Displacement Capacity Design Fundamentals - Buffalo deflection generation –Pushover analysis...

Displacement Capacity Design Fundamentals

Roy A. ImbsenPrincipalImbsen Consulting

1

Displacement Capacity Design Approachp p y g pp

Brief comparison of current LRFD and guide specificationsspecificationsDamping modification & Displacement

fmagnificationImplicit Deflection Capacity – SDC B and CMoment Curvature model generationLoad deflection generation – PushoverLoad deflection generation Pushover analysis capacity

2

AASHTO Adopted 2007 Guide Spec.

3

Project Phases2002 AASHTO T‐3 Committee Meeting 2003 MCEER/FHWA

Task F3‐4 Road MapTask F3‐5 Suggested Approach

2004 NCHRP 20 07/Task 193 AASHTO Guide Specifications2004 NCHRP 20‐07/Task 193 AASHTO Guide Specifications for LRFD Seismic Bridge DesignAASHTO T‐3 Committee and Volunteer States

2006 Trial Designs2007 Technical Review

2007 AASHTO Adoption as a Guide Specification with the2007 AASHTO Adoption as a Guide Specification with the continuous support and guidance of the T‐3 Committee

4

Overall T‐3 Project Objectivesj j

Assist T‐3 Committee in developing a LRFD p gSeismic Design Specification using available specifications and current research findings

f fDevelop a specification that is user friendly and implemental into production designComplete six tasks specifically defined by theComplete six tasks specifically defined by the AASHTO T‐3 Committee, which were based on the NCHRP 12‐49 review comments

5

Summary

Single Level Hazard for 1000 year return period applicable to all regions of the USperiod applicable to all regions of the USSingle Performance Criteria for “No Collapse”pUniform Hazard Design Spectra using Three Point Method with the new AASHTO/USGS M f h PGA 0 2 d 1 0Maps for the PGA, 0.2 sec, and 1.0 secNEHRP Site Class Spectral Acceleration CoefficientCoefficient

6

Summary (continued)Partition of Seismic Design Category (SDC) into four groups (A,B,C & D) of increasing level offour groups (A,B,C & D) of increasing level of design requirement and specifications complexityyIdentification of Earthquake Resistant System for SDC C and D based on three types of lateral load path mechanismsDetailed flow charts to guide the bridge designer through the new Guide Specifications with reference to specific articles

7

Summary (continued)

Displacement Based Approach with design factors calibrated to prevent collapse andfactors calibrated to prevent collapse and reflect both the inherent reserve capacity to deform under imposed seismic loads and to

d l i di l haccommodate relative displacements at the supports and articulated connections. Use of closed form equations for implicitUse of closed form equations for implicit displacement capacity for SDC B and CPushover Analysis for Displacement CapacityPushover Analysis for Displacement Capacity of SDC D

8

Summary (continued)

New support width equations consistent with current LRFD Specifications for SDC A B and C withcurrent LRFD Specifications for SDC A,B and C with SDC D calibrated to the Guide Specification Capacity Protection of column shear, superstructure p y pand substructure for SDC C and DCapacity Protection of column shear for SDC B Steel Superstructure Design Option based on Force Reduction Factors including the use of ductile end‐diaphragmsdiaphragms Liquefaction Design Minimum Requirements for SDC C &D (recommended for B)

9

( )

Comparison of Current and Proposed LRFD Specifications

F F

LRFD Specifications

FE

Elastic Response

E Forced Based Method

13

FElastic Response

F

FE

F

FE

R

Plastic Hinge

FDD

E

Ductile Design Responsepd

14

FElastic Response

F

FE

F

FE

Di l

FSER

Plastic Hinge Capacity Determined using Section Analysis with Limiting

RD Rd

Displacement Capacity

y gSteel and Concrete Strains

pd E

Performance Based Design

pd

pc

15

Design ApproachesForce Displacement‐Force‐ ‐Displacement‐

Division 1A and Current New 2007 Adopted Guide LRFD SpecificationComplete design for service load requirements

pSpecificationComplete design for service load requirements

Elastic demand forces w/ applied prescribed ductility factors “R” for anticipated d f bilit

qDisplacements demands w/ displacement capacity evaluation for deformability d d f l ddeformability

Ductile response is assumed to be adequate w/o verification

as designed for service loadsDuctile response is assured with limitations prescribed f h SDCw/o verification

Capacity protection assumed

for each SDCCapacity protection assured

16




17

Damping Modification (4.3.2)The following characteristics may be considered as justification for

the use of higher damping (maximum 10%).

• Total bridge length is less than 300 ft.

• Abutments are designed for sustained soil mobilization.

S l li h k (l h 20°)• Supports are normal or slight skew (less than 20°).

• The superstructure is continuous without hinges or expansion joints.j

18

Displacement Mag. (4.3.3)p g ( )

1)-(4 3 301*for01111*

>≥+⎟⎞

⎜⎛ −=

TTR

2)-(4.3.3 0.1*for 0.1

1)(4.3.3 0.1for 0.11

≤=

>≥+⎟⎠

⎜⎝

=

TTR

TRTRR

d

d

BSDCfor 2 structure theofductility nt displaceme expected maximum

3)-(4.3.3 25.1:in which

*S

RTT

===

(sec )3 4 1Articlefromdeterminedperiod:where

D SDCfor C SDCfor 3

S

D

T

μ

=

==

6. as taken bemay , analysis, detailed a oflieu In 4.9. Article with accordancein determined

demandductility nt displacememember local maximum (sec.)3.4.1Articlefromdeterminedperiod

D

D

ST

μ

μ =

19




20

Requirements for Ductile Design

Ability to deform inelastically without “serious” d d i f hdegradation of strength.Careful selection of locations of ductility (e.g., where plastic hinges will form)where plastic hinges will form).Detailing of plastic hinges to ensure adequate ductility.yNon‐ductile modes of failure (e.g., shear or bond) are to be avoided. This can be done h h i d i hthrough a capacity design approach.

