THE RESPONSE SPECTRUMA Technical Seminar on the Development and Application of the Response Spectrum Method for Seismic Design of Structures LECTURE #1 1-2 June 2007 Response Spectra Mihailo D. Trifunac University of Southern California http://www.usc.edu/dept/civil_eng/Earthquake_eng/ NOTES Response Spectrum M.D. Trifunac CSCE Vancouver Section UBC - Vancouver, June 1st, 2007 NOTE I I. INTRODUCTION - A Sketch on Early History of Response Spectrum The first practical steps, which initiated the engineering work on the design of earthquake resistant structures, accompanied the introduction of the seismic coefficient (shindo in Japan, and rapporto sismico in Italy, for example), and started to appear following the destructive earthquakes in San Francisco, California, in 1906, Messina-Reggio, Italy, in 1908 , and Tokyo, Japan, in 1923. The first seismic design code was introduced in Japan in 1924 following the 1923 earthquake. In California the work on the code development started in 1920s, but it was not after the Long Beach earthquake in 1933 that the Field Act was finally adopted in 1934 . In early 1900s, at most universities, engineering curricula did not include advanced mathematics and mechanics, both essential for teaching analysis of the dynamic response of structures. This lack in theoretical preparation is reflected in the view of C. Derleth, civil engineering professor and Dean of Engineering at U.C. Berkeley, who commented after the 1906 earthquake: Many engineers with whom the writer has talked appear to have the idea that earthquake stresses in framed structures can be calculated, so that rational design to resist earthquake destruction can be made, just as one may allow for dead and live loads, or wind and impact stresses. Such calculations lead to no practical conclusions of value (Derleth 1907). A comment by A. Ruge (1940), the first professor of engineering seismology at Massachusetts Institute of Technology, that “the natural tendency of average design engineer is to throw up his hands at the thought of making any dynamical analysis at all..”, made three decades later, shows that the progress was slow. In 1929, at University of Michigan, in Ann Arbor, first lectures were organized in the Summer School of Mechanics, by S. Timoshenko (1878-1972), with participation of A. Nádai, R. V. Southwell and H. M. Westergaard. “After the first session of the summer school in 1929, the number of doctoral students in mechanics….started rapidly to increase” (Timoshenko 1968). In the summer of 1933, M.A. Biot was among the young post-doctoral students, who took part in Timoshenko’s summer school (Mindlin 1989, Bolley 2005). In southern California, studies of earthquakes, and the research in theoretical mechanics, were expanded by R.Millikan, who became the first president of Caltech, in 1921. Millikan completed his Ph.D. studies in Physics, at Columbia University, in 1895, and following recommendation of his advisor M. Pupin spent a year in Germany. This visit to Europe appears to have influenced many of Millikan’s later decisions while recruiting the leading Caltech faculty two decades later. In 1921 H.O. Wood invited Millikan to serve on the Advisory Committee in Seismology. The work on that committee and Millikan’s Response Spectrum Seminar Lecture # 1 1 Response Spectrum M.D. Trifunac CSCE Vancouver Section UBC - Vancouver, June 1st, 2007 interest in earthquakes were also significant for several subsequent events. In 1926 C. Richter, and in 1930 B. Gutenberg joined the seismological laboratory. In the area of applied mechanics, Millikan invited Theodor von Karman, and in 1930 von Karman became the first director of the Guggenheim Aeronautical Laboratory. It was Millikan’s vision and his ability to anticipate future developments, which brought so many leading minds to a common place of work, creating environment, which made the first theoretical formulation of the concept of the response spectrum method possible. This year, 2007, marks the 75-th anniversary of the formulation of the concept of the Response Spectrum Method (RSM) in 1932. Since 1932 the RSM evolved into the essential tool and the central theoretical framework, in short a conditio sine qua non, for Earthquake Engineering. The mathematical formulation of the RSM first appeared in the doctoral dissertation of M.A. Biot (1905-1985) in 1932, and in two of his papers (Biot 1933; 1934). Biot defended his Ph.D. thesis at Caltech, in June of 1932, and presented a lecture on the method to the Seismological Society of America meeting, which was held at Caltech, in Pasadena, also in June of 1932. Theodore von Karman, Biot’s advisor, played the key role in guiding his student, and in promoting his accomplishments. After the method of solution was formulated, Biot and von Karman searched for an optimal design strategy. A debate at the time was whether a building should be designed with a soft first floor, or it should be stiff throughout its height, to better resist earthquake forces. An excerpt from New York Herald Tribune, in June of 1932, illustrates this: Shock Proof Buildings Sought by Scientists. Rigid or Flexible Materials, Their Difference in Theory A building proof against earthquakes is the goal of Dr. Theodor von Karman and Dr. M. Biot, of California Institute of technology. Dr. von Karman described to the American Society of Mechanical Engineers, whose convention was held recently at Yale, studies of the amount of shock, which various types of buildings have undergone in Japan, South America and California. Their researches are being conducted at the Institute’s Guggenheim Aeronautical Laboratory. One of the principal problems is to decide whether a rigid or flexible structure is better. Some scientists contend the first is preferable; others would make the ground floor of tall buildings flexible. Pointing out that the reinforced concrete is superior to steel in absorbing the shocks, Dr. von Karman’s personal belief is that the building should be constructed to shake “with the rhythm of the earth’s movements.” Biot’s interest in the maxima of the transient response in solids and in fluids preceded, and extended beyond earthquake engineering. After he formulated the concept of the RSM, he extended it to other vibrational problems, as in analysis of aircraft landing gear, for example. Biot briefly returned to the subject of earthquake engineering almost ten years later, presenting response spectral amplitudes of several earthquakes, which he calculated using the torsional pendulum at Columbia University (Biot 1941). In 1942 he presented a review of response spectrum method, discussed the effects of flexible soil on the rocking period of a rigid block (Biot 2006), and described the spectrum superposition method based on the sum of absolute modal maxima (Biot 1942). After 1942 Biot moved on to other subjects, making fundamental contributions to many other fields. He did not Response Spectrum Seminar Lecture # 1 2 second. Before the digital computer age. 1947). there were only a few wellrecorded accelerograms that could be used for response studies. the computation of response was time consuming. earthquake.D.Response Spectrum M. and. the digitization of analog accelerograph records and the digital computation of ground motion and of the response spectra were developed completely and tested for accuracy. A complete list of Biot’s publications can be found in Theodor von Karman (left) and Maurice A.14 of the Journal of Mathematical and Physical Sciences. the modern era of RSM was launched. Trifunac CSCE Vancouver Section UBC . There were two main reasons for this. This started to change in 1960s with arrival of digital computers and with commercial availability of strongmotion accelerographs. June 1st. First. 2007 write papers on earthquake engineering (Trifunac 2005). India. Biot (right) at professor von Karman’s house in Pasadena (circa 1932) Trifunac (2006). The RSM remained in the academic sphere of research for almost 40 years. gaining engineering acceptance during the early 1970s. and of his patents and awards. in 1980.Vancouver. in the introduction to Vol. published in Madras. By the late 1960s and early 1970s. on the occasion of his seventieth birthday anniversary. the computation of response to earthquake ground motion led to “certain rather formidable difficulties” (Housner. This earthquake was recorded by 241 Response Spectrum Seminar Lecture # 1 3 . and the results were unreliable (Trifunac 2003). in 1971. with the occurrence of the San Fernando. California. but followed closely and with interest the work of others. Then. W.W. pp.. (1907). pp. pp. Calif. in Poromechanics III. No. A. 14(4).” Zeitschrift für. “Theory of Elastic Systems Vibrating Under Transient Impulse With an Application to Earthquake-Proof Buildings. 7-9. V. M. Soil Dynamics and Earthquake Engineering. Biot. Biot. 213–223. Lee. (1932). Influence of Foundation on Motion of Blocks. Soc. "Theory of Vibration of Buildings During Earthquakes. Indian Society of Earthquake Technology journal. 262–268. Biot.A. Amer. 259. M. it become possible to launch the first comprehensive empirical scaling analyses of response spectral amplitudes (Lee 2002. 44(1). Vol. Vol. “Empirical Scaling of Strong Earthquake Ground Motion-Part I: Attenuation and Scaling of Response Spectra.. pp. (in press). Lee. “Analytical and Experimental Methods in Engineering Seismology.A. (1947).” Bull..Response Spectrum M. 19–31. 1906 on Engineering Constructions.A. F." Aeronautics Department..486-490. The Effects of the San Francisco Earthquake of April 18-th.Vancouver. Seism. (2007). Y. Edited by Abousleiman. entitled "Transient Oscillations in Elastic System. and Ulm. Soc. M.J. “Vibrations of Buildings During Earthquakes.N. Housner. V. Thesis No. References Biot. Trifunac CSCE Vancouver Section UBC .1. 31.A. Transactions of the American Society of Civil Engineers. (2005).S. M. 2007). Amer. Pasadena. “A Mechanical Analyzer for the Prediction of Earthquake Stresses.H. 26(6-7). December. M. (1934).39. Boley.” ASCE Transactions. (2002). (2006). Vol. G.D. Empirical Scaling and Regression Methods for Earthquake StrongMotion Response Spectra – A Review. “Characteristics of Strong Motion Earthquakes. Norman...” Chapter II in Ph. 2007 accelerographs. (1933). Biot. 365–408.” Indian Society of Earthquake Technology Journal. B. Response Spectrum Seminar Lecture # 1 4 . Biot. National Academy of Sciences. Maurice Biot – He is one of us. No.4. C. LIX. Cheng.D. By combining the data from the San Fernando earthquake with all previous strong-motion records. Proc. Inst. pp. Seism. 19(2). 219–254.A. June 1st. 108. Derleth. of Tech. 151–171.” Proc. (1942). pp. M. pp. Angewandte Matematik und Mechanik.A.” Bull. 37. Oklahoma. Biot centennial conference.W. pp. (1941). California. D. pp. 19–50. and Ulm. Paper 431. Scientific citations of M. Vol. Boston. 31–35. Soil Dynamics and Earthquake Engineering. A. Edited by Abousleiman. 23rd Annual ISET Lecture. M.H. 1. Vol. Trifunac. Ruge. Princeton. T. Timoshenko. A. Trifunac. Biot.105. (2006). 26(6-7).D..Response Spectrum M. (1967). Indian Society of Earthquake Technology Journal. Y. Norman.. A. Cheng. Proc. Ruge on Earthquakes and Structures. L. (1940). Van Nostrand Co.. R. Biot. Trifunac CSCE Vancouver Section UBC .D. Biographical Sketch and Publications of M. (1968). 40. S.. Brown and Co. 2007 Mindlin. 11-17. As I Remember. in Poromechanics III.. Response Spectrum Seminar Lecture # 1 5 . . A.J.P. in Memorial Tributes: Natl. Transactions of the American Society of Civil Engineers. 718-724. (2003) 70th Anniversary of Biot Spectrum.D.D. (1989) Maurice Anthony Biot. F. pp. (2005). Vol. Academy of Engineering.Vancouver. 3. Von Karman.N. The Wind and Beyond. June 1st. Oklahoma. and Edson. pp. M. pp. No. Biot centennial conference. Trifunac. M. Little. (A.3) (A. the fraction of critical damping ζ.of -freedom system. 1940) . RESPONSE SPECTRA . June 1st.4) For any general motion of the support z(t).1) && where x is the relative displacement between the structure and its base and −mz represents the inertial force applied to the mass m. For zero initial conditions.2) (A.D. 1932. 2007 NOTE A A. 2 km Equation (A. and the nature of the base acceleration.1.1). the relative displacement x(t) can be computed from the Duhamel integral (Biot.Vancouver. the differential equation of linear motion is && & && mx + cx + kx = − mz .An Outline of the Method Single -degree. which has the equivalent spring constant k. The total shear force VB (Figure A. for example. z (A. supported on two columns. For vibration z(t) of its base. & x z (A. Trifunac CSCE Vancouver Section UBC . The natural frequency ωn and the fraction of critical damping ζ are defined as k 2 ωn = m and c ς= . exerted by the columns on the ground is Response Spectrum Seminar Lecture # 1 6 . the linear relative response of the structure is characterized by its natural period Tn = 2π ωn .5) Thus.Response Spectrum M. von Karman and Biot. z The relative displacement x(t) is important for earthquake-resistant design because the strains in the structure are directly proportional to the relative displacements. 1933.1) then becomes 2 && + 2ω nς x + ω n x = − && . with damping force proportional to the relative velocity between the mass and its foundation. and c being the damping constant. 1934. the expression for x(t) takes the form x(t ) = −1 ωn 1 − ς 2 ∫ t 0 &&(τ )e −ςωn (t −τ ) sin ω n 1 − ς 2 (t − τ )dτ . A single-degree-of-freedom system corresponding to a one-story structure is schematically shown in figure A. &&(τ ). Energy dissipation is assumed to be of viscous nature. noting that &&(t ) = &&(t ) + &&(t ).7) The absolute acceleration &&(t ) of the mass m is obtained by further differentiation of y & x(t ) . It is: y x z &&(t ) = ω n y (1 − 2ς ) 2 1− ς 2 ∫ t 0 t 0 &&(τ )e −ςωn (t −τ ) sin ω n 1 − ς 2 (t − τ )dτ z &&(τ )e −ςωn (t −τ ) cos ω n 1 − ς 2 (t − τ )dτ .Response Spectrum M. for engineering applications.8) The absolute acceleration && is important for experimental measurements.9) (A. 2007 VB (t) = kx(t).D.11) Response Spectrum Seminar Lecture # 1 7 . June 1st.Vancouver. It may be concluded that of primary interest.1). are the & maximum absolute values of x(t). (A. an accelerograph located at that point records a close approximation y to &&(t ) .6) & The exact relative velocity x(t ) follows directly from (A.5): t & x(t ) = − ∫ &&(τ )e−ςωn ( t −τ ) cos ω n 1 − ς 2 (t − τ )dτ z 0 + ς 1− ς 2 ∫ t 0 &&(τ )e−ςωn (t −τ ) sin ω n 1 − ς 2 (t − τ )dτ . which is the most simple to measure during strong.10) (A. Trifunac CSCE Vancouver Section UBC . z + ς 1− ς 2 ∫ (A. z (A. x(t ) . and &&(t ) experienced during the earthquake y response. because it is y the quantity. earthquake-induced vibrations. That is. The absolute acceleration also defines the seismic force on the mass m (figure A. y (A. These quantities are commonly defined as: SD ≡ |x(t)|max & SV ≡ | x(t ) |max SA ≡ | &&(t ) |max . 