dynamic analysis with examples – seismic analysis

OpenSees Days in Portugal @FEUP

p. 1 Dynamic Analysis Notes

Dr. André R. Barbosa July 03, 2014

OpenSees Days in Portugal @Faculdade de Engenharia at Univ. do Porto

DYNAMIC ANALYSIS (Seismic and Tsunami loadings)

André R. Barbosa, Ph.D., P.E.

July 03, 2014




2

Outline

•  Moment-‐interac8on diagrams as an applica8on of sec8on analysis using OpenSees.exe

•  Modeling a 1-‐bay, 2-‐story RC concrete frame – Nonlinear material and nonlinear geometry

•  What can else can we do using OpenSees? – Building example – Bridge example – Soil-‐structure-‐fluid-‐interac8on?




Moment interacOon diagrams RC sec8on behavior under Combined

Bending and Axial Load

https://www.dropbox.com/s/evzcz6er3ep0jen/Ex1_MP_Interaction_Diagram.zip




Development of M-‐N interac8on diagrams

Concrete crushes before steel yields

Steel yields before concrete crushes

Moment

Axia

l Loa

d, P

Failure Criterion: ecu = 0.003

Interaction Diagram

(Failure Envelope)




General Procedure –  For various levels of axial load, increase curvature of the sec8on un8l a concrete strain of 0.003 is reached.

–  Files used:

• model.tcl • Mp.tcl

– Output:

• mp.out

Moment = f(c)

Axia

l Loa

d, P

P

M




Zero Length Sec8on Element for RC Sec8on Analysis

y

z

y

x

1Lu uL

L

ε

θχ θ

≡Δ= = Δ

Δ= = Δ

Zero-Length Section




Concrete01

2*$fpc/$epsc0

$fpc $fpcu

$epsU $eps0

strain

stre

ss

$E0

$b*E0 $Fy

strain

stre

ss

$Fy $b*E0

As1 = 4 No. 8 bars

As2 = 4 No. 8 bars

y

z y1

-y1

-z1 z1

cover

Fiber sec8on

Steel01

Concrete01

Core concrete

Cover concrete




Interac8on Diagram

c si1=

= +∑n

ni

P C F

( )c si12 =

⎛ ⎞= − + −⎜ ⎟⎝ ⎠∑n

n ii

aM C y F y d

Reinforced Concrete: Mechanics and Design (4th Edi8on) by James G. MacGregor, James K. Wight

1 11

0.003 ; where = 0.003

ε εε

⎛ ⎞= ⎜ ⎟−⎝ ⎠

s ys

c d Z

= 0.003ε −⎛ ⎞⎜ ⎟⎝ ⎠

isi

c dc

= ; ε ≤si si s si yf E f f

1 = 1.05 0.051000

β′⎛ ⎞

− ⎜ ⎟⎝ ⎠

cfpsi

( )( ) 1 = 0.85 ; β′ =c cC f ab a c

= (positive in compression)si si siF f A

( ) = 0.85 ′−si si c siF f f A

if < ia d

else

for symmetric sections2

= hy




Interac8on Diagram




Modeling a 1-‐bay, 2-‐story RC frame Beam column element with (elas8c) RC

fiber sec8on

https://www.dropbox.com/s/ove56qgu7dqg54r/Ex2_ElasticFrame.zip




P P

H/2

1 2

3 4

5 6 H

A A

P P

(1) (2)

(3)

(4) (5)

(6)

Linear Elas8c

Steel

Concrete

Lbeam = 42 _

Lcol = 36 _

Lcol = 36 _

Cross-‐sec8ons




Pushover Analysis

Linear geometry

PDelta

Corotational




13

Time-history Response Analysis Linear

geometry

PDelta

Corotational




Modeling a 1-‐bay, 2-‐story RC frame Beam-‐column element with RC

nonlinear fiber sec8on

https://www.dropbox.com/s/geigqdn3dsrvbyb/Ex3_NonlinearFrame.zip




e

P P

H/2

1 2

3 4

5 6 H

A A

P P

(1) (2)

(3)

(4) (5)

(6)

Lbeam = 42 _

Lcol = 36 _

Lcol = 36 _

Cross-‐sec8ons

s e s

Steel02 Concrete02

Material models




Pushover Analysis Time History Analysis




Concrete stress-strain response

Steel stress-strain response

z

y Fiber 2

Fiber 1

Pushover Analysis

Element 1 Section 5




z

y Fiber 2

Time History Analysis Concrete stress-strain response

Steel stress-strain response

Element 1 Section 5




Building Example Barbosa, Conte, Restrepo (2011)




q  NEHRP design example (FEMA 451)

