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Simulating creep induced moment redistribution in prestressed concrete bridges constructed by the balanced cantilever method: ad hoc traditional formulae versus real timedependent analysis
Journal of Engineering and Applied Science volume 70, Article number: 28 (2023)
Abstract
Assessing available numerical techniques adopted to determine the design timedependent moment for prestressed concrete segmental bridges constructed by the balanced cantilever method is of utmost importance to the bridge design community. In essence, despite some apparent diversity, there are basically two key conventional approaches to compute the design timedependent moment accounting for creep effects for this type of bridges. The first is a family of varied simplified methods typically known to practicing designers and with preconsensus on their reliability and effectiveness. Timedependent moments retrieved from these classical methods always reside in an intermediate state falling between the results from “two” timeindependent analysis cases, namely, (a) sequentially adding all partial permanent loads and prestressing pertaining to various construction steps using the partbridge structural system corresponding to each step, and (b) assuming all loads and prestressing forces to be applied atonce to the final completed bridge. The second approach is through performing real sophisticated stepbystep timedependent analysis using a specialized software. The research primary objective is to assess the validity/reliability of commonly used adhoc approaches that evolved over the years relying on simplified analyses/formulae to cater for timedependent creep effects for this type of bridges. Aiming at realistic conclusions, three casestudy realworld segmental balanced cantilever bridges over the Nile River in Egypt are elected. Midas Civil commercial package is used to perform timedependent finite element analyses for the three bridges. Main parameters considered are, inter alia, timedependent effects of creep and shrinkage of concrete, relaxation of prestressed steel, losses due to friction and anchor setting of prestressing tendons, sequence of construction, and constructiondriven temporary change of support conditions (where applicable). The study concludes that creepinduced moment redistribution from simplified traditional formulae typically adopted in the literature may lead to a considerable error in estimating the design timedependent moment in balanced cantilever bridges.
Introduction
Significant progress has been made over the years in the development of stepbystep methodologies to account for the timedependent effects in the analysis of segmental prestressed concrete bridges. A bunch of diversified articles on the design, analysis, and construction of segmental bridges have been published by many researchers. Prime attention is given to estimating reliable values for the deformations and internal moment redistribution due to creep and shrinkage of concrete as the structural system is changing during successive construction phases [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. In particular, Bishara and Papakonstantinou [11] investigated the timedependent deformation of cantilever construction bridges before and after closure. Cruz et al. [15] introduced a nonlinear analysis method to calculate the ultimate strength of bridges. Simplified formulae for estimating the internal moment redistribution due to creep and shrinkage of concrete have been always eagerly awaited by the bridge design community. Trost and Wolff [14] proposed an intuitive formula (namely, Eq. 1) that started since then to gain worldwide reputation and acceptance. It is based on combining elastic moments, ΣM_{s,i}, that occur at successive construction steps (see Fig. 1) on the one hand, and the moment obtained by assuming the entire structure constructed at one same stage, M_{E}, at the other hand:
where ϕ_{t} and ρ represent the creep and corresponding relaxation factors, respectively. A similar approach has been presented by the Prestressed Concrete Institute and PostTensioning Institute [16]. Furthermore, Kwak and Son [17], yet adopting the same format traditionally introduced in the literature to combine preelected elastic moments from different statical systems of the bridge under consideration, namely Eq. (1), proposed a slight alteration as reflected in Eq. (2), in order to determine the design moments in bridges constructed by the balanced cantilever method to account for timedependent creep effects. They claim that there should be some limitations if applying Eq. (1) to balanced cantilever bridges since the creepinduced moments grow continuously from each time the structural system changes throughout consecutive construction steps (t = c days). Maintaining the basic form of Eq. (1) suggested by Trost and Wolff [14], yet considering the actual construction sequence while calculating the internal moments at an arbitrary time t, Kwak and Son [17] proposed another simplified ad hoc equation as follows:
where f(ϕ_{t}) = χϕ_{t}/(1 + χϕ_{t}). χ is the concrete ageing coefficient [18,19,20,21,22] which accounts for the effect of ageing on the ultimate value of creep for stress increments or decrements occurring gradually after application of the original load. As explicitly revealed in Kwak and Son [17], the following primary differences can be sorted out when comparing Eq. (2) with Eq. (1): (i) to simulate the cantilevered construction, the term (1 − e^{−(ϕt−ϕc)}) describing the creep behavior of a cantilevered beam has been included in Eq. (2); and (ii) the term ϕ_{t}/(1 + ρϕ_{t}) in Eq. (1) is replaced by the functional f(ϕ_{t}) = χϕ_{t}/(1 + χϕ_{t}) in Eq. (2) on the basis of the relaxation phenomenon.
