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Numerical modelling of continuous concrete box girder bridges considering construction stages
Applied Mathematical Modelling 35 (2011) 3809–3820
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Numerical modelling of continuous concrete box girder bridges considering construction stages Sevket Ates ⇑ Karadeniz Technical University, Department of Civil Engineering, 61080 Trabzon, Turkey
a r t i c l e
i n f o
Article history: Received 17 December 2009 Received in revised form 27 January 2011 Accepted 2 February 2011 Available online 23 February 2011 Keywords: Bridges Construction stage Numerical modelling Time dependent material properties The balanced cantilever method
a b s t r a c t The aim of this study is to analyse the concrete continuous box girder bridges by considering segmentally construction stages through balanced cantilever method. Timedependent material properties of concrete and steel are also taken into account. Budan Bridge is selected as a numerical example. The Bridge constructed with balanced cantilever method and located on Artvin–Erzurum highway, Turkey, at 55 + 729.00–56 + 079.00 km is modelled using SAP2000 program. Geometric nonlinearities are taken into consideration in the analysis using P-Delta and large displacement criterion. Time-dependent material properties are considered as compressive strength, aging, shrinkage and creep for concrete, and relaxation for steel. The structural behaviour of the bridge at different construction stages is examined. Variation of internal forces such as bending moment, shear forces and axial forces, and displacements for bridge deck and pier are given with detail. Analyses show that, to obtain real behaviour of concrete bridges, segmentally construction stage analysis using time dependent material properties and geometric nonlinearity should be considered, because construction period continue along time and loads may change during this period and after. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction The use of continuous concrete box girder bridges has increased recently. In construction of this type bridges having constant or variable section height, the cantilever method can be applied. Box girder section forms consist of single or more box girder based on bridge wide. The cantilever method is considered as the natural and logical solution in construction of box girder bridges. There are two basic alternatives in the cantilever method: one is single cantilever method and the other is the double cantilever method. In the former, the side span girders of the bridge are constructed on interim piers and afterwards the stiffening girder in main span is constructed by one-sided free cantilevering until the span centre or the anchor pier on the far end is reached. In the latter, the bridge girder is constructed from both side of the tower towards the anchor piers and the main span centre by double-sided free cantilevering. The double cantilever method is also called as the balanced cantilever method. The method is especially recommended where scaffolding is difficult or impossible to construct over deep valleys, wide rivers or in case of expensive foundation conditions for scaffolds. In this method, bridges are built from one or more piers by means of formwork carriers. Normally the structure advances from a short stub on top of a pier symmetrically in segments of about 3–6 m length to the mid span or to an abutment, respectively. Each cantilevered part of the superstructure is tied to a previous one by concreting a key segment and post-tensioning tendons. The prestressing tendons are arranged based on the moment diagram of a cantilever. ⇑ Tel.: +90 462 377 26 69; fax: +90 462 377 26 06. E-mail address: [email protected] 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.02.016
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In recent years, many interesting research topics have arisen such as to be taken in to account segmentally construction stages in the analysis. Normally, structures are analysed by assuming that they are instantly built in a time. However, this type of analyses may be give unreliable results which compared with those obtaining from that construction stage is considered. In the construction stage analysis, time dependent material properties should be taken into account. Several studies have dealt with the analysis of segmentally constructed bridges, as long as a few studies have been struggled the analysis of the deflection and internal moment redistribution in bridges [1–3]. Abbas and Scordelis [4] achieved nonlinear geometric, material and time dependent analysis of segmentally erected three-dimensional cable stayed bridges. Cruz et al. [5] presented a general step by step model for the nonlinear and time dependent analysis of reinforced, prestressed concrete, and composite steel-concrete planar frame structures. An analytical model for the nonlinear and time dependent analysis of segmentally constructed three dimensional concrete frames was studied by Mari [6]. Kwak and Son [7,8] studied on span ratios in bridges constructed using the balanced cantilever method and reported that moment variation due to the change in structural system during construction requires a rigorous time dependent analysis that considers the construction stages. Wang et al. [9] carried out the analysis of cable stayed bridges at different stages during construction by the cantilever method. Somja and Goyet [10] came up with an efficient numerical procedure for materially and geometrically nonlinear finite element analysis of segmentally erected structures including time dependent effects due to load history, creep, shrinkage and aging of concrete. In that study, it was observed that time have a strong influence, especially, on concrete type structures. Therefore, it was emphasized that these effects must be taken into account in the design process. Cho and Kim [11] evaluated the risks in a suspension bridge by considering an ultimate limit state for the fracture of main cable wires. They examined the results compared with the conventional safety indices and allowable error for the control of deformations during construction. Altunisik et al. [12] carried out the construction stage analysis of Komurhan Bridge using time dependent material properties of concrete and steel. They reported that the construction stage analysis had remarkable effect on the structural behaviour of the bridge. Malm and Sundquist [13] studied the time-dependent analyses of segmentally constructed balanced cantilever bridges. For concrete bridges, time dependent effects make analyses even more complex. The effects develop at the early stages of the construction process and continue to evolve considerably after the concrete bridges are built. Depending on the construction method, the time dependent effects can appear and induce important stress redistribution in the structure. As seen in literature, studies on construction stage analysis are meagre and need to be enlarged by inserting new studies. In the light of aforementioned researches, construction stage analysis of a bridge constructed with balanced cantilever method using time dependent material properties is performed in this paper. Time dependent material properties are considered as compressive strength, aging, shrinkage and creep for concrete, and relaxation for steel. 2. Description of example bridge
6.00m
3.00m
3.00m
Fig. 1. The Budan Bridge and its dimensions.
56+079.00
92.50m
97.17m
55+986.50
55+821.50
165.00m
91.77m
92.50m
6.00m
55+729.00
The Budan Bridge, shown in Fig. 1, is a reinforced continuous concrete box girder type bridge located on between Artvin and Erzurum highway, Turkey, at 55 + 729.00–56 + 079.00 km. Construction of the bridge started in 2007 and completed recently. The configuration of the bridge is a three-span, cast-in-place concrete box girder superstructure supported on reinforced concrete piers. The bridge deck consists of a main span of 165.00 m and two side spans of 92.50 m each. The total bridge length is 350.00 m and width of bridge is 15.00 m. The structural system of the bridge consists of a continuous deck, two piers, two abutments and a closure segment. The piers have the heights of 91.77 m at the 55 + 821.50 km and of 97.17 m at the 55 + 986.50 km. The cross-sections of the piers and the deck are given in detail in Fig. 2. The cross-section of the deck
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15.00m
6.00m
2.50m-9.50m
1.20m
1.20m
8.00m
1.20m
1.20m 4.57m
a
5.86m
4.57m
b
Fig. 2. Cross-sections of (a) piers and (b) deck of the Budan Bridge.
