4 CE ELEC2 Earthquake Dynamic Lateral Force Procedure

CE ELEC2 EARTHQUAKE ENGINEERING

MARK ELSON C. LUCIO, LUCIO, MSCE (Structures) Association of Structural Engineers of the Philippines (ASEP) Philippine Institute of Civil Engineers (PICE) American Society of Civil Engineers (ASCE) American Concrete Institute (ACI)

EARTHQUAKE Dynamic Lateral Force Procedure

Structural damage during an earthquake is caused by the response of the structure to the ground motion input at its base. The dynamic forces produced in the structure are due to the inertia of its vibrating elements. The magnitude of the eff effective ective peak acceleration reached by the ground vibration directly affects affects the magnitude of the dynamic forces observed in the structure. •

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EARTHQUAKE Dynamic Lateral Force Procedure Accelerograph -An instrument that records the acceleration acceleration of the ground during an earthquake, also commonly called an accelerometer

EARTHQUAKE Dynamic Lateral Force Procedure Accelerogram - graphical output of an accelerograph

Peak Ground Acceleration (PGA)

EARTHQUAKE Dynamic Lateral Force Procedure The response of the structure exceeds the ground motion and the dynamic magnification depends on the following: •

a. b. c. d.

Ground vibration Soil properties at the site Distance from the epicenter Dynamic characteristics of the structure

EARTHQUAKE Dynamic Lateral Force Procedure Response Spectra A response spectrum is simply a plot of the peak or steady-state response (displacement, velocity or acceleration) of a series of oscillators of   varying natural frequency, that are forced into motion by the same base vibration. The resulting plot can then be used to pick off the response of  any linear system, given its natural frequency of oscillation.

EARTHQUAKE Dynamic Lateral Force Procedure Response Spectra – NSCP 2010

T o

= 0.2T S

EARTHQUAKE Structural Dynamics Dynamic Model A dynamic model of the structure consists of a single column with stiffness k supporting a mass of magnitude m to give the inverted pendulum, or lollipop structure shown. If the mass is subjected to an initial displacement and released, with no external forces acting, free vibration occur about the static position.

EARTHQUAKE Structural Dynamics Undamped Free Vibration Oscillations continue forever and the idealized structure will never come to rest The same maximum displacement occurs oscillations after oscillations Intuition suggests that this is unrealistic. •

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EARTHQUAKE Structural Dynamics Damped Free Vibration The process by which vibration steadily diminishes in amplitude is called damping. In damping, the energy of the vibrating system is dissipated by various mechanisms. In a vibrating building these includes friction at steel connections, opening and closing of microcracks in concrete, friction between the structure itself  and nonstructural elements such as partition walls. •

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EARTHQUAKE Structural Dynamics Dampers

EARTHQUAKE Structural Dynamics Dampers

EARTHQUAKE Structural Dynamics Dampers

EARTHQUAKE Structural Dynamics Equation of Motion : External Force The external force applied on the structure is resisted by the inertia force, elastic force, and damping force. •

Where:

- the velocity or the first derivative of dispalcement u - the acceleration or the second derivative of dispalcement u

EARTHQUAKE Structural Dynamics Equation of Motion : Earthquake Excitation The relative displacement or deformation of the structure due to ground acceleration will be identical to the displacement of the structure if its base was stationary and was subjected to an external force. •

Where:

- ground acceleration

EARTHQUAKE Structural Dynamics Equation of Motion : Undamped Free Vibration •

The equation of motion for systems without damping

The solution to the homogeneous differential equation is

Natural circular frequency of vibration =

in radians/sec

EARTHQUAKE Structural Dynamics Equation of Motion : Undamped Free Vibration The time required for the undamped system to complete one cycle of free vibration is the natural period of vibration of the system, which we denote as Tn , in units of seconds. It is related to ωn whose unit is in radians per second.

