Deformation Analysis of Prestressed Continuous Steel-Concrete Composite Beams

Transcription

1 Deformation Analysis of Prestressed Continuous Steel-Concrete Composite Beams Jianguo Nie 1 ; Muxuan Tao 2 ;C.S.Cai 3 ; and Shaojing Li 4 Abstract: Deformation calculation of prestressed continuous steel-concrete composite beams accounting for the slip effect between the steel and concrete interface under service loads is analyzed. A simplified analytical model is presented. Based on this model, formulas for predicting the cracking region of concrete slab near the interior supports and the increase of the prestressing tendon force are derived. A table for calculating the midspan deflection of two-span prestressed continuous composite beams is also proposed. It is found that the internal force of the prestressing tendon under service loads can be accurately calculated using the proposed formulas. By ignoring the increase of the tendon force, the calculated deflection are overestimated, and considering the increase of the tendon force can significantly improve the accuracy of analytical predictions. As the calculated values show good agreement with the test results, the proposed formulas can be reliably applied to the deformation analysis of prestressed continuous composite beams. Finally, based on the formulas for calculating the deformation of two-span prestressed continuous composite beams, a general method for deformation analysis of prestressed continuous composite beams is proposed. DOI: / ASCE ST X CE Database subject headings: Prestressed concrete; Composite beams; Deformation; Deflection; Cracking; Concrete slabs; Continuous beams. Introduction Continuous steel-concrete composite beams are widely used in buildings and bridges for higher span/depth ratios and less deflection etc., which results in superior economical performance compared with simply supported composite beams. For continuous composite beams, negative bending near interior supports will result in early cracking of concrete slab and reduction of stiffness. When beams are designed for span lengths and loads greater than usual, the requirement of serviceability limit state due to unacceptable deflection and crack width would require using prestressing technique. Compared with conventional steel-concrete composite beams, prestressed steel-concrete composite beams have a few major advantages: 1 extending the elastic range of structural behavior; 1 Professor, Dept. of Civil Engineering, Key Laboratory of Structural Engineering and Vibration of China Education Ministry, Tsinghua Univ., Beijing , China. 2 Ph.D. Candidate, Dept. of Civil Engineering, Key Laboratory of Structural Engineering and Vibration of China Education Ministry, Tsinghua Univ., Beijing , China corresponding author. dmh03@mails.tsinghua.edu.cn 3 Associate Professor, Dept. of Civil and Environmental Engineering, Louisiana State Univ., Baton Rouge, LA, 70803; presently, Adjunct Professor, School of Civil Engineering and Architecture, Changsha Univ. of Science and Technology, Changsha, China. 4 Formerly, Graduate Student, Dept. of Civil Engineering, Key Laboratory of Structural Engineering and Vibration of China Education Ministry, Tsinghua Univ., Beijing , China. Note. This manuscript was submitted on August 10, 2008; approved on April 20, 2009; published online on October 15, Discussion period open until April 1, 2010; separate discussions must be submitted for individual papers. This paper is part of the Journal of Structural Engineering, Vol. 135, No. 11, November 1, ASCE, ISSN /2009/ /$ increasing the ultimate loading capacity; 3 decreasing the deformation under service loads; 4 being favorable in crackwidth control; 5 fully using the materials and thus reducing the structural height and overall dead load; and 6 improving the fatigue and fracture behavior. Since Szilard 1959 suggested a method for the design and analysis of prestressed steel-concrete composite beams considering the effects of concrete shrinkage and creep, many researchers have developed methods for analyzing the behavior of simply supported prestressed composite beams Hoadley 1963; Klaiber et al. 1982; Dunker et al. 1986; Saadatmanesh 1986; Saadatmanesh et al. 1989a,b,c; Albrecht et al. 1995, Nie et al However, continuous prestressed composite beams have not been researched until the late 1980s Troitsky and Rabbani 1987; Troitsky 1990; Dall Asta and Dezi 1998, Ayyub et al. 1990, 1992a,b; Dall Asta and Zona As a result, prestressed continuous composite beams have not widely been used partly due to the lack of design theory. In fact, the behavior of prestressed continuous composite beams depends on the interaction between four main components: the reinforced concrete slab, the steel profile of beams, the shear connections, and the prestressing tendons, which makes prestressed continuous composite beams more complex than conventional ones. Dall Asta and Zona 2005 proposed a nonlinear finite element model simulating the behavior of prestressed continuous composite beams accurately. This numerical approach is a very powerful research tool for analyzing the externally prestressed structures, but it perhaps is too complicated for a routine design practice. As prestressing technique is an effective way to reduce deformation and crack width under service loads, particular attention has to be paid to the deformation calculation of prestressing continuous composite beams. The main objective of this research is to develop calculation methods for the deformation of prestressing continuous composite beams based on the reduced stiffness JOURNAL OF STRUCTURAL ENGINEERING ASCE / NOVEMBER 2009 / 1377

