PAST, PRESENT AND FUTUREOFEARTHQUAKE ANALYSIS OF STRUCTURESBy Ed Wilson Draft dated 8/15/14September 22 and 24 2014SEAONC Lectureshttp://edwilson.org/History/Slides/Past%20Present%20Future%202014.ppt
1964Gene’s Comment – a true story . Ed developed a new program for the Analysis of Complex Rockets Ed talks to Gene ------ Two weeks later Gene calls Ed ------ Ed goes to see Gene ----- The next day, Gene calls Ed and tells him “Ed, why did you not tell me about this program. It is the greatest program I ever used.”
Summary of Lecture Topics • Fundamental Principles of Mechanics and Nature • Example of the Present Problem – Caltrans Criteria • The Response Spectrum Method • Demand Capacity Calculations • Speed of Computers– The Last Fifty Years • Terms I do not Understand • The Load Dependant Ritz Vectors - LDR Vectors • The Fast Nonlinear Analysis Method – FNA Method • Recommendations edwilson.org • Questions email@example.com
Fundamental Equations of Structural Analysis • Equilibrium - Including Inertia Forces - Must be Satisfied • Material Properties or Stress / Strain or Force / Deformation • Displacement Compatibility Or Equations or Geometry • Methods of Analysis • Force – Good for approximate hand methods • Displacement - 99 % of programs use this method • Mixed - Beam Ex. Plane Sections & V = dM/dz • Check Conservation of Energy
My First Earthquake Engineering Paper October 1-5 1962
THE PRESENTSAB Meeting on August 28, 2013 Comments on the Response Spectrum Analysis MethodAs Used in the CALTRANS SEISMIC DESIGN CRITERIA
Topics • Why do most Engineers have Trouble with Dynamics? • Taught by people who love math – No physical examples • Who invented the Response Spectrum Method? • Ray Clough and I did ? – by putting it into my computer program • Application by CalTrans to “Ordinary Standard Structures” • Why 30 ? Why reference to Transverse & Longitudinal directions • Physical behavior of Skew Bridges – Failure Mode • Equal Displacement Rule? • 6 Quote from George W. Housner
Who Developed the Approximate Response Spectrum Method of Seismic Analysis of Bridges and other Structures? • 1. Fifty years ago there were only digital acceleration records for 3 earthquakes. • 2. Building codes gave design spectra for a one degree of freedom systems with no guidance of how to combine the response of of the higher modes. • At the suggestion of Ray Clough, I programmed the square root of the sum of the square of the modal values for displacements and member forces. However, I required the user to manually combine the results from the two orthogonal spectra. Users demanded that I modify my programs to automatically combine the two directions. I refused because there was no theoretical justification. • The user then modify my programs by using the 100%+30% or 100%+40% rules. • Starting in 1981 Der Kiureghian and I published papers showing that the CQC method should be used be used for combining modal responses for each spectrum and the two orthogonal spectra be combined by the SRSS method. • We now have Thousands or of 3D earthquake records from hundred of seismic events. Therefore, why not use Nonlinear Time-History Analyses that SATISFIES FORCE EQUILIBREUM.
Nonlinear Failure Mode For Skew Bridges F(t) F(t) Abutment ForceActing on Bridge Tensional Failure u(t) Contact at Right Abutment u(t) Tensional Failure F(t) Abutment ForceActing on Bridge Contact at Left Abutment
Possible Torsional Failure Mode Design Joint Connectors for Joint Shear Forces?
Seismic Analysis Advice by Ed Wilson • All Bridges are Three-Dimensional and their Dynamic Behavior is governed by the Mass and Stiffness Properties of the structure. The Longitudinal and Transverse directions are geometric properties. All Structures have Torsional Modes of Vibrations. • The Response Spectrum Analysis Methodis a very approximate method of seismic analysis which only produces positive values of displacements and member forces which are not in equilibrium. Demand / Capacity Ratios have Very Large Errors • A structural engineer may take several days to prepare and verify a linear SAP2000 model of an Ordinary Standard Bridge. It would take less than a day to add Nonlinear Gap Elements to model the joints. If a family of 3D earthquake motions are specified, the program will automatically summarize the maximum demand-capacity ratios and the time they occur in a few minutes of computer time.
Convince Yourself with a simple test problem • Select an existing Sap 2000 model of a Ordinary Standard Bridge with several different spans – both straight and curved. • Select one earthquake ground acceleration record to be used as the input loading which is approximately 20 seconds long. • Create a spectrum from the selected earthquake ground acceleration record. • Using a number of modes that captures a least 90 percent of the mass in all three directions. • At a 45 degree angle, Run a Linear Time History Analysis and a Response Spectrum Analysis. • Compare Demand Capacity Ratios for both SAP 2000 analysis for all members. • You decide if the Approximate RSA results are in good agreement with the Linear time History Results.
Educational Priorities of an Old Professor • on Seismic Analysis of Structures • Convince Engineers that the Response Spectrum Method • Produces very Poor Results • Method is only exact for single degree of freedom systems • It produces only positive numbers for Displacements and Member Forces. • Results are maximum probable values and occur at an “Unknown Time” • Short and Long Duration earthquakes are treated the same using “Design Spectra” • Demand/Capacity Ratios are always “Over Conservative” for most Members. • The Engineer does not gain insight into the “Dynamic Behavior of the Structure” Results are not in equilibrium. More modes and 3D analysis will cause more errors. • Nonlinear Spectra Analysis is “Smoke and Mirrors” – Forget it
Convince Engineers that it is easy to conduct • “Linear Dynamic Response Analysis” • It is a simple extension of Static Analysis – just add mass and time dependent loads • Static and Dynamic Equilibrium is satisfied at all points in time if all modes are included • Errors in the results can be estimated automatically if modes are truncated • Time-dependent plots and animation are impressive and fun to produce • Capacity/Demand Ratios are accurate and a function of time – summarized by program. • Engineers can gain great insight into the dynamic response of the structure and may help in the redesign of the structural system.
