Technical Note Generalized Approach for Determining the Maximum Moment under HL-93 Loading on Simple Spans Downloaded from ascelibrary.org by University Of Florida on 10/26/17. Copyright ASCE. For personal use only; all rights reserved. Steven Lowinger, A.M.ASCE1 Abstract: With the institution of AASHTO HL-93 loading, calculating the maximum live-load moment on simple spans became more complex than it had been previously. In the past, to ﬁnd the maximum live-load moment from the HS-20 design truck, the middle axle was offset from midspan by 0.711 m (2 ft 4 in.), and the moment under that axle was the maximum moment on the beam. However, with the combination of the design vehicle and the distributed design lane load, this general approach became obsolete. The purpose of this study was to develop a new generalized approach to determine the location and value of the maximum moment under HL-93 loading on simple spans. This was accomplished by creating moment envelopes and studying their extremum. Subsequently, equations, charts, and tables were developed that describe the location and value of the maximum moment on a beam under HL-93 loading, consisting of design truck or design tandem loading in combination with the design lane load for a simple span of any length. DOI: 10.1061/(ASCE)BE.1943-5592.0001160. © 2017 American Society of Civil Engineers. Author keywords: HL-93; AASHTO; Maximum moment. Introduction Method Before the introduction of HL-93 loading, to determine the value of the maximum moment on a simply supported beam under an HS-20 design truck, the middle axle load was placed 0.711 m (2 ft 4 in.) from the center of the span, with the center of mass of the vehicle located on the other half of the span. However, because HL-93 loading also contains a distributed lane load, this approach is no longer valid. Shipman (2014) developed a method for determining the location and value of the maximum moment of the HL-93 loading consisting of a design truck and a distributed lane load for simply supported spans greater than 12.192 m (40 ft). However, the goal of this work was to propose a different and more general formulation for ﬁnding the location and value of the maximum moment under all AASHTO (2012) HL-93 loading on simple spans of any length. Additionally, this work provides simple equations, tables, and charts that display these quantities as functions of span length for both the design truck combined with the design lane load and for the design tandem combined with the design lane load. This was accomplished by generating a boundary curve, or envelope, that encompasses multiple moment diagrams and identifying the maximum moment in the beam. The value and location of the maximum moment is kept in a general form so that it can be applied for a span of any length, any loads, and any load spacing. However, results speciﬁc to the HL-93 loading are provided as well. The value and location of the maximum moment are tabulated as a function of span length for both the HL-93 design truck and HL-93 design tandem loading. Before showing the results, the author will introduce the method by which the moment diagrams were analyzed. The method will be explained using the design truck; as will be shown, the generality of the method can easily be extended to the design tandem. To ﬁnd the worstcase moment at any point on the beam, the moment diagrams are tracked as the design vehicle moves along the beam. To better explain the method of analysis, the author deﬁnes certain variables and their set values under AASHTO (2012) HL-93 design truck loading: d = distance between the left support to the rear axle of the design vehicle; s = spacing between the rear axle and the middle axle [HL-93: 4.267–9.144 m (14–30 ft)]; t = spacing between the middle axle and front axle [HL-93: 4.267 m (14 ft)]; ‘ = span length; P1 = axle load from the rear axle [HL-93: 1.33 (142.34 kN or 32 k)] with impact factor as per AASHTO (2012, Section 3.6.2); P2 = axle load from the middle axle [HL-93: 1.33 (142.34 kN or 32 k)] with