9_315

Journal of Advanced Concrete Technology Vol. 9, No. 3, 315-326, October 2011 / Copyright © 2011 Japan Concrete Institute 315

Scientific paper

Elongation of Plastic Hinges in Ductile RC Members: Model Development Brian H. H. Peng1, Rajesh P. Dhakal2, Richard C. Fenwick2, Athol J. Carr3 and Des K. Bull3

Received 22 March 2011, accepted 8 July 2011

Abstract This paper describes the development of an analytical plastic hinge element that can predict flexural, axial, shear and elongation deformation in plastic hinges. The element consists of layers of longitudinal and diagonal springs that repre-sent the behavior of concrete and reinforcement. These springs are modeled using nonlinear path-dependent cyclic stress-strain relationships of either concrete or reinforcing bars depending on their position in the cross section and also depending on the mechanisms they are representing. Through a preliminary scrutiny of the analytical results, in-depth qualitative and quantitative information on the causes and consequences of plastic hinge elongation are discussed. An extensive experimental verification of the plastic hinge element is presented in a companion paper.

1. Introduction

Performance assessment of structures is a key step in performance based seismic design. As conducting ex-periments on each design alternative, or its scaled model, to assess its seismic performance requires significant time and resources, analytical methods are usually em-ployed to check if the conceived structural system satis-fies the seismic performance requirements. Common analysis procedures used to assess the seismic perform-ance of structures include pushover analysis, quasi-static analysis and time-history analysis. In these proce-dures, the structure and the actions representing differ-ent limit states need to be modeled appropriately. Al-though different limit states and their corresponding hazard levels are quantified in relevant standards/codes, structural modeling is somewhat subjective. While some simplifications in the adopted structural model are ac-ceptable, it is imperative from the perspective of struc-tural safety that major mechanisms affecting the seismic performance of a structure are accounted for in the ana-lytical model.

To assess seismic performance of reinforced concrete (RC) frames, the common analytical approach uses frame elements to represent the beams and columns where lumped plasticity models in the form of moment-curvature relationships are fed into the program and the length of plastic hinges is implicitly assumed. Such

models can be calibrated to capture the hysteresis re-sponse of frames, but they cannot capture elongation of the beams when subjected to inelastic cyclic actions. Hence, analysis using such models cannot be used for reliable seismic performance assessment as they fail to capture the additional inter-story drift demand and the beam strength enhancement from floor-frame interac-tions, which are important bi-products of beam elonga-tion.

Experimental studies over the last two decades have shown that RC plastic hinges in beams elongate be-tween 2 and 5 percent of beam depth under repetitive cyclic loading before strength degradation occurs (Fen-wick et al. 1981; Issa 1997; Matti 1998; Liddell et al. 2000; Walker and Dhakal 2009). This elongation occurs mainly due to an irrecoverable plastic strain in the rein-forcing bars. Even though elongation has been shown to have significant influence on the seismic performance of RC frames (Fenwick and Davidson 1993; Kim et al. 2004), it is generally overlooked in design and analysis due to a lack of a satisfactory analytical model for pre-dicting elongation response of plastic hinges.

Methods for predicting elongation of plastic hinges have been proposed in the past (Fenwick and Megget 1993; Lee and Watanabe 2003; Matthews et al. 2004). Most of these methods were developed empirically from experimental results. While some of these predict the elongation behavior satisfactorily, they cannot be read-ily incorporated into time-history analysis programs. An analytical elongation model for RC beams was proposed by Douglas et al (1996) and later refined by Lau et al (2003). These models employ a filament type element to represent plastic hinge region. Although the models have shown to have promising elongation prediction, they require the members to be calibrated prior to analy-sis. An elongation model for precast RC beams contain-ing prestressed tendons has also been proposed by Kim et al (2004). However, this could not be applied directly to predict elongation of plastic hinges in monolithic

1Structural Engineer, Holmes Consulting Group, Wellington, New Zealand. 2Associate Professor, Department of Civil and Natural Resources Engineering, University of Canterbury, Christchurch, New Zealand. E-mail: [email protected] 3Professor, Department of Civil and Natural Resources Engineering, University of Canterbury, Christchurch, New Zealand.

316 B. H.H. Peng, R. P. Dhakal, R. C. Fenwick, A. J. Carr and D. K. Bull / Journal of Advanced Concrete Technology Vol. 9, No. 3, 315-326, 2011

moment resisting frames where the behavior is more complex.

While detailed finite element analysis, using a fiber model implementing fully path-dependent cyclic mate-rial models for reinforcing bars and concrete fibers, may be able to predict the elongation response, such models are seldom used to analyze large scale structures. There-fore, it is desirable to have a fiber-model based plastic hinge element that can be used together with the tradi-tional frame elements to predict seismic performance of RC frames. This paper describes the development of a generic plastic hinge element that can account for elon-gation of plastic hinges in ductile RC beams.

