Plastic Hinge Length of Reinforced Concrete Columns

ACI STRUCTURAL JOURNAL

TECHNICAL PAPER

Title no. 105-S28

Plastic Hinge Length of Reinforced Reinforced Concrete Columns by Sungjin Bae and Oguzhan Oguzh an Bayrak As a column experiences earthquake-induced lateral displacements while supporting gravity loads, severe damage is observed in regions subjected to large moments. These regions are commonly referred to as plastic hinges and they experience large inelastic curvatures. The inelastic curvatures in plastic hinges are typically assumed to be constant over the plastic hinge length, lp , to simplify the estimation of the tip displacement of a column. Therefore, if the plastic hinge length is known, the tip displacement of a column can easily be obtained by integrating curvatures, and vice versa. As part of the research reported in this paper, the effects of axial load and shear span-depth ratio ( L / h) on lp are evaluated experimentally. Based on the experimental observations, a new analytical approach that can be used to estimate lp is presented. Finally, the research findings are synthesized into a simple expression that can be used to estimate lp. Keywords: axial load; deformation capacity; plastic hinge length.

INTRODUCTION

Numerous researchers 1-10 have suggested various expres exp ressions sions that can be used to estimate the plastic plastic hinge length l p of concrete members. Large differences exist among different expressions, however, and the performance of different expressions in estimating l p has not been evaluated systematically. This paper presents test results from four full-scale concrete columns. In addition, the effect of axial load and shear span-depth ratio ( L / h, where L is the height of a cantilever column and h is the overall depth of a column) on the l p is investigated. A concrete compression strain method to estimate l p is proposed. Finally, a simple l p expression is proposed and the influence of the estimated l p on the accuracy of lateral load-drift predictions is studied.

concrete column. The length of a plastic hinge depends on many factors. The following is a list of important factors that influence the length of a plastic hinge: 1) level of axial load; 2) moment gradient; 3) level of shear stress in the plastic hinge region; 4) mechanical properties of longitudinal and transverse reinforcement; 5) concrete strength; and 6) level of confinement and its effectiveness in the potential hinge region. In the 1950s and 1960s, researchers1-5 studied l p to estimate the flexural deformation capacity of reinforced concrete beams. To estimate the flexural deformation capacity, the plastic rotation capacity and the l p are used

θ p

εcu – εce = -------------------c

× l p

(1)

Park and Paulay11 extended this concept to a cantilever column. They simplified the curvature distribution along the length of a column using a plastic hinge (Fig. 1). Using the second moment area theorem, they calculated the tip displacement of a column (Eq. (2))

Δti p

=

Δ y + Δ p

φ y L 2

= ----------- + ( φ – φ y ) l p ( L – 0.5 l p ) 3

(2)

By further simplifying Eq. (2), Park and Paulay 11 obtained the relationship between curvature and displacement ductilities (Eq. (3))

μΔ

l ⎛ l ⎞ = 1 + 3 ( μ φ – 1 ) - p -- 1 – 0.5 - p - L ⎝ L ⎠

(3)

RESEARCH SIGNIFICANCE Research conducted over the past three decades has resulted in various constitutive relationships for concrete and reinforcing bars. As a result, the sectional behavior of reinforced rein forced concrete columns can be reliably estimated. For a given sectional performance, the member behavior of a column can be estimated if l p is known. As such, estimating the length of a plastic hinge establishes a key step in predicting the lateral load-drift response of a column. In this context, the work presented herein advances the state of knowledge: 1) by providing experimental evidence on the effects of axial load and L / h on l p; 2) by introducing a new concrete compression strain-based method to estimate l p; and 3) by synthesizing the research findings into a simple expression that can be use to estimate l p.

DEFINITION OF PLASTIC HINGE LENGTH Plastic hinges form at the maximum moment regions of reinforced concrete columns. If l p is known, the tip displacement of a column can be easily obtained by integrating curvatures. Therefore, accurate assessment of l p is important in relating section-level response to member-level response of a

29 0

Equations (2) and (3) have been commonly used to estimate the l p of concrete columns. 6-9,12,13 It is interesting to note that the curvature profile along the column is often related to the tip displacement rather than flexural displacement. 6-9,12,13

Fig. 1—Definition of plastic hinge length.11 ACI Structural Journal, Journal , V. 105, No. 3, May-June 2008. MS No. S-2006-224.R3 received April 19, 2006, and reviewed under Institute publication policies. Copyright © 2008, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published in the MarchApril 2009 ACI 2009 ACI Structural Journal if the discussion is received by November 1, 2008.

ACI Structural Journal/May-June 2008

ACI member Sungjin Bae is a Structural Engineer at Bechtel Corporation, Frederick, MD. He received his BS and MS from Hanyang University, Seoul, Korea, and his PhD from the University of Texas at Austin, Austin, TX. He is a Member of ACI Committee 209, Creep and Shrinkage of Concrete, and Joint ACI-ASCE Committee 441, Reinforced Concrete Columns. His research interests include the behavior of concrete columns subjected to combined axial load and flexure, testing of large-scale reinforced concrete columns, and performance-based design of concrete columns. ACI member Oguzhan Bayrak is an Associate Professor of civil, architectural, and environmental engineering at the University of Texas at Austin, Austin, TX, and a Fellow of the Clyde E. Lee Endowed Professorship. He is Chair of Joint ACI-ASCE Committee 441, Reinforced Concrete Columns; a member of ACI Committees 341, Earthquake-Resistant Concrete Bridges; E803, Faculty Network Coordinating Committee; and Joint ACI-ASCE Committee 445, Shear and Torsion.

