Seismic Pushover Analysis

(1)

Seismic'Pushover'Analysis:'Using'

AASHTO'Guide&Speciﬁca,ons&for&

LRFD&Seismic&Bridge&Design!

Michael!D.!Keever,!California!Department!of!Transporta8on! Elmer!E.!Marx,!Alaska!Department!of!Transporta8on!and!Public!Facili8es! WenChuei!Phillip!Yen,!Federal!Highway!Administra8on! Jeﬀrey!Ger,!Federal!Highway!Administra8on! !

TRB!AFF!50!–!Seismic!Design!

1!

(2)

Part!1!Overview!

• Fundamentals,!hand!methods,!simple!computer!

approaches!for!AASHTO'Guide&Speciﬁca,ons&for&

LRFD&Seismic&Bridge&Design&(SGS)!methodology!

!

• Workshop!focus!is!pushover!analysis!not!modeling!

and!seismic!/!capacity!design!

!

2!

(3)

Part!1!Overview!

• Flexural!mechanics!

• Genera8on!of!MomentCCurvature!(!M"#"

φ

")!

• Material!models!and!failure!strains!

• Hand!checking!M"#"

φ"

• Analy8cal!plas8c!hinge!length,!L

_p"

• ForceCdisplacement!example!

• Hand!checking!forceCdisplacement"

• Implicit!SGS!displacement!equa8ons!

• Design!example!1!–!Two!circular!column!bent!

• Other!design!considera8ons!

• Design!example!2!–!Two!square!column!bent!

3!

(4)

But!Before!We!Begin!

• _Name!

• _Organiza8on!

• _{Primary!job!func8on!(design,!manage,!etc.)!}

• _{Seismic!design!experience!}

• _{Workshop!expecta8ons!and!objec8ves!}

4!

(5)

Why!Pushover?!

• SGS!is!primarily!a!displacement!based!approach!

!

• More!ra8onal!approach!than!forceCbased!

method!of!AASHTO!LRFD!

• Seismic!Design!Category!D!(SDC!D)!requires!

pushover!analysis!

• Provides!for!a!beaer!understanding!of!bridge!

response!and!behavior!

! 5!

(6)

Flexural!Mechanics!

• _{Rela8onship!between!force,!shear,!moment,!}

curvature,!slope!and!deﬂec8on!

!

– 

_Integra-on"

– 

_Moment!area!

– 

_{Energy!methods!}

– 

_{S8ﬀness!methods!}

!

! 6!

(7)

Force!C!Displacement!

∫

⋅

=

P

dx

V

∫

⋅

=

V

dx

M

load

P =

7!

(8)

Force!C!Displacement!

∫

⋅

=

V

dx

M

EI

M

=

ϕ

8!

(9)

Force!C!Displacement!

I

E

M

⋅

=

ϕ

∫

⋅

=

ϕ

dx

θ

∫

⋅

=

Δ

θ

dx

9!

(10)

Example!C!Can8lever!

P

_{V =}

_P

L

P

M

_o

₌

_⋅

I

E

L

P

I

E

M

_o o

⋅

=

⋅

=

ϕ

2 L

o L

⋅

=

ϕ

θ

3

2

L

o L

⋅

=

Δ

ϕ

L

10!

(11)

Compa8bility!

• BernoulliCEuler!assump8on!–!sec8ons!that!are!plane!

(linear)!before!bending!remain!plane!ager!bending!

!

• Perfect!bond!between!the!steel!reinforcing!bars!and!

surrounding!concrete!

!

• _{Material!(cons8tu8ve!σCε)!models!are!}

representa8ve!of!actual!material!response!

(conﬁnement,!strain!hardening,!buckling,!spalling,!

cracking,!creep,!strainCrate,!shear,!etc.)!

!

! 11!

(12)

Moment!–!Curvature!(!M"#"

φ

)!

c

Park and Paulay 1975

d

(13)

Moment!–!Curvature!(!M"#"

φ

)!

•  Curvature,!φ,!is!calculated!as:!

" "φ ="ε_c"/"c"="ε_t"/"(D/2"–"c)""

(14)

Moment!–!Curvature!(!M"#"

φ

)!

•  From!equilibrium:! P!=!F_SC!+!F_CC!+!F_CU!–!F_ST" •  Summing!moments!about!the!neutral!axis! M!=!CG_SC*F_SC!+!CG_CC*F_CC!+!CG_CU*F_CU!+!CG_ST*F_ST!+!P*(D/2!–!c)! 14!

(15)

Moment!–!Curvature!(!M"#"

φ

)!

(16)

Moment!–!Curvature!(!M"#"

φ

)!

SGS 2011 Predicted Response Elastic perfectly-plastic used in SGS 16!

(17)

Moment!–!Curvature!(!M"#"

φ

)!

• Actual!rela8onship!versus!idealized!

• First"yield"Moment!and!Curvature,!M

_y"

and!

φ

_y"

• Eﬀec8ve!s8ﬀness,!E

_ce

*!I

_eﬀ"

=!M

_y"

/!

φ

_y"

• Idealized"yield!Moment!and!Curvature,!M

_p"

and"

φ

_yi"

• Expected!nominal!moment,!M

_ne"

at!

ε

_c

"=!0.003"

• Balance!the!area!above!and!below!the!curve!

(18)

SGS!Material!Models!

• Concrete:!Mander!et!al!!model!for!unconﬁned!

and!conﬁned!condi8on!

!

• Reinforcing"steel:!elas8c!C!perfectly!plas8c!with!

strain!hardening!

!

• Other:!prestressing!steel!(see!SGS)!

!

• Any!func8on!/!model!that!accurately!captures!

the!material!

σ

#

ε

!rela8onship!is!acceptable!

!

! 18!

(19)

• _{StressCstrain!rela8onship!for!concrete!}

!

where:!

SGS!Material!Models!C!Concrete!

r cc c x r xr f f + − = 1 ' ! ! " # $ $ % & − − + = 2.254 1 7.94 2 _' 1.254 ' ' ' ' ' ce l ce l ce cc f f f f f f cc c x

ε

= sec E E E r ce ce − =

(20)

SGS!Material!Models!C!Concrete!

! " # $ % & '' ( ) ** + , − + = 0.002 1 5 _' 1 ' ce cc cc f f

ε

' 1900 _ce ce f E = cc cc f E

ε

' sec =

(con8nue)!

(21)

• _{Conﬁnement!of!circular!columns!}

SGS!Material!Models!C!Concrete!

s D A_sp s _' 4 =

ρ

l l e l K f f f ' ₌ _≈ 0.95 2 2 ' yh s yh sp l f s D f A f ₌ ₌

ρ

(22)

• _{Conﬁnement!of!rectangular!columns!}

SGS!Material!Models!C!Concrete!

(

)

2 ) ( 8 . 0 2 ' e x y yh x y yh l f f K f _≈

ρ

+

ρ

_≈

ρ

+

ρ

c sx x sH A =

ρ

c sy y sB A =

ρ

(23)

SGS!Material!Models!C!Concrete!