2. Causes of elongation in plastic hinges

In seismic design of ductile structures, strengths of dif-ferent members in a frame are proportioned such that in the event of a major earthquake brittle failure modes, such as shear failure or column sway mechanisms, will not occur. Moreover, plastic hinges in RC structures are carefully positioned and detailed to ensure that struc-tures behave in a ductile manner. These plastic hinges are designed to sustain large inelastic rotations while maintaining their strengths. This action induces large plastic strains in flexural reinforcement and conse-quently plastic hinges grow in length under cyclic load-ing. Even in the presence of an axial compression, plas-tic hinges are found to elongate, albeit to a lesser extent.

Elongation behavior differs significantly in two dif-ferent types of plastic hinges, namely uni-directional and reversing plastic hinges (Fenwick and Megget 1993). Uni-directional plastic hinges typically form in gravity dominated frames where the gravity-induced moments are relatively large when compared to the seismic moments. In these frames, the maximum posi-tive (tension on the bottom) and negative (tension on the top) moments occur at different locations in the beam as the frame sways in both directions. On the other hand, reversing plastic hinges form in seismic dominated frames where the seismic moments are relatively large compared to the gravity moments. In this case, both the maximum positive and maximum negative moments in

beams occur next to the column faces. It has been found that the rotational demand in reversing plastic hinges is much smaller than that incurred in uni-directional plas-tic hinges (Fenwick and Megget 1993).

Elongation in reversing plastic hinges arises due to two main factors (Fenwick and Megget 1993). One is related to inelastic extension of tension reinforcement required to accommodate the inelastic rotation of plastic hinges. The other is due to irrecoverable extension in the compression reinforcement. These are illustrated in Fig. 1 where the extension of top and bottom reinforce-ment over the plastic hinge region in a beam test is plot-ted. In this figure, LT represents the extension of tension reinforcement associated with inelastic rotation and LC represents the irrecoverable extension of the compres-sion reinforcement.

The irrecoverable extension of the compression rein-forcement observed in Fig. 1 was found to arise due to two main actions:

1. Intersecting diagonal cracks in the plastic hinge region greatly reduce the shear resistance by ag-gregate interlock and dowel action of reinforce-ment. Consequently, truss like actions develop in the plastic hinges where the shear force is resisted by the shear reinforcement and diagonal compres-sion struts in the webs, as illustrated in Fig. 2a. In this figure, T and C are the flexural tension and compression forces in the reinforcement; V is the shear force acting in the beam, θ is the angle of the diagonal struts to the horizontal plane, s is the stirrup spacing, (d – d’) is the distance between the centroids of top and bottom reinforcement and LP is the horizontal projection of the diagonal strut. To satisfy force equilibrium at a given sec-tion, the flexural compression force in the rein-forcement must be smaller than the flexural ten-sion force as the horizontal component of the di-agonal compression force in the web also con-tributes to the total compression force in the sec-tion. Consequently, the compression reinforce-ment tends not to yield back; that is the plastic tensile strain developed in the compression rein-forcement in the previous loading half-cycle does

58

250

442

-1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Elongation (mm)

Ver

tical

hei

ght u

pth

e be

am (m

m)

LT

Level of top reinforcement

Level of bottom reinforcement

LC

Fig. 1 Components of reinforcement extension in a typical plastic hinge region leading to beam elongation.

B. H.H. Peng, R. P. Dhakal, R. C. Fenwick, A. J. Carr and D. K. Bull / Journal of Advanced Concrete Technology Vol. 9, No. 3, 315-326, 2011 317

not recover completely. As a result, inelastic ex-tension in the reinforcement tends to accumulate as the cyclic loading continues.

2. As shown in Fig. 2b, aggregate particles become dislodged from crack surfaces and these prevent the cracks from closing fully when the load is re-versed, particularly when large shear displace-ment develops across the cracks. This phenome-non is also known as ‘contact stress effect’ where the concrete member experiences compressive stress before the strain reverses into compression.

For uni-directional plastic hinges, as reinforcement on one side never yields in tension, elongation arises only due to extension in the tension reinforcement from cumulative inelastic plastic hinge rotation as described in more detail elsewhere (Fenwick and Davidson 1993; Fenwick and Megget 1993).

3. Development of plastic hinge element

A schematic representation of the proposed plastic hinge element is shown in Fig. 3 where LP is the length of the plastic hinge element and D is the effective width of the diagonal struts. The plastic hinge element consists of a

series of longitudinal and diagonal axial springs con-nected between the rigid links at the two ends. The lon-gitudinal springs are used to represent flexural and axial response of the plastic hinge, and the diagonal springs are used to represent the diagonal compression struts in the web, which provide shear resistance. The moment and shear are evaluated at the centre of the plastic hinge element and are extrapolated to obtain the nodal mo-ment and forces at the two ends of the plastic hinge element.

In the element, two steel springs are located at the centroids of the top and bottom reinforcement to repre-sent the reinforcing bars, two concrete springs are lo-cated at the centre of the top and bottom covers to rep-resent unconfined concrete, eight concrete springs are distributed evenly between the centroids of the tensile and compressive reinforcing bars to represent the con-fined core concrete and two diagonal concrete springs are connected between the ends of the top and bottom steel springs to represent the diagonal compression struts.

