Research objectives: To form a direct, simple and easy-to-use methodology to determine the redundancy of bridge’s structure. Form a finite element method in order to take into account the nonlinear behavior of the structure, when one or some of the main structure part is failure.
1 INTRODUCTION In the bridge design specifications of Vietnam (22TCN27205), the redanduncy is an important input data, which can strongly effect the dimension of structure and scope of design from increasing or reducing the value of structure behavior in the checking equation. However, there is no research on how to determine this coefficient nor a simple methodology to select the right redundancy coefficient in designing bridge in Vietnam. Therefor, it is necessary to create and improve a guideline to calculate the redundancy, which has to be simple and useful enough, to be appled by the engineer. That is the reason why we choos the topic “Redundancy of bridges in Vietnam” Aims of research: To form a direct, simple and easytouse methodology to determine the redundancy of bridge’s structure. Form a finite element method in order to take into account the nonlinear behavior of the structure, when one or some of the main structure part is failure Scope of research: The research considers the nonlinear behavior of structure, the scope of research is the superstructure and the substructure of bridge construction in Vietnam Methodology: We firstly proposal a theoretical model then verifying the theoretical model with experimental result Scientific and practical meaning This study has clearly explain the notation “redundancy” in designing bridge, introduced methods to determine the redundancy in bridge structure. The scientific meanings of the study are improving the “direct” method to determine the redundancy of bridge design, which has to be more easetouse in order to be used by the engineer. This research also contributes to the calculation of redundancy for typical types of bridges in Vietnam CHAPTER 1 OVERVIEW OF REDUNDANCY AND AIMS OF RESEARCH 1.1 Literature review In recent years, the overall trends of designing bridge in Vietnam is the using of more and more complex structure However, there is not many reseaches on the determination of redundancy of bridge structure in Vietnam, except several research of Prof. Duc Nhiem Tran on reability of bridge structure as a foundation of determining the redundancy. In over the world, Michel Ghosn, Fred Moses and Jian Yang are the pioneer researchers who study on the redundancy of bridge structure. Their study has define the redundancy of bridge structure and proposed several method to determine the redundancy in bridge structure 1.1.1 System resever ratio (R) The redundancy of bridge is defined as the capability of a bridge to continue to carry loads after damage or the failure of one or more of its member. In other words, the redundancy of a bridge is its maximum loading capacity Limit states using to determine the redundancy of bridge includes: Member failure Ultimate limit state Functionality limit state Damage condition limit state 1.1.2 System factor The system factoris the coefficient relates to the safety, redundancy and ductivity of the structure system 1.1.3 Redundancy in Specification 22TCN 27205 According to Specification 22TCN 27205, redundancy is considered based on load modifier. Each component and connection shall satisfy Equation (1.5) for each limit state, unless otherwise specified. All limit states shall be considered of equal importance Yi Qi Rn = Rr (1.) where: = D R l > 0.95 = load modifier: a factor relating to ductility, redundancy and operational importance D = a factor relating to ductility R = a factor relating to redundancy I = a factor relating to operational importance Multipleloadpath and continuous structures should be used unless there are compelling reasons not to use them Main elements and components whose failure is expected to cause the collapse of the bridge shall be designated as failurecritical and the associated structural system as nonredundant Alternatively, failurecritical members in tension may be designated fracturecritical Those elements and components whose failure is not expected to cause collapse of the bridge shall be designated as nonfailurecntical and the associated structural system as redundant For the strength limit state:: R ≥ 1.05 for nonredundant members = 1.00 for conventional levels of redundancy ≥ 0.95 for exceptional levels of redundancy For all other limit states:: R= 1.00 1.2 Remains in study of redundancy of bridges The bridge design specification AASHTO has been defined the redundancy and other related factors. AASHTO introduces the redundancy factor , which should be applied in R designing bridge The Vietnamese bridge design specifications 22 TCN 272 05 has also defined and using the same factor as AASHTO LRFD in take into account the redundancy in design bridge. Hovewer, the remaining problem is that the missing of the “direct” procedure to determine the redundancy of the structure. Michel Ghosn et al have been studied on determining the system reserve ratio (Rn); the reability index and the system factor s. However, the proposed procedure of these authors is nor capable of directly calculating the redundancy factor of the bridge structure 1.3 Problem statement Bese on the literature review, in this research we are plan to: 1) Clearly explain the notation “redundancy” and the factors relate to redundancy in the bridge design specifications 22TCN27205 of Vietnam 2) Introduce a “direct” procedure to calculate the redundancy factor of the structure. 