7) the following approximate relationships exist between the spectral quantities defined in (A.Response Spectrum M. 2π ωn . Therefore.8) may be neglected.10). 2007 SINGLE .DEGREE .9). the fraction of critical damping ζ is small. Plots of SD. for In typical engineering structures. and (A.Vancouver. It is approximately 2% .D.7) and (A. SV. are called earthquake response spectra. Trifunac CSCE Vancouver Section UBC . 1 − ς 2 ≈ 1 and the terms of order ζ and ζ 2 in equations (A.8% for buildings and 5% –10% for soil structures.11): Response Spectrum Seminar Lecture # 1 8 .FREEDOM SYSTEM x y m m = mass c = damping constant VB = base shear c VB z Ground x = relative motion of m y = absolute motion of m z = absolute motion of ground Fig. A. (A.OF . June 1st.1 Single-degree-of-freedom system. Furthermore. if the cosine term is replaced by a sine term in Equation (A. and SA versus the undamped natural period of vibration Tn = various fractions of critical damping ς . For engineering applications. 1956. In the engineering literature.18b) From definition (A. June 1st. (A. 1962).” Fourier Spectra and Response Spectra. and PSA can be conveniently plotted on the common tripartite logarithmic plot versus period.18) it follows that: 2 2 tmax ⎧ ⎡ tmax ⎫ ⎪ ⎪ &&(τ ) cos ωτ dτ ⎤ + ⎡ ∫ &&(τ ) sin ωτ dτ ⎤ ⎬ SV = ⎨ ⎢ ∫ z z ⎥ ⎢ 0 ⎥ ⎦ ⎣ ⎦ ⎪ ⎪⎣ 0 ⎩ ⎭ 1/ 2 . z 0 t (A.10) and from (A.12) 2π SV .Response Spectrum M. (A.19) Response Spectrum Seminar Lecture # 1 9 .18a) Expanding. Rubin.15) ⎟ SD ⎝ T ⎠ because SD. For ζ = 0. 1961. PSV. Hudson.D. 2007 SD ≈ and T SV 2π (A. Equation (A.7) (dropping the subscript n on ωn ) reduces to: & x(t ) = − ∫ &&(τ ) cos ω (t − τ )dτ . z z 0 0 t t (A. these approximations can be made plausible (Hudson. Trifunac CSCE Vancouver Section UBC .17) ⎦ ⎣ 0 ⎦ ⎪ ⎪⎣ 0 ⎩ ⎭ For an undamped oscillator there is a close relationship between the Fourier amplitude spectrum and the exact relative velocity response spectrum (Kawasumi.13) T For earthquake-like excitations.16) The Fourier amplitude spectrum is then given by the square root of the sum of the squares of the real and imaginary parts of F(ω) 2 2 ⎧⎡ T ⎪ ⎪ ⎤ ⎡ T ⎤ ⎫ FS ≡| F (ω ) |= ⎨ ⎢ ∫ &&(τ ) cos ωτ dτ ⎥ + ⎢ ∫ &&(τ ) sin ωτ dτ ⎥ ⎬ . 2 The Fourier spectrum of an input acceleration shows the significant frequency characteristics of recorded motion. z z (A. PSV and PSA are frequently referred to as "pseudo velocity" and "pseudo absolute acceleration.14) T and SA ≈ ⎛ 2π ⎞ PSA = ⎜ (A.Vancouver. For an accelerogram differing from zero in the time interval 0 < t < T the Fourier spectrum is defined as: F (ω ) = ∫ &&(τ )e− iωτ dτ z 0 T (A. & x(t ) = − cos ωτ ∫ &&(τ ) cos ωτ dτ + ∫ &&(τ ) sin ωτ dτ . it is convenient to use the following approximations 2π PSV = SD (A. 1962). Multi-Degree-of-Freedom Systems An example of a fixed base multi-degree-of freedom system used to study vibrations of tall buildings is shown in figure A. Similarly. whereas SV is the maximum velocity during both the earthquake and the subsequent free vibrations. 1993. while {⋅} represents a column vector with n components. and the stiffness matrix [k] may have all elements different from zero. and ζ = 0. The shear force at each floor level is designated by Vi. In the special case in which the time tmax coincides with the total duration of the earthquake T. like the one shown in figure A. Trifunac CSCE Vancouver Section UBC .17) and (A. however. the relative velocity spectrum SV characterizes the earthquake ground motion in terms of its influence on engineering structures.2. can be written in a compact matrix form as: & x z [m]{&&} + [c]{x} + [k ]{x} = − &&[m]{I } . one of which corresponds to each mass mi. For most structural models. for 0 ≤ tmax ≤ T. Response Spectrum Seminar Lecture # 1 10 . the two spectra SV and FS become identical. Dashpots characterized by the damping constants ci model the energy dissipation in this system.Response Spectrum M.19) show the similarity between the Fourier amplitude spectrum and the exact relative velocity spectrum. The system of equilibrium equations. [⋅] designates a square n × n matrix. masses mi are lumped at the floor levels and are interconnected with massless columns.2. the damping matrix [c].D.Vancouver. In general. 1995a.b). (A. June 1st. The above relation between SV and FS links the two measurements of the same physical phenomenon from different points of view. Vn corresponds to the base shear acting between the structure and the rigid soil. Equations (A. SV is always greater than FS. In this model. In physical terms. these matrices have only the diagonal and few off-diagonal terms that differ from zero. FS is the maximum velocity of the undamped oscillator in the free vibration following the earthquake.20) Here. In general. The Fourier amplitude spectrum FS is the quantity used in investigations of the earthquake mechanism as it relates to the amplitudes of recorded waves (Trifunac. the mass matrix [m]. 2007 where 0 ≤ tmax ≤ T and tmax is the time at which the maximum response occurs. which have the equivalent spring constants ki. 2 Multi-degree-of-freedom system. A.Response Spectrum M. Trifunac CSCE Vancouver Section UBC . Response Spectrum Seminar Lecture # 1 11 .D. 2007 MULTI-DEGREE-OF-FREEDOM SYSTEM x1 m1 c1 x2 k1 h1 m2 c2 k2 V2 h2 x3 m3 c3 xn k3 V n-1 h3 mn cn kn Vn hn z Ground Fig.Vancouver. June 1st. 23) (A.24) (A.22) becomes && & [ I ]{ξ } + [C ]{ξ } + [ω 2 ]{ξ } = − &&[ A]T [m]{I } . 1937.22) (A. a set of uncoupled coordinates {ξ} and the corresponding transformation (A. Response Spectrum Seminar Lecture # 1 12 . z where the constant { Ai }T [m]{I } .29) is the mode participation factor. June 1st. [C] ≡ [A]T[c] [A]. Caughey.21) into (A. ⎥ M ⎥ 2ς 1ω1 ⎥ ⎦ (A. ⎡ 2ς 1ω1 ⎢ 0 ⎢ [C ] = ⎢ 0 ⎢ ⎢ M ⎢ 0 ⎣ 0 2ς 1ω1 0 M 0 L 0 0 O O L L L 0 ⎤ M ⎥ ⎥ M ⎥.21) {x} = [A] {ξ} can be found in such a way that the classical normal mode solution is possible (Rayleigh..3.Vancouver.Response Spectrum M.25) [A]T[k][A] = [ω2].D.26) (A.27) then equation (A. It shows the extent to which the ith mode is excited by the earthquake. each describing a single-degree-of-freedom system response in terms of a normal coordinate ξi : && ξi + 2ς iωiξ&i + ωi2ξi = − &&α i .. (A. by substituting (A. i = 1. 1959). Trifunac CSCE Vancouver Section UBC .. && It is possible to normalize the coefficients of {ξ } and {ξ} in such a way that [A]T[m][A] = [I]. and Denoting equation (A. 2. 2007 In the special case in which the damping matrix [c] is a linear combination of the mass matrix and the stiffness matrix.20) by the transpose of [A] there follows: && & z [ A]T [m][ A]{ξ } + [ A]T [c][ A]{ξ } + [ A]T [k ][ A]{ξ ] = &&[ A]T [m]{I } . Then.20) and pre-multiplying (A. n { Ai }T [m]{ Ai } (A..16) represents a set of n independent equations.28) αi = (A. where [l] is the unit matrix. z If [C] is a diagonal matrix. 36) Response Spectrum Seminar Lecture # 1 13 . A. (A. where [H] is the lower triangular matrix ⎡ h1 ⎢h ⎢ 1 ⎢h [H ] = ⎢ 1 ⎢M ⎢ h1 ⎢ ⎢ ⎣ 0 h2 h2 M h2 0 L 0 L h3 M O h3 L hn −1 L L 0⎤ 0⎥ ⎥ M⎥ ⎥ .D. the ith equation (A. z (A.5) for a singledegree-of-freedom oscillator. (A.28) is the same as Equation (A. The shear forces Vi (see fig.33) (A.2) are: {M(t)}=[H][S][k]{x(t)}.31) When the displacements {x(t)} of masses are determined the earthquake forces Fi(t) acting on each mass. mi are given by {F(t)} = [k] {x(t)}.35) (A. ⎥ O M⎥ L 1⎥ ⎦ (A.34) Similarly. 2007 As may be seen. June 1st. {x} can be calculated from {x(t)} = [A] {ξ(t)}. where [S] is the lower triangular matrix ⎡1 ⎢1 ⎢ [ S ] = ⎢1 ⎢ ⎢M ⎢1 ⎣ 0 1 1 M 1 0 0 1 M 1 L 0⎤ L 0⎥ ⎥ M⎥ .Response Spectrum M.2) are then: {V(t)} = [S] {x(t)}.Vancouver. M⎥ hn ⎥ ⎥ ⎥ ⎦ (A.32) (A. the moments at each level of the structure (fig. Its solution therefore becomes: ξi (t ) = −α i ωi 1 − ς 2 i ∫ t 0 &&(τ )e −ς iωi (t −τ ) sin ωi 1 − ς i2 (t − τ )dτ .30) for all i the displacements.30) After solving (A. Trifunac CSCE Vancouver Section UBC . The latter was already defined in Equation (A. The sum of maximum modal responses (Biot. the dynamic response of the r-th mode alone may be described by the equation that corresponds to a single-degree-of-freedom system (see Equation’s (A. does not define the response condition at any one time but the maximum response experienced by the different levels of a structure during the entire time history of the analysis. in terms of displacement and velocity spectra (equations (A. Here. from the engineering point of view.15)) equation (A. 2007 The ultimate objective of this analysis.D.4)..9).38) can be written as {| x ( r ) (t ) |}max = {| A( r ) |}α r SD ( r ) = {| A( r ) |}α r Tr SV ( r ) . 1942) Response Spectrum Seminar Lecture # 1 14 . the contribution of the r-th mode to the total displacement of a multi-degree-of-freedom system would be: {x ( r ) (t )} = { A( r ) }α r ωi 1 − ς 2 r ∫ t 0 &&(τ )e−ς rωr (t −τ ) sin ω r 1 − ς r2 (t − τ )dτ . for example. and that the total displacement response may be computed by adding contributions of each individual mode (eq. z (A.28). and (A. For example. 2π (A. corresponding to the r-th mode of vibration.Vancouver. {x}max. which are to be used for design purposes.37) ⎪ M ⎪ ⎪ g nmax ⎪ ⎩ ⎭ It may be noted that the maximum response vector. {·}max defines the following operation: ⎧ g1max ⎫ ⎪g ⎪ ⎪ 2 ⎪ {g}max ≡ ⎨ max ⎬ .5) and its absolute value SD was given by Equation (A. shears {V}max .9) through (A. is to calculate the envelope of maximum response {x}max. Tr However. and moments {M}max. Thus.39) where the superscripts (r) on SD and SV indicate that these spectral values are computed 2π for damping ζr and frequency ω r = . Trifunac CSCE Vancouver Section UBC . (A. June 1st. forces {F}max.Response Spectrum M. Response Spectrum Superposition In the preceding sections it was demonstrated that for multi-degrees-of-freedom. the different modal maxima do not occur at the same time. and therefore the individual modes do not simultaneously contribute their peak values to the maximum total response.38) This contribution is seen to be directly proportional to 1/ ω r 1 − ς r2 times the integral term and will depend on the integral's maximum absolute value.31)). (A. 41). 1958). et al. It can often provide reasonable results for design purposes. but at the same time may be too conservative. based on statistical considerations (Goodman.b. 1985. Gupta and Trifunac. and the conditions under which a meaningful degree of conservatism can be achieved have been determined.D.c. The question of mode superposition has been studied extensively (Amini and Trifunac. 1962).Vancouver. Merchant and Hudson. . When the maxima of each response quantity have been determined from equations analogous to equation (A. 2007 ⎧ SD (1) ⎫ ⎪ (2) ⎪ n ⎪ SD ⎪ (r ) (A. 1960) for structures in which the main contributions come from the lowest few modes. the RMS approximation is given by: ⎧ ⎫ 1/ 2 ⎪ ⎪ n 2⎤ i ⎪⎡ ⎪ ⎢ ∑ x1max ⎥ ⎪ ⎪ ⎣ i =1 ⎦ ⎪ ⎪ 1/ 2 ⎪⎡ n i 2 ⎤ ⎪ (A. An alternative approach.Response Spectrum M. This method (RMS) has been shown to give reasonable results (Jennings and Newmark. The advantage of this method is that it avoids lengthy computations associated with the exact method and at the same time takes into account the dynamic nature of the problem. is to take the square root of the sum of the squares of the individual modal maxima. Response Spectrum Seminar Lecture # 1 15 .41) {x}max ≈ ⎨ ⎢ ∑ x2max ⎥ ⎬ ⎦ ⎪ ⎪ ⎣ i =1 ⎪M ⎪ ⎪ ⎪ 1/ 2 ⎪⎡ n i 2 ⎤ ⎪ ⎪ ⎢ ∑ xnmax ⎥ ⎪ ⎪ ⎪ ⎦ ⎭ ⎩ ⎣ i =1 ( ) ( ) ( ) Analogous expressions for {V}max and {M}max follow from equation (A. The above-described response spectrum superposition method provides only an approximate indication of the maximum response in the multi-degree-of-freedom systems.. 1987a. Trifunac CSCE Vancouver Section UBC .39).40) ∑{| x (t ) |}max = [| A |] ⎨ M ⎬ r =1 ⎪ ⎪ ⎪ SD ( n ) ⎭ ⎪ ⎩ would clearly give an upper bound to the total system response. June 1st. 7) (B. for ti ≤ t ≤ ti+1. The solution of equation (B. For a(t). an approach based on the exact analytical solution of the Duhamel integral for the successive linear segments of excitation appears to be the most practical. Trifunac CSCE Vancouver Section UBC .2) (B.4) where and ∆ t = ti+1 . given by a segmentally linear function for ti ≤ t ≤ ti+1.Vancouver. then becomes: x(t ) = e −ςω (t −ti ) [C1 sin ω d (t − ti ) + C2 cos ω d (t − ti ) ] − ω ai 2 + 2ς ∆ai 1 ∆ai − (t − ti ). ζ = fraction of critical damping and ω = the natural frequency of vibration of the oscillator.8) Substituting C1 and C2 into equation (B. (B. C1 and C2 become: 1 ⎛ 2ς 2 − 1 ς ⎞ & + ai ⎟ C1 = ⎜ ςω xi + xi − ωd ⎝ ω2 ω ⎠ 2ς ∆a a C2 = xi − 3 i + i2 .5) and setting t=ti+1 leads to the recurrence & relationship for xi and xi .1) becomes: && + 2ως x + ω 2 x = − ai + & x ∆ai (t − ti ) .ti = const. given by: Response Spectrum Seminar Lecture # 1 16 .1) where. & & Setting x =xi and x = xi at t = ti.2). et al. ∆t (B. which are available at equally spaced time intervals ∆t . The important features of this method are briefly summarized here. June 1st.5) where ωd ≡ ω 1 − ς 2 . 