CAPACITY SPECTRUM METHOD

Purdue UniversityCE571 - Earthquake Engineering

Spring 2002

Mete A. Sozen and Luis E. García

Reference:

Freeman, S. A., (1990), On the Correlation of Code Forces to Earthquake Demands, Proceedings of the 4th US-Japan Workshop on Improvements of Building Structural Design and Construction Practices (ATC 15-3), Kailua-Kona, Hawaii.

Acceleration-Displacement Spectra

If values for the same period T of the acceleration response spectra and the displacement response spectra, obtained for the same damping coefficient, are plotted with the value of Sd(T,ξ) in the abscissa axis, and the value of Sa(T,ξ) in the ordinates axis, an acceleration-displacement response spectra is obtained.

Acceleration-Displacement Spectra

Push-Over AnalysisThe objective of the Push-Over is to establish the lateral displacements of the structure as the applied base shear is increased monotonically. The relative distribution of the lateral loads that compose the base shear is maintained fixed during the analysis. Lateral load distribution employed usually follow the shape of the fundamental mode of vibration, but may be set arbitrarily to any type of distribution; inverted triangle, parabolic, and uniform have been employed, the result being sensitive to a certain extent to the distribution employed.

Push-Over AnalysisAs the base shear is increased during the process, the response of each individual element of the structure is evaluated for stiffness changes and failure modes. Using the component load-deformation data and the geometric relationships among components and elements, a global model of the structure relates the total seismic forces on a building to it overall lateral displacement to generate the capacity curve. During the pushover process of developing the capacity curve as brittle elements degrade, ductile elements take over the resistance and the result helps visualize the overall performance of the structure.

Push-Over AnalysisBackbone relationships are employed in most cases.

The properties of interest of such elements are relationships between the forces and the corresponding inelastic displacements.

Force

Displacement

Backbone curve

0

A

C

B D

E F G

Push-Over AnalysisDuring the procedure, once a point of behavior change is detected for a particular element, appropriate changes in stiffness properties are made, and a new stage of the analysis is performed increasing the base shear until reaching a new point of change of behavior in any of the elements. This process is carried out iteratively until critical strength failure of one or several elements is detected or a collapse mechanism is reached. Results from the push-over analysis are presented in different forms; with a base shear vs. roof lateral displacement plot being the more popular.

Push-Over Analysis

0

20

40

60

80

100

120

140

0.00 0.05 0.10 0.15 0.20 0.25Story Drift (%h)

Stor

y Sh

ear

(ton

)

AnalyticalTest BTest 0

Push-Over Example

������������������������������

������������������������������

������������������������������

������������������������������

�������������������������������

�������������������������������

3 m

6 m

A

B C

D E

F wall section 0.15 x 1 m

beam section 0.3 x 0.4 m

column section 0.5 x 0.5 m

Vs

Push-Over Example

0

50

100

150

200

250

300

0 10 20 30 40 50 60 70

Curvature φ (10-3/m)

Mom

ent (

kN .

m)

ColumnBeam PositiveBeam NegativeWall

Push-Over Example

�������������

�������������

�������������

������������������������������

�����������������

Step 1 Wall cracks

��������������

��������������

�������������

�������������

����������������

����������������

Step 2 Wall yields

�������������

�������������

�������������

������������������������������

�����������������

Step 3

Beam yields in positive moment

�������������

�������������

�������������

������������� �����������������

Step 4

Column yields

��������������

��������������

�������������

������������� ����������������

Step 5

Column yields

�������������

�������������

�������������

������������� �����������������

Step 6

Beam yields in negative

moment

Push-Over Example

0

50

100

150

200

250

300

350

0 5 10 15 20 25

Roof Deflection (mm)

Bas

e Sh

ear

(kN

)

Stage 2

Collapse mechanism

forms

Stage 6

Stage 5

Stage 4

Stage 3

Stage 1

Capacity Spectrum Method

Freeman developed a procedure for finding the displacement demand on a system in the inelastic range by employing the response spectra of the ground motion, presented using the acceleration-displacement scheme, simultaneously with the capacity of the structure as obtained in a push-over analysis plotted in the same spectra by dividing the base shear by the weight of the structure (V/W).

