A model for the practical nonlinear analysis of reinforced-concrete frames

including joint ﬂexibility

Anna C.Birely

⇑

,Laura N.Lowes,Dawn E.Lehman

Civil and Environmental Engineering,University of Washington,Seattle,WA 98195,United States

a r t i c l e i n f o

Article history:

Received 5 January 2011

Revised 1 September 2011

Accepted 1 September 2011

Keywords:

Beam–column joints

Analytical models

Rotational springs

Ductility

a b s t r a c t

A model is developed to simulate the nonlinear response of planar reinforced-concrete frames including

all sources of ﬂexibility.Conventional modeling approaches consider only beam and column ﬂexibility

using concentrated plasticity or springs to model this response.Although the joint may contribute the

majority of the deformation,its deformability is typically not included in practice.In part,this is because

few reliable,practical approaches for modeling all sources of frame nonlinearity are available.The

research presented herein was undertaken to develop a practical,accurate nonlinear model for reinforced

concrete frames.The model is appropriate for predicting the earthquake response of planar frames for

which the nonlinearity is controlled by yielding of beams and/or non-ductile response of joints and is

compatible with the ASCE/SEI Standard 41-06 nonlinear static procedure.The model was developed to

facilitate implementation in commercial software packages commonly used for this type of nonlinear

analysis.The nonlinearity is simulated by introducing a dual-hinge lumped-plasticity beam element to

model the beams framing into the joint.The dual-hinge comprises two rotational springs in series;

one spring simulates beamﬂexural response and one spring simulates joint response.Hinge parameters

were determined using data from45 planar frame sub-assemblage tests.Application of the model to sim-

ulate the response of these sub-assemblages shows that the model provides accurate simulation of stiff-

ness,strength,drift capacity and response mechanismfor frames with a wide range of design parameters.

2011 Elsevier Ltd.All rights reserved.

1.Introduction

Under seismic loading,the beams and columns in a concrete

moment frame typically experience moment reversals at the

beam–column joint.To ensure that strength is maintained under

multiple large drift cycles,design guidelines for frames in regions

of high seismicity are intended to result in ﬂexural yielding in

the beams at the face of the joint and essentially elastic response

in the columns above the base (e.g.,ACI 318-08 [1]).This can result

in high joint shear demand and high bond stress demand for beam

longitudinal reinforcement anchored in the joint;design guidelines

seek to limit both to ensure that joint damage does not reduce

frame toughness.The results of experimental tests on frame

subassmblages [2–13] show that joint damage can reduce frame

strength and stiffness and,in some cases,result in premature loss

of load-carrying capacity.

Prediction of frame response,as part of a performance-based

seismic design of a new structure or evaluation of an existing

structure,requires modeling of all sources of ﬂexibility in the

frame as well as stiffness and strength loss under earthquake load-

ing.Thus,to conduct a nonlinear analysis of a reinforced concrete

moment-resisting frame,engineers require not only accurate mod-

els for beams and columns but also models that simulate joint re-

sponse.For these models to be practical for use in the design ofﬁce,

they must (1) be compatible with commonly employed commer-

cial software packages,(2) support rapid model building,(3) be

computationally efﬁcient and robust,and (4) provide acceptable

accuracy over a range of design conﬁgurations.Several practical

nonlinear modeling approaches are available for beams and col-

umns (e.g.,[14,15]).These models,which have been validated by

others,are incorporated into the fame model proposed herein.

However,these models do not simulate the response of the

beam–column joint.Here,the focus is on developing an appropri-

ate approach for simulating joint ﬂexibility and degradation in

frame strength due to joint response mechanisms.Although non-

linear joint models are found in the literature,fewof these models

meet the requirements for widespread use by practicing structural

engineers.A model is proposed which uses conventional nonlinear

frame elements and is easily implemented in commercial struc-

tural analysis software.The model is developed and validated

using experimental data to simulate the full frame response,

including the joint.

0141-0296/$ - see front matter 2011 Elsevier Ltd.All rights reserved.

doi:10.1016/j.engstruct.2011.09.003

⇑

Corresponding author.

E-mail address:birely@uw.edu (A.C.Birely).

Engineering Structures 34 (2012) 455–465

Contents lists available at SciVerse ScienceDirect

Engineering Structures

j ournal homepage:www.el sevi er.com/l ocat e/engst ruct

1.1.Previously proposed beam–column joint models

The simplest approach for modeling joint response within the

context of a nonlinear frame analysis is to introduce a spring at

the intersection of the beamand column line elements.Often rigid

offsets are included in the beamand column elements to deﬁne the

physical size of the joint and ensure that the joint spring is the sole

source of simulated deformation in the joint region.The load-

deformation response of the joint spring is typically deﬁned on

the basis of the expected shear stress–strain response of the joint

volume and/or the bond slip of longitudinal reinforcement within

the joint.Models of this type were developed by El-Metwally and

Chen [16],Kunnath [17],Ghobarah and Biddah [18],and Anderson

et al.[19].The advantage of this type of model is its overall sim-

plicity.One disadvantage of the model is the need to deﬁne dupli-

cate nodes at the center of the joint,a process that is typically not

well supported in commercial software and can hinder the model-

building process.The primary disadvantage of this type of model,

however,is the challenging and time-consuming process of cali-

brating the model to provide an accurate simulation of joint re-

sponse.Often,model calibration is accomplished by simplifying

the assumed primary response of the joint.Speciﬁcally,the model

parameters are speciﬁc to a limited set of design parameters,a lim-

ited set of data,or both.Typically the modeling assumptions de-

pend on the design parameters,and applying these assumptions

to generate the model response history typically requires signiﬁ-

cant computation by the engineer.In some cases,models are cali-

brated directly fromexperimental data characterizing the response

of joints in frames with design parameters and details that are sim-

ilar to those in the structure of interest;in this case,the accuracy of

the simulations depends entirely on the similarity of the two

frames.

Macro-element joint models are a secondapproachfor modeling

joint behavior.Altoontashand Deierlein[20],Lowes and Altoontash

[21],and Mitra and Lowes [22] have proposed models that connect

beamandcolumncenterline elements toﬁnite-volume joint macro-

elements.These models comprise a shear-panel component and

rotational springs or zero-length springs to represent bar slip and

interface shear.Other macro-models have been developed by

Youssef and Ghobarah [23],Elmorsi et al.[24],Shin and LaFave

[25],Uma and Prasad [26],and Tajiri et al.[27].Relative to a single

concentrated spring,a macro-element model typically allows for a

simpler,more objective calibration and enables simulation of joints

with a wide range of design parameters.Aprimary drawback of this

modeling approach,however,is that macro-element models are not

easily implemented in commercial software.