1971).3) (B.Response Spectrum M. 2007 NOTE B B. This approach is described in Nigam and Jennings (1968). Computation of Response Spectra For the standard corrected accelerograms (Trifunac.ai . The differential equation for the relative motion x(t) of a single-degree-of-freedom oscillator subjected to base acceleration a(t) is given by: && + 2ως x + ω 2 x = − a(t ) & x (B.5) (B. ω ∆t ω (B.D. ∆ ai = ai+1 . ω 3 ∆t ω 2 ∆t (B. and are constant during the calculation of the response.Vancouver. & & ⎩ xi +1 ⎭ ⎩ xi ⎭ ⎩ai +1 ⎭ (B. the complete response can be computed by a step-by-step application of equation (B.D. The advantage of this method lies in the fact that for a constant time interval ∆ t. Trifunac CSCE Vancouver Section UBC . if the displacement and velocity of the oscillator are known at ti . ω −ςω∆t ⎪ a21 = − e sin ω d ∆t ⎪ 1− ς 2 ⎪ ⎛ ⎞⎪ 1 a22 = e −ςω∆t ⎜ cos ω m ∆t − − sin ω d ∆t ⎟⎪ ⎜ ⎟⎪ 1− ς 2 ⎝ ⎠⎭ ⎛ 1 ⎞ sin ω d ∆t + cos ω d ∆t ⎟ a11 = e −ςω∆t ⎜ 2 ⎜ ⎟ ⎝ 1− ς ⎠ e −ςω∆t a12 = sin ω d ∆t (B. ω .10) ⎡⎛ 2ς 2 − 1 ς ⎞ sin ωd ∆t ⎛ 2ς ⎤ 2ς ⎫ 1 ⎞ + ⎟ + ⎜ 3 + 2 ⎟ cos ωd ∆t ⎥ − 3 ⎪ b11 = e −ςω∆t ⎢⎜ 2 ⎝ ω ∆t ω ⎠ ⎣⎝ ω ∆t ω ⎠ ωd ⎦ ω ∆t ⎪ ⎪ ⎡⎛ 2ς 2 − 1 ⎞ sin ωd ∆t ⎤ 1 2ς 2ς ⎪ + 3 cos ωd ∆t ⎥ − 2 3 b12 = e −ςω∆t ⎢⎜ 2 ⎟ ⎪ ω ∆t ⎣⎝ ω ∆t ⎠ ωd ⎦ ω ω ∆t ⎪ ⎪ ⎡⎛ 2ς 2 − 1 ς ⎞ ⎛ ⎞ ς ⎪ + 2 ⎟ ⎜ cos ωd ∆t − b21 = e −ςω∆t ⎢⎜ 2 sin ωd ∆t ⎟ ⎟ ⎪ ⎢⎝ ω ∆t ω ⎠ ⎜ 1− ς 2 ⎪ ⎝ ⎠ ⎣ ⎬. velocity SV =| x(t ) |max and absolute acceleration SA =| &&(t ) + a (t ) |max are stored x Response Spectrum Seminar Lecture # 1 17 . June 1st.⎜ 2 + 2 ⎟ (ωd sin ωd ∆t + ςω cos ωd ∆t ) ⎥ + 2 ⎪ ⎝ ω ∆t ω ⎠ ⎦ ω ∆t ⎪ ⎪ ⎡ 2 ⎛ ⎞ ς ⎪ −ςω∆t 2ς − 1 ⎢ 2 ⎜ cos ωd ∆t − b22 = −e sin ωd ∆t ⎟ ⎪ ⎟ ⎢ ω ∆t ⎜ 1− ς 2 ⎝ ⎠ ⎣ ⎪ ⎪ 2ς 1 ⎤ ⎪ .9). ∆t ) ] ⎨ ⎬ + [ B(ς . ∆t ) ] ⎨ ⎬ .Response Spectrum M.11) Therefore. maximum values of displacement SD = & |x(t)|max. matrices A and B depend only upon ζ and ω . ω .2 (ωd sin ωd ∆t + ςω cos ωd ∆t ) ⎥ − 2 ω ∆t ⎪ ⎦ ω ∆t ⎭ B.9) The elements of matrices A and B are: ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ωd ⎪ ⎬. 2007 ⎧ xi +1 ⎫ ⎧ xi ⎫ ⎧ ai ⎫ ⎨ ⎬ = [ A(ς . 2 ⎤ ⎪ ⎛ 2ς 1 ⎞ 1 . To calculate and plot complete response spectra. (1958). Goodman. The calculation of these & maxima is approximate because the displacement x(t). Biot.02 s.D. (1942). L. “Statistical Extension of Response Spectrum Superpositions. Biot. E.A. and &&1 + ai for i = & x x 1. “Classical Normal Modes in Damped Linear Systems.D. Trifunac CSCE Vancouver Section UBC . T is the period of the oscillator for which the spectrum point is calculated. entitled "Transient Oscillations in Elastic System. 4..D.” Proc. of Tech. 2007 for each period T = 2π ω and a fraction of critical damping ζ . 59-A-62. 365-408. velocity x(t ). “Theory of Elastic Systems Vibrating Under Transient Impulse With an Application to Earthquake-Proof Buildings. 782-802.” Journ. 262-268. “A Mechanical Analyzer for the Prediction of Earthquake Stresses.” Chapter II in Ph. Biot. M.2.. where the values are xi. 19(2). Soc. Seism. Soc. 31.Vancouver. For such a choice of integration interval the discretization error is always less than 5%. Calif. (1933). (1985). “Seismic Design of Firmly Founded Elastic Structures.N (where N is the total number of discrete. xi . For standard spectrum calculations. and Trifunac. K.A. Biot. N. National Academy of Sciences.” Zeitschrift für. 2. 151-171.A. Thesis No. 259. "Theory of Vibration of Buildings During Earthquakes. Caughey. Vol. pp. pp. Mechs.. M. pp. App.” ASCE Transactions.12) 10 but it is always less than or equal to ∆ tmax = 0. “Analytical and Experimental Methods in Engineering Seismology. Amer.M. Civil Eng. M. A. References for NOTES A and B Amini. Here. (1959). June 1st. Biot.. Angewandte Matematik und Mechanik. E. 120. equally spaced points at which the input accelerogram is given).Response Spectrum M..S. (1941). (1932).. M. 108.. Pasadena. Response Spectrum Seminar Lecture # 1 18 . pp. “Vibrations of Buildings During Earthquake. M. T. California. M. (B.A. 213-223." Aeronautics Department.” Trans. pp. No.” Bull. Amer. and Newmark.…. and acceleration &&(t ) are found only at discrete points. (1934). Inst. 14(4). the choice of the interval of integration ∆ T is selected to be: T ∆t ≤ .. 54-63. Rosenblueth.” Soil Dynamics and Earthquake Eng. I. “Statistical Analysis of Response Spectra Method in Earthquake Engineering. of Indian Society of Earthquake Technology. of Civil Eng. pp. ASCE. 15-50.Vancouver. Earthquake Res. pp. and Hudson. Soc. Response Spectrum Seminar Lecture # 1 19 . 7.” Earthquake Eng. Seism. (1993).” New York: MacMillan.” Earthquake Eng. 52. and Trifunac. Univ. 24. C.” Bull. Engineering Mechanics. J.D.D. Amer.. Vol. Los Angeles.” Proc. Merchant. Trifunac. Gupta. “Investigation of Building Response to Translational and Rotational Earthquake Excitations. Gupta. M. Vibration. “Order Statistics of Peaks in Earthquake Response of Multi-Degree-of-Freedom Systems. M. Vol. Rubin. of Civil Eng. 4. (1956). lnst. Jennings. pp. “Mode Superposition in Multi-Degree-ofFreedom System Using Earthquake Response Spectrum Data. and Trifunac. California Institute of Technology. S. and Newmark. Hudson. L. On Earthquake Eng. H.E. V.. “Some Problems in the Application of Spectrum Techniques to Strong-Motion Earthquake Analysis. Tokyo University. 2007 Gupta. M. CE 93-01. “Order Statistics of Peaks in Earthquake Response. Res. 3-4. “Notes on the Theory of Vibration Analyzer. of Southern California. California. (1987b). “Concepts in Shock Data Analysis. Los Angels. “Elastic Response of Multi-Story Shear Beam Type Structures. Gupta. 10. No. Second World Conf.” J. Los Angeles. and Trifunac. I.D. “The Theory of Sound. Vol. Report No.Response Spectrum M.D.. I. (1961). Trifunac CSCE Vancouver Section UBC .” Bull. of Southern California.C. 699-717. and Eng.M. Vol. and Trifunac. No.E. Gupta. Kawasumi. (1988). Report No.” Dept. CE 87-03 Univ.” Dept. of Southern California. Part I. I. Pasadena. Seism.D.D. P.. pp. (1968). D.. (1989).. H. 405-416. Soc. (1937). XXXIV.. California. M. (1962). Report No. II. Rayleigh.. 417-430. CE 89-02.K. California.. “Order Statistics of Peaks of Response to MultiComponent Seismic Excitation.M. R.” Chapter 23 of Shock and Vibration Handbook (ed. and Jennings. N. M.E. 52.. No. 135-159. “Digital Calculation of Response Spectra From Strong-Motion Earthquake Records.D. (1960).C. and Trifunac.D. New York: McGraw-Hill. S. Crede).. Univ.D. Nigam. C. N. 114.D.. Tokyo. of Civil Eng.. Amer. D.. June 1st. (1987a). “Broad Band Extension of Fourier Amplitude Spectra of Strong Motion Acceleration” Dept.W. (1987c). pp. (1962). Harris and C.D.” Bull. Lab. 1605-1537. M.” Bull. Vol. . Corrected Accelerograms and Integrated Velocity and Displacement Curves. and Structural Dynamics. 2007 Trifunac. Pasadena. M.” Earthquake Engineering Research Laboratory. (1995b). Report EERL 71-51.. Response Spectrum Seminar Lecture # 1 20 . Hudson.D. pp. D. pp. Brady A. 331-346. Von Kármán. California Institute of Technology. California. . Trifunac CSCE Vancouver Section UBC . 14(5). (1940) “Mathematical Methods in Engineering.Response Spectrum M. and Vijayaraghavan.D. The Wind and Beyond. “Pseudo-Relative velocity spectra of Earthquake Ground Motion at Long Periods. (1967).. and Earthqu. and Edson. London...” Earthqu. “StrongMotion Earthquake Accelerograms.Vancouver. M.D. A.” Soil Dynam. M. T. 24(8).E. Boston. Toronto.A. Engng. Brown and Co. Trifunac. M. “Pseudo-Relative velocity spectra of Earthquake Ground Motion at High Frequencies. (1971). L. New York. Von Kármán. II.D. and Biot. Trifunac. (1995a).G. Little. June 1st. T. Engng. 1113-1130.” McGraw-Hill. . Response spectra were scaled in terms of peak amplitudes of strong ground motion approximately. and 0. 1988.10.. when modern digital computers became available. Veletsos et al.b). The data were then distributed on magnetic tapes and computer cards. 0. the 1952 Kern County. USAEC. Response Spectrum Seminar Lecture # 1 21 . velocity. empirical regression analyses of spectral amplitudes were not possible because there were only a few significant processed earthquake records (e. and the corresponding response spectral amplitudes were calculated at 91 periods between 0.. who analyzed 33 records.Vancouver. and displacement of each record.b. manual process.c). 1990a. 2006). EMPIRICAL SCALING OF RESPONSE SPECTRA (modified from Lee 2007) By the mid-1960s. Spectrum shape was also studied by Blume et al.04 to 15 seconds for damping ratios of 0. then started to select. 1970. the 1989 LomaPrieta. and by 1975 all of the records had been processed. Gupta and Trifunac.S. 1988a. 1994a. led by D. Nuclear Regulatory Commission. 1993). and the 1994 Northridge earthquakes.D. Todorovska. the 1940 Imperial Valley. The earthquake strong-motion data processing program at the California Institute of Technology in Pasadena. those recorded during the 1933 Long Beach. These equations were later used for the computation of uniform hazard PSV spectra in the probabilistic site-specific analyses for seismic microand macro-zonation (Trifunac. 2007 NOTE C C..S. Also. The joint recommendations of the Newmark and Blume studies of the shape of the response spectra (Newmark et al.E.b. California. 1989d. peak velocity. June 1st. Those equations were also needed in the probabilistic determination of the envelopes of shear forces and of bending moments in engineering design (Amini and Trifunac. 0..02. Atomic Energy Commission (now the U.20 (Hudson et al.Response Spectrum M.b. (1972). For example. 1971 1972a. The San Fernando earthquake of February 9. More than 250 analog accelerometers in Southern California were triggered and recorded many excellent acceleration traces. 1985. Trifunac CSCE Vancouver Section UBC . digitize. and the 1968 Borrego Mountain earthquakes). various groups started to develop regression equations for the empirical scaling of response spectral amplitudes.05. With an ever-increasing digitized database. During the following 30 years. the 1966 Parkfield. 1990b).0. 1965) noted that the shape of response spectra can be determined by specifying peak acceleration. 1973) for use in the design of nuclear power plants.g. 0. the digitization and processing of strong-motion records from analog instruments was a slow. A series of reports were published detailing the corrected and processed acceleration. including the three in California: the 1987 Whittier-Narrows. and in the estimation of losses for buildings exposed to strong shaking (Jordanovski et al. Hudson. requiring many hours of hand digitization (Trifunac. 1973) were later adopted by the U. Newmark and coworkers (Newmark and Veletsos. 1964. 1971 changed all that. and process all significant records. many well-recorded strong-motion earthquakes occurred worldwide. and peak displacement of strong ground motion. By the early 1980s. 1994. The following is a brief summary of all their work on the empirical scaling of response spectral amplitudes only. The data had been selected. Trifunac and Novikova.b. 1978. 1973.b.0 to 7. 1990. 1991.. Semi-theoretical extrapolation functions for extension of these empirical equations to both high and low frequencies had also been presented (Trifunac. 1979. The examples include: 1970s: Trifunac.b.D.. Three generations of empirical regression equations for the scaling and attenuation of spectral amplitudes have been developed so far. 1989a. 1975a. June 1st.b. 2001a) continued. 1993. Lee and Trifunac. 1979b. Trifunac and Todorovska.c. The magnitudes of the earthquakes in the database ranged from 3.d.e. 2007 Summary of the contributions of the Strong-Motion Group at USC The Strong-Motion Earthquake Research Group at the University of Southern California contributed papers and reports on the empirical scaling of strong-motion spectra. Trifunac.S. Todorovska. digitized.b. and the work on the collection of strong-motion records (Anderson et al. Trifunac et al.b.c.S. 1995a. 1993. 1993a.c. and processed while Trifunac and Lee were at the Engineering Research Laboratory of the California Institute of Technology in Pasadena.7. Lee. This corresponds to 558 acceleration components of data from 57 earthquakes in the western U. 1995. 1995a. 1991. 2000s: Trifunac and Todorovska.. Trifunac and Lee.c. The Database and Data Processing Procedures The database for the first generation of scaling equations of spectral amplitudes in the 1970s consisted of 186 free-field recordings. Trifunac and Zivcic. Trifunac and Lee.Vancouver. 1987. Lee and Trifunac. 1985. In 1976.d. 1994a. 1981.c. 1976a.c. Response Spectrum Seminar Lecture # 1 22 .b.b). 1971. 1989a. the Strong-Motion Group moved to the University of Southern California in Los Angeles. 1989. Trifunac. 1980s: Trifunac and Lee. 1979).b. 1991a.. 1978.b. 1977. 1992. Lee and Trifunac.b. All of these strong-motion records are documented in a series of USC reports entitled the Earthquake Strong-Motion Data Information System (EQINFOS) (Trifunac and Lee.b. 1987). 1980.). Trifunac CSCE Vancouver Section UBC . 1990s: Lee. Most of the contributing earthquakes were from northern and southern California. the secondgeneration database was expanded to 438 free-field records from 104 earthquakes.c. 1988. 1978a. 1985a. 1991.b.Response Spectrum M.. 1972a.d.c. 1970. Trifunac and Todorovska. 1989a. 2001a.e. 1990a. and all data were hand digitized from analog records using a manually operated digitizer (Hudson et al. Trifunac and Brady.b.d. 1990.b. The automatic digitization and data processing of strong-motion records by a mini-computer were developed and introduced in 1979 (Trifunac and Lee. Trifunac et al.d. The earthquakes included in the list of contributing events started with the 1933 Long Beach earthquake and ended with the San Fernando earthquake of 1971. A review of and further details on the contributions of this group can be found in Lee (2002). 1979a. Lee et al. Trifunac and Anderson. and all were from the western U. 1977a. 1994a.b. Vancouver. Those included the records from the main shock and aftershocks of both the 1987 Whittier Narrows and the 1994 Northridge earthquakes in Southern California. s. in contrast to the geotechnical site characterization viewed for the top several tens of meters only (Trifunac. the strong-motion database (third generation) grew to over 1. Other records included were mainly those from the Strong-Motion Instrumentation Program (SMIP) of the California Division of Mines and Geology (CDMG) and from the United States Geological Survey. Many accelerograms in Southern California were recorded by the USC strong-motion array (Trifunac and Todorovksa. 