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An optimum criterion for eccentrically loaded R. C. columns
Build. Sci. Vol. 6. pp. 83-88. Pergamon Press 1971. Printed in Great Britain
[
I (28) I Gf2 I (E4) [
An Optimum Criterion for Eccentrically Loaded R. C. Columns P R A K A S H DESAYI* A. S E T H U R A T H N A M t
An analysis of eccentrically loaded short reinforced concrete columns using a variable failure strain criterion is presented. The method dispenses with the usual procedure of assuming a fixed value for the ultimate strain in concrete. The analysis is based on the use of a simple, single equation for the complete stress-strain curve of concrete and the adoption of a process of maximisation of moment with respect to extreme fibre concrete compressive strain. Columns of rectangular section and loaded eccentrically along one axis only are considered in this paper. A good agreement is observed between the theoretical and experimental values of some test results.
parabola for ascending portion and a straight line for the descending part. Hognestad, also adopted an ultimate concrete strain of 0.0038. Two typical investigations in which an assumption of a magnitude for ultimate concrete strain was avoided are due to Lee[4] and Kriz[5]. While evaluating the ultimate load, Lee[4] adopted a process of maximising the resisting moment with curvature and used a second degree parabola for the stress-strain curve of concrete. In his analysis, Kriz[5] did not employ a quantitative definition of stress-strain relationship; he adopted a criterion of maximising the moment with respect to the extreme fibre strain. His equations, however, finally require the magnitude of extreme fibre stress in concrete which is to be taken from a relation between the cylinder strength and extreme fibre stress. Kabila[6] also noted that the necessity of the assumption of ultimate strain value can be dispensed with, by a procedure of maximisation of moment for ultimate condition. Pfrang et al.[7] pointed out that a variation of the limiting strain of concrete does not affect the magnitude of ultimate moment significantly and that the influence of ultimate strain on moment may be minimal in many cases. However, the limiting strain values exercise great influence on the moment-curvature ratio at failure.
INTRODUCTION T H E I M P O R T A N C E of the study of eccentrically loaded short R.C. columns is well known. The combination of axial compression and flexure is a stress condition that is typical of the cases occurring in real structural members. For determining the ultimate load of a reinforced concrete section subjected to bending or bending and compression, the important properties of concrete needed are the stress block and the ultimate strain. From a knowledge of the various stress blocks used in the ultimate strength studies[l], it is noticed that, (a) assumption of rectangular or trapezoidal stress blocks is empirical, since the block does not represent the actual stress distribution, (b) assumption of curves such as parabola, sine wave or ellipse does not represent correctly the complete stress-strain curve of concrete, hence, also, the actual stress distribution and (c) combinations of two curves or a curve and a straight line (for the ascending and descending portions respectively) are better approximations but lack convenience in the analysis when compared to that of using a single equation. The ultimate concrete strain, being influenced by a large number of parameters, is found to vary in a wide range and many analyses assume an average value based on experimental evidence[l]. Some of the available ultimate load theories for reinforced concrete sections subjected to combined bending and compression are due to Whitney[2] who used a rectangular stress block, Cowan[3] who adopted a trapezoidal stress block and Hognestad[1] who used a stress block consisting of a second degree
S C O P E OF T H E I N V E S T I G A T I O N It is seen that the shortcomings mentioned regarding the form of stress-blocks used can be overcome by adopting a single equation for the complete stress-strain curve of concrete. Assigning of an approximate magnitude for the ultimate concrete strain can be avoided by assuming it to be an unknown and variable in the analysis and deter-
* t Assistant Professor and former Post Graduate Student respectively, Civil and Hydraulic Engineering Department, Indian Institute of Science, Bangalore, India. 83
Prakash Desayi and A. .S'ethuratlmant
X4
mining the same by a maximising process. Such a method will result in arriving at the values of ultimate strength as well as the ultimate concrete strain. In addition, if required, it will permit the inclusion of the influence of rate of loading which can be incorporated through the stress-strain curve. It should, however, be noted that a very exact representation of the experimentally determined stressstrain curves may often lead to equations of a complex form (for instance those due to Kriz and Lee[8] and Saenz[9]), whereas an easy analysis requires the equations to be as simple as possible. Hence a balance is to be struck between accuracy and simplicity: the form of the stress-strain curve used should lead to simple equations and also should satisfactorily represent the experimental stressstrain curves. Keeping the above points in view, in this investigation, a simple equation for the stress-strain curve of concrete[10] is used and an optimum criterion consisting of maximising the resisting moment with respect to extreme fibre concrete strain in compression is adopted to determine the moments and strains at ultimate. The resulting equations are found to be fairly simple and it may be mentioned that this procedure when used for the analysis of singly[11] and doubly[12] reinforced beams, has been found to give good results.
The analysis has been subdivided into t~o section, depending on the position of the neutral axb. Section
l" Neutra[ axis lies within the cros.~ section
This is generally the case when the eccentricit~ of the load is large. With reference to tigure 1, the tbllowing expressions can be developed. p t
.......... i t
~s Cross section
T:Asfs
Fig. I. Strain and stress t~aHalions wJlh ,'lelllro! a.vi,~ within the cross section.
C,. = ~ n b d./~)
C~ bdjo-
The usual assumptions, viz, plane sections remain plane, tensile strength of concrete is negligible, perfect bond between steel and concrete and perfect elastic-plastic behaviour of steel with definite yield point, are assumed. The stress-strain curve of concrete in compression for a given rate of straining, as given by reference
Ego , ( n - r ) 2fore,, < e~, ./~. q = q' for e'~ _-> ~;,~.