2 Fig. 1. Sketch of two-span prestressed continuous composite beam method that was developed for conventional continuous composite beams Nie and Cai The proposed method, verified by test results, is suitable for design practice. Theoretical Study Analytical Model Prestressed continuous composite beams discussed in this paper are shown in Fig. 1 where the prestressing tendons are laid out as fold lines or straight lines for the convenience of construction. The straight lines can be considered as a special case of the foldline type with =0 in calculation. The position of tendons can be either internal or external, which will not influence the method of analysis. Thus, the research interest in this paper is concentrated on a two-span prestressed continuous composite beam with foldline tendons as shown in Fig. 1, and the methodology can be applied to other kinds of prestressed continuous composite beams. The calculation model of prestressed steel-concrete composite beams is shown in Fig. 2. The process of loading can be divided into two stages. In the first stage shown in Fig. 2 a, beams are initially prestressed by tendons and the equivalent loads applied to the continuous beams by tendons are composed of two parts. The first part includes axial compression force T 0 and moment T 0 e 0 at the beam ends, where e 0 =distance from the beam anchor to the neutral axis of the transformed section, positive below neutral axis. The second part includes vertical concentrated loads applied by tendons. Force equilibrium shown in Fig. 3 gives the value of the equivalent concentrate force F applied by the tendons as T 0 sin, which equals to T 0 approximately as is very small. Fig. 2. Calculation model of prestressed continuous steel-concrete composite beam: a first loading stage; b second loading stage Fig. 3. Equivalent load applied to the beam by tendons The downward concentrated force applied by tendons at the interior support is not shown in the figure as the force is applied directly on the support. The rigidity along the beam can be considered as unchanged in this stage since the cracking of concrete usually does not occur. The section properties can be calculated by the transformed section method ignoring the slip effect between steel and concrete interface at this stage. It is assumed that the distribution of moment along the beam due to the prestressing force keeps unchanged. Once all the parameters have been determined, deformation in the first stage f 1 can be directly calculated by methods of structure mechanics. In the second stage shown in Fig. 2 b, application of the external force P results in the increase of downward deflection f 2 and a change of prestressing tendon force T. In the region of sagging moment, the reduced flexural stiffness B=E 1 I 1 / 1+ is used due to the slip effects, where is stiffness reduction coefficient according to the reduced stiffness method Nie and Cai 2003, and the axial stiffness EA is calculated by the transformed section method. In the region of hogging moment in the range of nl near each side of the interior supports, concrete is considered no longer in service due to cracking. In this case the bending rigidity E 2 I 2 and axial rigidity E 2 A 2 can only include the contribution of the reinforcement and steel materials, and parameter and are defined as =B/ E 2 I 2, and =EA/E 2 A 2. Actually, in the second stage, concrete in the hogging moment may still contribute to stiffness because of the prestressing force. Therefore, the partial interaction between the steel and concrete should be considered for a rational analysis. For simplicity, this kind of interaction effect is considered in the present study by adjusting the value of nl instead of actually modifying the stiffness of composite beams near the supports, which results in only small errors as will be verified by the experiments and discussed later. In order to obtain the deflection of the composite beams in this stage, the length of cracking region of concrete slab at interior supports, defined by n, should be determined first. For conventional continuous composite beams, it is found in previous studies and experiments that taking 0.15 for the n value will be accurate enough for design Nie et al However, for prestressed continuous composite beams, the length of cracking region of concrete slab is smaller than the convention ones. Furthermore, n is related to the prestressing degree directly, which has been verified by tests. The other parameter T is also very essential for calculating the deflection. Since the materials are generally linear elastic under service load conditions, the principle of superposition can be used to obtain the total deflection as f 1 + f 2, where f 1 can be calculated