21

Ductile Responsep

22

Non‐ductile Responsep

23

Design for SDC B, C, & D (4.7)

0.60.4 ≤≤ DμConventional – Full ductility structures with a plastic mechanism having for a0.60.4 ≤≤ Dμplastic mechanism having for a bridge in SDC DLimited ductility – For structures with a Plastic ymechanism readily accessible for inspection having for a bridge in SDC B or C0.4<DμLimited Ductility – For structures having a plastic mechanism working in concert with a protective

Th l i hi fsystem. The plastic hinge may or may not form. This strategy is intended for SDC C or D

24

Plastic Hinging Forces (4.11.2)g g ( )

Center of GravityCenter of Gravity

φMuφMu

VL + VR

From columninteraction diagram

AXIAL

h L

φM VRVLAXIALLOAD MAXIMUM

PROBABLEMOMENT = φMu

PDL + PPHPDL

PDL - PPH

φMuφMu

PPH PPHa

MOMENTCOLUMN INTERACTION DIAGRAM

DL PH

25

Performance CriteriaPerformance Criteria

One design level for life safetyOne design level for life safetySeismic hazard level for 7% probability of

d i 75 (i 1000 texceedance in 75 years (i.e., 1000 year return period)Low probability of collapseMay have significant damage and disruption to y g g pservice

26

Displacement Capacity SDC B & C (4.8.1)

27

Displacement Capacity SDC B & C (4.8.1)p p y ( )

⎛ ⎞⎛ ⎞ Spalling Equation by11.6 1 110( )g c

PDH A f H

⎛ ⎞⎛ ⎞Δ ⎜ ⎟= − +⎜ ⎟⎜ ⎟′ ⎜ ⎟⎝ ⎠⎝ ⎠

Spalling Equation by Berry and Eberhard

The drift capacity for all curves are shown as afunction of the slenderness ratio

FbFbLwhere:

F = Fixity Factor ranging from 1 to 2 b = column width or diameterL = column clear height

29

Reinforced Concrete Spalling from Berry and Eberhard

6.00

4.00

5.00

city

(%) Experimental (C1)

2.00

3.00

Drif

t Cap

ac

0.00

1.00

0 1 0 15 0 2 0 25 0 3

D

0.1 0.15 0.2 0.25 0.3

D/H

30

Analytical Data Base of Columns

31

Drift Capacity vs. D/H for the Columns Designed in

Accordance with the Design SpecificationsAccordance with the Design Specifications

6.00

4.00

5.00

city

(%)

Yield (C2)Spalling (C3)Ductility 4 (C4)

2.00

3.00

Drif

t Cap

ac

0.00

1.00

0.1 0.15 0.2 0.25 0.3

D/H

33

Determination of the Drift Capacity for SDC B.

5.00

6.00Experimental (C1)Yield (C2)

3 00

4.00

5.00

acity

(%)

( )Spalling (C3)Ductility 4 (C4)SDC B (C5)

2.00

3.00

Drif

t Cap

a

0.00

1.00

0 1 0 15 0 2 0 25 0 30.1 0.15 0.2 0.25 0.3

D/H

34


35

Determination of the Drift Capacity for SDC C

5.00

6.00Experimental (C1)Yield (C2)

3 00

4.00

paci

ty (%

) Spalling (C3)Ductility 4 (C4)SDC C (C6)

2.00

3.00

Drif

t Cap

0.00

1.00

0 1 0 15 0 2 0 25 0 30.1 0.15 0.2 0.25 0.3

D/H

36


37

Comparison of SDC B and C with Ductility 2 and Ductility 3Ductility 2 and Ductility 3

3 5

4Ductility 2

2.5

3

3.5

(%)

Ductility 3

SDC B

SDC C

1.5

2

Drif

t Cap

acity

0.5

1

D

00.1 0.15 0.2 0.25 0.3

D/H

38

3.50

4.00 Experimental (C1)Lap Spliced Poor Conf. (C7)Cont. Reinf. Poor Conf. (C3)

2.50

3.00

acity

(%) SDC B

SDC C

1 00

1.50

2.00

Drif

t Cap

a

0 00

0.50

1.00D

0.000.1 0.15 0.2 0.25 0.3

D/H

39

SDCSDC

40

SDC Core Flowchart

41

SDC B (3.5)( )

42

SDC CSDC C

43


44


Ho = Height of column measured from top of footing to top of column (ft )top of column (ft.)

Bo = Column diameter or width measured parallel to the direction of the displacement under pconsideration (ft.)= factor representing column end restraint

ditiΛcondition:= 1.0 for fixed‐free column boundary conditions2 0 for fixed fixed column boundary conditions= 2.0 for fixed‐fixed column boundary conditions

For partial restraint at the column ends, interpolation is permitted.

45

is permitted.