T
-
E go --
q
./;.
(l-n)
(4a
2 lbr r, < e~.
(4bl
,1
= q for c~ >= e~.
P = C,.+C,-T
(5~
Pe' = C , , ( I - k 2 n ) d + C ~ ( l - r ) d
(61
and
where
(1)
\go/ is used.* When using equation (1) as a stress block for eccentrically loaded columns,./'[, will be replaced by./o ( = 0.85 f'c), the factor 0.85 having been substantiated by the results of tests on concentrically loaded columns[l 3]. A plot of% vs. cylinder strength of concrete[Ill from test data of a number of authors[4, 9, 14, 15, 16] shows that an average value of 0-002 can be assumed for concretes of all strengths and especially for those above 2400 psi (170 kg/cm2). The ultimate concrete strain, %, corresponds to the optimum criterion dM/dg,. = 0 when the moment M is expressed in terms of the extreme fibre strain ~¢.
14c)
For static equilibrium
t
* Notation is defined in the Appendix.
(3c1
r = A~./I,
2 1-~tan e,o i + ( _e, '~ 2
(3bl
or
it0] ,
(3a)
or
ANALYSIS
--
(2/
C.~ = A; / ;
bd[o
f=
Stress diogrOrr,
Strata diagram
k2 = I -
~1
1 "/
(7~
log (1 +221
In writing equations (3a), (3b) and (3c), the effect of concrete area displaced by compression steel is neglected. The possible combinations of the stress situations at ultimate in the steels As and A's give rise to the following cases. Case 1.
~ => e.~yand s~ > ~,~. (tension failure)
From equations (5), (3c) and (4c), n is given b~, 1
P
n =7(~fo+q-q)
(Sa)
Then, from equations (6), and (3c), Pe'
• = :t n(I - k a n ) + q ' ( I - r )
db2f;
(Sb)
An Optimum Criterion for Eccentrically Loaded R.C. Columns Case 2. e~ > e,y and ds < &y (tension failure) From equations (5), (3b) and (4c), n is given by,
2.nfEeo ff_._~ Eeoq'2r P n +-~--f/-yCt q'2-q . bdfo} . . fr. ~
d = Od
(14a)
where
1
0 (9a)
es eo
-~- - - ,
~s -~-
0 =
Pe' E eo q' = ~ n(l-k2n)+-~-_. -- ( n - r ) (1 - r ) 2 bd2fo n Jy
n-I n -
-
8c
and
From equations (6), and (3b),
(14b)
2-i
with (9b)
Case 3. es < e~y and e's > esy (compression failure) From equations (5), (3c) and (4b), n is given by,
=
2[(2- I ) - (tan- ~ 2 - tan- i I)] 1+22 log t l-7Z-q
For statical equilibrium (figure 2), P = Cc+C~+T~
n2+n(q'+fff---°q2-~\ Jr ~ o P ) - -"~Ee°q2=O f7
(15)
and
(lOa) From equations (6) and (3c),
Pc' bd2f° = ct n ( l - k 2 n ) + q ' ( 1 - r )
85
(lOb)
Pc'= ccd+cs(1-r)d
(16)
Two cases, case (4) and case (5), arise depending on the compressive stress level in A s.
Case 4. es < esy (compression), and ds > &y
Section 2. Neutral axis falls outside the cross-section This is associated with loads acting at small eccentricities. Figure 2 shows the strain and stress distributions for this case. Both steels As and A'~ are in compression. At ultimate A~ would have yielded irrespective of the stress level of As.
From equations (15), (11), (12a) and (13), n is given by
p
1 , (n-K) 2 bdfo - (2-)T----~l o g \ 1 - ' - ~ } +q + ~ . v q - - 7 (17a) From equations (16), (11), (13a) and (14a),
Pc' 0 lo / 1 + 2 2 \ , bdEfo = 2-,~ g ~ l ~ ) + q ( 1 - r )
(17b)
Case 5. es > esy (compression), and e~ > esy ---J
[
/To:As f s
From equations (15), (11), ( l l a ) and (13), n is given by, P
i/ / III .~
,
I/I/ _~__
S t r a i n diagram
Cross section
1 /
/// A
Stress diagram
Fig. 2. Strain and stress variations with neutral axis outside the cross-section. For this case we can formulate the following expressions: Cs
bdfo
= q' as e~ > &y
r~ E fo (n-- 1) • = -- q - 2 for e, < &y bdfo fr n
(11)
(12a)
or, = q for & > esy
cc
(12b)
1
bdfo - 2 - ~ log t k l ~ }
1
/1 +22"~
,
bdfo = X_ "~ log ~ l - - ~ ) + q +q
(13)
(18)
and the expression for Pe'/bd2fo remains the same as in equation (17b). In writing the equations in cases (4) and (5), the concrete area replaced by both the steels As and A's are not accounted for the sake of simplicity. This is compensated to some extent by neglecting the concrete area forming the cover of the steel As. The ultimate moment M,, = Pe where e is the eccentricity of P from the plastic centroid of the column section and can be obtained when once Pc' is determined from the above equations.