Given the complexity of beam–column joint response,contin-

uummodeling offers the greatest potential for accurately simulat-

ing the nonlinear response of joints with a range of design

parameters.Continuum modeling has been the focus of a number

of research studies [28–30].However,it has not been validated

using large data sets,is computationally demanding,requires sig-

niﬁcant model-building effort on the part of the engineer,and can-

not be accomplished using the analysis software employed

typically by practicing structural engineers.

1.2.Proposed model

The objective of this study was to develop a model to accurately

simulate the nonlinear response (including initial stiffness,

strength,and deformation capacity) of a reinforced concrete frame.

This requires simulation of all important sources of deformation

including the beam,column,and joint response mechanisms.Pre-

viously proposed joint models do provide accurate representation

of joint behavior,and some have been calibrated to explicitly ac-

count for speciﬁc response modes,such as bond slip and shear

deformation.However,these models require the addition of a sep-

arate element,typically concentrated springs or macro-elements,

for every joint,making their use in commercial nonlinear struc-

tural analysis software packages time consuming and cumber-

some.However,modeling a frame using only standard nonlinear

beam–column elements neglects simulation of joint response and

typically produces inaccurate results.The proposed model seeks

to provide an intermediate modeling alternative between the sim-

ple use of frame elements without joint representation and the

more time consuming use of detailed joint elements.

One of the most common approaches to modeling nonlinear re-

sponse of reinforced concrete frames is to use lumped-plasticity

elements for the beams and columns.This approach is used here,

with the moment-rotation history of the beam plastic hinges (for

the systemconsidered in this study,columns do not exhibit signif-

icant nonlinearity) modiﬁed to include simulation of not only the

beam response,but also the joint response.In the modiﬁed ele-

ment,the conventional single-hinge representation is replaced by

a dual-hinge,which comprises two rotational springs in series.

One of the springs simulates the nonlinear ﬂexural response of

the beamand is referred to as the beamspring.The second spring

simulates the response of the joint,and is referred to as the joint

spring.Rotation limits are provided for each spring to simulate

the onset of loss of load carrying capacity.Rigid offsets are in-

cluded in the beams and columns that frame into the joint to en-

sure that joint ﬂexibility is deﬁned entirely by the joint spring in

the dual hinge.

The model was developed with the objective of satisfying prac-

tical nonlinear modeling needs.The proposed model can be used in

existing commercial software with an effort that is equivalent to

using a conventional nonlinear beam–column element.By modify-

ing a traditional method for modeling frames,the intent of the

model is to provide increased accuracy over models commonly

used in practice.Consequently,the model should not be used to

extract local response of a joint.If such information is needed,

other models,such as those identiﬁed above,are more appropriate.

2.Experimental data set

The proposed joint model was developed using data from45pla-

nar frame sub-assemblages,tested by 11 research groups [2–13],as

listed in Table 1.The data set used in this study is a subset of that

assembled by Mitra and Lowes [22],where a detailed description

of the individual specimens canbe found.Specimens are bare-frame

sub-assemblages without slabs.The data set does not include spec-

imens constructedof light-weight concrete,veryhigh-strengthcon-

crete (over 16 ksi,or 110 MPa),or plain reinforcing bars.For the

specimens included in the data set,the lateral capacity was limited

by ﬂexural yielding of beams and/or damage in the joint.The pro-

posed model is appropriate for use in simulating the response of

frames with these characteristics and failure modes.

The data set spans a wide range of design parameters and in-

cludes specimens that are representative of newconstruction in re-

gions of high seismic hazard as well as construction that pre-date

modern seismic detailing requirements.Fig.1 shows histograms

for select design parameters,including concrete compressive

strength (f

0

c

),beamsteel yield strength (f

y

),ratio of the sumof col-

umn ﬂexural strengths to the sum of the beam ﬂexural strengths

(

R

M

nc

/

R

M

nb

),as well as bond demand (

l

) computed

l

¼

a

f

y

d

b

2h

c

ﬃﬃﬃﬃ

f

0

c

p

ð1Þ

where d

b

is the maximum diameter of the beam longitudinal rein-

forcement and all other variables are previously deﬁned,joint rein-

forcement ratios (

q

j

) computed

456 A.C.Birely et al./Engineering Structures 34 (2012) 455–465

q

j

¼

A

t

s

t

b

j

ð2Þ

where A

t

is the area of one layer of joint transverse reinforcement

passing through a plane normal to the axis of the beams,s

t

is the

vertical spacing of hoops in the joint region,and b

j

is the out-of-

plane dimension of the joint,and axial load ratio (p) computed

p ¼

P

A

g

f

0

c

ð3Þ

where P is the applied column axial load,A

g

is the gross area of the

column,and f

0

c

is the measured concrete compressive strength.

Additionally,Table 1 lists for each specimen,the following shear

stress values used in the current study:

Designshear stress demand,

s

design

,computedusing anapproach

that is consistent with ACI Com352 [31] recommendations:

s

design

¼

1

h

c

b

j

a

ðf

y

ðA

top

s

þA

bottom

s

Þ V

n

Þ ð4Þ

where h

c

is the height of the column and b

j

is the out-of-plane

dimension of the joint.The column shear,V

n

,corresponds to

the average nominal moment strength of the beam.The vari-

ables A

top

s

and A

top

s

are the areas of steel in the top and bottom

of the beam,respectively,and f

y

is the measured strength of

the reinforcement.To account for hardening of the steel under

earthquake loading and over-strength in the nominal value,f

y

is typically multiplied by a factor of 1.25 [1].However,as the ac-

tual strength of the steel was available for all specimens in the

data set,the 1.25 factor was reduced by 1.1 as recommended

by ACI Com.352 [31].Thus,

a

= 1.25/1.1 was used in Eq.(4).

Maximum measured shear stress demand,

s

max

s

max

¼

1

h

c

b

j

M

L

þM

R

jd

V

max

ð5Þ

Table 1

Ductility classiﬁcation data.