3.1 Length of plastic hinge element The length of the plastic hinge element, LP, (see Fig. 3)

C = T – V / tan θ

T

V d - d’

θ

s

LP

Fig. 2 Causes of irrecoverable extension of the compression reinforcement.

(a)Truss-like action in the plastic hinge region

(b) Wedging action of cracked concrete

Aggregate particles fall into cracks

Deformed bar


is chosen to represent the inclination of the diagonal compression struts, θ, in the plastic hinge region as il-lustrated in Fig. 2a. This length does not represent the length of actual plastic hinge or ductile detailing length as used in concrete structures codes in different coun-tries (ACI318 2005; NZS3101 2006). Using the truss analogy in this study, it is hypothesized that the diagonal cracks will form in an angle such that it crosses just enough stirrups to resist the shear demand in the web. Consequently, LP is equal to the number of stirrups re-quired to resist the shear force multiplied by the spacing of the stirrups, s. This can be expressed as Equation 1 where Vyc is the shear force corresponding to the flex-ural strength of the beam, Myc, given by Equation 2, and Vc is the shear resistance of concrete.

( ) yc cP

v vy

V V sL

A f−

= (1)

( ) ( )''2yc s y

d dM A f d d P

−= − + (2)

The theoretical flexural strength, Myc, where the com-pression steel has been previously yielded in tension, is used in Equation 2 instead of the nominal flexural strength. This is because under cyclic loading where the compression reinforcement has been yielded in tension in the previous cycles, majority of the compression force is resisted by the compression reinforcement unless compression bars yield back and the cracks close fully. It should also be noted that Vyc is generally smaller than the maximum shear force sustained in the beam due to strain hardening of the longitudinal reinforce-ment. In Equation 2, (d - d’) is the distance between the centroids of reinforcing bars, Av, As, fvy, fy are the area and yield stress of the shear and longitudinal reinforce-ment, respectively and P is the applied axial force.

In current concrete structures codes (ACI318 2005; NZS3101 2006), it is commonly assumed that the shear resistance of concrete in beam plastic hinges is negligi-

ble. However, as the axial compression force increases the shear resistance of concrete should also increase. Unfortunately, there are no appropriate guidelines avail-able in literature that specify the shear resistance of con-crete in plastic hinges with different levels of axial force. Therefore, despite acknowledging that the concrete con-tribution to shear resistance may not be insignificant in the presence of axial compression, the concrete shear resistance is taken as zero in this study.

3.2 Stiffness of steel springs As the flexural or bending stiffness of the plastic hinge element is governed by the axial stiffness of the longitu-dinal springs, it is imperative to feed accurate axial stiffness values for these springs to correctly model the flexural response of the plastic hinges. The axial stiff-ness of these springs is calculated as the product of the tangent modulus and cross-section area divided by the length of the spring. As concrete cracks at an early stage and the post-cracking tangent modulus of concrete and therefore the stiffness of the concrete springs is insig-nificant, the overall flexural behavior of the plastic hinge element is largely governed by the axial behavior of the steel springs, which represents the average behav-ior of the reinforcing bars over the yielded length. As illustrated in Fig. 4, the actual length over which the reinforcement yields, Lyield, is given by Equation 3 (Dhakal and Fenwick 2008) where M/V is the moment to shear ratio, Mmax is the maximum moment sustained by the beam, Lts is the length of tension shift and Le is the length of yield penetration into the support.

max

max

ycyield ts e

M MML L LV M

−= + + (3)

It should be noted that the length of the plastic hinge element, LP, does not correspond to the yielding length of the reinforcing bars and the stiffness of the steel spring calculated using LP would lead to inaccurate pre-dictions. The length of steel spring in the plastic hinge element is set as Lyield to represent the correct stiffness.

Confined concrete spring

Steel spring (Length = Lyield)

Rigid link

Diagonal concrete spring

Covered concrete spring

LP

θ

D

Fig. 3 Schematic illustration of the plastic hinge model.


The yielding length of the reinforcing bars changes with the level of loading. However, as the length of a spring in the plastic hinge element cannot be readily varied during different stages of an analysis, a constant yield length corresponding to the maximum flexural capacity of the member is used in the element. There-fore, Equation 3 requires the maximum moment, Mmax, to be predetermined. This can be assessed from the ex-perimental results. However, for analytical prediction this can be estimated manually by giving due considera-tion to the strain hardening of reinforcing bars, or by using appropriate section analysis packages such as Re-sponse 2000. In general, the length of yield penetration, Le, is assumed to be a portion of the development length. For a beam with no axial force, the length of tension shift Lts can be approximated using Equation 4 (Paulay and Priestley 1992).