ε_c!=!strain!in!concrete!(IN/IN)! f_c!=!stress!in!concrete!corresponding!to!strain!ε_c!(KSI)! f _ce!=!expected!nominal!compressive!strength!(KSI)! f_yh!=!nominal!yield!stress!of!transverse!reinforcing!(KSI)! εR_su!=!reduced!ul8mate!tensile!strain!of!transverse!bars!(IN/IN! A_sx!=!total!area!of!transverse!bars!in!the! x !axis!(IN2_)! A_sy!=!total!area!of!transverse!bars!in!the! y !axis!(IN2_)! A_sp!=!area!of!hoop!/spiral!bar!for!circular!column!(IN2_)! H_c!=!conﬁned!core!dimension!in!the! y !axis!(IN)! B_c!=!conﬁned!core!dimension!in!the! x !axis!(IN)! D !=!diameter!of!spiral!or!hoop!for!circular!column!(IN)! s!=!pitch!of!spiral!or!spacing!of!hoops!or!8es!(IN)!

(24)

SGS!Material!Models!C!Concrete!

(25)

• _{Eﬀec8ve!conﬁnement!of!circular!columns!}

!

• _{Eﬀec8ve!conﬁnement!of!rectangular!columns!}

SGS!Material!Models!C!Concrete!

cc c c N i d c i e d s b s d b w K

ρ

− ## $ % && ' ( − ## $ % && ' ( − ) * + , -. − =

∑

= 1 2 ' 1 2 ' 1 6 ) ( 1 1 2 ' cc n e D s K

ρ

− # $ % & ' ( − = 1 ' 2 ' 1 Chen 2003 25!

(26)

SGS!Material!Models!C!Concrete!

s’!=!clear!distance!between!transverse!bars!=!s!–!d_bh"!(IN)! d_bh!=!diameter!of!transverse!reinforcing!bar!(IN)! D’!=!centerline!diameter!of!hoop!or!spiral!(IN)! ρ_cc!=!A_st!/!A_cc" A_st!=!total!area!of!longitudinal!reinforcement!(IN2_)! A_cc!=!area!of!confined!concrete!core!(IN2_)! n!=!1!for!con8nuous!spiral! n!=!2!for!individual!hoops!! w_i’!=!clear!distance!between!adjacent!8ed!longitudinal!bars(IN)! b_d!=!confined!core!dimension!in!the!longer!direc8on(IN)! d_c!=!confined!core!dimension!in!the!shorter!direc8on!(IN)! N!=!number!of!spaces!between!longitudinal!bars! Chen 2003 26!

(27)

SGS!Failure!Strain!C!Concrete!

• Conﬁned!concrete!crushing!strain!limit,!

ε

_cu"

!

"

!

where:!

"ρ

_s

!=!transverse!reinforcement!ra8o!(

ρ

_x

+

ρ

_y

!for!rect.)!

"f

_yh

"=!nominal!yield!stress!of!transverse!steel!

"ε

_su

"=!rupture!strain!of!transverse!steel:!use!

ε

R_su!

=!0.09

_!

"f

_cc

!=!conﬁned!concrete!compressive!stress!

02 . 0 4 . 1 004 . 0 ₊ _' _< = cc su yh s cu f f

_ε

ρ

ε

important!

(28)

SGS!Material!Models!C!Concrete!

• _{StressCstrain!curves!for!concrete!}

(29)

SGS!Material!Models!C!Concrete!

• _{StressCstrain!curves!for!concrete!}

!

• _Limit!

_ε

_cu

_{!<!0.02!for!design!purposes!}

Spalling! strain,!ε_sp,!!not! to!exceed! 0.005! Conﬁned!concrete! crushing!strain,!ε_cu,!! cri8cal!component!

ε

_cu 29!

(30)

SGS!Material!Models!C!Steel!

• _{Actual!stressCstrain!curve!for!reinforcing!steel!}

(31)

SGS!Material!Models!C!Steel!

• _{Design!stressCstrain!curve!for!reinforcing!steel!}

(32)

SGS!Material!Models!C!Steel!

• _{ASTM!A!706!Grade!60!v.!ASTM!A!615!Grade!60!}

(33)

SGS!Material!Models!C!Steel!

• _{StressCstrain!rela8onship!for!reinforcing!steel!}

ye s

ε

≤ sh s ye

ε

≤ ≤ R su s sh

ε

≤ ≤ s s s E f ₌

_ε

ye s f f = ! ! " # $ $ % & '' ( ) ** + , − − − − = 2 ) ( 1 sh su s su ye ue ue s f f f f

ε

(34)

SGS!Failure!Strains!C!Steel!

• _{Reinforcing!steel!failure!strain,!}

_ε

R

su

!!

Property Notation Bar Size ASTM A706 ASTM A615 Grade ₆₀

Specified minimum yield stress (ksi) f_y #3 - #18 60 60

Expected yield stress (ksi) f_ye #3 - #18 68 68

Expected tensile strength (ksi) f_ue #3 - #18 95 95

Expected yield strain εye #3 - #18 0.0023 0.0023

Onset of strain hardening #3 - #8 0.0150 0.0150

#9 0.0125 0.0125

εsh #10 - #11 0.0115 0.0115

#14 0.0075 0.0075

#18 0.0050 0.0050

Reduced ultimate tensile strain #4 - #10 0.090 0.060

#11 - #18 0.060 0.040

Ultimate tensile strain εsu #4 - #10 0.120 0.090

#11 - #18 0.090 0.060

R su ε

(35)

Moment!–!Curvature!(!M"#"

φ

)!

• QuasiCsta8c!cons8tu8ve!models!for!cyclic!response"

Kowalsky et al. 2010

(36)

• Find!the!M#

φ

!for!the!column!shown!below!

• ASTM!A!706!Grade!60 !

!!f

_c

!=!4!KSI"

Moment!–!Curvature!Example!

D = 48 IN L =H o = 20 F T #5 hoop @ 4IN 20 #11 2 IN clr. 940 KIP 36!

(37)

Moment!–!Curvature!Example!

• Diameter,!D!=!48!IN!

• Gross!Area,!A

_g

!=!πD*

2

_{/4!=!1810!IN^2!}

• 20!#11!!!A

_st

"=!201.56!=!31.2!IN^2!*

• _ρ

_l

!=!A

_st

!/!A

_g

"=!0.01724!=!1.724%!

• #5!spiral!(d

_sp

!=!0.625!IN!and!A

_sp!

=!0.31!IN^2)!!

• Hoop!spacing!=!4!IN!pitch!(s!=!4!IN)!

• Clear!cover,!cov!=!2!IN!over!transverse!bars!

• Core!diameter,!D =!D!–!2cov!–!d*

_sp"

=!43.375!IN!

• _ρ

_s

"=!4A*

_sp

!/!(D !s)!=!0.00715!=!0.715%!*

• ASTM!A!706!Grade!60!so!f

_ye

!=!68!KSI,!f

_ue

!=!95!KSI!

• f

_c

!=!4!KSI!so!f

_ce

!=!5.2!KSI!

(38)

Moment!–!Curvature!Example!

• Conﬁned!concrete!crushing!strain!limit,!

ε

_cu""

"

""

!

where:!

"ρ

_s

!=!4A*

_sp

/(D s)!=!0.00715!*

"f

_l"

=!K

_e

*

ρ

_s

*f

_yh

/2!~!0.950.0071560/2!=!0.204!KSI!

"f

_yh

"=!60!KSI!(use!nominal!for!horizontal!steel)!

"ε

_su

"=!

ε

R_su"

=!0.09!for!#5!hoops!

!

• Spreadsheet"demo"then"check"with"commercial"

!

01232 . 0 4 . 1 004 . 0 ₊ _' ₌ = cc su yh s cu f f

_ε

ρ

ε

49 . 6 254 . 1 2 94 . 7 1 254 . 2 _' ' ' ' ' ' = ! ! " # $ $ % & − − + = ce l ce l ce cc f f f f f f 38!