46

Plastic Moment Capacity for Ductile Concrete b f S C C d (8 )Members for SDC B, C, and D (8.5)

Equal Areas

Figure 8.5‐1 Moment‐Curvature Model

47

Moment‐Curvature Diagramg

48

Plastic Moment CapacitySDC B, C & D (8.5)

Moment‐Curvature Analyses M φ−yExpected Material PropertiesAxial Dead Load Forces with Overturning

φ

Axial Dead Load Forces with OverturningCurve Idealized as Elastic Perfectly Plastic

Elastic Portion of the Curve Pass through the point ofM φ−Elastic Portion of the Curve Pass through the point of marking the first reinforcing bar yieldPlastic moment capacity determined from equal areasPlastic moment capacity determined from equal areas of idealized and actual

49

Moment‐Curvature Diagram

εφ =

yφ

50

Pushover of a Single Columnf g

51

From the Equivalent Curvature Diagramq g

At Yield Condition:At Yield Condition:The Elasto‐Plastic Yield Displacement can b l l t dbe calculated asΔy = φy (L/2)(2L/3)

1/3 φ L2= 1/3 φy L2

52

From the Equivalent Curvature Diagram (cont.)Diagram (cont.)

At Ultimate Condition:φθp = φp Lp(φu ‐ φy)Lp

Therefore, the Plastic displacement can beTherefore, the Plastic displacement can be calculated as

Dp = θp (L ‐ 1/2Lp)(φ φ ) L (L 1/2L )= (φu ‐ φy) Lp (L ‐ 1/2Lp)

And finally,Du = Dy + Dpu y p

= 1/3 φy L2 + (φu ‐ φy) Lp (L ‐ 1/2 Lp)

53

Concept of Ductility based on Idealized Elasto‐Pl ti R t R l RPlastic Response to Real Response

Indicates real responseIndicates idealized response

Δy´ First yield displacementΔy Equivalent Elasto-Plastic yield displ.

Vu Ultimate lateral loadVy First yield of rebar

Δu Ultimate displacmentDisplacement ductility Factor:μ = where Δy = Δy´ (Vu/Vy)

ΔuΔy

54

y

Properties and Applications of Reinforcing Steel, Prestressing Steel and Concrete for SDC

B, C, and D (8.4)

For SDC B and C the expected materialFor SDC B and C, the expected material properties shall be used to determine the section stiffness and overstrengthsection stiffness and overstrength capacities.For SDC D the expected material propertiesFor SDC D, the expected material properties shall be used to determine section stiffness, overstrength capacities and displacementoverstrength capacities, and displacement capacities.

55

Reinforcing Steel (8.4.1)Reinforcing bars, deformed wire, cold‐drawn wire, welded plain wire fabric, and welded deformed wire fabric shall conform to the material standards as specified in the AASHTO LRFD Bridge Design SpecificationsSpecifications.Use of high strength high alloy bars with an ultimate tensile strength of up to 250 ksi shall beultimate tensile strength of up to 250 ksi, shall be permitted for longitudinal column reinforcement provided the low cycle fatigue properties are not p y g p pinferior to normal reinforcing steels with yield strengths of 75 ksi or less.

56

Reinforcing Steel (8.4.1)‐continuedcontinued

Use of wire rope or strand shall be permitted for spirals in columns if it can be shown through testing that the modulus of toughness exceeds 14 k iksi For SDC B and C, use of ASTM A 706 or ASTM A 615 Grade 60 reinforcing steel shall be permitted615 Grade 60 reinforcing steel shall be permitted.For SDC D, ASTM A 706 reinforcing steel in members where plastic hinging is expected shallmembers where plastic hinging is expected shall be used.

57

Reinforcing Steel Stress‐Strain (8.4.2)

58

Reinforcing Steel Modeling (8.4.2)

59

Reinforcing Steel (8.4.2) (cont.)Reinforcing Steel (8.4.2) (cont.)

60

Concrete Modeling (8.4.4)g ( )

61

Mander’s Concrete Model (8.4.4)

62

Mander’s Model (8.4.4) continued

⎟⎟⎠

⎞⎜⎜⎝

⎛−

′′

−′

′+′=′ 254.1

ff2

ff94.71254.2ff

co

l

co

lcocc

ccsec

co5.1

c

fE

fw33E

′′

=

′=

coco

cccc 11

ff5 ε′⎥

⎦

⎤⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛−

′′

=ε′

secc

c

ccsec

EEEr−

=

ε′

cc

cxε′ε

=r

ccc

secc

x1rrxff

+−

′=

The value of εcu is obtained from balancing the strain energy capacity of the transverse steel with the strain energy required to change the concrete from an unconfined state to a confined state.

