Report

Damage Detection in a Simple Shear Beam Model: Modal Analysis and Seismic Interferometry Vanessa Heckman Monica Kohler Tom Heaton May 25, 2012 University of Memphis Overview • Modeling a building as a continuous beam • Seismic Interferometry of UCLA Factor Building • Experimental Analysis Using Uniform Shear Model Flexible Building as a Continuous Cantilevered Beam • Dynamic motions of a continuous beam can provide insight into the deformation of buildings. • Example: MRF • If the building is considered as a continuum, it would often be • • • • anisotropic. Stiffness associated with inter-story drift (ε13 and ε23) much less than the stiffness associated with shearing the actual floor slabs (ε12). Stiffness associated with extension along columns (ε33) is different than for extension along the floor slabs (ε11 and ε22). Assume that only parts of the strain tensor which are important to describe the deformation of the building are inter-story shear strain and extensional strain along the columns. Can thus approximate the building as being isotropic, since the other elastic moduli are not important. Heaton, T. CE 151 Class Material, 2010. Shear Beam • When the ground beneath the building moves horizontally, this is identical to the problem of having an SH wave propagate vertically in a layer of building; the bending is approximately zero in this case. • Total stiffness against shear is equal to the shear modulus times the cross sectional area. • While actual buildings are neither a true bending beam nor a shear beam, we can gain some useful insight by looking at these approximate modes of deformation. • It is identical to the problem of a vertically propagating SH wave in a plate with a rigid boundary at the bottom and a free boundary at the top. Heaton, T. CE 151 Class Material, 2010. Shear Beam • Solution to this problem can be written as a sum of reflecting pulses. The motion in the building is given by u1g(t) c=β= • Drift is given by m/r Horizontal motion of the ground at the base of the building Shear wave velocity in the building • Base of the building: zero displacement, double the drift • Top of the building: zero drift, double the displacement • Sequence repeats with periodicity 4h/c, which is the fundamental period of the building oscillation. Heaton, T. CE 151 Class Material, 2010. Shear Beam • Mode Shapes for the first four modes • Natural Frequencies Background: UCLA Factor Building • UCLA Factor Building • 17-story, moment-resisting steel-frame structure • Embedded 72-channel accelerometer array • N-S modal frequencies: 0.59 Hz, 1.8 Hz, 3.1 Hz (1:3:5 ratio for shear beam) • E-W modal frequencies: 0.55 Hz, 1.6 Hz, 2.8 Hz (1:3:5 ratio for shear beam) • Experimental data and numerical model (ETABS) Kohler, M. Heaton, T., Bradford, C. 2007. BSSA. Background: UCLA Factor Building • Impulse Response Function • • • Experimental Hammer Test Deconvolution is used to extract the transfer function Bandpass filtered between 0.5 Hz and 10 Hz Stack over small EQs to stay in linear response • Shear beam: waves travel nondispersively throughout the lower floors of the building (v = ~160 m/sec) • For bending beams, the waves would disperse with the wave velocity increasing as the square root of the frequency Impulse Response Functions Kohler, M., Heaton, T., Bradford, C. 2007. BSSA. Laboratory Example: Uniform Shear Beam • Experimental Setup • Five-story uniform shear model • Piezoelectric accelerometers, DAQ • Force transducer hammer • Shake table, signal generator • Damaged/undamaged configuration Laboratory Example: Uniform Shear Beam • Consider the structure modeled by a multi-degree-of-freedom system æ ç ç ç x(t) = ç ç ç ç è x1 (t) ö ÷ x2 (t) ÷ ÷ x3 (t) ÷ x4 (t) ÷ ÷ x5 (t) ÷ø Mx(t)+Cx(t)+ Kx(t) = f (t) æ ç ç M = mç ç ç ç è 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 ö ÷ ÷ ÷ ÷ ÷ ÷ ø æ ç ç K = kç ç ç ç è 2 -1 0 0 0 ö ÷ -1 2 -1 0 0 ÷ 0 -1 2 -1 0 ÷ 0 0 -1 2 -1 ÷ ÷ 0 0 0 -1 1 ÷ø • Compute constants m, k from mass, geometry, and material properties • m: Mass of each floor • k: Stiffness due to bending plate (column) • Determine the mode shapes and frequencies of the system from the eigenvalues and eigenvectors of M-1K • Damping is found experimentally • 1st mode: logarithmic decrement method • Higher modes: half-power bandwidth method • Neglected: ζn< 0.005, n = 1,…,5 Kohler, M., CE 181 Class Notes, 2010. MDOF System • Transfer function used to find mode shapes • Relates the response at DOF k to simple harmonic excitation at DOF m frkfrm H km (wr ) = å2 2 M (w + 2i z w + w rr r r r ) r=1 N frk zr wr kth component of the rth mode shape modal damping frequency of the rth mode • Consider the transfer function at the rth modal frequency frkfrm H km (wr ) = i+0 2 M (2z rwr ) H km (w r ) frkfrm frk = m m= m H mm (w r ) fr fr fr Kohler, M., CE 181 Class Notes, 2010. Damping Coefficient Logarithmic Decrement Method • 1st Mode damping ratio • Free vibration Half-Power Bandwidth Method • Higher Mode damping ratio • Based on the response of a SDOF system, but can be used for MDOF system as long as there is no coupling between the modes • Can be problems obtaining low damping ratio due to freq resolution Mode Shapes, Frequencies, and Wave Propagation • Calculated vs. Observed: Good agreement • Damage: Introduced at the 4th floor • Damaged vs. Undamaged Calculate d Observed Damaged ω1 3.2 3.3 2.6 ω2 9.5 9.6 6.2 • Significant decrease in frequencies ω3 14.9 15.2 10.8 • Changes in mode shapes ω4 19.2 20.0 14.6 ω5 21.9 22.4 21.2 • Changes in wave propagation SH Plane Waves • Ray Path Diagram • Continuous displacements transmitted θT β2< β1 β2=sqrt(μ2/ρ2) • Continuous tractions • Reflection/Transmission Coefficients x1 β1=sqrt(μ1/ρ1) incident • Vertical incidence θR θI x3 reflected • Fixed Surface: Rss = -1 • Free Surface: Rss = 1 • Higher to Lower Velocity/Stiffness: Rss > 0; Tss > 1 Heaton, T. CE 151 Class Material, 2010. Experimental Results: 1st Floor Experimental Results: 2nd Floor Experimental Results: 3rd Floor Experimental Results: 4th Floor Experimental Results: 5th Floor Seismic Interferometry • Seismic interferometry may aid in damage detection by comparing post-event waveforms with pre-event waveforms • Changes in wave speed • Floor-to-floor reflections • Forward modeling using a numerical model (ETABS) will be used to test the application to civil structures Impulse Response Function Kohler, M., Heaton, T., Bradford, C. 2007. BSSA. Hayashi, Y., Sugino, M., Yamada, M., Takiyama, N., Onisha, Y. 2012. STESSA Acknowledgements • Thomas Heaton, Professor of Seismology, Caltech • Dr. Monica Kohler, Researcher, Caltech • Ming Hei Cheng, PhD Candidate, Caltech • Rob Clayton, Professor of Geophysics, Caltech • Hartley Fellowship • Housner Fund THANK YOU! Questions? Seismic Interferometry • Cross-Correlation of Ambient Noise vs Deconvolution • Synthetic Example: Simple Shear Beam • D’Alembert Solution • Cross-Correlation CrossCorrelation IRF Floor # • Deconvolution Deconvolution 12 12 12 10 10 10 8 8 8 6 6 6 4 4 4 2 2 2 0 0 0 0 0.5 1 0 0.5 Time (s) 1 0 0.5 1 Seismic Interferometry • IRF from Cross-correlation of Ambient Noise Prieto, G., Lawrence, J..Chung, A., Kohler, M. 2010