(39)

Moment!–!Curvature!Example!

• _{Spreadsheet!summary!}

(40)

Moment!–!Curvature!Example!

•  How!is!idealized!M"#"φ!rela8onship!calculated?!

!

•  Elas8cCperfectly!plas8c!rela8onship!is!deﬁned!as:! !

M_i!=!idealized!moment!values!=!minimum!of!E_ceI_eﬀ*φ!or!M_p"

M_p"=!idealized!plas8c!moment! φ!!=!actual!calculated!curvature!from!M#φ!results!! E_ceI_eff!=!effec8ve!s8ffness!at!first!yield!=!M_y!/!φ_y! Δ(Mφ)!=!difference!in!actual!and!idealized!=!(ΔM)*(Δφ) _!! ! ! Σ!Δ(Mφ)!=!sum!of!difference!=!0!by!changing!M_p"! ! •  Spreadsheet"demo"then"check"with"commercial! ! 40!

(41)

Moment!–!Curvature!Example!

• _{Idealized!M"#"}

_φ

_{!rela8onship!(balanced!area)!}

(42)

Moment!–!Curvature!Example!

(43)

Moment!–!Curvature!Example!

P = 940 K

φyi = 0.0001109 1/IN 2%

φu = 0.001073 1/IN 0.2%

M_p = 50460 K-IN 2%

E_ceI_eff = 454.85E6 K-IN2 _0.3%

Comparison"

φyi = 0.0001132 1/IN φu = 0.001075 1/IN

M_p = 51626 K-IN

E_ceI_eff = 456.3E6 K-IN2

(44)

Approximate!Methods!

• the!use!of!curvature!(!

φ

)!in!design!is!uncommon!

!

• don t!have!a!good! feel !for!curvature!values!

!

• simple!method!to!check!the!computer!results!

!

• can!help!with!itera8ve!processes!

!

• use"these"approxima-ons"with"due"skep-cism!

! 44!

(45)

Approximate!Methods!C!

φ

_yi

"

• For!a!rough!check!of!conven8onal!circular!

reinforced!concrete!column!sec8ons:

! ! ! !φ_yi"~!2.25*ε_ye/12B_o!~!1/2300B_o!! " " " φ_yi"~!2.25*ε_ye/D_!~!1/190D!! ! where:! ! ! &φ_yi"=!idealized!yield!curvature!(1/IN)_! ! ! !B_o"=!column!diameter!(FT)! " " "D"=!column!diameter!(IN)! ! ! !ε_ye"=!expected!yield!strain!~!0.002345!(IN/IN)!! ! 45!

(46)

Approximate!Methods!C!

φ

_yi

"

• For!a!rough!check!of!conven8onal!rectangular!

reinforced!concrete!column!sec8ons:

! ! ! ! φ_yi"~!2.1*ε_ye/12B_o"~!1/2500B_o!! ! where:! ! ! &φ_yi"=!idealized!yield!curvature!(1/IN)_! ! ! !B_o"=!column!width!in!loaded!direc8on!(FT)! ! ! !ε_ye"=!expected!yield!strain!~!0.002345!(IN/IN)!! ! 46!

(47)

Approximate!Methods!C!

φ

_u

"

• And!for!a!very"rough"check"of!conven8onal!

circular!or!rectangular!reinforced!concrete!

column!sec8ons!with!good!conﬁnement:

! ! ! !φ_u!=!min!(ε_cu/c_c"",!!ε_suR/d#c)!!~!ε_suR"/12B_o! where:! ! &φ_u!=!ul8mate!curvature!(1/IN)! ! ! !ε_cu"=!ul8mate!conﬁned!concrete!strain!(1/IN)! ! ! !ε_suR"=!reduced!ul8mate!tensile!strain!(IN/IN)_! ! ! !c_c"=!neutral!axis!to!edge!of!conﬁned!core!(IN)! ! ! !d#c"=!neutral!axis!to!extreme!tension!bar!(IN)! ! ! !B_o"=!column!diameter!/!width!(FT)! 47!

(48)

Eﬀec8ve!S8ﬀness!C!E

_ce

*I

_eﬀ"

" E_ce*I_eff"=!M_y"/"φ_y" where:!! !E_ce!=!Expected!modulus!of!elas8city!for!concrete! !M_y!=!moment!at!first!yield!(expected!materials!at!first!yield)! !φ_y!=!curvature!at!first!yield!(expected!materials!at!first!yield)! ! So,! I_eff"=!M_y"/("φ_y"*E_ce)" And!typically,!!

!0.7!<!I_eﬀ!/!I_g!!<!0.3!

(49)

Approximate!Methods!–!I

_eﬀ"

/!I

_g"

!

! I_eﬀ/I_g!~!0.2!+!0.1*ρ_l"+!0.5*(P/f _ce*A_g)! where:!! ! !ρ_l!=!longitudinal!reinforcement!ra8o!in!%!

! !!!!!=!A_st!/!A_g"*!100!<!3%!prac8cal!limit! ! !P!!=!axial!load!–!keep'below'0.2*A_g*f _ce&

"

(50)

Approximate!Methods!C!M

_p"

M_p"~"D3_{"*"[0.05!+!0.2*}_ρ l"+!P/(f ce*Ag)]! where:!" " "M_p"!=!idealized!plas8c!moment!for!circular!sec8on!(KCIN)!! ! !D!!!!!=!column!diameter!(IN)!!>!30!IN! ! !ρ_l!!!!=!longitudinal!reinforcement!ra8o!in!%!

! !!!!!!!!=!A_st!/!A_g"*!100!!<!4%!but!3%!is!prac8cal!limit! ! !P!!!!!=!axial!load!on!column!(K)!!!!<!!0.2!*"f _ce*"A_g! ! !f _ce!!=!expected!concrete!strength!(KSI)! ! !A_g!!!!=!gross!area!of!column!(IN^2)! ! •  Assumes!!ASTM!A!706!Grade!60!reinforcing!steel!and!f _ce!=!5.2! " 50!

(51)

Approximate!Methods!C!M

_p"

M_p"~"b*d2_{"*"[0.15!+!0.25*}_ρ l"+!1.5*P/(f ce*Ag)]! where:!" " "M_p"!=!idealized!plas8c!moment!for!rectangular!sec8on!(KCIN)!! ! !d!!!!!=!column!depth!in!direc8on!of!loading!(IN)!!>!30!IN! ! !b!!!!!=!column!width!(IN)!!>!30!IN!

! !ρ_l!!!!=!A_st!/!A_g"*!100!!=!longitudinal!reinforcement!ra8o!in!%!

! !P!!!!!=!axial!load!on!column!(K)!!!!<!!0.2!*"f _ce*"A_g! ! !f _ce!!=!expected!concrete!strength!(KSI)! ! !A_g!!!!=!b!*!d!=!gross!area!of!column!(IN^2)! ! •  Assumes!!ASTM!A!706!Grade!60!reinforcing!steel!and!f _ce!=!5.2! " 51!

(52)

Moment!–!Curvature!Check!

• Diameter,!D!=!48!IN!

• I

_g

!=!D

4

_{π/64!=!260576!IN}

4!

• _ρ

_l

!=!A

_st

!/!A

_g

"=!0.01724!=!1.724%!

• Axial!Load!Ra8o,!ALR!=!P!/!(f

_ce

*A

_g

)!=!0.1!![P"~!940!K]!

• _ε

_ye

!=!f

_ye

/E

_s

"=!68/29000!=!0.002345!!!!!!

ε

R_su

"=!0.06!(#11)!