OPTIMISATION CRITERION For a given cross-section with specified amounts of reinforcement, for different values of P starting from the case of pure flexure (P = 0) to the maximum axial load [P = Po = (1 + q + q ' ) bdfo ] values of corresponding Pe'/bd2fo are determined from the
x6
Praka.@ Desayi and A. Sethurathnam
above equations as the highest values, when /, is given incremental values in each case (say starting from ). = 1 and with increments o1'0.2). The values of ,:. corresponding to the highest value of Pe'/bd21{~ is designated as ).. and it gives the limiting or ultimate strain e.. From a knowledge of Pe'/bd2l~), the ttltimate moment M,, ( = Pe') can be calculated.
ILLUSTRATIVE EXAMPLE AND INTERACTION DIAGRAMS
in this study, the computational work based on the above method was done and the interaction diagrams of load and moment at ultimate were determined for the following data: E = 30x 106 psi (21.2x 105 kg/cm2), ./~, = 40,000 psi (2820 kg/cm2),.// = 3000 psi (210 kg/cm2), % = 0.002 and r = 0.15. The assigned values o f q ranged from 0.2 to 1.2 at steps of 0.2 and q' was provided as 0, 25, 50, 75 and 100 per cent of q for each of its values. All the computational results and all interaction diagrams are presented elsewhere [17] in detail. Figure 3 represents a typical interaction diagram obtained in the analysis.
30
25
2.0
Magnitude of 2. (and hence ~:,,) is dependem on
q, q'. P/Po and the stress levels in the steel at failure. For the case of large eccentricities, it is observed that for most of the values ofq+q', e,, equals about 0.0028 for a tension failure with compression sled also yielding: e.. is between 0"0028 and 0"0036 for tension failures with compression steel not yielding: and % is in the range 0.0032 to 0-0036 for compression failures. For small eccentricities, 6, is between 0.0024 and 0.0028 when A~ does not yield lin compression), and ~, is about 0"002 when both the steels yield (in compression), which indicates that this last case is close to the pure axial compression. A smooth variation of 2,, was not obtained in this study which could be due to the fact that 2 is given discreet incremental values. The results, however, are indicative of the general qualitative behaviour.
COMPARISON
WITH TEST RESULTS
Some of the test results, the data of which fairly approximate to that used in the determination of the interaction diagrams, have been examined in the light of the results of this analysis. For this purpose, 45 column test results of Hognested (22 of Group I1 and 23 of Group III tests) [1], have been utilised. The comparison of the theoretical and the experimental values of M,,/bd2fo is shown in Fig. 4 in which most of the points are found to lie within + 10~0 agreement lines. On an average, the ratio of theoretical to experimental values of M,, is 0.956. Computed results of ultimate moment also agreed fairly well with those based on Hognestad's method.
,,2
i 5
09
L: 8
///
i O (3 z
05
o
s 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
,'J -o x~
O5
0 a
Mu/bdZf0
F~,,, 3. Interaction diagram J o t q ' = 0 ' 7 5 q.
02
0 t
V A R I A T I O N OF U L T I M A T E STRAIN
To study the variation of ultimate strain, 2, vs. P/Po has been plotted for different combinations o t ' q and q'. From these plots [17], the following points are noted.
0 I
0 2
03
0.4
0 5
06
07
08
M~/bd2f (experimentol)
Fig. 4. Comparison o f the theoretical ultimate moment with test results.
09
An Optimum Criterion.for Eccentrically Loaded R.C. Columns SUMMARY AND CONCLUSIONS 1. A simple equation (equation 1) which fairly represents the stress-strain curve o f concrete in compression has been used for the development o f necessary stress blocks in the analyses o f eccentrically loaded columns. 2. An o p t i m u m criterion by way o f maximising the resisting m o m e n t with respect to the extreme fibre strain has been used for determining the ultimate stage and this analysis is found to determine the ultimate strength quite satisfactorily. 3. The strain at ultimate is found to be dependent on tension and compression steel ratios, on the
87
steel stress-levels at ultimate and on the ratio P/Po as well as on the strength properties of the materials used. This indicates that the usual assumption o f a fixed ultimate concrete strain or assuming it to be a function o f concrete strength only, requires reexamination in the ultimate strength studies.
Acknowledgements--The authors are grateful to Prof K. T. S. Iyenger, Professor of Civil Engineering, Indian Institute of Science, for his keen interest and constant encouragement in this research project. The authors are also thankful to Sri. P. Vasudeva Rap for his willing assistance in the computational work which was done on the 803 National Elliot electronic digital computer.
