Title Stability of Slender Reinforced Concrete Columns ... · A total of 36 column specimens with...

Title Stability of Slender Reinforced Concrete Columns Subjected toBiaxially Eccentric Loads

Author(s) IWAI, Satoshi; MINAMI, Koichi; WAKABAYASHI, Minoru

Citation Bulletin of the Disaster Prevention Research Institute (1986),36(3-4): 137-157

Issue Date 1986-12

URL http://hdl.handle.net/2433/124940

Right

Type Departmental Bulletin Paper

Textversion publisher

Kyoto University

Bull. Disas. Prey. Res. Inst., Kyoto Univ., Vol. 36, Parts 3, 4, No. 321, December, 1986 137

Stability of Slender Reinforced Concrete Columns

Subjected to Biaxially Eccentric Loads

By Satoshi IWAI, Koichi MINAMI and Minoru WAKABAYASHI

(Manuscript received October 11, 1986)

Abstract

The elastic-plastic behavior of pin-ended reinforced concrete slender columns subjected to bi-axially eccentric loads is investigated experimentally and theoretically. The ultimate loads, longi-tudinal and transverse deformations and the behavior up to failure of the columns are examined in detail.

A total of 36 column specimens with rectangular cross sections including square sections were tested. The ratios of column length to minimum depth ranged from 6 to 26. Loads were applied monotonically at each column end with equal eccentricities at various angles from an axis of sym-metry. The ultimate load carrying capacity of a slender column is reduced by additional eccentri-city due to lateral deflection even in a column having a length to depth ratio of 15. The numeri-cal analysis to solve the load-deformation response of the column predicts the test behavior very

well. In the case of square columns, there is not much difference in the ultimate loads of the col-umns in spite of variation of the angle of eccentricity of the applied load. However, the remark-able effects of biaxial bending on the elastic-plastic behavior occurs with the rectangular columns. When the rectangular columns are subjected to biaxially eccentric load to cause bending about the near-strong axis, the ultimate loads and the deformation behavior are significantly affected by the bending about the weak axis of the section.

1. Introduction

Columns in building are designed conventionally on the basis of a structural anal-

ysis of frames in the planes in which the principal axes of columns are constructed. However, since almost all columns are subjected to biaxial bending, the effects of bi-

axial bending on the behavior of columns should be investigated. A number of

experiments and analyses have been done to investigate the ultimate strength of short

reinforced concrete columns under biaxially eccentric loading, and many strength

formulations have been proposed for their design.

Investigations on instability problems of relatively slender concrete columns have

been extensively carried out in the area of no fear of strong earthquakes, while the

shear resistance of short columns has been more investigated in Japan. In AIJ Stand-

ard1), slender reinforced concrete columns are regulated by the following limitations

of Art. 15.7; The ratio of the minimum depth or diameter of a member to the distance between

the main supporting points shall be not less than 1/15 and 1/10 in case of using ordinary concrete and light weight concrete, respectively. These limitations, how-

ever, are not applied when the structural safety of the column is certified by struc-

138 S. IW AI, K. MINAMI and M. WAKABAY ASHI

tural calculation considering the effective slenderness ratio.

In recent years, as the material strengths of concrete and steel increase more and

more, the cross sections are able to be designed to compact and the members be-

come to be slender. Moreover the reinforced concrete braces with slender propor-

tions are sometimes used in a frame to resist earthquake forces. Therefore, clarifica-

tion of the elastic-plastic behavior of slender reinforced concrete columns is very im-

portant. The column behavior under biaxial loading becomes further complex due to the

slenderness effects. There is very little available experimental information of the sta-

bility of slender columns. In this paper, fundamental data from biaxially eccentric

loading tests of slender columns with square and rectagular cross sections2).3) are re-

ported. The objective of this study is to investigate the slenderness effects under biaxial bending on the ultimate strength and the deformation behavior up to the fail-

ure of reinforced concrete columns.

2. Outline of Experimental Work

01 1(.5.

.

L- 600 -41 ri--:74611 680

(a) C series (LID = 5.7) + Load Pokft

1201-4:1-40-12o1 .1 120 " 4-71 2 0

,1(d) Square cross section 1800 4 1880 (A, B, C series)

(b) B series (L/D . 15.7)

01 1, 7120

3000 1 3080

(c) A series (LID . 25.7) 22.."

1S•• Strain 1111

g

.ioniammiamninlIMMIL vorcaumwou1111"

-

.1!m" ipf

(e) RS series (i/b = 10) (g) Rectangular cross section

(RS, RL series)

,1 I Strain Gages Strain Gages

IiiIIIRIBMIIIIIIIIIIMMI11•111•1111 II 1 1111111•••111••••••••••111111••11•1111, _

8-010454060 40141301 110011241241201201 1100 1120140 3000

(f) RL series (lib = 25) Fig. 1 Geometry of test specimen (units in mm).