In such a case, the l p in Eq. (2) and (3) can include the effects of bar slip and shear displacements.

Corley5

To expand Mattock’s 3 work, Corley5 tested 40 simply supported concrete beams subjected to single point loads. The confinement and size effects were the primary variables investigated. In addition, the effects of moment gradient and amount of tension reinforcement were studied. Corley 5 reported that the length of the plastic hinge region was primarily a function of the geometry of a concrete beam and that the size of a beam did not have a significant influence on the rotational capacity. Based on the scatter in the measured values of the plastic hinges, he concluded that the effect of ( q – q′)/ qb in Eq. (6) could be ignored. Corley 5 suggested the use of a simple expression for calculating l p d z l p = --- + 0.2 ------2 d

TECHNICAL LITERATURE: PLASTIC HINGE LENGTH Baker1 To investigate the moment-curvature relationship of beams and columns, 94 beam/column tests were conducted by six laboratories under the auspices of CEB in the 1960s. The main test variables included concrete strength, yield strength or amount of tension reinforcement, amount of compression reinforcement, single or double concentrated loads, and axial load. Based on the test results, Baker 1 proposed the following equation to calculate the l p z 0.25 l p = k 1 k 2 k 3 ⎛ ---⎞ ⎝ d ⎠ d

(5)

Mattock 3 conducted 37 beam tests and investigated the effect of various parameters on the behavior of reinforced concrete beams. The parameters studied by Mattock 3 were concrete strength ( f c′ = 4000 to 6000 psi [28 to 41 MPa]), effective depth of beam ( d = 10 to 20 in. [254 to 508 mm]), moment gradient ( z / d = 2.75 to 11), amount (ρl = 1 to 3%), and yield strength ( f y = 47 to 60 ksi [324 to 414 MPa]) of tension reinforcement. Based on the test results, Mattock 3 concluded that the spread of plasticity along a beam length increased as z / d increased, and as the net tension reinforcement (q – q′) / qb decreased. Mattock 3 proposed the following empirical relationship to calculate l p.


Mattock 4 simplified Eq. (6) in 1967. He stated that even though there was considerable scatter, the trend in the observed l p could be represented reasonably well by the following simple expression d l p = --- + 0.05 z 2

(8)

Park et al. 6 tested four full-scale concrete columns with square sections of 22 x 22 in. 2 (550 x 550 mm2) and an L / h of 2. The axial loads applied to the column specimens were in the range of 0.2 f c′ Ag to 0.6 f c′ Ag. Park et al.6 estimated the l p of the test specimens using Eq. (2). They concluded that the experimentally obtained l p were comparatively insensitive to the axial load level and had an average value of 0.42 h, where h is the overall depth of the column. They suggested using a simple l p of 0.4h for concrete columns. Using a similar approach, Priestley and Park 7 proposed the following equation for the calculation of l p in reinforced concrete columns l p = 0.08 L + 6d b

Mattock 3

where both d and z are in inches.

Mattock 4

Park et al.6

where c is the neutral axis depth at collapse.

⎧ d z q – q′ d ⎫ l p = --- 1 + ⎛ 1.14 --- – 1⎞ ⎨ 1 – ⎛ ------------- ⎞ ---------- ⎬ ⎝ ⎠ ⎝ ⎠ d qb 2 16.2 ⎭ ⎩

where both d and z are in inches.

(4)

where z is the distance between the critical section and the point of contraflexure and d is effective depth of a beam. Baker1 reported that the l p ranged from 0.4 d to 2.4d for practical values of z / d . The z / d ratio is a similar parameter to L / h and was used to represent the effect of moment gradient. Baker and Amarakone 2 simplified Eq. (4) to obtain the following z l p = 0.8 k 1 k 3 ⎛ ---⎞ c ⎝ d ⎠

(7)

(6)

(9)

The suggested l p (Eq. (9)) has two components. Priestley and Park 7 stated that the first term mainly accounted for column bending, while the second accounted for bar slip due to the elongation of longitudinal bars beyond the theoretical base (tensile strain penetration into the joint or foundation). Paulay and Priestley8 revised Eq. (9) to account for different grades of flexural reinforcement. The revised expression is given in Eq. (10) l p = 0.08 L + 0.15 d b f y ( f y in ksi) l p = 0.08 L + 0.022 d b f y ( f y in MPa)

(10)

Paulay and Priestley 8 reported that Eq. (10) resulted in l p ≈ 0.5h for typical columns.

Sakai and Sheikh14 Based on an extensive review of the literature, Sakai and Sheikh14 reported that the l p increased as the aspect ratio ( L / h or

291

L / D), which is nearly equivalent to the L / h, increased. Bilinear curves illustrating the relationship between the l p and the aspect ratio were reported as trend lines by the authors. They concluded that the l p was affected by the amount of transverse reinforcement, axial load level, and aspect ratio.

Sheikh and Khoury9

Sheikh and Khoury, 9 Sheikh et al., 12 and Bayrak and Sheikh13 reported that the measured l p were approximately equal to 1.0h in their column tests. It is important to note that most of their column tests were conducted under high axial loads.

Mendis10

Mendis10 conducted tests on 13 simply supported concrete beams subjected to single point loads and examined their l p. He reported that the l p increased as the L / h or the longitudinal reinforcement ratio increased, but decreased as the amount of lateral reinforcement increased. Based on four f c′ Ag = column tests where the axial load levels were low ( P / 0.06 to 0.20), Mendis concluded that the l p was not sensitive to the level of axial load.

