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Summary of Lecture Topics Methods for Nonlinear Analysis

PAST, PRESENT AND FUTURE OF EARTHQUAKE ANALYSIS OF STRUCTURES By Ed Wilson Draft dated 8/15/14 September 24 2014 SEAONC Lectures #2. Summary of Lecture Topics Methods for Nonlinear Analysis New and Super Fast Nonlinear Analysis Method The Load Dependent Ritz Vectors – LDR Vectors

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Summary of Lecture Topics Methods for Nonlinear Analysis

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  1. PAST, PRESENT AND FUTUREOFEARTHQUAKE ANALYSIS OF STRUCTURESBy Ed Wilson Draft dated 8/15/14September 24 2014SEAONC Lectures #2

  2. Summary of Lecture Topics Methods for Nonlinear Analysis New and Super Fast Nonlinear Analysis Method The Load Dependent Ritz Vectors – LDR Vectors The Fast Nonlinear Analysis Method – FNA Method Error Analysis The Retrofit of the Richmond-San Rafael Bridge

  3. Load-Dependent Ritz Vectors LDR Vectors – 1980 - 2000

  4. MOTAVATION – 3D Reactor on Soft Foundation Dynamic Analysis - 1979 by Bechtel using SAP IV 200 Exact Eigenvalues were Calculated and all of the Modes were in the foundation – No Stresses in the Reactor. The cost for the analysis on the CLAY Computer was $10,000 3 DConcrete Reactor 3 D Soft Soil Elements 360 degrees

  5. Simple Example of Dynamic Response

  6. Simple Example of Free Vibration Columns in bending Rigid Foundation For a Static Initial Condition of The Solution of the Force Equilibrium Equation is Check Solution: Since Therefore

  7. Physical Analysis of Free Vibration Displacement Velocity Strain Energy Kinetic Energy Total Energy Without Energy Dissipation the System would Vibrate Forever Energy Pump

  8. DYNAMIC EQUILIBRIUM EQUATIONS • M a + C v + K u = F(t) • a = Node Accelerations • v = Node Velocities • u = Node Displacements • M = Node Mass Matrix • C = Damping Matrix • K = Stiffness Matrix • F(t) = Time-Dependent Forces

  9. PROBLEM TO BE SOLVED M a + C v + K u = fi g(t)i = - Mx ax - My ay - Mz az For 3D Earthquake Loading THE OBJECTIVE OF THE ANALYSIS IS TO SOLVE FOR ACCURATE DISPLACEMENTS and MEMBER FORCES

  10. METHODS OF DYNAMIC ANALYSIS For Both Linear and Nonlinear Systems ÷ STEP BY STEP INTEGRATION - 0, dt, 2 dt ... N dt USE OF MODE SUPERPOSITION WITH EIGEN OR LOAD-DEPENDENT RITZ VECTORS FOR FNA For Linear Systems Only TRANSFORMATION TO THE FREQUENCY DOMAIN and FFT METHODS ÷ RESPONSE SPECTRUM METHOD - CQC - SRSS

  11. STEP BY STEP SOLUTION METHOD 1. Form Effective Stiffness Matrix 2. Solve Set Of Dynamic Equilibrium Equations For Displacements At Each Time Step 3. For Non Linear Problems Calculate Member Forces For Each Time Step and Iterate for Equilibrium - Brute Force Method

  12. MODE SUPERPOSITION METHOD 1. Generate Orthogonal Dependent Vectors And Frequencies 2. Form Uncoupled Modal Equations And Solve Using An Exact Method For Each Time Increment. 3. Recover Node Displacements As a Function of Time 4. Calculate Member Forces As a Function of Time

  13. GENERATION OF LOAD DEPENDENT RITZ VECTORS 1. Approximately Three Times Faster Than The Calculation Of Exact Eigenvectors 2. Results In Improved Accuracy Using A Smaller Number Of LDR Vectors 3. Computer Storage Requirements Reduced 4. Can Be Used For Nonlinear Analysis To Capture Local Static Response

  14. STEP 1. INITIAL CALCULATION A. TRIANGULARIZE STIFFNESS MATRIX B. DUE TO A BLOCK OF STATIC LOAD VECTORS, f, SOLVE FOR A BLOCK OF DISPLACEMENTS, u, K u = f C. MAKE u STIFFNESS AND MASS ORTHOGONAL TO FORM FIRST BLOCK OF LDL VECTORSV 1 V1T M V1 = I

  15. STEP 2. VECTOR GENERATION i = 2 . . . . N Blocks A. Solve for Block of Vectors, K Xi = M Vi-1 B. Make Vector Block, Xi , Stiffness and Mass Orthogonal - Yi C. Use Modified Gram-Schmidt, Twice, to Make Block of Vectors, Yi , Orthogonal to all Previously Calculated Vectors - Vi

  16. STEP 3. MAKE VECTORS STIFFNESS ORTHOGONAL A. SOLVE Nb x Nb Eigenvalue Problem [ VT K V ] Z = [ w2 ] Z B. CALCULATE MASS AND STIFFNESS ORTHOGONAL LDR VECTORS VR = V Z =

  17. DYNAMIC RESPONSE OF BEAM 100 pounds 10 AT 12" = 120" • FORCE = Step Function TIME

  18. MAXIMUM DISPLACEMENT Numberof Vectors Eigen Vectors Load Dependent Vectors 1 0.004572 (-2.41) 0.004726 (+0.88) 2 0.004572 (-2.41) 0.004591 ( -2.00) 3 0.004664 (-0.46) 0.004689 (+0.08) 4 0.004664 (-0.46) 0.004685 (+0.06) 5 0.004681 (-0.08) 0.004685 ( 0.00) 7 0.004683 (-0.04) 9 0.004685 (0.00) ( Error in Percent)