directly by methods of structural mechanics. In this study, we are more concerned about the increase of deflection under service loads, i.e., f 2. Therefore, this paper will only investigate the increase of deflection in the second stage, and for convenience, f 2 will be rewritten as f hereafter. According to the discussion made above, the core of deformation calculation is to determine the values of n and T, which will be discussed further in the following parts. The cable slip at the saddle points is a complex behavior of the externally prestressed composite beams. The slip friction at the saddle points can influence the behavior of beams under service loads. Negligible friction occurs by using individually coated single-strand tendons Conti et al and the assumption of negligible friction can be found in the previous model Dall Asta and Zona This assumption is also used in the following analytical studies / JOURNAL OF STRUCTURAL ENGINEERING ASCE / NOVEMBER 2009

3 M k = 0.85M ek = m 1 m P kl 6 where M ek =moment due to P k ignoring the moment redistribution. The relationship between the service load and the initial prestressing force can be derived using Eqs. 5 and 6 as P k = 40T 51m 1 m L e A + W + 20 T Under the application of external force and prestressing force, the distribution of moment along the beam is shown as Figs. 4 b and c, respectively. The tension stress at the top of concrete at the boundary of the cracking region equals to zero, which leads to Fig. 4. Theoretical analysis of the length of cracking region of concrete slab: a calculation model of two-span prestressed continuous composite beams; b moment distribution due to prestressing tendon force; and c moment distribution due to external loads Prediction of Cracking Region of Concrete Slab In this part, the length of cracking region of concrete slab over interior supports will be theoretically analyzed based on the calculation model shown in Fig. 4 a. After the initial force T 0 is prestressed, a structural analysis gives the sagging moment at the interior support as M T0 = T 0e m 1 m T 0 L 1 Accordingly, the initial compressive stress at the top of concrete slab at the interior support is calculated as pc = M T0 W + T 0 A = T 0e 0 2W + 3m 1 m T 0 L + T 0 2 2W A where W = section modulus of transformed composite section at the top of concrete flange and A=cross-sectional area of transformed section. The moment needed to eliminate the compressive stress at the interior support is obtained as M 0 = pc W = 1 2 T 0e m 1 m T 0 L + T 0 A W The prestressing degree is defined as = M 0 4 M k where M k =moment at the interior support due to service load P k excluding prestressing effect. Introducing Eq. 3 into Eq. 4 gives M k = T 0e m 1 m T 0 L + T 0W 5 2 A It is found in experiments that the moment redistribution coefficient a at the interior support can reach about 15% under service load conditions. Therefore, 15% is used to calculate the moment at the interior support under service loads approximately as 3 M T x = nl + M P x = nl T W A =0 8 where T = tendon force under service load conditions. Compared with the initial prestressing force, the increase of tendon force is relatively small, and T can be taken proximately as T 0 ; M T x = moment distribution along the beam due to the prestressing force, and M P x =moment distribution along the beam due to the service load. They are calculated as M T x = 3 Te 0 2 L x 1 2 Te m m +1 T x 3 m 1 m T L 0 x nl 2 9 M P x = m m 1 P kx m 1 m P kl 0 x nl 10 Introducing Eqs. 7, 9, and 10 into Eq. 8 leads to the expression of n as a function of A 1 n = 11 B CA where A, B, and C can be calculated as A = W Ae m m L e 0 B = m m m L e 0 C = 51m2 51m 40 51m 2 51m From Eq. 11 we can see that the main factors influencing the range of concrete cracking region include the prestressing degree, the parameter W/ Ae 0, the parameter m L/e 0, and the loading position m. Their effects on n are plotted in Figs From Figs. 5 7 we can see that the length of concrete cracking region falls more and more quickly as the prestressing degree rises. When the prestressing degree is taken as 1, the length of concrete cracking region is zero, referred to as fully prestressed composite beams. Similarly, a zero of the prestressing degree results in the length of concrete cracking region being as 1/C, which depends only on the loading position m and corresponds to conventional composite beams. Fig. 5 indicates how n varies within the usual range of parameter W/ Ae 0 when the other parameters are fixed. It is JOURNAL OF STRUCTURAL ENGINEERING ASCE / NOVEMBER 2009 / 1379