PREVIOUS RESEARCH INTO PLASTIC HINGE LENGTH: DISCUSSION Various expressions, recommended for use in l p estimations, were briefly discussed in the previous section. A comparison of the previously reported l p expressions is provided in Fig. 2. This figure clearly illustrates that large variations exist among the suggested l p. It is interesting to note that most of the l p expressions do not consider axial load as a parameter, except for the

Fig. 2—Comparison of plastic hinge length expressions.

expressions proposed by Baker 1 and Baker and Amarakone. 2 This is because the behavior of concrete beams was the major focus in investigating the l p in early research. 1-5 In the more recent research, 6-9 the behavior of concrete columns has been the focus of l p investigations. In addition, although the effect of axial load on the l p of concrete columns has been reported by several researchers, the reported results can be considered contradictory. As discussed previously, Park et al. 6 and Mendis10 concluded that the l p were insensitive to axial load levels based on their test results. In contrast, Atalay and Penzien 15 reported that the spread of plasticity or the length of the damaged region increased as the axial load level increased. Tanaka and Park 16 reported that the l p increased from 0.46 h to 0.75h as the applied axial load level increased from 0.1 f c′ Ag to 0.3 f c′ Ag. The test results of Thomson and Wallace 17 and Légeron and Paultre 18 indicated that the increase of l p with the axial loads could also be observed for high-strength concrete columns. Therefore, an investigation into the l p of reinforced concrete columns is needed to: 1) reconcile differences encountered in the previous research; and 2) develop an expression that can be used to estimate l p more accurately for various axial load levels. As discussed previously, Eq. (2) and (3) have been commonly used to estimate the l p of concrete columns. 6-9,12,13 Using these equations, the l p can be estimated from measured tip displacements and curvatures. Tip displacements of a concrete column include flexural, bar slip, and shear displacement components. Therefore, the l p estimated using Eq. (2) and (3) includes the effect of bar slip and shear displacements. As shear deformations can be ignored for columns with moderate L / h, the l p measured in this manner will be influen ced mainly by flexural and bar slip displacements. The influence of various displacement components other than flexural displacements can be observed in Eq. (9) and (10), which were proposed by Priestley and Park 7 and Paulay and Priestley. 8 These equations have two separate terms that take into account the effect of flexural and bar-slip displacements. By relating the curvature distribution along the column shown in Fig. 1 to the tip displacement rather than flexural deformations, the l p in Eq. (2) and (3) includes the effects of bar slip and shear displacements. Conversely, the displacements calculated through the use of the l p expression derived in this manner implicitly account for displacement components other than the flexural displacements. As flexural displacements are directly related to curvatures , it is more appropriate to relate the l p to the flexural displacement component only. Bar slip and shear displacements have to be estimated independently. Subsequently, the tip displacement can be estimated by summing these displacement components. Following this logic, a new l p expression is proposed in this study.

PLASTIC HINGE LENGTHS: COLUMN TESTS

Fig. 3—Test setup.

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To investigate the effect of L / h and axial load ( P / Po), four full-scale concrete columns were tested under moderate to high axial load levels and reversed cyclic displacement excursions. Figure 3 illustrates the test setup used in this research. The test frame, shown in Fig. 3, can be used to apply and maintain an axial load up to 2000 kips (8900 kN) and reversed cyclic moments as large as 4000 ft·kips (5400 kN·m). White arrows in Fig. 3 illustrate the axial force applied by the hydraulic ram and the loads applied by two actuators that were used to impart reversed cyclic base


Fig. 4—Strain distribution.

Table 1—Details of test specimens Longitudinal steel Specimen

b x h, in. x in. (mm x mm)

f c′ , ksi (MPa)

S24-2UT

24 x 24 (610 x 610)

63 (43.4) No. 7-1 (22M-1) 1.25

S17-3UT

17.25 x 17.25 (440 x 440) 6.3 (43.4)

Size No. 5 (16M)

Transverse steel

f yl, ρl , % ksi (MPa)

Size

f yh, sh, in. (mm) ρs,* % ksi (MPa) Ash / Ash,ACI P / Po†

73 (50.3) No. 4-1 (13M-1)

3-3/4 (95)

2.04

62 (42.7)

1.09

0.5

1.25

72 (49.6) No. 3-1 (10M-1)

3-3/8 (86)

1.76

72 (49.6)

1.12

0.5

24-4UT

24 x 24 (610 x 610)

5.3 (36.5) No. 7-2 (22M-2) 1.25

58 (40.0) No. 3-2 (10M-2)

6 (152)

0.72

66 (45.5)

0.44

0.2

S24-5UT

24 x 24 (610 x 610)

6.0 (41.4) No. 7-2 (22M-2) 1.25

58 (40.0) No. 4-2 (13M-2)

6 (152)

1.30

63 (43.4)

0.74

0.2

*

ρs = volumetric ratio of transverse reinforcement to concrete core.

†

Po = 0.85 f c′ Ac + f yl As.

rotations. The shear forces at hinges are shown with black arrows. Table 1 shows the details of the specimens tested in this research program and the magnitude of the axial load that was applied to each specimen during testing. A detailed discussion of this experimental research can be found elsewhere 19 and is considered to be beyond the scope of this paper. The damage observed within the plastic hinge region of each test specimen and the corresponding tie strains along the columns are shown in Fig. 4. Specimens S24-2UT and S24-3UT were tested under an axial load level of 0.5 Po, while Specimens S24-4UT and S24-5UT were tested under a lower axial load level of 0.2Po. Specimens S24-2UT, S24-4UT, and S24-5UT had 24 in. (609.6 mm) square sections ( L / h = 5), whereas Specimen S17-3UT had a 17.25 in.