consists of a single cell box girder with cantilevered slabs. The girder depth varies from a maximum of 9.50 m over the piers to a minimum of 2.50 m at the mid span and abutments. The closure segment in the middle of the central span is 1.50 m long. The bottom slab thickness is variable, as well. In order to investigate the construction stage response effects on the Budan Bridge, three-dimensional finite element model is used. The primary objective of this study is to perform a parametrical study associated with the construction stages and its effects on the response of continuous concrete box girder bridges. However, soil–structure interaction is not considered. The finite element model of the Budan Bridge is shown in Fig. 3. Analyses are performed by using SAP2000 [14]. The finite element model of the bridge consists of solid elements having 3° of freedom at each nodal point. They are in two horizontal and vertical translational directions. The three dimensional finite element models are represented with view in Fig. 3. The model has 3764 nodal points and 1710 solid area elements. In addition, 20 frame elements are used to represent piers. Additionally, the modulus of elasticity of concrete is taken according to Fig. 6. The relative humidity is 60% and the cement type is normal. The prestressing steel yielding point is fyk = 500 MPa and the steel modulus of elasticity is Es = 200,000 MPa. The parameters related to time dependent material properties is given in Table 1. Geometric nonlinearity is taken into consideration using P-Delta and large displacement criterion in the staged construction analysis. 3. Construction stages Bridges constructed with the balanced cantilever method consist of main structural elements such as deck, piers and side abutments. The balanced cantilever segmental construction has long been known as one of the most efficient methods due to
Plan
Three-dimensional view Fig. 3. Plan and three-dimensional view of the finite element model of the Budan Bridge.
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Table 1 Parameters used in the analysis for time dependent material properties. Parameters
Deck
Pier
Steel
Cement type, s Relative humidity, RH (%) Notional size, h (mm) Shrinkage coefficient, Bsc Relaxation
0.25 60 0.600 5 –
0.25 60 1.306 5 –
– – – – Class 1
Closure Segment
2@3m=6m 3@4m=12m
12@5m=60m
Constructed deck segments
Abutment
7.5m
3@4m=12m
12@5m=60m
1.5m
55+821.00
55+729.00
1.5m 10.75m
Pier
Constructed deck segments
Pier
Abutment
Fig. 4. Construction stages of the Budan Bridge and construction stages as per the balanced cantilever method.
the fact that the method does not need any falsework to build a bridge. That is why among the construction methods, the mentioned balanced cantilever construction of reinforced concrete box girder bridges has been preferred. In this method, precast segmental construction and cast-in-place construction techniques can be executed. There are some advantages and disadvantages of these techniques; Cast in-place construction may permit a rate of one pair of segments 3–6 m long to be constructed and stressed every 4– 7 days. On the average, a pair of travellers permits the completion of approximately 50 m of bridge deck within a month, excluding the transfer from pier to pier and fabrication of the pier table. Cast-in-place cantilever construction is basically a slow process, while precast segmental construction with matching joints is among the fastest. Precast segmental construction for long, repetitive structures may be more economical than a cast-in-place solution. Precast segmental solutions are limited by the capacity of transportation and placing equipment. Segments exceeding 2500 kN are seldom economical. Cast-in place construction does not have the same limitation, although the weight and cost of the travellers are directly proportional to the weight of the heaviest segment. In the segmental construction, some limitations arise in the form of the curvature of the bridge and the size and weight of the segment. Both precast and cast-in-place segmental construction permits all work to be performed at the top. The Budan Bridge was constructed with cast-in-place construction technique is selected as an example. Firstly, piers and small part of bridge deck are constructed over substructure using suitable formwork. Then, segments of 3–6 m length as shown in Fig. 3 are erected on opposite sides of each pier to balance the loads by using a movable form carrier. After the concreting, prestressing tendons are inserted into the segments and stressed with post-tension. Finally, form carrier is moved to the next position and a new cycle starts. This sequence is completed at one week on average and is going on until bridge decks meet at mid span. At the mid span, closure segment is established to complete one span. Because of the fact that maximum displacements are occurred at this point after finite element analysis, construction of this segment is very important. The typical operation sequence is summarized as: carrier is firstly set up and adjusted, and aligned form of deck, after the stage placing reinforcement and tendon ducts are achieved. Concreting is fulfilled and prestressing tendons in the seg-
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ment are then inserted. In due course, the formwork is removed. The form carrier to the next position is moved in order to start a new cycle. The construction process of each segment is repeated until the bridge is completed. The schematic view mentioned above of balanced cantilever construction of the prestressed concrete highway bridge is given in Fig. 4. In the construction stage analysis, added and removed loads for each construction stages should be determined. In order to obtain the reliable solution, each stage results should be added to end of the each stage and next stage analysis is done. Additionally, nonlinear solution parameters should be selected depending on the literature. In the finite element analysis of the Budan Bridge, a total of 39 segments at the deck and 60 segments at the two piers are considered. To close the centre span, a segment of 1.50 m is used, while two end segments of the side spans of 10.75 m are required to reach the abutments. Over the piers, segments of 7.50 m lengths are placed. A total of 100 prestressing tendons, each of 19 strands with 15.2 mm, stressed with an initial force of 4052.45 kN is used. Additionally, the reinforcing steel is uniformly distributed over the cross-sections with an amount of 0.02% in the girder and 0.1% in the piers. A time dependent analysis of the balanced cantilever construction procedure and 10,000 days (27.4 years) after completion of construction of the selected bridge is executed. However, the construction period was of nearly eight months. During the construction of the deck; concreting of a pair of new segments, a waiting period of three days, and at the rest of 4 days stressing of tendons and shifting of travelling formworks are simulated. The iterative calculations at each construction stage considering added stiffness from the initial equilibrium state. The element stiffness matrix is given the following equation
½K ¼ ½K e þ ½K g ;
ð3:1Þ
where ½K e is the elastic stiffness matrix and ½K g is the geometric stiffness matrix. The expression for ½K g is derived in detail by Przemieniecki [15] and Abbas [16]. The finite element analysis is performed at each construction stages of the bridge by using SAP2000 [14]. In the analysis of the bridge, the following load cases are considered; (a) Dead load: Weight of all elements.