63

Analytical Plastic Hinge Length Framing into a Footing or a Cap (4 11 6)Framing into a Footing or a Cap (4.11.6)

64

Analytical Plastic Hinge Length for Non‐cased Prismatic Shafts (4.11.6)

65

Analytical Plastic Hinge Length for Isolated Flare to Cap (4 11 6)Isolated Flare to Cap (4.11.6)

66

Displacement Capacity Design Approachp p y g ppBrief comparison of current LRFD and guide specificationsspecificationsDamping modification & Displacement

ifi timagnificationImplicit Deflection Capacity – SDC B and CMoment Curvature model generation (go to 146)Load deflection generation – Pushover analysisLoad deflection generation Pushover analysis capacity

67

Local Displacement Capacity for SDC D (4.8.2) Inelastic quasi‐static pushover analysis (IQPA) shall be used toInelastic quasi‐static pushover analysis (IQPA), shall be used to determine the reliable displacement capacity of a structure or frame. IQPA i i t l li l i hi h t th llIQPA is an incremental linear analysis, which captures the overall nonlinear behavior of the elements, including soil effects, by pushing them laterally to initiate plastic action. Each increment of loading pushes the frame laterally, through all possible stages, until the potential collapse mechanism is achieved. IQPA is expected to provide a more realistic measure of behavior than may be obtained from elastic analysis procedures.The effect of seismic load path on the column axial load andThe effect of seismic load path on the column axial load and associated member capacities shall be considered in the simplified model

68

Member Ductility Requirement for SDC D (4 9)SDC D (4.9)

69

LRFD ‐Member Ductility Requirement for SDC D (4 9)

(4.9-5)

Requirement for SDC D (4.9) pd

D

Δ+=1μ

Where:yi

D Δ+1μ

= plastic displacement demand (in.)pdΔ

= idealized yield displacement corresponding to the idealized yield curvature, ,yiφ

yiΔy

shown in figure 8.5-1 (in.)Pile shafts should be treated similar to columns.

yiφ

70

Shear Demand & Capacity (8.6.1)Shear Demand & Capacity (8.6.1)

SDC B is the lesser of :VSDC B is the lesser of :Force obtained from linear elastic seismic analysisForce, , corresponding to plastic hinging with

uV

poVForce, , corresponding to plastic hinging with overstrength

SDC C and D is the force associated with the uV

po

overstrength moment poMu

71

Concrete Shear Capacity SDC B, C & D (8 6 2)(8.6.2)

(8.6.2‐1)V v A= (8.6.2 1)

(8 6 2‐2)0 8A A=

c c eV v A=

(8.6.2‐2)If Pc is compressive then

0.8e gA A=

0 11 fP ⎧ ⎫′⎧ ⎫⎪ ⎪ ⎪ ⎪(8.6.2‐3)

Other ise (i e not compression)

0.110.032 {1 }

2 0.047cu

c cg c

fPv fA f

αα

⎧ ⎫⎧ ⎫⎪ ⎪ ⎪ ⎪′ ′= + ≤⎨ ⎬ ⎨ ⎬′⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭

Otherwise (i.e., not compression)(8.6.2‐4)0cv =

72

Concrete Shear Capacity SDC B, C & D (8.6.2)

Concrete Shear Capacity V within the plastic hingeConcrete Shear Capacity, Vc within the plastic hinge

μSDC EquationDμSDC Equation

B 2 -

Dμ

B 2 -

C 3 -

D 4.9-5yi

pd

Δ

Δ+1

73

#18, tot 24

75

Strain Compatibility Analysis of the Cross‐SectionSection

First Bar Yield

Ultimate

76

Load‐Deflection Diagramf g

Idealized Yield

77

Load Deflection Diagram (cont )(cont.)

Plastic hinge length: .08 x L + x db = 50 infy40

Gross Moment of Inertia = 30.72 ft4

Effective Moment of Inertia = 19 9 ft4

40

Effective Moment of Inertia = 19.9 ft

Stage Deflection (in) Curvature (rad/in) Moment (k-ft)Yield 3 68 0 000053 7357Yield 3.68 0.000053 7357Ultimate 21.51 0.000839 10349Idealized Yield 4.96 0.000072 9927Id li d Ulti t 21 51 0 000839 9927

displacement ductility = = 4.321.514.96

Idealized Ultimate 21.51 0.000839 9927

78

Problem Assignment 4‐4 ‐ SolutionMulti‐Column Pushover AnalysisMulti Column Pushover Analysis

Construct the Capacity Load‐Deflection diagram for the frame shown below.

Long. Reinforcement #11 tot 15

D.L.

VP1+VP2+VP3 21.3’ 21.3’

2” CL 48”28’

MP2 =Plastic moment using PD.L. axial force

Spiral Reinforcement #5 @ 4”

f´c = 5,000 psify = 68,000 psi

VP1 VP2VP3

P T i P P C i#5 @ 4 y , p

COLUMN SECTIONPD.L.–Tension PD.L. PD.L.+ Compression

Tension or Compression =21.3'2

3M2P

×

79

Idealized Values:Yield Curvature (1/in) = .000128e d Cu vatu e ( / ) 000 8Plastic Moment (k‐ft) = 3652Ultimate Curvature (1/in) = 01409

Axial Load 933 kips

40004500

20002500300035004000

500100015002000

00 0.0005 0.001 0.0015

Curvature (1/in)

80

Idealized Values:Yield Curvature (1/in) = 00013Yield Curvature (1/in) = .00013Plastic Moment (k‐ft) = 3898Ultimate Curvature (1/in) = 01287

Axial Load 1220 kips

40004500

2000250030003500

500100015002000

00 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014

Curvature (1/in)

81

Idealized Values:Yield Curvature (1/in) = .000132( / )Plastic Moment (k‐ft) = 4129Ultimate Curvature (1/in) = .001188

Axial Load 1507 kips

40004500

2000250030003500

0500

100015002000

00 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014

Curvature (1/in)

82

Calculate Elastic Stiffness “Ki” for Columns 1, 2 and 3 based on effective EIand 3 based on effective EI