Recently, following in previous researchers’ footsteps, Gendy and Rashed [23] adopted similar format of formerly established simplified formulae but introduced some optimizationderived combination factors. Timedependent moments along the bridge length are still extracted from ‘two’ conventional timeindependent analysis models. The first consists of a family of models representing the construction sequence, and a combined moment, ΣM_{O}, is then computed by superposing all partial moments, M_{Oj}, obtained from various successive construction steps using the statical system corresponding to each step ignoring timedependent effects. The second model then assumes all relevant loads and prestressing forces to be applied to the final structural system of the completed bridge after installing the closure segment, and the corresponding ‘onego’ moment, M_{C}, is hence calculated. The final timedependent moment, \({M}_{T,est.}\), can be thus estimated from linearly combining the two contributions of this couple of timeindependent (and simultaneously conducted) analyses as per Eq. (3):
where α and β are preassumed combination parameters purposely concocted to approximately simulate the timedependent effects. These linear combination coefficients α and β are always classically constrained to a unit sum, thus combining the two timeindependent moment terms in a manner similar to antecedent traditional formulae as illustrated by Eqs. (1) and (2). An optimization algorithm has been introduced by Gendy and Rashed [23] to cast the estimation of these parameters considering realworld two balanced cantilever prestressed concrete bridges. α and β are found sensitive to the time schedule of construction, and accordingly some recommended values for these parameters are given for different construction scenarios.
It has to be noted that the format traditionally adopted when linearly combining the two timeindependent moment terms–featuring either (i) some implicit combination coefficients that add up to a unity (viz. Eqs. (1) and (2)), or (ii) explicit combination coefficients with a unit sum such as α and β of Eq. (3)—results in final design timedependent creepaccounting moment “always” bounded by the values of the moments retrieved from the two timeindependent analyses illustrated above. Such anticipated trend as predictated by the classical linear combination format with “unit sum constrained” combination coefficients has been refuted by Pimanmas [24] for a realworld case study prestressed concrete segmental bridge in Thailand constructed by the balanced cantilever method. Contradicting general consensus among designers, the results presented in Pimanmas [24] show that the creep can increase the magnitude of hogging moment, rather than decreasing it as widely anticipated. As further illustrated by the author, this is usually the case if the prestressing amount in the top cables exceeds circa 70% of the ultimate tensile strength of the cables and is, as such, beyond a certain appreciable threshold value relative to the bridge dead weight effects. Accordingly, all simplified formulae (adopting the supposedly intuitive ‘constrained linear combination’ format) commonly known to practicing designers to estimate the longterm creepinduced moment as a fraction of (a) sequentially retrieved and superimposed dead load and prestressing moments from successive construction stages up to completion, and (b) moment in the continuous state from onego timeindependent analysis of the complete bridge, have been rebutted for balanced cantilever construction.
It is finally worth highlighting that simplified methods for determining timedependent moment redistribution in segmental prestressed concrete bridges through adhoc formulae combining results from different statical systems (viz. a series of ‘open’ bridge successive configurations literally following consecutive construction steps; and a ‘closed’, i.e., completed bridge) using some reasonably preselected/optimized weighting factors has emanated in the early literature for spanbyspan construction using movable scaffolding systems (Schlaich and Scheef [25], and Trost and Wolff [14], among others). Tweaking this simplified approach by numerous researchers and designers through fine tuning the ‘unit sumconstrained' combination coefficients to suit segmental prestressed concrete bridges constructed by the balanced cantilever method might be a bold extrapolation. Such step would have needed further careful investigations prior to considering the technique as sound and reliable, and hence holding the approach as the design method of choice in absence of specialized software able to conduct sophisticated stepbystep timedependent analysis accounting for creep effects. As a longawaited effort, and building upon earlier work by Pimanmas [24], the current research contributes towards assessing the robustness and validity of such simplified ‘linear combination’ formulae to determine design timedepended moments in realworld balanced cantilever prestressed concrete bridges.
Methods
The research primary aim is to assess the validity of typically adopted adhoc approaches that evolved over the years relying on simplified analyses and formulae to cater for timedependent creep effects for this type of bridges. Targeting realistic conclusions, three casestudy actual segmental balanced cantilever bridges over the Nile River in Egypt (namely, El Warrak, Girga and Talkha bridges) are considered. Midas Civil commercial package is used to perform timedependent finite element analyses for the three bridges. Main parameters considered are, inter alia, timedependent effects of creep and shrinkage of concrete, relaxation of prestressed steel, losses due to friction and anchor setting of prestressing tendons, sequence of construction, and constructiondriven temporary change of support conditions (where applicable).
Application to segmental prestressed concrete bridges constructed by the balanced cantilever method