impact factor; P3 = axle load from the front axle [HL-93: 1.33 (35.59 kN or 8 k)] with impact factor; and w = distributed lane load applied to the beam [HL-93: 9.34 kN/m (0.64 k/ft)]. All of these variables are depicted in Fig. 1 for design truck loading combined with the design lane load. The value of s can vary from 4.267 to 9.144 m (14–30 ft). However, in this work, s will always be taken as 4.267 m (14 ft). Additionally, the impact factor, IM, that will be used in this work is 1.33. To generate a general moment diagram, the left vertical support reaction, Ay, is determined for a generalized d. Using basic statics, Ay is represented by Eq. (1). P1 ðdÞ þ P2 ðd þ sÞ þ P3 ðd þ s þ tÞ w‘ Ay ¼ ðP1 þ P2 þ P3 Þ þ ‘ 2 (1) 1 Assistant Engineer, WSP, 1 Penn Plaza, New York, NY 10119. E-mail: Lowingersteven@gmail.com Note. This manuscript was submitted on February 1, 2017; approved on July 13, 2017; published online on October 26, 2017. Discussion period open until March 26, 2018; separate discussions must be submitted for individual papers. This technical note is part of the Journal of Bridge Engineering, © ASCE, ISSN 1084-0702. © ASCE Therefore, using singularity functions, the moment on the beam is represented by Eq. (2). H[x] is the Heaviside step function, also 06017008-1 J. Bridge Eng., 2018, 23(1): 06017008 J. Bridge Eng. known as the unit step function, and x represents the location along the beam. ( P1 d þ P2 ðd þ sÞ þ P3 ðd þ s þ tÞ MðxÞ ¼ ðP1 þ P2 þ P3 Þ ‘ ) w‘ þ x P1 ðx dÞH ½x d P2 ½x ðd þ sÞH ½x ðd þ sÞ 2 Downloaded from ascelibrary.org by University Of Florida on 10/26/17. Copyright ASCE. For personal use only; all rights reserved. P3 ½x ðd þ s þ tÞH ½x ðd þ s þ tÞ wx2 2 (2) In every moment diagram, the moments under the axle loads are studied because these locations will yield the maximum moment. The values of the moment at the locations of the axle loads are referred to as potential maximums. These values are tracked to understand how they change as the design truck moves along the span. To properly track these potential maximums, the locations of the axles are studied with respect to d, the distance between the left support and the rear axle. The rear axle is located at x1 = d, the middle axle at x2 = d þ s, and the front axle at x3 = d þ s þ t. The moment diagram is therefore evaluated at these three locations, x = {x1, x2, x3}, to generate the values of the potential maximums, {M(x1), M (x2), M(x3)}, respectively. However, to keep the dependent variable as x, d will be replaced with a function of x that is unique to each axle under consideration. As an example, M(x2), where x = x2 = d þ s, is the moment at the location x = x2. Within M(x2), all d must be substituted with x – s. A similar process can be followed for M (x1) and M(x3). After these substitutions are made, the values of the moment functions under the axle loads (the potential maximums) are functions of x, the location on the beam; ‘, the span length of the beam; and the loading variables. Eqs. (3)–(5) are the envelope curves and bound the moment diagrams [Fig. 2(a)]. ( Mðx1 ¼ dÞ ¼ Fig. 1. Design truck on a simply supported span with all applicable variables displayed ðP1 þ P2 þ P3 Þ ) P1 x þ P2 ðx þ sÞ þ P3 ðx þ s þ tÞ w‘ wx2 x þ ‘ 2 2 (3) Fig. 2. Plot of multiple moment diagrams and the envelope curves corresponding to each axle of the design truck for different span lengths: (a) span length of ‘ = 25 m (82.02 ft) with all three design truck axle loads on the span; (b) span length of ‘ = 8 m (26.25 ft) with the two rear design truck axle loads on the span © ASCE 06017008-2 J. Bridge Eng., 2018, 23(1): 06017008 J. Bridge Eng. Mðx2 ¼ d þ sÞ ¼ P1 ðx sÞ þ P2 x þ P3 ðx þ tÞ w‘ ðP1 þ P2 þ P3 Þ þ x ‘ 2 P1 s wx2 2 (4) M ðx3 ¼ d þ s þ tÞ Downloaded from ascelibrary.org by University Of Florida on 10/26/17. Copyright ASCE. For personal use only; all rights reserved. ¼ ðP1 þ P2 þ P3 Þ P1 ðs þ tÞ P2 t P1 ðx s tÞ þ P2 ðx tÞ þ P3 x w‘ þ x ‘ 2 wx2 2 (5) The maximum value of all of the envelopes corresponds to the maximum