'2ts

d dL −= (4)

This is based on the assumption that the diagonal crack extends over a distance (d - d’) along the member at the low moment end of plastic hinges as illustrated in

Fig. 5, where C1 is the flexural compression force at section 1, T2 is the flexural tension force at section 2 and Vs is the shear force resisted by the shear reinforcement. As the shear force in the plastic hinge region is assumed to be carried solely by the stirrups crossing the crack, Vs = V. From Fig. 5, the moment at section 1 can be ex-pressed as

1 2 ( ') 0.5( ')M T d d d d V= − + − (5)

2 ( ')M V d d= + − (6)

Rearranging the above equations; a relationship be-tween the tension force and moment at section 2 can be derived as

22 0.5

'MT V

d d= +

− (7)

In Equation 7, the term 0.5V implies that the flexural tension force at section 2 is proportional to the moment at a distance 0.5(d – d’) to the left of the section. For beams with axial compression force, the diagonal crack angle would decrease and the length of tension shift Lts would increase. Unfortunately, the relationship between

Mmax

Myc

Actual tension force

Theoretical tension force

Support interface

Lts Le

Lyield

M / V

Fig. 4 Tension force in the beam.

C1

1 2

T2

M d - d’

d - d’

V Vs

Fig. 5 Internal forces in a reinforced concrete member at low moment end of a plastic hinge.


the crack angle and the applied axial force in the low moment end of plastic hinges under reversing cyclic actions is not readily available in literature. Therefore, the length of tension shift is assumed to be 0.5(d – d’) for any axial load case.

3.3 Stiffness and strength of diagonal struts The initial stiffness of the diagonal struts is calculated as the product of its cross sectional area and the elastic modulus of concrete divided by the length. The area of the diagonal concrete spring is simply the width of the beam, b, multiplied by the effective depth of the com-pression strut, D, commonly used in strut and tie analy-sis. The effective depth is taken as the perpendicular distance from the diagonal strut to the end-point of the reinforcement spring as illustrated in Fig. 3 and can be calculated using Equation 8.

sinPD L θ= (8)

The strength of the diagonal spring is different from the uni-axial compressive strength of concrete due to the presence of diagonal cracks and tensile strains in the orthogonal direction. It has been found that transverse tensile strain in concrete members reduces the longitu-dinal concrete compressive strength (Vecchio and Collins 1986; Schlaich et al. 1987; Pang and Hsu 1995, 1996). For regions where significant inelastic strain oc-curs in the neighboring reinforcement, To et al (2001) found that the effective concrete compressive strength of the diagonal struts can be taken as 0.34f ’c. Hence, the effective compressive strength of the diagonal springs is set as 0.34f ’c.

3.4 Material models As the extension of tensile reinforcement due to inelas-tic rotation is related to the cyclic stress strain behavior of reinforcing steel and the irrecoverable extension of the compression reinforcement is related to the contact stress effect, which is a unique feature of cyclic stress strain relationship of concrete, the ability of the pro-posed plastic hinge element to capture the cyclic re-sponse and to predict the elongation depends mainly on how accurately the path-dependent cyclic behavior of the axial springs is modeled. The material models adopted for the concrete and steel springs are based on uni-axial averaged (over the spring length) stress-strain relationship of concrete and reinforcing steel. A para-metric study was conducted by changing the compres-sive strength and peak strain of the confined concrete springs, and it was found that the difference in the pre-dicted response was very small (Peng 2009). This was expected because most of the concrete springs are gen-erally in tension due to the bending and elongation. Therefore, the same stress-strain relationship was used for both cover and confined concrete springs.

The envelope of the constitutive relationship used for the concrete springs comprises a tension stiffening model in tension and an Elasto-Plastic and Fracture

(EPF) model in compression. Detailed information on these models can be found elsewhere (Shima et al. 1987; Maekawa et al. 2003). The relationships used to trace the cyclic unloading and reloading loops are also important for the plastic hinge element. A schematic illustration of the path-dependent cyclic loops adopted for the concrete springs and the response of a concrete spring during an arbitrary loading path are shown in Fig. 6a. As can be seen in the figures, the loss of stiffness due to fracture of concrete is taken into account in the cyclic model. Also, the unloading loop from tension into compression envelope includes an allowance for contact stress effect where axial compression stress develops before the cracks fully close. More details on the fea-tures of the concrete cyclic behavior can be found in Okamura and Maekawa (1991).

The level of contact stress effect in the original model was based on the results of uni-axial cyclic tests. How-ever, for RC members under combined axial and shear actions, contact stress effect would be more pronounced than that observed in the uni-axial tests. The level of contact stresses would depend on the level of axial force and shear displacement sustained across the cracks, the aggregate size, concrete compressive strength, and dis-placement histories applied to the specimen. As there is currently no literature available to quantify these effects, an intuitive approach is used to qualitatively take this difference into account. In the original concrete cyclic stress-strain relationship, the tensile strain at which the contact stress effect starts during the unloading from a large tensile strain is multiplied by a factor of 1.5.

The cyclic constitutive model for steel springs is based on path-dependent averaged stress-strain relation-ship developed for reinforcing steel (Dhakal and Maekawa 2002a). Although the original steel cyclic model accounted for buckling of reinforcing bars inside RC members (Dhakal and Maekawa 2002b and 2002c), buckling related features were switched off for the steel springs in the plastic hinge element. This is justified because incorporating buckling renders the model more complex without adding significant value to it because as shown in Fig. 1, elongation results from extension of reinforcing bars which has little relation with buckling. Figure 6b schematically illustrates the cyclic model used for the steel springs and it also shows the response of a steel spring during an arbitrary loading. The path-dependent cyclic model for steel consists of compres-sion and tension envelopes, which can represent any combination of yield plateau and strain hardening, and the unloading/reloading loops represented by Giuffre-Menegotto-Pinto model which takes into account the Bauschinger effect.