Three case studies of realworld segmental prestressed concrete bridges over the Nile River in Egypt, namely, ElWarrak Bridge, Girga Bridge and Talkha Bridge constructed by the balanced cantilever method are considered herein. ElWarrak Bridge is located in Cairo Governorate, the capital of Egypt and constructed in 2001; Girga Bridge is located in Sohag Governorate in upper Egypt and constructed in 2012; and Talkha Bridge is located in Dakahlia governorate in the Nile delta area to the north of Cairo and constructed in 2011. The key target is to accurately determine the timedependent bending moments along these bridges accounting for all relevant longterm factors. Midas Civil commercial software is first used to perform a sophisticated timedependent finite element analysis for the three bridges. The actual construction schedule of each bridge available to the authors has been followed while performing the construction stages analysis. Eurocode models [26] for creep, shrinkage, compressive strength, relaxation of prestressed steel, and losses in prestressing tendons have been utilized for the timedependent analysis. In a complementary step and for comparison purposes, the simplified approach in Kwak and Son [17] has been adopted to determine some estimates for the timedependent moments along each of the case study bridges considering all relevant actual parameters, namely, properties of crosssections, ambient relative humidity, age at loading of each segment, time of closure of the structural system, etc. The main objective of the study is to critically compare results obtained from the simplified (i.e., combination formuladriven timeindependent) versus sophisticated (viz. real timedependent) analysis approaches, and accordingly assess the suitability of applying simplified approaches to this type of bridges constructed by the balanced cantilever method.
Case study Bridge 1: ElWarrak Bridge
Bridge description
ElWarrak Bridge is among the longest segmental prestressed balanced cantilever bridges over the Nile River in Egypt. The bridge has a total length of 380 m, with a maximum span length of 120 m (Fig. 2). The bridge consists of four spans, two 120 m navigational (center) spans and two 70 m side spans. It has eight lanes, i.e., four lanes for each traffic direction. The crosssection of the bridge consists of a fourcell hunched box girder; each direction has a doublecell box girder completely separated in the longitudinal direction of the bridge. The doublecell box has slanted webs and two side cantilevers providing a roadway of 22.65 m wide, as shown in Fig. 3. The girder depth varies over the length following seconddegree parabola from the maximum of 6.0 m over the piers to a minimum of 3.5 m at the middle of the main spans. The thickness of the bottom slab varies also parabolically from a maximum of 1.20 m at the piers to a minimum of 0.30 m at the middle of the main span. The bridge is built using the cantilever method. Three double cantilever parts are constructed independently, i.e., around the O, P, and Q axes. Each double cantilever is constructed in segments starting from the pier towards the center spans, i.e., one segment from the right end followed by another one from the left end, etc. Each segment is 5.0 m long except for three segments of length 2.5 m (i.e., one near each of piers O, P, and Q) to maintain/limit the unbalanced length between the two arms of the cantilevers during construction equal to 2.5 m. Two travelers, one at each tip of the two cantilevers, are utilized to erect segments. After the erection of each segment, it is posttensioned to the previous segment (i.e., the one located on the other end of the double cantilever) with “cantilever” tendons located in the top slab of the crosssection referred to as top prestressing cables. After closure at all transition piers (viz. piers at axes N & R) and at midspan, the entire bridge is prestressed with several additional “continuity” tendons in zones with high sagging (i.e., positive) moments. The longitudinal prestressing layout is shown in Fig. 4, each tendon consists of 12, 0.6inch (15.7 mm) diameter strands and is stressed from one end.
Actual construction sequence
The bridge is incrementally constructed as a balanced cantilever from piers O, P, and Q, as schematically depicted in Table 1. A set of 21 castinplace segments are installed sequentially at both sides of each of the two piers O and Q, while 22 segments are constructed around axis P. During construction, piers O and Q are temporarily fixed to the superstructure by means of vertical prestressing to avoid instability produced by unbalanced cantilever moment. Each segment is cast utilizing conventional formwork in 7 days, while the construction of each pier takes 90 days in addition to 30 days for the stump (i.e., pier table segments). On the two sides of the river, a 15.0mlong portion located at one end of each side span is cast on temporary scaffolding and then connected to the segmentally erected double cantilevers located at axes O and Q. After the side span hardens, 8 bottom tendons are locally tensioned in each side span. The tips of the cantilevers in the navigation spans are then connected by 5.0mlong closure segments. After the two closure segments harden, 24 bottom tendons are prestressed in each navigation span along with 6 more bottom tendons in each side span. The construction of the entire bridge takes 311 days. After all bottom continuity tendons are prestressed, the structure achieves its ‘final’ continuous form. Any load applied afterwards, namely, superimposed load including asphaltic wearing surface and safety barriers, traffic loads and other live loads will act on the closed bridge having a continuous structural system.
Finite element analysis of the bridge structure
A finite element analysis is conducted using Midas Civil (2021) commercial software. Figure 5 shows the threedimensional model of the bridge built to simulate the behavior during construction and throughout the bridge full service life accounting for the longterm timedependent creep effects. Numerical analysis is performed adopting a stepbystep strategy for segmental prestressed concrete bridges, considering each increase in the applied loads introduced by successively erecting each new segment and tensioning new tendons. The actual construction sequence of the bridge has been respected in the constructionstage analysis of the FE model. Eurocode specifications as per EN 199211(2004) and EN 19922 (2005) [26, 27], as well as CEBFIB [28] and AASHTO LRFD [29] provisions/models for comparison purposes, have been utilized to account for all relevant parameters that affect the timedependent analysis such as creep, shrinkage, compressive concrete strength, and losses of prestressing tendons.
Results and discussion
Sophisticated real stepbystep timedependent FE analysis
