Stay Tuned! Practical Cable Stayed Bridge Design

Transcription

1 midas Civil Stay Tuned! Practical Cable Stayed Bridge Design 2017 Francesco Incelli

2 I. Introduction II. Modeling of the cable-stayed bridge a. Bridge wizard b. Girder Cross Section III. Nonlinear Effect a. Sag effects of long cables b. P-Delta effects c. Large deformations d. Material nonlinearity IV. Initial Cable Forces a. The Unknown Load Factor function - Constraints - Influence matrix b. Tuning of cables midas Civil

3 1. Introduction Major Characteristics of Cable Stayed Bridge The deck acts as a continuous beam with a number of elastic supports with varying stiffness. The deck and pylon are both in compression and therefore bending moment in these elements will be increased, due to second order effects. Application of these moments will be non-linear. The use of influence lines, which rely on the principles of linear superposition, can only be used as an approximate method of determining the stay loads. Nonlinear material properties (Creep and shrinkage) will also influence the design. 3

4 1. Introduction Design Process in Cable Stayed Bridge (Forward or Backward Construction Stage) Determine Back span to main span ratio Determine Cable Spacing Determine Deck Stiffness Determine Pylon Height Determine Preliminary Cable Force Deck Form (Concrete / Composite / Hybrid) Deck Design Deck Erection (Backward / Forward Stage Analysis) Unknown Load Factor Static Analysis Dynamic Analysis Lack of Fit Force Unknown Load Factor Cable Force Tuning 4

5 1. Introduction Design Step 1. Back span to main span ratio The ratio between back span and the main span should be less than 0.5. It influences the uplift forces at the anchor pier and the range of load within the back stay cables supporting the top of the pylon. The optimum length: between 0.4 ~ 0.45 of the main span. Design Step 2. Cable spacing a b The spacing of the stay anchors along the deck should be compatible with the capacity of the longitudinal girders and the limiting stay s size. The spacing should also be small enough so that the deck may be erected using cantilevering method. 5

6 1. Introduction Design Step 3. Deck stiffness The deflection of the longitudinal girders is primarily determined by the stay layout. Depth of girders should be kept to minimum, subject to sufficient area and stiffness being provided to carry the large compressive forces without buckling. Design Step 4. Pylon height The height of the pylon will determine the overall stiffness of the structure. As the stay angle increases, the required stay size will decrease as will the height of the pylon. However, the deflection of the deck will increase as each stay becomes longer. The most efficient stay is that with a stay inclination of 45. In practice the efficiency of the stay is not significantly impaired when the stay inclination is varied within 25 ~ 65. This implies an optimum ratio of pylon height above the deck (h) to main span (l) is between 0.2 and h l 6

7 1. Introduction Design Step 5. Preliminary stay forces The main span stay forces resist the dead loads such that there is no deflection of the deck or pylon. An initial approximation of the main span stay forces can be determined by considering the structure as a simple truss ignoring bending stiffness of both the pylon and the deck. Ignoring bending stiffness of the pylon will be a valid assumption as the bending stiffness of the pylon is usually small when compared to the axial stiffness of the stays. The back stay anchoring forces can be calculated assuming the horizontal component of the main span and back span stay forces are balanced at the pylon. Design Step 6. Deck form The primary factors influencing the choice of deck will be the length of the main span and deck width. Concrete deck section is the most economic for the span range m and the composite deck above 400m. Above 600m a hybrid combination is economic with the back span as concrete and the main span in an all steel construction. 7

8 1. Introduction Design Step 7. Deck design It is possible to minimize the moments in the deck under the dead load by tuning the loads in the stays to the small local moments arising from the span between stays. The balance between positive and negative live load moments at any section along the girder will not be equal. In most cases the properties of the deck section will be more favorable when resisting positive moments. Design Step 8. Deck erection The common method of deck erection is the cantilever method. The stay forces that are compatible with the final distribution of dead load moment and the defined structure geometry are known. However the initial stay forces introduced at each stage of the erection are not. Backward stage analysis: the completed structure is dismantled stage by stage. Forward stage analysis 8

9 1. Introduction Design Step 9. Static analysis For the final analysis, the most common approach is to model either a half or the entire structure as a space frame. The pylon, deck and the stays will usually be represented within the space frame model by truss elements. The stays can be represented with a small inertia and a modified modulus of elasticity that will mimic the sag behavior of the stay. Design Step 10. Dynamic analysis The seismic analysis of the structure Response of the structure to turbulent wind Time history transient analysis of vibrations 9

10 2. Modeling of Cable Stayed Bridge (1) Bridge Wizard Modeling symmetric or Asymmetric bridge truss & Cable element Box sloped girders Vertical station of Girder Cable Stayed Bridge Wizard 10

11 2. Modeling of Cable Stayed Bridge Truss Element Uniaxial tension-compression line element Used to model space trusses or diagonal braces Undergoes axial deformation only Equivalent truss element Tension-only line element Capable of transmitting axial tension force only This will consider decreased axial stiffness of cable due to sagging effect. Cable element is simulated as Equivalent truss element in linear analysis. h h element length Lh: horizontal projection length of the cable element 11

12 2. Modeling of Cable Stayed Bridge Elastic Catenary Cable Element Capable of transmitting axial tension force only Reflects the change in stiffness varying with internal tension forces (sagging effect) Tangent stiffness of a cable element applied to a geometric nonlinear analysis (Large displacement effect) 12