moment on the beam under the HL-93 loading. This method can also be applied to loading scenarios in which one or two axles have exited the span. These circumstances will be addressed later in the article. Based on Fig. 2(a) and the author’s studies, the maximum moment always occurs under the middle axle load. The function for the envelope curve of this axle is found in Eq. (4). Therefore, ﬁnding the location and value of the maximum of this curve would provide an equation that explains the worst-case moment on the beam. Taking the derivative of Eq. (4) with respect to x, setting it equal to zero, and solving for x results in x¼ ‘ P1 s P3 t þ 2 2ðP1 þ P2 þ P3 Þ þ w‘ (6) The ﬁrst term of Eq. (6) refers to the location of the midspan, and the second term represents the offset associated with the design truck combined with the lane load. When w = 0 kN/m, corresponding to a loading scenario in which there is only a HS-20 design truck on the span, but no lane load, the value of the offset coincides with the offset of 0.711 m (2 ft 4 in.) that was used prior to HL-93 loading. Eq. (6) represents xmax the location of the maximum moment on the beam of a span ‘. Plugging Eq. (6) into Eq. (4) and simplifying results in ( 1 2‘ðP1 þ P2 þ P3 Þ 4ðP1 s þ P3 tÞ þ w‘2 Mmax1 ¼ 8 4ðP1 s P3 tÞ2 þ ‘½2ðP1 þ P2 þ P3 Þ þ w‘ Table 1. Value and Location of the Maximum Moment under HL-93 Design Truck and HL-93 Design Tandem Loading HL-93 design truck Span m (ft) Maximum moment kN·m 1 (3.28) 48.5 2.094 (6.87) 104.2 3 (9.84) 152.5 6 (19.69) 326.1 7.408 (24.31) 414.7 9 (29.53) 585.7 10.115 (33.19) 710.9 12.329 (40.45) 1,000.9 15 (49.21) 1,367.5 18 (59.06) 1,800.2 21 (68.90) 2,254.6 24 (78.74) 2,730.5 27 (88.58) 3,227.6 30 (98.43) 3,746.0 35 (114.8) 4,656.9 40 (131.2) 5,626.5 45 (147.6) 6,654.6 50 (164.0) 7,741.2 55 (180.4) 8,886.2 60 (196.9) 10,089.7 Location HL-93 design tandem Maximum moment Location k-ft (xmax =‘) kN·m k-ft (xmax =‘) 35.8 76.9 112.5 240.5 305.9 432.0 524.3 738.2 1,008.6 1,327.8 1,662.9 2,013.9 2,380.6 2,762.9 3,434.8 4,149.9 4,908.2 5,709.6 6,554.2 7,441.8 0.500 0.500 0.500 0.500 0.500 0.607 0.594 0.551 0.541 0.533 0.528 0.523 0.520 0.518 0.515 0.512 0.511 0.509 0.508 0.507 38.1 82.5 150.9 399.8 525.1 672.6 779.7 1,000.9 1,283.3 1,620.4 1,978.7 2,358.0 2,758.4 3,179.8 3,928.9 4,736.3 5,602.2 6,526.5 7,509.2 8,550.2 28.1 60.9 111.3 294.8 387.3 496.1 575.0 738.2 946.5 1,195.2 1,459.4 1,739.2 2,034.5 2,345.3 2,897.8 3,493.3 4,132.0 4,813.7 5,538.5 6,306.3 0.500 0.500 0.597 0.546 0.537 0.530 0.526 0.521 0.516 0.513 0.511 0.509 0.508 0.507 0.506 0.505 0.504 0.503 0.503 0.503 Note: The bolded values correspond to the controlling maximum moment for that span length. ) (7) Eq. (7) is a general equation for the maximum moment on a simply supported beam under HL-93 design truck loading. For HL-93 design truck loading, Eqs. (6) and (7) are applicable to all spans of ‘ > 10.115 m (33.19 ft). When dealing with spans smaller than 10.115 m (33.19 ft), the loading scenario in which the front axle of the design truck has already exited the span and only the two rear axles are on the span leads to a larger maximum moment. The equation for the maximum moment for this loading scenario is shown later in the article. The value 10.115 m (33.19 ft) was determined by comparing the maximum moment under design truck loading with three axles on the span and design truck loading with only the two rear axles on the span. With spans in the range of 4.267 m (14 ft) < ‘ ≤ 8.534 m (28 ft), a maximum of two axles of the design truck can ﬁt on the span. For spans 7.408 m (24.31 ft) < ‘ ≤ 8.534 m (28 ft), the loading scenario © ASCE in which the rear two axle loads are on the span will govern. However, for span lengths ‘ ≤ 7.408 m (24.31 ft), the loading scenario where only the one rear axle remains on the span will govern. For these spans, the rear axle is able to reach midspan after the middle and front axles have exited the span. The magnitude of this moment is larger than the maximum moment when the rear and middle axles are both on