The design timedependent moment at any section along the bridge length is captured by a 3D timehistory stepbystep FE analysis literally following the construction sequence in Table 1. Final moments are obtained from the summation of the bending moments due to all permanent loads such as the selfweight of the bridge deck, the primary moment of prestressing steel and the secondary (i.e., parasitic/indeterminate) moment caused by the prestressing action, as well as the longterm effects due to creep and shrinkage of concrete and losses in the prestressing steel. For illustration purposes, Fig. 6 shows a breakdown of the bending moments for all applicable loads after 10,000 days. This duration has been assumed to appropriately capture and scrutinize the longterm effects. The ‘full’ (viz. combined) design timedependent moment (viz. summation of all itemized components depicted in Fig. 6) is provided in Fig. 8.
Simplified method per available linear combination formula by Kwak and Son (2002) [17]
The simplified Eq. (2) introduced by Kwak and Son [17] is used to find some estimate for the design timedependent moment along the bridge length. The bending moment values in the two combined terms of Eq. (2) refer to the results obtained from two conventional timeindependent analysis models as described above. The first series/family of models follow the actual construction sequence considering the structure’s statical system at each construction stage and then superposing all partial moments retrieved from successive steps/phases in order to compute ΣM_{s,i}. The second model assumes that all relevant loads and prestressing forces are applied in onego to the final statical system of the complete, i.e., ‘continuous’, bridge to calculate the moment, M_{E}. These timeindependent FE analyses have been also conducted using Midas Civil software package to estimate reasonable values of the timeindependent moments (ΣM_{s,i} and M_{E}) referred to in Eq. (2). In addition, relevant creep factors for each segment along the bridge length have been calculated: namely, ϕ_{c}, at the time at which the closure segment is installed, i.e., just after closing the bridge deck (t = c days) to achieve a continuous structural system; and ϕ_{t}, at the time considered herein to capture and inspect the longterm effects (i.e., t = 10,000 days). On the other hand, the functional f(ϕ_{t}) = χϕ_{t}/(1 + χϕ_{t}) in Eq. (2), as addressed in Kwak and Son [17], assumes values of 0.62, 0.64, and 0.65 at 1, 10, and 100 years, respectively [17,18,19, 22]. A value of 0.645 for f(ϕ_{t}) has been adopted herein corresponding to 10,000 days. For completeness, Table 2 illustrates the calculation of creep factors for all segments along the bridge length. Finally, Fig. 7 provides the estimated design timedependent moment, M_{T}, along the bridge length according to the simplified formula in Eq. (2), as well as its combinatory constituents (namely, ΣM_{s,i} and M_{E}) resulting from the two conventional timeindependent FE analyses. It is worth highlighting that as mathematically anticipated by the linear combination format of Eq. (2) with implicit combination coefficients summing up to 1.0, M_{T} values along the whole bridge length are always bounded by the two combined terms ΣM_{s,i} and M_{E}.
Comparative assessment of results
The main purpose of this section is to compare the results of the timedependent moments obtained from (i) sophisticated real stepbystep timedependent FE analysis versus (ii) simplified formula (viz. Eq. (2)) combining results from conventional timeindependent FE analyses. Figure 8 shows the design timedependent bending moments along the bridge length for the two approaches.
Results reveal, unexpectedly to the design community, a significant difference between the two approaches. In the twobalanced cantilever construction, as the amount of top prestressing increases for a particular bridge configuration, the magnitude of creep redistribution is highly likely to get reduced. This may be explained by the fact that this top (i.e., cantilever) prestressing introduced during the openbridge construction stages acts opposite to the bridge deck own weight and hence restricts creep movement produced by sustained selfweight stresses in the closed bridge structural system. Further increase in top prestressing amount could nullify creep induced by selfweight and may even reverse the creep redistribution since it moves up the cantilever tip against gravity. Accordingly, the prestressinginduced creep increases the design hogging (i.e., over support) moment magnitude (irrespective of the creep model used) in contradiction with traditional simplified formulae as shown in Fig. 8. For further demonstration, the moment due to creep along the prestressed bridge deck has been isolated and presented in Fig. 9 for various construction stages from the time of installation of the closure segment up to 10,000 days in order to display development of creep effects with time. The trend of results in figure seems to contradict widely anticipated timedependent creep effects in typical segmentally constructed RC bridges or in archetypal prestressed concrete bridges constructed by the spanbyspan erection methods. For completeness, a separate timedependent FE analysis is conducted but excluding prestressing in order to obtain the creep moment distribution along the bridge length due to the sustained selfweight stresses only such as for typical segmentally constructed RC bridges, and results are shown in Fig. 10. When comparing trends in Figs. 9 and 10, it is obvious that the top prestressing in balanced cantilever construction is responsible for limiting selfweight induced creep in the closed bridge configuration, and that some amount of such ‘cantilever’ prestressing beyond a given threshold could even reverse the bending moment redistribution trend along the bridge deck length due to creep.
On the other hand, the sagging (or loosely speaking ‘all withinspan’) moments are appreciably reduced relative to those estimated through the linear combination simplified formulae available in the literature as also revealed in Fig. 8. Another important note is that the final design moment including creep due to selfweight and prestressing longterm effects lies outside the range of values bounded by the two (referring to light grey curves in Fig. 8) timeindependent moment terms of the simplified formula, namely Eq. (2), at all crosssections along the whole length of the bridge.