4 Fig. 5. Influence of parameter W/Ae 0 on n found that the influence of parameter W/ Ae 0 on n is very slight and can be ignored. In most cases, the neutral axis in the region of positive moment is adjacent to the steel top flange, and the prestressing tendons are adjacent to the steel bottom flange. According to the sketch shown in Fig. 1, m L represents the vertical distance from the beam anchor to the center of tendons taken proximately as the position of the steel bottom flange, leading to the following: m L + e 0 h s m L e 0 h s e Fig. 8. Comparison among test results, theoretical results and simplified theoretical results Since the height of the steel beam h s is about 4 to 8 times of the anchor eccentricity e 0, the parameter m L/e 0 varies from 3 to 7. Within this range we can conclude from Fig. 6 that the variation of parameter m L/e 0 will not significantly influence the value of n. Fig. 7 shows how the loading position m influences the n value. The n approaches to unity in high prestressing degree region, and in low prestressing degree region it varies from 0.15 to 0.20 approximately when m varies within the usual range. Since the actual length of concrete cracking region is slightly shorter than the theoretical result due to the assumption that the tensile strength of concrete and the increase of tendon force are negligible, Eq. 11 should be modified to a certain extent. Furthermore, except for, the other three parameters all slightly influence the n value. Thus, Eq. 11 can be simplified considering the following factors: 1. Relationship format between n and as defined by Eq. 11 is maintained by adjusting only the coefficient in the equation. 2. The new equation can reduce to conventional nonprestressed case, i.e., when =0, n= The parameter W/ Ae 0 and m L/e 0 can be taken as the average values within the usual range. Consequently, Eq. 11 is simplified as 3 1 n = The comparison between test results discussed later, theoretical results, and simplified modified theoretical results is shown in Fig. 8, which proves that Eq. 13 is reasonable and accurate for the calculation. Prediction of Tendon Force Fig. 6. Influence of parameter L/e 0 on n Fig. 7. Influence of parameter m on n The increment of tendon force due to external loads can be predicted by developing the equilibrium equation, the deformation compatibility equation, and the physical equation for the structure system. The external loads mainly result in beam moments whose distribution depends on the loading conditions. The change of prestressing tendon forces mainly result in axial forces and moments, which can be solved by a simple structural analysis as shown in Fig. 9. The effect of prestressing force increment T is resolved into two parts in Fig. 9 a, namely the equivalent vertical forces and horizontal axial forces at beam ends. The position change of neutral axis in the region of hogging moment near the interior support influences the moment distribution due to prestressing force. As a result, a coefficient =e 02 /e 0 is defined here to describe it, where e 02 represents the vertical distance from the prestressing tendon to the elastic neutral axis in the region of hogging moment as shown in Fig. 9 a with positive for being below the neutral axis. In order to solve the expression of R 1 and R 2 in Fig. 9 a, 1380 / JOURNAL OF STRUCTURAL ENGINEERING ASCE / NOVEMBER 2009

5 Fig. 9. Distribution of internal forces due to prestressing tendon forces: a distribution of moments due to the increase of prestressing tendon forces; b distribution of axial forces due to the increase of prestressing tendon forces deformation compatibility equations at Point 1 as shown in Fig. 9 a under the two load cases can be developed as following Eqs. 14 and 15, respectively = 1, T 2R 1 T 11 = = 1, T 2R 2 Te 0 11 =0 L where 1 1 =actual vertical displacement of Point 1 under Load Case 1 ; 2 1 =actual vertical displacement of Point 1 under Load 1 Case 2 ; 1, T =vertical displacement of Point 1 under Load Case 2 1 with the interior support removed; 1, T = vertical displacement of Point 1 under Load Case 2 with the interior support removed; and 11 =vertical displacement of Point 1 produced by unit vertical force applied on Point 1 with the interior support removed. Solving Eqs. 14 and 15 gives the expression of R 1 and R 2 in Fig. 9 a as in the following: P = TL P 19 E P A P where L P =length of the prestressing tendons; E P A P =axial stiffness of the prestressing tendons; and Young s modulus of tendons E p can be taken proximately as Young s modulus of steels b. For the convenience of further derivations, the following definitions are introduced: e = e 1, = e 2 20 e 1 where e 1 and e 2 are shown in Fig. 10 a with a positive value being for below the neutral axis. Accordingly, fold-line of tendons has a negative and straight-line a positive value. In the region of hogging moment, since only the top reinforcement bars and the steel beam are considered in the calculation, the slip effect can be excluded, which results in the strain of the steel fiber at the height level of tendons as R 1 = 3 1 n n 3m 2 +3m n n n n n +3 R 2 = 2 1 n n n +2 The externally prestressed composite beams do not deform compatibly with prestressing tendons at all sections. However, in addition to the two end anchors, several intermediate connections or called deviators are provided for the prestressing tendons so that the deformation compatibility at these points is maintained during the loading process, ensuring the global deformation compatibility between the prestressing tendons and the composite beam. From the global deformation compatibility conditions, the total deformation of the prestressing tendons equals to the integration of the strain of the steel fiber at the height level of tendons, thus leading to the deformation compatibility equation as P = 0 2L b x dx 18 where p =total deformation of the prestressing tendons and b x =strain of the steel fiber at the height level of tendons at section x from the side support. The deformation of prestressing tendons can also be calculated as Fig. 10. Distribution of eccentricity of the tendons to the elastic neutral axis of transformed section along the beam: a position of the elastic neutral axis; b actual distribution of e x ; and c simplified distribution of e x JOURNAL OF STRUCTURAL ENGINEERING ASCE / NOVEMBER 2009 / 1381