the span. Therefore, in the span range of 7.408 m (24.31 ft) < ‘ ≤ 10.115 m (33.19 ft), the loading scenario of having the rear two axle loads on the span will govern. The maximum moment can be determined by taking Eq. (7) and making the simple substitutions of P3 = 0 kN, t = 0 m, which results in Eq. (8). Fig. 2(b) Fig. 3. Plot representing the maximum moment under HL-93 design truck and design tandem loading for any span ‘ 06017008-3 J. Bridge Eng., 2018, 23(1): 06017008 J. Bridge Eng. shows a moment envelope for a span of 8 m (26.25 ft), in which the two rear axle loads are on the span. Downloaded from ascelibrary.org by University Of Florida on 10/26/17. Copyright ASCE. For personal use only; all rights reserved. Mmax2 ¼ 2‘ðP1 þ P2 Þ 2P1 s þ w‘2 8‘ð2P1 þ 2P2 þ w‘Þ 2 In the span range ‘ ≤ 7.408 m (24.31 ft), in which the loading scenario of having only one axle of the design truck on the span controls, the value of the maximum moment can be expressed by Eq. (9). (8) Mmax3 ¼ P1 ‘ w‘2 þ 4 8 (9) In summary, to properly express the maximum moment on a beam under HL-93 design truck loading for any span, three span ranges must be analyzed: ‘ > 10.115 m (33.19 ft), 7.408 m (24.31 ft) < ‘ ≤ 10.115 m (33.19 ft), and ‘ ≤ 7.408 m (24.31 ft). Applying Eqs. (7)–(9) to their respective ranges will result in a function that displays the maximum moment on a span of any length ‘ under HL-93 design truck loading. Table 1 contains the values of this piecewise function evaluated at selected span lengths and the corresponding location of this maximum moment. Because of the generality of the approach, this method can very easily be applied to the design tandem by simply using the previous equations and making the following replacements: P3 = 0 kN, P1 = P2 = 1.33(111.21 kN or 25 k), s = 1.22 m (4 ft), t = 0 m. As an example, Eq. (7) becomes Eq. (10). Fig. 4. Plot representing the location of the maximum moment under HL-93 design truck and design tandem loading for any span ‘ > 10.115 m (33.19 ft). The vertical axis corresponds to the location of the maximum moment, xmax, divided by the span length, ‘ Mmax 2‘ðP1 þ P2 Þ 2P1 s þ w‘2 ¼ 8‘ð2P1 þ 2P2 þ w‘Þ 2 (10) This equation represents the maximum moment under HL-93 design tandem loading for spans of length ‘ > 2.093 m (6.87 ft). Additionally, the location of the maximum moment is Table 2. Equations for the Location and Value of Maximum Moment under HL-93 Design Truck and Design Tandem Loading Loading type HL-93 design truck Mmax kN·m (k-ft) xmax m (ft) 504:91 þ 106:49‘ þ 1:168‘2 þ 19; 651:89 91:21‘ þ ‘2 156; 016:98 372:40 þ 23:94‘ þ 0:08‘ þ 299:25‘ þ ‘2 2 403:93 þ 94:66‘ þ 1:168‘2 þ 297:92 þ 21:28‘ þ 0:08‘2 þ 34; 936:59 81:08‘ þ ‘2 277; 363:52 266‘ þ ‘2 47:33‘ þ 1:168‘2 ‘ 0:711 þ 2 1 þ 0:011‘ 1 0 7 C B‘ 3 C B þ @2 4 A ‘ 1þ 1197 ‘ 1:067 þ 2 1 þ 0:012‘ 1 0 7 C B‘ 2 C B þ @2 1 A ‘ 1þ 266 ‘ 2 90:16 þ 73:95‘ þ 1:168‘2 þ 66:50 þ 16:63‘ þ 0:08‘2 þ 1; 740:7 63:34‘ þ ‘2 13; 819:53 207:81‘ þ ‘2 © ASCE 06017008-4 J. Bridge Eng., 2018, 23(1): 06017008 7.408 m < ‘ 10:115 m (24.31 ft < ‘ 33:19 ft) ‘ 7:408 m ‘ :3048 þ 2 1 þ 0:016‘ ‘ > 2:094 m 0 ‘ @2 þ 1 1 16 A ‘ 1þ 3325 ‘ 2 (8:31‘ þ 0:08‘2 ) (‘ > 33:19 ft) (‘ 24:31 ft) ‘ 2 36:98‘ þ 1:168‘2 ‘ > 10:115 m ‘ 2 (10:64‘ þ 0:08‘2 ) HL-93 design tandem Span m (ft) (‘ > 6:87 ft) ‘ 2:094 m (‘ 6:87 ft) J. Bridge Eng. Downloaded from ascelibrary.org by University Of Florida on 10/26/17. Copyright ASCE. For personal use only; all rights reserved. x¼ ‘ P1 s þ 2 2ðP1 þ P2 Þ þ w‘ (11) Similar to the earlier description, with spans ‘ ≤ 2.093 m (6.87 ft), the loading scenario in which only one axle remains on the span controls. Therefore, for HL-93 design tandem loading, two span ranges must be analyzed: ‘ > 2.093 m (6.87 ft), in which two axles on the span lead to a larger moment, and ‘ ≤ 2.093 m (6.87 ft), in which having only one axle on the span leads to a larger moment. After making the aforementioned replacements to the equations and following the same procedure, a piecewise function can also be generated for HL-93 design tandem