REFERENCES 1. E. HOGNESTAD,A study of combined bending and axial load in reinforced concrete members, Bulletin No. 399, Univ. of Ill. Engng Expt. Station (1951). 2. C. S. WHITNEY, Plastic theory of reinforced concrete design, Trans. Am. Soc. Cir. Engrs. 107, 251 (1942). 3. H. J. COWAN, Ultimate strength theory for eccentrically loaded reinforced concrete columns, Mag. Concr. Res., 3 (7), 19 (1951). 4. L. H. N. LEE, Inelastic behaviour of reinforced concrete members, Trans. Am. Soc. Civ. Engrs. 120, 181 (1955). 5. L.B. Kmz, Ultimate strength criteria for reinforced concrete, Trans. Am. Soc. Cir. Engrs. 126, Part 1,439 (1961). 6. A. KABILA,Discussion on Equation for the stress-strain curve of concrete, J. Am. Concr. Inst., 61 (9), 1227 (1964). 7. E. O. PFRANG, C. P. SIESS, and M. A. SOZEN, Load moment curvature characteristics of reinforced concrete cross-sections, J. Am. Concr. Inst., 61 (7), 763 (1964). 8. L. B. KRIZ and S. L. LEE, Ultimate strength of over-reinforced beams, J. of Engng Mech. Div., Proc. Am. Soc. Civ. Engrs, 86 (EM3), 95 (1960). 9. L. P. SAENZ, Discussion on Equation for the stress-strain curve of concrete, J. Am. Concr. Inst., 61 (9), 1229 (1964). 10. PRAKASn DESAYI and S. KR~SIaNAN,Equation for the stress-strain curve of concrete, J. Am. Concr. Inst., 61 (3), 345 (1964). 11. PRAKASHDESAYI and C. S. VISWANATnA,Ultimate flexural strength of reinforced concrete sections, Proc. of Syrup. on Ultimate Load Design of Concrete Structures, PSG College of Technology, Coimbatore, India, Session II, 62 (1967). 12. N. GOPAL, Strength and behaviour in flexure of doubly reinforced concrete beams, M.E. Dissertation, Civ. and Hyd. Engg. Dept, Indian Institute of Science, Bangalore, India (1966). 13. F:. E. RICHARTand R. L. BROWN,An investigation of reinforced concrete columns, Bulletin No. 267, Univ. of Ill. Engg. Expt. Station (1934). 14. D. RAMLEYand D. MCHENRY, Stress-Strain curves for concrete strained beyond ultimate load, Lab. Report No. SP-I 2, U.S. Bureau of Reclamation, Denver (1947). 15. E. HOGNESTAD,N. W. HANSONand D. McHENRV, Concrete stress distribution in ultimate strength design, J. Am. Concr. Inst., 52 (4), 455 (1955). 16. G. M. SMITHand L. E. YOUNC, Ultimate flexural analysis based on stress-strain curves of cylinders, J. Am. Concr. Inst., 53 (4), 597 (1956). 17. A. SETViURATHNAM,Analysis of eccentrically loaded short reinforced concrete columns, M.E. Dissertation, Civ. and Hyd. Engg. Dept, Indian Institute of Science, Bangalore, India (1967).
S8
P r a k a s h D e s a v i a n d A. S e t h u r a t h n a m NOMENCLATUR E
,4~ ,4"~ h
(',. (, d
d" d E e e' f
f',. .1~ f~ .["~ ~.
k2c M
area of tension reinforcement area of compression reinforcement width of c o l u m n section distance of neutral axis from extreme compression fibre compressive force in concrete force in c o m p r e s s i o n steel effective depth of c o l u m n section distance from extreme compression fibre to centroid o f A ' ~ ( = rd) distance of C,. from A., Y o u n g ' s m o d u l u s of steel eccentricity of load measured from plastic centroid eccentricity of load measured from centre o f gravity of tension steel stress in concrete at a n y strain ~: cylinder strength of concrete m a x i m u m stress in stress block ( = 0.85 .[',.) corresp o n d i n g to a strain eo stress in tension steel stress in compression steel yield strength o f reinforcement distance of C,. from extreme compression fibre resisting m o m e n t
~1,. n P P,,
ultimate m o m e n t capacit} ratio c'd load on the c o l u m n ultimate strength of c o l u m n under axial loading
p
p"
4. hd 4 "~,,hd
q
tension reinforcement index = - - - -
q"
compression reinforcement index = ~ - -
r T T,
ratio d'/d tensile force in steel As compressive force in steel As 1 a factor defined as --: log ( I + 2 2) ,t factors given by equations (14b), (14c) respectively strain in concrete at any stress f strain in extreme concrete fiber in compression strain in concrete at stress [', or J~, o f the stress block strain in steel A~ strain in steel A'~ yield point strain o f steel ultimate concrete strain ratio e,.,/eo ratio ~deo ratio e,,l~,,
:~ eL d ~:
c,. ~:o ~:~ ~:', ~, e,, ,:. ). ).,,
A, J;, hd Jo
A', L.
hd l~
On pr6sente ici une analyse de poutres en b6ton renforc6, courtes et charg6es excentriquement. On utilise pour cela un critSre variable d'effort. Cette m~thode 6vite la proc6dure habituelle qui consiste b. attribuer une valeur fixe fi l'effort ultime du b6ton. L'analyse est bas6e sur une 6quation du premier degr8 pour la courbe totale tensioncompression du b6ton et pour l'adoption d'un proc6d6 de maximisation du moment par rapport fi l'extrfime contrainte de compression du b6ton. On examine ici des poutres verticales de section rectangulaire et chargfes excentriquement sur un seul axe. Une bonne concordance apparait entre les valeurs th6oriques et exp&imentales de certains r~sultats de tests. Es wird eine Analyse kurzer, exzentrisch belasteter StahlbetontrSger dargestellt, welche ein Kriterium ve#inderlicher Bruchspannungen benutzt. Die Methode verzichtet auf den gewtihnlichen Weg, einen festen Wert fiir die ~iusserste Spannung in Beton anzunehmen. Die Analyse ist aufdem Gebrauch einer einfachen, einzelnen Gleichung fiir die komplette Spannungs-Dehnungskurve fiir Beton begrfindet und auf der Annahme eines Verfahrens der H/Schststeigerung des Moments in Bezug auf~iusserste Druckspannung der Betonfaser. Nur Tr/iger mit rechteckigem Querschnitt und mit exzentrischer Belastung entlang einer Achse werden in diesem Bericht behandelt. Die theoretischen und experimentellen Werte stimmen in einigen Testresultaten gut iiberein.