4. Analytical results

A generic beam was set up to examine the proposed analysis approach using the developed plastic hinge element. The beam has a shear span of 1500 mm, a


width of 200 mm and a depth of 500 mm. It contains 5 deformed 20 mm diameter longitudinal bars at the top and bottom. The distance between the centroids of the top and bottom reinforcement is 384 mm. The shear reinforcement comprised of 2 legs of 10 mm diameter and one leg of 6 mm diameter plain round bars at 100 mm spacing. The effective depth of the section is 442 mm. The concrete compressive strength is 38 MPa and the yield stress of the steel is 300 MPa for the 10 mm and 20 mm bars and 350 MPa for the 6 mm round bars. Using these geometrical and mechanical properties, the flexural strength of the beam, Myc, is 185 kNm, and the maximum moment, Mmax, is calculated as 217 kNm. The corresponding plastic hinge length, Lp, is approximately 220 mm, the steel yield length, Lyield, is 463 mm and the effective depth of the diagonal spring is 190 mm. Gradually increasing displacement cycles were applied to the beam at the loading point (the free end); the re-versed cyclic displacement history comprised of two cycles each of amplitude 6 mm, 16 mm, and 32 mm which correspond to a displacement ductilities of 0.75, 2 and 4 (μ0.75, μ2 and μ4 respectively).

The cantilever beam, as shown in Fig. 7, was mod-

eled in RUAUMOKO2D (Carr 2004), an inelastic time history analysis program. In this figure, Δ is the applied displacement, L is the length of the shear span and LP is the length of the plastic hinge element. The beam is divided into two parts; namely elastic and plastic re-gions. The elastic region is modeled using elastic beam element and the plastic region is modeled using the newly-developed plastic hinge element. The shear modulus of the elastic beam is taken as 0.4Ec calculated according to Clause 6.9.1 in the New Zealand Concrete Structures Standard NZS3101 (2006). The shear area is taken as bd, where b is the width of the section and d is the effective depth. The effective moment of inertia of the elastic beam, Ieff, is taken as 0.4Ig given by Table C6.6 in NZS3101 (2006) where Ig is the gross section moment of inertia.

The hysteresis and elongation responses of the beam predicted from the proposed analytical approach are shown in Fig. 8. The letters A to O in the different plots correspond to each other and they indicate different stages of the beam’s cyclic response. Figure 8a shows the force-displacement relationship at the beam tip, Fig. 8b plots the shear force vs. shear displacement of the

No buckling

Bauschinger effect (Giuffre-Menegotto-

Pinto model)

Bauschinger effect

Tension envelope

fy

Es E=Es

Stress

Strain

fy

Compression envelope

-500-400-300-200-100

0100200300400500

-0.1 -0.05 0 0.05 0.1Strain

Stre

ss (M

Pa)

Cyclic

Monotonic

(b) Steel hysteresis

Fig. 6 Material hysteretic loops.

Tension stiffening model

EPF model

Contact stress effect

fc’

ft

Ec

E < Ec

Strain

Stress

-40-35-30-25-20-15-10-505

-0.004 -0.002 0 0.002 0.004Strain

Stre

ss (M

Pa)

(a) Concrete hysteresis


plastic hinge element and Fig. 8c shows the moment-rotation response of the beam. It can be seen that the force vs. total displacement loops show some pinching, which arises from shear response of plastic hinges. For example, points L and M almost coincide in the mo-

ment-rotation curve (Fig. 8c), but in Fig. 8b they refer to the start and end of the shear pinching in that cycle which arises due to opening and closing of the diagonal cracks. Comparison between Fig. 8a and Fig. 8b reveals that the shear deformation accounts for about 10% to

L LP

Δ D20

R10

R6 500

200

Fig. 7 Analytical model for cantilever beam.

-150

-100

-50

0

50

100

150

-40 -20 0 20 40

Applied Displacement (mm)

Shea

r For

ce (k

N)

A

B

C

DE

J

F

G

H

I K

L

M

N

O

-150

-100

-50

0

50

100

150

-6 -4 -2 0 2 4 6

Shear Displacement (mm)

Shea

r For

ce (k

N)

A

B

C

DE J

F

G

H

I K

M

N

O

L

Vyc

-300

-200

-100

0

100

200

300

-0.03 -0.02 -0.01 0 0.01 0.02 0.03Rotation (rads)

Mom

ent (

kNm

)

A

B

C

D E

J

F

G

H

I K

L

M

N

O

0

2

4

6

8

10

12

-40 -20 0 20 40

Applied Displacement (mm)

Elon

gatio

n (m

m)

BA CD

EH

GF

I J

KLM

N

OMyc

(a) Force-displacement relationship (b) Force-shear displacement relationship

(c) Moment-rotation relationship (d) Elongation history

Fig. 8 Analytical global response of the beam.