3) In order to have the “direct” procedure, it is necessary to develop a structural model and the coressponding finite element method to determine the ultimate bearing of structure under the ultimate limite state and the functionality limit state. This model should be able to model the nonlinear behavior of the structure, especially when one or more structural parts have been failure. CHAPTER 2 REDUNDANCY OF BRIDGE STRUCTURE AND THE “DIRECT” PROCEDURE TO DETERMINE THE REDUNDANCY OF BRIDGE The study consists of several main steps: The first step is to category the typical structure of bridge, including the substructure and the superstructure. The second step is to define the limit states: the state when the structure loose its working capacity The next step is using the “direct” method with nonlinear modeling to determine the ultimate load of structure, taken from the corresponding limit states and the typical bridge structure Finally, calculate the redundancy factor from the ultimate loads The redundancy factor is represented through: the system reserve factor ®, the reability factor βmemberor the system factor s 2.1 2.1.1 Evaluation and Classification of bridge substructures Classification of typical substructures According to survey, shapedsubstructure systems of bridge are classified as following : Flexual shapedstructure : pier wall , kết cấu uốn đơn cột, kết cấu uốn hai cột và kết cấu uốn nhiều cột Foundations: strip foundations, pile foundation and cassion foundation Geological condition : stone, sand and clay Connection: motholithic, continous and simple 2.1.2 2.1.3 2.1.4 2.1.5 Hypothesis of structure working condition and related limit state 2.1.6 2.1.7 2.2 The system resever factor Ru of some typical substructure Method of analysis redundancy Computing redundancy Relation between resistance factor resever factor Ru s , reability factorβ member and the system Direct procedure to determine the redundancy for substructure Evaluation and determination of bridge superstructures The redundancy of bridge superstructure is the capacity of to continue to carry out loads after the damage or failure of one of its members. The method of determining redundancy factor is« direct » analysis method .The method includes : (a) define the limit states; (b) calculate the behavior of the structure at limit states and the corresponding ultimate load; (c) From the ultimate load calculate the redundancy factor of the superstructure 2.2.1 Safety level of superstructure 2.2.2 Limit States 2.2.3 Life Cycle and load model, reliability index 2.2.4 Reliability mothology 2.2.5 Determination of reliability index 2.2.6 « Direct » method to determine the redundancy factor 2.2.7 Step by step method to detemine the redundancy factor 2.2.8 System reserve factor 2.2.9 System reserve factor for typical bridge superstructure 2.2.10 Bridge rating 2.3 Conclusion in chappter 2 Proposal of direct procedure for determining the redundancy factor: Indentify internal force of structure according to Specifications. (Ptk) Structure modeling, applying design load for structure Increase the design load to determine load factor corresponding to limit states: Serviceability limit state: Psd Strength limit state: Pcd Determine the redundancy factor corresponding to limit states.The redundancy factor is the smallest one. If redundancy factor >1 the bridge is redundant,and vice versa CHAPTER 3 CALCULATE THE REDUNDANCY OF THE STRUCTURE BASE ON NONLINEAR MODEL AND EMBEDDED DISPLACEMENT FINITE ELEMENT METHOD 3.1 General Nonlinear model which taking into account the bending and shear failure of the structure was proposed by Armero, Ibrahimbegovic, Ngo, Pham, Bui and some other researchers for Timoshenko beam element (in the framework of embed displacement finite element method, EDFEM) In this chapter, the author proposes the procedure to apply this nonlinear model to calculate the load corresponding to the ultimate limit state and the serviceability limit state for reinforce concrete structure, which is necessary for the proposed “direct” method to determine the redundancy factor proposed in chapter 2 3.2 Summary of Timoshenko beam theory combinned with "jump" in displacement Using Timoshenko beam element (with taking into accout the