[
I (28) I Gf2 I (E4) [
An Optimum Criterion for Eccentrically Loaded R. C. Columns P R A K A S H DESAYI* A. S E T H U R A T H N A M t
An analysis of eccentrically loaded short reinforced concrete columns using a variable failure strain criterion is presented. The method dispenses with the usual procedure of assuming a fixed value for the ultimate strain in concrete. The analysis is based on the use of a simple, single equation for the complete stress-strain curve of concrete and the adoption of a process of maximisation of moment with respect to extreme fibre concrete compressive strain. Columns of rectangular section and loaded eccentrically along one axis only are considered in this paper. A good agreement is observed between the theoretical and experimental values of some test results.
parabola for ascending portion and a straight line for the descending part. Hognestad, also adopted an ultimate concrete strain of 0.0038. Two typical investigations in which an assumption of a magnitude for ultimate concrete strain was avoided are due to Lee[4] and Kriz[5]. While evaluating the ultimate load, Lee[4] adopted a process of maximising the resisting moment with curvature and used a second degree parabola for the stress-strain curve of concrete. In his analysis, Kriz[5] did not employ a quantitative definition of stress-strain relationship; he adopted a criterion of maximising the moment with respect to the extreme fibre strain. His equations, however, finally require the magnitude of extreme fibre stress in concrete which is to be taken from a relation between the cylinder strength and extreme fibre stress. Kabila[6] also noted that the necessity of the assumption of ultimate strain value can be dispensed with, by a procedure of maximisation of moment for ultimate condition. Pfrang et al.[7] pointed out that a variation of the limiting strain of concrete does not affect the magnitude of ultimate moment significantly and that the influence of ultimate strain on moment may be minimal in many cases. However, the limiting strain values exercise great influence on the moment-curvature ratio at failure.
INTRODUCTION T H E I M P O R T A N C E of the study of eccentrically loaded short R.C. columns is well known. The combination of axial compression and flexure is a stress condition that is typical of the cases occurring in real structural members. For determining the ultimate load of a reinforced concrete section subjected to bending or bending and compression, the important properties of concrete needed are the stress block and the ultimate strain. From a knowledge of the various stress blocks used in the ultimate strength studies[l], it is noticed that, (a) assumption of rectangular or trapezoidal stress blocks is empirical, since the block does not represent the actual stress distribution, (b) assumption of curves such as parabola, sine wave or ellipse does not represent correctly the complete stress-strain curve of concrete, hence, also, the actual stress distribution and (c) combinations of two curves or a curve and a straight line (for the ascending and descending portions respectively) are better approximations but lack convenience in the analysis when compared to that of using a single equation. The ultimate concrete strain, being influenced by a large number of parameters, is found to vary in a wide range and many analyses assume an average value based on experimental evidence[l]. Some of the available ultimate load theories for reinforced concrete sections subjected to combined bending and compression are due to Whitney[2] who used a rectangular stress block, Cowan[3] who adopted a trapezoidal stress block and Hognestad[1] who used a stress block consisting of a second degree
S C O P E OF T H E I N V E S T I G A T I O N It is seen that the shortcomings mentioned regarding the form of stress-blocks used can be overcome by adopting a single equation for the complete stress-strain curve of concrete. Assigning of an approximate magnitude for the ultimate concrete strain can be avoided by assuming it to be an unknown and variable in the analysis and deter-
* t Assistant Professor and former Post Graduate Student respectively, Civil and Hydraulic Engineering Department, Indian Institute of Science, Bangalore, India. 83
Prakash Desayi and A. .S'ethuratlmant
X4
mining the same by a maximising process. Such a method will result in arriving at the values of ultimate strength as well as the ultimate concrete strain. In addition, if required, it will permit the inclusion of the influence of rate of loading which can be incorporated through the stress-strain curve. It should, however, be noted that a very exact representation of the experimentally determined stressstrain curves may often lead to equations of a complex form (for instance those due to Kriz and Lee[8] and Saenz[9]), whereas an easy analysis requires the equations to be as simple as possible. Hence a balance is to be struck between accuracy and simplicity: the form of the stress-strain curve used should lead to simple equations and also should satisfactorily represent the experimental stressstrain curves. Keeping the above points in view, in this investigation, a simple equation for the stress-strain curve of concrete[10] is used and an optimum criterion consisting of maximising the resisting moment with respect to extreme fibre concrete strain in compression is adopted to determine the moments and strains at ultimate. The resulting equations are found to be fairly simple and it may be mentioned that this procedure when used for the analysis of singly[11] and doubly[12] reinforced beams, has been found to give good results.
The analysis has been subdivided into t~o section, depending on the position of the neutral axb. Section
l" Neutra[ axis lies within the cros.~ section
This is generally the case when the eccentricit~ of the load is large. With reference to tigure 1, the tbllowing expressions can be developed. p t
.......... i t
~s Cross section
T:Asfs
Fig. I. Strain and stress t~aHalions wJlh ,'lelllro! a.vi,~ within the cross section.
C,. = ~ n b d./~)
C~ bdjo-
The usual assumptions, viz, plane sections remain plane, tensile strength of concrete is negligible, perfect bond between steel and concrete and perfect elastic-plastic behaviour of steel with definite yield point, are assumed. The stress-strain curve of concrete in compression for a given rate of straining, as given by reference
Ego , ( n - r ) 2fore,, < e~, ./~. q = q' for e'~ _-> ~;,~.