Terminology commonly used in nonlinear analysis that do not have a unique definition • Equal Displacement Rule – can you prove it? • Pushover Analysis • Equivalent Linear Damping • Equivalent Static Analysis • Nonlinear Spectrum Analysis • Onerous Response History Analysis
Equal Displacement Rule In 1960 Veletsos and Newmark proposed in a paper presented at the 2nd WCEE For a one DOF System, subjected to the El Centro Earthquake, the Maximum Displacement was approximately the same for both linear and nonlinear analyses. In 1965 Clough and Wilson, at the 3rd WCEE, proved the Equal Displacement Rule did not apply to multi DOF structures. http://edwilson.org/History/Pushover.pdf
1965 Professor Clough’s Comment . “If tall buildings are designed for elastic column behavior and restrict the nonlinear bending behavior to the girders, it appears the danger of total collapse of the building is reduced.” This indicates the strong-column and week beam design is the one of the first statements on Performance Based Design
The Response Spectrum Method Basic Assumptions I do not know who first called it a “response spectrum,” but unfortunate the term leads people to think that the characterize the building’s motion, rather than the ground’s motion. George W. Housner EERI Oral History, 1996
Integration will produce Earthquake Ground Displacement – inches These real Eq. Displacement can be used as Computer Input
Relative Displacement Spectrum for a unit mass with different periods • These displacements Ymaxare maximum(+ or -) values versus period for a structure or mode. • Note: we do not know the time these maximum took place. • Pseudo Acceleration Spectrum • Note: S = w2Ymax has the same properties as the Displacement Spectrum. Therefore, how can anyone justify combining values, which occur at different times, and expect to obtain accurate results. • CASE CLOSED
General Horizontal Response Spectrum from ASCE 41- 06
Where did the Hat go - on the Response Spectrum ? As I Recall -------
Demand-Capacity Ratios The Demand-Capacity ratio for a linear elastic, compression member is given by an equation of the following general form: If the axial force and the two moments are a function of time, the Demand-Capacity ratio will be a function of time and a smart computer program will produce R(max) and the time it occurred. A smart engineer will hand check several of these values.
RSM Demand-Capacity Ratios If the axial force and the two moments are produced by the Response Spectrum Method the Demand-Capacity ratio may be computed by an equation of the following general form: A smart computer program can compute this Demand-Capacity Ratio. However, only an idiot would believe it.
SPEED and COST of COMPUTERS 1957 to 2014 to the Cloud You can now buy a very powerful small computer for less than $1,000 However, it may cost you several thousand dollars of your time to learn how to use all the new options. If it has a new operating system
1957 My First Computer in Cory Hall IBM 701 Vacuum Tube Digital Computer Could solve 40 equations in 30 minutes
1981 My First Computer Assembled at Home Paid $6000 for a 8 bit CPM Operating System with FORTRAN. Used it to move programs from the CDC 6400 to the VAX on Campus. Developed a new program called SAP 80 without using any Statements from previous versions of SAP. After two years, system became obsolete when IBM released DOS with a floating point chip. In 1984, CSI developed Graphics and Design Post-Processor and started distribution of the Professional Version of Sap 80
Floating-Point Speeds of Computer Systems Definition of one Operation A = B + C*D64 bits - REAL*8
Computer CostversusEngineer’s Monthly Salary $1,000,000 c/s = 1,000 c/s = 0.10 $10,000 $1,000 $1,000 1963 Time 2013
Fast Nonlinear Analysis With Emphasis On Earthquake Engineering BY Ed Wilson Professor Emeritus of Civil Engineering University of California, Berkeley edwilson.org May 25, 2006
Summary Of Presentation 1. History of the Finite Element Method 2. History Of The Development of SAP 3. Computer Hardware Developments 4. Methods For Linear and Nonlinear Analysis 5. Generation And Use Of LDR Vectors and Fast Nonlinear Analysis - FNAMethod 6. Example Of Parallel Engineering Analysis of the Richmond - San Rafael Bridge
From The Foreword Of The First SAP Manual "The slang name S A P was selected to remind the user that this program, like all programs, lacks intelligence. It is the responsibility of the engineer to idealize the structure correctly and assume responsibility for the results.” Ed Wilson 1970
The SAP Series of Programs 1969 - 70 SAP Used Static Loads to Generate Ritz Vectors 1971 - 72 Solid-Sap Rewritten by Ed Wilson 1972 -73 SAP IV Subspace Iteration – Dr.Jűgen Bathe 1973 – 74 NON SAP New Program – The Start of ADINA 1979 – 80 SAP 80 New Linear Program for Personal Computers Lost All Research and Development Funding 1983 – 1987 SAP 80 CSI added Pre and Post Processing 1987 - 1990 SAP 90 Significant Modification and Documentation 1997 – Present SAP 2000 Nonlinear Elements – More Options – With Windows Interface
Load-Dependent Ritz Vectors LDR Vectors – 1980 - 2000
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
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 = 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
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
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
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
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
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 =