shear strain of the beam) in order to better model the behavior of the beam, in which, after subjecting to loading, the crosssection of the beam remains plain but not perpendicular to the neutral axis of the beam. The incline angle is φ Figure0 Beam element subjected to external loading Note u(x) is the displacement vector of point x, x ϵ [0,l]: (1) The deformation vector of the point is equal to: (2) NoteN, V andM is the axial load, shear force and bending moment of the beam at the position x, the equilibrium equations are: (3) Equation (3.3) is the analytical form of the equilibrium equation. Form of thee quation (3.3) can be rewriten as the following (4) Where σ is the vector of inner force () When considering the relation of internal force and deformation, three equations of the finite element method can be described: (1). Approximating the displacement of beam by normal functions for two node beam (5) N(x) is a normal matrix d is the displacement vector: (2). From the above equation, the deformation approximated equation can be shown as: in which, N and B are normalfunction matrices for twonode Timoshenko beam , (3). Modifying the analytical form of the equilibrium equation (3) base on the virtual work principle (6) (4). Approximating virtual displacement function w(x): d is the virtual displacement vector at the nodes of element * (5). Replacing the virtual displacement and virtual deformation equations into equation (6): (7) Traditional form of finite element equation: 3.3 (8) Forcedeformation curve (momentcurvature curve and shear force/shear strain curve) for reinforced concrete beam When considering the “jump” in displacement, the displacement vector at a crosssection of the beam is: (9) whereis the “jump” of displacement at point andis the Heaviside function, defined by the equation:withand with In this thesis, we useequal to 0 atx = 0 and equal to 1 atx = l. The displacement vector becomes the composition of two ingredients: the continuum part and the irregular part: (10) Wherecan be represented byand: The deformation vector becomes: (11) Where, is the Dirac function, represent the trend of the “jump” in displacement. Equation (3.12) can be rewritten as: (12) In which, G equals to , L is transformation from displacement to deformation. Applying the interpolation function for displacement , the interpolation of displacement at equation (3.7) can ber ewritten: (13) We have already chosen the form of at (3.10) is the interpolation function. With this interpolation function for the displacement vector, the finite element equation becomes : (14) where 3.4 Using "multilayer" method to account the stress and deformation statement in beam Figure 2. Stressstrain at a layer Divide the crosssection into layers, the depth of each layer is small enough sothat the stress – strain state at each layer can be considered to be uniform. The inner forces can be computed from the stress at each layer from the following equation: `(15) Where: : normal stress at layer i yi : distance from neutral axis to layer i ai : diện tích lớp thứ i Nc, Ns: number of concrete layers and steel bar layers The relation between innerforce (moment, shear, axial loading) and strains (curvature, elongation, shear deformation) of the Timoshenko beam can be calculated by the following procedure 10 Figure 3. Flowchart to calculate relation between inner forces and strain of Timoshenko beam element base on the multilayer method 10 11 3.5 Establish the table to determine bending curves (M κ ) depend on axial force and shear force Base on the flowchart on the Figure 3, we can determine the momencurvature curve for an example of reinforce concrete beam which takine into account the effect of shear force and axial force as the following figure: Figure 4. Depending of M к curve on axial load in the beam Figure 5.Depending of M к curve on axial load in the beam 3.6 3.6.1 Pilot test for validity of the proposal model Configuration of the test beam * Concrete:35MPa (base on compressive strength test) * Reinforcement: CB400V base on TCVN 16512:2008 Diameter D = 12mm * Dimension: Total length of the beam is 2.4m, calculated length of the beam is 2.2m, the depth of the beam is 0.2m, the width is 0.14m Two steel plates 200mx140mmx3mm are attached at the bottom face of the beam at two ends in order to subjected to the local reaction. Two other steel place 200mmx140x3mm is placed at the top face of the beam, with distance equals to 0.8m from the beam ends in order to subjected to jacking force * Reinforcement Two D12 reinforcement is placed at the top, two other D12 reinforcement is place at the bottom fiber with the thickness of the cover layer is 40mm. The stirrups are in diameter 12mm, spacing between stirrup is 200mm. The beam is designed to meet the minimum and the maximum reinforcement ratio due to 22 TCN 272 05 11 Figure 7. Layout of reinforcement in test beam 3.6.2 12 Loading procedure * Loading