Stability of Slender Reinforced Concrete Columns Subjected to Biaxially Eccentric Loads 139

Table 1 Mix proportion of concrete and material properties (unit : 1 Vein' =0.098 kN/mm2).

Test Series C B A

Mixing Ratios by Weight Water 0. 67 0. 65 0. 66 0. 62

Cement 1 1 1 1 Sand 1.97 1.91 1.91 1.90 Gravel 2. 74 2. 76 2. 76 2. 75

Specific Gravities

Cement 3. 16 3. 16 3. 16 3. 16 Sand 2. 61 2. 62 2. 45 2. 48 Gravel 2. 58 2.60 2. 50 2.54

Concrete Compressive Strength Fc (kg/cm2) 275 281 311 316

Strain at Compressive StrengthEP(19-3) 2.59 2.30 2.41 2. 43

Splitting Tensile Strength (kg/cm2) - 30. 1 26. 2 28. 9

Reinforcement Upper Yield Strength (t/cm2) 3. 72 3. 72 3.63 3.77

Lower Yield Strength (t/cm2) 3.54 3. 62 3. 49 3. 68 Ultimate Strength (t/cm2) 5. 27 5. 17 5. 50 5. 18 Elongation (10-2) 18. 2 16. 9 22. 6 27. 2

A total of 36 column specimens were tested. Their column length to minimum

section depth rations, 1/D or i/b, ranged from 6 to 26. The geometry of test speci-mens is shown in Fig. 1. The columns had a 12 cm (b) x 12 cm (D) square cross sec-

tion or a 12 cm (b) X 18 cm (D) rectangular cross section with 8-D10 longitudinal bars

(gross reinforcement ratios of 0.0396 and 0.0264, respectively) and 4.5 mm hoops spac-ed 6 cm apart. The mix proportion of concrete and the properties of materials are

listed in Table L The concrete was cast in the horizontal position using a machin-

ed metal-form. The compressive strength of the concrete was 270-340 kgf/cm2 (26-

33 N/mm2) and the tensile yield strength of main reinforcement was 3,500-3,700 kgf/

cm2 (340-360 N/mm2). Age of specimens at test was 40-210 days.

Photo 1 and Fig. 2(a) show a specimen placed in the loading apparatus. Both

ends of the column were supported by universal bearings to simulate pin-ended con-

ditions. The bearings were greased to minimize friction. Load was applied mono-

tonically at each column end with equal eccentricities, e, at various angles, 0, from an axis of symmetry, as given in Fig. 1. Transverse deflections, u and v in the prin-

cipal axes of the section, denoted by x and y, respectively, and torsional rotation, a,

at the mid-height of the column (Fig. 2(b)), rotations of both column ends, 0, and Ow (where added subscripts, u and 1, represented upper end and lower end, respec-

tively), about the x and y axes of the section, respectively, and axial deformation, w,

140 S. IW AI, K. MINAMI and M. W AKABAY ASHI

, . ,

, ,.

,,.

1

10?.ia:v::.,:3,1,,,i:::.;:,... k,ft,..' ....;1',?!.,..,.::;,::"P....e.,;.Ve..:7 01.4Y.V...,lifk,',,,.t.-.k.*:,-.`,....,h,.:1,,,..,,,,,.

E : ..:.'00..,:-!),5:::-Ve;:::fiv,p,L'11.''::::;:,.F.,,..,...,,,..-AileRi.i':,;P; ..!'./..:.:e5V4ftt'::-.,:-.?:,:it' ..fi:....-!:r:A.:::::::,....:.:E:,,,:...,•?.:v...:.:::.:,:::::,:::T.:t6':,,,,:::'..,,:u'f.1-;;;f40.11 ,, ,

7 ,w4t,,. , ... ;., .• . ;-• ,

.. ,.....: - -' • ,,,, . „ , , .. ,:

- 6 I.:- '.. - # k — ,,. - .. . 4rif ,'

-

i_ -.,,.. i, ,..„.,:....,z. ....., .... „........._. ,...

,.4..., .. „

. .,

. .

' '.1. ,,• i

-, - -.. , .

.,...:.. ..,...,..,, lig ̂.: , , -• '

' r ., .: '• ' — ..,,,, , -, .

' ,.-''',1..4,....,,...'^.r..iI,-tii,

t.., ik 021: 1 ' 1

i, - ''''' ' - . i - '0'.ti

1.1 1 . rij 1 '1-; , .! , !...,7-.4,

r rm .rk r - : — - , .::111; \'1—\.,,,, , --,- :

..!...:1121.1.!IFIk.,:.:, ,,,. ,. , 1,', - ',.. • :..