1979b. However. representative of the top 100 ~ 200 m of soil (Trifunac. The first is the local soil type. 1984). In the 1980s.926 free-field records from 297 earthquakes and aftershocks. h. At each stage of the database processing. h. This geological site classification is: Table 1 Geological Site Classification 0 1 2 Description alluvial and sedimentary deposits intermediate sites basement rock Ideally. This new parameter was used in the second generation of empirical scaling equations in the 1980s. and from the 1989 Loma Prieta earthquake in Northern California. all data were treated uniformly. If the analog records were available. Table 2 Soil Type. Site Classification The first geological site classification was introduced (Trifunac and Brady. those were digitized and processed by the automatic digitization system using a PC in the strong-motion laboratory at USC (Lee and Trifunac. 1975b) to describe the broad environment of the recording station and was based on geologic maps.D. according to this approach. Lee and Trifunac. June 1st. using the standard software for image processing developed at USC (Trifunac and Lee. 1990a). a site should be classified either as being on sediments (s = 0) or on the basement rock (s = 2). sL 0 1 Description “rock” soil site stiff soil site Lecture # 1 Response Spectrum Seminar 23 . The recording sites were to be viewed on a scale measured in terms of kilometers. and the depth of sediments. 1990). Trifunac CSCE Vancouver Section UBC . for some sites having a complex environment. II. 1990a). 2007 By late 1994. as a site characteristic. in km.Response Spectrum M. an “intermediate” classification (s = 1) was assigned. sL . Trifunac and Lee (1979a) later refined the above classification and used the depth of sediment beneath the recording site. additional parameters were introduced to refine the characterization of the local site beyond the geological site condition. 2001b). 1979. 767 . M . detailed physical description of such a function. was used as follows: Table 3 Soil Velocity Type. T ). the reader is referred to the above reference.b) developed the first magnitude-frequencydependent attenuation function.36 0.8sec and a constant beyond that. the functional form of the attenuation with epicentral distance R followed the definition of local magnitude scale (Trifunac. the source-to-station distance. M . approximated by a parabola for T < 1. 1991b). Trifunac CSCE Vancouver Section UBC . Trifunac. a function of the “representative” distance ∆ from the source to the site.526 . (C.2) with A 0 (T ) . 1958. of the soil in the top 30 m. Hence. the velocity type was represented by indicator variables.Vancouver. where log A0 ( R ) together with a term linear in epicentral distance at each period was intended to account for the average correction for anelastic attenuation.D. ∆ = S ⎜ ln 2 2 2 ⎟ ⎝ R + H + S0 ⎠ Response Spectrum Seminar Lecture # 1 (C. The soil velocity type variable.75 ≥ VL > 0. Trifunac and Lee (1985a. VL .272 and c = −0. Distance Definition Used for Attenuation Relation In the 1970s. the functional form of attenuation. log A0 ( R) + K g (T ) R .732025 ⎪ ⎩ T < 1. T ) = A 0 (T ) log10 ∆ . where A 0 (T ) = ⎨ ⎧a + b log10 T + c ( log10 T )2 ⎪ − 0.75 km/s 0. A detailed description of this attenuation function can be found in Trifunac (1976b).3) 24 . In 1980s. For a complete. for magnitude M and for period T of strong motion. Att (∆.18 VL ≤ 0. 1976b).8sec T ≥ 1.8sec (C. which states that the logarithm of the corrected peak amplitude on a standard instrument is equal to the earthquake magnitude (Richter. ST .1) was used. June 1st. was defined as in Gusev (1983): ⎛ R2 + H 2 + S 2 ⎞ . 2007 2 deep soil site The second parameter added to site characterization was the average shear wave velocity. Att (∆. a function in T. III. where a = −0. Briefly. ST A B C D Description VL > 0. b = 0. ∆ .Response Spectrum M.18 km/s In the scaling equations.36 ≥ VL > 0. Response Spectrum M. due to the limited amount of data. 2. L = L(M). 4. where cS is the shear wave velocity in the rocks surrounding the fault. In this. 8. almost horizontally rock-to-rock (100%) rock-to-rock through sediments. r is the ratio (or percentage) of wave path through geological basement rock relative to the total path. S = 0. almost vertically rock-to-sediments. only eight such categories could be identified with a sufficient number of recordings to be included in the regression analyses. Response Spectrum Seminar Lecture # 1 25 . Table 4 Path Type 1. 1995. ∆ is thus a dimensionless L representative source-to-station distance. M .b).01× 100. T ) = ⎨ ( R − Rmax ) ∆ ⎪ A 0 (T ) log10 ⎛ max ⎞ − ⎜ L⎟ 200 ⎝ ⎠ ⎩ ( ) R ≤ Rmax R > Rmax (C. S.5 M km (Trifunac. It was approximated by L = .. as a percentage) was introduced.2 + 8. almost vertically rock-to-sediments through rock and sediments. IV.51( M − 3) is the size of the earthquake source at magnitude M. 5. a generalized path type classification was also used. It describes the characteristic types of wave paths between the sources and stations for the strong-motion data available up to the early 1990s in the western U. Trifunac and Lee (1990) modified this attenuation function to the following form: ⎧ A 0 (T ) log10 ∆ L ⎪ Att (∆.D. H is the focal depth. 0 ≤ r ≤ 1 (or 100r. a new term (Lee et al. r. ∆ max and Rmax represent the distances beyond which Att (∆. almost horizontally.b). measured along the surface from the earthquake epicenter to the recording site. In the 1990s. The new parameter. Description sediments-to-sediments (100%) rock-to-sediments. M .. 2007 where R is the surface distance from epicenter to the site. June 1st. and S0 is the correlation radius of the source function. Trifunac CSCE Vancouver Section UBC . It was approximated by S0 = cS T / 2 . Lee and Trifunac. At that time. 1993a. almost vertically rock-to-sediments through rock and sediments. Alternately.Vancouver.4) with ∆ and R defined as above. 3. The Source-to-Station Path Types In the third generation of regression studies of spectral amplitudes in the 1990s. almost horizontally rock-to-rock through sediments. was introduced to model the length of the earthquake fault. 7. 1995a. T ) has a slope defined by the Richter’s local magnitude scale M L . 6. Figure C. “4” 4. The eight path types in this figure can further be grouped into four path groups: “1”. 5. and “4”. as described in Table 5. Path Group Path Types Description “1” 1 Earthquake source and recording site within the same sediment.D. 8 Earthquake source in basement rock. “2” 2.Response Spectrum M. recording site on the same basement rock. The Scaling Equations Lecture # 1 Response Spectrum Seminar 26 . “3”. Trifunac CSCE Vancouver Section UBC . “3” 3. 6 Earthquake source in basement rock. Table 5 Eight path types from source to recording station. V.1. recording site almost vertically above. with or without sediments between.Vancouver. almost horizontally. Recording site on nearby sediment.1 shows a schematic representation of the “geometry” of these path types. 2007 Figure C. “2”. June 1st. 7 Earthquake source in basement rock close to the surface. 5) where R< = min ( R.Vancouver. M . M . M . A description of the complete set of scaling relations of all three generations can be found in Lee (2002). T ) + b1 (T ) M + b2 (T ) s + b3 (T )v + b4 (T ) + b5 (T ) M 2 ( ( + ∑ i b6i ) (T ) S6i ) + ( b70 (T )r + b71 (T )(1 − r ) )R< . 1995a. and illustration of the results and comparison with the actual data can be found Response Spectrum Seminar Lecture # 1 27 . Trifunac CSCE Vancouver Section UBC . Model (i): log PSV (T ) = Mag-site + soil + % rock path multi-step model M + Att (∆. M .7) log PSV (T ) = M + Att (∆. Substituting Att (∆. 4). M .Response Spectrum M.6) ( ) Model (ii): Mag-depth + soil + % rock path multi-step model (C. C.5 gives: ⎧ M + A (T ) log ∆ + b (T ) M + b (T ) s + b (T )v + b (T ) + b (T ) M 2 0 10 1 2 3 4 5 L ⎪ (i ) (i ) ⎪ + ∑ i b6 (T ) S6 + ( b70 (T )r + b71 (T )(1 − r ) )R R ≤ Rmax ⎪ ⎪ log PSV (T ) = ⎨ ∆ M + A 0 (T ) log10 ⎛ max ⎞ + b1 (T ) M + b2 (T ) s + b3 (T )v + b4 (T ) + b5 (T ) M 2 ⎜ L⎟ ⎪ ⎝ ⎠ ⎪ (R − R ) ⎪ + ∑ i b6(i ) (T )S6(i ) + ( b70 (T )r + b71 (T )(1 − r ) )Rmax − 200max R > Rmax ⎪ ⎩ (C.D. T ) into Eqn. T ) + b1 (T ) M + b2 (T )h + b3 (T )v + b4 (T ) + b5 (T ) M 2 ( ( + ∑ i b6i ) (T ) S6i ) + ( b70 (T )r + b71 (T )(1 − r ) )R< Model (iii): Mag-site + no soil + % rock path multi-step model (C.b) for spectral amplitudes will be illustrated here. The following regression equations illustrate the four scaling models.8) log PSV (T ) = M + Att (∆.9) log PSV (T ) = M + Att (∆. T ) + b1 (T ) M + b2 (T ) s + b3 (T )v + b4 (T ) + b5 (T ) M 2 + ( b70 (T )r + b71 (T )(1 − r ) ) R< Model (iv): Mag-site + no soil + % rock path multi-step model (C. (C. Rmax ) . June 1st. T ) + b1 (T ) M + b2 (T )h + b3 (T )v + b4 (T ) + b5 (T ) M 2 + ( b70 (T )r + b71 (T )(1 − r ) ) R< Description of the detailed steps required for the development of these regression equations. and Att (∆. 2007 Only the most recent (the third generation) scaling equations (Lee and Trifunac. T ) were defined in the previous section (Eqn. M . b). 1979a): log10 [ PSV (T ) ] = b(T ) I MM + c(T ) + d (T )h + e(T )v (C. Scaling With Modified Mercalli Intensity (MMI) Instead of using magnitude and distance to describe the strong earthquake motions. so estimated MMI levels were used instead. (1985). In the second generation. J. in the 1980s. the (first generation) scaling equations can take the following form (Trifunac and Lee.11) The scaling models of thee other research groups were compared by Lee (2007) with those outlined above.10). the regression parameters. Int. Soil Dynam.b) (as did all previous USC equations) considered the horizontal and vertical response spectral amplitudes simultaneously in the same equation. Trifunac CSCE Vancouver Section UBC . pp.S. VI.. a brief summary of which can also be found in Lee (2002). In the U. These estimated MMI levels were calculated using the equation (Lee and Trifunac. the reported MMI levels were not available. the Modified Mercalli Intensity Scale (MMI) is used. M. Vol. 1985): ) I MM = 1. Response Spectrum Seminar Lecture # 1 28 . For some of the free-field sites.5M − A − B ln ∆ − C ∆ /100 − Ds . and Trifunac. and all other scaling parameters. References Amini.Vancouver. 2007 in Lee and Trifunac (1995a. the analysis was carried out on the database of 438 free-field records from 104 earthquakes. “Statistical Extension of Response Spectrum Superposition”. 54–63. another alternate scaling parameter is the site intensity. being the same as before.Response Spectrum M. where v = 0 for the horizontal components and v = 1 for the vertical components. h and v.. The reader can further peruse those comparisons in Lee (2007). and their differences and similarities discussed. I MM . Earthquake Engg.10) with I MM being the discrete levels of the MMI scale at the recording site.D. To scale PSV spectra in terms of MMI. These are differentiated by the term b3 (T )v. No.D. 4. A. ) I MM was then used in place of I MM in the above scaling equation (C. and the scaling equation used in regression. all these for all the studies have been examined. 2. the dependent scaling variables. Note 1:The equations of Lee and Trifunac (1995a. June 1st. Since a typical scaling model involves the database. Comparison with the Work of Several Other Groups and Conclusions (C. A. University of Southern Calif.. Hudson.. “Attenuation of Intensity with Epicentral Distance in India”.R. Corrected Accelerograms and Integrated Velocity and Displacement Curves”. California Institute of Technology. Jordanovski. AEC Report No.D. Europ. II. (1983).D. Part I: The Model. M.D. 14–32.K. Department of Civil Engineering. “Order Statistics of Peaks in Earthquake Response”.I. D. Manic. No. Mech.D. Vol. J.R. “Strong Earthquake Ground Motion Data in EQINFOS: Yugoslavia. 2007 Anderson. Brady.. “Strong-Motion Earthquake Accelerograms.S.. Trifunac.D. and Trifunac. Part II: A Hypothetical Example”. Blume and Associates. Blume. Mech. ASCE J. “Response of Multistoried Buildings to Ground Translation and Rocking During Earthquakes”. A. M. (1988a). A. A.A. “Los Angeles Vicinity Strong Motion Accelerograph Network”. Olumceva.. and Moslem. Prob.D. 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Source to Station Distance and Recording Site Conditions”. M. Vol. Univ. 3. E. M. “A Note on the Range of Peak Amplitudes of Recorded Accelerations. of Civil Eng.A. M. (1995b). 2007 Lee. C.. Dept. Vol. Int. J. M. Univ. Soil Dynam. Amer. Trifunac CSCE Vancouver Section UBC . Seism. M. Soc. Trifunac. M. CA. and Novikova. of Southern California.D. 1.” Vol.K.D. Blume. Engng Struct. “Empirical Equations Describing Attenuation of the Peaks of Strong Ground Motion.. Response Spectra of SingleDegree-of-Freedom Elastic and Inelastic Systems. (1958). 13. and Veletsos. Earthqu. Report CE 95-04. ASCE.. 211–217. San Francisco. Newmark. N. “Analysis of Strong Earthquake Ground Motion for Prediction of Response Spectra”. (1976b). N. Vol. Distance. CA. Soil Dynam.I. “Comparison of Response Spectrum Amplitudes from Earthquakes with Lognormally and Exponentially Distributed Return Period”. M. (1976a).S. Amer. Todorovska. M. Dept. 66. Vol. Epicentral Distance and Recording Site Conditions”. Seism. (1964). Los Angeles. Freeman and Co.D. June 1st. Path Effects. 14. Bull.I.Vancouver. (1995). “Preliminary Empirical Model for Scaling Fourier Amplitude Spectra of Strong Ground Acceleration in Terms of Earthquake Magnitude. K. and Trifunac..D.. 3. Trifunac. Response Spectrum Seminar Lecture # 1 31 . Vol. No. Engng. (1973).. “A Note on Distribution of Amplitudes of Peaks in Structural Response Including Uncertainties of the Exciting Ground Motion and of the Structural Model”. Lee. M. 189–219. 97–116. Report CE 95-02. III.” J.D. of Southern California.I. Velocities and Displacements with Respect to the Modified Mercalli Intensity”. “Pseudo Relative Velocity Spectra of Strong Earthquake Ground Motion in California”.M.I. Earthqu. A. Earthqu. 287–303. “Preliminary Analysis of the Peaks of Strong Earthquake Ground Motion—Dependence of Peaks on Earthquake Magnitude. 13.” Report for Air Force Weapons Laboratory. D. (1989d). June 1st. 2. “Empirical Scaling of Fourier Spectrum Amplitudes of Recorded Strong Earthquake Accelerations in Terms of Modified Mercalli Intensity. M. Mech. Vol. (1989b).D. Trifunac. M. Response Spectrum Seminar Lecture # 1 32 . Earthqu. J. Earthqu. M.D. M. VII.D. Vol. Int.. Vol. Vol. 9th World Conf. 217–224. Soil Dynam. J. (1989a). (1990a). Soil Dynam. Trifunac. 63–74. Earthqu. Soil Dynamics Earthquake Eng.D. Geofizika. Earthqu. Int. No. Vol. Engng. 9. 1081–1097. 10.. No.. 6. Dynam. (1978). M. M. Trifunac. 