  19. MAXIMUM MOMENT Number of Vectors Eigen Vectors Load Dependent Vectors 1 4178 ( - 22.8 %) 5907 ( + 9.2 ) 2 4178 ( - 22.8 ) 5563 ( + 2.8 ) 3 4946 ( - 8.5 ) 5603 ( + 3.5 ) 4 4946 ( - 8.5 ) 5507 ( + 1.8) 5 5188 ( - 4.1 ) 5411 ( 0.0 ) 7 5304 ( - .0 ) 9 5411 ( 0.0 ) ( Error in Percent )

  20. LDR Vector Summary After Over 20 Years Experience Using the LDR Vector Algorithm We Have Always Obtained More Accurate Displacements and Stresses Compared to Using the Same Number of Exact Dynamic Eigenvectors. SAP 2000 has Both Options

  21. The Fast Nonlinear Analysis Method The FNA Method was Named in 1996 Designed for the Dynamic Analysis of Structures with a Limited Number of Predefined Nonlinear Elements

  22. EVALUATE LDR 1. VECTORS WITH NONLINEAR ELEMENTS REMOVED AND DUMMY ELEMENTS ADDED FOR STABILITY FAST NONLINEARANALYSIS 2. SOLVE ALL MODAL EQUATIONS WITH NONLINEAR FORCES ON THE RIGHT HAND SIDE 3. USE EXACT INTEGRATION WITHIN EACH TIME STEP 4. FORCE AND ENERGY EQUILIBRIUM ARE STATISFIED AT EACH TIME STEP BY ITERATION

  23. BASE ISOLATION Isolators

  24. BUILDING IMPACT ANALYSIS

  25. FRICTION DEVICE CONCENTRATED DAMPER NONLINEAR ELEMENT

  26. GAP ELEMENT BRIDGE DECK ABUTMENT TENSION ONLY ELEMENT

  27. P L A S T I C H I N G E S 2 ROTATIONAL DOF DEGRADING STIFFNESS ?

  28. Mechanical Damper F = ku F = f (u,v,umax ) F = C vN Mathematical Model

  29. LINEAR VISCOUS DAMPING DOES NOT EXIST IN NORMAL STRUCTURES AND FOUNDATIONS 5 OR 10 PERCENT MODAL DAMPING VALUES ARE OFTEN USED TO JUSTIFY ENERGY DISSIPATION DUE TO NONLINEAR EFFECTS IF ENERGY DISSIPATION DEVICES ARE USED THEN 1 PERCENT MODAL DAMPING SHOULD BE USED FOR THE ELASTIC PART OF THE STRUCTURE - CHECK ENERGY PLOTS

  30. 103 FEET DIAMETER - 100 FEET HEIGHT NONLINEAR DIAGONALS BASE ISOLATION ELEVATED WATER STORAGE TANK

  31. 92 NODES 103 ELASTIC FRAME ELEMENTS 56 NONLINEAR DIAGONAL ELEMENTS 600 TIME STEPS @ 0.02 Seconds COMPUTER MODEL

  32. COMPUTER TIME REQUIREMENTS PROGRAM ( 4300 Minutes ) ANSYS INTEL 486 3 Days ANSYS CRAY 3 Hours ( 180 Minutes ) (2 Minutes ) SADSAP INTEL 486 ( B Array was 56 x 20 )

  33. Nonlinear Equilibrium Equations

  34. Nonlinear Equilibrium Equations

  35. Summary Of FNA Method 1. Calculate LDR Vectors for Structure With the Nonlinear Elements Removed. 2. These Vectors Satisfy the Following Orthogonality Properties

  36. 3. The Solution Is Assumed to Be a Linear Combination of the LDR Vectors. Or, Which Is the Standard Mode Superposition Equation Remember the LDR Vectors Are a Linear Combination of the Exact Eigenvectors; Plus, the Static Displacement Vectors. No Additional Approximations Are Made.

  37. 4. A typical modal equation is uncoupled. • However, the modes are coupled by the • unknown nonlinear modal forces which • are of the following form: • 5. The deformations in the nonlinear elements • can be calculated from the following • displacement transformation equation:

  38. 6. Since the deformations in the nonlinear elements can be expressed in terms of the modal response by Where the size of the array is equal to the number of deformations times the number of LDR vectors. The array is calculated only once prior to the start of mode integration. THE ARRAY CAN BE STORED IN RAM

  39. 7. The nonlinear element forces are calculated, for iteration i , at the end of each time step 7.

  40. 8. Calculate error for iteration i , at the end of each time step, for the N Nonlinear elements – given Tol

  41. FRAME WITH UPLIFTING ALLOWED UPLIFTING ALLOWED

  42. Four Static Load Conditions Are Used To Start The Generation of LDR Vectors EQ DL Left Right

  43. NONLINEAR STATIC ANALYSIS • 50 STEPS AT dT = 0.10 SECONDS DEAD LOAD LOAD LATERAL LOAD 0 1.0 2.0 3.0 4.0 5.0 TIME - Seconds

  44. Advantages Of The FNA Method 1. The Method Can Be Used For Both Static And Dynamic Nonlinear Analyses 2. The Method Is Very Efficient And Requires A Small Amount Of Additional Computer Time As Compared To Linear Analysis 2. The Method Can Easily Be Incorporated Into Existing Computer Programs For LINEAR DYNAMIC ANALYSIS.

  45. PARALLEL ENGINEERING AND PARALLEL COMPUTERS

  46. ONE PROCESSOR ASSIGNED TO EACH JOINT 3 2 1 3 1 2 ONE PROCESSOR ASSIGNEDTO EACH MEMBER

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