' ' ,.. .., , !;.• .i. , , 1 '

- .

, - .„

; .., ,..:-'1. ' , .- • ..:.,*t-

,

.,„

....--,M1P111140111.141°'

Photo. 1 Overall view of test of A000b.


Loading Beam

x

OilJack e -----

.

-1

i 1 ---

Y

i^ 1 M1. p yiOn=r7ill . Load Cell1u P.I L ^

. ; . • .. . 7. .", ; . . . , = 0 i ; - -

1 1 1 il n

Test Specimen

V la (b)

IIIII

82i..10- z 2 -,

0 eXU 0 ISY:

I

iti ill.ID.--- eyiP "1°1'

W

U e- -t1r-----vr- NI

..--•--x

3' mi...-... 9,1C I "Pe exi -Load Cell

lP it.. Test Bed (c)

(a) Fig. 2 Specimen in loading apparatus (units in mm).

(Fig. 2(c)) were measured by potentio-meters. Those potentio-meters were support-ed on a rigid frame running parallel to the line of action of the applied load. Typi-

cal arrangements of the instrumentation devices are shown in Fig. 3. Load was meas-

ured by load cells installed at the ends of the columns. The strains on concrete sur-

face and in reinforcing bars were measured by potentio-meters and strain gages at

some selected locations.

142 S. IWAI, K. MINAMI and M. WAKABAY ASHI

.;;_.

1^Ir. it Thrust Bearing

IOW _'.1w:A II'I':, Thrust Bearing -.,,,0

I

Load

7lIk.i ...”.....“.. 4)- 4Pin Bolt ATI:i,lcet....',.1:--1A s..‘Ill

.L_..j-,,.=ailiszi=

,

ffiialm^

A , Thrust Bearing 1r (a) A-A Section - S

train Gage . r-i

IITIWI.M111111 Specimen Speclimen111 I raL;s1-ir

I

Br__-.---'....mr:- (

IVII 1.1.il -.iNip041-1

2,ii..12V.=IliZ e

hi \ '7. .. =Zriii 2

i...---- DisplocementmeterIimri

'lumic'

-11.11

IIMMEINIIIMEI Dis• at ementrneter ,,

(b) B-B Section

! i .

F— numeolimen.— 1

i r umil. a -1 i , row .. 1

c

1 .

i • ._ g 1 ERIMiur.r^wEll

1 1 1,--,;,L.,-:',,,1 "de•Unit I 11117Slide Unit Wil A =57 mrn

!" lei Tcbi Track Roil lir Universül Hinge

.Track Rail . r .: (d) Elevation

• , .] 1 • -..,...irir g .

1.rill

gi11 L4, L1111111614110/6 -', _.1 -iThrust Bearing

(c) C-C Section

Fig. 3 Measuring device of long column.

3. Method of Theoretical Analysis

The load-deformation response of the column was solved by the numerical meth-

od of a second-order analysis') taking into account the effects of deflections and

changes in stiffness of the columns on moments and forces. The following basic as-

sumptions have been made:

(1) lateral deflections of the column are small compared to its length and the cur-

vatures are represented by the second derivatives of the deflections;


Oc

E $5cep(2 -ET) IP

Ep Eu E ey ex

(a) Concrete • ,

• , • , •

ay ---

V

A tan1Es -Ey

y4111, exo Ey eyo

(b) Reinforcing Bar Fig. 4 Assumption of stress- Fig. 5 Notation relating to

strain relationships. column analysis.

(2) plane sections remain plane after deformation;

(3) effects of shear on lateral deflection and rotation of the cross section due to torsional deformation are negligible;

(4) the assumed stress-strain relationships for the concrete and the reinforcing bars are shown in Fig. 4(a) and 4(b), respectively. Strain reversal is not considered.

The concrete is unable to sustain any tensile stress.

The effect of the strain reversal on the analytical results is discussed later.

A symmetrically reinforced rectangular cross section is considered and an x-y

axis system is taken with the origin at the centroid of the section (Fig. 1). A force

P is assumed to act at the point which is defined in the x-y axis by the eccentrici-

ties e, and ey, as shown in Fig. 5. If the assumptions of perfect bonding and plane

distribution of strains are made, the strain, E, in the section can be defined by the

three quantities E0, 0, and Ov.

5=Ea+s'Sbv+Y'Sbx (1)

in which 50=strain at the centroid and tbr, Ov=curvatures produced by bending mo-ment components, Mr and My, about the x and y axes, and are positive when they

cause compression in the positive y and x directions, respectively. Knowing the strain distribution, the axial force, N, and the bending moments, Mx and My, are

calculated from the assumed stress-strain relationships. In the present study, the con-

crete and the reinforcing bars are partitioned into many small elemental areas. The

total force and the moments in the section are then evaluated by summation of the

144 S. IW AI, K. MIN AMI and M. W AKABAY ASHI

elemental forces acting on the elemental areas and the moments of the elemental forces, respectively.