(438.15 in.) square section ( L / h = 7). Figure 4 shows that the severely damaged regions of Specimens S24-2UT and S17-3UT are longer than those of Specimens S24-4UT and S24-5UT. The tie strains measured along the column specimen s at the 20th loading cycle in Fig. 4 also illustrate that Spec imens S24-2UT and S17-3UT experienced inelastic tie strains over a longer length than Specimens S24-4UT and S24-5UT. Many ties in Specimens S24-2UT and S17-3UT experienced larger inelastic strains after the 20th loading cycle than the measurement limits of strain gauges. Therefore, it can be concluded that, for the specimens tested in this study, l p increased as the level of axial load increased. Figure 5 illustrates the sectional and member performances of Specimens S24-2UT and S24-4UT. Specimen S24-2UT and S24-4UT have the same section size ( h = 24 in. [610 mm]) and

293

L / h = 5. These specimens were tested under different axial load levels (P / Po = 0.5 for Specimen S24-2UT and P / Po = 0.2 for Specimen S24-4UT). Figure 5(a) illustrates that the sectional performance of Specimen S24-4UT is somewhat better than that of Specimen S24-2UT with respect to deformation capacity. The opposite trend can be observed from the member performances, however, as shown in Fig. 5(b). The reversal in the trends is attributed to the difference observed in the l p of Specimens S24-2UT and S24-4UT.

PLASTIC HINGE LENGTH: CONCRETE COMPRESSION STRAIN METHOD Estimating the length of a plastic hinge that forms in a reinforced concrete column through the use of a sophisticated computer program is complicated. Because of this difficulty, the estimation of l p has typically been based on experimental data. A plastic hinge is defined as the zone where severe damage of concrete sections occurs. By observing that large cracks typically occur at curvatures greater than the yield curvature, some researchers 20,21 attempted to relate the yield moment to the l p. This approach is practical when the curvature increases with moment (that is, in the ascending branch of the response), but the application of this approach to members that experience post-peak strength degradation is questionable. In this section, a concrete compression strain method that can be used to estimate the l p of concrete columns is introduced. Following the description of this concrete compression strain method, the key factors that influence the length of plastic hinges are identified by conducting sensitivity analyses. Based on the results of the sensitivity analyses, a simple expression that can be used to estimate the length of the plastic hinges is proposed. Finally, the influence of the l p estimations on the lateral load-displacement response predictions is discussed.

Compressive strain profile of core concrete As reinforced concrete columns are subjected to earthquake-induced lateral displacements while supporting

Fig. 5—Comparison of Specimens S24-2UT and S24-4UT.

Fig. 6—Sectional analysis for Specimen S24-2UT.

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gravity loads, plastic hinges form within the core concrete. The magnitude of damage in concrete members is closely related to the intensity of concrete compressive strains measured at critical sections. Under increasing curvatures, concrete strains on the compression side of the member increase. As the concrete compressive strains increase and reach a critical value, the cover concrete spalls off. Subsequently, yielding of longitudinal bars on the compression side occurs. This is followed by crushing of the core concrete. With accumulation of damage under increasing compressive strains, plastic hinges form. Experimental observations6,19 indicate that the plastic hinges start to form after the maximum moment capacity is reached. Therefore, the magnitude of concrete compressive strains experienced by the core concrete when the maximum moment capacity is reached at the critical section of a column can give a good indication on the formation of a plastic hinge. To estimate the length of a plastic hinge that may form at the base of a reinforced concrete column, the following procedure is proposed: 1. The moment-curvature response of a typical column section within the potential plastic hinge region is obtained from a sectional analysis (Fig. 6(a)). In the sectional analysis, core concrete is modeled as confined concrete, while cover concrete is modeled as unconfined concrete. Bar buckling behavior in compression reinforcement is not considered because the ascending branch of the moment-curvature relationship is of concern. Reinforcing bar buckling is not experienced within the ascending branch of the response; 2. Neglecting the second order effects, the bending moment diagram is drawn along the column height (Fig. 7(c)). The moment value at the center of the potential plastic hinge (~0.5h away from the base) is assumed to be equal to the maximum moment capacity of the section obtained from the sectional analysis in Step 1. This assumption would imply that the sections between the center of the plastic hinge and the base of the column have a larger flexural capacity than predicted using the sectional analysis. The shift in the location of the critical section (Fig. 7(d)) is due to the confinement provided by the concrete stub to the neighboring sections. This effect, commonly referred to as the stub confinement effect, has been observed in previous experimental research 6,9,13,18,22 ; 3. Using the bending moment diagram from Step 2 and the sectional analysis’ results (Fig. 6(b)) from Step 1, the compressive strain experienced by the outer fiber of core concrete can be determined for a sufficient number of sections along the length of a column. If these compressive strains are plotted along the length of a column, the compressive concrete strain profile along the length of a column can be obtained (Fig. 7(d)); 4. The compressive strains at the outer fiber of the core concrete are equal to the reinforcing bar strains. In effect, the compressive strain profile obtained in Step 3 represents the variation of compressive longitudinal bar strain along the length of a column. By examining the compressive strain profile, such as the one shown in Fig. 7(d), the length of the region in which longitudinal bars are yielding in compression can be estimated. The identification of this length establishes a key step in estimating the l p of a column; and 5. As indicated in Step 2, the critical section shifts away from the face of the stub due to additional confinement effects provided by the stub. Because of the additional confinement provided by the stub to adjacent sections, sections within a distance of approximately 0.25 h from the stub remain nearly undamaged. Therefore, to estimate the


Fig. 7—Estimation of plastic hinge length (Specimen S24-2UT).