15% of the total deformation in this case. Figure 8b also shows that the shear deformation is noticeably lar-ger during the second cycle than in the first cycle of the same displacement amplitude (see points E and J). Ob-viously, the rotation in the second cycle has to decrease slightly to accommodate this increase in shear deforma-tion, which can be observed in Fig. 8c.

Figure 8d shows the predicted elongation response of the plastic hinge. The plotted elongation refers to the axial displacement of the plastic hinge element, which is calculated at the mid-height of the beam section. During the elastic loading, between points B and C, the beam elongates during the loading phase due to the neutral axis not coinciding with the nodal line at the mid-height; however there is no residual/permanent elonga-tion in this elastic loading range. As the beam starts to undergo inelastic response, after point D, elongation starts to accumulate. During the loading phase of a cycle, the elongation increases rapidly, but only a small por-tion of this is recovered during the unloading phase until the displacement is reversed to zero, after which the elongation starts to increase again. Another noteworthy point in this figure is the significant increase in the elongation in the second cycle of the same displacement amplitude compared to the first cycle. However, this increase in elongation between the second and the third cycles is significantly less than that between the first and the second cycles. For example, compare points E and J with points J and a point at the same displacement in the line immediately above it in Fig. 8d.

The above mentioned predictions are natural out-comes of the plastic hinge element, which do not require any calibration. This is a significant advantage over conventional frame analysis where many model parame-ters need to be calibrated to get the desired cyclic re-sponse and in the absence of experimental results the analyst has to use subjective judgment to assign values for these parameters. Moreover, the conventional frame analysis does not have the capability to predict elonga-tion of the plastic hinge and it would not be possible to account for and extract the inelastic shear deformation as shown in Fig. 8b. Hence, the proposed plastic hinge element provides an objective and more rational ap-proach to conduct analytical performance assessment of RC frame structures. An extensive experimental verifi-cation of the plastic hinge element is presented in a companion paper (Peng et al 2010).

5. Discussions: Contribution of the plastic hinge element

To facilitate discussion on qualitative and quantitative contribution of the newly developed plastic hinge ele-ment on the predicted hysteretic and elongation re-sponses of the beam; axial responses of four longitudi-nal springs (two representing reinforcing bars and two representing cover concrete at the top and bottom) and the two diagonal springs are extracted from the analyti-

cal results and are plotted in Fig. 9. In these figures, the letters A to O correspond to each other and also with the plots in Fig. 8.

Points B and C are the two extremes of elastic re-sponse within which the behavior is linear. Point D represents the onset of yielding. During the inelastic response phase (e.g. from D to E), the compression force sustained in the top cover concrete increased elas-tically (see Fig. 9c) and the reinforcing bars in the ten-sion side (i.e. bottom) experienced large inelastic exten-sion while maintaining the yield force (see Fig. 9b). To accommodate the aforementioned behaviors of cover concrete and tensile steel, the neutral axis had to move up during this inelastic loading phase, and the beam underwent extension at the mid-depth due to the addi-tional rotation. This indicates that the elongation from inelastic rotation (i.e. inelastic extension of tension rein-forcement) is captured in the analysis.

When the load is returned back to zero after inelastic loading (i.e. point F), the bottom bars released all tensile stresses; but this elastic recovery left plastic extension unrecovered (see Fig. 9b). From there, the beam sus-tained permanent elongation when the applied dis-placement reversed back to zero at point G. Conse quently, the crack in the bottom cover concrete did not close fully as shown in Fig. 9d. As the displacement reversed to 16 mm downwards, at point H, the top bars underwent a significant inelastic extension (see Fig. 9a) but the absolute strain in the bottom steel was still posi-tive (i.e., tensile, see Fig. 9b). Obviously, the cracks at the bottom from the previous loading were still open, and only minor compression force was sustained by the bottom cover concrete due to contact stress effect (see Fig. 9d). As a result, the bottom bars sustained much higher compressive force compared to that in point C (see Fig. 9b). Note that the compression force in the bottom reinforcement was smaller than the tension force in the top tension reinforcement at point H due to the diagonal strut contribution. This highlights the impor-tance of diagonal struts in modeling elongation of plas-tic hinges.

From point I to J (i.e. inelastic loading in the upward direction), the response of top bars did not change no-ticeably (see Fig. 9a) whereas the bottom bars sustained further inelastic extension (see Fig. 9b), which resulted in additional elongation. From Fig. 8c, it can be seen that despite the inelastic rotation at point J being similar to that at point E (i.e. the same displacement in the pre-vious cycle), elongation continued to increase. This is a clear indication that the analysis is capturing the elonga-tion mechanism associated with the irrecoverable exten-sion of the compression reinforcement.