Case study bridge 2: Girga Bridge
Girga Bridge is one of the longest segmental prestressed balanced cantilever bridges over the Nile River in Egypt. The bridge is 410 m in length, with a maximum span of 130 m (Fig. 11). The bridge consists of four spans, two 130 m navigational (center) spans and two 75 m side spans. It has four lanes, i.e., two lanes for each traffic direction. The superstructure consists of two parallel identical box girders which are completely separated along the longitudinal direction of the bridge. The crosssection of the box girder has vertical webs and two side cantilevers providing a roadway of 10.5 m wide, as shown in Fig. 12. The girder depth varies over the length following seconddegree parabola with a maximum depth of 6.9 m over the piers to a minimum depth of 3.5 m at the middle of the main spans. The thickness of the bottom slab varies also parabolically from a maximum of 1.23 m at the piers to a minimum of 0.25 m at the middle of the main spans. The bridge is built using the cantilever method. Three double cantilever parts are constructed independently, i.e., around the N13, N14, and N15 axes. Each double cantilever is constructed in segments starting from the pier towards the center spans, i.e., one segment from the right end followed by another one from the left end, etc. Two travelers, one at each tip of the two cantilevers, are utilized to erect segments. After the erection of each segment, it is posttensioned to the previous segment (i.e., on the other end of the double cantilever) with “cantilever” tendons located in the top slab of the crosssection referred to as top prestressing cables. After closure at all transition piers (viz. piers at axes N12 & N16) and at midspan, the entire bridge is prestressed with several additional “continuity” tendons in zones with high sagging (i.e., positive) moments. The longitudinal prestressing layout is shown in Fig. 13, each tendon consists of 12, 0.6inch (15.7 mm) diameter strands and is stressed from one end.
Actual construction sequence
The bridge is incrementally constructed as a balanced cantilever from piers N13, N14, and N15. A set of 23 castinplace segments are installed sequentially at both sides of each pier. The bridge is supported on sliding bearings at edge piers (namely, N12 and N16) while it is cast integrally (i.e., monolithically) with intermediate piers (viz. N13, N14, and N15). Each segment is cast utilizing conventional formwork in 7 days, while the construction of piers takes 90 days in addition to 30 days for the stump (i.e., pier table segment). On the two sides of the river, a 12.5mlong portion located at one end of each side span is cast on temporary scaffolding and then connected to the segmentally erected double cantilevers located at axes N13 and N15. After the side span hardens, 6 bottom tendons are locally tensioned in each side span. The tips of the cantilevers in the navigation spans are then connected by 5.0mlong closure segments. After the two closure segments harden, 26 bottom tendons are prestressed in each navigation span along with 10 more bottom tendons in each side span. The construction of the entire bridge takes 325 days. After all bottom continuity tendons are prestressed, the structure achieves its ‘final’ continuous/complete form Fig. 14 shows the threedimensional model of the bridge built to simulate the behavior during construction and throughout the bridge full service life accounting for the longterm timedependent creep effects.
Results of the two analysis approaches and comparative assessment of results
Same analysis schemes adopted for ElWarrak Bridge have been also followed for Girga Bridge. In addition, relevant creep factors (ϕ_{c} and ϕ_{t}) for each segment along the bridge length have been calculated according to EN 199211:2004 [26] but not included herein for space limitations; refer to Rashed [30] for a set of complete data. Figure 15 provides the estimated design timedependent moments along the bridge length according to the simplified formula in Eq. (2), as well as its constituents (namely, ΣMs,i and M_{E}) resulting from the two conventional timeindependent FE analyses as demonstrated before. On the other hand, the second approach (actual timedependent analysis) is adopted by performing a sophisticated time history FE analysis using Midas Civil considering the actual construction sequence of the bridge as well as all relevant parameters that affect the timedependent analysis. The timedependent moments along the bridge after 10,000 days are accordingly obtained.
Finally, Fig. 16 illustrates the design timedependent bending moments along the bridge length according to the two approaches for comparative purposes. Results come in line with those presented for ElWarrak Bridge despite differences in concrete dimensions, boundary conditions, prestressing arrangement, etc. between Girga and ElWarrak bridges. As it is clear from Fig. 16, there are again some remarkable variations in the results recuperated from the simplified (through combining timeindependent results) versus the sophisticated (via real timedependent) analysis approaches.
Case study bridge 3: Talkha Bridge
Talkha Bridge is constructed across Damietta branch of the Nile River to connect the north and south parts of the Nile at Talkha City which is located in the Delta of Egypt. The final bridge structure consists of three continuous spans, i.e., two side spans and one main (navigation) span as shown in Fig. 17. Each side span is 65.0 m long, and the main span is 100.0 m long. The bridge hosts four lanes, two lanes for each traffic direction. The superstructure consists of two parallel identical box girders which are completely separated along the longitudinal direction of the bridge. The crosssections of the box girder at mid navigation span and at piers are shown in Fig. 18. The width of the top slab is 10.5 m, while the width of the bottom slab varies from 4.70 m at the midspan to 3.80 m at the piers. The height of the crosssection varies parabolically from 3.0 m at midspan to 5.50 m at the piers. The bridge deck is supported on longitudinally sliding bearings at edge piers while it is cast integrally with intermediate piers. The longitudinal prestressing layout is shown in Fig. 19. Each prestressing tendon consists of 12 strands. The strand is a 15.2mm diameter 7wire low relaxation type. The number of top “cantilever” tendons is 38 around each pier, while the number of bottom “continuity tendons is 18 in the main span and 14 in each side span. The tendons are initially stressed to 75% of the ultimate tensile strength (UTS).
