Karthik Mander 2010
Karthik Mander 2010
Karthik Mander 2010
Abstract: Equations to obtain equivalent rectangular stress-block parameters for unconfined and confined concrete are derived for rapid
共hand兲 analysis and design purposes. To overcome a shortcoming of existing commonly used stress-strain models that are not easy to
integrate, a new stress-strain model is proposed and validated for a wide range of concrete strengths and confining stresses. The efficacy
of the equivalent rectangular stress-block parameters is demonstrated for hand calculations in predicting key moment-curvature results for
a confined concrete column. Results are compared with those obtained from a computational fiber-element analysis using the proposed
stress-strain model and another widely used existing model; good agreement between the two is observed.
DOI: 10.1061/共ASCE兲ST.1943-541X.0000294
CE Database subject headings: Concrete; Stress strain relations; Parameters.
Author keywords: Concrete; Stress strain; Stress blocks; Confined concrete; Moment curvature.
customarily uses cu = 0.003 to define the nominal strength. How- (a) Equivalent rectangular stress-block analysis.
ever, as pointed out by Park and Paulay 共1975兲, stress blocks may f'cc Unconfined concrete
Confined concrete
be used across a spectrum of maximum strains. Indeed a stress- Esh
Concrete stress (fc)
fcu
1 (sh ,fy)
Graduate Assistant Researcher, Zachry Dept. of Civil Engineering,
Texas A&M Univ., College Station, TX 77843-3136. fc1
2
Zachry Professor 1, Zachry Dept. of Civil Engineering, Texas A&M
Univ., College Station, TX 77843-3136 共corresponding author兲. Ec Es
Note. This manuscript was submitted on December 2, 2009; approved co c1cc sp cu f
Concrete strain (c) Steel strain (s)
on August 23, 2010; published online on August 24, 2010. Discussion
(b) Concrete model (c) Reinforcing steel
period open until July 1, 2011; separate discussions must be submitted for
individual papers. This technical note is part of the Journal of Structural
Fig. 1. Stress-block approach and constitutive models for moment-
Engineering, Vol. 137, No. 2, February 1, 2011. ©ASCE, ISSN 0733-
9445/2011/2-270–273/$25.00. curvature analysis
e
Failure stress= 0 for all cases.
these are the peak strength 共co , f ⬘c 兲, at the termination of the customarily adopted to provide a dependable estimate for design.
postpeak branch 共c1 , f c1兲, and the failure strain 共sp , 0兲. Similarly, For “exact” analysis of existing reinforced concrete members, a
for confined concrete the corresponding principal control coordi- realistic stress-strain model should be adopted using expected val-
nates are 共cc , f ⬘cc兲, 共cu , f cu兲, and 共 f , 0兲. Using these coordinates ues of the control parameters. Fig. 1共c兲 represents such a model
as commencement and termination points, the proposed stress- and is conveniently cast in the form of a single equation as fol-
strain model has three branches—an initial power curve up to the lows:
peak stress, followed by a bilinear relation in the postpeak region.
E s s
再 冏 冏冎
The expressions representing concrete stresses as a function of fs =
strain are E s s 20 0.05
1+
0 ⱕ x ⬍ 1; f c = Kf ⬘c 共1 − 兩1 − x兩n兲 共1兲 fy
1 ⱕ x ⬍ xu ; f c = Kf ⬘c − 冉 冊
Kf ⬘c − f cu
xu − 1
共x − 1兲 共2兲
冋
+ 共f su − f y兲 1 −
兵兩su − sh兩
兩su
20P
− s兩 P
+ 兩su − s兩20P其0.05
册 共4兲
where
xu ⱕ x ⬍ x f ; f c = f cu 冉 冊
x − xf
xu − x f
共3兲 P=
Esh共su − sh兲
共f su − f y兲
共5兲
in which f cu = stress corresponding to hoop fracture strain cu; K in which f s, s = stress and strain in steel; Es and Esh = Young’s
= confinement ratio and for confined concrete 共K ⬎ 1兲; x modulus of elasticity and strain hardening modulus, respectively;
= normalized strain, where x = c / cc; xu = cu / cc; and x f = f / cc, f y and f su = yield strength and ultimate strength of reinforcing
where cc and f = strain at maximum confined strength of con- steel; and sh and su = strain hardening strain and ultimate strain,
crete f ⬘cc = Kf ⬘c and final failure strain of confined concrete, respec- respectively.