Test program Test specimen

s

design

ﬃﬃﬃ

f

0

c

p

(MPa/

p

MPa)

a

s

meas

max

ﬃﬃﬃ

f

0

c

p

(MPa/

p

MPa)

a

s

model

max

ﬃﬃﬃ

f

0

c

p

(MPa/

p

MPa)

a

l

D

b

Ductility

Test Model

Durrani and Wight [7] DWX1 11.10 1.27 1.06 4.7 D D

DWX2 1.12 1.35 1.07 5.5 D D

DWX3 0.87 1.12 0.85 6.4 D D

Otani et al.[3] OKAJ1 1.22 0.97 1.01 5.3 D D

OKAJ2 1.26 1.03 1.04 5.6 D D

OKAJ3 1.26 1.15 1.04 6.7 D D

OKAJ4 1.22 0.95 1.01 5.1 D D

OKAJ5 1.15 0.90 0.96 4.4 D D

Meinheit and Jirsa [10] MJ1 2.24 1.32 1.70 – B B

MJ2 1.78 1.52 1.57 – B B

MJ3 2.23 1.47 1.70 – B B

MJ5 1.92 1.57 1.60 – B B

MJ6 1.89 1.69 1.59 – B B

MJ12 1.93 2.04 1.62 3.5 LD B

MJ13 1.79 1.49 1.57 – B B

Alire [5] and Walker [6] PEER14 0.87 0.94 0.89 3.6 LD D

PEER22 1.55 1.17 1.49 – B B

PEER0850 0.61 0.70 0.64 4.3 (E) D D

PEER0995 0.93 1.04 0.95 5.3 D D

PEER4150 3.40 1.90 1.97 – B B

Park and Ruitong [12] PR1 0.46 0.51 0.47 29.5 (E) D D

PR2 0.72 0.81 0.72 6.8 D D

PR3 0.51 0.53 0.52 13.2 (E) D D

PR4 0.69 0.73 0.69 7.1 D D

Noguchi and Kashawazaki [11] NKOKJ1 1.88 1.52 1.67 – B B

NKOKJ3 1.90 1.54 1.63 – B B

NKOKJ4 1.88 1.60 1.67 – B B

NKOKJ5 2.37 1.59 1.88 – B B

NKOKJ6 2.02 1.61 1.75 – B B

Oka and Shiohara [13] OSJ1 1.45 1.44 1.34 3.0 LD B

OSJ2 3.60 1.55 1.84 – B B

OSJ4 1.55 1.55 1.39 3.8 LD B

OSJ5 2.05 1.75 1.72 – B B

OSJ6 1.57 1.55 1.41 3.0 LD B

OSJ7 1.16 1.24 1.17 5.0 (E) D D

OSJ8 1.95 1.76 1.59 2.9 LD B

OSJ10 2.35 1.60 1.80 – B B

OSJ11 2.83 1.84 1.87 – B B

Kitayama et al.[9] KOAC1 0.91 1.01 0.87 116 (E) D D

KOAC3 0.91 0.97 0.87 42.3 (E) D D

Park and Milburn [4] PM1 1.25 1.15 1.22 3.1 (E) LD D

Endoh et al.[8] HC 1.02 1.10 0.96 8.2 (E) D D

A1 2.24 1.45 1.64 – B B

Beckingsale [2] B11 0.76 0.83 0.78 No Loss D D

B12 0.76 0.86 0.79 No Loss D D

a

Conversion factor:1 MPa = 145 psi;1 MPa/

p

MPa = 12.04 psi/

p

psi.

b

(E):Displacement ductility at 10% strength loss extrapolated;(–):brittle joint,no displacement ductility.

A.C.Birely et al./Engineering Structures 34 (2012) 455–465

457

where V

max

is the maximumexperimental column shear,M

L

and

M

R

are the corresponding moments in the left and right beamat

the joint face,respectively.The termjd is the moment lever arm

for the beam,where d is the depth from the extreme compres-

sion ﬁber to the tension reinforcement in the beam,assumed

to be 0.9h

b

if unknown,and j is an empirically derived parameter

typically taken equal to 0.85 [32].

Fig.2 shows a typical test specimen and setup.Test specimens

represent a sub-assemblage froma two-dimensional building frame,

comprising a segment of a continuous beam extending from mid-

spanof one frame bay tomid-spanof the next,a segment of a contin-

uous column extending frommid-height of one story to mid-height

of the next,and the beam–column joint at the intersection of these

two members.The mid-length points are assumed to approximate

points of inﬂection in an actual frame and therefore are a suitable

location to apply the shear forces and sustain the reactions.Frame

sub-assemblages were subjected to reversed cyclic lateral loading

under displacement control in the laboratory,and load was applied

either as a shear load at the top of the column or as a pair of equal-

and-opposite shear loads at the beamends.Some sub-assemblages

were also subjected to simulated gravity load applied as a constant

axial load to the top of the column.

2.1.Force–displacement data

The primary data used from the experimental tests were the

column shear versus lateral displacement envelope for each spec-

imen.In some cases,data characterizing the response of the indi-

vidual frame components (beams,columns and joints) were

available,in other cases they were not.Therefore individual com-

ponent responses were not directly considered;instead,the overall

sub-assemblage response was used.This is consistent with previ-

ous research addressing performance-based seismic design of rein-

forced concrete frames that has shown frame drift to be the most

practical engineering demand parameter [33].

The response envelope for each specimen was determined from

the experimental cyclic load–displacement history.Points on the

envelope correspond to the following load–displacement pairs:

(i) theoretical initial ﬂexural yielding of beams,(V

y

,

D

y

),(ii) maxi-

mumcolumn shear,(V

max

,

D

V

max

),(iii) 20% loss of strength follow-

ing maximum load,(V

80%

,

D

80%V

),and (iv) any additional points

required to accurately represent the shape of the load–displace-

ment envelope.The theoretical yield load,V

y

,was computed using

a moment–curvature analysis of a ﬁber cross-section of the beams

with ﬂexural yield strength of the beams deﬁned by initial tensile

yielding of the beam longitudinal reinforcement.Fig.3 shows an

example of the cyclic response and the response envelope for Spec-

imen PEER0995 [5–6].

2.2.Ductility classiﬁcation

To support the model calibration effort,specimens were classi-

ﬁed as brittle,ductile or limited-ductility.Brittle specimens exhibit

a maximum strength that was less than that required to yield all

beam longitudinal reinforcement in tension.Ductile specimens

reach the yield shear force and exhibit displacement ductility,

l

D

,greater than four.Limited-ductility specimens are specimens

not classiﬁed as brittle or ductile.Displacement ductility,

l

D

,was

deﬁned as

l

D

¼

D

90%

D

y

ð6Þ

20

40

60

80

100

120

0

10

20

30

(a) f’

c

(MPa)

# Specimen

250

500

750

1000

1250

1500

0

10

20

30

(b) f

y

(MPa)

# Specimen

0.5

0.75

1

1.25

1.5

1.75

2

2.25

0

5

10

15

20

(c) Σ M

nc

/Σ M

nb

# Specimen

10

20

30

40

50

60

0

5

10

15

(d) μ

# Specimen

0

0.5

1

1.5

2

2.5

3

3.5

4

0

5

10

15

(e) ρ

j

(%)

# Specimen

0

0.1

0.2

0.3

0.4

0.5

0

5

10

15

20

(f) P/(A

g

f’

c

)

# Specimen

Fig.1.Histogram of basic beam–column joint sub-assemblage properties.