(b) Additional mass: Weight of the asphalt, cobble, pipeline and its supports, scarecrow. (c) Gantry: Load of the form carrier. This load is implemented to previous one before the construction one segment and slide next one after construction of the segment. Then, the gantry load is wholly removed from the constructed parts. (d) Diaphragm: Weight of the reinforced concrete walls at the abutments and at both sides of the expansion joints. (e) Pre-stress: Post-tension cables are modelled using cable elements. Post-tension loads are considered as strain. (f) Jack: Load of the jack applied to the side segments before fixed to the side abutments. (g) Temperature: This load is applied to consider temperature variation due to the climate. +30 °C and 20 °C temperature (according to the bridge region) are considered separately in the analysis. 4. Time dependent properties for concrete 4.1. Compressive strength The compressive strength of concrete at an age t depends on the type of cement, temperature and curing conditions. The relative compressive strength of concrete at various ages may be estimated by the following formula [17]:
fcm ðtÞ ¼ bcc ðtÞfcm ;
ð4:1Þ
in which bcc ðtÞ is a coefficient with depends on the age of concrete and is calculated by:
("
28 bcc ðtÞ ¼ exp s 1 t=t 1
1=2 #) ;
ð4:2Þ
fcm ðtÞ is the mean concrete compressive strength at an age of t days; fcm is the mean compressive strength after 28 days; t is the age of concrete in days; t1 = 1 day; and s is a coefficient with 0.20, 0.25 and 0.38 which type of cement. In the light of the Eqs. (4.1) and (4.2), the mean concrete compressive strength is given in Fig. 5. 4.2. Aging of concrete The modulus of elasticity of concrete changes with time. For this reason, the modulus at an age t – 28 days may be estimated as below equation [17]:
Eci ðtÞ ¼ Eci
pffiffiffiffiffiffiffiffiffiffiffiffi bcc ðtÞ;
ð4:3Þ
where Eci ðtÞ is the modulus of elasticity at age of t days; Eci is the modulus of elasticity at an age of 28 days; bcc ðtÞ is a coefficient which depends on the age of concrete. For the deck and the piers of the example bridge, the aging of concrete is plotted in Fig. 6.
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Mean Strength (kN/m²)
4E+004
3E+004
2E+004
1E+004
0E+000 0E+000
2E+003
4E+003
6E+003
8E+003
1E+004
Time (Day)
Modulus of Elasticity (kN/m²)
Fig. 5. Variation of the mean concrete compressive strength with days.
5E+007 4E+007 3E+007 2E+007 Deck Pier
1E+007 0E+000 0E+000
2E+003
4E+003
6E+003
8E+003
1E+004
Time (Day) Fig. 6. Aging of concrete in days.
4.3. Shrinkage of concrete The CEB-FIP Model Code [17] gives the following equation of total shrinkage strain of concrete:
ecs ðt; ts Þ ¼ ecso bs ðt ts Þ;
ð4:4Þ
where ecso is notional shrinkage coefficient; bs is the coefficient to describe the development of shrinkage with time; t is the age of concrete in days and t s is the age of concrete in days at the beginning of shrinkage. The notional shrinkage coefficient may be obtained from:
ecso ¼ es ðfcm ÞbRH ;
ð4:5aÞ
es ðfcm Þ ¼ 160 þ 10bsc 9
fcm fcmo
ð4:5bÞ
;
where fcm is the mean compressive strength of concrete at the age of 28 days in MPa; fcmo is taken as 10 MPa; bsc is a coefficient ranging from 4 to 8 which depends on the type of cement.
bRH ¼ 1:55bsRH
40% 6 RH < 90%
bRH ¼ 0:25
RH P 99%
;
ð4:6Þ
where
bsRH ¼ 1
RH RHo
3 ;
ð4:7Þ
with RH is the relative humidity of the ambient atmosphere (%) and RHo is 100%. The development of shrinkage with time is given by:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðt ts Þ=t1 ; bs ðt t s Þ ¼ 350ðh=ho Þ þ ðt t s Þ=t 1
ð4:8Þ
where h is the notional size of member (mm) and is calculated by h ¼ 2Ac =U in which Ac is the cross-section and U is the perimeter of the member in contact with the atmosphere; ho = 100 mm and t1 = 1 day. For the deck and the piers of the
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4E-004
Shrinkage Strain
3E-004
2E-004 Deck Pier
1E-004
0E+000 0E+000
2E+003
4E+003
6E+003
8E+003
1E+004
Time (Day) Fig. 7. Time dependent shrinkage strain of concrete.
example bridge, the shrinkage strain of concrete depending on relative humidity, notional size and shrinkage coefficient is depicted in Fig. 7. 4.4. Creep The effect is calculated using creep model [17]. For a constant stress applied at time to, this leads to:
ecc ðt; to Þ ¼ in which
rc ðto Þ
/ðt; t o Þ;
Eci
ð4:9Þ
rc ðto Þ is the stress at an age of loading to; /ðt; to Þ is the creep coefficient and is calculated from:
/ðt; t o Þ ¼ bc ðt to Þ/o ;
ð4:10Þ
where bc is the coefficient to describe the development of creep with time after loading; t is the age of concrete in days at the moment considered; to is the age of concrete at loading in days. The creep coefficient is explained by:
/o ¼ /RH bðfcm Þbðt o Þ; 1
/RH ¼ 1 þ
0:46
RH RHo
1=3 ;
ð4:11bÞ
h ho
5:3 bðfcm Þ ¼ qffiffiffiffiffiffi ; fcm fcmo
bðto Þ ¼
ð4:11aÞ
1 0:2 : 0:1 þ tt1o
ð4:11cÞ
ð4:11dÞ
All parameter is defined above. The development of creep with time is given by:
bc ðt t o Þ ¼
ðt to Þ=t 1 ; bH þ ðt t o Þ=t 1
( 18 ) RH h bH ¼ 150 1 þ 1:2 þ 250 6 1500; RHo ho
ð4:12aÞ
ð4:12bÞ
where t1=1 day; RHo=100 and ho=100 mm. In the analysis, the creep coefficient of concrete is given in Fig. 8 for the deck and the piers having different notional size. 5. Time dependent properties for steel According to CEB-FIP Model Code [17], relaxation classes referring to the relaxation at 1000 h are divided into three groups for prestressing steels. The first relaxation class is defined as the normal relaxation characteristics for wires and strands, the second class is defined as improved relaxation characteristics for wires and strands, and the last one is defined as relaxation characteristics for bars. For an estimate of relaxation up to 30 years the following formula may be applied:
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Creep Coefficient
3
2
1
Deck Pier
0 0E+000
2E+003
4E+003
6E+003
8E+003
1E+004
8E+003
1E+004
Time (Day) Fig. 8. Time dependent creep coefficient.
Relaxation Coefficient
0.25 0.20 0.15 0.10 0.05 0.00 0E+000
2E+003
4E+003
6E+003
Time (Day) Fig. 9. Time dependent relaxation coefficient of prestressing steel.
qt ¼ q1000
k
t 1000
ð4:13Þ
;
where qt is the relaxation after t hours; q1000 is the relaxation after 1000 h; k logðq1000 =q100 Þ in which k to be 0.12 for relaxation class 1, and 0.19 relaxation class 2; q100 is the relaxation after 100 h. Normally, the long-term values of the relaxation are taken from long-term tests. However, it may be assumed that the relaxation after 50 years and more is three times the relaxation after 1000 h. In the example bridge, all tendons compose of 19 strands with u15.2 mm ant they are categorized as the normal relaxation characteristics for wires and strands, so the relaxation class is taken as class1. The relaxation coefficient as per mentioned properties is given in Fig. 9.