Actual construction sequence
The bridge is incrementally constructed as a balanced cantilever from piers P2 and P3 such that two parts with equal weight can be constructed at the same time around each pier. A total of four castinplace segments are located at each side of the two piers P2 and P3. Each segment is cast utilizing conventional formwork in 30 days, while the construction of each pier takes 90 days. After constructing each segment, sufficient tendons are tensioned in the top slab (known as ‘cantilever’ prestressing) to carry its own weight, and then the formwork is removed. At the two sides of the river, a 17.0mlong stretch at the end of each side span is cast on temporary scaffolding and is hence connected to the near end of the double cantilevers extending from piers P2 and P3. After the integrated side span hardens, 6 bottom tendons are tensioned in each. The construction of each side span takes 30 days. On the other hand, the tips of the cantilevers in the navigation span are then connected by a 5.0 m closure segment. After the closure segment hardens, 18 bottom ‘continuity’ tendons are prestressed in the navigation span, along with 8 more bottom tendons in each of the two side spans. The construction of the entire bridge takes 255 days. After all bottom tendons are prestressed, the structure achieves a continuous form and is ready to accept any superimposed load including asphaltic wearing surface and safety barriers, traffic, and other live loads expected on the completed bridge Fig. 20 shows the threedimensional model of the bridge built to simulate the behavior during construction and throughout the bridge full service life accounting for the longterm timedependent creep effects.
Results and comparative assessment
Being majorly different in various aspects from previously analyzed ElWarrak and Girga (quite comparable) bridges, Talkha Bridge has been deliberately selected by the authors to serve as a third casestudy to scrutinize earlier observations made on results obtained for the former two bridges. Talkha Bridge has the following distinctive characteristics when compared to both ElWarrak and Girga bridges, namely: (a) it consists of three instead of four spans with only one main (navigation) span; (b) it has different construction conditions and sequence, where each segment is constructed on a movable scaffolding system mounted on removable steel piles which made it possible to use segments with longer length (viz. 12.5 m) relative to the typical castinsitu segments of about 5.0 m commonly adopted in balanced cantilever construction; and (c) it features different amount and arrangement of prestressing along the bridge length dependent on its unique segments’ length. Results of the simplified approach according to Eq. (2) are shown in Fig. 21 in the form of the estimated final timedependent moment, M_{T}, as well as its constituents (namely, ΣMs,i and M_{E}) resulting from the two simplified timeindependent FE analyses. Reference could be made to Rashed [30] for relevant creep factors for each segment along the bridge length. On the other hand, the second more precise approach (conducting a real timedependent analysis) is also accomplished through a sophisticated time history FE analysis using Midas Civil based on actual construction sequence as well as relevant timedependent parameters. Final timedependent design moments along the bridge after 10,000 days from end of construction are accordingly computed. Figure 22 illustrates the design timedependent bending moments along the bridge length according to the two approaches. Despite the prime differences between the characteristics of Talkha Bridge and those of ElWarrak and Girga bridges, results of the three casestudy bridges are still fairly alike. Accordingly, one could ascertain that simplified analysis approaches commonly adopted in the literature through combining results from timeindependent analyses would definitely lead to considerable and serious errors in estimating design timedependent moments in bridges constructed using the balanced cantilever technique.
For a useful visualization of the results, a final quantitative comparison of main values of the timedependent bending moments due to permanent loads at selected key critical sections for the three studied bridges according to the two (simplified versus sophisticated) design approaches adopted herein is illustrated in Fig. 23. It may be clear from results presented that by adding the bending moments from superimposed dead loads and live loads acting on the final completed bridge system, the expected design “total” sagging moment near mid span and the design “total” hogging moment near pier would be on the unconservative side if the creepinduced permanent loads’ moments (referring to Fig. 23) are computed based on the simplified timedependent analysis approach.
Conclusions
The primary focus of this research is to assess the validity and reliability of adhoc approaches readily available in the literature and commonly used by the design community for the analysis of prestressed concrete segmental bridges constructed by the balanced cantilever technique. Such approaches have evolved over the years relying on simplified analyses/formulae to cater for timedependent creep effects on the moment redistribution along the bridge length in light of the lack—until very recently—of specialized commercial analysis software packages able to conduct rigorous and robust actual stepbystep timedependent analyses. To provide realistic conclusions, three distinct casestudy realworld segmental balanced cantilever bridges over the Nile River in Egypt are elected. Relying on critical and comparative review of the results captured by both (a) simplified analyses/formulae and (b) real timedependent time history analysis, some valuable observations have been pinpointed as in the body of the paper, and the following key conclusions have been drawn:

1.
Simplified formulae commonly validated and endorsed by several researchers and designers for creepinduced moment redistribution in typical segmentally constructed RC bridges or prestressed concrete bridges constructed by the spanbyspan erection methods may lead to considerable errors if instead applied to estimate the design timedependent longterm moments in balanced cantilever bridges.

2.
Unexpectedly to the design community, the final design moment including creep due to selfweight and prestressing longterm effects at all crosssections along the whole length of a prestressed bridge constructed by the balanced cantilever technique lies outside the range of values bounded by the two timeindependent moment terms referred to in the simplified formulae readily available in the literature.

3.
The exceptionally large amount of top (i.e., cantilever) prestressing typically used in the balanced cantilever segmental bridges to carry the segments’ own weight in the ‘open’, i.e., during construction, configuration generally restricts creep movement produced by sustained selfweight stresses in the ‘closed’ bridge structural system. Accordingly, the prestressinginduced creep increases the design hogging (i.e., over support) moment magnitude in contradiction with traditional simplified formulae. This actual behavior (and the subsequent rise in the hogging moment) could never be captured by the explicit linear combination format of these formulae since as mathematically anticipated, such equation form shall reveal an estimated final design timedependent moment along the bridge length always bounded by the two timeindependent moment terms linearly combined through combination coefficients constrained to a unit sum.