φ

_yi

"~2.250.00234/48!=!0.000110!1/IN!v.!*

0.0001132"1/IN"

φ

_u"

~!0.06/48!=!0.00125!1/IN!v.!

0.001075"1/IN"

I

_eﬀ

!~!(0.2+.11.724+0.50.1)260576!=!0.42260576!

"

"I

_eﬀ

!~!109963!IN

4

_!v.!

_105307"IN

4""""""""""

_4%"

M

_p

!~!48

3

_{(0.05+0.2*1.724+0.1)!}

"

"M

_p

!~!54713!KCIN!v.!

51626"K#IN""""""

6%

!

(53)

Moment!–!Curvature!Check!

Predicted Response Approximate Method Idealized Response XTRACT Response 53!

(54)

• Calculate!deﬂec8ons!using!MC

φ

!result!

• Elas8c!deforma8on!component,!

Δ

_yi"

• Plas8c!deforma8on!component,!

Δ

_p"

• Founda-on"deforma-on"component"–"important"

but"not"speciﬁcally"addressed"in"this"workshop"

• Conserva8ve!to!neglect!shear!deforma8ons"

!

Force!C!Displacement

_"

54!

(55)

Force!C!Displacement

_"

Park and Paulay 1975

Strain penetration

(56)

Force!C!Displacement

_"

(57)

Force!C!Displacement

_"

Caltrans SDC Version 1.7

(58)

Analy8cal!Plas8c!Hinge!Length!C"L

_p"

• Approxima8on!used!to!simplify!analysis!

!

• Converts!(integrates)!curvature!to!rota8on!!

!

• Includes!a!moment!gradient!part!(integra8on)!

and!a!strain!penetra8on!part!(yielding!into!cap,!

foo8ng!or!shag)!

!

• Calibrated!to!the!failure!condi8on!only!and!

modiﬁca8on!may!be!needed!for!full!strainC

displacement!response!

! 58!

(59)

Analy8cal!Plas8c!Hinge!Length!C"L

_p"

•  Researchers!have!proposed!a!plas8c!hinge!length!mechanism! and!a!recommended!ε_cu! ! Baker: ! !L_p!=!0.33*(L/D)*c" " Maaock: ! !L_p=0.5*D+0.05*L "" " Sawyer: ! !L_p=0.25*D+0.075*L" " •  The!SGS!(and!others)!uses!an!analy8cal!plas8c!hinge!length! based!upon!the!work!of!Dr.!Priestley" 59!

(60)

Analy8cal!Plas8c!Hinge!Length!C"L

_p"

L

_p

!=!k!!L!+!L*

_sp

!>!2!!L*

_sp"

!

L

_p

!=!0.08!!L!+!0.15!!f

_ye

!!d*

_bl

!>!0.3!f

_ye

!!d*

_bl

!

_" !

where:!

!

k!=!0.2(f*

_ue

/f

_ye

!–!1)!≤!0.08!

!L!=!length!of!column!from!point!of!maximum!moment!!

! !!!!!!!to!the!point!of!moment!contraCﬂexure!(IN)!

!L

_sp

!=!strain!penetra8on!component!=!0.15f*

_ye

*d

_bl

!!(IN)!

!f

_ye

!=!expected!yield!stress!of!longitudinal!bars!(KSI)!

!f

_ue"

=!expected!tensile!strength!(KSI)!

!d

_bl

"=!diameter!of!longitudinal!column!bars!(IN)!

60! SGS 2011

(61)

Force!C!Displacement

_"

Δ_crack""~!!1/3*_! φ_cr*L2! Δ_y""~!!1/3*_!φ_y*(L+L_sp)2! Δ(M,"φ)_"~!!Δ_y*(M/M_y)+!(φ_!C!φ_y!)*L_p*(LCL_p/2)! ! where:! ! ! !φ_y !=!curvature!at!ﬁrst!yield! ! ! !L !=!column!height! ! !" "L_p !=!analy8cal!plas8c!hinge!length! " " "L_sp !=!strain!penetra8on! " " " φ !=!curvature!at!point!of!interest! " " "M_y !=!moment!at!ﬁrst!yield! " " "M !=!moment!associated!with φ" " " "F !=!force!associated!with!Δ!=!M!/!L"

(62)

Force!C!Displacement

_"

Δ_yi""~!!1/3*φ_!_yi*(L+L_sp)2!

ΔL_C""=!Δ_yi!!+!Δ_p!!~!!Δ_yi!!+!(φ_u!C!φ_yi!)*L_p*(LCL_p/2)!

! where:! ! ! !φ_yi !=!idealized!yield!curvature! ! ! !Δ_yi" "=!idealized!yield!displacement! ! ! !Δ_p" "=!plas8c!displacement!capacity! ! ! !L !=!column!height! ! !" "L_p !=!analy8cal!plas8c!hinge!length! " " "L_sp !=!strain!penetra8on! " " " φ_u !=!ul8mate!curvature! ! ! !ΔL_C" "=!ul8mate!displacement! " " "F_p !=!plas8c!force!=!M_p!/!L!! "!

!

(63)

Force!C!Displacement

_"

•  So!the!deforma8on!values!for!the!example!problem!are:! " Δ_yi""=!!1/3*0.0001132*(240+14.38)2!=!2.44!IN! ! ΔL_C"=!!2.44!+!(0.001075_!–!0.0001132_!)*33.58*(240C33.58/2)!=!9.62!IN! ! where:! ! ! !φ_yi !=!0.0001132!1/IN! ! ! !L !=!20!FT!=!240!IN! " " "L_sp !=!0.15*68*1.41!=!14.38!IN! ! ! !"L_p !=!0.08*240!+!14.38!=!33.58!IN!>!28.8!IN! " " "φ_u !=!0.001075!1/IN! " " "M_p !=!M_yi""=!M_u!=!51626!KCIN! ! ! !F_p !=!M_p!/!L!=!51626!/!240!=!215!KIP! ! ! !µ_D !=!ΔL_C"!/!Δ_yi""=!9.62!/!2.44!=!3.94!

!

63!

(64)

• The!idealized!response!for!the!single!column!example!

Force!C!Displacement

_"

Approximate! M_p,!φ_yi"and!φ_u" Predicted! Response!from! Spreadsheet! calculated!M-φ _" Idealized! Spreadsheet! M_p,!φ_yi!and!φ_u" 64!

(65)

Approximate!Methods"

• As!with!M#

φ

,!we!prefer!a!simpliﬁed!method!to!

check!the!computer!results!

• simple!method!to!check!the!computer!results!

!

• could!use!the!closedCform!AASHTO!equa8ons!to!

start!but!they!are!developed!for!speciﬁc!target!

duc8lity!/!strain!limits!

• use"these"approxima-ons"with"due"skep-cism!

65!

(66)

Approximate!Methods!C!

Δ

_yi

"

• For!a!rough!check!of!conven8onal!circular!

reinforced!concrete!column!sec8ons:

! " " "! !"""Δ_yi""~!!1/3*φ_yi*(12*L+0.15*f_ye*d_b)2!!~!!L2!/!42B_o!! ! !where:! !Δ_yi!=!idealized!yield!displacement!(IN)! ! !L!=!contraﬂexure!to!plas8c!hinge!distance!(FT)! ! !d_b!=!diameter!of!longitudinal!column!bar!(IN)! ! !φ_yi!=!idealized!yield!curvature!~!2.25*ε_ye/B_o!(1/IN)_! ! !B_o!=!column!diameter!(FT)! ! !f_ye!=!expected!yield!stress!(KSI)! 66!