6. Numerical results Fig. 10 gives the bending moments along the bridge deck in case the construction stage procedure is considered and is not considered, respectively. As seen the figure, the bending moments obtained from consideration of the construction stage are significantly bigger than these of inconsideration of the construction stage. The deck bending moments, if the construction stage is considered, increase around 80% if compared with the results obtained the construction stage is not considered in
-4E+005
Bending Moment (kNm)
Construction Stage
-3E+005
Considered Not Considered
-2E+005 -1E+005 0E+000 1E+005
0
50
100
150
200
250
Length (m) Fig. 10. Bending moments of the deck of the bridge.
300
350
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the analysis. The results clearly show that, in the event of consideration of construction stage analysis, the deck bending moments substantially increase. Besides, if the variation of the bending moment related to stage by stage is drawn at the support points at the 55 + 821.00 km on the deck as in Fig. 11, the same results can be comprehended. On the other hand, shear forces and axial forces have not the same differences like bending moments. This is clearly seen in Figs. 12 and 13. The deck displacements are compared in Fig. 14. The vertical displacements considerably increase in case of the fact that construction stage analysis is taken into account in the time dependent analysis. It is also produced that displacements have an increasing trend towards the middle of the bridge deck. Variation of maximum displacements along the height of the piers of the selected bridge is shown in Fig. 15. It can easily be seen that the horizontal displacement increase with the pier height of bridge in view of differences between the two methods. The differences are reached to approximately 18 cm at the top of bridge pier. Bending moments change with height of the bridge pier as shown in Fig. 16. It can be easily seen that the bending moments obtained from that the construction stage analysis is considered have bigger results than the responses for that construction stage analysis is not considered. Fig. 17 points out the shear forces on the piers of the bridge for the mentioned two methods. The values of the axial forces are nearly equal along the height of the bridge piers as shown in Fig. 18. However, the values of the shear forces have a difference as 30%.
Bending Moment Variation (kNm)
-4E+005 -3E+005 -2E+005 -1E+005 Construction Stage
0E+000
Considered Not Considered
1E+005 0
50
100
150
200
250
Stage Fig. 11. Variation of the bending moments at the support.
3E+004
Construction Stage
Shear Force ( kN)
2E+004
Considered Not Considered
1E+004 0E+000 -1E+004 -2E+004 -3E+004 0
50
100
150
200
250
300
350
300
350
Length (m) Fig. 12. Shear forces of the deck of the bridge.
4E+004
Axial Force ( kN)
0E+000 -4E+004 -8E+004 Construction Stage
-1E+005
Considered Not Considered
-2E+005
0
50
100
150
200
250
Length (m) Fig. 13. Axial forces of the deck of the bridge.
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Displacement (m)
0.00
-0.10
-0.20
Construction Stage
-0.30
Considered Not Considered
-0.40 0
50
100
150
200
250
300
350
Length (m) Fig. 14. Vertical displacements of the deck of the bridge.
100
Pier Height (m)
75
50
25
Construction Stage Considered Not Considered
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0
Displacement (m) Fig. 15. Horizontal displacements of the piers of the bridge.
7. Conclusions This study outlines an investigation about the construction stage analysis of the concrete continuous box girder bridge including time dependent effects such as creep, shrinkage and aging of the concrete and relaxation of the steel. The time dependent analyses are carried out with and without considering construction stages of the bridge model, separately. The assumed material and cross sectional properties are taken from a real bridge. The maximum and minimum response values of the bridges are compared with each other. The results obtained from this study can be given as: The bending moments, while the construction stage is considered, are significantly greater than those of inconsideration of the construction stage. The deck bending moments are increased around 80% in case of consideration of construction stages in the analysis. The results clearly show that consideration of construction stage in the analysis substantially increases the deck bending moments. Besides, the bending moment variation in case of considering stage by stage construction gives the same results. When the results of the construction stage analysis are compared to inconsideration of the construction stage analysis, it is seen that there are large differences between some internal forces and displacements for the deck and the piers. It means, the analysis in case of inconsideration of construction stages, cannot give the reliable solutions. Large differences observed between the results with and without considering construction stages. It can be stated that the analysis without construction stages cannot give the reliable solutions.
S. Ates / Applied Mathematical Modelling 35 (2011) 3809–3820
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100 Construction Stage Considered Not Considered
Pier Height (m)
75
50
25
40000
20000
0
-20000
-40000
-60000
-80000
0
Bending Moment (kNm) Fig. 16. Bending moments of the piers of the bridge.
100
Pier Height (m)
75
50
25 Construction Stage Considered Not Considered
1000
800
600
400
200
0
Shear Force (kN) Fig. 17. Shear forces of the piers of the bridge.
To obtain real behaviour of engineering structures, construction stage analysis using time dependent material properties and geometric nonlinearity should be considered. It specifically is very important for bridges, because construction period continue along time and loads may change during the construction period and after.
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100
Pier Height (m)
75
50
25
Construction Stage Considered Not Considered
120000
100000
80000
60000
40000
20000
0
Axial Force (kN) Fig. 18. Axial forces of the piers of the bridge.