Use EI = and K =Plastic MomentYield Curvature

3EIL3

Watch for units!!Yield Curvature L3

Column EI (k-ft2) "Ki" (k/ft)#1 2377604 325#2 2498718 342#2 2498718 342#3 2606692 356

83

Calculate Plastic Shear Vpi and the Yield Deflections for Columns 1 2 and 3Deflections for Columns 1, 2 and 3

Use V = and Δ =Mp VpiUse Vpi = and Δyi =pL

pKi

Col mn V (kips) (in)Column Vpi (kips) Δyi (in)#1 130.4 4.8#1 130.4 4.8#2 139.2 4.9#3 147 5 5#3 147.5 5

84

Calculate Plastic Hinge Lengthg g

= 08L+6fy db p

= .08L+640

db

08 x 28' x 12 in/ft + 6 x x 1 41"68

p =.08 x 28' x 12 in/ft + 6 x x 1.41"

= 26.9" + 14.4" = 41.3"

6840

85

Calculate Plastic Displacement Δp for all Three Col mnsall Three Columns

Use θ = (φ – φ ) x l Δ = θ (L – /2)Use θp = (φu – φy) x lp , Δp = θp (L – p/2)

C l ( d) (i )Column φu – φy θp (rad) Δp (in)#1 0.00128 0.0529 16.7#2 0.00116 0.0479 15.1#3 0.00106 0.0438 13.8#3 0.00106 0.0438 13.8

86

Construct Capacity Load‐Deflection DiagramDeflection Diagram

500

Δcapacity = 18.8 in 450417

415 400409

350

ps)

18.8

300

250

200Tota

l She

ar (k

ip150

100

50

T

50

02 4 6 8 10 12 14 16 18 20

deflection (in)

5.04.8

4.9

87

Calculate Shear Demand to Capacity Ratio f G i C lfor Governing Column

If shear D/C ratios < 1 than the frame displacement If shear D/C ratios 1 than the frame displacement capacity is governed by flexural deformation.If shear D/C ratios > 1 then the frame displacement / pcapacity should be revised to reflect the fact that shear is governing.

88

Example: Pushover Analysis of a slender two‐column bent

W=600 kips W=600 kips

Long. Reinforcement#10 tot 68Ratio = 2.525% P, Δ

W=600 kips W=600 kips

=5.5

’

3 ft

D=

2” CL

45.3

Spiral transverseReinforcement #[email protected]”

fc’=4 ksify = 60 ksi 16ft

89

Variations in axial load18000

14000

16000

18000P=2000 kips

P=600 kipsP= 0 kips

10000

12000

kips

-ft)

P= -800 kips

P= -1800 kips

6000

8000

Mom

ent (

k

P(kips) Mp(k-ft) φ y(1/ft) (EI)eff K(k/ft)

-1800 7908 0.000879 8993518 290

800 9426 0 000870 10831993 350

2000

4000-800 9426 0.000870 10831993 350

0 10530 0.000876 12026039 388

600 11330 0.000888 12759009 412

2000 13060 0 000890 14675806 47400.000 0.005 0.010 0.015 0.020 0.025

Curvature (1/ft)

2000 13060 0.000890 14675806 474

90

Variations in Long. SteelMoment (kips-ft)

200003.5%

4.0%

Moment (kips ft)Case with Axial Force P = -700 kips (Tension)

150002.5%

3.0%

100001.5%

2.0%

5000 1.0%

Using ultimate strain

00.000 0.005 0.010 0.015 0.020 0.025

Curvature (1/ft)

91

Variations in Long. SteelMoment (kips-ft) Case with Axial Force P = 1900 kips (Compression)

20000

3 0%

3.5%

4.0%

( p ) Case with Axial Force P = 1900 kips (Compression)

15000

1 5%

2.0%

2.5%3.0%

100001.0%

1.5%

0

5000 Using ultimate strain

00.000 0.005 0.010 0.015 0.020 0.025

Curvature (1/ft)

92

Force Demands on Capacity Protected Members (4.11.2)Protected Members (4.11.2)

po mo pM Mλ= (4.11.2‐1)

where:= idealized plastic moment capacity of reinforcedM idealized plastic moment capacity of reinforced

concrete member based upon expected material properties (kip‐ft)

pM

M = overstrength plastic moment capacity (kip‐ft)poM

λ = overstrength magnifier1.2 for ASTM A 706 reinforcement1.4 for ASTM A 615 Grade 60 reinforcement

moλ

93

Design Forces resulting from Plastic Hinging‐Single Column & Pier(4 11 3)Single Column & Pier(4.11.3)

Single Column Bent• Weak Direction of Pier• Weak Direction of a Bent

i h l i iSTEP 1: Determine Over-strength Plastic Moment CapacityConcrete: MPH = 1.3 MuSteel: MPH = 1.25 Mu

STEP 2: Determine Corresponding Shear Force,

VPH =MPH

top +MPHbot

hwhere:h = column height between plastic hingesNo adjustment in Axial Force

PH h

94

j

Plastic Hinging – Column I t ti DiInteraction Diagram

Center of GravityCenter of Gravity

φMuφMu

VL + VR

From columninteraction diagram

AXIAL

h L

φM VRVLAXIALLOAD MAXIMUM

PROBABLEMOMENT = φMu

PDL + PPHPDL

PDL - PPH

φMuφMu

PPH PPHa

MOMENTCOLUMN INTERACTION DIAGRAM

(NOTE: φ = 1.30 for concrete and φ = 1.25 for steel)

DL PH

95

Design Forces resulting fromPlastic Hinging Two or More Columns (4 11 4)Plastic Hinging‐Two or More Columns (4.11.4)

Bents with Two or More ColumnsSTEP 1: Determine the Overstrength Moment CapacitySTEP 2 D t i th C di Sh FSTEP 2: Determine the Corresponding Shear ForceSTEP 3: Determine the Total Shear Force in the Bent and Corresponding Overturning Axial ForcesCo espo d g O e tu g a o cesSTEP 4: Determine Revised Overstrength Moments and Shears Corresponding to Newly Calculated Axial Forces. C l S 3 U il Sh F i i hi 10% f P iCycle to Step 3 Until Shear Force is within 10% of Previous Value.