Ø  Demonstrate the design procedures (ASCE7-‐05, ACI318-‐08) Ø  Building was re-‐designed to account for latest Seismic Design Maps and

common prac8ces in California

Latitude: 37.87N

Longitude: -122.29W

Plan View Eleva,on Loca,on

Barbosa, Conte, Restrepo (2011)




Elevation Location

Ø  Walls: Nonlinear truss modeling approach Ø  Columns and beams: Force-‐based beam-‐column elements Ø  Diaphragms: Flexible diaphragms allowing for plas8c hinge

elonga8on

q First…. CHECK AND VALIDATE YOUR MODEL… q The model

Barbosa, Conte, Restrepo (2011)




Bridge Example Soil-structure interaction: Barbosa, Mason, Romney (2013)

Tsunami following earthquake: Carey, Mason, Barbosa, Scott (2014)




Soil-‐Founda8on-‐Bridge Model q Type-‐I sha\ q California, Oregon, Washington, USA

boundary is modeled in OpenSees by specifying a horizontal “compliant bedrock” dashpot [7], as shown in Figure 1b. The dashpot coefficient, c, is calculated as c = ȡE Vs A, where ȡE is the mass density of the bedrock layer, Vs is the shear wave velocity of the bedrock layer, and A is the out-of-plane cross sectional area of the quadrilateral element. The earthquake motion is applied to the model as an equivalent force-time series, which is coupled with the dashpot. The equivalent force-time series, FE, is calculated as FE = 2ȡE Vs uլ g A, where uլ g is the velocity-time series of the input earthquake motion. Applying the force-time series at the soil-bedrock interface requires the soil column to have unconstrained horizontal degrees of freedom, which is accomplished by modeling soil column bedrock interface with rollers (i.e. the vertical degree-of-freedom is constrained).

(a)

(b)

Figure 1. (a) Generalized elevation schematic of the bridge deck, column, pile, and interface springs, and (b) generalized view of the fair-field soil column modeled using 9-4 quadrilateral elements. The soil mesh is connected to the concrete pile with a series of interface springs, as

shown conceptually in Figure 1a. The interface springs – lateral (p-y), vertical (t-z), and end bearing (q-z) – capture the dynamic response of the surrounding soil and its interaction with the concrete pile foundation. The interface spring coefficients are calculated in accordance with API recommendations [20]. Each interface spring is defined by its depth varying ultimate resistance (pult, qult, and tult) and its expected displacement when 50% of the ultimate strength is mobilized. At larger depths, subgrade reaction moduli, which are used to define the shape of load-displacement curves (i.e., p-y, t-z, and q-z curves), are adjusted for overburden effects by the factor (50 kPa/ıүv)0.5, where ıүv is the effective vertical stress at the depth of interest, in accordance with Boulanger et al. [21]. In OpenSees, the interface springs are implemented using PySimple1, TzSimple1 and QzSimple1 [21] for the p-y, t-z, and q-z springs, respectively. More details about the spring coefficients are given in Barbosa et al. [13].

Barbosa, Mason, Romney (2013)




Bridge Deck & Abutments •  Linear elas8c beam-‐column

•  10.36 m W x 1.67 m T x 63.4 m L •  Area = 4.56 m2

•  Ixx = 5.98x1012 mm4




Bridge Deck & Abutments •  Abutment (Shamsabadi et al., 2007, Caltrans SDC)

•  Silty sand •  S8ffness, K = 307 kN/cm/m •  Yield Force, Fy = 1397 kN •  Ini8al Gap Opening = 2.54 cm




27

Pile and Column •  Moment-‐Curvature Analysis

•  f’ c = 28 MPa •  Fy = 475 MPa •  E = 200 GPa •  Long. steel ra8o = 1.0%

•  Fiber-‐sec8on: •  Varied number of theta wedges and

radial rings




Analysis Methodology •  Development of SFB Model

•  Step 1: Define soil •  Step 2: Define structural nodes and elements

the seismic resiliency of bridges is an important topic of investigation for earthquake engineers. To this end, a two-dimensional finite element model of a soil-bridge system was developed to evaluate the effects of earthquake motions from shallow crustal and subduction zone sources on the seismic response of bridges. Within the developed model, soil-bridge interaction is accounted for by connecting the soil to the bridge pile with soil-interface springs. Furthermore, the direct method [1] for analyzing soil-bridge interaction was employed. A thorough literature review is outside the scope of this study. Readers can consult Barbosa et al. [2] for more information about soil-bridge interaction. The works by Khosravifar [3], Chiaramonte et al. [4], Zhang et al. [5], Shamsabadi et al. [6], Chang [7], Brandenberg et al. [8] and Boulanger et al. [9] were instrumental for creating the soil-bridge models and informing the research work.