T
-
E go --
q
./;.
(l-n)
(4a
2 lbr r, < e~.
(4bl
,1
= q for c~ >= e~.
P = C,.+C,-T
(5~
Pe' = C , , ( I - k 2 n ) d + C ~ ( l - r ) d
(61
and
where
(1)
\go/ is used.* When using equation (1) as a stress block for eccentrically loaded columns,./'[, will be replaced by./o ( = 0.85 f'c), the factor 0.85 having been substantiated by the results of tests on concentrically loaded columns[l 3]. A plot of% vs. cylinder strength of concrete[Ill from test data of a number of authors[4, 9, 14, 15, 16] shows that an average value of 0-002 can be assumed for concretes of all strengths and especially for those above 2400 psi (170 kg/cm2). The ultimate concrete strain, %, corresponds to the optimum criterion dM/dg,. = 0 when the moment M is expressed in terms of the extreme fibre strain ~¢.
14c)
For static equilibrium
t
* Notation is defined in the Appendix.
(3c1
r = A~./I,
2 1-~tan e,o i + ( _e, '~ 2
(3bl
or
it0] ,
(3a)
or
ANALYSIS
--
(2/
C.~ = A; / ;
bd[o
f=
Stress diogrOrr,
Strata diagram
k2 = I -
~1
1 "/
(7~
log (1 +221
In writing equations (3a), (3b) and (3c), the effect of concrete area displaced by compression steel is neglected. The possible combinations of the stress situations at ultimate in the steels As and A's give rise to the following cases. Case 1.
~ => e.~yand s~ > ~,~. (tension failure)
From equations (5), (3c) and (4c), n is given b~, 1
P
n =7(~fo+q-q)
(Sa)
Then, from equations (6), and (3c), Pe'
• = :t n(I - k a n ) + q ' ( I - r )
db2f;
(Sb)
An Optimum Criterion for Eccentrically Loaded R.C. Columns Case 2. e~ > e,y and ds < &y (tension failure) From equations (5), (3b) and (4c), n is given by,
2.nfEeo ff_._~ Eeoq'2r P n +-~--f/-yCt q'2-q . bdfo} . . fr. ~
d = Od
(14a)
where
1
0 (9a)
es eo
-~- - - ,
~s -~-
0 =
Pe' E eo q' = ~ n(l-k2n)+-~-_. -- ( n - r ) (1 - r ) 2 bd2fo n Jy
n-I n -
-
8c
and
From equations (6), and (3b),
(14b)
2-i
with (9b)
Case 3. es < e~y and e's > esy (compression failure) From equations (5), (3c) and (4b), n is given by,
=
2[(2- I ) - (tan- ~ 2 - tan- i I)] 1+22 log t l-7Z-q
For statical equilibrium (figure 2), P = Cc+C~+T~
n2+n(q'+fff---°q2-~\ Jr ~ o P ) - -"~Ee°q2=O f7
(15)
and
(lOa) From equations (6) and (3c),
Pc' bd2f° = ct n ( l - k 2 n ) + q ' ( 1 - r )
85
(lOb)
Pc'= ccd+cs(1-r)d
(16)
Two cases, case (4) and case (5), arise depending on the compressive stress level in A s.
Case 4. es < esy (compression), and ds > &y
Section 2. Neutral axis falls outside the cross-section This is associated with loads acting at small eccentricities. Figure 2 shows the strain and stress distributions for this case. Both steels As and A'~ are in compression. At ultimate A~ would have yielded irrespective of the stress level of As.
From equations (15), (11), (12a) and (13), n is given by
p
1 , (n-K) 2 bdfo - (2-)T----~l o g \ 1 - ' - ~ } +q + ~ . v q - - 7 (17a) From equations (16), (11), (13a) and (14a),
Pc' 0 lo / 1 + 2 2 \ , bdEfo = 2-,~ g ~ l ~ ) + q ( 1 - r )
(17b)
Case 5. es > esy (compression), and e~ > esy ---J
[
/To:As f s
From equations (15), (11), ( l l a ) and (13), n is given by, P
i/ / III .~
,
I/I/ _~__
S t r a i n diagram
Cross section
1 /
/// A
Stress diagram
Fig. 2. Strain and stress variations with neutral axis outside the cross-section. For this case we can formulate the following expressions: Cs
bdfo
= q' as e~ > &y
r~ E fo (n-- 1) • = -- q - 2 for e, < &y bdfo fr n
(11)
(12a)
or, = q for & > esy
cc
(12b)
1
bdfo - 2 - ~ log t k l ~ }
1
/1 +22"~
,
bdfo = X_ "~ log ~ l - - ~ ) + q +q
(13)
(18)
and the expression for Pe'/bd2fo remains the same as in equation (17b). In writing the equations in cases (4) and (5), the concrete area replaced by both the steels As and A's are not accounted for the sake of simplicity. This is compensated to some extent by neglecting the concrete area forming the cover of the steel As. The ultimate moment M,, = Pe where e is the eccentricity of P from the plastic centroid of the column section and can be obtained when once Pc' is determined from the above equations.