Fig.2.Experimental set-up for Park and Milburn [4] and Alire [5] sub-assemblages

(equal and opposite forces applied to beams).

0

40

80

120

0

150

300

450

Dis

p

lacement, [mm]

Column Shear, [kN]

Experimental

Envelope

Second Beam Yield

90% Maximum Shear

Δ

Y

V

Y

Δ

Vmax

V

max

Δ

80%Vmax

V

80%

(V

fy

, Δ

fy

)

(V

90%

, Δ

90%Vmax

)

Fig.3.Positive force–displacement history for specimen PEER 0995 with envelope.

458 A.C.Birely et al./Engineering Structures 34 (2012) 455–465

where

D

90%

is the displacement at which 10% strength loss

occurred,as determined fromthe column lateral load–displacement

envelope,and

D

y

is as deﬁned previously.A solid circle in Fig.3

shows

D

90%

and the corresponding column shear force for Specimen

PEER0995.The ductility classiﬁcation of each specimen is provided

in Table 1.For the complete data set,18 specimens were classiﬁed

as brittle,20 as ductile,and 7 as limited-ductility.

3.Model deﬁnition

The proposed dual-hinge model is incorporated in a lumped-

plasticity beam–column element,where two rotational springs

are combined in series to form the dual-hinge that represents the

inelastic deformations of the beam and the joint.One of the rota-

tional springs simulates the beam response.The second spring

simulates the joint response,including that caused by joint shear

deformation and bond slip.

Fig.4 shows a model of a typical frame sub-assemblage in

which the proposed dual-hinge beam–column element is used to

model the beams.Beams are modeled as elastic outside the hinge

region.Columns are modeled using elastic beam–column ele-

ments;elastic elements were considered adequate because,for

the sub-assemblages in the data set,columns did not yield.The

effective elastic stiffness values recommended in ASCE/SEI Stan-

dard 41-06 [34] were used for beams and columns.Rigid offsets

equal to the joint dimensions were included at the ends of the

beamand column elements to deﬁne the joint volume and ensure

that the joint spring was the only source of joint deformation.

Fig.5 shows the moment–curvature response of the two com-

ponents of the dual-hinge.The beamresponse is similar to a tradi-

tional nonlinear beam–column hinge,but has a rotation limit,

h

fail

beam

,beyond which a loss of strength occurs.The joint spring

has a bilinear response,with stiffness K

1

,to the yield moment of

the beam,and K

2

beyond yield.For the joint spring,strength loss

initiates at a rotation demand of h

fail

joint

.The following sections dis-

cuss the calibration of these values.

3.1.Rotational spring simulating nonlinear beam ﬂexural response

Typically,the moment-rotation response for a lumped-plasticity

beam or column element is determined from the moment–

curvature response of the member cross section and a plastic-

hinge length.As described previously,the proposed dual-hinge

model uses two springs in series.The rotational spring represent-

ing the nonlinear ﬂexural response of the beam was calibrated to

simulate the nonlinear response of the beam plastic-end zone

including a rotation limit deﬁned to simulate the onset of ﬂexural

strength degradation.The beam ﬂexural moment-rotation

response prior to strength loss was determined by performing a

moment–curvature analysis (for both positive and negative

bending) using a ﬁber-type model of the beam section.The beam

depth was discretized into 32 concrete ﬁbers;ﬁber thickness was

approximately one-half inch (13 mm).Concrete was assumed to

have no tensile strength.A parabolic compressive stress–strain

response was assumed for unconﬁned concrete,with maximum

strength corresponded to a strain of 0.002 mm/mm.The modiﬁed

Kent–Park model [35] was used to deﬁne the compressive

response of conﬁned concrete.A bilinear stress–strain response

was assumed for reinforcing steel in tension and compression;

the hardening stiffness was deﬁned to be 0.01% of the initial

stiffness,essentially creating an elastic–plastic model.

The resulting moment–curvature response was transformed to

a moment-rotation response by multiplying curvatures by a plastic

hinge length equal to one-half the beam depth.This hinge-length

model was proposed by Corley [36] and has been found by several

research groups [16–19] to provide good estimates of member dis-

placement.Fig.5a shows the moment-rotation relation for the

beam spring,including the proposed rotation limit.Calibration of

the rotation limit is discussed in Section 3.3.

3.2.Rotational spring simulating nonlinear joint response

In the proposed model,joint nonlinearity was explicitly mod-

eled using a second rotational spring in series with the beamspring

discussed in Section 3.1.The properties of the joint spring were

Rigid Offsets

h

b

h

c

L

c

L

b

BeamSpring

Joint Spring

Dual-hinge

Lumped-plasticity

beamelement

Fig.4.Proposed model of sub-assemblage using rigid offsets and dual-hinge at

beam-joint interface.

(a)

(b)

M

hinge

θ

bea

m

M

pos

yield

M

neg

yield

θ

fail

beam

−

θ

fail

beam

M

hinge

θ

joint

M

min

yield

−

M

min

yield

θ

fail

joint

−

θ

fail

joint

K

1

K

2

K

1

K

2

Fig.5.Dual-hinge components:(a) beam spring and (b) joint spring.

A.C.Birely et al./Engineering Structures 34 (2012) 455–465

459

determined assuming that joint response may be deﬁned by a joint

shear stress versus shear strain model and related to a moment-

rotation response using frame geometry.This approach to model

calibration is similar to that used in previous studies [16–19]

employing a single joint hinge.The joint spring in the dual-hinge

includes a rotation limit beyond which the strength drops signiﬁ-

cantly.Discussion of the spring follows,with calibrated values

listed in Table 2.Calibration of the rotation limit is discussed in

Section 3.3.

To calibrate the spring stiffness,equations were derived relating

joint shear strain to rotation at the beam-joint interface and joint

shear stress to the moment at the beam-joint interface for a frame

sub-assemblage.Detailed derivations are provided in Appendix A.