A pin ended reinforced concrete column shown in Fig. 5 is considered for the case of monotonic loads P applied with biaxial eccentricities at each end. The later-al deflections, u and v, of the column are assumed to be small, so that the curvatures

0x. and 0„ can be expressed in the form of second derivatives of deflections

d'v °I.-- de (2a)

d'u de (2b —)

For the given load P=N, the deflections u and v also have to satisfy conditions of external equilibrium, namely

Mx(0„,95y)-=-P(v+ey) (3a)

My (0,z, Chy) = P(u +ez) (3b)

The equilibrium shape can be determined by the solution of simultaneous equations for the various unknowns P, u, v, Mx, My, 0, and 0,, using equations (2) and (3). Using the first derivatives of deflections about the x and y axes, 0, and By, the cur-vatures are given by

dOx (4a) dz

Oy =dOy (4b) dz

Therefore the second-order simultaneous differential equations (2) and (3) can be re-

placed by the first-order simultaneous equations expressed as

du =Oy (5a) dz

dv —0 dz, (5b)

( dO, d0y dz 'dz) =P(v+ey) (5c) )

My dOx 'ddOz) =P(u+e x) (5d) dz

The column length 1 is divided into several segments of the same length. For

equations (5), application of the Runge-Kutta-Gill numerical integration method and

the Newton-Raphson iteration technique yields the deflections u and v corresponding

to the load, P. In this numerical analysis, the column length was divided into 10

segments, and the square and the rectangular cross sections were divided into 10 x

10 and 10 x 15 elements, respectively. As the condition of convergence of calculation,

deviation of deflection at the end of a column was limited to less than 1/10,000 of

the column length. Convergence for the given load level was achieved within two or


three applications of the Newton-Raphson technique.

4. Test Results Compared with Analytical Results

The test results are summarized in Table 2.

Figure 6 shows the experimental strain distributions across the section for sever-

al load levels for square short columns C 000, C 020 and C 210. The concrete strains

and the reinforcing bar strains may be seen to conform closely to the plane strain

distribution. It is only near maximum load level that there occurs some deviation

from the plane strain distribution. The strain in concrete just before crushing was

0.004-0.005 for the columns subjected to biaxially eccentric loads. Final collapse was

( \

5000 , . E x10-6 - i 1 Pmax

,/ _-----' 4000 -

„? 3000 - //

2000 Appli P:501 ..."'‘..

140INP.301P30 ,4101 1000 OpV--,,... P.401 )4"411s•'- 1.1.4"-t glopoii,4.,,,,P=20tAlp=20t

041410111°-.40111 PP:10°tt allitlx.$17."71.1 ="3t '111101gVir1111441R�. 0 t

0

5000(a) c000 : 1 E X10-6

. 1 „ 4000 -, . . , . 14 1,

1 1 1 1,

3000 -.J P.301 . , \ P=301 , .

2000 - / • ^ , . . , . 4.,..P=20t 1000 /,/,A/ P=201 ,.irP.10t„.....iv,,,,,,,.

,..., o_,:.P=Ot -‘4111111F=Ot

(b) CO20 5000 - /'

E X10-6

FPrnax ! 4000 - I / =40 t / I P=401

3000 i /

2000 i P.30 t 1 1Fig. 6 Strain distributions across the P.m tsection at increasing load

1000-:. P.204 1,,.-; levels; concrete strain (left) A1.SPIIII7,;",,,, ,,P=20t ,,,,A,42.1440=1010ItliiiR

P= OtP=I0t and steel strain (right). AMPu o -- riimpitfrP. 0 tAls100000100,. A'(a) Concentric loading. i (b) Uniaxial loading.

(c) C210 (c) Biaxial loading.

146 S. .1W AI, K. MINAMI and M. WAKABAY ASHI

Table 2 Summary of test results (unit: 1 t =9. 8 kN).

1 Specimen0e P=

Pmax U V W a 0.z.0 Oyu ex/ Oyl Demax GE max cm name deg cm t mm mm mm deg deg deg deg deg x 10-3 x 10-3

Square columns

C 000 0 0 57.1 4.04 5.64

C205 22.5 0.6 51.4-- 4.16 5.23

C 210 22. 5 1. 2 43. 3_--- 4. 23 5. 15

68. 0 C 220 22. 5 2. 4 30. 0 4. 75 4. 71

C 250 22.5 6.0 18. 0 5.41 -

C 020 0 2.4 33.4 4.64 6.98

C 420 45 2.4 33.3 5.01 6.42

B 000 0 0 47. 9 5. 09-0. 18 3.40 0.01 0. 00 0.50 0.03 0. 51 2.56 2. 73

B 205 22. 5 0.6 46. 2 2.46 4. 45 3.03 -0. 02 0. 43 0. 43 0. 51 0. 39 2.71 2. 88

B 210a 22. 5 1. 2 35. 8 7. 73 6. 45 2.41 - 0. 02 0. 64 0. 83 0. 67 0. 84 3. 13 3. 13