References [1] M.A. Ketchum, Redistribution of stresses in segmentally erected prestressed concrete bridges UCB/SESM-86/07. Department of Civil Engineering, University of California, Berkeley, 1986. [2] A.G. Bishara, N.G. Papakonstantinou, Analysis of cast-in-place concrete segmental cantilever bridges, J. Struct. Eng., ASCE 116 (5) (1996) 1247–1268. [3] H.I. Chiu, J.C. Chern, K.C. Chang, Long-term deflection control in cantilever prestressed concrete bridges I: Control method, J. Eng. Mech., ASCE 12 (6) (1996) 489–494. [4] S. Abbas, A.C. Scordelis, Nonlinear geometric, material and time-dependent analysis of segmentally erected three-dimensional cable stayed bridges, UCB/SESM-93/09, University of California, Berkeley, 1993. [5] P.J.S. Cruz, A.R. Mari, P. Roca, Nonlinear time-dependent analysis of segmentally constructed structures, J. Struct. Eng., ASCE 124 (3) (1998) 278–288. [6] A.R. Mari, Numerical simulation of the segmental construction of three dimensional concrete frames, Eng. Struct. 22 (2000) 585–596. [7] H.G. Kwak, J.K. Son, Determination of design moments in bridges constructed by balanced cantilever method, Eng. Struct. 24 (2002) 639–648. [8] H.G. Kwak, J.K. Son, Span ratios in bridges constructed using a balanced cantilever method, Eng. Struct. 18 (2004) 767–779. [9] P.H. Wang, T.Y. Tang, H.N. Zheng, Analysis of cable-stayed bridges during construction by cantilever methods, Comput. Struct. 82 (2004) 329–346. [10] H. Somja, V.V. Goyet, A new strategy for analysis of erection stages including an efficient method for creep analysis, Eng. Struct. 30 (2008) 2871–2883. [11] T. Cho, T.S. Kim, Probabilistic risk assessment for the construction phases of a bridge construction based on finite element analysis, Finite Elem. Anal. Des. 44 (6-7) (2008) 383–400. [12] A.C. Altunisik, A. Bayraktar, B. Sevim, S. Adanur, A. Domanic, Construction stage analyses of Kömürhan highway bridge using time dependent material properties, Struct. Eng. Mech. 36 (2) (2010) 207–224. [13] R. Malm, H. Sundquist, Time-dependent analyses of segmentally constructed balanced cantilever bridges, Eng. Struct. 32 (2010) 1038–1045. [14] SAP2000, Integrated finite element analysis and design of structures, Computers and Structures Inc, Berkeley, California, USA, 2008. [15] J.S. Przemieniecki, Theory of Matrix Structure Analysis, McGraw-Hill, New York, 1967. [16] S. Abbas, Nonlinear analysis of segmentally erected reinforced and prestressed concrete cable stayed bridges, UCB/SESM-90/24, University of California, Berkeley, 1990. [17] CEB-FIP Model Code, Thomas Telford, London, 1990.
Contents lists available at ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Numerical modelling of continuous concrete box girder bridges considering construction stages Sevket Ates ⇑ Karadeniz Technical University, Department of Civil Engineering, 61080 Trabzon, Turkey
a r t i c l e
i n f o
Article history: Received 17 December 2009 Received in revised form 27 January 2011 Accepted 2 February 2011 Available online 23 February 2011 Keywords: Bridges Construction stage Numerical modelling Time dependent material properties The balanced cantilever method
a b s t r a c t The aim of this study is to analyse the concrete continuous box girder bridges by considering segmentally construction stages through balanced cantilever method. Timedependent material properties of concrete and steel are also taken into account. Budan Bridge is selected as a numerical example. The Bridge constructed with balanced cantilever method and located on Artvin–Erzurum highway, Turkey, at 55 + 729.00–56 + 079.00 km is modelled using SAP2000 program. Geometric nonlinearities are taken into consideration in the analysis using P-Delta and large displacement criterion. Time-dependent material properties are considered as compressive strength, aging, shrinkage and creep for concrete, and relaxation for steel. The structural behaviour of the bridge at different construction stages is examined. Variation of internal forces such as bending moment, shear forces and axial forces, and displacements for bridge deck and pier are given with detail. Analyses show that, to obtain real behaviour of concrete bridges, segmentally construction stage analysis using time dependent material properties and geometric nonlinearity should be considered, because construction period continue along time and loads may change during this period and after. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction The use of continuous concrete box girder bridges has increased recently. In construction of this type bridges having constant or variable section height, the cantilever method can be applied. Box girder section forms consist of single or more box girder based on bridge wide. The cantilever method is considered as the natural and logical solution in construction of box girder bridges. There are two basic alternatives in the cantilever method: one is single cantilever method and the other is the double cantilever method. In the former, the side span girders of the bridge are constructed on interim piers and afterwards the stiffening girder in main span is constructed by one-sided free cantilevering until the span centre or the anchor pier on the far end is reached. In the latter, the bridge girder is constructed from both side of the tower towards the anchor piers and the main span centre by double-sided free cantilevering. The double cantilever method is also called as the balanced cantilever method. The method is especially recommended where scaffolding is difficult or impossible to construct over deep valleys, wide rivers or in case of expensive foundation conditions for scaffolds. In this method, bridges are built from one or more piers by means of formwork carriers. Normally the structure advances from a short stub on top of a pier symmetrically in segments of about 3–6 m length to the mid span or to an abutment, respectively. Each cantilevered part of the superstructure is tied to a previous one by concreting a key segment and post-tensioning tendons. The prestressing tendons are arranged based on the moment diagram of a cantilever. ⇑ Tel.: +90 462 377 26 69; fax: +90 462 377 26 06. E-mail address: [email protected] 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.02.016
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In recent years, many interesting research topics have arisen such as to be taken in to account segmentally construction stages in the analysis. Normally, structures are analysed by assuming that they are instantly built in a time. However, this type of analyses may be give unreliable results which compared with those obtaining from that construction stage is considered. In the construction stage analysis, time dependent material properties should be taken into account. Several studies have dealt with the analysis of segmentally constructed bridges, as long as a few studies have been struggled the analysis of the deflection and internal moment redistribution in bridges [1–3]. Abbas and Scordelis [4] achieved nonlinear geometric, material and time dependent analysis of segmentally erected three-dimensional cable stayed bridges. Cruz et al. [5] presented a general step by step model for the nonlinear and time dependent analysis of reinforced, prestressed concrete, and composite steel-concrete planar frame structures. An analytical model for the nonlinear and time dependent analysis of segmentally constructed three dimensional concrete frames was studied by Mari [6]. Kwak and Son [7,8] studied on span ratios in bridges constructed using the balanced cantilever method and reported that moment variation due to the change in structural system during construction requires a rigorous time dependent analysis that considers the construction stages. Wang et al. [9] carried out the analysis of cable stayed bridges at different stages during construction by the cantilever method. Somja and Goyet [10] came up with an efficient numerical procedure for materially and geometrically nonlinear finite element analysis of segmentally erected structures including time dependent effects due to load history, creep, shrinkage and aging of concrete. In that study, it was observed that time have a strong influence, especially, on concrete type structures. Therefore, it was emphasized that these effects must be taken into account in the design process. Cho and Kim [11] evaluated the risks in a suspension bridge by considering an ultimate limit state for the fracture of main cable wires. They examined the results compared with the conventional safety indices and allowable error for the control of deformations during construction. Altunisik et al. [12] carried out the construction stage analysis of Komurhan Bridge using time dependent material properties of concrete and steel. They reported that the construction stage analysis had remarkable effect on the structural behaviour of the bridge. Malm and Sundquist [13] studied the time-dependent analyses of segmentally constructed balanced cantilever bridges. For concrete bridges, time dependent effects make analyses even more complex. The effects develop at the early stages of the construction process and continue to evolve considerably after the concrete bridges are built. Depending on the construction method, the time dependent effects can appear and induce important stress redistribution in the structure. As seen in literature, studies on construction stage analysis are meagre and need to be enlarged by inserting new studies. In the light of aforementioned researches, construction stage analysis of a bridge constructed with balanced cantilever method using time dependent material properties is performed in this paper. Time dependent material properties are considered as compressive strength, aging, shrinkage and creep for concrete, and relaxation for steel. 2. Description of example bridge
6.00m
3.00m
3.00m
Fig. 1. The Budan Bridge and its dimensions.