(67)

Approximate!Methods!C!

Δ

_yi

"

• For!a!rough!check!of!conven8onal!rectangular!

reinforced!concrete!column!sec8ons:

! ! !"""Δ_yi""~!!1/3*φ_yi*(12*L+0.15*f_ye*d_b)2!!~!!L2!/!45B_o!! ! !where:! !Δ_yi!=!idealized!yield!displacement!(IN)! ! !L"=!contraﬂexure!to!plas8c!hinge!distance!(FT)! ! !d_b!=!diameter!of!longitudinal!column!bar!(IN)! ! !φ_yi"=!idealized!yield!curvature!~!2.1*ε_ye/12B_o!(1/IN)_! ! !B_o!=!column!width!in!direc8on!of!loading!(FT)! ! !f_ye!=!expected!yield!stress!(KSI)! 67!

(68)

Approximate!Methods!C!

Δ

_c

L"

• And!for!a!very"rough"check"of!conven8onal!

reinforced!concrete!columns!with!L/B

_o

!>!4!:

!

!"""""ΔL_C"~!"Δ_yi"+!(φ_u"C!φ_yi!)*L_p*(12*LCL_p/2)!!~!!L2!/!10B_o"!

! !where:! !ΔL_C!=!local!displacement!capacity!(IN)! ! !L"=!contraﬂexure!to!plas8c!hinge!distance!(FT)! ! !L_p!=!analy8cal!plas8c!hinge!length!(IN)! ! !φ_u!=!ul8mate!curvature!~"ε_suR"/12B_o!(1/IN)!

& &φ_yi!=!idealized!yield!curvature!(1/IN)_!

! !B_o"=!column!diameter!/!width!(FT)!

(69)

ForceCDisplacement!Check

"

• Diameter,!D"=!48!IN!=>"B

_o

!=!4!FT!!

!

• Height,!L"=!H

_o

"="20!FT!

!

• M

_p

!~!54713KCIN/12!=!4560!KCFT!(see!previous!check)!

!

Δ

_yi

""~!L

2!

/!42B

_o!

=!20

2

/(42*4)!=!2.4!IN!v.

!2.4"IN

!

Δ

L_C"

~!L

2!

/!10B

_o"

=!20

2

/(10*4)!=!10!IN!v.

!9.6"IN

!

!F

_p

!=!M

_p"

/!L"=!4560/20!=!228!KIP!v.!

215"KIP

!

(70)

• Idealized!response!for!the!single!column!example!

Force!C!Displacement

_"

Approximate! M_p,!Δ_yi"and!ΔL_C" Spreadsheet! calculated! !M_p,!φ_yi!and!φ_u" 70!

(71)

What!about!Double!Curvature?!

• Eﬀec8ve!column!height,!L,!is!taken!from!the!maximum!

moment!loca8on!to!the!contraﬂexure!point!

• Then!add!the!displacement!results!for!each!part!

L_1" L_2" SGS 2011 71!

(72)

What!about!Double!Curvature?!

Caltrans!SDC!Version!1.7!

(73)

What!about!Pile/Shag!Extensions?!

• For!ﬁxed!head!condi8on,!the!eﬀec8ve!column!

height!values,"L

₁

!and!L

₂

,"will!not!be!of!equal!length!

above!and!below!the!contraﬂexure!point!

!

• Calculate!

Δ

_yi

!from!the!point!of!eﬀec8ve!ﬁxity,!L

_S

,!for!

s8ﬀness!calcula8ons!(typically!3B

_o

!<!L

_S

!<!7B

_o

)!

!

• Calculate!

Δ

_p

"from!the!plas8c!hinge!loca8on,!L

_M

,!

below!the!ground!line!(typically!1B

_o

!<!L

_M

!<!3B

_o

)!

(74)

What!about!Pile/Shag!Extensions?!

F_! F_! F_!

For Δ_yi For Δ_p

(75)

Concrete!Filled!Steel!Pipe!Piles!

(76)

What!about!PCΔ!Eﬀects?!

Caltrans!SDC!Version!1.7! From!equilibrium!and! summing!moments! about!the!base!of!the! column:! ! M_p!=!F_p"*!H_o!+!P"*!Δ& 76!

(77)

• The!moment!leg!to!resist!lateral!forces!becomes,!!

!

M

_p

!=!M

_p

!–!P""*

Δ "

"

F

_P

"="M

_P

"/"H

_o" "

• But!when!using!an!idealized!elas8c!perfectlyCplas8c!

forceCdisplacement!rela8onship!check!

!

P

_dl

"*

Δ

_r

≤!0.25!!M*

_p"

What!about!PCΔ!Eﬀects?!

77!

(78)

What!about!PCΔ!Eﬀects?!

(79)

What!about!PCΔ!Eﬀects?!

• Adjusted!response!for!the!single!column!example!

Unadjusted! for!P-Δ" Adjusted!for! P-_Δ" 79!

(80)

• Rearranging!the!P#

Δ

!limit!the!maximum!permissible!

deﬂec8on!for!the!single!column!example!

!

Δ

_r

≤!0.25!!M*

_p!

/!P

_dl

!

"

Δ

_r

≤!0.25!*!51626!KCIN

_!

/!940!K!=!13.7!IN!

• In!this!case,!the!P#

Δ

!limit!is!greater!than!the!

calculated!

Δ

L_C"

!value!

!

• With!more!conﬁnement,!the!P#

Δ

!limit!may!govern!

What!about!PCΔ!Eﬀects?!

80!

(81)

Local!Duc8lity!v.!Global!Duc8lity!

• Use!local"member!displacements!

!

Δ

_DL"

"<!

Δ

_CL

!!

!

µ

_D

=!

Δ

_DL

!!/!

Δ

_yi""

!!

where:!

"Δ

_DL"

=!Local!member!deforma8on!demand!

"Δ

_CL"

=!Local!member!deforma8on!capacity!

"

µ

_D

=!local!member!displacement!duc8lity!demand

"

"Δ

_yi""

=!idealized!yield!deforma8on!

81!

(82)

Local!Duc8lity!v.!Global!Duc8lity!

(83)

Local!Duc8lity!v.!Global!Duc8lity!

(84)

Implicit!SGS!Equa8ons

_"

•  For!SDC!B!and!C,!closed#form!member!displacement!capacity! equa8ons!are!available! ! !SDC!B: ! !ΔL_C"=!0.12H_o(C1.27*ln(x)C0.32)>0.12H_o" ! !SDC!C: ! !ΔL_C"=!0.12H_o(C2.32*ln(x)C1.22)>0.12H_o!! ! !where:! ! !x!=!ΛB_o/H_o! ! !Λ!=!fixity!factor,!pinCfix!=!1,!fixCfix!=!2! ! !H_o!=!clear!height!of!column!(FT)! ! !B_o!=!column!diameter!/!width!in!direc8on!of!loading(FT)! ! 84! SGS 2011

(85)

Implicit!SGS!Equa8ons

_"

(86)

Implicit!SGS!Equa8ons

_"

(87)

Implicit!SGS!Equa8ons

_"

(88)

Implicit!SGS!Equa8ons

_"

(89)

Implicit!SGS!Equa8ons

_"

•  Assuming!SDC!B,!determine!the!permissible!deforma8on!for! the!example!problem_" ! !ΔL_C""=!0.12H_o(C1.27*ln(x)C0.32)!>!0.12*H_o! " "ΔL_C""=!0.12*20"*"(C1.27*ln(0.2)C0.32)!=!4.1!IN! ! !where:! ! !x!=!ΛB_o/H_o!=!1*4/20!=!0.2! ! !Λ!=!ﬁxity!factor!for!pinCﬁx!=!1!for!this!example! ! !H_o!=!founda8on!to!top!of!pier!=!20!FT! ! !B_o!=!column!diameter!=!4!FT! ! 89!