188.0 B 210b 22. 5 1. 2 34. 4 9. 76 6. 76 2.65 -O. 01 0. 73 0. 99 0.73 1. 00 3.97 4. 80

B 220 22. 5 2. 4 25. 4 14. 09 9. 19 1.80 -0. 01 0. 92 1. 46 0. 97 1. 49 4. 62 3. 84

B 250 22. 5 6. 0 13. 4 20. 45 11. 13 0.56 - 0. 02 1. 21 2. 32 1. 18 2. 28 4. 34 4. 36

B 020 0 2. 4 27. 7 14. 09-0. 82 1. 66 -0. 03 - 0.09 1. 52 - 0.09 1. 49 2. 96 3. 39

B 420 45 2.4 23.2 11.78 14.64 1.72 -0. 02 1.55 1. 20 1.52 1. 18 5.08 3.83

A 000a 0 0 55. 4 5. 2 4. 4 5.87 -0. 04 0. 06 0. 08 0. 21 0. 16 3. 32 2. 89

A 000b 0 0 54.9 5.8 5. 4 - - - - - - - -

A 205 22. 5 0. 6 32. 6 12. 1 14. 6 3.42 - 0. 07 0. 93 0. 74 0. 89 0. 85 2. 59 2.98

308. 0 A210 22. 5 1. 2 28. 0 13. 5 10. 9 2. 50 -0. 02 0. 65 0. 78 0. 71 0. 90 2. 13 2. 41

A 220 22. 5 2. 4 18. 9 25. 7 17. 4 2. 19 0. 07 1. 06 1. 58 1. 05 1. 59 2. 87 2. 57

A250 22. 5 6. 0 10. 9 42. 3 22. 5 1. 65 -0. 07 1. 45 2. 73 1. 48 2. 79 3. 23 3. 39

A020 0 2.4 18.8 33.7 -1.2 2.16 - 0.07 - 0. 12 2.00 -0. 04 2.06 2.24 2.41

A420 45 2. 4 17.2 20. 6 25. 4 2.08 - 0.04 1. 52 1.26 1.54 1. 24 2.75 2.73

Rectangular columns

RS300 0 3. 0 61. 4 3. 77 0. 49 1. 76 0. 01 0. 07 0. 60 0. 13 0. 65 3. 54 4. 38

RS322 22. 5 3. 0 59. 1 3. 23 4. 28 1. 67 -0. 02 0. 74 0. 59 0. 81 0. 62 4. 14 4. 48

120.0 RS345 45. 0 3. 0 50. 0 2. 11 6.00 0.97 - 0.04 1. 26 0. 36 1. 08 0.40 4. 12 4. 42

RS367 67. 5 3. 0 43.4 0. 58 8. 36 1.45 - 0.01 1. 54 0. 17 1. 46 0.09 4. 22 5. 25

RS390 90. 0 3.0 42. 8 - 0. 18 8.78 1.56 -0. 02 1. 43 - 0. 13 1. 54- O. 08 4. 44 4. 18

RL300 0 3. 0 48. 4 20. 57 1. 50 3.88 0. 02 - 0. 02 1. 30 0. 18 1. 36 2. 70 3. 13

RL322 22. 5 3. 0 35. 5 10. 29 19. 57 2.64 - 0.06 1. 31 0. 68 1. 35 0. 75 2. 54 3. 15

RL345 45. 0 3. 0 28. 4 6. 08 27. 49 2. 14 - 0.05 1. 64 0.39 1. 74 0. 45 2. 32 2. 94

300. 0 RL367 67. 5 3. 0 24. 1 2. 46 29. 32 1. 66 0. 02 1. 78 0. 21 1. 83 0. 22 1. 97 2. 27

RL390 90. 0 3. 0 23. 5 0. 40 39. 42 1. 83 0.01 2. 37 - 0.03 2. 47 0. 02 2. 19 3. 12

RL622 22. 5 6. 0 18. 7 15. 16 26. 94 1. 49 0. 03 1. 65 1. 00 1. 71 1. 04 2. 79 2. 97

RL645 45. 0 6. 0 14. 5 8. 73 36. 78 1. 23 -0. 07 2. 43 0. 64 2. 48 0. 68 2. 33 2. 86

RL667 67. 5 6. 0 12. 5 2. 92 43. 59 0. 95 -0. 07 3. 04 0. 22 3. 10 0. 33 2. 34 2. 93


N \ A

I P(t)