96

Capacity Protection Philosophyp y p y

Expected material propertiesOver‐strength forces with expected material and resistance factorsMaximum and minimum axial load effects

97

Properties and Applications of Reinforcing Steel, Prestressing Steel and Concrete for SDC B, C, and D (8.4)

For SDC B and C the expected materialFor SDC B and C, the expected material properties shall be used to determine the section stiffness and overstrengthsection stiffness and overstrength capacities.For SDC D the expected material propertiesFor SDC D, the expected material properties shall be used to determine section stiffness, overstrength capacities and displacementoverstrength capacities, and displacement capacities.

98

SDC B, C, and D (8.3)General‐ Initial sizing of columns should be performed using Strength and Service load combinations defined in the AASHTO LRFD Bridge Design Specifications.g g p fForce Demands on SDC B‐ The design forces shall be the lesser of the forces resulting from the over‐strength plastic hinging moment capacity or unreduced elastic seismic forceshinging moment capacity or unreduced elastic seismic forces in columns or pier walls. Force Demands on SDC C and D‐The design forces shall be b d f lti f th t th l tibased on forces resulting from the over‐strength plastic hinging moment capacity or the maximum connection capacity following the capacity design principles.Local Ductility Demands SDC D‐The local displacement ductility demands, , of members shall not exceed the maximum allowable displacement ductilities.

Dμ

99

Earthquake Resisting Systems (ERS, 3.3) q g y ( )

Required for SDC C and Db d f bl h h b dMust be identifiable within the bridge system

Shall provide a reliable and uninterrupted load thpath

Shall have energy dissipation and/or restraint to control seismically induced displacementscontrol seismically induced displacementsComposed of acceptable Earthquake Resisting Elements (ERE)Elements (ERE)

100

ERS (3.3) ( )

Permissible Earthquake Resisting S t (ERS)Systems (ERS)

101


Expected material properties Overstrength forces with expected material and resistance factorsMaximum and minimum axial load effects

102

Design for SDC B, C, & D (4.7)g f ( )Conventional – Full ductility structures with a plastic mechanism having for a bridge 0.60.4 ≤≤ Dμp g gin SDC DLimited ductility – For structures with a Plastic

Dμ

mechanism readily accessible for inspection having for a bridge in SDC B or C0.4<DμLimited Ductility – For structures having a plastic mechanism working in concert with a protective system The plastic hinge may or may not formsystem. The plastic hinge may or may not form. This strategy is intended for SDC C or D

103

Member Ductility Requirement for SDC D (4 9)SDC D (4.9)

104

Member Ductility Requirement for SDC D (4 9)

(4.9-5)SDC D (4.9)

pdΔ1

Where: yi

pdD Δ

+=1μ

= plastic displacement demand (in.)pdΔ

= idealized yield displacement corresponding to the idealized yield curvature, ,

yiΔyiφ

shown in figure 8.5-1 (in.)Pile shafts should be treated similar to columns.

y

105

Capacity Design Requirement for SDC C & D (4 11)SDC C & D (4.11)

Capacity protection is required for all membersCapacity protection is required for all members that are not participating as part of the energy dissipating systemCapacity protected members include:

SuperstructuresJoints and cap beamsSpread footingsPile capsFoundations

106

LRFD – Over‐strength Capacity Design Concepts f SDC C & D L (4 11)for SDC C & D Long. (4.11)

107

LRFD – Over‐strength Capacity Design C t f SDC C & D T (4 11)Concepts for SDC C & D Trans. (4.11)

108

Plastic Moment CapacitySDC B, C & D (8.5)

Moment‐Curvature Analyses M φMoment‐Curvature AnalysesExpected Material PropertiesAxial Dead Load Forces with Overturning

M φ−

Axial Dead Load Forces with OverturningCurve Idealized as Elastic Perfectly Plastic

El ti P ti f th C P th h th i tM φ−Elastic Portion of the Curve Pass through the point of marking the first reinforcing bar yieldPlastic moment capacity determined from equalPlastic moment capacity determined from equal areas of idealized and actual

109

Minimum Support Length Requirements SDC A, B, C & D (4.12)SDC A, B, C & D (4.12)

110

Plastic Moment Capacity for Ductile Concrete b f S C C d (8 )Members for SDC B, C, and D (8.5)

Equal Areas

Figure 8.5‐1 Moment‐Curvature Model

111

Force Demands on Capacity Protected Members (8 5)Members (8.5)

(8.5‐1)M Mλ=where:

= idealized plastic moment capacity of reinforced

po mo pM Mλ

M = idealized plastic moment capacity of reinforced concrete member based upon expected material properties (kip‐ft)

pM

M = over‐strength plastic moment capacity (kip‐ft)poM

λ = over‐strength magnifier1.2 for ASTM A 706 reinforcement1.4 for ASTM A 615 Grade 60 reinforcement

moλ

112

Shear Demand & Capacity (8.6.1)Shear Demand & Capacity (8.6.1)

SDC B is the lesser of :VSDC B is the lesser of :Force obtained from linear elastic seismic analysisForce, , corresponding to plastic hinging with over‐

uV

VForce, , corresponding to plastic hinging with overstrength

SDC C and D is the force associated with the

poV

uVover‐strength moment poM

u

113


Expected material properties Over‐strength forces with expected material and resistance factorsMaximum and minimum axial load effects

114

Max Axial Load in Ductile Member SDC C & D (Sec 8 7 2)SDC C & D (Sec 8.7.2)

115

Max Axial Load in Ductile Member ( )SDC C & D (Sec 8.7.2)

A higher axial load value, Pu, may be used provided that a moment-curvature pushover analysis isthat a moment-curvature pushover analysis is performed to compute the maximum ductility capacity of the membercapacity of the member.

116

Max. Longitudinal Reinforcement (Sec 8 8 1)(Sec 8.8.1)

117

Min. Longitudinal Reinforcement For Col mns (Sec 8 8 2)For Columns (Sec 8.8.2)

118

Min. Longitudinal Reinforcement For Pier Walls (Sec 8 8 2)For Pier Walls (Sec 8.8.2)

119

Concrete Shear Capacity SDC B and C (8.6.2)

For SDC B the concrete shear capacity of aVFor SDC B, the concrete shear capacity, , of a section within the plastic hinge region shall be determined using:

cV

2μ =g 2Dμ =

For SDC C, the concrete shear capacity of a section within the plastic hinge region shall be determined p g gusing: 3Dμ =

120

Concrete Shear Capacity SDC B, C & D (8.6.2)

Concrete Shear Capacity V within the plastic hingeConcrete Shear Capacity, Vc within the plastic hinge

SDC EquationμSDC Equation

B 2 -

Dμ

B 2 -

C 3 -

D 4.9-5yi

pd

Δ

Δ+1

121

Lateral Reinf. in Plastic Hinges SDC C & D (Sec 8 8 7)C & D (Sec 8.8.7)

The volume of lateral reinforcement, ρs or ρw, specified in Article 8.6.2 provided inside the plastic hinge region as specified in Article 4.11.7 shall be sufficient to ensure that the column or pier wall has adequate shear capacity and confinement le el to achie e the req ired d ctilitand confinement level to achieve the required ductility capacity.

122

Lateral Reinf. in Plastic Hinges SDC C & D (Sec 8 8 7)C & D (Sec 8.8.7)

F l d i d t hi di l t d tilit• For columns designed to achieve a displacement ductility demand greater than 4, the lateral reinforcement shall be either butt-welded hoops or spirals.

• Combination of hoops and spiral shall not be permitted except in the footing or the bent cap. Hoops may be placed around the

l i li f i i l i f i hcolumn cage in lieu of continuous spiral reinforcement in the cap and footing.

At i l h t i l di ti iti th i l h ll• At spiral or hoop to spiral discontinuities, the spiral shall terminate with one extra turn plus a tail equal to the cage diameter.

123

Lateral Reinf. outside Plastic Hinges SDC C & D (Sec 8 8 8)SDC C & D (Sec 8.8.8)

• The volumetric ratio of lateral reinforcement requiredThe volumetric ratio of lateral reinforcement requiredoutside of the plastic hinge region shall not be less than50% of that determined in accordance with Articles 8.8.7and Article 8.6.

• The lateral reinforcement type outside the plastic hingeyp p gregion shall be the same type as that used inside theplastic hinge region.

124

Lateral Reinf. outside Plastic Hinges SDC C & D (Sec 8 8 8)SDC C & D (Sec 8.8.8)

• At spiral or hoop to spiral discontinuities, splices shall beprovided that are capable of developing at least 125% of thespecified minimum yield stress, fyh, of the reinforcing bar.

• Lateral reinforcement shall extend into footings to thebeginning of the longitudinal bar bend above the bottom mat.

• Lateral reinforcement shall extend into bent caps a distance which is as far as is practical and adequate to develop the reinforcement for development of plastic hinge mechanisms.

125

Lateral Reinforcement in Plastic Hinge Region (4 11 7)Region (4.11.7)

Plastic Hinge Region-the maximum of:maximum of:

•1.5 sectional dimension in the direction of bending

M t d d d•Moment demand exceeds 75% of the maximum plastic moment

A l ti l l ti hi•Analytical plastic hinge length

126

REQUIREMENTS FOR CAPACITY PROTECTED MEMBERS (8 9)PROTECTED MEMBERS (8.9)

Capacity‐protected members such as footings, p y p gbent caps, oversized pile shafts, joints, and integral superstructure elements that are adjacent to the plastic hinge locations shall be designed to remain p g gessentially elastic when the plastic hinge reaches its over‐strength moment capacity, Mpo.The expected nominal capacity M is used inThe expected nominal capacity, Mne, is used in establishing the capacity of essentially elastic members and should be determined based on a strain compatibilit anal sis sing a φMstrain compatibility analysis using a diagram as illustrated in Figure 8.5‐1 and outlined in Article 8.5.