length of the plastic hinge region, where columns are expected to dissipate large amounts of inelastic energy by undergoing large inelastic deformations, 0.25 h is subtracted from the overall length in which compressive reinforcing bar strains greater than the yield strain are calculated. The curvature profile depicted in Fig. 7(e) also shows a similar trend in the compressive strain profile shown in Fig. 7(d). Hence, the use of curvature distribution along the height of a column can be considered as an alternative for identifying the zone in which inelastic curvatures are localized. While this curvature profile has been used by some researchers,20,21 the use of the compressive strain profile facilitates a clearer observation of the effect of axial loads on the l p. In addition, because the damage in concrete is more closely related to the magnitude of the concrete strains than the curvatures, compressive strains were used in the method described previously. In this approach, it is assumed that plastic hinges start to form when the maximum moment capacity is reached and, therefore, important information on the plastic hinge formation can be obtained by analyzing the compressive strain profile when the maximum moment is reached. It is important to note that the l p predicted in this manner can be used to estimate flexural deformations only because sectional analysis is used as the basis of calculations. To calculate the tip d eformation of columns, deformations due to bar slip and shear deformations need to be calculated separately and added to the flexural deformations. The lengths of the potential plastic hinge regions of Specimens S24-2UT through S24-5UT are estimated by using the analysis procedure described previously. By using the compressive strain profiles along heights of the test specimens, the l p were estimated and the results of these analyses are shown in Fig. 8. This figure illustrates that the proposed method provides good estimations for the l p of the column specimens tested in this study.

PARAMETRIC STUDY ON PLASTIC HINGE LENGTH It was shown that the proposed concrete compression strain method provided good estimations for the l p of the columns tested in this study. Using the proposed method, the influence of various parameters on the l p is studied. The


Fig. 8—Comparison of predicted plastic hinge lengths.

effect of axial load level ( P / Po), L / h, and the amount of Ag) are investigated. longitudinal reinforcement ( ρl = As /

Axial load level To study the effect of axial load on the l p, a square column (24 x 24 x 120 in. 3 [610 x 610 x 3048 mm3]) is studied. The percentage of longitudinal reinforcement is kept constant at 1% (ρl = 0.01) and the ratio of the center-to-center distance of extreme reinforcement layers to the overall column depth ( γ) is assumed to be 0.8. A concrete strength of 6000 psi (41.4 MPa) is used. The yield and ultimate strength of reinforcing bars are assumed to b e 60 and 90 ksi (4 14 and 621 MPa), respectively. The curvature and compressive concrete strain profiles along the length of the columns are investigated for various axial load levels. Figure 9 illustrates a summary of the results of the analyses. As can be observed in this figure, the curvature profiles do not effectively show the effect of axial load level. Unlike moment-curvature relationships, the curvature profiles in Fig. 9(a) show curvature profiles along the column height. In moment-curvature relationships, the effect of axial load influences the strength and stiffness. As these curvature profiles provide only

295

Fig. 9—Effect of axial load on curvature and compressive strain profiles.

Fig. 10—Relationship between plastic hinge length and axial load.

curvatures but not corresponding moments, however, the effect of axial load is not reflected in these curvature profiles. In contrast, the compressive strain profiles clearly illustrate the influence of axial load. For each case studied in Fig. 9, the length of the plastic hinge is estimated using the procedure described previously. In other words, the spread of reinforcing bar yielding in compression is examined to establish the length of the plastic hinges. Figure 10 summarizes the results of this analysis, shown in Fig. 9(b). As can be seen in this figure, the length of the plastic hinge is nearly constant for low axial loads (P ≤ 0.2Po). For low axial loads, the l p is approximately equal to 0.25h. Starting at an axial load of approximately 0.2 Po, the l p increases with increasing axial loads. It is interesting to note that Mendis 10 reported that the l p measured in the columns tested in his research were not sensitive to the axial load level. Considering the fact that all of the columns studied by Mendis 10 were tested under low axial load levels ( P ≤ 0.2Po) and bearing in mind the findings of this parametric study (summarized in Fig. 10), the conclusio n reported by Mendis 10 can be better appreciated. The l p estimate of 0.25 h (Fig. 10) can be compared with 0.4h recommended by Park et al. 6 and 0.5h recommended by Paulay and Priestley. 8 The differences observed in the l p estimates can be attributed to the displacement components used to estimate the l p. Only flexural displacements are considered in the proposed analysis, as the strains experienced by compression bars are obtained from the moment-curvature relationships. On the other hand, the tip displacements are used in estimating the l p by Park et al.6 and Paulay and Priestley.8 As discussed previously, tip displacements include bar slip and shear deformation components in addition to the flexural deformations.

296

Fig. 11—Relationship between plastic hinge length and shear span-depth ratio.

Chapter 21 of the ACI Building Code 23 requires closelyspaced transverse reinforcement be provided over a length (lo) from a joint face where flexural yielding is likely to occur as a result of large inelastic lateral displacement of columns. The required length ( lo) of the ACI Code is the largest of the overall depth of a column, 1/6 of the clear height of a column, or 18 in. (457 mm). Figure 10 shows that the estimated l p approaches 0.8 h at an axial load of 0.52 Po (= φPn,max = φ × 0.8Po = 0.65 × 0.8Po), which is the maximum axial load permitted in the ACI Building Code. 23 It is important to note that in the process of estimating the l p, a distance of 0.25h from the face of the stub is subtracted, as discussed in the procedure of concrete compression strain method. Therefore, the total length from a joint face has to be considered as the sum of the l p and a distance of 0.25 h from the joint face. Accordingly, the total length from a joint face is 1.05h at an axial load of 0.52 Po, which is close to the ACI required length ( lo) of 1.0h. An examination of Fig. 1, however, shows that the length of the region where large inelastic curvatures develop will be longer than the equivalent plastic hinge region. Therefore, the required length of closely-spaced transverse reinforcement needs to be larger than the estimated plastic hinge region of 1.0 h from the joint face. In summary, the potential l p specified by ACI 318-05 23 is likely to be unsatisfactory for columns supporting high axial loads.