During the next unloading phase from point K to L, the top and bottom bars deformed elastically as the Bauschinger effect was not significant in this range. The pinching behavior from point L to M arises as a result of the diagonal cracks in both directions remaining open, which facilitates the shear displacement associated with


-180-160-140-120-100-80

-60-40-20

020

-2 0 2 4 6 8 10 12

Axial Displacement (mm)

Axi

al F

orce

(kN

)

A

B

C

D E

J

M N

I

L

F G

O H

K -180-160-140-120-100-80-60-40-20

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(e) Top left to bottom right diagonal spring (f) Bottom left to top right diagonal spring

Fig. 9 Predicted deformation history of the main springs.

(a) Top reinforcing bars (b) Bottom reinforcing bars

(c) Top cover concrete (d) Bottom cover concrete


closing and opening of the diagonal cracks. In this case, one diagonal spring extended by 4 mm (see Fig. 9e) and the other contracted by 4 mm (see Fig. 9f). Conse-quently, the beam deflected vertically by about 5 mm without a significant change in the shear force (see Fig. 8b). It can be seen from the force-displacement relation-ship of the diagonal struts in Fig. 9e and Fig. 9f that the diagonal struts remained elastic in the compression re-gion through out the loading history.

It is also interesting to note that as the applied dis-placement was reversed back to zero at point N, the ro-tation in the plastic hinge calculated based on the exten-sions of the top and bottom reinforcement in Fig. 9 was still positive. Nevertheless, the total rotation at point N is close to zero in Fig. 8c because it is the combination of elastic beam rotation and plastic hinge rotation. In this case, the elastic beam rotation at point N is negative thereby reducing the total rotation in the system.

For further clarification, the top and bottom rein-forcement deformation history over the plastic hinge region is plotted in Fig. 10. It can be seen that the pre-dicted elongation has two components; inelastic exten-sion of the tension reinforcement from inelastic rotation and permanent extension in the compression reinforce-ment (see Fig. 1 for the definition of these components). It can be seen in this figure that during the repeated dis-placement cycles, the increase in elongation is due to the additional extension of the compression (and ten-sion) reinforcement without any noticeable change in the inelastic rotation, whereas the increase of elongation during further inelastic displacement in the same direc-tion is due to the additional inelastic rotation. This matches with the mechanisms observed in previous studies.

The proposed plastic hinge element has shown its ability to predict the cyclic and elongation response of the plastic hinges. However, further research into the effects of axial load on the shear resistance of concrete and the length of tension shift as well as parameters affecting the level of contact stresses in concrete should improve the accuracy of the proposed element.

6. Conclusions

In order to capture important mechanisms such as ine-

lastic shear deformation and elongation of plastic hinges in cyclic analysis of RC structures, a nonlinear inelastic plastic hinge element has been developed and incorpo-rated into a structural analysis package. The plastic hinge element is a two-node element comprising two rigid plates which hold the opposite ends of two diago-nal springs representing the diagonal compression struts and several longitudinal springs representing reinforcing bars and concrete depending on their positions over the depth of the member. The axial behavior of these springs is modeled using path-dependent cyclic stress-strain relationships of reinforcing steel and concrete, which take into account important material mechanisms likely to make significant influence on the predicted response, such as Bauschinger effect of reinforcing bars and contact stress effect of concrete.

A scrutiny of the cyclic behavior of a cantilever beam predicted by the proposed model indicates that the plas-tic hinge element developed in this study can predict the flexural, shear and elongation response of beam plastic hinges when subjected to inelastic cyclic loading. The investigation has revealed three advantages of the pro-posed analysis approach over the traditional approach; they are (i) the approach does not require calibration of any model parameters; instead generic nonlinear path-dependent material models automatically lead to realis-tic prediction of cyclic response; (ii) the inelastic shear deformation due to elongation can be accounted for and can be easily separated from the flexural deformation in the proposed analysis; and most importantly, (iii) elon-gation of plastic hinges during reversed cyclic actions can be captured. Hence, the developed plastic hinge element offers a more objective tool for conducting ana-lytical performance assessments of RC structures.

References American Concrete Institute, (2005). “Building code

requirements for structural concrete and commentary (ACI 318M-05).” Farmington Hills, MI.

Carr, A. J., (2004). “RUAUMOKO2D - Inelastic dynamic analysis.” Department of Civil Engineering, University of Canterbury, Christchurch, New Zealand.

Dhakal R. P. and Fenwick R. C., (2008). “Detailing of plastic hinges in seismic design of concrete structures.” ACI Structural Journal, 105(6), 740-749.

Dhakal R. P. and Maekawa, K., (2002a). “Path-dependent cyclic stress-strain relationship of reinforcing bar including buckling.” Engineering Structures, 24(11), 1383-1396.

Dhakal, R. P. and Maekawa, K., (2002b). “Modeling for postyield buckling of reinforcement.” Journal of Structural Engineering, 128(9), 1139-1147.

Dhakal, R. P. and Maekawa, K., (2002c). “Reinforcement stability and fracture of cover concrete in reinforced concrete members.” Journal of Structural Engineering, 128(10), 1253-1262.