6 Fig. 11. Analytical model for calculating the strain of steel beam under moment in the sagging moment regions b = e B M P x + M T x + N T x 21 EA where M P x, M T x, and N T x represent the distribution of internal forces along the beam due to both the external loads and the increase of prestressing tendon forces. In the region of sagging moment, the distribution of strains across the height of a composite beam section is shown as Fig. 11. Due to the slip effect, the slip strain which results in the additional curvature exists in the adjacent fibers at the interface of concrete and the steel beam. Therefore, it is necessary to modify the elastic results of the steel strains. It can be seen from Fig. 11 that the strain of the steel fibers where the prestressing tendons are positioned in the region of sagging moment can be calculated as Fig. 12. Calculation diagram of midspan deflection of prestressed continuous composite beams T = 2L 1 e B A, M,P EA 0 + e B C 1 L + C 2 e 0 27 where C 1 and C 2 are coefficients related to,, m, n, and, calculated as follows: C 1 = 1 2 R 1n 2 R 1 1 n 1 2 R 1 + m 2 m 1 2 b + = be b where b =additional strain due to the slip effect and be = strain calculated by beam theory and the transformed section method be = M P x + M T x e 23 B 1+ Assuming that the steel beam and concrete flange have the same curvature and the distribution of stresses and strains across the height of a section due to the slip effect is linear, the additional strain can be obtained as b = y e The reduced stiffness method Nie and Cai 2003 gives 24 = M EI = M P x + M T x 25 B 1+ Based on the analysis above and considering the steel strain due to axial forces, the strain of the steel fiber at the height level of steel tendons in the region of sagging moment can be derived as + b = B 1 e y P x + M T x + 1+ e M N T x EA = e B M P x + M T x + N T x 26 EA where reflects the slip effect between steel and concrete interface in the region of sagging moment. Considering the equilibrium conditions, deformation compatibility conditions, and physical conditions altogether, the increase of tendon force can be predicted from simultaneous equations from Eq. 18 to Eq. 26 as C 2 = 1 2 R 2n 2 R 2 1 n 1 2 R where A 0 =transformed sectional area of the composite beam and prestressing tendons, calculated as A P A A 0 = 30 1 n + n A P + L P /2L A A, M,P =area between the graph of moment distribution due to the external loads and the coordinate axis, multiplied by in the hogging moment and in sagging moment, positive for sagging moment and negative for hogging moment. For the loading case shown in Fig. 1, A, M,P =2C 1 P k L 2 ; then putting this expression of A, M,P into Eq. 27, we can finally obtain the increase of tendon force due to the two concentrated loads symmetric to the midspan, as a special case of Eq. 27, as T = ec 1 P k L B EA 0 + e C 1 L + C 2 e 0 31 The increase of tendon force due to other loading cases can be obtained using the same methodology. Deformation Calculation Once the length of concrete cracking region at the interior support and the increase of prestressing tendon force are determined, the deformation can be obtained following the same procedure as conventional continuous composite beams Nie and Cai For the loading case shown as Fig. 1, the midspan deflection of prestressed continuous composite beams can be calculated from the calculation diagram as Fig / JOURNAL OF STRUCTURAL ENGINEERING ASCE / NOVEMBER 2009

7 M = M P + T M T 34 Fig. 13. Deformation calculation of three-span prestressed continuous composite beam: a sketch of three-span prestressed continuous composite beams; b calculation model of three-span prestressed continuous composite beams; c M P graph; and d M T graph The expression for the midspan deflection can be derived as f = f 1 + f 2 + f 3 + f 4 32 where f 1 4 can be calculated using the formulas in Fig. 12. General Method for Deformation Calculation Using the developed methodology for calculating the deformation of two-span prestressed continuous composite beams, the deformation of prestressed continuous composite beams with any number of spans at serviceability limit states can be obtained. As an example Fig. 13 a shows the sketch of a three-span prestressed continuous composite beam. First, the length of concrete cracking region at every support n i can be determined according to Eq. 13, based on which the calculation model for prestressed continuous composite beams can be developed as shown in Fig. 13 b. Second, the moment diagrams due to external loads M P and due to the unit change of prestressing tendon force M T can be solved