Shear span-depth ratio (L / h) Researchers1-5,7,8,10 recognized that the length of a plastic hinge is influenced by the L / h. To investigate the influence of the L / h on the l p, a series of analyses were conducted. At this stage of the parametric study, a 24 x 24 in. 2 (610 x 610 mm2) square column with varying L / h is considered. The longitudinal reinforcement is kept constant ( ρl = 0.01) and a γ value of 0.8 is used. The results of the analyses are summarized in Fig. 11. As can be observed in the figure, the l p increases with increasing L / h for a given axial load level. For low axial loads (≈ 0.2Po), however, the increases observed in l p with increasing L / h are insignificant. For a given L / h, the l p increases with increasing axial loads. The increases in l p observed at small L / h (2 < L / h < 3) are less pronounced than those observed at a large L / h. It is interesting to note that Park et al. 6 tested a series of columns with an L / h of 2.2. Based on the test results, they concluded that the l p was not affected by the axial loads and recommended a l p of 0.4h. Figure 11 clearly illustrates that for an L / h of 2.2, the effect of axial load on the l p is relatively small in comparison with the effects seen at high L / h.


Amount of longitudinal reinforcement (ρl = As / Ag )

Mattock 3 reported that the l p increased with a decrease in Ab, where As is the the net tension reinforcement (( As – As′ )/ area of tension reinforcement, As′ is the area of compression reinforcement, and Ab is the area of balanced reinforcement). The effect of net tension reinforcement was further studied by Corley.5 In contrast, recently, Mendis 10 reported that the l p increased with an increasing amount of tension reinforcement. It is important to note that these conclusions 3,5,10 were primarily based on the results from tests conducted on reinforced concrete beams. Due to the lack of consensus among various researchers, the amount of longitudinal reinforcement has been ignored in most previous proposals for estimating the l p. To study the effect of longitudinal reinforcement on l p, a 24 x 24 in. 2 (610 x 610 mm2) square column with a column height of 120 in. (3048 mm), a L / h of 5, and a γ value of 0.8 is studied. The results of the parametric study conducted on a 24 x 24 in. 2 (610 x 610 mm 2) square column are summarized in Fig. 12 and 13. As can be observed in Fig. 12(a), the post-cracking stiffness and strength of the column sections increase with increasing reinforcement ratios. The curvatures and reinforcing bar strains on the compression side are evaluated at maximum lateral load capacity and plotted along the height of the column (Fig. 12(c) and (d)). An examination of the curvature and compressive strain profiles along the height of the column indicates that with increasing longitudinal reinforcement ratio ( ρl), the length of the plastic hinge increases. Figure 13 also indicates this trend for various axial load levels. For all of the axial load levels, l p increases with increasing ρll values.

various displacement components and summing them together. Those displacement components considered in calculating the tip displacement of a column are: 1) flexural displacements; 2) fixed-end rotations resulting from the slip of longitudinal bars out of joints; and 3) shear displacements. The additional displacements due to secondary moments generated by the P-Δ effect are also included. The l p estimated by using Eq. (11) are used in calculating flexural displacements to approximate the descending part of the lateral load response of concrete columns. A detailed description of Eq. (11) can be found elsewhere. 19

PLASTIC HINGE LENGTH: NEW EXPRESSION The previous discussion using the proposed concrete compression strain method shows that axial load, L / h, and the amount of longitudinal reinforcement are main parameters for estimating the length of a plastic hinge. Based on this observation, a linear relationship between these parameters (P / Po, L / h, and As / Ag) is assumed in developing the l p expression for simplicity. Least squares analyses were conducted to identify a coefficient for each parameter using the UW/PEER column database. Key features of the column database used for this purpose can be found elsewhere. 19 A new l p expression of Eq. (11) is proposed from a series of least squares analyses l p --- = h

A P L 0.3 ⎛ -----⎞ + 3 ⎛ -----s⎞ – 0.1 ⎛ ---⎞ + 0.25 ≥ 0.25 ⎝ P o⎠ ⎝ Ag⎠ ⎝ h⎠

Fig. 12—Moment and curvature curves with various amount of longitudinal reinforcement (P = 0.3Po).

(11)

In the process developing the proposed l p expression, a computer program is used to estimate the response of column specimens. The drift capacities of columns are estimated and compared with experimentally measured drift capacities. The computational algorithm used in this computer program is illustrated in Fig. 14. As seen in Fig. 14, the tip displacements of columns are calculated by estimating

Fig. 13—Effect of amount of longitudinal reinforcement.

Table 2—Predicted plastic hinge lengths Specimen

Baker1

Corley 5

Mattock 4

Park et al.6

S24-2UT

0.60h

0.49 h

0.70 h

0.40 h

0.80h

1.00 h

0.66h

0.69h

S17-3UT

0.65h

0.52 h

0.80 h

0.40 h

0.96h

1.00 h

0.91h

0.86h

S24-4UT

0.62h

0.49 h

0.70 h

0.40 h

0.72h

1.00 h

0.49h

0.25h

S24-5UT

0.56h

0.49 h

0.70 h

0.40 h

0.72h

1.00 h

0.47h

0.25h


Paulay and Priestley8 Sheikh and Khoury9 Measured

Eq. (11)

297

PLASTIC HINGLE LENGTH: ESTIMATIONS AND EXPERIMENTS

Fig. 14—Flowchart of computer program.

