R EVIEW OF L ITERATURE
2.7 P LASTIC H INGE L ENGTH
2.7.1 Beams and Columns
In case of isolated RC column, the plastic hinges form at the maximum moment region. If Lp is known, the tip displacement of a column can be easily obtained by integrating curvatures. Therefore, accurate assessment of Lp is important in relating section-level response to member-level response of a concrete column. Many researchers (Baker, 1956; Baker and Amarakone, 1964; Mattok, 1964;
Mattok, 1967; Corley, 1966) studied Lp to estimate the flexural deformation capacity of RC beams.
To estimate the flexural deformation capacity, the plastic rotation capacity (θp) and the Lp are used as,
p ce cu
p L
c
(2.12)
where, cuis the maximum concrete compressive strain, ceis elastic concrete compressive strain and c is distance from extreme compression fiber to neutral axis (Figure 2.21).
Figure 2.21: Estimation of curvature at any section along the height of wall.
Lw
hs
floor slab
Baker (1956) investigated the plastic deformation of hinges and members in concrete frames. It was proposed that the length of plastic hinge in column is between 0.5h and h, where, h is the depth or height of the member. Baker and Amarakone (1964) proposed the following expression to determine the plastic hinge length of members with unconfined concrete as,
4 / 1 3 2
1
d k L k k
Lp s (2.13)
4 / 1
3
2 1 0.5
Pu
k P
(2.14)
92 8 . 9 13
.
3 0
fck k
(2.15) where, Lsis the distance from the section of maximum moment to the section of zero moment, k1 is the factor for the influence of the tension reinforcement (considered as 0.7 for mild steel or 0.9 for cold worked steel). k2is the factor for the influence of axial load, k3is the factor for the influence of concrete strength, d is the effective depth of the member, Pis the axial load, Puis the axial compressive strength of the member without any bending moment and fck is the concrete cube strength in MPa.
Sawyer (1964) developed a design methodology for RC frames based on a bilinear moment- curvature relationship. Three assumptions were made to develop the method, namely (a) the maximum moment at any section is equal to the ultimate moment (Mu), (b) the ratio My Muis taken as 0.85 based on the previous test results obtained for beams, and (c) the spread of plasticity
upto a distance of d/4 from the end in which the bending moment is equal to the yield moment (My). The plastic hinge length proposed on the basis of the assumption is given by,
s
p d L
L 0.25 0.075 (2.16)
Mattock (1964) performed tests on simply supported beams subjected to a concentrated load at the midspan to investigate the effect of confinement of the concrete in compression and the effect of the size of the member in their rotational capacity. 37 beams were tested to investigate how the rational capacity is influenced by the concrete strength, the effective depth, the distance from the section of maximum moment to the point of contraflexure and the amount and yield stress of reinforcement.
On the basis of these experiments, the following equation was proposed to calculate the plastic hinge length as,
1 1.14 1 1 0.411
2
' d
d L
Lp d s (2.17)
where,
and 'are the tension and compression reinforcement index respectively.On the basis of the previously mentioned experiments, the following equation was proposed by Corley (1966) to obtained Lp as,
d d L
Lp 0.5 0.032 s
(2.18) Based on the previous experiments carried by Mattock (1964) and Corley (1966), the simplified equation proposed by Mattock (1967) was given as,
s
p d L
L 0.5 0.05
(2.19) The ACI-ASCE Committee 428 (1968) proposed lower and upper bounds for the plastic hinge length in beams and frames with the expression as,
s m p d LsRm
R L d R R d L
R
Min 0.10
, 2 03 .
4 0
(2.20)
where,
ce cu
R ce
0.004 ,
y u
y
m M M
M R M
max , Ris the strain ratio, Rmis the moment ratio, ceis
the concrete strain in the extreme compression fiber at yield curvature, cu is the concrete strain in the extreme compression fiber at ultimate curvature (neglecting the effects of confinement, loading rate and the strain gradients) and Mmaxis the maximum moment in the length of the member.
Priestly and Park (1987) performed experiments on concrete bridge columns with different cross- sections subjected to combined axial load and bending to study their strength and ductility. The influence of axial load, the amount and yield strength of the transverse reinforcement and the aspect ratio on the seismic behavior of concrete bridge column was investigated. Based on these test results the expression for the plastic hinge length was proposed as,
b s
p L d
L 0.08 6 (2.21)
The equation was used to compute the plastic hinge length of the columns with different aspect ratios that were tested by other researchers, and the obtained values of Lp were compared. It is
observed that the average value plastic hinge length calculated for all the test was approximately as h
5 . 0 .
In one experimental study, 14 RC bridge columns with different cross sections were subjected to combined axial load and bending action for assessing the strength and ductility of the columns (Zahn et al., 1986). As the equation of Lp, proposed by Priestly and Park (1987), was valid for RC bridge columns with axial load, the validity of the same equation in presence of axial compression was also checked as part of the study. The test observations showed that the inelastic curvatures spread over a longer portion of the column due to concrete spalling when under large axial compression.
Considering the effect of axial load Lp was proposed as,
0.08 6
0.5 1.67 ' ; ' 0.3
g c g
c b
s
p f A
for P A
f d P
L
L (2.22)
0.08 6
; ' 0.3
g c b
s
p f A
for P d
L
L (2.23)
For circular columns with no confinement and one ring of reinforcement, Lp was recommended as,
b s
p L d
L 0.06 4.5
(2.24) Paulay and Priestly (1992) proposed the following expression to estimate the plastic hinge length.
b y b
y s
p L f d f d
L 0.08 0.022 0.044
(2.25) where, fyis the yield strength of reinforcement in MPa. The authors indicated that for commonly used beam and column dimensions, Eq. 2.25 gave plastic hinge lengths of approximately as0.5h.
Moehle (1992) indicated that the equivalent plastic hinge length in RC columns depends on the section depth, aspect ratio, bar diameter and the axial and shear force. He stated that good agreement with experimental results may be obtained when the plastic hinge length is equal to0.5h.