Capacity Spectrum Method

The point where both the demand and the capacity curves intersect corresponds to the expected displacement demand during the ground motion.

Capacity Spectrum Method

The damping to use, ξeff, in order to define the demand spectral value correspond to the damping that occurs when the structure is pushed into the inelastic range and is viewed under this procedure as a combination of viscous and hysteretic damping.

Capacity Spectrum Method

V/W

Sa

Sd

Teff Tinitial

ξeff

ξ=2% ξ=20%

δ displacement demand

CAPACITY push-over curve

ξ=5%

DEMAND Acceleratio-displacement spectra

Ve/W

strength demand

elastic demand

Capacity Spectrum Method

The effective damping is obtained from:

Where λ is a modification factor to account for the approximation involved in describing the hysteretic response of the system by a bilinear representation in the capacity curve. λ ranges with values as low as 0.3 for systems with poor and unreliable hysteretic behavior to a value of one for well-detailed elements with stable hysteresis loops. The 0.05 accounts for the viscous damping inherent in the system.

05.00eff +ξ⋅λ=ξ

Damping in the Capacity Spectrum Method

Sa

Sd Sdm Sdy

Sam Say

Area = Energy dissipated by hysteretic damping

Area = Maximum strain energy

Bilinear representation of capacity

Kinitial Keff

Capacity Spectrum Method

The value of ξ0 can be obtained from

Where ED corresponds to the energy dissipated by the hysteretic damping corresponding to the area of the shaded parallelogram in previous figure.

And Es corresponds to the maximum strain energy absorbed by the structure, equal to the area of the shaded triangle.

D0

S

E14 E

ξ = ⋅π

( )D ay dm am dyE 4 S S S S= ⋅ ⋅ − ⋅

( )S am dm1E S S2

= ⋅ ⋅

Capacity Spectrum Method

In order to obtain the displacement demand on the structure, an iterative procedure must be employed. The initial stiffness and an arbitrary value of effective damping, say ξeff = 5%, are used to initiate the process. With these values, a displacement demand is obtained from the demand acceleration-displacement spectra for 5% damping, corresponding to point (0). The displacement demand for this period and damping is obtained, marked as δ0.

Capacity Spectrum Method

Sa

Sd

Tinitial

ξ=2%

ξ=20%

δ0

ξ=5% ξ=10%

(0)

Capacity Spectrum Method

From the capacity curve the effective period, Teff (1), compatible with this displacement is obtained, and the effective damping, ξeff (1), is computed. A new cycle is initiated by using this period and damping thus obtaining a new displacement demand δ1. This procedure is repeated until the displacement demand δm matches the spectral value for the Teff and ξeff employed.

Capacity Spectrum Method

Sa

Sd

Teff (1) Tinitial

ξeff (1)

ξ=2%

ξ=20%

δ0

ξ=5% ξ=10%

(0)

δ1

(1)

Capacity Spectrum Method

Sa

Sd

Teff (1) Tinitial

ξeff (1)

ξ=2%

ξ=20%

δ0

ξ=5% ξ=10%

(0)

δ1

Teff (2)

(1) ξeff (2)

(2)

δ2

Capacity Spectrum MethodSa

Sd

Teff (1) Tinitial

ξeff (1)

ξ=2%

ξ=20%

δ0

ξ=5%

ξ=10%

(0)

δ1

Teff (2)

(1) ξeff (2)

(2)

δ2 δm

Teff

Capacity Spectrum Method

Sa

Sd

Teff (1) Tinitial

ξeff (1)

ξ=2%

ξ=20%

δ0

ξ=5% ξ=10%

(0)

δ1

Teff (2)

(1) ξeff (2)

(2)

δ2 δm

Teff

This is the expecteddisplacement compatiblewith the strength, stiffness,and ground motion