To examine the accuracy of Eq. (11), the l p of four fullscale concrete column specimens are estimated using various expressions and compared with the measured l p, as shown in Table 2. This table illustrates that the use of Eq. (11) results in good estimates for all test specimens. For specimens tested under low axial loads (Specimens S24-4UT and S24-5UT), Eq. (11) gives somewhat smaller l p than the measured values. This is due to the fact that only flexural displacements are considered in deriving Eq. (11). The comparison of damaged specimens after tests (Fig. 8) shows that the estimated l p by Eq. (11) are similar to sizes of severely damaged regions. For further evaluation, responses of test specimens are estimated by using a computer program, which is described in Fig. 14. The sectional and member responses of the test specimens are used as bases of comparison. The l p estimated by various expressions including Eq. (11) are used in generating the member responses. The measured l p are also used for this purpose. It is important to note that only the flexural displacements and the magnified displacements due to the P-Δ effect are used to get member responses when l p expressions other than the proposed are used (as these l p expressions include the effect of other displacement components and these displacements should not be accounted for twice). The estimated moment-curvature and lateral load-drift plots of Specimens S24-2UT and S17-3UT (tested under high axial loads) are illustrated in Fig. 15. As can be seen in

Fig. 15—Experimental and predicted response ( P = 0.5Po).

298


Fig. 16—Experimental and predicted response ( P = 0.2Po).

Fig. 15, the use of the l p expression proposed by Park et al. 6 (l p = 0.4h) results in considerable underestimation of the drift capacity. In contrast, the expressions proposed by Paulay and Priestley 8 (l p = 0.8h) and Sheikh and Khoury9 (l p = 1.0h) result in satisfactory predictions. The same trends can be observed for Specimen S17-3UT, as shown in Fig. 15. Results of the analyses conducted on Specimens S24-4UT and S24-5UT are included in Fig. 16. These specimens were tested under low axial load levels. Figure 16 shows that the use of the l p expression proposed by Park et al. 6 (l p = 0.4h) results in a reasonably good estimation for the drift capacity. The expressions proposed by Paulay and Priestley 8 (l p = 0.72h) and Sheikh and Khoury 9 (l p = 1.0h), however, result in overestimated predictions of the drift capacity. Therefore, it can be concluded that the use of some l p expressions work for high axial loads, whereas they fail to provide satisfactory estimates for low axial loads and vice versa. The use of Eq. (11), however, results in satisfactory predictions both for high and low axial load levels. In performance-based design, prediction of the deformation capacity of reinforced concrete columns is important. An analysis of Fig. 15 and 16 clearly shows the influence of the l p estimations on the drift capacity predictions. This analysis indicates the value of an expression that can provide reasonable estimations for l p forming in columns supporting high or low axial loads.

SUMMARY AND CONCLUSIONS Based on the experimental and analytical research into the seismic behavior of concrete columns reported in this study, the following conclusions can be drawn:


1. In light of the experimental data on four full-scale concrete column tests, ACI 318-05 23 provisions for the length of the potential plastic hinge regions were found to be slightly unconservative for columns supporting high axial loads. Although the experimental evidence is not exhaustive, it may be prudent to increase the length of the region in which closely-spaced ties are used from 1.0 h to 1.5h from the joint face until further experimental evidence is gathered; 2. The level of axial load influenced the length of the plastic hinges that formed in the full-scale column specimens tested in this research. Specimens tested under high axial loads developed longer plastic hinges than those tested under low axial loads; and 3. The following equation, developed in this research, can be used to estimate the length of the plastic hinges forming in columns supporting a wide range of axial loads. l p --- = h

A P L 0.3 ⎛ -----⎞ + 3 ⎛ -----s⎞ – 0.1 ⎛ ---⎞ + 0.25 ≥ 0.25 ⎝ P o⎠ ⎝ Ag⎠ ⎝ h⎠

NOTATION Ab Ag As As′ b c d d b f c′

= = = = = = = = =

area of balanced reinforcement gross area of concrete section area of tension reinforcement area of compression reinforcement width of compression face of member distance from extreme compression fiber to neutral axis effective depth of beam diameter of longitudinal reinforcement compressive strength of concrete

299

f y h k 1 k 2 k 3 L l p P Po q q′ qb z Δ p Δ y εce εcu φ φ y θ p ρl

= = = = = = = = = = = = = = = = = = = = = = = =

yield stress of reinforcement overall depth of column 0.7 for mild steel 0.9 for cold worked steel 1 + 0.5( P / Pu) 0.9 – (0.3/23.5)( f c′ – 11.7) ( f c′ in MPa) distance from critical section to point of contraflexure plastic hinge length applied axial force 0.85 f c′ ( Ag – As) + f y As nominal axial load capacity as per ACI 318-0523 f c′ ) tension reinforcement index (= As / bd × f y / compressive reinforcement index (= As′ / bd × f f c′ ) y / f c′ ) balanced tension reinforcement index (= Ab / bd × f y / distance of critical section to point of contraflexure plastic displacement yield displacement elastic concrete compressive strain maximum concrete compressive strain curvature (Eq. (2)) yield curvature plastic rotation As / Ag longitudinal reinforcement ratio