as shown in Fig. 13 c and Fig. 13 d, respectively. Then the change of prestressing tendon force can be calculated as T =, A M,P 2LB, A eea M, T 0 33 where A, M,P =area between the diagram of M P and the coordinate axis, multiplied by for the hogging moment and for the, sagging moment, positive for the sagging moment and A M, T =area between the diagram of M T and the coordinate axis, multiplied by for the hogging moment and for sagging moment, positive for the sagging moment. The other parameters are the same as those in the formulas for two-span prestressed continuous composite beams. The principle of superposition gives the moment distribution along the beam as Finally, select the related formulas from Fig. 12 and the calculation diagram for conventional continuous beams Nie and Cai 2003, and obtain the midspan deflection solutions for the middle and side spans. In the design practice, we always hope that the introduction of internal moment by the prestressing tendons can act in opposite direction to that induced by the externally applied loads. As a result, the tendon profile is usually designed according to the moment diagram of external forces as summarized in Fig. 13 a, in which two applied loads coincide with the point of change in angle in the prestressing tendons. In fact, the proposed general method for deformation calculation can also be used for the application of any given force at any arbitrary location along the beam span since Eqs. 33 and 34 do not containing the assumption that the diagram of M P and M T should satisfy some certain relationship. M P and M T can be obtained according to the actual load location and tendon profile, respectively, when using Eqs. 33 and 34 for more general analysis. Experimental Program Description of Tests In order to validate the developed analytical procedures, one nonprestressed CCB-1 and six prestressed PCCB-1 to PCCB-6 continuous composite beams were tested Li The details of these seven specimens are given in Table 1 and the layout is shown in Fig. 14. The tested beams are 8 m long with two equal spans, and the cross section Fig. 15 consists of a steel box beam and a concrete slab of 500 mm 70 mm. CCB-1 is a regular steel-concrete continuous composite beam without prestressing tendons. For PCCB series, the prestressing tendons were anchored at the two beam ends with two intermediate connections within each span and one at the interior support so that they could deform compatibly with the steel beam at these connecting points during the loading process. The main factors influencing the behavior of prestressed continuous composite beams are the form of tendons, the number of tendons, and the position of tendons. In order to make a comprehensive investigation on the prestressed continuous composite beams, the PCCB series were designed as Fig. 14 PCCB-1: straight-line, one-tendon, internal prestressed beam; PCCB-2: straight-line, two-tendon, internal prestressed beam; PCCB-3: fold-line, one-tendon, internal prestressed beam; PCCB-4: fold-line, two-tendon, internal prestressed beam; PCCB-5: straight-line, two-tendon, external prestressed beam; and PCCB-6: fold-line, two-tendon, external prestressed beam. The specimens were tested with two servo controlled hydraulic jacks, with each force being spread into two symmetric point loads as shown in Figs. 16 and 17. The test setup also included deflection measurements at the midspan and strain measurements at critical sections by strain gauges glued on the longitudinal reinforcement, steel beam, concrete slab, and prestressing tendons. Strains and deflections were measured automatically by a data acquisition system Isolated Measurement Pod system controlled by a computer. The arrangement of measuring devices is summarized in Fig. 18 in detail. The height of beam supports was adjustable, hinged for the interior one and sliding for the side ones. Before the beams were JOURNAL OF STRUCTURAL ENGINEERING ASCE / NOVEMBER 2009 / 1383

8 Table 1. Specimen Parameters Specimens Items CCB-1 PCCB-1 PCCB-2 PCCB-3 PCCB-4 PCCB-5 PCCB-6 Prestressing force kn Elastic eccentricity at beam end e Elastic eccentricity in the region of sagging moment e Elastic eccentricity in the region of hogging moment e One tendon area mm Shear stud number per row n s Shear stud spacing within 1.36 m from the side support Shear stud spacing within 2.64 m from the interior support Thickness of concrete slab Transverse reinforcement ratio % Longitudinal reinforcement ratio % Strength of concrete slab MPa Yield strength of steel top flange MPa Yield strength of steel bottom flange MPa Yield strength of steel web MPa Ultimate strength of steel top flange MPa Ultimate strength of steel bottom flange MPa Ultimate strength of steel web MPa Note: 1 Shear capacity of studs V u =0.43A us Ec f c 0.7A us f u, where f u =500 MPa, A us = d us /4, d us =8 mm, E c =10 5 / 22+33/ f cu, f c =0.8f cu, and f t =0.26f cu ; 2 the concrete strength was measured with three 150-mm-cubes per beam the same day as the beam test; and 3 the steel and reinforcement specimens were both tested in tension to determine the yield strength and ultimate strength. Fig. 14. Design of steel beams and prestressing tendons 1384 / JOURNAL OF STRUCTURAL ENGINEERING ASCE / NOVEMBER 2009