To determine the relationship between joint shear strain and hinge

rotation,the story drift resulting froma joint shear strain of

c

j

,was

constrained to be equal to that resulting from equal but opposite

rotations,h

js

,in the hinges at the beam-joint interface.Imposing

this constraint results in

h

js

¼

v

c

c

j

ð7Þ

where

v

c

¼ 1

h

b

L

c

1

h

c

L

b

ð8Þ

L

c

and L

b

are the lengths of the column and beam (Fig.4),respec-

tively,and h

c

and h

b

are the depth of the column and beam,respec-

tively.Similarly,a relationship between joint shear stress,

s

j

,and

moment in the hinge,M

hinge

,was determined by relating the beam

moment at the joint face for a particular column shear to the joint

shear stress for the same column shear.For a particular column

shear load,the moment at the beam-joint interface is assumed to

transfer into the joint as a tension–compression couple,with ten-

sion and compression forces determined assuming a moment arm

of jd,where d is the depth in the beam cross-section from the ex-

treme compression ﬁber to the tension reinforcement (assumed

to be 0.9h

b

if unknown) and j is an empirically derived parameter

taken equal to 0.85 [32].The joint shear,V

j

,is deﬁned equal to

the sumof the couple forces (one tension and one compression) less

the column shear acting on the top half of the joint.The joint shear

stress is the joint shear force divided by the cross-sectional area of

the joint,A

j

.The resulting relationship between joint moment and

shear stress is

M

hinge

¼

v

s

s

j

ð9Þ

where

v

s

¼

A

j

1

h

c

L

b

jd

2 1

jd

L

c

h

c

L

b

ð10Þ

and all parameters in Eq.(10) are as previously deﬁned.

The joint shear stress is related to the shear strain by an effec-

tive shear modulus,

j

G:

j

G ¼

s

c

ð11Þ

where G is the shear modulus of the concrete.Using Eqs.(7)–(11)

the stiffness of the joint rotational spring,K,can be written:

K ¼

M

hinge

h

js

¼

j

G

v

s

v

c

ð12Þ

A bilinear shear stress–shear strain response,with a change in stiff-

ness occurring at initial beam yielding,was used.Prior research

shows that this is an approximate yet accurate model of the nonlin-

ear response of the joint [5].The initial joint spring stiffness,K

1

,was

established using the relationship (Eq.(13)) to

j

1

,the effective

shear stress–strain stiffness parameter.

K

1

¼

j

1

v

s

v

c

G ð13Þ

The parameter

j

1

,was computed as that which minimized the

cumulative error in the predicted initial yield displacement for

the data set.The Matlab (www.mathworks.com) fminbnd function

was used for the optimization,and the cumulative error was

deﬁned

e

yield

¼

X

N

i¼1

D

y

D

analytical

y

D

y

!

2

i

ð14Þ

where

D

analytical

y

is the simulated initial yield displacement corre-

sponding to the initial yield load,V

y

,

D

y

is the measured yield dis-

placement,and only specimens that reached the yield load,V

y

,

were included in the summation.The optimumeffective shear stiff-

ness factor,

j

1

,is listed in Table 2.

The post-yield tangent stiffness of the joint spring was deﬁned

as:

K

2

¼

j

2

v

s

v

c

G ð15Þ

The parameter

j

2

was determined using the simulated,

D

analytical

fy

,

and measured,

D

fy

,displacements at the column shear correspond-

ing to full yield,V

fy

.Full yield was deﬁned as the point at which both

beams had yielded,with yielding for a beamdeﬁned by initial yield-

ing of longitudinal reinforcement as determined from the moment

curvature analysis of the beamsection.The experimental displace-

ment at full yield,

D

fy

,was taken fromthe sub-assemblage response

envelope.For most sub-assemblages,the full yield force is larger

than the initial yield force due to differences in conﬁguration and

strength of the reinforcement in the top and bottom of the beams.

The full yield point,rather than the point of maximum load,was

used to determine

j

2

because the full yield point was found to pro-

vide a good representation of the post-yield stiffness while the low

tangent stiffness of sub-assemblages in the vicinity of maximum

load was found to result in a high level of variation and poor repre-

sentation of the post-yield stiffness.Only tests in which the V

fy

shear force level was achieved or exceeded were used to calibrate

the post-yield stiffness ratio.Fig.3 shows the full yield point for

specimen PEER0995.

Initially,post-yield stiffness parameter

j

2

was calibrated using

the same error-minimization approach that was used to calibrate

j

1

;however,this resulted in an unreasonable value for

j

2

.The fail-

ure of this approach was attributed to two factors.First,the post-

yield response of the dual-hinge was dominated by the response of

the beamspring and,as a result,the simulated response was fairly

insensitive to changes in the post-yield stiffness of the joint hinge.

Second,because the calibrated initial stiffness of the joint spring

underestimated the stiffness of some of the specimens,application

of the model resulted in under-prediction of stiffness and over-

prediction of displacement at full yield of the beams for these

specimens.

Table 2

Summary of model parameters.

Parameter Calibrated value

j

1

0.14

j

2

0.038

c

fail

joint

0.0069

/

fail

beam

0.0056

460 A.C.Birely et al./Engineering Structures 34 (2012) 455–465

Ultimately,

j

2

was determined by averaging the

j

i

2

values com-

puted for each sub-assemblage,i,with an imposed upper-bound

limit on

j

i

2

equal to the initial stiffness coefﬁcient

j

1

.The following

error function was minimized for each specimen to ﬁnd the

j

i

2

values

e

i

post-yield

¼

D

i

fy

D

i analytical

fy

D

i

fy

!

ð16Þ

where

D

i

fy

and

D

i analytical

fy

are as deﬁned previously.The optimal value

for

j

2

is listed in Table 2.

3.3.Spring rotation limits to determine onset of strength degradation

To simulate strength loss,rotation limits were included in both

components of the dual-hinge;beyond the rotation limit,the

strength of the springs was reduced.The characteristics of strength

loss and simulation of post-peak response were not considered,

and post-peak response is shown in Fig.5 as a dashed line.ASCE/

SEI Standard 41-06 [34] provides recommendations for simulation

of strength degradation.To determine appropriate rotation limits

for use in the model,a pushover analysis was performed for each

specimen using the dual-hinge model (without rotation limits).