56+079.00
92.50m
97.17m
55+986.50
55+821.50
165.00m
91.77m
92.50m
6.00m
55+729.00
The Budan Bridge, shown in Fig. 1, is a reinforced continuous concrete box girder type bridge located on between Artvin and Erzurum highway, Turkey, at 55 + 729.00–56 + 079.00 km. Construction of the bridge started in 2007 and completed recently. The configuration of the bridge is a three-span, cast-in-place concrete box girder superstructure supported on reinforced concrete piers. The bridge deck consists of a main span of 165.00 m and two side spans of 92.50 m each. The total bridge length is 350.00 m and width of bridge is 15.00 m. The structural system of the bridge consists of a continuous deck, two piers, two abutments and a closure segment. The piers have the heights of 91.77 m at the 55 + 821.50 km and of 97.17 m at the 55 + 986.50 km. The cross-sections of the piers and the deck are given in detail in Fig. 2. The cross-section of the deck
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15.00m
6.00m
2.50m-9.50m
1.20m
1.20m
8.00m
1.20m
1.20m 4.57m
a
5.86m
4.57m
b
Fig. 2. Cross-sections of (a) piers and (b) deck of the Budan Bridge.
consists of a single cell box girder with cantilevered slabs. The girder depth varies from a maximum of 9.50 m over the piers to a minimum of 2.50 m at the mid span and abutments. The closure segment in the middle of the central span is 1.50 m long. The bottom slab thickness is variable, as well. In order to investigate the construction stage response effects on the Budan Bridge, three-dimensional finite element model is used. The primary objective of this study is to perform a parametrical study associated with the construction stages and its effects on the response of continuous concrete box girder bridges. However, soil–structure interaction is not considered. The finite element model of the Budan Bridge is shown in Fig. 3. Analyses are performed by using SAP2000 [14]. The finite element model of the bridge consists of solid elements having 3° of freedom at each nodal point. They are in two horizontal and vertical translational directions. The three dimensional finite element models are represented with view in Fig. 3. The model has 3764 nodal points and 1710 solid area elements. In addition, 20 frame elements are used to represent piers. Additionally, the modulus of elasticity of concrete is taken according to Fig. 6. The relative humidity is 60% and the cement type is normal. The prestressing steel yielding point is fyk = 500 MPa and the steel modulus of elasticity is Es = 200,000 MPa. The parameters related to time dependent material properties is given in Table 1. Geometric nonlinearity is taken into consideration using P-Delta and large displacement criterion in the staged construction analysis. 3. Construction stages Bridges constructed with the balanced cantilever method consist of main structural elements such as deck, piers and side abutments. The balanced cantilever segmental construction has long been known as one of the most efficient methods due to
Plan
Three-dimensional view Fig. 3. Plan and three-dimensional view of the finite element model of the Budan Bridge.
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Table 1 Parameters used in the analysis for time dependent material properties. Parameters
Deck
Pier
Steel
Cement type, s Relative humidity, RH (%) Notional size, h (mm) Shrinkage coefficient, Bsc Relaxation
0.25 60 0.600 5 –
0.25 60 1.306 5 –
– – – – Class 1
Closure Segment
2@3m=6m 3@4m=12m
12@5m=60m
Constructed deck segments
Abutment
7.5m
3@4m=12m
12@5m=60m
1.5m
55+821.00
55+729.00
1.5m 10.75m
Pier
Constructed deck segments
Pier
Abutment
Fig. 4. Construction stages of the Budan Bridge and construction stages as per the balanced cantilever method.
the fact that the method does not need any falsework to build a bridge. That is why among the construction methods, the mentioned balanced cantilever construction of reinforced concrete box girder bridges has been preferred. In this method, precast segmental construction and cast-in-place construction techniques can be executed. There are some advantages and disadvantages of these techniques; Cast in-place construction may permit a rate of one pair of segments 3–6 m long to be constructed and stressed every 4– 7 days. On the average, a pair of travellers permits the completion of approximately 50 m of bridge deck within a month, excluding the transfer from pier to pier and fabrication of the pier table. Cast-in-place cantilever construction is basically a slow process, while precast segmental construction with matching joints is among the fastest. Precast segmental construction for long, repetitive structures may be more economical than a cast-in-place solution. Precast segmental solutions are limited by the capacity of transportation and placing equipment. Segments exceeding 2500 kN are seldom economical. Cast-in place construction does not have the same limitation, although the weight and cost of the travellers are directly proportional to the weight of the heaviest segment. In the segmental construction, some limitations arise in the form of the curvature of the bridge and the size and weight of the segment. Both precast and cast-in-place segmental construction permits all work to be performed at the top. The Budan Bridge was constructed with cast-in-place construction technique is selected as an example. Firstly, piers and small part of bridge deck are constructed over substructure using suitable formwork. Then, segments of 3–6 m length as shown in Fig. 3 are erected on opposite sides of each pier to balance the loads by using a movable form carrier. After the concreting, prestressing tendons are inserted into the segments and stressed with post-tension. Finally, form carrier is moved to the next position and a new cycle starts. This sequence is completed at one week on average and is going on until bridge decks meet at mid span. At the mid span, closure segment is established to complete one span. Because of the fact that maximum displacements are occurred at this point after finite element analysis, construction of this segment is very important. The typical operation sequence is summarized as: carrier is firstly set up and adjusted, and aligned form of deck, after the stage placing reinforcement and tendon ducts are achieved. Concreting is fulfilled and prestressing tendons in the seg-
S. Ates / Applied Mathematical Modelling 35 (2011) 3809–3820
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ment are then inserted. In due course, the formwork is removed. The form carrier to the next position is moved in order to start a new cycle. The construction process of each segment is repeated until the bridge is completed. The schematic view mentioned above of balanced cantilever construction of the prestressed concrete highway bridge is given in Fig. 4. In the construction stage analysis, added and removed loads for each construction stages should be determined. In order to obtain the reliable solution, each stage results should be added to end of the each stage and next stage analysis is done. Additionally, nonlinear solution parameters should be selected depending on the literature. In the finite element analysis of the Budan Bridge, a total of 39 segments at the deck and 60 segments at the two piers are considered. To close the centre span, a segment of 1.50 m is used, while two end segments of the side spans of 10.75 m are required to reach the abutments. Over the piers, segments of 7.50 m lengths are placed. A total of 100 prestressing tendons, each of 19 strands with 15.2 mm, stressed with an initial force of 4052.45 kN is used. Additionally, the reinforcing steel is uniformly distributed over the cross-sections with an amount of 0.02% in the girder and 0.1% in the piers. A time dependent analysis of the balanced cantilever construction procedure and 10,000 days (27.4 years) after completion of construction of the selected bridge is executed. However, the construction period was of nearly eight months. During the construction of the deck; concreting of a pair of new segments, a waiting period of three days, and at the rest of 4 days stressing of tendons and shifting of travelling formworks are simulated. The iterative calculations at each construction stage considering added stiffness from the initial equilibrium state. The element stiffness matrix is given the following equation
½K ¼ ½K e þ ½K g ;
ð3:1Þ
where ½K e is the elastic stiffness matrix and ½K g is the geometric stiffness matrix. The expression for ½K g is derived in detail by Przemieniecki [15] and Abbas [16]. The finite element analysis is performed at each construction stages of the bridge by using SAP2000 [14]. In the analysis of the bridge, the following load cases are considered; (a) Dead load: Weight of all elements.