layout The beam is test follow the fourpoit bending test as the following: Figure.8 Loading layout * Loading velocity The force applying into the beam with the velocity equal to 2.5kN/m, slowly enough to not result in the dynamic response in the beam * Test result The vertical displacement at the bottom of the beam is captured by displacement meter LVDT. Other LVDT is placed at the middle section of the beam in order to measure the crack opening width 3.6.3 Comparison between test result and modeling result The first model: using the pure bending model with the input variables shown in the Table 3.6 (shear force equals to zero). Table 3.6. Input variables for pure bending model Curvature Moment Beam state Tangent Modulus (1/m) (kNm) Begin 0 Name Value “Crack” moment 0.0001 2.953 EI 295309.148 “Yield” moment 0.0003 11.148 K1I 26050.5 “Ultimate” moment 0.0007 19.328 K2I 3449.11 Remaining moment after failure 0.0012 19.240 Kbar 11250 The second model: using bending which taking into account the effects of shear force. The input variables are shown in the Table 3.7 Table 3.7. Input variable (taking into account shear force) Curvature Moment Bema state Tangent Modulus (1/m) (kNm) Nam Begin Value e 2.784 278446.6 “Crack” moment 0.0001 EI 12 13 “Yield” moment 0.0003 “Ultimate” moment 0.0007 Remaining moment after failure 0.0012 10.919 18.523 18.314 K1I K2I Kbar 25260.2 3281.01 11350 (normal line : the first hypothesis of modeling, dashline: the second hypothesis of modeling (taking into account the shear force) Figure 8. Force/deflection curve due to modeling result Figure 9. Model results vs Experimental result In can be seen from the figure that the model results good follow the experiment result of the reinforce beam. The “ultimate” loading in reinforced concrete beam base on the first and the second assumption is 67.95 kN and 67.90 kN, respectively, where as the “ultimate” loading due to experimental result is 77.14kN, making the difference is about 10%. This difference occurs due to the perfectly elastoplastic model for the steel bar, which ignore the “hardening” of rebar after yielding 3.6. Conclusion of chapter 3 A the flowchart which allows to determine the dependence of bending model to the shear force and the axial force was proposed in this chapter. This model can be applied to determine the loads due to “ultimate” loading state and “serviceability” loading state in the “direct” method of determining the redundancy of structure in Chapter 2 CHAPTER 4 APPLIED EXAMPLES OF NONLINEAR MODEL AND DIRECT PROCESS IN ANALYSING AND CALCULATING THE BRIDGE REDUNDANCY 4.1 4.1.1 Twocolumn pier Operation analysis of pier under the effect of horizontal thrust following nolinear model Consider to a frame pier with the height is 4.6m, distance between 2 columns is 3.6m. Vertical forces transfer from bearing to piers at column centerline. The value of a vertical force is 700kN Figure 10. Twocolumn frame pier Figure 10 shows the dimension of columns, transverse beams , pier caps and reinforcement arrangement. Table 4.1 presents the material characteristics. Table 3. The material characteristics of twocolumn pier 13 14 Comcrete Elastic modulus Ec 26889.6 N/mm2 Compressive strength f’c 30 N/mm2 Steel Yield strength fsy 400 N/mm2 Elastic modulus Es 20000 N/mm2 The forces from superstructuer is direct transferred to two columns, load value acting to each column is 700 kN. Horizontal loads Q applied to frame pier systems on pier caps (Figure 4.1). Application to suggested model for concrete reinforcement structure in Article 3.3, chapter 3, determined torelation curve between momentbending of column and transverse beam: Figure 11. Relation moment bending of column and transverse beam Note that bending resistance of column is increased significantly compared to the transverse beams due to compression acting to columns (compression force is 700 kN) Relation shear force deformation of column is determined : Figure 12. Relation shear force deformation of column Application, Figure 4.4 shows the relation between horizontal force and horizontal displacement of pier cap Figure 13. Theorelation between horizontal force and horizontal displacement of pier cap From the figure 13, we can infer: Horizontal forces referred to service limit state (displacement) Horizontal forces referred to strength limit state is 242.46kN Ultimate horizontal forces reached when 2 cross sections on the pier are damaged, this is the left column stub section and pier cap section that is close to the left column (Figure 14) Figure 14. Displacement of pier when the value of horizontal displacement is 160mm 4.1.2 Determine of redundancy factor of twocolumn pier base on “direct” procedure Step 1: Determine of load factor following linear analysis of design specification 14 15 According to design specification, the ultimate moment of the column is M req=161 kNm Standard horizontal forces acting to pier cap referred