Inelastic Displacement Surface Method

Inelastic Displacement Surface Method

Tom Shantz CALTRANS- Division of Research and Innovation

Basic Approach Uses peak displacement response of an elastic-perfectly plastic

oscillator as the primary intensity measure used for record scaling and selection.

uy ue ui

Fy

Fmax

Basic Approach Uses peak displacement response of an elastic-perfectly plastic

oscillator as the primary intensity measure used for record scaling and selection.

uy ue ui

Fy

Fmax

R = Fmax

Fy

ue

uy

=

ui

uy

Basic Approach Evaluate displacement surface for a record (precalculated)

0 1 2 3 4 5Periods 2

4

6

8

0

2

4

6

8

Dmaxcm

0 1 2 3 4 5Periods

Basic Approach Evaluate displacement surface for a record (precalculated)

Basic Approach Evaluate displacement surface for a record (precalculated) Define target displacement surface (will discuss details in a minute)

0 1 2 34 5

Periods 2

4

6

8

0

2

4

6

8

Dmaxcm

0 1 2 34 5

Periods

Basic Approach Evaluate displacement surface for a record (precalculated) Define target displacement surface (will discuss details in a minute) Define region for comparison

0 1 2 34 5

Periods 2

4

6

8

0

2

4

6

8

Dmaxcm

0 1 2 34 5

Periods

Basic Approach Evaluate displacement surface for a record (precalculated) Define target displacement surface (will discuss details in a minute) Define region for comparison Determine optimum scale factor

0 1 2 34 5

Periods 2

4

6

8

0

2

4

6

8

Dmaxcm

0 1 2 34 5

Periods

Basic Approach A record’s displacement surface scales uniformly with scale factor as long as the yield force of the oscillator is scaled as

well. Thus, is selected so that after scaling, the yield force is consistent with the pushover curve. ( i.e. = Dtarget / uyield ) Select records for nonlinear analysis by the quality of their fit to the target region

0 1 2 34 5

Periods 2

4

6

8

0

2

4

6

8

Dmaxcm

0 1 2 34 5

Periods

Can also impose secondary criteria such as how close the scaled record matches 2nd mode elastic displacements.

Determining Target Displacement Surface Case 1: What is the distribution of peak displacement response

given M, r, and Fy?

(M, r) San Andre

as

Determining Target Displacement Surface

Calculate the response of the oscillator for a grid of and T for each of over 1800 records of the NGA dataset.

Define the response in terms of “Normalized Inelastic Displacement Demand” (NIDD) where

NIDD(, T) j = ui, jSd, j

and Sd , j

is the median spectral displacement as estimated

using A&S97 and M , rjMotivation for normalization: elimination of (M, r) dependence

Determining Target Displacement Surface

A very convenient form is …

NIDD(, T) = uiSd =

uiSd

* SdSd

= C' Ca

The ‘ in C reflects that constant strength is imposed with the reference strength defined for Ca =1.

Determining Target Displacement Surface

Statistics on NIDD response:

Since NIDD is a function of Ca, and since Ca is influenced by earthquake event terms, a 2-step regression procedure is required determine the median.

The median Target Displacement Surface is then calculated as..

DTarget (, T) = NIDDMedian (, T) * Sd

(T) D84Target (, T) = NIDDMedian (, T) * Sd

(T) *e NIDD

Determining Target Displacement Surface

An approximate functional form looks something like…:

N ID D (T , ) = 1 .374 - 0 .787 T + 0.326 T2

- 0 .0526 T3 +0 .00365 T

4

- 0 .000142 T5 + 0 .0165

Ln(T , ) = 0 .808 - 0 .425 T + 0 .297 T2

- 0 .0912 T

3 + 0 .0128 T4 - 0 .000679 T

5

+ 0 .0336 - 0 .00231 2

0 1 2 3 4 5P e rio d (s)

0.7

0.8

0.9

1

1.1

1.2

Me

dian

NID

D

0 1 2 3 4 5Period (s)

0.6

0.64

0.68

0.72

0.76

0.8

Ln

(NID

D)

Case 2: What is the distribution of peak displacement response given M, r, Fy, and (or a specified Sa)?

Determining Target Displacement Surface

(or Sa) can easily be related to Ca. Both terms in NIDD have Ca dependence.

NIDD(, T) = C', TCa (T)

Determining Target Displacement Surface

-3 -2 -1 1 2LogCa

-3

-2

-1

1

2

LogNIDD

One can plot NIDD vs Ca (in this example for =4 and T=1s)…

Take median and from a data bin centered on the desired Ca

Determining Target Displacement Surface

-3 -2 -1 1 2LogCa

-3

-2

-1

1

2

LogNIDD

One can plot NIDD vs Ca (in this example for =4 and T=1s)…

Take median and from a data bin centered on the desired Ca

One can plot NIDD vs Ca (in this example for =4 and T=1s)…

Take median and from a data bin centered on the desired Ca

Determining Target Displacement Surface

-3 -2 -1 1 2LogCa

-3

-2

-1

1

2

LogNIDD