,

V Crushing of Concrete C C 0 B A P(t) 1

60C0 B A), ___

50 .,.1 -‘"Iris--Y\ ..„,,

_.,

,\2___

+

40r31

, , V%)

30 - Analysis -0.1 0 0.1 0.2 0.3 aa 05 ^ - -- Experiment 4) C220

'

20i (Clisplacementmeters) P(t)

10 C 0 20- e • • A ______ E(144____

-0.1 0 0.1 0.2 0.3 04 0.5 I) 0000C(%)

P(1) -d.3 -02 -d.1 00.I1(1!20.31 0.14d5 C D .B A 5) C250 50 --- _,

------11..----s'.'..."-• 40/1 P(t)

,,,--------- 41„- C40 -30 --1 B 0 A .. ....30...-;)---------------------

20, ..,

20 1---1

10 EN.) '01/

1', -0.1 0 0.102030.405'E(%)

2) 0205 -0.1 0 01 0.2 0.3 DA 0.5 P(t) 6) CO20

-C D B A P(t) 40 r,,.......'....'40 ' '..............,, C 0 B A

----TO--' - -_ ' 1

20 2th .

'

, ,'

$0 " io I , E CO IE (%)

-01 0 0.1 0.2 0.3 04 0.5 -01 0 0.1 02 0.3 0.4 0.5 3) 0210 7) 0420

Fig. 7 Load versus longitudinal strain relationships (C series) (unit: 1 t=9. 8 kN).

accompanied by spalling of concrete in the compression zone. This immediately re-

sulted in the buckling of longitudinal reinforcement between ties at the compression

corner. All columns except one case, specimen C 220, deflected symmetrically with

respect to the column mid-height and failed at the mid-height section. In Fig. 7,

the test results of load-strain relationships obtained from the longitudinal strain data

at the selected positions (A, B, C and D) of the mid-height section are compared

with the analytical results. The broken lines and the solid lines denote the test data and the analytical data, respectively. By this analytical method, the test behavior of

load and strain of the short columns is seen to be well predicted.

Figures 8 and 9 show the load-deflection curves and the load-strain curves, re-

spectively, for the square long columns of l/D = 25.7. The solid lines and the broken

lines in Fig. 8 indicate deflections at mid-height, u and v, respectively. The thin

lines correspond to the test results and the thick line to the analytical results. In the

test, crushing of concrete occured at the descending branch after the peak load.

148 S. IW AI, K. MINAMI and M. W AKABAY ASHI

U V 40 P(t) Experiment-- - - - 40fP(I)Experiment -- --

Analysis--- C . D B AAnalysis -

(II?, t '''''. V First Crack in Tension Side li Crushing of Concrete

20 -,,,...,,--"---201! ss„. B(10-3) 0 2 4 6-4 -202 4

u(cm),v(cm)(a) A205 (a) A205

P(t) P(t) CD

iiB,..„

.A

''

20 /1 .. 20 , ...

E(10-3)

. . .

0 2

(b) A210 6 -4 -20 2 4 u(cm),v(cm) (b) A 210 20 P(t) 20 PO) - - = B -- ---- - -. A

. - - -

E00'3) ...---

O • 2 6 -4 -2 2 4 u(cm). v(cm)(c)A220

20- P(t)(c) A220 20- P(t)

.._ -- --k . _ _0

: 8_ __ A _ .4r c(10-3) O • i 4 i -4 -2 2 4. u(cm). v(cm) (d)A250

(d) A250 P(t) P(t)

20.4C _20 D BA ---.., ... IT. x'------- __,.._ Ec1031 6'i'4' i' - -4 -2 0 2 4

u(cm), v(cm) (e) A020 (e) A020

20. P(t)cD P(t) B A

-4-----.-------"' ____,_-_-,, EN-3) O 26 -4 -20 2 4 u(cm), v(cm) (f) A420 (f) A420

Fig. 8 Load-deflection relationships Fig. 9 Load versus longitudinal (A series) strain relationships

(unit: 1 t =9. 8 kN). (A series) (unit: 1 t =9. 8 kN).

The load decreased suddenly just after the crushing of concrete and the column was

failed. The numerical analysis adopted herein need not take the strain reversal into

consideration. The reason for this can be seen in the following. The experimental

strains at the compression and tension zone increase monotonically with increasing load in each zone and strain reversal occurs in only a range of small strain near the

neutral axis of the section, as shown in Figs. 7 and 9. Therefore it is regarded that

the ultimate strength and the deformation behavior of the columns are scarcely affect-

ed by the strain reversal.