tively; and n = Ecco / f ⬘c and n = Eccc / f ⬘cc for unconfined and
confined concrete, respectively, where Ec = 5 , 000冑 f ⬘c 共MPa兲
= 60, 000冑 f ⬘c 共psi兲 = concrete modulus. For unconfined concrete Equivalent Rectangular Stress-Block Parameters
共K = 1兲, cc = co, cu = c1, f = sp, and f cu = f c1 in all of the above
equations 关refer to Fig. 1共b兲兴. Stress-block parameters can be easily derived using Eqs. 共1兲–共3兲
In the present widely used Mander model 共Mander et al. for a wide range of maximum strains as shown in Fig. 2. The
1988b兲, the governing stress-strain relation lacks the necessary force in concrete 共Cc兲 for a known value of maximum strain can
control over the slope of the postpeak branch. This is particularly be expressed in terms of equivalent stress-block parameters ␣ and
the case for high-strength concrete as pointed out by Li et al.  such that Cc = ␣f ⬘c cb, where c = depth to the neutral axis from
共2001兲. It is for this reason and also the ease of algebraic manipu- the top concrete fiber in compression and b = breadth of the sec-
lation the above equations are proposed. Proposed default values tion as shown in Fig. 1共a兲. The stress-block parameters can be
for the parameters in Eqs. 共1兲–共3兲 are defined in Table 1. The found from taking the first and second moments of area of the
expression for f cu was calibrated using data from full-scale ex- stress-strain relations which lead to the following results:
冕
perimental results of Mander et al. 共1988a兲 and Li et al. 共2000兲 c
and adjusted to essentially conform to the widely used original f cdc
Mander model. 0
For the stress-strain model of unconfined concrete in tension, ␣ = 共6兲
f ⬘c c
the same model as described above for unconfined concrete in
compression can be used by replacing the terms 共to , f t⬘兲, 共t1 , f t1兲, and
冕
and 共u , 0兲 for their corresponding terms in compression. Mea- c
sured values or as a good approximation the values from Table 1 f ccdc
may be used. 0
=2−2 共7兲
冕
In the analysis of moments and axial loads, two different mod- c
els of the stress-strain performance of the reinforcing steel may be c f cdc
adopted. For nominal design capacities, an elastoplastic model is 0
50
0.4 0.6
30 MPa 40 MPa k=2.0 k=1.75
k=1.50 k=1.25
0.2 50 MPa 60 MPa
0.4
600
70 MPa 80 MPa k=1.05 12mm @ 100
0
mm c/c
0.2
1.1 25mm bars
0.0
1
0 0.02 0.04 0.06
0.9 D = Dimensionless Curvature
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0.8
Fig. 3. Comparison of moment-curvature results
0.7
冉 冊
0.6
0 1 2 3 4 5 0 1 2 3 4 5 1 f cu
c/co c/cc A2 = 1+ 共17兲
2 Kf ⬘c
Fig. 2. Stress-block parameters for unconfined and confined concrete
n共n + 3兲
B1 = 共18兲
Carrying out the integration in Eqs. 共6兲 and 共7兲 using the 共n + 1兲共n + 2兲
stress-strain relations 共1兲–共3兲 gives the stress-block relations as
follows: B2 = 共x2u − 1兲 共19兲
1. For 0 ⱕ x ⬍ 1
␣ = 1 +
共1 − x兲n+1 − 1
x共n + 1兲
共8兲 B3 = 冉 f cu
Kf ⬘c
−1 冊冉 2x3u − 3x2u + 1
3共xu − 1兲
冊 共20兲
=2−
2
x2␣ 2
+ 冋
x2 x共1 − x兲n+1 共1 − x兲n+2 − 1
共n + 1兲
+
共n + 1兲共n + 2兲
册 共9兲 B4 =
Kf ⬘c
冉
f cu 3x f x2u − 2x3u − x3f
3共xu − x f 兲
冊 共21兲
␣ =
A1
−
x−1
x x共xu − 1兲
x
A2 − xu + 1 −
2
冋
f cu
Kf ⬘c
冉 冊册 共10兲
are shown in Fig. 2. Note that calculation of the strength enhance-
ment factor 共K兲 should be performed in accordance with any ac-
ceptable concrete model, for example, Mander et al. 共1988b兲 and
冋 冉 冊冉 冊册
Li et al. 共2001兲.
1 f cu 2x3 − 3x2 + 1
=2− B1 + 共x2 − 1兲 − 1 −
x ␣
2
Kf ⬘c 3共xu − 1兲
Numerical Example
共11兲
An example of the proposed stress-block approach for generating
3. For xu ⱕ x ⬍ x f moment-curvature results for a square column with the following
␣ =
A1
x
+ A2
xu − 1
x
+冉 冊 冉
f cu 共x − xu兲共x + xu − 2x f 兲
Kf ⬘c 2x共xu − x f 兲
冊 properties is performed. The section properties are breadth and
height= 600 mm, clear cover= 50 mm to the hoops of diameter
ds = 12 mm, and stirrup spacing s = 100 mm containing 12 sym-
共12兲 metrically placed longitudinal rebars of diameter db = 25 mm.
Concrete properties are 关refer to Fig. 1共b兲兴 f ⬘c = 30 MPa, co
=2−
1
x ␣
2 冋
B1 + B2 + B3
= 0.0019, sp = 0.009, f ⬘cc = 45 MPa, cc = 0.00675, and Ec
= 27387 MPa 共the above parameters were calculated using the
冉 冊册
expressions presented earlier兲. The longitudinal reinforcing steel
f cu 3x f x2u − 2x3u + x2共2x − 3x f 兲 properties are 关refer to Fig. 1共c兲兴 f y = 430 MPa, Es
+ 共13兲
Kf ⬘c 3共xu − x f 兲 = 200, 000 MPa, f u = 650 MPa, u = 0.12, sh = 0.008, and Esh
= 8 , 000 MPa. The axial load on the column is 2 , 000 kN
4. For x ⱖ x f = 0.185f ⬘c Ag.
␣ =
A1
x
+ A2
xu − 1
x
+ 冉 冊 冉 冊
f cu x f − xu
Kf ⬘c 2x
共14兲
In the hand computations for the example considered above,
for a value of K = 1.5, and at the first yield strain of steel y 共“Y”
in Fig. 3兲, the values of the stress blocks were ␣ = 0.828 and
 = 0.724 for unconfined concrete and ␣ = 0.470 and  = 0.702 for
B1 + B2 + B3 + B4
=2− 共15兲 confined concrete. Hand computations were also performed at the
x2␣ following values of the strain: strain at the extreme cover concrete
In the above, the following coefficients are used: fiber max = 0.003 and sp 共“N” and “S1,” respectively in Fig. 3兲
and strain at the extreme confined concrete fiber max = sp and
n 2cc 共“S2” and “U,” respectively, in Fig. 3兲. In order to implement
A1 = 共16兲
共n + 1兲 the iterative computational procedure to obtain the moment-