4 CE ELEC2 Earthquake Dynamic Lateral Force Procedure

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CE ELEC2

EARTHQUAKE ENGINEERING

MARK ELSON C.

MARK ELSON C. LUCIO

LUCIO, MSCE (Structures)

, MSCE (Structures)

Association of Structural Engineers of the Philippines (ASEP) Association of Structural Engineers of the Philippines (ASEP)

Philippine Institute of Civil Engineers (PICE) Philippine Institute of Civil Engineers (PICE)

American Society of Civil

American Society of Civil Engineers (ASCE)Engineers (ASCE) American Concrete Institute (ACI)

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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

ective peak acceleration reached by

the ground vibration directly aff

ects the

magnitude of the

dynamic forces observed in the

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EARTHQUAKE

Dynamic Lateral Force Procedure

Accelerograph

-An instrument that records the accelera

-An instrument that records the acceleration of the tion of the ground during anground during an

earthquake, also commonly called

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EARTHQUAKE

Dynamic Lateral Force Procedure

Accelerogram

- graphical output of an accelerograph

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EARTHQUAKE

Dynamic Lateral Force Procedure

•

The response of the structure exceeds the ground motion and

the dynamic magnification depends on the following:

a. Ground vibration

b. Soil properties at the site

c. Distance from the epicenter

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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.

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EARTHQUAKE

Dynamic Lateral Force Procedure

Response Spectra – NSCP 2010

S o

T

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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.

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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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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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Structural Dynamics

Dampers

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Structural Dynamics

Dampers

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Structural Dynamics

Dampers

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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

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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.

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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

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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

T_n_{, in units of seconds. It is related to} ω_n whose unit is in radians per second.

The natural cyclic frequency of vibration is 1/Tn

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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.

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Structural Dynamics

Solution: rad/sec 8 . 21 485 . 0 1 . 236 = = =

m

k

n ω North-South Direction:

in

kips

in

kips

g

W

m

/ sec 485 . 0 sec / 4 . 386 5 . 187 2 2 − = = = . sec 287 . 0 8 . 21 2 2 = = = π ω π n n

T

Hz

T

f

n 48 . 3 1 = =

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EARTHQUAKE

Structural Dynamics

Solution: rad/sec 2 . 27 485 . 0 7 . 358 = = =

m

k

n ω East-West Direction: . sec 23 . 0 2 . 27 2 2 = = = π ω π n n

T

Hz

T

f

n 3 . 4 1 = =

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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

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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:

Assume free vibrations and ignore damping:

Assume harmonic motion:

The roots of this equation are ω

i2, 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.

[ ]

M

{ }



u



+

[ ]

C

{ }

u



+

[ ]

K

{ } ( )

u =

{ }

F t

[ ]

M

{ }



u



+

[ ]

K

{ } { }

u = 0

[ ] [ ]

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Structural Dynamics

Mode Shape – 3D

Mode 1: T = 1.82s

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Structural Dynamics

Mode Shape -3D

Mode 2: T = 1.59s

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Structural Dynamics

Mode Shape - 3D

Mode 3: T = 1.08s

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Structural Dynamics

Mode Shape - 3D

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Structural Dynamics

Mode Shape - 3D

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Structural Dynamics

Mode Shape - 3D

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Structural Dynamics

Modal Analysis

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Structural Dynamics

Modal Analysis

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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

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Structural Dynamics

Scaling of Results static dynamic

V

_≥ 0.90 static dynamic

V

_≥ 0.80 static dynamic

V

_≥1.00

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Structural Dynamics

Scaling of Results

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Structural Dynamics

Example: Determine the base shear from modal analysis of the seven storey

building.

Spectral Acceleration from Response Spectrum:

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Structural Dynamics

Solution:

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Structural Dynamics

Static vs. Dynamic

• Static analysis are used for regular and irregular structures with height less

than 20m.

• 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