13 2. Modeling of Cable Stayed Bridge (2) Stiffened Girder using SPC The Import function permits the use of AutoCAD DXF Import files. CAD data Simple data entry using various modeling functions or The section property calculations are provided for Define the sections input section in SPC configuration by generating fully automated optimum meshes. The properties of hybrid sections composed of different material properties can be calculated. Define Section Shape in CAD Import SPC Section using Value Type of PSC Section Composite Section imported from SPC 13

14 3. Nonlinear Effect (1) Sag Effects of Long Cables h h element length Lh: horizontal projection length of the cable element 14

15 3. Nonlinear Effect (2) P-Delta Effect 15

16 3. Nonlinear Effect (3) Large deformations 16

17 4. Initial Cable Forces Unknown Load Factor in midas Civil This function optimizes tensions of cables at the initial equilibrium position of a cable structure. The program can calculate the initial cable force by inputting the restrictions such as displacement, moment, etc. and satisfying the constraints. Copy & Paste 21

18 4. Initial Cable Forces Unknown Load Factor in midas Civil Object Function type: Select the method of forming an object function consisted of unknown load factors. Linear: The sum of the absolute values of Load factor x scale factor Square: The linear sum of the squares of Load factor x scale factor Max Abs: The maximum of the absolute values of Load factor x scale factor Sign of Unknowns: Assign the sign of the unknown load factors to be calculated. Negative: Limit the range of the calculated values to the negative (-) field. Both: Do not limit the range of the calculated values. Positive: Limit the range of the calculated values to the positive (+) field. Simultaneous Equations Method Using linear algebraic equations, the equality conditions are solved. If the numbers of the unknown loads and equations are equal, the solution can be readily obtained from the matrix or the linear algebra method. 22

19 4. Initial Cable Forces Unknown Load Factor in midas Civil Inequality condition Object Function type midas Civil finds a solution to Inequality conditions, which uses variables that minimizes the given object functions. T 2 Numerous solutions satisfying the inequality conditions Linear Square Square Linear T 1 Max. Abs Max. Abs 23

20 4. Initial Cable Forces Unknown Load Factor in midas Civil Influence Matrix Moment/Displacement at the corresponding element/node ID due to a unit load applied for each load case. Ti δi Value = Σ(Ti * δi) 24

21 4. Initial Cable Forces Unknown Load Factor in midas Civil Tip to enter Constraint Constraint Position: Vertical Deformation of Span Center Node Horizontal Deformation of Pylon Top Node Once it is converged, try to increase Constraint condition. Once it is converged, try to decrease Constraint range. 25

22 4. Initial Cable Forces Unknown Load Factor in midas Civil Note when using Cable elements The unknown load factors obtained by using the Unknown Load Factor feature for the final stage model do not include the change in stiffness of the cable due to the change in pretension. Therefore the user must use truss element in Unknown Load Factor. In order to determine the pretension in the truss element to satisfy constraints, iteration will be required. The following procedure can be adopted: 1. Define the constraints and obtain the Unknown Load Factors for the Pretension Forces. 2. Determine the Pretension Force by multiplying those factors with the assigned Pretension Loads 3. Change the Pretension Forces with the new ones ( obtained in step 2) 4. Perform the Analysis. 5. Check whether the constraints are satisfied with modified pretensions 6. If not then determine the Unknown load factors again and keep repeating steps 2 to 5 till you get the constraints satisfied after static analysis ( step 5) 26

23 4. Initial Cable Forces Unknown Load Factor in midas Civil Girder Bending Moment before Cable Force Tuning Girder Bending Moment after Cable Force Tuning 27

24 4. Initial Cable Forces Tuning of Cables Reduce the repetitive computation process to obtain the optimum cable pretension. Calculates the effects of the cable pretension (or load factor) on the displacements/ member forces/ stresses through influence matrix and updates the results graph in real time. The process of Cable Force Tuning 1. Adjust the cable pretension (or load factor) using the table or bar graph. 2. Select the result item for which the effects of the cable pretension are to be checked. 3. Produce the results graph for the result item selected from step 2. If the pretension (or load factor) is adjusted in step 1, it is reflected in the results graph in real time. 4. Save the adjusted pretension forces in a load combination or apply the new pretension forces to the cables directly using the pre-programmed buttons. 28

25 5. FAQ in Cable Stayed Bridge (1) When Nonlinear Analysis is required? In the cable stayed bridge or suspension bridge, engineers will determine the initial cable force in the complete state(final shape) without construction stage. After that, construction stage analysis will be performed. If this initial cable force is correctly found, the cable force will be above 70% of its yielding force and it will behave very similar to the truss element. Therefore in the most of the general cablestayed bridge, the engineers can assume the cable to act like truss element and there is no need to consider large deformation analysis (=nonlinear geometric analysis). However, if the bridge span is very large (ex. Larger than 600m) and shape is complex (like stonecutter bridge or sutong bridge), engineers will perform large deformation analysis. There is no clear criteria when exactly the engineer need to perform nonlinear analysis. However, in the general case for cable bridge, it is not very common to perform nonlinear geometric analysis if they have correct value of initial cable force. One way to determine it clear will be performing both analysis, linear and nonlinear. By comparing the results, if the difference in these two analysis are very large, nonlinear analysis will be needed. 29

26 For any questions, feedback and suggestions send an at: Follow us on

27 Thank you