φ−M

127

Superstructure Capacity for Integral Caps for L Di ti f SDC C d D (8 10)Long. Direction for SDC C and D (8.10)

Moment demand caused by dead load or secondary preMoment demand caused by dead load or secondary pre‐stress effects shall be distributed to the entire width of the superstructure. Th l h M i ddi i hThe column over‐strength moment, Mpo, in addition to the moment induced due to the eccentricity between the plastic hinge location and the center of gravity of the

h ll b di ib d h f i isuperstructure shall be distributed to the spans framing into the bent based on their stiffness distribution factors. This moment demand shall be considered within the effective width of the superstructure.

128

Superstructure Capacity Design for Transverse

Direction (Integral Bent Cap) for SDC C and DDirection (Integral Bent Cap) for SDC C and D

(8.11)Bent caps are considered integral if they terminate at the outside of the exterior girder and respond monolithically with the girder system during dynamic excitation M d d d b d d l d dMoment demand caused by dead load or secondary prestress effects shall be distributed to the effective width of the bent cap Bwidth of the bent cap, Beff

131

Superstructure Capacity Design for Transverse Direction (Integral Bent Cap) for

SDC C and D (8.11)The column over‐strength moment, Mpo, and the moment induced due to the eccentricity between the plastic hingeinduced due to the eccentricity between the plastic hinge location and the center of gravity of the bent cap shall be distributed based on the effective stiffness characteristics of the frame The moment shall be considered within thethe frame. The moment shall be considered within the effective width of the bent cap. The effective width, Beff, shall be taken:

132

Integral B t CBent Cap (8.11) ( )

133

Superstructure Design for Non‐integral Bent Caps for SDC C and D (8.12)

Non‐integral bent caps shall satisfy all requirements stated g p y qfor frames with integral bent cap in the transverse direction.For superstructure to substructure connections that areFor superstructure to substructure connections that are not intended to fuse, a lateral force transfer mechanism shall be provided at the interface that is capable of transferring the maximum lateral force associated withtransferring the maximum lateral force associated with plastic hinging of the ERS For superstructure to substructure connections that are intended to fuse the minimum lateral force at the interfaceintended to fuse, the minimum lateral force at the interface shall be taken as 0.40 times the dead load reaction plus the overstrength shear key(s) capacity, Vok

134

Joint Design for SDC C and D (8.13)g f ( )Moment resisting connections shall be designed to transmit the maximum forces produced when the column has reached its overstrength capacity, MpoMoment‐resisting joints shall be proportioned so that the principal stresses satisfy the requirementsthat the principal stresses satisfy the requirements of Eq. 1 and Eq. 2.Transverse reinforcement in the form of tied l i f t i l hcolumn reinforcement, spirals, hoops, or

intersecting spirals or hoops shall be provided. The joint shear reinforcement may also be provided in h f f l l ithe form of column transverse steel or exterior transverse reinforcement continued into the bent cap.

135

Joint Design for SDC C and D (Continued)(8 13)(Continued)(8.13)

Where the principal tension stress in the joint, pt, p p j ptas specified in Article 8.13.2 is less than , the transverse reinforcement in the joint, rs, shall satisfy Eq. 1 and no additional reinforcement

cf ′11.0

y qwithin the joint is required.Where the principal tension stress in the joint, pt, is greater than then transversef ′110is greater than , then transverse reinforcement in the joint, rs, shall satisfy Eq. 2 and additional joint reinforcement is required as indicated in Article 8 13 4 for integral bent cap

cf11.0

indicated in Article 8.13.4 for integral bent cap beams or Article 8.13.5 for non‐integral bent cap beams.

136

Joint Performance (C8.13.1)f ( )A “rational” design is required for jointis required for joint reinforcement when principal tension stress levels becomestress levels become excessive. The amounts of reinforcementreinforcement required are based on a strut and tie mechanism similarmechanism similar to that shown.

137

Joint Proportioning (C8.13.2)p g ( )The figure illustrates theillustrates the forces acting on the joint as jwell as the associated principal stresses.

138

Column Shear Key Design for SDC C and D (8 15)and D (8.15)

• Column shear keys shall be designed for the axial andColumn shear keys shall be designed for the axial andshear forces associated with the column’s over-strengthmoment capacity, Mpo, including the effects ofpooverturning. The key reinforcement shall be located asclose to the center of the column as possible to minimized l i f l i hi h k i fdeveloping a force couple within the key reinforcement.

• Moment generated by the key reinforcing steel should be considered in applying capacity design principles.

139

Influence of Column Base Detail on the Contribution of Steel Shear Resisting Mechanism

140

P‐Δ Requirementsq

Displacement check for column design

141

Retrofit Evaluation Method Cf

Plastic curvature capacity determinationImplicit deflection capacity for poorly detailed elements

146

Retrofit Evaluation Method Cf

Plastic curvature capacity determinationImplicit deflection capacity for poorly detailed elements

149

Learning OutcomesExplain the Displacement Capacity Design ApproachDefine full and limited ductile response and explain the correlation to the SDC’sDefine implicit and pushover displacement capacityDefine expected and over strength material properties. Define the Component Capacity/Demand Retrofit Method (Method C)

158

Displacement Capacity Design Fundamentals - Buffalo deflection generation –Pushover analysis...

Documents

Transcript of Displacement Capacity Design Fundamentals - Buffalo deflection generation –Pushover analysis...