to pulsating load of pier cap is F = 50 kN According to linear analysis, the value leadin to the maximum bending moment at column section is 56.7 kN Hence, the value of load factor due to linear analysis is: LFreq = 161/56.7= 2.82 Step 2. Determine the limit horizontal load at serviceability limit state. Horizontal load at serviceability limit state is the horizontal force caused large displacements on structure make structural inability to use. For pier structures, this displacement is H / 50 = 4600mm / 50 = 92mm. Consider to diagrams between force displacements (Figure 13), the value of ultimate horizontal load is 230kN. Hence, the value of load facor is: LFf= 230/50=4.6. The value of redundancy facor referred to service limit state: r f = (4.6/2.82)/1.2) =1.358 Step 3: Determine of ultimate horizontal force due to strength limit state From the analysis result in figure 13, the ultimte horizontal force at strength limit state is 242,46kN. Hence, the value of load factor is: Lfu = 242.46/50 =4.04 The value of redundancy coefficient referred to strength limit state: r u = 4.04/2.82/1.2= 1.193 So, the value of redundancy coefficient of the structure is the smaller of the two values: value due to strength limit state and due to serviceability limit state, redundancy factor is 1.193 4.2 4.2.1 Threecolumn pier Operation analysis of horizontal forcebearing threecolumn To increase the reserve level of pier under the effect of horizontal force, we consider to the problem of horizontal forcebearing threecolumn pier. The dimension of the pier is similar to the case of twocolumn pier above. However, adding a column in between two old columns with dimension and reinforcement arrangement similar to the case of twocolumn pier. Figure 15. Threecolumn frame pier 15 16 Figure 15 shows the dimension of columns, transverse beams , pier caps and reinforcement arrangement The material characteristics is similar to the case of two coumn pier (Table 3). The forces from superstructuer is direct transferred to two end columns, load value acting to each column is 700 kN. Figure 16 presents the relation diagram between horizontal force and horizontal displacement of pier cap Figure 16. The relation between horizontal load displacement of threecolumn pier Hence, ultimate horizontal load of threecolumn pier is 330.22 kN Pier deformation referred to horizontal displacement of pier cap is 16cm that shown in figure 17. Note: In this moment, there is only one failure section, this is the right column stub section. Figure 17. Pier deformation when horizontal displacement of pier cap is 160mm 4.2.2 Determine of the redundancy factor of threecolumn pier using “direct” procedure Step 1. Determine load factor base on linear analysis of design specification According to design specification, the limiting moment of column section is Mreq=161 kNm Standard horizontal forces acting to pier cap referred to pulsating load of pier cap with value following is F = 50 kN. The force causes to the maximum bending moment on pier and the value is 37.5 kNm. Hence, the value of load coefficient following linear analysis is: LFreq = 161/37= 2.82 Step 2. Determined toredundancy coefficient referred to service limit state. Horizontal forces referred to service limit state cause to displacements, this displacement H/50 = 4600mm/50 = 92mm), F=320kN (figure 4.7) The load coefficient referred to service limit state is: LFf= 320/50=6.4. The redundancy coefficient referred to service limit state is: rf= (6.4/4.29)/1.2) =1.2435 Step 3. Determined toredundancy coefficient referred to strength limit state. Horizontal forces referred to sthength limit state is: F =330,22kN The load coefficient referred to strength limit state is: LFu= 330.22/50 =6.6044. 16 17 The redundancy coefficient referred to strength limit state is: ru = 6.6044/4.29/1.2= 1.2833 Như vậy hệ số tinh dư của kết cấu bằng 1.2435 4.3 4.3.1 Twospan continuous beam Vertical bearing capacity analysis of twospan continuous girder Consider to twospan continuous girder is shown in figure: Figure 18. Vertical load bearing twospan continuous girder Girdercrosssection structure is shown in figure 4.10 Figure 19. Girder crosssection structure Girder materials are shown in table 4: Table 4. The material characteristics of twospan continuous girder Concrete Elastic modulus Ec 26889.6 N/mm2 Compression strength f’c 30 N/mm2 fsy 400 N/mm2 Steel Yield strength Elastic modulus Es 20000 N/mm2 Analysis results following theory gives the forcedisplacement curve and behavior of girder during failure Figure 20. The relation of force and deflection at midspan when load increase Figure 21. Failure behavior of beam in strength limit state 4.3.2 Determine the redundancy of twospan continuous girder Step 1. Determine the ultimate load following elastic analysis :According to design specification, the ultimate moment of girder section M req = 161 kNm. External force causing to