Figure 10 shows the change of mid-height deflections with increasing load. In

the case of square columns under biaxial loading at the angle of 0 =22.5° (Fig. 10

(a)), the direction of deflection corresponds closely to the plane including the load-


'4 6v(cm).•„, j" 1 / ,f/ /.. , 6 v (cm) 0 0 e• • • e, /2. / // eet'•:r.'ri?r/f - --- Experiment ,-. ./..0,•,,,-- -- Experiment• •4),,,/4, ••c o,_..,, — Analysis ,'„.•/,,• ,,"—Analysis,',// ,,:• •///

,e'r4 , •",' • •• / / • //

A210/•,• ./A420/•,' .------11-5 ,•, / A205/,/ ,/..././-e,<7.15.e)"-,----

,

..------

2 ,' ., ....-- 2 , • ,,' ..----- • ,o' ----- / ,.' • ,------

• • , I A250 Maximum Load...•,--I' " ° Maximum Load

,eA220g/,,'::---- A220 A020 u (cm) , • u (cm) 0 2 4 6 0 2 ' 4 -6

(a-1) (a-2)

, , , — Analysis v (cm) I 1 i --- Experiment/ v (cm) i / I

t,r r,1 i

rii i I , 1 1 I I I, ,t, 0 Maximum Load r ,i

6 1 1 I6 I i i r1 I i r 1 1

1

I 1/ RL 645

I 1 1 I II / R L 367I

RL667 ! I

/

RL 390 111 / Ng!I . 4\ i '107 RL 345/

/

i 4 . 1.71 RL622 i /„

1„.„/1 2 ri iRL322//II.1,7—Analysis

2

-- — Experiment

, 1

I

/ / 1 / 0 Maximum Load RL300 rt i

__L.' u (cm) u (cm) 0 2 4 0 2 4

(b-1) (b-2)

Fig. 10 Change of mid-height deflections with increasing load.

(a-1, a-2) A series. (b-1, b-2) RL series.

ing points and the centroids of the end section, before reaching the maximum load.

However, after the attainment of the maximum load, the column has a tendency to

deflect away from that plane. In the case of rectangular columns (Fig. 10(b)), the

lateral deflections significantly depend on bending about the weak axis of the section.

The numerical analysis to solve the load-deformation response of the column is seen

to predict the test behavior very well. The torsional deformation a in biaxially ec-

centric loads was very small, as shown in Table 2, and the effect is negligible. The

maximum torsional angle was measured to be only 0.6°. From these test results, it is

confirmed that the assumption of no torsional deformation is useful to the analysis.

150 S. IW AI, K. MINAMI and M. WAKABAY ASHI

5. Ultimate Loads of Biaxially Loaded Short Columns

The ratios of the maximum load by the test to that by the numerical analysis,

Ptest/P.1, for every square short columns are listed in Table 3 and displayed by bars

in Fig. 11. Analytical results agree well with the test results. In the same figure,

predicted values obtained by Bresler's method') and Ramamurthy's method} are rep-

resented. Bresler and Ramamurthy proposed expressions for the shape of the inter-

action surface for a reinforced concrete column section with biaxial bending from

P test

1 2Pcal.Analysis • • Bresler Ramornurthy 10

^ffig,!!7afa

0 8 0c 0.85Fc Fc Fc Fc Eu 0.003 0.003 0.004 0005

C000 C205 C210 C220 C250 CO20 C420

Fig. 11 Comparison of ultimate loads of short columns (C series). *: failed at column end .

Table 3 Summary of analytical results of square short columns (unit: 1 t =9. 8 kN).

Peal(t) (Pte.,./Peat)

Specimen Post =0. 85F, 6c =Fe

Name ( t) Analysis Eu =0. 003 eu= 0. 003 004 Eu =0. 005

I* I II

C 000 57. 1 56. 5 53. 9 53. 9 59. 8 59. 8 59. 8 59. 8 59. 8 59. 8 (1.011) (1.059) (1.059) (0.955) (0.955) (0.955) (0.955) (0.955) (0.955)

C 205 51. 4 49.6 45. 7 47. 2 50. 8 52. 6 51. 3 52. 9 51. 4 53. 0 (1.036) (1. 125) (1.089) (1.012) (0.977) (1. 002) (0.972) (1. 000) (0. 970)

C 210 43. 3 43. 3 39.6 41.6 44. 1 46.3 44. 8 46, 7 44. 8 46.7 (1.000) (1. 093) (1. 041) (0. 982) (0. 935) (0. 967) (0.927) (0. 967) (0. 927)

C 220 30. 0 33. 3 30. 0 31. 9 33. 4 35. 4 34. 2 36. 0 34. 5 36. 6 (0.901) (1.000) (0.940) (0.898) (0.847) (0.877) (0.833) (0.870) (0.820)

