PAST, PRESENT AND FUTURE OF EARTHQUAKE ANALYSIS OF STRUCTURES By Ed Wilson Draft dated 8/15/14 September 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 The Fast Nonlinear Analysis Method – FNA Method Error Analysis The Retrofit of the Richmond-San Rafael Bridge

Load-Dependent Ritz Vectors LDR Vectors –

MOTAVATION – 3D Reactor on Soft Foundation 3 D Concrete Reactor 3 D Soft Soil Elements 360 degrees Dynamic Analysis 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

Simple Example of Dynamic Response

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

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

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

PROBLEM TO BE SOLVED M a + C v + K u = f i g(t) i For 3D Earthquake Loading THE OBJECTIVE OF THE ANALYSIS IS TO SOLVE FOR ACCURATE DISPLACEMENTS and MEMBER FORCES = - M x a x - M y a y - M z a z

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

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

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

G ENERATION OF L OAD D EPENDENT RITZ V ECTORS 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

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 VECTORS V 1 V 1 T M V 1 = I

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

STEP 3. MAKE VECTORS STIFFNESS ORTHOGONAL A.SOLVE Nb x Nb Eigenvalue Problem [ V T K V ] Z = [ w 2 ] Z B.CALCULATE MASS AND STIFFNESS ORTHOGONAL LDR VECTORS V R = V Z =

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

MAXIMUM DISPLACEMENT Number of Vectors Eigen Vectors Load Dependent Vectors (-2.41) (+0.88) (-2.41) ( -2.00) (-0.46) (+0.08) (-0.46) (+0.06) (-0.08) ( 0.00) (-0.04) (0.00) ( Error in Percent)

MAXIMUM MOMENT Number of Vectors Eigen Vectors Load Dependent Vectors 14178( %)5907( ) 24178( )5563( ) 34946( )5603( ) 44946( )5507( + 1.8) 55188( )5411( 0.0 ) 75304( -.0 ) 95411( 0.0 ) ( Error in Percent )

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

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

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

Isolators BASE ISOLATION

BUILDING IMPACT ANALYSIS

FRICTION DEVICE CONCENTRATED DAMPER NONLINEAR ELEMENT

GAP ELEMENT TENSION ONLY ELEMENT BRIDGE DECK ABUTMENT

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

Mechanical Damper Mathematical Model F = C v N F = ku F = f (u,v,u max )

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

103 FEET DIAMETER FEET HEIGHT ELEVATED WATER STORAGE TANK NONLINEAR DIAGONALS BASE ISOLATION

COMPUTER MODEL 92 NODES 103ELASTIC FRAME ELEMENTS 56 NONLINEAR DIAGONAL ELEMENTS 600TIME 0.02 Seconds

COMPUTER TIME REQUIREMENTS PROGRAM ( 4300 Minutes ) ANSYSINTEL 4863 Days ANSYSCRAY3 Hours( 180 Minutes ) SADSAPINTEL 486 (2 Minutes ) ( B Array was 56 x 20 )

Nonlinear Equilibrium Equations

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

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.

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:

6.Since the deformations in the nonlinearelements 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

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

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

FRAME WITH UPLIFTING ALLOWED

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

TIME - Seconds DEAD LOAD LATERAL LOAD LOAD NONLINEAR STATIC ANALYSIS 50 STEPS AT dT = 0.10 SECONDS

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.

PARALLEL ENGINEERING AND PARALLEL COMPUTERS

ONE PROCESSOR ASSIGNED TO EACH JOINT ONE PROCESSOR ASSIGNEDTO EACH MEMBER

PARALLEL STRUCTURAL ANALYSIS DIVIDE STRUCTURE INTO "N" DOMAINS FORM AND SOLVE EQUILIBRIUM EQ. FORM ELEMENT STIFFNESS IN PARALLEL FOR "N" SUBSTRUCTURES EVALUATE ELEMENT FORCES IN PARALLEL IN "N" SUBSTRUCTURES NONLINEAR LOOP TYPICAL COMPUTER

FIRST PRACTICAL APPLICTION OF THE FNA METHOD Retrofit of the RICHMOND - SAN RAFAEL BRIDGE 1997 to 2000 Using SADSAP

S T A T I C A N D D Y N A M I C S T R U C T U R A L A N A L Y S I S P R O G R A M

TYPICAL ANCHOR PIER

MULTISUPPORT ANALYSIS ( Displacements ) ANCHOR PIERS RITZ VECTOR LOAD PATTERNS

SUBSTRUCTURE PHYSICS JOINT REACTIONS ( Retained DOF ) MASS POINTS and MASSLESS JOINT ( Eliminated DOF ) Stiffness Matrix Size = 3 x 16 = 48 "a" "b"

SUBSTRUCTURE STIFFNESS REDUCE IN SIZE BY LUMPING MASSES OR BY ADDING INTERNAL MODES

ADVANTAGES IN THE USE OF SUBSTRUCTURES 1.FORM OF MESH GENERATION 2.LOGICAL SUBDIVISION OF WORK 3.MANY SHORT COMPUTER RUNS 4.RERUN ONLY SUBSTRUCTURES WHICH WERE REDESIGNED 5.PARALLEL POST PROCESSING USING NETWORKING

ECCENTRICALLY BRACED FRAME

FIELD MEASUREMENTS REQUIRED TO VERIFY 1.MODELING ASSUMPTIONS 2.SOIL-STRUCTURE MODEL 3.COMPUTER PROGRAM 4.COMPUTER USER

MECHANICAL VIBRATION DEVICES CHECK OF RIGID DIAPHRAGM APPROXIMATION

FIELD MEASUREMENTS OF PERIODS AND MODE SHAPES MODET FIELD T ANALYSIS Diff. - % Sec.1.78 Sec

15 th Period T FIELD = 0.16 Sec. FIRST DIAPHRAGM MODE SHAPE

At the Present Time Most Laptop Computers Can be directly connected to a 3D Acceleration Seismic Box Therefore, Every Earthquake Engineer C an verify Computed Frequencies If software has been developed

Final Remark Geotechnical Engineers must produce realistic Earthquake records for use by Structural Engineers

Errors Associated with the use of Relative Displacements Compared with the use of real Physical Earthquake Displacements Classical Viscous Damping does not exist In the Real Physical World

Comparison of Relative and Absolute Displacement Seismic Analysis

Shear at Second Level Vs. Time With Zero Damping Time Step = 0.01

Illustration of Mass-Proportional Component in Classical Damping.