Methodology A two-dimensional (2-D) finite element model of a double-span reinforced concrete bridge and foundation connected to a nonlinear soil column by nonlinear soil springs was developed using the Open System for Earthquake Engineering Simulations (OpenSees) finite element framework [10]. The seismic response of the soil-bridge system was analyzed by subjecting the model to seven shallow crustal earthquake motions and seven subduction zone earthquake motions. Figure 1 shows a schematic of the overall soil-bridge system and a cross-section of the modeled pile and bridge column. Barbosa et al. [2] contains more details about the soil-bridge model, though some of the modeling details have changed, which are documented herein.

(a) (b)

Figure 1. (a) Soil-bridge system analyzed, and (b) cross-section of bridge pile and column (all dimensions are in meters) [2].

Earthquake Motion Selection In the United States, the Pacific Northwest (PNW) and Alaska are prone to subduction zone earthquakes. Traditionally, bridges have been designed to withstand shallow crustal earthquakes, because these bridges are predominant in California. However, the subduction zone earthquake




Analysis Methodology •  Step 3: Gravity (self-‐weight) loads

•  Soil self-‐weight loading •  Connect pile to soil column •  Structural self-‐weight loading




Analysis Methodology •  Step 4: Nonlinear dynamic analysis using earthquake mo8ons




column, and accordingly, the out-of-plane contact width for the tsunami wave impact is 31.7 m, which is equal to the span length in the longitudinal direction of the bridge. Within the PFEM procedure, the mass and body forces are increased to represent the out-of-plane contact width. For numerical stability, the column contact width (1.1 m) is not considered. Neglecting the column width when considering tsunami impact is acceptable, because the soil-bridge system response will be dominated by tsunami impact on the much larger deck width. The length of the tsunami bore is set as twice the open length (defined in Figure 3). The accuracy and stability of the PFEM solution depends on the mesh density of the fluid and structure under consideration [9]. To ensure accuracy and stability, the mesh size for the tsunami bore simulation was selected as 175 mm x 175 mm.

Figure 3. Schematic of the tsunami bore simulation (Note: pile and soil column are not shown).

Analysis Framework

The soil-bridge model analysis framework is divided into three stages. The stages simulate the earthquake and tsunami wave loading. Furthermore, each stage is divided into sub-stages for analytical and conceptual reasons. The sub-stages presented for the seismic analysis were adapted from Barbosa et al. [13], and the tsunami stages from Zhu and Scott [9]. Figure 4 shows a flow chart of the stages and sub-stages described below. Stage 1: Earthquake Simulation x Stage 1.1: The global geometry (nodes, connectivity, and boundary conditions) of the soil-

bridge model is established. The bridge column, bridge pile, and soil mesh are created. Additionally, the 9-4 quadrilateral elements and soil interface springs are created. The interface-springs are connected to the soil column, but not to the bridge pile.

x Stage 1.2: The bridge pile and bridge column fiber sections are defined. The bridge pile and column is created using nonlinear beam column elements. The self-mass of the pile and column elements is lumped at the end nodes.

x Stage 1.3: The linear elastic rigid bridge deck is created and attached to the bridge column created in stage 1.2. The bridge deck elements are discretized to the same dimension as the

Tsunami following Earthquake Modeling q Type-‐I sha\ q California, Oregon, Washington, USA q  In OpenSees, use PFEM:

v  Zhu and Scott (2014)

Carey, Mason, Barbosa, Scott (2014)




Tsunami-‐Bridge Interac8on Model

PFEM procedure. At the conclusion of each time step, the fluid, wall, flume and bridge column were re-meshed for the subsequent time steps. Figure 5 shows a schematic of the tsunami simulation.

Figure 4. Flow Chart of the three stages comprising the analysis framework.

(a)

(b)

Figure 5. (a) Tsunami bore at the end of Step 3.1, and (b) tsunami bore during Step 3.2.

−20 −15 −10 −5 0 5 10−5

0

5

10

15

20

25

30Initial Bore Time 0

−20 −15 −10 −5 0 5 10−5

0

5

10

15

20

25

30Analysis at 0.575 Sec




Tsunami-‐Bridge Interac8on Model q Type-‐I sha\ q California, Oregon, Washington, USA

Carey, Mason, Barbosa, Scott (2014)



(a)

(b)


−20 −15 −10 −5 0 5 10−5

0

5

10

15

20

25


−20 −15 −10 −5 0 5 10−5

0

5

10

15

20

25




(a)

(b)


−20 −15 −10 −5 0 5 10−5

0

5

10

15

20

25


−20 −15 −10 −5 0 5 10−5

0

5

10

15

20

25


Units in meters




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dynamic analysis with examples – seismic analysis

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si si c si f f f

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