9. Soil Dynam.D. Dynam. Engng. 34–43. 2007 Trifunac.. 6. Int. No. Engng and Struct. M. M. Vol. JSCE. Structural Dyn. 10. Vol. Eng. Trifunac. Earthqu. (1990b).D. 18. 6. No. Vol. 75–80. (1989c). “Seismic Microzonation Mapping via Uniform Risk Spectra”. No. “MSM ”. Trifunac. 12. “A Microzonation Method Based on Uniform Risk Spectra”. Vol.. Engng. 1. “Dependence of Fourier Spectrum Amplitudes of Recorded Strong Earthquake Accelerations on Magnitude. 7. 104./Earthquake Eng. (1993a). 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Proc. “Scaling Strong Motion Fourier Spectra by Modified Mercalli Intensity. 19. “Empirical Scaling of Fourier Spectrum Amplitudes of Recorded Strong Earthquake Accelerations in Terms of Magnitude and Local Soil and Geologic Site Conditions”. 65–72. (1991a).Vancouver. Vol. 2. Structural Eng. “Preliminary Empirical Model for Scaling Fourier Amplitude Spectra of Strong Motion Acceleration in Terms of Modified Mercalli Intensity and Geologic Site Conditions”. Japan.. J. No. 999–1016. “Response Spectra of Earthquake Ground Motion”.D. Dept. M. Soil Dynam. (1978b). (1995b). Austria. No. Trifunac..D. “Earthquake Source Variables for Scaling Spectral and Temporal Characteristics of Strong Ground Motion”. “Pseudo Relative Velocity Spectra of Earthquake Ground Motion at High Frequencies”. Earthqu.D. (1994c). Southern California. Trifunac CSCE Vancouver Section UBC . No. Trifunac. Vol. Report CE 78-04. Los Angeles. Trifunac. Vol. M.Vancouver. Dept. 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(1979a). 139–162. of Southern California. J. “Correlations of Peak Acceleration. and Lee. Los Angeles. Soc. Response Spectrum Seminar Lecture # 1 34 . (1975c). and Brady. Los Angeles. Dept. Earthquake Engineering. Trifunac. (1978). Univ. CA.. Seism.. Dept. 65. Part I: Automatic Digitization”. V. A. Proc. M. Trifunac. and Lee. A.W. MI. Report CE 79-02. CA. CA. Los Angeles. Dynamics.D. Int. A. Vol. (1985c). M. (1985a). and Brady.D. of Southern California. Soc.D. and Brady. Seism. M. Dept. “Estimation of Relative Velocity Spectra”. V. Vol.Response Spectrum M. and Lee. 65. Dept.W. “Frequency Dependent Attenuation of Strong Earthquake Ground Motion”. and Brady. Epicentral Distance and Site Conditions”. V. Site Intensity and Recording site Conditions”. Trifunac. Trifunac. M. “On the Correlation of Seismoscope Response with Earthquake Magnitude and Modified Mercalli Intensity”.. “A Study on the Duration of Strong Earthquake Ground Motion”. A. Amer.W. Soc.D. (1975a). 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(1979b). 65. Bull.G.. M.D. Trifunac. Bull. Trifunac. Trifunac.S. India. Report CE 79-15 I. M. Source to Station Distance. “Automatic Digitization and Data Processing of Strong-Motion Accelerograms. M. Trifunac. Report CE 78-14. “Dependence of Pseudo Relative Velocity Spectra of Strong Motion Acceleration on the Depth of Sedimentary Deposits”. of Civil Eng. of Civil Eng. M. Report CE 85-03.D and Lee. 6th Symp.. A. National Conference Earthquake Engineering. Univ. M..D. Trifunac CSCE Vancouver Section UBC . M. Trifunac. Dept.W. Report CE 85-02. and Brady. Seism. of Southern California. 455–471. (1975e). (1975d). 581–626. V. Proc.D. No. Site Intensity and Recording Site Conditions”. Univ. 10th Europ. 5. Vol. and Todorovska. Trifunac. M. No. of Southern Calif.. Trifunac... M. M. “A Note on Scaling Peak Acceleration. No.D.. H... “Attenuation of Seismic Intensity in Balkan Countries”. CA. 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Vienna. Trifunac. Engng..D. Los Angeles. Report CE 85-04. Vol. and Lee.. of Civil Eng. M.I. 1.D.. Report NUREG/CE-4903. June 1st. Earthqu. Vol.W. 1. M. 27–33.Vancouver. Trifunac CSCE Vancouver Section UBC . M. V. CA. Response Spectrum Seminar Lecture # 1 35 . V. Cao. 1. “Evolution of Accelerographs. of Southern California. ” Regulatory Guide No. Response Spectrum Seminar Lecture # 1 36 .S.Response Spectrum M. “Design Response Spectra for Seismic Design of Nuclear Power Plants.C.D..60.Vancouver. 1. 2007 United States Atomic Energy Commission (1973). Washington. June 1st. Atomic Energy Commission. . Trifunac CSCE Vancouver Section UBC . U. D. 1. D.5.” 2 0. 1970). M = 6.seconds 10 Fig. M = 6. 1952. and Tehachapi.2g 5 simple hyperbola A = T Biot (1941.2). 1970). Housner averaged and smoothed the response spectra of three strong-motion records from California (El Centro. he continued: “These standard curves … could be made to depend on the nature and magnitude of the damping and on the location.D.2 1 0 0. 1959. Response Spectrum Seminar Lecture # 1 37 . Housner.4).1 through D. A = g(4T + 0.Vancouver. Biot stated: “If we possessed a great number of seismogram spectra we could use their envelope as a standard spectral curve for the evaluations of the probable maximum effect on buildings.005 of gravity (this standard spectrum is plotted in Figures D. M = 7.2 s. Fifteen years later.05 0.2 s. 1940.7.1942) and for T < 0.2 s.02 investigations. D.” In Biot (1941a).1) He advocated the use of this average spectrum shape in earthquake engineering design (Fig. Although the do not lead to final results. DESIGN SPECTRA Fixed-Shape Response Spectra Normalized Spectral Acceleration In his 1934 paper. 1942) “standard spectrum” (heavy line) with average spectrum of Housner (1959.7) and one from Washington (Olympia. where T is the period in 4 seconds and g the acceleration ζ = 0.1 1 Period . could very well be the 0. Whether 3 this function would fit other earthquakes can only be decided by further 0. 1934. El Centro. 2007 NOTE D D.1 0. M = 7.1 Comparison of Biot (1941. Trifunac CSCE Vancouver Section UBC .Response Spectrum M. A standard curve 6 for earthquakes of the Helena and Ferndale … for values T > 0. June 1st. we … conclude that the spectrum will generally be a function decreasing with the period for values of the latter greater than about 0. 1949. The joint 0.Response Spectrum M. and peak displacement of strong 5 Biot (1941.02 using 14 strong-motion records and by Blume et al..Vancouver.05 3 recommendations of the Newmark and Blume studies of the shape of 0. Nuclear Regulatory Commission. who analyzed 33 records. Normalized Spectral Acceleration 0. Atomic Energy Commission (now the 1 U.005 velocity. Veletsos et al. 1965) noted that the shape of response spectra can be 6 determined by specifying peak acceleration.S. 1 Period . Trifunac CSCE Vancouver Section UBC .seconds 10 Response Spectrum Seminar Lecture # 1 38 . 1964.2). 1942) “standard spectra. normalized to spectrum” (heavy line) with regulatory guide 1. spectrum (USAEC. USAEC. were scaled by selecting the “design” peak acceleration. peak ζ= 0..1942) ground motion. 2007 Newmark and co-workers (Newmark and Veletsos. emerged as the “standard” scaling procedure for determination of design spectra in the late 1960s and early 1970s. D. 1973) for use in the design of nuclear power 0 plants (Fig. (1972) 4 0. (1972).D. June 1st.2 Comparison of Biot (1941.60 unit peak acceleration.1 In engineering design work. 1973). the fixed shapes of Housner and Newmark Fig. D.S. which was first systematically used in the design of nuclear power plants. Spectrum shape was further studied by Mohraz et al. This procedure. 1973) were later adopted by the U.1 the response spectra 2 (Newmark et al. seconds 10 Mohraz et al.3). and showed that the soil site condition has an effect 6 on the shape of ζ = 0. A major and persistent problem in the evaluation of site-dependent spectra of strong earthquake motion is the lack of generally accepted procedures on how to characterize a site. B – soils with intermediate characteristics. 2007 Site-Dependent Spectral Shapes Normalized Spectral Acceleration In one of the first studies to consider the site-dependent shape of earthquake response spectra. Hayashi et al.3 Comparison of Biot (1941. Trifunac CSCE Vancouver Section UBC . Deep cohesionless soil 3 Soft to medium clay 2 1 0 0./s for "alluvium" sites and d = 12 in. June 1st. v. who considered 104 records and four site conditions (rock.D.1942) Seed et al. deep cohesionless Stiff soil "Rock" soil. conclusive recommendations on how to describe the dependence of spectra on site conditions did Fig. D. according to the recording site conditions (A – very dense sands and gravels.Response Spectrum M. (1972) suggested that the peak ground displacement. This result was later confirmed by 5 Biot (1941. and v = 48 in. d. and v = 28 in. both corresponding to a 1g peak ground Because of the small number of recorded accelerograms on rock in 1972. Gutenberg (1957) studied the amplification of weak earthquake motions in the Los Angeles area and published the results on average trends and amplification of peak wave Response Spectrum Seminar Lecture # 1 39 .1 1 Period . not appear possible at that time. Fig.Vancouver. 1942) “standard spectrum” (heavy line) with average (heavy lines) and average plus standard deviation spectra (light lines) of Seed et al. and C – very loose soils).05 average response spectra. and soft-tomedium clay and sand./s for "rock" sites. (1971) averaged spectra from 61 accelerograms in three groups. stiff 4 soil. were d = 36 in. and peak ground velocity. D. (1976) for four soil site conditions. (1976). which depend on magnitude (M = 4.D. 1976) opened a new chapter in the empirical studies of earthquake response spectra. 5 R=0 While it is clear today p = 0. at zero epicentral distance (R = 0) and for 2 percent of critical damping (ζ = 0.5 s.5 that both geotechnical and geological site characterizations must 4 be considered simultaneously (Trifunac.5). and DistanceDependent Spectra 1 M = 4. s=2 2 M = 7. 1976).02.1 1 Period . D.02 of 186 records Biot (1941.5 Site-. Trifunac. for average spectral amplitudes (p = 0.seconds 10 Fig. 1990. 1942) “standard spectrum” (heavy line) with spectral shapes. 2007 Normalized PSV Spectrum motions in sedimentary basins for periods of motion longer than about 0. Twenty years later. 1995). there is so far no 3 s=0 general consensus on how to do this.” because he considered the “site” on the scale of kilometers and used the term “rock” to represent geological basement rock. recorded strong-motion data. It became possible to show how The occurrence of the San Fernando.Vancouver.1942) (Trifunac.5 and 7.4 Comparison of Biot (1941. Lee and Trifunac. Trifunac CSCE Vancouver Section UBC . His site characterization could be termed “geological. 1978). June 1st. California earthquake in 1971 and the large number of new recordings it contributed to the strong-motion database (Hudson. Gutenberg’s results were shown to be in excellent agreement with the empirical scaling of 6 Fourier amplitude spectra of strongHORIZONTAL motion accelerograms ζ = 0. Magnitude-.Response Spectrum M. it became possible to consider multi-parametric regressions and to search for the trends in spectral amplitudes and Response Spectrum Seminar Lecture # 1 40 .5 0 0. For the first time.5) and geological site conditions (s = 2 for basement rock and s = 0 for sediments). 6. 151–171. but also with earthquake magnitude and source-to-station distance (Fig. direct-scaling equations for spectral amplitudes in terms of almost every practical combination of scaling parameters. References Biot. D. pp. “Analytical and Experimental Methods in Engineering Seismology. Soc. (Fig. During the following 20 years. June 1st. “Effects of Ground on Earthquake Motion. D. and Seed et al.1 through D.2.” Bull. NRC (Fig. Sharpe.2). contributing numerous detailed improvements and producing a family of advanced.4.A. Blume.W. the subsequent regression studies evolved into advanced empirical scaling equations.L. (1957). 2007 spectrum shape change.” S. 1935. Angewandte Matematik und Mechanik. 365-408.0. Biot. Figures D. and 104. It was based on the spectra of two earthquakes only (Helena. Housner (Fig. but it does show clearly how the spectra broaden with increasing magnitude and how larger magnitudes contribute larger long-period spectral amplitudes. (2002).S. No. Gutenberg. 213–223. Blume and Associates. pp.4) spectral shapes. 3. Therefore they have broader spectral shapes.” Indian Society of Earthquake Technology Journal. M = Lee. Montana.. not only with local soil and geologic site conditions. “A Mechanical Analyzer for the Prediction of Earthquake Stresses. pp.” Zeitschrift für. Seism. 108.4.4 shows only the dependence of spectral shape (normalized to 1-g acceleration) on magnitude and geological site conditions. Amer. pp. D.” ASCE Transactions.D.Response Spectrum M. M.3) and variable (Fig.. M = 6. D. V. Biot. R. (1972).4 compare Biot’s “standard” spectrum shape with other examples of fixed (Figs. 1978). B. (1942). 47. Trifunac CSCE Vancouver Section UBC . 14(4).1). 1254. Vol. The variable shape spectrum in Figure D. “Recommendations for Shape of Earthquake Response Spectra.. and Ferndale. (1934). “Empirical Scaling of Strong Earthquake Ground Motion-Part I: Attenuation and Scaling of Response Spectra.Vancouver. 1934. The literature on this subject is voluminous. J. These comparisons are only qualitative. and Dalal. J. Response Spectrum Seminar Lecture # 1 41 . Vol. because the methods used in their development and the intended use of the spectral shapes differ. 33. D. California. Biot’s spectrum was originally thought to correspond to zero damping.3) spectra were based on progressively larger numbers of recorded accelerograms (4.4).” Bull. No. Soc.A. respectively) and on recordings during large earthquakes. M. 31. Seism.: J. and its review is beyond the scope of this paper. Amer.39. 221–250. probably less than 3 percent of critical. D. and D. M. The readers can find many examples and a review of this subject in Lee (2002). AEC Report No. D.1. "Theory of Vibration of Buildings During Earthquakes. Trifunac. (1941).A. but it was later discovered that it has small variable damping.A. Francisco.. (1959).” Index Volume.K.M. and Veletsos.” N. CE 95–04. (1978).A. Vol.” Bull Seismol. Response Spectra of SingleDegree-of-Freedom Elastic and Inelastic Systems.. UJNR. Mohraz. No. (1990). Newmark.” Earthquake Eng. Housner.M. Hansen and Associates. 1081–1097.M.W. pp. pp. B. “Seismic Design Criteria for Nuclear Power Plants.. Am. Newmark Consulting Engineering Services. and Lysmer.Vancouver. and Kapur. EMS. Vol. Response Spectrum Seminar Lecture # 1 42 . 