(90)

Implicit!SGS!Equa8ons

_"

•  Assuming!SDC!C,!determine!the!permissible!deforma8on!for! the!example!problem_" ! !ΔL_C""=!0.12H_o(C2.32*ln(x)C1.22)!>!0.12*H_o!! " "ΔL_C""=!0.12*20"*"(C2.32*ln(0.2)C1.22)!=!6.0!IN! ! !where:! ! !x!=!ΛB_o/H_o!=!1*4/20!=!0.2! ! !Λ!=!ﬁxity!factor!for!pinCﬁx!=!1! ! !H_o!=!20!FT! ! !B_o!=!4!FT! ! 90!

(91)

Implicit!SGS!Equa8ons

_"

• Prescrip8ve!detailing!for!SDC!B

! !

"ρ_s!>!0.003! "ρ_w!>!0.002!

"0.03"*!A_g"">"A_l!>!0.007!*!A_g"

• Prescrip8ve!detailing!for!SDC!C

! !

"ρ_s!>!0.005! "ρ_w!>!0.004!

"0.03"*!A_g"">"A_l!>!0.007!*!A_g"

! 91!

(92)

Other!Considera8ons

_"

• Hollow!column!sec8ons!and!conﬁnement!

• NonCprisma8c!and!ﬂared!columns!

• Essen8ally!elas8c!response!

• Direc8onal!load!combina8ons!

• Uncoupling!the!deforma8on!veriﬁca8on!

• Cold!climate!eﬀects!on!material!proper8es!

• Frozen!soil!s8ﬀness

_" ! 92!

(93)

Design!Example!1!

D = 48 IN H o = 24 F T #5 hoop @ 4IN 20 #11 2 IN clr. 940 KIP 940 KIP Z = 32 FT d = 4 FT

Rigid cap and footings

(94)

Design!Example!1!

•  Assume!that!the!point!of!contraﬂexure!is!at!column!midCheight! L_1""=L"="H_o"/"2_" L_2""="L"="H_o"/"2_" Caltrans!SDC!Version!1.7! 94!

(95)

Design!Example!1!

•  Use!approximate!methods!to!predict!expected!response! •  Diameter,!D"=!48!IN!=>"B_o!=!4!FT!! •  Height,!H_o"=!24!FT!so!for!the!transverse!direc8on,!L!=!12!FT! •  P!=!P_DL!±!!P_EQ"! •  P_EQ!=!(M_p#L"+!M_p#R)!*!(1!+!d"/!H_o)!/!Z"""!"why?" •  Use!P!=!P_DL!=!940!K!for!ﬁrst!itera8on! •  M_p!~!54720KCIN/12!=!4560!KCFT!(see!previous!calcula8ons)! ! ! Δ_yi""~!2!*!L2/42B_o!=!2!*!122/(42*4)!=!1.71!IN! ! !ΔL_C"~!2!*!L2/10B_o"=!2!*!122/(10*4)!=!7.2!IN! ! ! !F_p!=!(2*M_p#L!+!2*M_p#R)!/!H_o"~!4!*!M_p!/24!=!760!KIP"!"why?! ! 95!

(96)

Design!Example!1!

d /2 L = H o / 2 Moment'Diagram'–'slope'of'line'is'the'shear'–'shear'in'cap'beam'is'axial'force'in'column' M_p-L M_p-R M_p-L Mp-R M_p-L*(1 + d/H_o) M_p-R*(1 + d/H_o) F_p 96!

(97)

Design!Example!1!

•  Perform!a!second!itera8on!to!verify!ini8al!assump8ons! ! •  P_EQ!=!(M_p#L+M_p#R)*(1+d/H_o)/!Z"~"2*54720*1.167!/12/32!~!332!K! •  P!=!P_DL"±!P_EQ""=!940!C/+!332!=!608K!and!1272K![compression]! •  M_p#L!~!(0.05+0.2*1.724+0.0643)!*!483!_{=!50808!KCIN!} •  M_p#R!~!(0.05+0.2*1.724+0.130)!*!483!_{=!58612!KCIN!!} •  P_EQ!=!(M_p#L"+!M_p#R)*(1!+!d"/!H_o)!/!Z"" •  P_EQ!=!(50808+58612)(1.167)/32/12!=!332!K!!! •  F_p!=!(2!*!M_p#L!+!2!*!M_p#R)!/!H_o"=!760!KIP!!! _97!

(98)

Design!Example!1!

•  Now!use!the!reﬁned!analysis!to!determine!the!forceC displacement!response!of!the!pier! ! •  First!calcula8on!idealized!M#φ" •  Then!calculate!L_sp!and!L_p" •  Then!calculate!the!forceCdisplacement!for!each!column! •  Then!add!the!results!of!each!column!to!ﬁnd!the!total!pier! response! ! _98!

(99)

Design!Example!1!

(100)

Design!Example!1!

•  Verify!that!the!axial!forces!are!reasonably!close!to!the!ini8al! es8mate!! ! "P_EQ!=!(M_p#L"+!M_p#R)!*!(1!+!d"/!H_o)!/!Z"" " "P_EQ!="(48172+55174)*1.167!/(12*32)!~!314!K!v.!332k! ! !P!=!P_DL"C!P_EQ""=!940!C!314!=!626K!v.!608K!–!close!enough!@!3%! ! !P!=!P_DL"+!P_EQ""=!940!+!314!=!1254K!v.!1272K!–!close!enough!@!2%! ! ! •  Use!the!ini8al!values!since!they!are!close! ! ! 100!

(101)

Design!Example!1!

• Order!or!hinge!forma8on!

H o = 24 F T 940 KIP 940 KIP Z = 32 FT Plastic Hinge Formation 101!

(102)

Design!Example!1!

• Order!or!hinge!failure!(maximum!strain!limit)!

H o = 24 F T 940 KIP 940 KIP Z = 32 FT Plastic Hinge Failure 102!

(103)

Design!Example!1!

•  For!the!leg!/!trailing!side!column:! " Δ_yi#L""=!!1/3*0.0001106*(144+14.38)2!=!0.92!IN! ! ΔL_C#L"=!!0.92!+(0.0011844–!0.0001106_!)*28.8*(144C28.8/2)!=!4.92!IN! ! where:! ! ! !φ_yi !=!0.0001106/IN! ! ! !L !=!H_o!/!2!=!24!FT!/!2!*!12!=!144!IN!! " " "L_sp !=!0.15*68*1.41!=!14.38!IN! ! ! !"L_p !=!0.08*144!+!14.38!=!25.9!IN!<!28.8!IN! " " "φ_u !=!0.0011844!1/IN! " " "M_p !=!48172!KCIN! ! ! !F_p#L !=!M_p!/!L!=!48172!/!144!=!335!KIP! 103!

(104)

Design!Example!1!