The natural cyclic frequency of vibration is 1/Tn

The units of f n are hertz (Hz) [cycles per second (cps)]; fn is related to ωn

EARTHQUAKE Structural Dynamics Example: Determine the natural period of  vibration and the natural cyclic frequency for the industrial building shown. Total Weight, W = 187.5 kips North-South (Moment Frames) Stiffness: k = 231.6 kips/in. East-West (Braced Frames) Stiffness: k = 358.7 kips/in.

EARTHQUAKE Structural Dynamics Solution:

m=

W  g

=

187.5kips 386.4in / sec

2

2

= 0.485kips − sec  / in

North-South Direction: ω 

n

=

T n

k  m =

236.1

=

2π  

0.485 =

ω 

n

 f 

=

2π   21.8 1

T n

= 21.8 rad/sec

= 0.287 sec .

= 3.48 Hz

EARTHQUAKE Structural Dynamics Solution: East-West Direction: ω 

n

=

T n

k  m =

=

2π  

358.7 0.485 =

ω 

n

 f 

=

2π   27.2 1

T n

= 27.2 rad/sec

= 0.23 sec .

= 4.3 Hz

EARTHQUAKE Structural Dynamics Modal Analysis A technique used to determine a structure’s vibration characteristics: Natural frequencies Mode shapes Mode participation factors (how much a given mode participates in a given direction) Gives engineers an idea of how the design will respond to different types of dynamic loads. •

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EARTHQUAKE Structural Dynamics Mode Shape A mode shape is a specific pattern of vibration executed by a structural system at a specific frequency. Different mode shapes will be associated with different frequencies. The experimental technique of modal analysis discovers these mode shapes and the frequencies. •

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EARTHQUAKE Structural Dynamics Modal Analysis General equation of motion:

[M ]{u}+ [C]{u }+ [K ]{u} = {F(t )} Assume free vibrations and ignore damping:

[M ]{u} + [K ]{u} = {0} Assume harmonic motion:

([K ]

−ω

2

[M ]){u} = {0}

The roots of this equation are ωi2, the eigenvalues, where i ranges from 1 to number of DOF. Corresponding vectors are {φ}i, the eigenvectors. The eigenvectors {φ}i represent the mode shapes - the shape assumed by the structure when vibrating at frequency f i.

EARTHQUAKE Structural Dynamics Mode Shape – 3D Mode 1: T = 1.82s Translational •

EARTHQUAKE Structural Dynamics Mode Shape -3D Mode 2: T = 1.59s Translational •

EARTHQUAKE Structural Dynamics Mode Shape - 3D Mode 3: T = 1.08s Torsional •

EARTHQUAKE Structural Dynamics Mode Shape - 3D Mode 1: T = 1.82s

Mode 4: T = 0.48s

EARTHQUAKE Structural Dynamics Mode Shape - 3D Mode 2: T = 1.59s

Mode 5: T = 0.34s

EARTHQUAKE Structural Dynamics Mode Shape - 3D Mode 3: T = 1.08s

Mode 6: T = 0.25s

EARTHQUAKE Structural Dynamics Modal Analysis •

Usually the lower modes are significant.

EARTHQUAKE Structural Dynamics Modal Analysis

EARTHQUAKE Structural Dynamics Modal Analysis •

Results from each mode are combined statistically using methods such as SRSS – Square Root of the Sum of Squares CQC - Complete Quadratic Combination

EARTHQUAKE Structural Dynamics Scaling of Results

V dynamic

≥ 0.90V static

V dynamic

≥ 0.80V static

V dynamic

≥ 1.00V static

EARTHQUAKE Structural Dynamics Scaling of Results

EARTHQUAKE Structural Dynamics Example: Determine the base shear from modal analysis of the seven storey building.

Spectral Acceleration from Response Spectrum:

EARTHQUAKE Structural Dynamics Solution:

EARTHQUAKE Structural Dynamics Static vs. Dynamic Static analysis are used for regular and irregular structures with height less than 20m. •

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The base shear may be equal but the distribution of storey forces will vary.

The structural response from dynamic analysis is from the combination of  response from several modes. In static analysis, only the fundamental mode is used. •

Dynamic analysis, being the more general approach, can be used for all types of structures. •