66.D. “Strong-Motion Earthquake Accelerograms. and Newmark. Inst.. M. WASH–1255. of the Power Division. “A Study of Vertical and Horizontal Earthquake Spectra. Newmark. Trifunac. 221–243. W. pp. and Trifunac.. of Civil Eng. No.. H.D. June 1st. V. New Jersey: R..S. Los Angeles.S. Eng. 287–303. Amer. Source to Station Distance and Recording Site Conditions. and Structural Dynamics. “Pseudo Relative velocity spectra of Strong Earthquake Ground Motion in California. Trifunac. Trifunac.” Third Joint Meeting of U. Housner. (1973). 104.J. Earthquake. (1964).” J. S.” J. Urbana. Hall. of Tech.” Vol. Research Lab. Japan Panel on Wind and Seismic Effects. M. III. (1970). “Site-Dependent Spectra for EarthquakeResistant Design. RTD TDR 63–3096. No. 6. N. (1976).W.” Dept. California. EERL 76–02. 85. Seed.T. Vol. (1976). pp.D. Tokyo.” ASCE.D. (1976). Pasadena. G.B. C. (1972). of Eng. Mechanics Division. California. M.Response Spectrum M.K. N. 4. Illinois: AEC Report No. and Kurata. Lee. Calif. Vol. 833–846. 19.. of Southern California. G.. “Design Procedures for Shock Isolation Systems of Underground Protective Structures. “Behavior of Structures During Earthquakes. by Newmark. Report No. N. 109–129.L Wiegel. (1971). pp. E. J. 1343–1373. Hudson. Prentice-Hall.D.. ASCE. Blume. Vol. J. D. pp. “Design Spectrum. Vol. Univ. H. 1. “Average Response Spectra for Various Subsoil Conditions. No. A. EM 4. “Preliminary Empirical Model for Scaling Fourier Amplitude Spectra of Strong Ground Acceleration in Terms of Earthquake Magnitude.. M. Soc. K. “How to Model Amplification of Strong Earthquake Motions by Local Soil and Geologic Site Conditions. 2007 Hayashi. Ugas.” Report for Air Force Weapons Laboratory. (1995)... 99.. “Response Spectra of Earthquake Ground Motion..” Bull Seism Soc. ASCE. Tsuchida. Trifunac CSCE Vancouver Section UBC .H. 66.W.” Chapter 5 in Earthquake Engineering.. 663–680. New Zealand. . (1965). U. Trifunac CSCE Vancouver Section UBC . on Earthquake Eng. Newmark.. of Third World Conf.. 1.S. pp. Response Spectrum Seminar Lecture # 1 43 . Vol. Veletsos.S.” Regulatory Guide No.D. Atomic Energy Commission.V.Vancouver.Response Spectrum M.60.” Proc. II. and Chelapati. June 1st.M. Washington. “Deformation Spectra for Elastic and Elasto-Plastic Systems Subjected to Ground Shock and Earthquake Motions. C. D. N. “Design Response Spectra for Seismic Design of Nuclear Power Plants. A. 2007 United States Atomic Energy Commission (1973)..C. This approach works reasonably well for intermediate and distant sites (say for distances greater than 50 km from an earthquake). which depend on the size and distance to the event. E. but on the judgment of the committees.Vancouver. Trifunac CSCE Vancouver Section UBC . surrounding Los Angeles area.1. spectrum is then chosen to envelope the spectra from all those contributing events. In one of the oldest methods for selection of design criteria for important structures. based on the scenario approach. on the propagation path. but can lead to unbalanced representation of strong shaking close to earthquake sources and for large Response Spectrum Seminar Lecture # 1 44 . a scenario approach. Major Quaternary Faults. are also. the shape and the amplitudes of spectral shapes used in design codes. The same concept of enveloping the spectra of the contributing events has been employed in the formulation of the classical design spectral shapes (see Note D). albeit indirectly. The design Fig. UNIFORM HAZARD SPECTRA For a given earthquake direct empirical scaling produces a response spectrum with amplitude and shape. June 1st. in Southern California. 2007 NOTE E E. and on the local geologic and soil conditions. After further amplitude and shape modifications. considers the possible contributing events (typically not more than 5 or 6). which are frequently not based on the physical properties of strong motion.D.Response Spectrum M. for many different periods. From a catalogue of figures on the right. The concept of Uniform Hazard Spectrum (UHS) was introduced in late 1970s (Anderson and Trifunac 1977). for probability of being exceeded equal to 0.Vancouver. due to the geological site conditions and proximity of the active faults. read from this map. are shown by open circles in the two figures on the left. 2007 strong motion amplitudes. E. These difficulties can all be eliminated by use of Uniform Hazard Spectrum methodology. spectral amplitudes. and by considering all known active earthquake sources within 350 to 450 km radius Response Spectrum Seminar Lecture # 1 45 . Spatial variations of Pseudo Relative Velocity Spectrum amplitudes.Response Spectrum M. June 1st. E. UHS is calculated from the distribution functions of all possible Fig. Trifunac CSCE Vancouver Section UBC . at T=0. and probabilities of being exceeded (Lee and Trifunac 1987) one can construct the two figures on the left. Furthermore the basic scenario approach typically does not consider the likelihood of occurrence of a given earthquake event.1).D. to Southern California seismicity (see Fig. and this creates a situation where the spectra which serve as a basis for development of the spectral envelopes do not have proper weighting factors. It is seen that the amplitudes and the shape of the Uniform Hazard Spectrum can vary appreciably over short distances. (right). during specified exposure time (typically 50 years).90. Spectral amplitudes.2. during exposure of 50 years.90 s. Response Spectrum M. Univ. and Trifunac. 15(5). The methodology was originally developed for use in the design of nuclear power plants. V. Todorovska. 44(1). Univ. J. (1995). Liquefaction opportunity mapping via seismic wave energy. of civil Eng. and M. of Civil Eng. but then gradually became the principal tool in seismic hazard mapping and for advanced site-specific estimates of the consequences of strong shaking (Lee and Trifunac 1987. I. Trifunac. 321-329.D. V. It is seen that the UHS of PSV can change amplitudes and shape over relatively small distances. Todorovska. (2007).D.. Los Angeles. M. Shaking Hazard Compatible Methodology for Probabilistic Assesment of Permanent Ground Displacement Across Earthquake Faults. M. Hazard mapping of normalized peak strain in soil during earthquakes: microzonation of a metropolitan area.. and Earthq. Indian Society of Earthquake Technology Journal. of Civil Eng. Trifunac (1999). Microzonation of A Metropolitan Area. Fig. and M.. of Southern California. ASCE.I. Dept. M. Gupta. Selected Topics in Probabilistic Seismic Hazard Analysis. Response Spectrum Seminar Lecture # 1 46 . USA. Lee.. Soil Dynamics and Earthquake Engineering. By calculating UHS. Trifunac (1987). V. On Uniform Risk Functionals which Describe Strong Ground Motion: Definition. I. Eng. Jour.1 shows the faults in Southern California. and for a given exposure period (50 years in this example). (in press). of Southern California. CE77-02..G. USA Todorovska.D. 87-02. M. Todorovska. M. and an Application to the Fourier Amplitude Spectrum of Acceleration. 2007).W. in this case less than 25 km (Lee and Trifunac 1987). and Lee. Gupta. Todorovska and Trifunac 1996.I. Todorovska et al. (2007). Probabilistic Seismic Hazard Analysis Method for Mapping of various Parameters to Estimate the Earthquake Effects on Manmade Structures. 586-597 . M. USA. 1032-1042. Report No. Trifunac CSCE Vancouver Section UBC . at a dense matrix of points. of Geotech and Geoenvironmental Eng. Univ.. 1999.D. and URS spectra can be plotted as illustrated in the left two examples in Fig. Soil Dyn. Los Angeles.I. 27(6).W.I.2. Trifunac (1996). 125(12). Todorovska et al (1995) and Gupta (2007) present comprehensive reviews of the method and many examples of how UHS method can be used. UHS amplitudes can be read from a series of maps. in this example for PSV amplitudes.Vancouver. Los Angeles. E. Gupta. June 1st.D. V. surrounding the Los Angeles metropolitan area..W. Dept.D. of Southern California.D.D. 2007 surrounding the cite.E. and M. References Anderson. Report CE 95-08.K..D. for different probabilities of being exceeded. Numerical Estimation.. Dept. Lee. and M. Report No. Trifunac (1977). Response Spectrum M.D. Trifunac CSCE Vancouver Section UBC - Vancouver, June 1st, 2007 NOTE F F. RESPONSE SPECTRUM AND DESIGN CODES (modified from Freeman 2007) Work on developing building codes began in Italy in 1908, following the Messina disaster in which more than 100,000 persons were killed; in Japan following the 1923 Tokyo disaster, in which more than 150,000 perished; and in California after the Santa Barbara earthquake of 1925 (Freeman, 1932, Suyehiro, 1932). In 1927, the “Palo Alto Code,” developed with the advice of Professors Willis and Marx of Stanford University, was adopted in Palo Alto, San Bernardino, Sacramento, Santa Barbara, Klamath, and Alhambra, all in California. It specified the use of a horizontal force equivalent to 0.1 g, 0.15 g and 0.2 g acceleration on hard, intermediate, and soft ground, respectively. “Provisions Against Earthquake Stresses,” contained in the Proposed U.S. Pacific Coast Uniform Building Code was prepared by the Pacific Coast Building Officials Conference and adopted at its 6th Annual Meeting, in October, 1927, but these provisions were not generally incorporated into municipal building laws (Freeman, 1932). The code recommended the use of horizontal force equivalent to 0.075, 0.075, and 0.10 g acceleration on hard, intermediate, and soft ground, respectively. Following the 1933 Long Beach earthquake, the Field Act was implemented. Los Angeles and many other cites adopted an 8 percent g base shear coefficient for buildings and a 10 percent g for school buildings. In 1943 the Los Angeles Code was changed to indirectly take into account the natural period of vibration. San Francisco’s first seismic code (“Henry Vensano” code) was adopted in 1948, with lateral force values in the range from 3.7 to 8.0 percent of g, depending upon the building height (EERI Oral History Series: Blume, 1994a; Degenkolb, 1994b). Vensano code called for higher earthquake coefficients than were then common in Northern California, and higher than those prescribed by the Los Angeles 1943 code. Continued opposition by San Francisco area engineers led to a general consensus-building effort, which resulted in the “Separate 66” report in 1951. “Separate 66” was based on Maurice Biot’s response spectrum calculated for the 1935 Helena, Montana earthquake (The EERI Oral History Series: Housner, 1997; Proc. ASCE, vol. 77, Separate No. 66, April 1951). In Los Angeles, until 1957 (for reasons associated with urban planning, rather than earthquake safety, and to prevent development of downtown “canyons”), no buildings higher than 150 feet (13-story height limit) could be built. In 1957, the fixed height limit was replaced by the limit on the amount of floor area that could be built on a lot. After the San Fernando, California earthquake of 1971, Los Angeles modified the city code in 1973 by requiring dynamic analysis for buildings over 16 stories high (160 feet). In 1978, the Applied Technology Council (ATC) issued its ATC-3 report on the model seismic code for use in all parts of the United States. This report, written by 110 volunteers working in 22 committees, incorporated many new concepts, including “more Response Spectrum Seminar Lecture # 1 47 Response Spectrum M.D. Trifunac CSCE Vancouver Section UBC - Vancouver, June 1st, 2007 realistic ground motion” intensities. Much of the current Uniform Building Code was derived from ATC-3 report. Influence of Response Spectra on Building Code Provisions (from Freeman 2007) The basis for the development of current seismic building code provisions had their beginnings in the 1950s. A Joint Committee of the San Francisco Section of ASCE and the Structural Engineers Association of Northern California prepared a “model lateral force provision” based on a dynamic analysis approach and response spectra (Anderson et al, 1952). The Proposed Design Curve, C = K / T , was based on a compromise between a Standard Acceleration Spectrum by M. A. Biot (Biot 1941, 1942) and the analysis of El Centro accelerogram by E. C. Robison (Figure F.1). It is interesting to note that the Biot curve PGA of 0.2g has a peak spectral acceleration of 1.0g at a period of 0.2 seconds. The curve then descends in proportion to 1/T (i.e., constant velocity). If the peak spectral acceleration is limited to 2.5 times the PGA, the Biot spectrum is very close to the 1997 UBC design spectrum for a PGA of 0.2g (dashed line without symbols in Figure F.1). The proposed design lateral force coefficient was C = 0.015 / T , with a maximum of 0.06 and minimum of 0.02 (line with dots in Figure F.2). These values were considered consistent with the current practice and the weight of the building included a percentage of live load. Building Code Development of Response Spectra (1952) 1.1 Standard Acceleration Spectrum (M. A. Biot) 1.0 0.9 El Centro Analysis (Robison) Proposed Design Curve C=K/T 1997 UBC Z=0.2, Soil Type B Spectral Acceleration, Sa (g) 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 Period, T (sec) Fig. F.1 1952 Joint Committee (Anderson et al. 1952) Response Spectrum Seminar Lecture # 1 48 Response Spectrum M.D. Trifunac CSCE Vancouver Section UBC - Vancouver, June 1st, 2007 In 1959, the Seismology Committee of the Structural Engineers Association of California published “Recommended Lateral Force Requirements” (generally referred to as the SEAOC bluebook) and included “Commentary” in 1960 (SEAOC 1960). Influenced by the Joint Committee (many of the members were on both committees), recommendations were proposed that were adopted for the 1961 Uniform Building Code (UBC) (ICBO 1961). The new recommended design lateral force coefficient was C = 0.05 / T 1/ 3 and the live loads were not included in the weight (except for a percentage in storage facilities). By using T to the one-third power, the equation could account for higher modal participation and give a larger load factor for tall buildings. In addition it avoided the need for a minimum cut-off. The maximum was set at C = 0.10 (Figure F.2). Also shown in Figure F.2 is a comparably adjusted version of the 1997 UBC. Building Code Development of Response Spectra (1959) 0.14 0.13 0.12 0.11 Lateral Force Coefficient, C 0.10 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 0.0 0.4 0.8 1.2 SEAOC 1959, UBC 1961 Proposed Design Coefficient (1952), with LL Lower limit for 1952 1997 UBC zone 4, adjusted for ASD Lower limits for 1997 UBC Lower limits for 1997 UBC 1.6 2.0 2.4 2.8 3.2 3.6 4.0 Period, T (sec) Fig. F.2 1959 code development (SEAOC 1960) Over the years, the SEAOC bluebook and the UBC went through many revisions, generally influenced by some event such as the 1971 San Fernando, 1989 Loma Prieta, and 1994 Northridge earthquakes and by data relating to soil effects. The comparable curves shown in Figure F.3 have been adjusted to represent strength design response spectra and include factors representing soil classification type D. At this level of design, the structures would be expected to remain linear-elastic with some reserve capacity before reaching yield. In order to survive major earthquake ground motion (e.g., PGA = 0.4 g ) the structure is expected to experience nonlinear post yielding response. Response Spectrum Seminar Lecture # 1 49 Response Spectrum M.20 0.M. Sedgwick. G. Blume. M.5 2.15 1988-94 UBC 1976-85 UBC 0. M. June 1st. Trifunac CSCE Vancouver Section UBC . F.A.. Stanley Scott Interviewer. Powers.5 1. CA 94612.11*Ca*I UBC: 0.W.25 0.0 0.O.B.A. 499 14th Street.” ASCE Transactions. J.00 0. (1942).5 4. Marchand. 