•  For!the!right!/!right!side!column:! " Δ_yi#R""=!!1/3*0.00011231*(144+14.38)2!=!0.94!IN! ! ΔL_C#R"=!!0.94!+(0.0009898_!–!0.00011231)*28.8*(144C28.8/2)!=!4.21!IN! ! where:! ! ! !φ_yi !=!0.00011231/IN! ! ! !L !=!H_o!/!2!=!24!FT!/!2!*!12!=!144!IN! " " "L_sp !=!0.15*68*1.41!=!14.38!IN! ! ! !"L_p !=!0.08*144!+!14.38!=!25.9!IN!<!28.8!IN! " " "φ_u !=!0.0009898!1/IN! " " "M_p !=!55174!KCIN! ! ! !F_p#R !=!M_p!/!L!=!55174!/!144!=!383!KIP! 104!

(105)

Design!Example!1!

LeH'Column' Right'Column'

Axial!compression,!P" 608! 1272!

Effec8ve!s8ffness,!(EI)_eff" 4.36E9! 4.91E9!

Plas8c!moment,!M_p" 48172! 55174! Idealized!yield!curvature,!φ_yi" 0.00011060! 0.00011231! Ul8mate!curvature,!φ_u" 0.0011844! 0.0009898! Plas8c!curvature,!φ_p" 0.001074! 0.000877! Analy8cal!Plas8c!Hinge!Length,!L_p" 28.8! 28.8! Δ_yi!(per!half!of!column!height)! 0.92! 0.94! Δ_p"(per!half!of!column!height)! 4.00! 3.27! Δ_u!(per!half!of!column!height)! 4.92! 4.21! Plas8c!shear,!F_p" 335! 383! 105!

(106)

Design!Example!1!

•  Adding!the!columns!together!

(107)

Design!Example!1!

•  Total!eﬀec8ve!lateral!force! ! ! ! !F_p!=!F_p#L!+!F_p#R""=!335K!+!383K!=!718!K!! ! ! •  Idealized!yield!displacement! !

! !Δ_yi"~!(2*Δ_yi#L!+!2*Δ_yi#R)!/"2!=!0.92!+!0.94!=!1.86!IN!

! ! •  Ul8mate!displacement!(ﬁrst!hinge!failure!on!right)! ! ! ! !ΔL_c!=!2!* ΔL_c#R!!=!2!*!4.21!IN!=!8.42!IN! 107!

(108)

Design!Example!1!

Combined_" Leg!/!Trailing! Column_" Right!/! Leading! Column_" 108!

(109)

Design!Example!1!

• Idealized!force!displacement!for!the!two!column!pier!

Approximate! M_p,!Δ_yi"and!ΔL_C" Spreadsheet! Idealized! !M_p,!φ_yi!and!φ_u" 109!

(110)

•  The!maximum!deﬂec8on!based!on!P#Δ!limits! " Δ_r ≤!0.25!*!48172!KCIN_!/!940!K!=!12.8!IN!!!(per!half)! •  And!as!a!check!using!SDC!C!deﬂec8on!(µ_D!!~!2.1!with!L_sp)! ! !ΔL_C"=!0.12*24*(C2.32*ln(0.33)C1.22)!=!3.83!IN! •  And!the!displacement!duc8lity!capacity! ! ! ! !µ_D!<!ΔL_c!/Δ_yi"=!8.42!/!1.86!=!4.5!

Design!Example!1!

110!

(111)

Capacity!Design

_"

•  Designer!dictates!the!bridge!response!(e.g.,!column!hinge! response!mechanism!–!Type!I)! •  Preclude!all!failure!modes!that!would!prevent!the!forma8on!of! the!predicted!forceCdeﬂec8on!response!including:! –  Shear!failure!inside!and!outside!the!plas8c!hinge!region!(not! the!same!as!analy8cal!plas8c!hinge!length,!L_p)! –  ColumnCcap!and!columnCfoo8ng!joint!failure! –  Cap!beam,!foo8ng!or!shag!moment,!shear,!and!axial! overload!/!hinging! –  Foo8ng!overturning,!sliding,!uplig!and!rocking!failure! –  Any!other!failure!that!occurs!prior!to!reaching!ΔL_C! ! _111!

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Capacity!Design

_"

•  Use!overstrength!plas8c!hinging!forces!for!connected!elements! ! M_po!=!λ_mo*!M_p"" " •  λ_mo!=!1.2!for!ASTM!A!706!Grade!60!(must!use!in!SDC!D)!!

•  Associated!shear!forces,!V_po,!and!axial/overturning,!λ_mo*!P_EQ"! ! •  Use!for!!joints,!shear!in!hinges,!foo8ngs,!cap!beams,!etc.! ! •  May!use!expected!nominal!moment!resistance,!M_ne!,for! capacity!design!of!caps,!foo8ngs,!shags!at!concrete! compression!strain!of!0.003!and!a!resistance!factor,!φ!,!of!1! 112!

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Shear!Resistance!in!Plas8c!Hinges

_"

! !

!

φ V_n!!=!φ (V_c"+!V_s)!>!V_po"

where:!

"φ!=!resistance!factor!of!0.9!for!shear!in!plas8c!hinge!region!

"P_u!=!P_DL!!!±!!λ_mo!*!P_EQ"

!If!P_u!<!0!!(tension)!then!v_c!=!0!otherwise! ! ! ! V_c!=!v_c!*!A_e!! ! A_e!=!0.8!*!A_g" ! !" ! # $ % % ⋅ % ≤ % ( ( ) * + + , -+ % ⋅ = c c c g u c f f f A P v α α 047 . 0 11 . 0 min 2 1 032 . 0 SGS 2011 113!

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Shear!Resistance!in!Plas8c!Hinges

_"

For!circular!columns!with!spirals!or!hoops! ! ! ! ! ! n!=!number!of!individual!interlocking!spiral!or!!hoop!core!sec8ons!! f_s!=!ρ_s*f_yh!<!0.35! A_sp!=!area!of!hoop!or!spiral!bar!(in.2_)! f_yh!=!yield!stress!of!spiral!or!hoop!reinforcement!(ksi)! D’_{!=!core!diameter!measured!from!center!of!spiral!or!hoop!(in.)!} s"=!spacing!of!spiral!or!hoop!reinforcement!(in.)! ! ! ! 3 67 . 3 15 . 0 3 . 0 _< ' ₌ fs ₊ ₋ _D _<

µ

α

e c yh sp s f A s D f nA V ' ' 25 . 0 2 "" ≤ # $ % % & ' = π SGS 2011 114!

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Shear!Resistance!in!Plas8c!Hinges

_"

For!rectangular!columns!with!8es! ! ! ! ! " A_v!=!area!of!shear!reinforcement!in!the!direc8on!of!loading!(in.2_)! f_w!=!2*ρ_w*f_yh!<!0.35! µ_D!=!displacement!duc8lity!demand! d"=!depth!of!sec8on!in!direc8on!of!loading!(in.)! f_yh!=!yield!stress!of!8e!reinforcement!(ksi)! s"=!spacing!of!8e!reinforcement!(in.)! e c yh v s f A s d f A V ₀_.₂₅ ' ≤ "" # $ %% & ' = 3 67 . 3 15 . 0 3 . 0 _< ' ₌ fw ₊ ₋ _D _<

µ

α

SGS 2011 115!

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Shear!Resistance!in!Plas8c!Hinges

_"

φ*V_n!plas8c! hinge!shear! resistance" V_po!overstrength! shear!in!plas8c! hinge!(λ_mo!=!1.2)" 116!

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Shear!Resistance!in!Plas8c!Hinges

_"

φ*V_n!plas8c! hinge!shear! resistance!–!limit!! 8.35!IN!on!right! column" V_po!overstrength! shear!in!leg! plas8c!hinge! (_λ_mo = 1.2)_" V_po!overstrength! shear!in!rights! plas8c!hinge!! (_λ_mo = 1.2)_" 117!