(b) Additional mass: Weight of the asphalt, cobble, pipeline and its supports, scarecrow. (c) Gantry: Load of the form carrier. This load is implemented to previous one before the construction one segment and slide next one after construction of the segment. Then, the gantry load is wholly removed from the constructed parts. (d) Diaphragm: Weight of the reinforced concrete walls at the abutments and at both sides of the expansion joints. (e) Pre-stress: Post-tension cables are modelled using cable elements. Post-tension loads are considered as strain. (f) Jack: Load of the jack applied to the side segments before fixed to the side abutments. (g) Temperature: This load is applied to consider temperature variation due to the climate. +30 °C and 20 °C temperature (according to the bridge region) are considered separately in the analysis. 4. Time dependent properties for concrete 4.1. Compressive strength The compressive strength of concrete at an age t depends on the type of cement, temperature and curing conditions. The relative compressive strength of concrete at various ages may be estimated by the following formula [17]:
fcm ðtÞ ¼ bcc ðtÞfcm ;
ð4:1Þ
in which bcc ðtÞ is a coefficient with depends on the age of concrete and is calculated by:
("
28 bcc ðtÞ ¼ exp s 1 t=t 1
1=2 #) ;
ð4:2Þ
fcm ðtÞ is the mean concrete compressive strength at an age of t days; fcm is the mean compressive strength after 28 days; t is the age of concrete in days; t1 = 1 day; and s is a coefficient with 0.20, 0.25 and 0.38 which type of cement. In the light of the Eqs. (4.1) and (4.2), the mean concrete compressive strength is given in Fig. 5. 4.2. Aging of concrete The modulus of elasticity of concrete changes with time. For this reason, the modulus at an age t – 28 days may be estimated as below equation [17]:
Eci ðtÞ ¼ Eci
pffiffiffiffiffiffiffiffiffiffiffiffi bcc ðtÞ;
ð4:3Þ
where Eci ðtÞ is the modulus of elasticity at age of t days; Eci is the modulus of elasticity at an age of 28 days; bcc ðtÞ is a coefficient which depends on the age of concrete. For the deck and the piers of the example bridge, the aging of concrete is plotted in Fig. 6.
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Mean Strength (kN/m²)
4E+004
3E+004
2E+004
1E+004
0E+000 0E+000
2E+003
4E+003
6E+003
8E+003
1E+004
Time (Day)
Modulus of Elasticity (kN/m²)
Fig. 5. Variation of the mean concrete compressive strength with days.
5E+007 4E+007 3E+007 2E+007 Deck Pier
1E+007 0E+000 0E+000
2E+003
4E+003
6E+003
8E+003
1E+004
Time (Day) Fig. 6. Aging of concrete in days.
4.3. Shrinkage of concrete The CEB-FIP Model Code [17] gives the following equation of total shrinkage strain of concrete:
ecs ðt; ts Þ ¼ ecso bs ðt ts Þ;
ð4:4Þ
where ecso is notional shrinkage coefficient; bs is the coefficient to describe the development of shrinkage with time; t is the age of concrete in days and t s is the age of concrete in days at the beginning of shrinkage. The notional shrinkage coefficient may be obtained from:
ecso ¼ es ðfcm ÞbRH ;
ð4:5aÞ
es ðfcm Þ ¼ 160 þ 10bsc 9
fcm fcmo
ð4:5bÞ
;
where fcm is the mean compressive strength of concrete at the age of 28 days in MPa; fcmo is taken as 10 MPa; bsc is a coefficient ranging from 4 to 8 which depends on the type of cement.
bRH ¼ 1:55bsRH
40% 6 RH < 90%
bRH ¼ 0:25
RH P 99%
;
ð4:6Þ
where
bsRH ¼ 1
RH RHo
3 ;
ð4:7Þ
with RH is the relative humidity of the ambient atmosphere (%) and RHo is 100%. The development of shrinkage with time is given by:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðt ts Þ=t1 ; bs ðt t s Þ ¼ 350ðh=ho Þ þ ðt t s Þ=t 1
ð4:8Þ
where h is the notional size of member (mm) and is calculated by h ¼ 2Ac =U in which Ac is the cross-section and U is the perimeter of the member in contact with the atmosphere; ho = 100 mm and t1 = 1 day. For the deck and the piers of the
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4E-004
Shrinkage Strain
3E-004
2E-004 Deck Pier
1E-004
0E+000 0E+000
2E+003
4E+003
6E+003
8E+003
1E+004
Time (Day) Fig. 7. Time dependent shrinkage strain of concrete.
example bridge, the shrinkage strain of concrete depending on relative humidity, notional size and shrinkage coefficient is depicted in Fig. 7. 4.4. Creep The effect is calculated using creep model [17]. For a constant stress applied at time to, this leads to:
ecc ðt; to Þ ¼ in which
rc ðto Þ
/ðt; t o Þ;
Eci
ð4:9Þ
rc ðto Þ is the stress at an age of loading to; /ðt; to Þ is the creep coefficient and is calculated from:
/ðt; t o Þ ¼ bc ðt to Þ/o ;
ð4:10Þ
where bc is the coefficient to describe the development of creep with time after loading; t is the age of concrete in days at the moment considered; to is the age of concrete at loading in days. The creep coefficient is explained by:
/o ¼ /RH bðfcm Þbðt o Þ; 1
/RH ¼ 1 þ
0:46
RH RHo
1=3 ;
ð4:11bÞ
h ho
5:3 bðfcm Þ ¼ qffiffiffiffiffiffi ; fcm fcmo
bðto Þ ¼
ð4:11aÞ
1 0:2 : 0:1 þ tt1o
ð4:11cÞ
ð4:11dÞ
All parameter is defined above. The development of creep with time is given by:
bc ðt t o Þ ¼
ðt to Þ=t 1 ; bH þ ðt t o Þ=t 1
( 18 ) RH h bH ¼ 150 1 þ 1:2 þ 250 6 1500; RHo ho
ð4:12aÞ
ð4:12bÞ
where t1=1 day; RHo=100 and ho=100 mm. In the analysis, the creep coefficient of concrete is given in Fig. 8 for the deck and the piers having different notional size. 5. Time dependent properties for steel According to CEB-FIP Model Code [17], relaxation classes referring to the relaxation at 1000 h are divided into three groups for prestressing steels. The first relaxation class is defined as the normal relaxation characteristics for wires and strands, the second class is defined as improved relaxation characteristics for wires and strands, and the last one is defined as relaxation characteristics for bars. For an estimate of relaxation up to 30 years the following formula may be applied:
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Creep Coefficient
3
2
1
Deck Pier
0 0E+000
2E+003
4E+003
6E+003
8E+003
1E+004
8E+003
1E+004
Time (Day) Fig. 8. Time dependent creep coefficient.