OPTIMISATION CRITERION For a given cross-section with specified amounts of reinforcement, for different values of P starting from the case of pure flexure (P = 0) to the maximum axial load [P = Po = (1 + q + q ' ) bdfo ] values of corresponding Pe'/bd2fo are determined from the
x6
Praka.@ Desayi and A. Sethurathnam
above equations as the highest values, when /, is given incremental values in each case (say starting from ). = 1 and with increments o1'0.2). The values of ,:. corresponding to the highest value of Pe'/bd21{~ is designated as ).. and it gives the limiting or ultimate strain e.. From a knowledge of Pe'/bd2l~), the ttltimate moment M,, ( = Pe') can be calculated.
ILLUSTRATIVE EXAMPLE AND INTERACTION DIAGRAMS
in this study, the computational work based on the above method was done and the interaction diagrams of load and moment at ultimate were determined for the following data: E = 30x 106 psi (21.2x 105 kg/cm2), ./~, = 40,000 psi (2820 kg/cm2),.// = 3000 psi (210 kg/cm2), % = 0.002 and r = 0.15. The assigned values o f q ranged from 0.2 to 1.2 at steps of 0.2 and q' was provided as 0, 25, 50, 75 and 100 per cent of q for each of its values. All the computational results and all interaction diagrams are presented elsewhere [17] in detail. Figure 3 represents a typical interaction diagram obtained in the analysis.
30
25
2.0
Magnitude of 2. (and hence ~:,,) is dependem on
q, q'. P/Po and the stress levels in the steel at failure. For the case of large eccentricities, it is observed that for most of the values ofq+q', e,, equals about 0.0028 for a tension failure with compression sled also yielding: e.. is between 0"0028 and 0"0036 for tension failures with compression steel not yielding: and % is in the range 0.0032 to 0-0036 for compression failures. For small eccentricities, 6, is between 0.0024 and 0.0028 when A~ does not yield lin compression), and ~, is about 0"002 when both the steels yield (in compression), which indicates that this last case is close to the pure axial compression. A smooth variation of 2,, was not obtained in this study which could be due to the fact that 2 is given discreet incremental values. The results, however, are indicative of the general qualitative behaviour.
COMPARISON
WITH TEST RESULTS
Some of the test results, the data of which fairly approximate to that used in the determination of the interaction diagrams, have been examined in the light of the results of this analysis. For this purpose, 45 column test results of Hognested (22 of Group I1 and 23 of Group III tests) [1], have been utilised. The comparison of the theoretical and the experimental values of M,,/bd2fo is shown in Fig. 4 in which most of the points are found to lie within + 10~0 agreement lines. On an average, the ratio of theoretical to experimental values of M,, is 0.956. Computed results of ultimate moment also agreed fairly well with those based on Hognestad's method.
,,2
i 5
09
L: 8
///
i O (3 z
05
o
s 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
,'J -o x~
O5
0 a
Mu/bdZf0
F~,,, 3. Interaction diagram J o t q ' = 0 ' 7 5 q.
02
0 t
V A R I A T I O N OF U L T I M A T E STRAIN
To study the variation of ultimate strain, 2, vs. P/Po has been plotted for different combinations o t ' q and q'. From these plots [17], the following points are noted.
0 I
0 2
03
0.4
0 5
06
07
08
M~/bd2f (experimentol)
Fig. 4. Comparison o f the theoretical ultimate moment with test results.
09
An Optimum Criterion.for Eccentrically Loaded R.C. Columns SUMMARY AND CONCLUSIONS 1. A simple equation (equation 1) which fairly represents the stress-strain curve o f concrete in compression has been used for the development o f necessary stress blocks in the analyses o f eccentrically loaded columns. 2. An o p t i m u m criterion by way o f maximising the resisting m o m e n t with respect to the extreme fibre strain has been used for determining the ultimate stage and this analysis is found to determine the ultimate strength quite satisfactorily. 3. The strain at ultimate is found to be dependent on tension and compression steel ratios, on the
87
steel stress-levels at ultimate and on the ratio P/Po as well as on the strength properties of the materials used. This indicates that the usual assumption o f a fixed ultimate concrete strain or assuming it to be a function o f concrete strength only, requires reexamination in the ultimate strength studies.
Acknowledgements--The authors are grateful to Prof K. T. S. Iyenger, Professor of Civil Engineering, Indian Institute of Science, for his keen interest and constant encouragement in this research project. The authors are also thankful to Sri. P. Vasudeva Rap for his willing assistance in the computational work which was done on the 803 National Elliot electronic digital computer.