108. Hammill. Oakland.D. (1952).1961 to 1997 References Anderson.” EERI.30 1997 UBC 1976-1985 UBC 1988-94 UBC 1997 UBC UBC: Cs=0. Amer. “A Mechanical Analyzer for the Prediction of Earthquake Stresses.A.A. Suite 320.J. 716-780. Volume 117. Biot. and H. Sjoberg.10 Pre-1976 C 0. The EERI Oral History Series: J. Soc. “Lateral Forces of Earthquake and Wind”.E. Soil D Equivalents ( design strength at φ =1) Pre-1976 UBC 0. Degenkolb.0 2. E. H. Knapik.8*Z*Nv*I/R Spectral Acceleration.Vancouver. H. H. A. Sa (g) 0.5 3.C.0 1. Response Spectrum Seminar Lecture # 1 50 . H. Seism. (1941). Earthquake Engineering Research Institute (1994a).3 UBC response spectra .0 Period T (sec) Fig.A. J.L.05 0. Blume. “Connections. “Analytical and Experimental Methods in Engineering Seismology.0 3. 2007 0.” Bull. 151–171. Transactions of ASCE. Rinne. 31.35 UBC Zone 4. 365–408. Biot.. Housner. Response Spectra as a useful Design and Analysis Tool for Practicing Structural Engineers. 4. J. (1932). Whittier. “Recommended Lateral Force Requirements and Commentary” (Blue Book). 1-110. (2007).D. Earthquake Engineering Research Institute (1997).R. CA 94612. S. 44(1).” by International Conference of Building Officials (ICBO). ISET Journal of Earthquake Technology. ICBO (1961 et al). SEAOC (1960).Vancouver. . (1932). 2007 Earthquake Engineering Research Institute (1994b). Stanley Scott Interviewer. Vol. Oakland.” Proc. 58. CA 94612. Degenkolb.A. Seismology Committee. California. Freeman.” EERI. The EERI Oral History Series: H.Response Spectrum M.W. K. Stanley Scott Interviewer.” EERI. Suite 320. Trifunac CSCE Vancouver Section UBC . (in press). Freeman. Suite 320. 1961-1997. 499 14th Street. Oakland. Structural Engineers Association of California. ASCE. Suyehiro. “Connections. “Uniform Building Code (UBC). “Earthquake Damage and Earthquake Insurance. June 1st. 499 14th Street.J. California. No. “Connections. “Engineering Seismology Notes on American Lectures. San Francisco. Response Spectrum Seminar Lecture # 1 51 . The EERI Oral History Series: G.” New York: McGraw-Hill. spatial and temporal variations of excitation. June 1st. and the system becomes metastable for ψ r smaller than its critical value.1) For any initial conditions. Rotation ψ r is restrained by a spring with stiffness K r . An example of a simple model. the overturning moment of gravity force is just balanced by the elastic moment in the restraining spring. dashpots. the ωr in Eq. G. ∆ x is the single-degree-of-freedom system (SDOF) experiencing rocking ψ r .Response Spectrum M.Vancouver. and assuming that the ground does not deform in the vicinity of the foundation. 2007 NOTE G G.1) was used originally to develop the System concept of the relative response spectrum.1). Eqn. ∆x B x In more advanced vibrational representations of the response.2. soil-structure interaction. At the critical value of ψ r . and continues to this day as the main vehicle in formulation of most earthquake engineering analyses of response (Trifunac 2003). neglecting the soil structure interaction (Fig. and by a dashpot with the rocking damping constant Cr . G. additional components of earthquake excitation. It has horizontal vertical and rocking excitations. differential motions at different support points.1 Single-Degree-of-Freedom (G. providing the fraction of critical mb ψr damping ς r . dynamic instability. The natural frequency of this h system is ωr = ( K r / h 2 mb ) . springs. and nonlinear behavior of the stiffness K r can be considered. relative to the normal to the ground surface. which includes instability is shown in Fig. Dynamic instability. G. Advanced Vibrational representations of Response The basic model employed to describe the response of a simple structure to only && horizontal earthquake ground motion.D. and with rigid mass mb .1) has to be reduced (Biot 2006). (G. Trifunac CSCE Vancouver Section UBC . which can result from Response Spectrum Seminar Lecture # 1 52 . but the structure usually continues to be modeled by mass-less columns. that is. If the gravity force is considered. Fig. and for small 1/ 2 z y O rocking angles it is governed by the linear ordinary differential equation && && & ψ r + 2ωrς rψ r + ωr2ψ r = − ∆ x / h (G. and for arbitrary excitation this system always leads to deterministic and predictable response. In the following we illustrate some of the above examples. June 1st. Response Spectrum Seminar Lecture # 1 53 . which consider soil-structure interaction. and h B O a φy ψr ∆x ∆ z z y x ∆ z .3). whose solution will require numerical analysis. Taking moments about B results in the equation of motion && && && && & φ y +ψ r + 2ωrς rψ r + ωr2ψ r = − ( ∆ x / a ) cos (φ y + ψ r ) + (ωr2ε g + ∆ z / a ) sin (φ y +ψ r ) / ε (G. with a concentrated mass mb at a height h above the foundation.2) is a differential equation coupling the rocking of the foundation and the structure with the horizontal and vertical motions of the foundation. (G. will be periodic. The degree-of-freedom in the model of a structure is chosen to correspond to the relative rocking ψ r . (G. φ y . It is a nonlinear equation. φ y will be determined directly by the rocking Fig. and therefore the coefficients of (G. and ∆ z will be determined by the recorded components of motion.3) φ y describes rocking of the foundation to which the structure is attached. ζ r is a fraction of the critical damping in 2ωrς r = Cr / ⎡ m ( h 2 + rb2 ) ⎤ . G. for example. Equation (G. φ y . The gravitational force mb g is considered. SV P θ In Eqn.Vancouver.3) is then a special form of Hill's equation.2). with frequency ω . Eqn.Response Spectrum M.2) 2 where ε = h 1 + ( rb / h ) / a . This rotation is restrained by a spring with rocking stiffness K r and by a dashpot with rocking damping Cr (both not shown in Fig. Wong and Trifunac 1979). and in the predictive analyses by simulated ground motions (Lee and Trifunac 1985. For general earthquake excitation ∆ x .D. In the analyses. φ y will be one of the variables to be determined by the analysis (Lee 1979). which do not consider soil-structure interaction. 2007 incident P and SV waves. Jalali and Trifunac 2007). Analysis of the stability of this equation can be found in the work of Lee (1979).3) { } { } mb Ib For steady-state excitation by incident P and SV waves. 1987. ωr2 = K r / ⎡ m ( h 2 + rb2 ) ⎤ is the natural frequency of rocking ⎣ ⎦ { } squared. In this example we will discuss only the case when φ y +ψ r is small. It has a radius of gyration rb and a moment of inertia I b = mb rb2 about O. ∆ x . Then && && && && && & ψ r + 2ωrς rψ r + ωr2 (1 − ε g / ε ) − ∆ z / ε a ψ r = −φ y + −∆ x / a + (ωr2ε g + ∆ z / a ) φ y / ε (G. In the studies. The structure is represented by an equivalent single-degree-of-freedom system. G. ⎣ ⎦ ( ) and ε g = 2 / ωr2 a . Trifunac CSCE Vancouver Section UBC .2 component of strong ground motion (Lee and Trifunac 1987. Fig. which is presumed to be immobile. where T ∆ x and ∆ y are horizontal translations. ∆ x .c). and {U 0 } is the relative displacement corresponding to the displacement of the foundation under the action of the external forces in the absence of incident wave excitation. The linear soil-structure interaction embodies the phenomena.3). φx and φ y are rotations about horizontal axes. G.4) {U } = {U *} + {U 0 } where {U *} is the foundation input motion corresponding to the displacement of the foundation under the action of the incident waves in the absence of external forces. Examples and a ∆y discussion of non-linear aspects of soilM bz φz F by structure interaction can be found in φy M Gicev (2005).Vancouver. It depends upon the material properties of the soil medium. and (2) from the vibration of the structure which is supported by the foundation and which exerts dynamic forces on the ∆ z F bz foundation (Lee 1979). the characteristics and shape of the foundation. and it is related to {U 0 } by where ⎡ K s (ω ) ⎤ is the 6 x 6 complex stiffness matrix of the ⎣ ⎦ embedded foundation. June 1st. ∆ z is vertical translation.D.φx .3) in the half space (Lee and Trifunac 1982). {Fs } = ⎡ K s (ω )⎤ {U 0 } . B F bx The dynamic response of a rigid M bx embedded foundation to seismic waves P θ SV can be separated into two parts. and in a review of by observations of response to earthquake φx ∆x shaking in full-scale structures in m0 a Trifunac et al. It corresponds to the force that the foundation exerts on the soil. (2001a.φ y . The first part corresponds to the SH determination of the restraining forces due to the rigid body motion of the Fig. The steady-state harmonic motion of the foundation having d frequency ω can be described by a vector {∆ x .φz } (Fig. G. The interaction force { Fs } generates the relative displacement {U 0 } . It describes the force-displacement relationship between the rigid foundation and the soil medium. displacement of the foundation is the sum of two displacements (G. ⎣ ⎦ Response Spectrum Seminar Lecture # 1 54 . ∆ y . 2007 Soil-Structure Interaction. G. Using superposition. This can be illustrates by considering a foundation embedded in an elastic medium and supporting an elastic superstructure.3 inclusion.Response Spectrum M. The second part deals with the evaluation of the driving forces due to the scattering of incident waves by the inclusion. and the frequency of the harmonic motion. which result from (1) the presence of an inclusion (foundation. Trifunac CSCE Vancouver Section UBC . and φz is torsion about the vertical axis.b. for example. excited by P and SV waves.D. where the input motion {U *} is measured relative to an inertial frame. if the effects of soil-foundation interaction in the presence of differential ground motions can be neglected (Bycroft. This. Common use of the response spectrum method and many dynamic analyses in earthquake engineering implicitly assume that all points of building foundations move synchronously and with the same amplitudes.Vancouver. G. those can be used to determine the foundation displacement {U } . and rocking Response Spectrum Seminar Lecture # 1 55 . Unless the structure is long (e. the stiffness matrix [ K s ] . The "driving force" is the force that the ground exerts on the foundation when the rigid foundation is kept fixed under the action of incident waves. M by . After the mass matrix [ M 0 ] . G. (5) becomes && [ M 0 ]{U } + [ K s ]{U } = F * + {Fext } { } s (G. the dynamic equilibrium equation is && [ M ]{U } = − {F } + {F } . and the force { Fs*} have all been evaluated. 2007 The driving force of the incident waves is equal to { Fs*} = [ K s ]{U *} .3). Fbz . For simplicity. (G. 1990). where a is wave amplitude and λ is the corresponding wavelength. consisting of SH and Love waves. in effect. the wave propagation effects on the response of simple structures can be neglected (Todorovska and Trifunac. the impedance matrix.5) {Fext } = {F bx . M bz } is the force the structure exerts on the foundation 0 s ext (Fig.3) Differential motions.g. For in-plane response.. June 1st. and Rayleigh waves. Fby . G.4-bottom) consisting of P.4-top). { Fext } .6) The solution of {U } requires the determination of the mass matrix. It depends upon the properties of the foundation and the soil and on the nature of excitation. {U } is related to the interaction and driving forces via For a rigid foundation having a mass matrix [ M 0 ] and subjected to a periodic external force. The inplane motion can further be separated into horizontal (longitudinal). Then Eqn. Figures G. The displacement [ K s ]{U } = { Fs } + { Fs*} . Simple analyses of two-dimensional models of long buildings suggest that when a/λ < 10-4. these simplifications are justified and can lead to a selection of approximate design forces. implies that the wave propagation in the soil is neglected. the driving forces and the external forces (Lee 1979). the relative response ψ r is then given by the Eqn. the incident wave motion has been separated into out-of-plane motion (Fig. (G. vertical.4a and b illustrate the “short” waves propagating along the longitudinal axis of a long building or a multiple-span bridge. 1980). a tunnel) or “stiff” relative to the underlying soil. M bx . a bridge with long spans.Response Spectrum M. and in-plane motion (Fig. a dam. SV. Trifunac CSCE Vancouver Section UBC . it is necessary to specify a relation between um and u0 (Fig. June 1st. in terms of the maximum linear response.D. Trifunac and Gicev (2006) showed how to modify the spectra of translational motions. um . for the same ground motion. into a spectrum which approximates the total H (translational and torsional) λ responses.5). SH or Love Wave x L H a λ Rayleigh Wave Fig.Vancouver. increase the dimensions of the governing differential equations. By defining the yield-strength reduction factor as Ry = u0 / u y . where u y is the yielding displacement of the SDOF system equivalent spring. G. These are all conditions that create environment in which even with the most detailed numerical simulations it is difficult to predict all complexities of the possible responses. As can be seen from the above examples the differential motions lead to complex excitation and deformation of the structural members (columns. and how this approximation is valid for strong c motion waves an order of magnitude longer than the a structure ( λ >> L ). shear walls. u0 .4(bottom).4 Differential out-of-plane response of a long structure excited by SH or Love waves (top). the ratio um / u0 is then equal to µ / Ry . and in-plane response during the passage of Rayleigh waves (bottom). beams. Trifunac and Todorovska (1997) analyzed the effects of the horizontal in-plane components of differential motion for buildings with models that are analogous to the sketch in Fig. lead to thee-dimensional dynamic instability problems. 