Tóm tắt Luận án Redundancy of bridges in Vietnam

INTRODUCTION In the bridge design specifications of Vietnam (22-TCN-272-05), 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 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. 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 ease-to-use in order to be used by the engineer. This research also contributes to the calculation of redundancy for typical types of bridges in Vietnam. OVERVIEW OF REDUNDANCY AND AIMS OF RESEARCH 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. 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 System factor The system factoris the coefficient relates to the safety, redundancy and ductivity of the structure system. Redundancy in Specification 22TCN 272-05 According to Specification 22TCN 272-05, 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. håYi Qi£F Rn = Rr (1.1) where: h= hDhRhl > 0.95 h = load modifier: a factor relating to ductility, redundancy and operational importance. hD = a factor relating to ductility hR = a factor relating to redundancy hI = a factor relating to operational importance Multiple-load-path 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 failure-critical and the associated structural system as nonredundant Alternatively, failure-critical members in tension may be designated fracture-critical.. Those elements and components whose failure is not expected to cause collapse of the bridge shall be designated as nonfailure-cntical and the associated structural system as redundant. For the strength limit state:: hR≥ 1.05 for nonredundant members = 1.00 for conventional levels of redundancy ≥ 0.95 for exceptional levels of redundancy For all other limit states::hR= 1.00 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 hR, which should be applied in 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 Db and the system factor fs. However, the proposed procedure of these authors is nor capable of directly calculating the redundancy factor of the bridge structure. 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 22-TCN-272-05 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. 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 fs. Evaluation and Classification of bridge substructures Classification of typical substructures According to survey, shaped-substructure systems of bridge are classified as following : Flexual shaped-structure : 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. Hypothesis of structure working condition and related limit state Method of analysis redundancy Computing redundancy Relation between resistance factorFs, reability factorβmember and the system resever factor Ru The system resever factor Ru of some typical substructure. 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. Safety level of superstructure Limit States Life Cycle and load model, reliability index Reliability mothology Determination of reliability index « Direct » method to determine the redundancy factor Step by step method to detemine the redundancy factor System reserve factor System reserve factor for typical bridge superstructure Bridge rating 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 CALCULATE THE REDUNDANCY OF THE STRUCTURE BASE ON NONLINEAR MODEL AND EMBEDDED DISPLACEMENT FINITE ELEMENT METHOD 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, ED-FEM) . 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. 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 cross-section of the beam remains plain but not perpendicular to the neutral axis of the beam. The incline angle is φ Q C F Γu f(x) Γq q(x) m(x) Figure0.1. 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 re-writen 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 normal-function matrices for two-node 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: (8) Force-deformation curve (moment-curvature curve and shear force/shear strain curve) for reinforced concrete beam When considering the “jump” in displacement, the displacement vector at a cross-section 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 re-written 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 e-written: (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 Using "multi-layer" method to account the stress and deformation statement in beam Ɛ yy =0 s yy =0 s xx τ xy τ yx γ /2 Ɛ xx γ /2 n-layer in vertical Figure 2. Stress-strain at a layer Divide the cross-section 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 inner-force (moment, shear, axial loading) and strains (curvature, elongation, shear deformation) of the Timoshenko beam can be calculated by the following procedure. NO:Modify và κ OK OK Calculate normal strain () due to the assuming curvature and neutral axis position Calculate stress state () from strain state () from stress-strain relation equations. Calculate normal stress () and stress stress () from Mohr circle. , ; Check if N= Nu, M = Mu END BEGIN Determine inner forces: Mu, Nu, Vu Assume the distribution of shear strain (for example parabolic distribution and maximum shear strain Calculate strain state () at layer “I” from , Check if V = Vu NO: Modify γxy Figure 3. Flow-chart to calculate relation between inner forces and strain of Timoshenko beam element base on the multi-layer method Establish the table to determine bending curves (M- κ) depend on axial force and shear force Base on the flow-chart on the Figure 3, we can determine the momen-curvature 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 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 1651-2:2008 - 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 Figure 7. Layout of reinforcement in test beam Loading procedure * Loading lay-out The beam is test follow the four-poit bending test as the following: 80cm 80cm 80cm 10cm 220cm 10cm Figure.8 Loading lay-out * 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. 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 Beam state Curvature (1/m) Moment (kNm) Tangent Modulus Begin 0 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) Bema state Curvature (1/m) Moment (kNm) Tangent Modulus Begin 0 0 Name Value “Crack” moment 0.0001 2.784 EI 278446.6 “Yield” moment 0.0003 10.919 K1I 25260.2 “Ultimate” moment 0.0007 18.523 K2I 3281.01 Remaining moment after failure 0.0012 18.314 Kbar -11350 (normal line : the first hypothesis of modeling, dash-line: 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 elasto-plastic model for the steel bar, which ignore the “hardening” of rebar after yielding. 3.6. Conclusion of chapter 3 A the flow-chart 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. APPLIED EXAMPLES OF NONLINEAR MODEL AND DIRECT PROCESS IN ANALYSING AND CALCULATING THE BRIDGE REDUNDANCY Two-column 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. 4.2m 0.4m 3.8m A-A A-A B-B 700kN 700kN Q Figure 10. Two-column 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 two-column pier 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 moment-bending 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 d