4.
The sagging (or generally speaking, ‘all within span’) timedependent bending moments precisely accounting for creep due to selfweight and prestressing longterm effects through real time history analysis are appreciably reduced relative to their counterpart values estimated through the linear combination simplified formulae available in the literature.
Further investigations involving other realworld prestressed concrete bridges constructed using the balanced cantilever techniques and featuring different span configurations, crosssection properties, amount and arrangement of prestressing cantilever and/or continuity tendons, construction duration, etc. shall be performed in order to decide whether the above conclusions shall be retained or further refined.
Availability of data and materials
The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.
Abbreviations
 M _{T} :

Estimated timedependent moment
 M _{E}, M _{C} :

Moment obtained by assuming the entire structure constructed at the same stage
 ΣMs,i, ΣM_{O j} :

Combined elastic moments that occur at successive construction steps
 ϕ :

Creep coefficient
 ρ:

Relaxation factor
 χ:

Concrete ageing coefficient
 α:

Contribution of ΣM_{s,i} in estimating timedependent moment
 β:

Contribution of ΣM_{E} in estimating timedependent moment
 t_{o} :

Age of segment at loading
 t_{1} :

Age at segment placement
 t_{2} :

10,000 Days (long term)
 h_{o} :

Notional size of the member (segment)
 ϕ _{ o } :

Notional creep coefficient
 ϕ _{ RH } :

Factor to allow for the effect of relative humidity on the notional creep coefficient
 β(t _{ o } ) :

Factor to allow for the effect of concrete age at loading on the notional creep coefficient
 β _{ H } :

Coefficient depending on the relative humidity
 βc(t,t_{0}):

Coefficient to describe the development of creep with time after loading
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Acknowledgements
The authors are very grateful to Dr. Samer Youakim of Dar AlHandasah Bridge Department, Cairo, Egypt, for the valuable discussions and for his fruitful comments throughout the research.
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MR collected the data, created the analysis models using Midas Civil, analyzed the results, and wrote the paper. SM guided, supervised, evaluated the results, and assisted MR in writing the paper. All authors read and approved the final manuscript.
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Rashed, M., Mehanny, S.S.F. Simulating creep induced moment redistribution in prestressed concrete bridges constructed by the balanced cantilever method: ad hoc traditional formulae versus real timedependent analysis. J. Eng. Appl. Sci. 70, 28 (2023). https://doi.org/10.1186/s44147023001994
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DOI: https://doi.org/10.1186/s44147023001994