2007 components. while out-of-plane motion consists of horizontal motion in the transverse direction and torsion along the vertical axis.Response Spectrum M. and can lead to non-linear boundary conditions. Veletsos and Response Spectrum Seminar Lecture # 1 56 . Trifunac CSCE Vancouver Section UBC . G. and ductility as µ = um / u y . braces). Nonlinear Vibrational Analyses of Response For estimation of the maximum nonlinear response of a SDOF system. and they showed how the response spectrum method can be modified to include the first-order L effects of differential motions. G. ψ The model we illustrate next is a relationship for bi-linear spring. we hope that the present examples will help to further understand the complexities of response in more realistic models of structures. and only horizontal (one component) representation of ground motion. For such a sytem the classical equal energy and equal displacement rules for SDOF system will not apply. when it is excited by synchronous horizontal ground motion at its two supports (1 and 2). 1995).example. um / u0 tends to 1. will further contribute to the complexities of the response. Complexities of simultaneous action of dynamic instability. vertical. Analyses of the consequences of the differences in ground motion at structural supports. excited by synchronous. 2007 Newmark (1960) showed that (1) for a long-period SDOF system when its natural period Tn = 2π / ωn becomes very long. for example.Vancouver.6. caused by non-uniform soil properties. Fig. even though it has been 75 years since the response spectrum method was formulated and about 40 years since it became the principal tool in engineering design (Trifunac 2003). The role of simultaneous action of all six components of ground motion (three translations and three rotations) is still rarely considered in modern engineering design (Trifunac 2006). Trifunac CSCE Vancouver Section UBC . June 1st. Response Spectrum Seminar Lecture # 1 57 .5 Force-displacement (moment-rotation) r u. (2) for the response amplitudes governed mainly by the peak excitation velocities. and lateral spreading. um / u0 can F(u). Because the response spectrum method has become an essential part of engineering design and of the description process of how future strong motion should be specified for a broad range of design applications (Todorovska et al. and kinematic boundary conditions. The purpose here is to describe the effects of differential motion on strength-reduction factors of the simple structure shown in Figure G. when it is subjected to all components of near-source ground motions and to illustrate the resulting complexities of nonlinear response. G.D. uy uo um SDOF. but will not be discussed here. and (3) for a highfrequency (stiff) system when Tn ~ 0 . soil-structure interaction. Φ(ψ r) Corresponding linear system Metastable region be approximated by µ / 2 µ − 1 and Ry by 2µ − 1 (equal strain fo fm fy Elastoplastic system α m bghsin ψ energy rule). but behaves like a three-degree-of-freedom (3DOF) system. The original response spectrum method has been formulated using a vibrational solution of the differential equation of a SDOF system. ψ r.cr nonlinearity. when excited by propagating horizontal.Response Spectrum M. Ry ~ 1 . and rocking ground motions. and Ry approaches µ (equal deformation rule). Response Spectrum M. the characteristics of the soil surrounding the foundation.Vancouver. kφ . Trifunac CSCE Vancouver Section UBC . and rocking components of motion of column foundations. is assumed Response Spectrum Seminar Lecture # 1 58 . with phase velocity Cx . It represents a one-story structure consisting of a rigid mass. and show selected results of the analysis for structures on isolated foundations only. which are connected at the top to the mass and at the bottom to the ground by rotational springs (not shown in Fig. G.6). The stiffness of the springs. such that at the base of each column the motion has six degrees of freedom.6 The system deformed by the wave. vertical.D. June 1st. 2007 The nature of relative motion of individual column foundations or of the entire foundation system will depend upon the type of foundation. G. and the direction of wave arrival.G. In this example we assume that the effects of soil-sructure interaction are negligible. the type of incident waves. supported by two rigid mass-less columns with height h. We suppose that the excitations at piers have the same amplitude with different phases. axis of the structure (X axis) coincides with the radial direction (r axis) of the propagation of waves from the earthquake source so that the displacements at the base of columns are different as a result of the wave passage only. m. The simple model we consider is described in Fig.6. for the case of + vgi (“up” motion). We assume that the structure is near the fault and that the longitudinal uT1 ν Τ1 T1 T1 m ν Τ2 T2 M1 uT2 T2 θ g1 ψ1 θ g2 ψ2 M2 h M′ 1 h θ g1 1 ν g1 Deformed Ground Surface 1 Cx M′ 2 ug1 L 2 ug2 2 ν g2 θ g2 Fig. propagating from left to right. with length L. consider only the in-plane horizontal. The phase difference (or time delay) will depend upon the distance between piers and the horizontal phase velocity of the incident waves. Fig. τ = L Cx . the effects of gravity force. Iterations are required to compute the inelastic deformation ratio for a specified ductility factor because different values of moments may lead to the same ductility. The convention is to choose the largest moment (Veletsos and Newmark 1964). For the structure in Fig. radial. and for a time delays. leads us to consider the geometric nonlinearity. 2007 to be elastic-plastic. G. G. G. the relative rotation for two columns at their top and bottom will be different. Then we plot Ry versus Tn for four corners of the system.6).6. with downward vertical ground displacement. and upward (+vgi ) . g. in a more accurate modeling. instead of one factor for the total system. τ . without hardening ( α = 0 ).5. Trifunac CSCE Vancouver Section UBC .05 s. In all calculations here we consider the actions of the horizontal. G. for the assumed model and because of differential ground motions and rotation of the beam. The yield-strength reduction factor for the system subjected to synchronous ground motion is Ry = f 0 / f y = u0 / u y .D. and rocking.7 illustrates typical results for Ry versus the oscillator period for near-source. magnitude M = 8. The functional forms of u gi . It shows the results for the top-left. G. Rotation of the columns. In this work. and is excited by differential horizontal. June 1st. vgi . assuming wave propagation from left to right (see Fig. θ g2 (t ) = θ g1 (t − τ ) . Therefore. G. for ductility ratio of eight. The mass-less columns are connected to the ground and to the rigid mass by rotational dashpots. and rocking components of the ground motion.Vancouver. providing a fraction of critical damping equal to 5 percent. For reference and easier comparison with the previously published results. and θ gi are defined by near-source ground motions (Jalali and Trifunac 2007). which is assumed not to be small.6) at two bases so that u g2 (t ) = u g1 (t − τ ) . while the associated rocking θ gi will be described by a superposition of rocking angles associated with incident body and dispersed surface waves (Lee and Trifunac 1987). with τ being the time delay between motions at the two piers and C x the horizontal phase velocity of incident waves. The rocking component of the & ground motion will be approximated by (Lee and Trifunac 1987) θ gi (t ) = −vgi (t ) / Cx . dynamic instability. ductility µ . d N (t ) = AN (1 − e−t /τ N ) / 2 (Jalali and Trifunac 2007). vertical.Response Spectrum M. cφ . The mass is acted upon by the acceleration of gravity. where all of the quantities are defined in Fig. vgi . φi = θ gi +ψ i for i=1. Of course. top-right. 2 (Fig. as in Fig. radial. vertical. fault-parallel displacement. and time delay τ = 0. we Response Spectrum Seminar Lecture # 1 59 . vg2 (t ) = vg1 (t − τ ) . it is necessary to define the R-factor and ductility for each corner of the system. and bottom-right corners of the system. & where vgi (t ) is the vertical velocity of the ground motion at the i-th column. and rocking ground motions. u gi . and θ gi . and geometric non-linearity. and rocking near-source differential ground motions corresponding to given earthquake magnitude.5. we calculate maximum linear and nonlinear relative rotations at four corners of the system under downward (−vgi ) . bottom-left. i = 1.2. the ratio of vgi to u gi amplitudes will depend on the incident angle and the character of incident waves. Nevertheless the above 60 Bottom-left corner 5 Top-left corner (R y)min 0 M=8 µ=8 τ = 0. design criteria can be formulated for design of simple structures to withstand near-fault differential ground motions (Jalali and Trifunac 2007). At or beyond these boundaries. Ry curves d N(t) Ry approach “collapse boundaries” (Jalali and Trifunac 2007).1 period for near-source. for different earthquake magnitudes. G.05 s. the nonlinear system collapses due to action of gravity loads and dynamic instability.7 can be simplified by keeping only ( Ry ) min .Vancouver. using piecewise straight lines (Jalali and Trifunac 2007). 2007). downward vertical ground Period . G. for ductility ratio of eight.s displacement. µ . magnitude M = 8. since it is only the minimum value of Ry that is needed for engineering design. This is implied in Fig.7 by rapid decrease of Ry versus period. 2007 also plot one of the oldest estimates of Ry versus period. for periods longer than about seven seconds. June 1st. Trifunac CSCE Vancouver Section UBC .7 R0. By mapping ( Ry ) min versus period of the oscillator. µ = 8. The complex results illustrated in Fig. For this set of parameters the curve ( Ry ) min shows the minimum values of Ry versus period. different ductilities. 10 y versus oscillator 1 fault-parallel displacement.D.05 s.Response Spectrum M.05 Fig. and time delay τ = 0. M. and for M = 8. The oldest estimates of Ry versus period are shown using piecewise straight lines (Jalali et al. and τ = 0. and different delay times τ . G. The curve ( Ry ) min shows the minimum values of Ry for d N ( t ) motion with −vgi . Response Spectrum Seminar Lecture # 1 . 10 Top-right corner Bottom-right corner For periods longer than 5 to 10 s. 1671. Soil Dynamics and Earthquake Eng. V. Rocking Strong Earthquake Accelerations. M. 952-964. Dept. 19-50. M. Soil Dynamics & Earthquake Engrg. Indian Society of Earthquake Technology. when differential ground motion with all components of motion is considered. N. Southern California. and Trifunac. (1985). Southern California. M. 8(5). V. Lee. 546–563.. G. 26(6–7). June 1st. D. Soil-Foundation Interaction and Differential Ground Motions.A. 6(2).W. 2007 shows how complicated the response becomes even for as simple structure as the one shown by the model in Fig. Los Angeles. 108(3). Jalali. Dept. M. Torsional Accelerograms. Los Angeles. (2006a). 132-139. and Tifunac. California. I. M. EMD.. J.W. Los Angeles. (2007). Investigation of Three-Dimensional Soil-Structure Interaction. References Biot. Earthquake Engineering and Structural Dynamics.Vancouver. V. Investigation of soil-flexible foundation-structure interaction for incident plane SH waves.W. Trifunac. and Trifunac. Note on excitation of long structures by ground waves.. Body Wave Excitation of Embedded Hemisphere. Dissertation.D. and Trifunac.D. M. and Errata in 116. Rotations in the Transient Response of Nonlinear Shear Beam.. Strength-Reduction Factors for Structures Subjected to Differential Near-Source Ground Motion. (2005). 70-th Anniversary of Biot Spectrum. (2006). 75-89. ASCE. (1979).W. Univ. Trifunac CSCE Vancouver Section UBC .D. J. Gicev. California.Response Spectrum M. 79-11. Univ. 4(3). & Trifunac. Int. 44(1). G. ASCE. 486–490. and kinematic boundary conditions.D. 40. of Civil Engineering. Soil Dynamics and Earthquake Engineering. Lee. In this example this complexity results from simultaneous consideration of material and geometric nonlinearities.D. California. 116(4). No. Univ. (2003). Lee..6. (1990). V. (in Press). Int. Lee. 397-404. Ph.. 23-rd Annual ISET Lecture. Response Spectrum Seminar Lecture # 1 61 . Department of Civil Engineering. V. (1980). V.D. (1982). Report CE. M.D. (1987). of Civil Engineering Report CE 06-02. M. Todorovska. dynamic instability. Vol. Gicev. Bycroft. of Southern California.D. and Trifunac. Influence of Foundation on Motion of Blocks. 1. 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Apparent Periods of a Building II: Time-Frequency Analysis. 517–526. (1979). 509-527. RTD-TDR-63-3096. Univ. (1964). Engrg. Paper II: Out-of-Plane Response. and Todorovska. Structural Media and Rotation Effects. N. (2001a). (1994). Ivanovic. Trifunac CSCE Vancouver Section UBC .Y. M. Teisseyre. Soil Dynamics and Earthquake Engineering. S. and E. 527–537. S. of Struct. M.I.M. ASCE.V.I. Report CE 01-02. M. Earthquake Eng. D.Response Spectrum M.D. Albuquerque.. Dynamics. Vol.). M. M. Response Spectra and Differential Motion of Columns. State of the Art Review on Strong Motion Duration. Dept. 26(12). Wong. Trifunac. of Civil Engrg. Apparent Periods of a Building I: Fourier Analysis.D.D. M. 26(2). 1149-1160. Germany. 251-268.D. Engrg. and Novikova.. Effects of Torsional and Rocking Excitations on the Response of Structures.D.. Takeo. Los Angeles. T. and Trifunac. Trifunac. California. R. (2001c). (2006).