The beam and joint spring rotations corresponding to the drift at

onset of degradation in lateral load-carrying capacity were re-

corded.Ductile and limited-ductility specimens were considered

to reach this limit state when 10% of the specimen’s peak strength

was lost,

D

90%

;brittle specimens were considered to reach this lim-

it state at maximum strength,

D

V

max

.Joint spring rotations were

converted to equivalent shear strains using the relationship pro-

vided by Eq.(7).Beam spring rotations were converted to equiva-

lent curvatures using an assumed hinge length (l

p

) equal to

one-half the beam depth.

To determine an appropriate approach for deﬁning rotation lim-

its for response prediction,the relationships between specimen de-

sign parameters and spring rotation limits were explored.

However,no signiﬁcant correlation was found.Comparing the

equivalent strains and curvatures at the onset of strength loss for

brittle,ductile and limited-ductility specimens did show correla-

tion between these values and sub-assemblage ductility.Fig.6

shows the equivalent shear strain and curvature values plotted

versus specimen number with different markers used to indicate

the ductility classiﬁcation of each specimen.Fig.6a shows that

the equivalent shear strains at strength loss were signiﬁcantly lar-

ger for brittle specimens than for ductile specimens;Fig.6b shows

that the equivalent beam curvatures were signiﬁcantly larger for

ductile specimens than for brittle specimens.Thus,by deﬁning

constant rotation limits for each spring in the dual-hinge model,

the onset of strength loss can be predicted without prior knowl-

edge or prediction of the response mechanism.Strength degrada-

tion is triggered by the joint spring component of the dual-hinge

for a brittle specimen and by the beam spring for a ductile

specimen.

Beam and joint spring rotation limits were deﬁned as the con-

stant values that minimized the sum of the error squared in the

simulated drift at strength loss.Equivalent beam curvatures

(/

fail

beam

) and equivalent joint shear strains (

c

fail

joint

) at initiation of

strength loss were calibrated using data from ductile and brittle

specimens,respectively.These limits were incorporated into the

dual-hinge using the relationships deﬁned by Eqs.(17) and (18).

A constant value for each parameter was found that minimized

the sumof the error squared in the simulated drift at strength loss.

The calibrated values are provided in Table 2.

h

fail

beam

¼/

fail

beam

l

p

ð17Þ

h

fail

joint

¼

c

fail

joint

v

c

ð18Þ

4.Application of the model

The proposed model is intended for use with the commercial

structural analysis software used commonly in design ofﬁces,

and can be implemented using the following procedure.

1.Determine the properties of the beam spring.Compute the

moment–curvature response for each beam cross section and

convert beam curvatures to rotations using a plastic hinge

length,l

p

,equal to one-half of the beam depth.Using the com-

puted cross-sectional response,identify yield moments for use

in creating the joint spring moment-rotation history.Compute

the beam rotation limits (positive and negative bending) for

each section,using Eq.(17) with

u

fail

beam

deﬁned in Table 2.

2.Determine the joint spring response history.Determine the ini-

tial and post-yield stiffness values using Eqs.(13) and (15),with

j

1

and

j

2

deﬁned in Table 2.The joint shear stress at which

stiffness change occurs is the joint shear stress corresponding

to initial beam yielding.The strength of the joint spring

degrades at an absolute rotation demand given by Eq.(16),with

c

fail

joint

deﬁned in Table 2.

3.Create a centerline model of the structure with elastic beam–

column elements used for the columns,lumped-plasticity

beam–column elements used for the beams,rigid offsets in

the beam and column elements deﬁning the physical volume

of the joint,and dual-hinges introduced in the lumped-plasticity

beamelements at the joint faces.The dual hinges consist of the

zero-length beamand joint springs placed in series.

4.Perform the desired analysis of the structure.

0

10

20

30

40

50

0

0.005

0.01

0.015

Joint Number

(b)

φbeam

pos

0

10

20

30

40

50

0

0.005

0.01

0.015

Joint Number

(a)

γjoint

pos

Brittle

Ductile

Limited Ductility

Fig.6.Engineering parameter values corresponding to strength drop for all data:

(a) joint spring shear strain limits and (b) beamspring curvature limits for all data.

A.C.Birely et al./Engineering Structures 34 (2012) 455–465

461

5.Comparison of simulated and observed response

For each of the sub-assemblages in the data set,a nonlinear

model was created using the steps described above and a push-

over analysis was performed.

Fig.7 illustrates the impact the joint spring has on the simu-

lated response by showing envelopes for (i) the experimental data,

(ii) an analytical model that includes a conventional beam hinge/

spring,and (iii) the proposed dual-hinge model,including termina-

tion when the rotation limits are reached.Envelopes are shown for

(a) PEER 0850,for which specimen drift is due primarily to beam

ﬂexure and (b) PEER 4150,for which specimen drift is due primar-

ily to joint shear.The addition of the joint spring to the model has a

greater impact on the response of the sub-assemblage with greater

measured shear deformation (PEER 4150).The proposed rotation

limits for the beamand joint springs result in the model accurately

simulating (a) the drift capacity of the ductile PEER 0850 specimen

and (b) the peak strength of the brittle PEER 4150 specimen.

Table 1 presents the ductility classiﬁcation for each specimen as

observed experimentally and as predicted using the model.Fig.8

shows the best and worst simulated versus observed response his-

tories for the ductile,limited-ductility and brittle specimens.

For the full data set,the model was evaluated ﬁrst by comparing

the simulated and experimentally determined ductility of each

specimen.Section 2.2 described classiﬁcation of specimen ductility

using experimental data.The simulated response of the specimen

was considered ductile if beam rotation demand exceeded the

beam spring rotation limit but not the joint spring rotation limit

and brittle if joint rotation demand exceeded the joint spring

rotation limit but not the beamspring rotation limit.Brittle behav-

ior was simulated for all 18 specimens experimentally classiﬁed as

brittle.Ductile behavior was simulated for all 20 specimens exper-

imentally classiﬁed as ductile.Of the seven limited-ductility spec-

imens,the simulated response of six was controlled by joint

failure;for these six specimens the rotation demand imposed on

the beam-ﬂexural spring was only 7% of the rotation limit.Of the

limited-ductility specimens,specimen PEER14 was the only speci-

men for which simulated strength loss was due to activation of the

beamspring rotation limit;this occurred at a joint spring rotation

demand equal to 45% of the rotation limit.Thus,the model was

considered to provide accurate simulation of specimen ductility.