9 Fig. 15. Dimensions of cross section tested, the height of the supports was adjusted so that the support reacting forces could agree with the elastic theoretical results under the self-weight. For CCB-1, the specimen was monolithically loaded to failure, and for the PCCB series, the specimens were initially prestressed first and then were tested to failure. Results of Tests Prestressing the entire composite section cambered the beam upwards in the midspan first. At the initial stage of loading, the behavior of the specimens was linear and no cracks were visible on the slab. The concrete slab at the interior support did not crack until the load reached the crack load mainly depending on the prestressing degree. As the load increased, the cracks extended slowly and new cracks appeared in succession on the top surface of the slab in the region of hogging moment. When the load exceeded about 80% of the ultimate load P u, the steel bottom flange at the interior support became compressive yielding followed by the tensile yielding of the longitudinal reinforcement with the cracks expanding rapidly. When the ultimate load of the specimen was approached, the concrete slab at the first loading point from one side support was crushed, then with the other side of concrete slab crushed, and finally the beam reached its ultimate load. Plastic hinges in the region of sagging moment and hogging moment were formed and a full ductile failure was developed. Fig. 19 shows a typical failure mode and Fig. 20 shows a typical cracking pattern. The main test results are shown in Table 2. Fig. 21 gives the load versus the midspan deflection curves where the load in the vertical axis is the load from one hydraulic jack. The curves show a large plateau after the yielding of steel and reinforcement at the interior support, indicating a good ductile failure with large deformation in the inelastic region. Comparisons between CCB-1 without prestressing and PCCB series with prestressing in Fig. 21 show that the prestressing force enhances significantly the stiffness and loading capacity of continuous composite beams. Comparisons between PCCB-1 with one tendon and PCCB-2 with two tendons in Fig. 21 a as well as PCCB-3 with one tendon and PCCB-4 with two tendons in Fig. 21 b show that the increase of the tendon number has relatively insignificant effect on the stiffness, Fig. 17. Test Setup but the enhancement of ultimate load can be observed. Fig. 21 c shows PCCB-3 with fold-line tendons has larger stiffness and ultimate load than PCCB-1 with straight-line tendons because the moment produced by the variable eccentric prestressing force in PCCB-3 nearly counteracts the moment produced by the applied load. Comparisons in Figs. 21 d and e give the same observation. Fig. 21 f shows that for beams with straight-line tendons, PCCB-5 with external tendons has almost the same stiffness and ultimate load as PCCB-2 with internal tendons. However, for beams with fold-line tendons, PCCB-6 with external tendons has larger ultimate load than PCCB-4 with internal tendons as shown in Fig. 21 g. In order to investigate the slip performance between the steel and concrete, twelve extensometers are arranged along the beam as shown in Fig. 22. Fig. 23 shows typical curves for slip performance. It is found that the distribution of the slip along the beam is nonlinear. The maximum slip occurs near the two zero-moment points, first one between the interior support and the loading point and the second one between the loading point and the side support. Fig. 24 shows the tendon force versus the load curve for PCCB-1. Before the load reaches 80% P u, the tendon force increases approximately linearly with the increase of load, which confirms the analytical result derived in Eq. 27. The nonlinear effect between the tendon force and the load becomes prominent when the load is high. Experimental Verification All specimens tested can be analyzed using the procedure developed in the present study. Based on the specimen parameters, the length of concrete cracking region in the region of hogging moment and the increase of prestressing tendon force can be obtained from Eqs. 13 and 31, respectively. Using the analytical Fig. 16. Configuration of beam tests JOURNAL OF STRUCTURAL ENGINEERING ASCE / NOVEMBER 2009 / 1385

10 Fig. 18. Arrangement of measuring devices model shown in Fig. 2 b and introducing the formulas selected from Fig. 12 into Eq. 32 will finally solve the midspan deflection f. Comparison of predictions and measurements for 50% of ultimate load is shown in Table 3. From Table 3 we can see that the internal force of the prestressing tendon under service loads can be accurately calculated using Eq. 31. In the deflection prediction, by ignoring the increase of the tendon force, the calculated deflection values are overestimated, meaning that this method is conservative for the engineering design, and considering the increase of the tendon force can improve the accuracy of analytical predictions. The calculated values are in good agreement with the experimental results. Conclusions Fig. 19. Typical failure mode A deformation calculation method for prestressed continuous steel-concrete composite beams under service loads has been in- Fig. 20. Typical cracking pattern 1386 / JOURNAL OF STRUCTURAL ENGINEERING ASCE / NOVEMBER 2009