Douglas, K. T., Davidson, B. J. and Fenwick, R.C., (1996). “Modelling reinforced concrete plastic

D2i -D2i D2ii -D2ii

D4i -D4i D4ii -D4ii

58

250

442

-1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Elongation (mm)

Ver

tical

hei

ght u

pth

e be

am (

mm

)

μ4i −μ4ii−μ4i μ4ii

μ2i −μ2ii−μ2i μ2ii

Fig. 10 Reinforcement extension within the plastic hinge region.


hinges.” Proceedings of the Eleventh World Conference on Earthquake Engineering, Acapulco, Mexico.

Fenwick, R. C. and Davidson, B. J., (1993). “Elongation in ductile seismic resistance reinforced concrete frames.” ACI Special Publication SP-157, 143-170.

Fenwick, R. C. and Megget, L. M., (1993). “Elongation and load deflection characteristics of reinforced concrete members containing plastic hinges.” Bulletin of the New Zealand National Society for Earthquake Engineering, 26(1), 28-41.

Fenwick, R. C., Tankut A. T. and Thom, C. W., (1981). “The deformation of reinforced concrete beams subjected to inelastic cyclic loading: experimental results.” Research Report, Department of Civil Engineering, University of Auckland, Auckland, New Zealand.

Issa, M. S., (1997). “The deformations of reinforced concrete beams containing plastic hinges in the presence of axial compression or tension load.” Master Thesis, Department of Civil Engineering, University of Auckland, Auckland, New Zealand.

Kim, J., Stanton, J. and MacRae, G., (2004). “Effect of beam growth on reinforced concrete frames.” Journal of Structural Engineering, 130(9), 1333-1342.

Lau, D. B. N., Davidson, B. J. and Fenwick, R. C., (2003). “Seismic performance of RC perimeter frames with slabs containing prestressed units.” Proceedings of the Pacific Conference on Earthquake Engineering, Christchurch, New Zealand.

Lee, J. Y. and Watanabe, F., (2003). “Predicting the longitudinal axial strain in the plastic hinge regions of reinforced concrete beams subjected to reversed cyclic loading.” Engineering Structures, 25(7), 927-939.

Liddell, D., Ingham, J. M. and Davidson, B. J., (2000). “Influence of loading history on ultimate displacement of concrete structures.” Research Report, Department of Civil Engineering, University of Auckland, Auckland, New Zealand.

Maekawa, K., Pimanmas, A. and Okamura, H., (2003). “Nonlinear Mechanics of Reinforced Concrete.” Spon Press, London, 721 pp.

Matthews, J. G., Mander, J. B. and Bull, D. K., (2004). “Prediction of beam elongation in structural concrete members using a rainflow method.” Proceedings of the New Zealand Society of Earthquake Engineering Conference. Rotorua, New Zealand.

Matti, N. A., (1998). “Effect of axial loads on the behaviour of reversing plastic hinges in reinforced

concrete beams.” Master Thesis, Department of Civil Engineering, University of Auckland, Auckland, New Zealand.

Okamura, H. and Maekawa, K., (1991). “Nonlinear analysis and constitutive models of reinforced concrete.” Gihodo Publication, Tokyo, Japan.

Pang, X. B. and Hsu, T. T. C., (1995). “Behavior of reinforced concrete membrane elements in shear.” ACI Structural Journal, 92(6), 665-679.

Pang, X. B. and Hsu, T .T. C., (1996). “Fixed angle softened truss model for reinforced concrete.” ACI Structural Journal, 93(2), 197-207.

Paulay, T. and Priestley, M. J. N., (1992). “Seismic design of reinforced concrete and masonry buildings.” Wiley, New York, 744 pp.

Peng, B. H. H. (2009). “Seismic performance assessment of reinforced concrete buildings with precast concrete floor systems” http://hdl.handle.net/10092/3103, PhD Thesis, Department of Civil and Natural Resources Engineering, University of Canterbury, Christchurch, New Zealand.

Peng, B. H. H., Dhakal, R. P., Fenwick, R. C., Carr, A. J. and Bull, D. K., (2011). “Elongation of plastic hinges in ductile RC members: model verification.” Journal of Advanced Concrete Technology, 9(3), 327-338.

Schlaich, J., Schaefer, K. and Jennewein, M., (1987). “Toward a consistent design of structural concrete.” PCI Journal, 32(3), 74-150.

Shima, H., Chou, L. and Okamura, H., (1987). “Micro and macro models for bond behaviour in reinforced concrete.” Journal of Faculty of Engineering, University of Tokyo, Tokyo, Japan, 39(2), 133-194.

Standards New Zealand. (2006). “Concrete Structures Standard: NZS 3101:2006.” Wellington, New Zealand.

To, N. H. T., Ingham, J. M. and Sritharan, S., (2001). “Monotonic non-linear analysis of reinforced concrete knee joints using strut-and-tie computer models.” Bulletin of the New Zealand Society for Earthquake Engineering, 34(3), 169-190.

Vecchio, F. J. and Collins, M. P., (1986). “Modified compression-field theory for reinforced concrete elements subjected to shear.” Journal of the American Concrete Institute, 83(2), 219-231.

Walker, A. and Dhakal, R. P., (2009). “Assessment of material strain limits for defining plastic regions in concrete structures.” Bulletin of the New Zealand Society of Earthquake Engineering, 42(2), 86-95.

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