To further quantify the accuracy and precision of the model,the

error in the simulated displacement/drift and load at critical points

of the response history was computed.The critical points consid-

ered were:initial yield of beam longitudinal steel (

D

y

,V

y

),maxi-

mum strength (

D

max

,V

max

),and 10% loss of lateral load carrying

capacity (

D

90%

,V

90%

).For each specimen,the error in displace-

ment/drift was computed

e

D

¼

D

measured

D

simulated

D

measured

ð19Þ

and the error in load was computed

e

V

¼

V

measured

V

simulated

V

measured

ð20Þ

Error data are reported in Table 3.

The data in Fig.8 and Table 3 indicate the following capabilities

of the model to simulate the ductile and limited-ductility

specimens.

1.Initial stiffness is simulated accurately for both the ductile spec-

imens (average displacement error of 8% at initial yield) and the

limited-ductility specimens (average error of 4%).

2.The standard deviation of the error for simulation of initial stiff-

ness for ductile and limited-ductility specimens is approxi-

mately 25%.This is considered to be acceptable for nonlinear

analysis of concrete components and is consistent with other

recently developed models for performance-based seismic

design and evaluation (e.g.,Berry et al.[37]).

3.The strength of ductile specimens is accurately and precisely

simulated with an average error of 8% and a standard deviation

of 7%.It should be noted that strength is determined by the

computed moment–curvature response history for the beam

section.

4.Displacements at maximum strength are not reported for the

non-ductile specimens.Specimens typically exhibit very low

stiffness once beam(s) reach full yield strength;thus,the dis-

placement at which maximum strength is achieved is not

meaningful.

5.For ductile specimens,the displacement at strength loss is sim-

ulated with a high level of accuracy and moderate precision

(average error of 8%;standard deviation of 35%).For limited-

ductility specimens,the accuracy of the model for simulation

of strength loss is much worse (average displacement error at

strength loss of 36%).This inaccuracy could be expected given

that the model is intended to simulate the onset of strength loss

in ductile specimens and was calibrated using data for these

specimens.

For brittle specimens,the data in Fig.8 and Table 3 support the

following conclusions.

1.Initial stiffness (same as stiffness at maximum load) for brittle

joints is,on average,fairly accurate (8% average error) but

lacks precision (55% standard deviation).

0

1

2

3

4

5

6

0

50

100

150

200

250

% Drift

(a)

Column Shear [kN]

Experimental

Conventional Hinge

Dual−Hinge

Dual−Hinge Limit

0

1

2

3

4

5

6

0

200

400

600

800

1000

% Drift

(b)

Column Shear [kN]

Experimental

Conventional Hinge

Dual−Hinge

Dual−Hinge Limit

Fig.7.Comparison of experimental response to simulated response using a

conventional beam hinge and the proposed dual-hinge for specimens (a) PEER

0850 and (b) PEER 4150.

462 A.C.Birely et al./Engineering Structures 34 (2012) 455–465

2.The model provides an accurate and precise simulation of max-

imumstrength (average error of 5%,standard deviation of 9%)

as well as accurate simulation of displacement at maximum

strength (average error of 8%).

3.The simulated displacement at maximum strength has a rela-

tively high level of uncertainty,with a standard deviation of

53%;this was attributed to the simplicity of the model.

As previously discussed,the model provides accurate simula-

tion of frame ductility and,thus,may be used in evaluation of

existing reinforced concrete frames to assess frame ductility under

seismic loading.To further explore the parameters that control

frame ductility,the computed joint shear stress demands for duc-

tile and brittle joints were compared.The simulated maximum

joint shear stress,

s

model

max

,was computed as follows:

s

model

max

¼

M

max

L

þM

max

R

2

v

s

ð21Þ

where M

max

L

and M

max

R

are the maximummoments developed in the

left and the right beams,respectively.Values are provided in Ta-

ble 1.The simulated maximum shear stresses are similar to those

computed directly from experimental data using Eq.(5);the aver-

age difference between simulated and experimental stresses was

approximately 10%.Joint failure controlled the response of brittle

specimens;thus,for these specimens,the stress computed using

Eq.(21) is the joint shear strength (capacity).For brittle specimens,

simulated joint shear stress demands at failure ranged from

17.9

ﬃﬃﬃﬃ

f

0

c

p

psi to 23.7

ﬃﬃﬃﬃ

f

0

c

p

psi,with an average value of 20.5

ﬃﬃﬃﬃ

f

0

c

p

psi

and a coefﬁcient of variation of 7%.For ductile specimens,specimen

response was controlled by beam yielding and the joint shear de-

mand,computed using Eq.(21),is less than the joint shear strength.

For ductile specimens,shear stress demand ranged from5.7

ﬃﬃﬃﬃ

f

0

c

p

psi

to 14.1

ﬃﬃﬃﬃ

f

0

c

p

psi with an average of 10.6

ﬃﬃﬃﬃ

f

0

c

p

psi and coefﬁcient of var-

iation of 22%.Because there is no overlap in the ranges of simulated

shear stress for the ductile and brittle specimens,and given the rel-

ative low coefﬁcient of variation on shear strength for brittle spec-

imen,a strength-based limit model for the joint spring could be

expected to provide reasonably accurate prediction of strength.

However,the strain-based model simulates the range of strengths

observed in the lab,provides accurate (5% error) and precise (9%

standard deviation) prediction of strength and provides accurate

prediction of the drop at maximum strength (8% error).

6.Summary and conclusions

A dual-hinge lumped-plasticity beamelement was developed to

provide a practical model capable of simulating the nonlinear re-

sponse of planar concrete frames.The dual-hinge consists of two

0

2

4

6

0

100

200

300

400

% Drift

(f)

Column Shear [kN]

0

2

4

6

8

0

50

100

150

% Drift

(e)

Column Shear [kN]

0

2

4

6

0

100

200

300

400

500

% Drift

(d)

Column Shear [kN]

Experimental

Dual−Hinge

Dual−Hinge Limit

Experimental

Dual−Hinge

Dual−Hinge Limit

Experimental

Dual−Hinge

Dual−Hinge Limit

0

2

4

6

0

100

200

300

% Drift

(c)

Column Shear [kN]

0

2

4

6

0

20

40

60

80

% Drift

(b)

Column Shear [kN]

0

2

4

6

0

50

100

150

% Drift

(a)

Column Shear [kN]

Experimental

Dual−Hinge

Dual−Hinge Limit

Experimental

Dual−Hinge

Dual−Hinge Limit

Experimental

Dual−Hinge

Dual−Hinge Limit

Fig.8.Experimental (open circles) and model (thick line) envelopes for best and worst predictions.Best (a) brittle,MJ3,(b) ductile,PR3,and (c) limited-ductility,OSJ.Worst

(d) brittle,PEER22,(e) ductile,HC,and (f) limited-ductility,OSJ8.