REFERENCES 1. E. HOGNESTAD,A study of combined bending and axial load in reinforced concrete members, Bulletin No. 399, Univ. of Ill. Engng Expt. Station (1951). 2. C. S. WHITNEY, Plastic theory of reinforced concrete design, Trans. Am. Soc. Cir. Engrs. 107, 251 (1942). 3. H. J. COWAN, Ultimate strength theory for eccentrically loaded reinforced concrete columns, Mag. Concr. Res., 3 (7), 19 (1951). 4. L. H. N. LEE, Inelastic behaviour of reinforced concrete members, Trans. Am. Soc. Civ. Engrs. 120, 181 (1955). 5. L.B. Kmz, Ultimate strength criteria for reinforced concrete, Trans. Am. Soc. Cir. Engrs. 126, Part 1,439 (1961). 6. A. KABILA,Discussion on Equation for the stress-strain curve of concrete, J. Am. Concr. Inst., 61 (9), 1227 (1964). 7. E. O. PFRANG, C. P. SIESS, and M. A. SOZEN, Load moment curvature characteristics of reinforced concrete cross-sections, J. Am. Concr. Inst., 61 (7), 763 (1964). 8. L. B. KRIZ and S. L. LEE, Ultimate strength of over-reinforced beams, J. of Engng Mech. Div., Proc. Am. Soc. Civ. Engrs, 86 (EM3), 95 (1960). 9. L. P. SAENZ, Discussion on Equation for the stress-strain curve of concrete, J. Am. Concr. Inst., 61 (9), 1229 (1964). 10. PRAKASn DESAYI and S. KR~SIaNAN,Equation for the stress-strain curve of concrete, J. Am. Concr. Inst., 61 (3), 345 (1964). 11. PRAKASHDESAYI and C. S. VISWANATnA,Ultimate flexural strength of reinforced concrete sections, Proc. of Syrup. on Ultimate Load Design of Concrete Structures, PSG College of Technology, Coimbatore, India, Session II, 62 (1967). 12. N. GOPAL, Strength and behaviour in flexure of doubly reinforced concrete beams, M.E. Dissertation, Civ. and Hyd. Engg. Dept, Indian Institute of Science, Bangalore, India (1966). 13. F:. E. RICHARTand R. L. BROWN,An investigation of reinforced concrete columns, Bulletin No. 267, Univ. of Ill. Engg. Expt. Station (1934). 14. D. RAMLEYand D. MCHENRY, Stress-Strain curves for concrete strained beyond ultimate load, Lab. Report No. SP-I 2, U.S. Bureau of Reclamation, Denver (1947). 15. E. HOGNESTAD,N. W. HANSONand D. McHENRV, Concrete stress distribution in ultimate strength design, J. Am. Concr. Inst., 52 (4), 455 (1955). 16. G. M. SMITHand L. E. YOUNC, Ultimate flexural analysis based on stress-strain curves of cylinders, J. Am. Concr. Inst., 53 (4), 597 (1956). 17. A. SETViURATHNAM,Analysis of eccentrically loaded short reinforced concrete columns, M.E. Dissertation, Civ. and Hyd. Engg. Dept, Indian Institute of Science, Bangalore, India (1967).
S8
P r a k a s h D e s a v i a n d A. S e t h u r a t h n a m NOMENCLATUR E
,4~ ,4"~ h
(',. (, d
d" d E e e' f
f',. .1~ f~ .["~ ~.
k2c M
area of tension reinforcement area of compression reinforcement width of c o l u m n section distance of neutral axis from extreme compression fibre compressive force in concrete force in c o m p r e s s i o n steel effective depth of c o l u m n section distance from extreme compression fibre to centroid o f A ' ~ ( = rd) distance of C,. from A., Y o u n g ' s m o d u l u s of steel eccentricity of load measured from plastic centroid eccentricity of load measured from centre o f gravity of tension steel stress in concrete at a n y strain ~: cylinder strength of concrete m a x i m u m stress in stress block ( = 0.85 .[',.) corresp o n d i n g to a strain eo stress in tension steel stress in compression steel yield strength o f reinforcement distance of C,. from extreme compression fibre resisting m o m e n t
~1,. n P P,,
ultimate m o m e n t capacit} ratio c'd load on the c o l u m n ultimate strength of c o l u m n under axial loading
p
p"
4. hd 4 "~,,hd
q
tension reinforcement index = - - - -
q"
compression reinforcement index = ~ - -
r T T,
ratio d'/d tensile force in steel As compressive force in steel As 1 a factor defined as --: log ( I + 2 2) ,t factors given by equations (14b), (14c) respectively strain in concrete at any stress f strain in extreme concrete fiber in compression strain in concrete at stress [', or J~, o f the stress block strain in steel A~ strain in steel A'~ yield point strain o f steel ultimate concrete strain ratio e,.,/eo ratio ~deo ratio e,,l~,,
:~ eL d ~:
c,. ~:o ~:~ ~:', ~, e,, ,:. ). ).,,
A, J;, hd Jo
A', L.
hd l~
On pr6sente ici une analyse de poutres en b6ton renforc6, courtes et charg6es excentriquement. On utilise pour cela un critSre variable d'effort. Cette m~thode 6vite la proc6dure habituelle qui consiste b. attribuer une valeur fixe fi l'effort ultime du b6ton. L'analyse est bas6e sur une 6quation du premier degr8 pour la courbe totale tensioncompression du b6ton et pour l'adoption d'un proc6d6 de maximisation du moment par rapport fi l'extrfime contrainte de compression du b6ton. On examine ici des poutres verticales de section rectangulaire et chargfes excentriquement sur un seul axe. Une bonne concordance apparait entre les valeurs th6oriques et exp&imentales de certains r~sultats de tests. Es wird eine Analyse kurzer, exzentrisch belasteter StahlbetontrSger dargestellt, welche ein Kriterium ve#inderlicher Bruchspannungen benutzt. Die Methode verzichtet auf den gewtihnlichen Weg, einen festen Wert fiir die ~iusserste Spannung in Beton anzunehmen. Die Analyse ist aufdem Gebrauch einer einfachen, einzelnen Gleichung fiir die komplette Spannungs-Dehnungskurve fiir Beton begrfindet und auf der Annahme eines Verfahrens der H/Schststeigerung des Moments in Bezug auf~iusserste Druckspannung der Betonfaser. Nur Tr/iger mit rechteckigem Querschnitt und mit exzentrischer Belastung entlang einer Achse werden in diesem Bericht behandelt. Die theoretischen und experimentellen Werte stimmen in einigen Testresultaten gut iiberein.