11 Table 2. Main Test Results Specimen Crack load P cr /kn Ultimate load P u /kn Ultimate sagging moment/kn m Ultimate hogging moment/kn m Yield deflection f y /mm Ultimate deflection f u /mm Ductile coefficient f u / f y CCB PCCB PCCB PCCB PCCB PCCB PCCB vestigated in this paper. Based on the simplified calculation model, formulas for calculating the length of cracking region of concrete slab in the region of hogging moment and the increase of prestressing tendon force have been derived. With the comparison between the predicted and measured results, the following conclusions can be drawn: 1. The internal force of the prestressing tendon under service loads can be accurately calculated by the formulas proposed in this paper. Fig. 22. Extensometer to measure the slip between steel and concrete Fig. 23. Typical slip performance Fig. 21. Load versus midspan deflection curves Fig. 24. Tendon force versus load curves JOURNAL OF STRUCTURAL ENGINEERING ASCE / NOVEMBER 2009 / 1387

12 Table 3. Comparison between Analytical Predictions and Test Results Specimens P k P u n T kn T kn T T PCCB PCCB PCCB PCCB PCCB PCCB Mean Standard deviation Note: T =analytical results of prestressing tendon force; T=measured prestressing tendon force; f N =measured midspan deflection of the north span; f S =measured midspan deflection of the south span; f =average measured midspan deflection; f 1 =analytical results of midspan deflection by ignoring the increase of tendon force; and f 2 =analytical results of midspan deflection considering the increase of tendon force. f N f S f f 1 f 1 / f f 2 f 2 / f 2. By ignoring the increase of the tendon force, the calculated deflection values are overestimated, meaning that this method is conservative for engineering design, and considering the increase of the tendon force can improve the accuracy of analytical predictions. As the calculated values are in good agreement with the experimental results, the proposed formulas can be reliably applied to routine design practice. 3. Based on the formulas for calculating the deformation of two-span prestressed continuous composite beams, a general methodology for deformation analysis of prestressed continuous composite beams has been proposed, which has expanded the application range of the analytical method developed in the present study. Acknowledgments The writers gratefully acknowledge the financial support provided by the National Natural Science Foundation of China Grant Nos and , Changjiang Scholars, and Innovative Research Team in University Grant No. IRT00736 Notation The following symbols are used in this paper: A cross-sectional area of transformed composite section; A 0 cross-sectional area of transformed composite and prestressing tendon section; B reduced stiffness of composite beam in the region of sagging moment; EA axial stiffness of composite section in the region of sagging moment; E P A P axial stiffness of prestressing tendons; E 2 A 2 axial stiffness of composite section in the region of hogging moment; EI bending stiffness of transformed composite section in the region of sagging moment; E 1 I 1 bending stiffness of transformed composite section in the region of sagging moment; E 2 I 2 bending stiffness of transformed composite section in the region of hogging moment; e elastic eccentricity in the region of sagging moment; elastic anchor eccentricity; e 0 e 02 vertical distance from the beam anchor to the elastic neutral axis in the region of hogging moment; e 2 elastic eccentricity in the region of hogging moment; F equivalent vertical force applied on the composite beam by the prestressing tendons; f deflection increment under service loads; f 1 initial deflection after prestressed; h s height of the steel beam; L span length; L P length of prestressing tendons; elastic internal moment under service loads; internal moment under service loads; M ek M k M P M T internal moment under loads; internal moment due to the increase of tendon force; M T internal moment due to the unit change of tendon force; m loading position; N T internal axial force due to the increase of tendon force; n coefficient of the length of concrete cracking region in the region of hogging moment; P external load; P k service load; T internal force of prestressing tendons; T 0 initial tendon force; W section modulus of transformed composite section at the top of concrete flange; y distance from the elastic neutral axis in the region of sagging moment to the bottom fiber of the steel; B/E 2 I 2 ; E E s /E c ; e/e 2 ; EA/E 2 A 2 ; P total deformation of the prestressing tendons; f 2 deflection increment under service loads; T increase of prestressing tendon force; additional curvature due to slip effect in the region of sagging moment; b strain of the steel fiber at the height level of tendons; parameter for slip effect in the region of sagging moment; 1388 / JOURNAL OF STRUCTURAL ENGINEERING ASCE / NOVEMBER 2009

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