Table 3

Error in proposed model for evaluation of experimental frame sub-assemblages.

Data subsets Normalized error Initial yield Max.load 10% loss

Stiffness Disp.Load Disp.Load

Ductile Average 8% – 8% 8% 2%

Stand.dev.25% – 7% 35% 9%

Limited ductility Average 4% – 11% 36% 5%

Stand.dev.24% – 9% 41% 11%

Brittle Average

1

8% 8% 5% – –

Stand.dev.

1

55% 55% 9% – –

1

For brittle joints,the stiffness at yield is the same as that at maximum load

A.C.Birely et al./Engineering Structures 34 (2012) 455–465

463

rotational springs in series to simulate the nonlinear ﬂexural re-

sponse of the beam and the nonlinear response of the joint.The

beam spring response is determined from the moment–curvature

response of the beam cross section and an assumed plastic hinge

length.A rotation limit for the beamspring,which deﬁnes the on-

set of strength loss,was determined using laboratory data from

frame tests in which beams yielded in ﬂexure.The rotation limit

was represented as an equivalent curvature value.The joint spring

response is deﬁned by a bilinear shear stress–strain relationship,

which is converted to a moment-rotation response using geomet-

ric transformations.Joint spring stiffnesses were determined using

laboratory data from frame sub-assemablage tests.Experimental

data from frame sub-assemblages exhibiting joint failure prior to

beamyielding were used to determine a joint spring rotation limit

at which strength loss initiates.This rotation limit was represented

as an equivalent shear strain value.

Frame models were constructed using standard beam column

elements and the dual-hinge spring model to simulate the re-

sponse of 45 experimental sub-assemblages.The interaction of

the beam and joint springs and the introduction of rotation limits

in both springs resulted in a model that accurately simulated the

ductile (beam-controlled) or brittle (joint-controlled) response ob-

served in the laboratory.Results show that the model provides

accurate simulation of initial stiffness,strength and displacement

at strength loss for ductile specimens as well as accurate simula-

tion of stiffness and strength for brittle specimens.

Acknowledgements

Support of this work was provided primarily by the Earthquake

Engineering Research Centers Program of the National Science

Foundation,under Award Number EEC-97015668 through the

Paciﬁc Earthquake Engineering Research Center (PEER).Any

opinions,ﬁndings,and conclusions or recommendations expressed

in this material are those of the authors and do not necessarily

reﬂect those of the National Science Foundation.

Appendix A

A.1.Derivation of shear force transformation coefﬁcient

Fig.A.1 shows the reaction forces of a frame sub-assemblage

subject to a column shear V

c

.The forces acting on the joint are

shown in Fig.A.2.For a moment M

jf

at the face of the joint,the col-

umn shear V

c

is:

V

c

¼

2M

jf

L

c

1

h

c

L

b

ðA:1Þ

and the moment produces the tension/compression couples:

T ¼ C ¼

M

jf

jd

ðA:2Þ

where jd is the beammoment armwith d equal to the distance from

the extreme compression ﬁber to the extreme tension ﬁber and j is

taken as 0.85.Fromthe column shear and tension/compression cou-

ples,the joint shear force can be calculated as:

V

j

¼ C þT V

c

ðA:3Þ

Because the joint shear force V

j

is equal to the shear stress times the

area of the joint,the shear stress can be written as a function of the

moment M

jf

:

s

¼

V

j

A

j

¼

2M

jf

A

j

1

jd

1

L

c

1

h

c

L

b

0

@

1

A

ðA:4Þ

The moment at the joint face can then be expressed as a function of

the shear stress and joint geometry:

M

jf

¼

v

s

s

ðA:5Þ

v

s

¼

A

j

2

1

1

jd

1

L

c

1

h

c

L

b

0

B

B

@

1

C

C

A

ðA:6Þ

The transformation coefﬁcient

v

s

can be simpliﬁed to:

v

s

¼

A

j

jd 1

h

c

L

b

2 1

jd

L

c

h

c

L

b

ðA:7Þ

A.2.Derivation of shear strain transformation coefﬁcient

Fig.A.3 shows an idealization of a frame sub-assemblage with a

ﬁnite volume joint and centerline elements for the beams and col-

umns framing into the joint.The joint is subject to a shear strain

c

,

resulting in displacements

D

b

and

D

c

of the beams and columns,

respectively.

D

b

¼

c

2

L

c

2

h

b

ðA:8Þ

D

c

¼

c

2

L

b

2

h

c

ðA:9Þ

In the proposed model,shown in Fig.A.4,the column is modeled

such that the column extends to the center of the joint,rather than

the face of the joint.The displacement

D

c

can be replicated using a

rotation of h

c

,measured from the vertical axis,at the center of the

joint:

L

c

1/2 (L

b

-h

c

)

L

b

V

b

V

c

V

c

V

b

M

jf

Fig.A.1.Sub-assemblage forces and reactions.

V

c

V

j

C

T

T

C

h

c

h

b

jd

M

jf

M

jf

Fig.A.2.Forces acting on joint.

464 A.C.Birely et al./Engineering Structures 34 (2012) 455–465

h

c

¼

c

2

1

2h

b

L

c

ðA:10Þ

This rotation causes a displacement in the beams of:

D

h

c

b

¼ h

c

L

b

2

¼

c

2

L

b

2

h

b

L

b

L

c

ðA:11Þ

The hinge at the beam–joint interface must account for the differ-

ence between the beam displacement of the physical system,

D

b

,

and the beamdisplacement due to the column rotation of the mod-

eled system,

D

hc

c

.This is represented in equation form as:

D

b

D

h

c

b

¼

1

2

ðL

b

h

c

Þh

b

ðA:12Þ

Inserting Eqs.(A.9) and (A.11) into Eq.(A.12) and solving for h

b

pro-

vides a relationship between the spring rotation h

b

and the shear

strain

c

that will ensure that the boundary conditions of the model

are satisﬁed.

h

b

¼

c

L

b

h

b

L

b

L

c

h

c

1

L

b

h

c

ðA:13Þ

h

b

¼

c

1

h

b

L

c

h

c

L

b

0

@

1

A

ðA:14Þ

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Fig.A.3.Deformations of sub-assemblage due to joint shear strain.

Fig.A.4.Displacements and rotations of proposed model.

A.C.Birely et al./Engineering Structures 34 (2012) 455–465

465

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