REFERENCES 1. Baker, A. L. L., Ultimate Load Theory Applied to the Design of Reinforced and Prestressed Concrete Frames , Concrete Publications Ltd., London, UK, 1956, 91 pp. 2. Baker, A. L. L., and Amarakone, A. M. N., “Inelastic Hyperstatic Frame Analysis,” Flexural Mechanics of Reinforced Concrete, SP-12, American Concrete Institute, Farmington Hills, MI, 1964, pp. 85-142. 3. Mattock, A. H., “Rotational Capacity of Hinging Regions in Reinforced Concrete Beams,” Flexural Mechanics of Reinforced Concrete, SP-12, American Concrete Institute, Farmington Hills, MI, 1964, pp. 143-181. 4. Mattock, A. H., discussion of “Rotational Capacity of Hinging Regions in Reinforced Concrete Beams,” Journal of the Structural Division, ASCE, V. 93, No. ST2, Apr. 1967, pp. 519-522. 5. Corley, W. G., “Rotational Capacity of Reinforced Concrete Beams,” Journal of the Structural Division, ASCE, V. 92, No. ST5, Oct. 1966, pp. 121-146. 6. Park, R.; Priestley, M. J. N.; and Gill, W. D., “Ductility of SquareConfined Concrete Columns,” Journal of Structural Division, ASCE, V. 108, No. ST4, 1982, pp. 929-950. 7. Priestley, M. J. N., and Park, R., “Strength and Ductility of Concrete Bridge Columns Under Seismic Loading,” ACI Structural Journal , V. 84, No. 1, Jan.-Feb. 1987, pp. 61-76. 8. Paulay, T., and Priestley, M. J. N., Seismic Design of Reinforced Concrete and Masonry Buildings , John Wiley and Sons, New York, 1992, 767 pp. 9. Sheikh, S. A., and Khoury, S. S., “Confined Concrete Columns with Stubs,” ACI Structural Journal , V. 90, No. 4, July-Aug. 1993, pp. 414-431. 10. Mendis, P., “Plastic Hinge Lengths of Normal and High-Strength Concrete in Flexure,” Advances in Structural Engineering , V. 4, No. 4, Oct. 2001, pp. 189-195.

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11. Park, R., and Paulay, T., Reinforced Concrete Structures , John Wiley and Sons, New York, 1975, 769 pp. 12. Sheikh, S. A.; Shah, D. V.; and Khoury, S. S., “Confinement of HighStrength Concrete Columns,” ACI Structural Journal, V. 91, No. 1, Jan.-Feb. 1994, pp. 100-111. 13. Bayrak, O., and Sheikh, S. A., “Confinement Reinforcement Design Considerations for Ductile HSC Columns,” Journal of Structural Engineering, ASCE, V. 124, No. 9, Sept. 1998, pp. 999-1010. 14. Sakai, K., and Sheikh, S. A., “What Do We Know about Confinement in Reinforced Concrete Columns? (A Critical Review of Previous Work and Code Provisions),” ACI Structural Journal , V. 86, No. 2, Mar.-Apr. 1989, pp. 192-207. 15. Atalay, M. B., and Penzien, J., “The Seismic Behavior of Critical Regions of Reinforced Concrete Components as Influenced by Moment, Shear and Axial Force,” Report No. EERC 75-19, University of California at Berkeley, Berkeley, CA, Dec. 1975, 226 pp. 16. Tanaka, H., and Rark, R., “Effect of Lateral Confining Reinforcement on the Ductile Behavior of Reinforced Concrete Columns,” Research Report 90-2, Department of Civil Engineering, University of Canterbury, Christchurch, New Zealand, June 1990, 458 pp. 17. Thomson, J. H., and Wallace, J. W., “Lateral Load Behavior of Reinforced Concrete Columns Constructed using High-Strength Materials,” ACI Structural Journal , V. 91, No. 5, Sept.-Oct. 1994, pp. 605-615. 18. Légeron, F., and Paultre, P., “Behavior of High-Strength Concrete Columns under Cyclic Flexure and Constant Axial Load,” ACI Structural Journal, V. 97, No. 4, July-Aug. 2000, pp. 591-601. 19. Bae, S., “Seismic Performance of Full-Scale Reinforced Concrete Columns,” Department of Civil, Architectural and Environmental Engineering, the University of Texas at Austin, Austin, TX, Dec. 2005, 312 pp. 20. Chan, W. W. L., “The Ultimate Strength and Deformation of Hinges in Reinforced Concrete Frameworks,” Magazine of Concrete Research, V. 7, No. 21, 1955, pp. 121-132. 21. Joint ACI-ASCE Committee 428, “Progress Report on Code Clauses for Limit Design,” ACI J OURNAL, Proceedings V. 65, No. 9, Sept. 1968, pp. 713-715. 22. Soesianawati, M. T.; Park, R.; and Priestley, M. J. N., “Limited Ductility Design of Reinforced Concrete Columns,” Research Report 86-10, Department of Civil Engineering, University of Canterbury, Christchurch, New Zealand, Mar. 1986, 208 pp. 23. ACI Committee 318, “Building Code Requirements for Structural Concrete (ACI 318-05) and Commentary (318R-05),” American Concrete Institute, Farmington Hills, MI, 2005, 430 pp. 24. Alsiwat, J. M., and Saatcioglu, M., “Reinforcement Anchorage Slip under Monotonic Loading,” Journal of Structural Engineering , ASCE, V. 118, No. 9, Sept. 1992, pp. 2421-2438. 25. Lehman, D. E., and Moehle, J. P., “Seismic Performance of WellConfined Concrete Bridge Columns,” PEER Report 1998-01, College of Engineering, University of California, Berkeley, CA, Dec. 2000, 286 pp.


Plastic Hinge Length of Reinforced Concrete Columns

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