this bending moment is Freq =162 kN Step 2. Determine redundancy factor referred to the serviceability limit state. Horizontal force referred to service litmit state (cause to displacement with value is L/100 = 5000mm/100 = 50mm) is F=210kN. The redundancy coeficient rf = (210/162)/1.1) =1.18 17 18 Step 3. Determine redundancy factor referred to the strength limit state Horizontal force referred to ultimate limit state for failure condition: F =229.78 kN. ru = 229.78/162/1.3= 1.06 So, the redundancy factor of twospan continuous girder in this example is 1.06 4.4 Conclusion of chapter 4 In this chapter, thesis has determined the ultimate load and redundancy factor for three cases: Twocolumn pier, threecolumn pier and twospan countinuous girder These are the typical cases for superstructure and substructure in Vietnam The obtained results: The redundancy factor of twocolumn pier is 1.193 The redundancy factor of threecolumn pier is 1.2453, greater than twocolumn pier case The redundancy factor of twospan continuous girder is 1.06 These coeficients can not be represented for all types of superstructure and substructure of bridge in Vietnam because they were not investigated for all type of dimensions and material properties. However,these examples have proved the validity and the usage of the proposed direct procedure and the application of nonlinear embed displacement finite element method in determining the redundancy factor of the bridge’s structure CONCLUSION AND RECOMMENDATIONS Conclusions on the contributions of the thesis The thesis has done the analysis, modeling assumptions, reliability analysis and redundancy coefficient norm By studying, applying materialnonlinear analytical theory and extended finite element method, the thesis has proposed the direct redundancy determining process simpler than the process of previous authors to apply in bridge design The thesis also proposes nonlinear analysis model by finite extended element method to expand, allowing consideration to the work of the structure after the first main components are collapsed The thesis mentions the identification of typical structure forms in bridge’structure to determine the redundancy, help to set up the redundancy coefficients table for the structures to convenience in application The results of this thesis was development a reasonable basis for a consideration to the redundancy of span strcuture and substructure in the design and bridge 18 19 structureevaluation, and necessary data development to supplement the bridge design standard 22TCN 27205 The research results are: (1) development of analytical procedures for the redundancy norm of span structure and substructure and (2) provide a method analytical for the redundancy which can be applied to typical form of superstructure and substructure. Perpectives In this thesis, we have proposed a simple process and attached analysis tools to analysis the redundancy for bridge structure, including the superstructure and substructural Analysis model considered to the nonlinear collapse state that previous models have not mentioned such as: Shear failure, general failure of structure after a component was collapse The authors summarized the types of typical structures in bridges and a preliminary analysis to the structure redundancy. In the next study, post graduatewill apply this model to analysis to the redundancy for the typical bridge structures in Vietnam. In particular: The analysis is presented in this thesis is done individually for span structures and substructures This method is reasonable to the case when the span structures are connected to substructures via bearings The future research is the bridge structure contiguous connected between the two structuresystems The results of the analysis in this thesis have just provided some limitof typicalbridge structures. In the future, it may be extended to other typical forms Load model was used in the thesis that corresponds to the mode of LRFD specifications. The load model was developed based on the linear reaction of the bridge system and use a statistical database about traffic and truck weight In the future, it will be further normalized to the load position. 19 ... determination of redundancy of bridge structure in Vietnam, except several research of Prof. Duc Nhiem Tran on reability of bridge structure as a foundation of determining the redundancy. ... determine the loads due to “ultimate” loading state and “serviceability” loading state in the “direct” method of determining the redundancy of structure in Chapter 2 CHAPTER 4 APPLIED EXAMPLES OF NONLINEAR MODEL AND DIRECT PROCESS IN ANALYSING AND CALCULATING THE BRIDGE ... The Vietnamese bridge design specifications 22 TCN 272 05 has also defined and using the same factor as AASHTO LRFD in take into account the redundancy in design bridge. Hovewer, the remaining problem is that the missing of the “direct” procedure to determine