Relaxation Coefficient
0.25 0.20 0.15 0.10 0.05 0.00 0E+000
2E+003
4E+003
6E+003
Time (Day) Fig. 9. Time dependent relaxation coefficient of prestressing steel.
qt ¼ q1000
k
t 1000
ð4:13Þ
;
where qt is the relaxation after t hours; q1000 is the relaxation after 1000 h; k logðq1000 =q100 Þ in which k to be 0.12 for relaxation class 1, and 0.19 relaxation class 2; q100 is the relaxation after 100 h. Normally, the long-term values of the relaxation are taken from long-term tests. However, it may be assumed that the relaxation after 50 years and more is three times the relaxation after 1000 h. In the example bridge, all tendons compose of 19 strands with u15.2 mm ant they are categorized as the normal relaxation characteristics for wires and strands, so the relaxation class is taken as class1. The relaxation coefficient as per mentioned properties is given in Fig. 9.
6. Numerical results Fig. 10 gives the bending moments along the bridge deck in case the construction stage procedure is considered and is not considered, respectively. As seen the figure, the bending moments obtained from consideration of the construction stage are significantly bigger than these of inconsideration of the construction stage. The deck bending moments, if the construction stage is considered, increase around 80% if compared with the results obtained the construction stage is not considered in
-4E+005
Bending Moment (kNm)
Construction Stage
-3E+005
Considered Not Considered
-2E+005 -1E+005 0E+000 1E+005
0
50
100
150
200
250
Length (m) Fig. 10. Bending moments of the deck of the bridge.
300
350
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the analysis. The results clearly show that, in the event of consideration of construction stage analysis, the deck bending moments substantially increase. Besides, if the variation of the bending moment related to stage by stage is drawn at the support points at the 55 + 821.00 km on the deck as in Fig. 11, the same results can be comprehended. On the other hand, shear forces and axial forces have not the same differences like bending moments. This is clearly seen in Figs. 12 and 13. The deck displacements are compared in Fig. 14. The vertical displacements considerably increase in case of the fact that construction stage analysis is taken into account in the time dependent analysis. It is also produced that displacements have an increasing trend towards the middle of the bridge deck. Variation of maximum displacements along the height of the piers of the selected bridge is shown in Fig. 15. It can easily be seen that the horizontal displacement increase with the pier height of bridge in view of differences between the two methods. The differences are reached to approximately 18 cm at the top of bridge pier. Bending moments change with height of the bridge pier as shown in Fig. 16. It can be easily seen that the bending moments obtained from that the construction stage analysis is considered have bigger results than the responses for that construction stage analysis is not considered. Fig. 17 points out the shear forces on the piers of the bridge for the mentioned two methods. The values of the axial forces are nearly equal along the height of the bridge piers as shown in Fig. 18. However, the values of the shear forces have a difference as 30%.
Bending Moment Variation (kNm)
-4E+005 -3E+005 -2E+005 -1E+005 Construction Stage
0E+000
Considered Not Considered
1E+005 0
50
100
150
200
250
Stage Fig. 11. Variation of the bending moments at the support.
3E+004
Construction Stage
Shear Force ( kN)
2E+004
Considered Not Considered
1E+004 0E+000 -1E+004 -2E+004 -3E+004 0
50
100
150
200
250
300
350
300
350
Length (m) Fig. 12. Shear forces of the deck of the bridge.
4E+004
Axial Force ( kN)
0E+000 -4E+004 -8E+004 Construction Stage
-1E+005
Considered Not Considered
-2E+005
0
50
100
150
200
250
Length (m) Fig. 13. Axial forces of the deck of the bridge.
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Displacement (m)
0.00
-0.10
-0.20
Construction Stage
-0.30
Considered Not Considered
-0.40 0
50
100
150
200
250
300
350
Length (m) Fig. 14. Vertical displacements of the deck of the bridge.
100
Pier Height (m)
75
50
25
Construction Stage Considered Not Considered
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0
Displacement (m) Fig. 15. Horizontal displacements of the piers of the bridge.
7. Conclusions This study outlines an investigation about the construction stage analysis of the concrete continuous box girder bridge including time dependent effects such as creep, shrinkage and aging of the concrete and relaxation of the steel. The time dependent analyses are carried out with and without considering construction stages of the bridge model, separately. The assumed material and cross sectional properties are taken from a real bridge. The maximum and minimum response values of the bridges are compared with each other. The results obtained from this study can be given as: The bending moments, while the construction stage is considered, are significantly greater than those of inconsideration of the construction stage. The deck bending moments are increased around 80% in case of consideration of construction stages in the analysis. The results clearly show that consideration of construction stage in the analysis substantially increases the deck bending moments. Besides, the bending moment variation in case of considering stage by stage construction gives the same results. When the results of the construction stage analysis are compared to inconsideration of the construction stage analysis, it is seen that there are large differences between some internal forces and displacements for the deck and the piers. It means, the analysis in case of inconsideration of construction stages, cannot give the reliable solutions. Large differences observed between the results with and without considering construction stages. It can be stated that the analysis without construction stages cannot give the reliable solutions.
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100 Construction Stage Considered Not Considered
Pier Height (m)
75
50
25
40000
20000
0
-20000
-40000
-60000
-80000
0
Bending Moment (kNm) Fig. 16. Bending moments of the piers of the bridge.
100
Pier Height (m)
75
50
25 Construction Stage Considered Not Considered
1000
800
600
400
200
0
Shear Force (kN) Fig. 17. Shear forces of the piers of the bridge.
To obtain real behaviour of engineering structures, construction stage analysis using time dependent material properties and geometric nonlinearity should be considered. It specifically is very important for bridges, because construction period continue along time and loads may change during the construction period and after.
3820
S. Ates / Applied Mathematical Modelling 35 (2011) 3809–3820
100
Pier Height (m)
75
50
25
Construction Stage Considered Not Considered
120000
100000
80000
60000
40000
20000
0
Axial Force (kN) Fig. 18. Axial forces of the piers of the bridge.
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