found that the distribution factors in continuous bridges are slightly higher than in the simply supported bridges. This difference is less than 5% for positive moments and less than 10% for negative moments. However, it was assumed that redistribution of moment will cancel this effect. As a result, it was decided that the formulas are directly applicable for negative moment. Therefore, it is suggested that the span length used in the formula be the average of the adjacent spans. Edge Girders: It was found that the edge girders are more sensitive to the truck placement than any other factor, i.e., the most important issue is how close can the trucks get to the exterior girder. This was determined using a parameter called edge distance (d e ). In some cases, the lever rule produced more accurate results than applying correction factors to the base formulas. The final proposed procedures are shown in Table 4. Skew Effect: It was found that the skewed supports change the load path slightly. In these cases, the load is transferred to the supports in their shortest path, i.e., towards the obtuse corners. Therefore the moments are smaller, and the shear at the obtuse corner is larger when compared to a non-skewed bridge of the same length and size. This effect is dependent on the amount of skew, and therefore, the correction factors must be TABLE 7. AASHTO-LRFD Formulas for Moment/Shear Distribution (g) to Interior Girders Beam and slab Bridge type Concrete box girders Bridge designed for one traffic lane S S K g 14 f L Lt s S f f L N c Slab 2f LW Multibox beam decks Beam and slab (slab on girder) Concrete box girder Multibox beams Note: f = 04.8 mm (1.0 ft) S f L L b I 1.2k L J k = 2.5(N b ) S 25f f L f L for S 11.5 use lever rule S I 10.8L J Bridge designed for two or more traffic lanes (a) Moment (b) Shear Range of applicability S S K g.5f S 16 f 20f L 240f 9.5 f L Lt s S f 2 N 5.8f L c 7f S 1f 60f L 240f N c if N c > 8 use N c =8.5f 0.06 LW 8f L 70f f W 100f S 2 6.f L L b b I 2k 25.4f L J k = 2.5(N b ) S S f 5f 2 7.f L 2 7.4f L b b I 2 1f L J 20f L 140f 1.5f d 5.4f 20f L 120f 2.92f b 5f.5f S 16f 0.75f t s 12.0f 0.48f 4 K g 7f 7f S 1f 60f L 240f N c 20f L 15.5f 20f L 105f 1.2f 4 J 29.5f f 4 I 29.5f 4 JOURNAL OF BRIDGE ENGINEERING / MAY 2000 / 17

8 dependent on the skew angle. The correction factors that were developed for this case are shown in Table 5. AASHTO-LRFD SPECIFICATION The formulas developed in NCHRP needed to be modified to be compatible with the LRFD specifications. Two issues are of particular importance in comparing the live load response calculation procedures of the AASHTO 16th edition and LRFD specifications: live load description and multiple presence factors. The live load truck in the 16th edition consists of an HS20 truck or a lane load. The live load in the LRFD consists of an HS20 truck in conjunction with a lane load. Both trucks have a 6 ft axle (gauge) width, which is the most important factor affecting the transverse distribution of live loads. Therefore, it was assumed that the difference in the live load configuration does not affect the live load distribution. The multiple presence factors for the two specifications are shown in Table 6. The formulas need to be revised to reflect this difference. Accurate distribution factors were calculated for the LRFD specifications using the finite-element models, and the formulas were revised (recalibrated) to these results. The new formulas were incorporated in the LRFD specifications. Table 7 presents these formulas for comparison with Table. Note that the formulas in Table 7 are presented in a slightly different format than the LRFD specifications (i.e., as wheel load distribution factors) to allow easier comparison. These formulas are based on unitless ratios of parameters, as explained for Table. LIMITATIONS AND SPECIAL CASES In order to apply the load distribution formulas to actual bridges, we should consider the limitations of the study and understand when accurate results can be expected. The models that were used to develop the formulas had uniform spacing, girder inertia, and skew. Continuous models had equal spans. Diaphragm effects were not included in the model. The results were calibrated against a database of real bridges with certain ranges of span length, inertia, spacing, and so on. Although these formulas are much more accurate than the simple S/D factors, they would be most accurate when applied to bridges with similar restraints. An engineer s judgment must be used when the parameters used in the formulas are determined. For example, when a girder has variable inertia, the average girder inertia may be used, or the maximum inertia can be used to be conservative. When girder spacings are different, the average of the spacings on the two sides of one girder may be a good estimate. When the ends of the span have different skews, resulting in different span lengths, the specific girder length and the average skew may be an acceptable approximation. The engineers should judge when the variations are too much, causing the formulas to be inapplicable. The live load analysis for permit trucks (by applying one lane of the truck to a beam model and adjusting that by the distribution factor) may be too conservative, since it assumed that all lanes are loaded by similar trucks. A simple grillage analysis can, in most cases, be performed to calculate more accurate distribution factors than the formula results if needed. One simple program, called LDFAC (Zokaie et al. 199), was developed as part of this NCHRP study to assist engineers in cases for which the formulas may not be applicable but for which a detailed analysis is not warranted. CONCLUSIONS The work that was done under NCHRP resulted in lateral load distribution factors for highway bridges. The formulas that were developed generally produce results that are within 5% of the results of a finite-element deck analysis. These formulas were calibrated against an extensive database of actual bridge decks to verify their applicability to real bridges. The formulas were developed for beam-and-slab bridges with steel, prestressed, or T-beam girders, multicell box girder bridges, side-by-side box beam bridges, solid slab decks, and spread box beam bridges. A grillage or finite-element analysis is recommended for cases in which the simple formula method is not applicable. ACKNOWLEDGMENTS The writer would like to acknowledge the invaluable contributions of Mr. Timothy Osterkamp, who performed much of the analysis. The efforts of the National Cooperative Highway Research Program in funding and monitoring the research are also appreciated. APPENDIX. REFERENCES AASHTO-LRFD bridge design specifications. (1994). 1st Ed., American Association of State Highway and Transportation Officials, Washington, D.C. Guide specifications for distribution of loads for highway bridges. (1994). American Association of State Highway and Transportation Officials, Washington, D.C. Powell, G. M., and Buckle, I. G. (1970). Computer programs for bridge deck analysis. Rep. No. UC SESM 70-6, University of California, Berkeley, Calif. Standard specifications for highway bridges. (1996). 16th Ed., American Association of State Highway and Transportation Officials, Washington, D.C. Zokaie, T., Mish, K. D., and Imbsen, R. A. (1995). Distribution of wheel loads on highway bridges, phase. NCHRP 12-26/2 Final Rep., National Cooperative Highway Research Program, Washington, D.C. Zokaie, T., Osterkamp, T. A., and Imbsen, R. A. (1991). Distribution of wheel loads on highway bridges. NCHRP 12-26/1 Final Rep., National Cooperative Highway Research Program, Washington, D.C. 18 / JOURNAL OF BRIDGE ENGINEERING / MAY 2000