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Joints!and!Connec8ons

_"

• _{Average!principal!joint!stresses!are!used!as!}

an!indicator!of!joint!performance!

!

• _{Take!cap!beam!forces!at!the!face!of!column!}

(not!centerline)!including!λ

_mo"

!

• _{Resultant!tensile!force!from!MC}

_φ

_{!analysis!or!}

use!approximate!value!of!T!~!0.7!!A*

_st!

*!f

_ye" 118!

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Splices!and!Transverse!Bars

_"

• _{Plas8c!hinge!region!larger!than!analy8cal!}

plas8c!hinge!length,!L

_p"

• _{“No!Splice!Zone”!in!Plas8c!hinge!regions!}

!

• _{May!reduce!transverse!reinforcing!outside!}

plas8c!hinge!region!

• _{Development,!anchorage,!and!other!details!}

119!

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Design!Example!2!

B_o = 36 IN H o = 18 F T 4 legs #5 stirrups @ 4IN 24 #10 2 IN clr. 500 KIP 500 KIP Z = 24 FT d = 3 FT

Rigid cap and footings

B o

=

36

IN

Very high transverse

reinforcing for this example – likely hard to construct

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Design!Example!2!

•  Use!approximate!methods!to!es8mate!forceCdisplacement! •  Width!and!depth,!B_o!=!3!FT!,!b!=!36!IN!and!d!=!36!IN!

•  Height,!H_o"=!18!FT!so!for!the!transverse!direc8on,!L!=!108!IN!

•  Longitudinal!bars!#10,!L_sp!=!13.0!IN,!L_p!=!25.9!IN,!ρ_l!=!2.35%!! " """""M_p"~"36*362_{"*"[0.15!+!0.25*2.35} "+!1.5*0.074]!~!39600!KCIN! !!!!!φ_yi"~!2.1*ε_ye/12B_o"=!2.1*0.002345/36!=!0.0001368!1/IN! "φ_u"~!ε_suR"/12B_o!=!0.09/36!=!0.002500!1/IN! !Δ_yi"~!1/3*φ_yi*(L+L_sp)2!=!0.0001368*(108+13)2/3!=!0.67!IN! ΔL_C"~!Δ_yi!!+!(φ_u!C!φ_yi!)*L_p*(LCL_p/2)! "ΔL_C"!~!0.67!+!0.002363*25.9*(108C13)!=!6.50!IN! ! "F_p!=!(2*M_p#L!+!2*M_p#R)!/!H_o"~!4!*!M_p!/18/12!=!733!KIP! ! 121!

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Design!Example!2!

•  Es8mate!the!axial!force!per!column! ! •  P_EQ!=!(M_p#L+M_p#R)*(1+d/H_o)/!Z"~"2*39600*1.167!/12/24!~!321!K! •  P!=!P_DL"±!P_EQ""=!500!C/+!321!=!179K!and!821K![compression]! •  M_p#L!~"36*362_{"*"[0.15!+!0.25*2.35} "+!1.5*0.027]!~!36300!KCIN! •  M_p#R!~"36*362_{"*"[0.15!+!0.25*2.35} "+!1.5*0.122]!~!42900!KCIN! •  P_EQ!=!(M_p#L"+!M_p#R)*(1!+!d"/!H_o)!/!Z"" •  P_EQ!=!(36300+42900)(1.167)/24/12!=!321!K!v.!321K!!!OK! •  F_p!=!(2!*!M_p#L!+!2!*!M_p#R)!/!H_o"=!733K!v.!733K!!!OK! 122!

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Design!Example!2!

Spreadsheet! Idealized&

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Design!Example!2!

LeH'Column' Right'Column'

Axial!compression,!P" 185! 815!

Effec8ve!s8ffness,!(EI)_eff" 2.44E8! 2.77E8!

Plas8c!moment,!M_p" 36442! 41280! Idealized!yield!curvature,!φ_yi" 0.0001494! 0.0001492! Ul8mate!curvature,!φ_u" 0.003611! 0.002682! Plas8c!curvature,!φ_p" 0.003461! 0.002533! Analy8cal!Plas8c!Hinge!Length,!L_p" 25.9! 25.9! Δ_yi!(per!half!of!column!height)! 0.73! 0.73! Δ_p"(per!half!of!column!height)! 8.52! 6.24! Δ_u!(per!half!of!column!height)! 9.25! 6.96! Plas8c!shear,!F_p" 337! 382! 124!

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Design!Example!2!

•  Total!eﬀec8ve!lateral!force! ! ! ! !F_p!=!F_p#L!+!F_p#R""=!337K!+!382K!=!720!K!!!OK! ! ! •  Idealized!yield!displacement! !

! !Δ_yi"~!(2*Δ_yi#L!+!2*Δ_yi#R)!/"2!=!0.73!+!0.73!=!1.46!IN!

! ! •  Ul8mate!displacement!(ﬁrst!hinge!failure!on!right)! ! ! ! !ΔL_c!=!2* ΔL_c#R!!=!2!*!6.96!IN!=!13.9!IN! 125!

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Design!Example!2!

Combined_" Leg!/!Trailing! Column_" Right!/! Leading! Column_" 126!

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•  The!maximum!deflec8on!based!on!P#Δ!limits" Δ_r ≤!0.25!*!36442!KCIN_!/!500!K!=!18.2!IN!(per!half)!>!13.9!IN!!! •  And!as!a!check!using!SDC!C!deflec8on!(µ_D!!~!2.1!with!L_sp)! !ΔL_C"=!0.12*18*(C2.32*ln(0.33)C1.22)!=!2.87!IN! •  And!the!displacement!duc8lity!capacity! ! !µ_D!<!ΔL_c!/!Δ_yi"=!13.9!/!1.46!=!9.5!>>!6!"!exceeds!limit! ! •  Thus,!limi8ng!deflec8on!due!to!maximum!duc8lity!(SGS!4.9)! " "ΔL_C"=!µ_D!*!Δ_yi""=!6!*!1.46!=!8.75!IN!"!maximum!

Design!Example!2!

127!

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Design!Example!2

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•  Now!check!the!shear!in!the!hinge!at!maximum!deﬂec8on" "" " " " "φ V_n!!=!φ (V_c"+!V_s)!>!V_po" where:! "φ!=!resistance!factor!of!0.9!for!shear!in!plas8c!hinge!region! ! "P_u!=!P_DL!!!±!!λ_mo!*!P_EQ"~!122K!and!878K! ! ! !

"A_e!=!0.8!*!A_g"=!0.8!*!36!*!36!=!1036!IN2!

! !" ! # $ % % ⋅ % ≤ % ( ( ) * + + , -+ % ⋅ = c c c g u c f f f A P v α α 047 . 0 11 . 0 min 2 1 032 . 0 128!

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Design!Example!2

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For!rectangular!columns!with!8es! ! ! ! ! " "A_v!=!4!*!0.31!=!1.24!IN2! "f_w!=!2*ρ_w*f_yh!=!1.17!>>!0.35!so!f_w!=!0.35!"!maximum! "µ_D!=!6!maximum!so!α’"=!0.3!"!minimum!value!governs! "d"=!36!IN! "f_yh!=!60!KSI! "s"=!4!IN! e c yh v s f A s d f A V ₀_.₂₅ ' ≤ "" # $ %% & ' = 3 67 . 3 15 . 0 3 . 0 _< ' ₌ fw ₊ ₋ _D _<

µ

α

129!