loading. The value and location of maximum moment for selected span lengths can be found in Table 1. Additionally, these piecewise functions for the value and location of the maximum moment under all HL-93 loading can be found in Figs. 3 and 4, respectively. To maintain clarity, Fig. 4 only includes spans ‘ > 10.115 m (33.19 ft). Based on Table 1, the span length after which the design truck loading begins to govern over the design tandem loading is 12.329 m (40.45 ft). The ﬁnal table and charts provide an efﬁcient way for engineers to determine the maximum moment under HL-93 loading for a simple span of any length. These results are produced using the HL-93 loading variables set forth previously. If one of these variables were modiﬁed, the results would be different, but the method would be the same. The results found for the value and location of the maximum moment have been conﬁrmed using structural analysis software. Additionally, for the design truck loading on spans greater than 12.192 m (40 ft), the results presented match those found by Shipman (2014). However, when comparing results to those of Shipman, the difference in orientation of x relative to the design vehicle must be noted. Additionally, in the beginning of his work, Shipman (2014) rounded 14/3 to 4.67, which leads to a small difference in results. Without this rounding, the results coincide exactly. HL-93 Equations The equations presented thus far have been expressed using generalized variables. The author will now present equations that result from replacing all variables with the values from HL-93 loading (as deﬁned in the earlier), s = 4.267 m (14 ft), and an impact factor of 1.33. The equations for the location and value of the maximum moment under HL-93 design truck or design tandem loading can be found in Table 2. © ASCE Example Find the location and value of the maximum moment under HL-93 loading on one lane on a 45-m (147.6-ft) simply supported beam. • Refer to Table 1 and recognize that the HL-93 design truck loading will govern. Then, using the same table, Fig. 3, Eq. (7), or the applicable equation from Table 2, one can ﬁnd that the maximum moment is 6,654.6 kN·m (4,908.2 k-ft). • Using Table 1, Fig. 4, Eq. (6), or the applicable equation from Table 2, one can determine that xmax/‘ = 0.511 or that the location of the maximum moment is xmax = 23 m (75.44 ft) = (0.511)(45) m. This corresponds to a 0.5-m (1.64-ft) offset from midspan. Conclusion The author presented a generalized method for determining the maximum moment on a simple span under HL-93 loading. This method can be applied to any design code, any vehicle, or any impact factor because the equations are all functions of variables that can be altered. The basis of the work was to develop a boundary, or envelope, curve that encompasses the moment diagrams for different locations of the design vehicle. Subsequently, the value and location of the maximum of this envelope curve were determined as functions of the span length, and generalized equations were presented. The author also developed tables and charts that explain the value and location of the maximum moment under HL93 loading for any span length. These results will aid in the understanding of the behavior of a beam under HL-93 loading and will signiﬁcantly improve the efﬁciency of the beam design process. Acknowledgments I thank Dr. Fredy R. Zypman for reviewing this paper and educating me on how to approach problems analytically; his courses informed the method developed in this paper. I also thank Dr. Tom T. Panayotidi for teaching me about all aspects of structural engineering; his courses encouraged me to approach this problem. References AASHTO. (2012). AASHTO LRFD bridge design speciﬁcations, 7th Ed., Washington, DC. Shipman, C. L. (2014). “Finding maximum moment: Determining Hl-93 truck position on simple spans.” J. Bridge Eng., 10.1061/(ASCE)BE .1943-5592.0000591, 06014003. 06017008-5 J. Bridge Eng., 2018, 23(1): 06017008 J. Bridge Eng.

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