dynamic analysis with examples – seismic analysis
DESCRIPTION
Presentation made by Dr André Barbosa @ University of Porto during the OpenSees Days Portugal 2014 workshopTRANSCRIPT
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
OpenSees Days in Portugal @FEUP
p. 2 Dynamic Analysis Notes
Dr. André R. Barbosa 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?
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p. 3 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 2014
Moment interacOon diagrams RC sec8on behavior under Combined
Bending and Axial Load
https://www.dropbox.com/s/evzcz6er3ep0jen/Ex1_MP_Interaction_Diagram.zip
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p. 4 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 2014
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)
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p. 5 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 2014
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
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p. 6 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 2014
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
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p. 7 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 2014
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
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p. 8 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 2014
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
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p. 9 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 2014
Interac8on Diagram
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p. 10 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 2014
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
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p. 11 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 2014
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
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p. 12 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 2014
Pushover Analysis
Linear geometry
PDelta
Corotational
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p. 13 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 2014
13
Time-history Response Analysis Linear
geometry
PDelta
Corotational
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p. 14 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 2014
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
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p. 15 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 2014
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
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p. 16 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 2014
Pushover Analysis Time History Analysis
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p. 17 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 2014
Concrete stress-strain response
Steel stress-strain response
z
y Fiber 2
Fiber 1
Pushover Analysis
Element 1 Section 5
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p. 18 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 2014
z
y Fiber 2
Time History Analysis Concrete stress-strain response
Steel stress-strain response
Element 1 Section 5
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p. 19 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 2014
Building Example Barbosa, Conte, Restrepo (2011)
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p. 20 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 2014
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)
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p. 21 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 2014
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)
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p. 22 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 2014
OpenSees Days in Portugal @FEUP
p. 23 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 2014
Bridge Example Soil-structure interaction: Barbosa, Mason, Romney (2013)
Tsunami following earthquake: Carey, Mason, Barbosa, Scott (2014)
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p. 24 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 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)
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p. 25 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 2014
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
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p. 26 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 2014
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
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p. 27 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 2014
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
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p. 28 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 2014
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
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p. 29 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 2014
Analysis Methodology • Step 3: Gravity (self-‐weight) loads
• Soil self-‐weight loading • Connect pile to soil column • Structural self-‐weight loading
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p. 30 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 2014
Analysis Methodology • Step 4: Nonlinear dynamic analysis using earthquake mo8ons
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p. 31 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 2014
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)
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p. 32 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 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
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p. 33 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 2014
Tsunami-‐Bridge Interac8on Model q Type-‐I sha\ q California, Oregon, Washington, USA
Carey, Mason, Barbosa, Scott (2014)
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
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
Units in meters
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p. 34 Dynamic Analysis Notes
Dr. André R. Barbosa July 03, 2014