C 250 18. 0 17. 2 15. 7 17. 2 17. 3 18. 9 17. 8 18. 9 18.0 18. 9 (1.047) (1.146) (1.047) (1.040) (0.952) (1.011) (0.952) (1.000) (0.952)

CO20 33.4 33.9 32.7 32. 7 36. 3 36. 3 36. 9 36. 9 37.4 37. 4 (0.985) (1.021) (1.021) (0.920) (0.920) (0.905) (0.905) (0.893) (0.893)

C 420 33. 3 33.0 29. 3 31. 0 32. 6 34. 3 33. 5 35. 0 33.5 35.6 (1.009) (1.137) (1.074) (1.021) (0.971) (0.994) (0.951) (0.994) (0.935)

* I: Bresler's Method II: Ramamurthy's Method


which, knowing the uniaxial strengths, the biaxial bending strengths may be calcu-lated.

An expression derived by Bresler5 for the strength of a Biaxially loaded column

is

1 1 1 1 P.— P .. PyA (6)

where Pu=ultimate load under biaxial bending, Pz=ultimate load under uniaxial

bending when only eccentricity e, is present, Py =ultimate load under uniaxial bend-

ing when only eccentricity ey is present, and Po=ultimate load when there is no ec-

centricity. Values of F3, and PP are obtained from the interaction functions that rep-

resent the section capacity under uniaxial loading conditions.

Ramamurthy6) proposed two equations to define approximately the shape of Ms-

My interaction curves in square and rectangular columns with eight or more bars,

equally distributed among the faces. The locus of the ultimate radial moment Mu at

any load level in square columns is approximately defined by

= Muyo — 0.1 (0 /45°) 1 (7)

where M.,0 =ultimate uniaxial moment about y axis corresponding to load P under

consideration and (deg.) =tan-1(evie3,) =inclination of the line joining the load

point to the centroid of the section to x axis. Equation (7) yields Mu=0.9 Muyo for B=45°. Mu for any value of 0 is calculated by linear interpolation between bending

about a major principal axis and bending about a diagonal.

Using the rectangular stress block with a constant ultimate stress Cr and a fail-

ure limit strain Eu to represent concrete in the analysis of cross section strength",

interaction curves under uniaxial loading conditions were calculated in four cases;

(1) 6c=0.85 (Fc=compressive strength of cylinder) and Eu=0.3%, (2) ac—F, and Eu=0.3%, (3) 6,=--F, and fu=0.4%, and (4) 6e=1', and eu=0.5%.

The majority of calculated values obtained from Bresler's method were less than

those obtained from Ramamurthy's method. All calculated values for the cases where

was assumed give good predictions of maximum load observed by test com-

pared with the cases where 6,=0.85 Fe. However, the value of extreme fiber strain, Fu, has slight influence on the prediction of load. The maximum load obtained by

the authors' numerical analysis corresponds to the case of ce—F, and Eu=0.5%.

6. Load-Moment Behavior of Square Long Columns

The behavior of the column under increasing load is illustrated by the axial load

versus bending moment diagram for the critical section of the column given in Fig.

12. The lateral coordinate gives the magnitude of the biaxial bending moment, M.

The points for the combination of axial load and nominal end moment at the maxi-

mum strength obtained experimentally are designated by small circles for each col-

umn. The three curves denoted by "1/D=15.7" or "1/D-25.7" represent the corre-

152 S. IW AI, K. MIN AMI and M. WAKABAY ASHI

N(t) 8)/ o N(t) — - G. 0' / ---"- 8=225° —.— 8=225'

60. ,C Q (1.60 1/D.0 ---(3 --.45' 0',//

..z. 000 bib!' B205 1,..//1. 40 0

11.41C; (Ia.&e\c)- •11,-----"ihn."Nal,220 1/0=15 74`8 -imi7Av.- oftla40i /‘.8220. ^ e A. ,..____......,,,B020

Nv.AVO ,,110,ev"°47

? B210 ;

20 10iaji20 V-ralleB250B42:11 *.!••-'''

I 'ice 1/0=15.7 M (t.cm) '...-- M(1 cm)

0 50 100 0 50 100

(a) k/D = 15.7, 0 = 22.5° (b) k/D = 15.7, e/D = 0.2

8) A000abg N(t)N(t) —•— 0. 0° 1/D

=0 0.22 5° — 8.22 5' 6o e60

I/0.0/ ---13.45°

,

, '1., 1/13=25.7 / eee*/

41:1 /1''A205 N.4/11111. 1/D=25.7 \ s\ .--

411111.1"--411kA210-5 if'A220 h a10-°4000%i (1.20\\ 1 A220,

2020•!IIP"I'llibk._t, ,1

mivioMP"17:-^i0w"---------','"` ...,14.......-4? N1/4 • 50 M (t•cm) .'-'"M(t.cm)

0 50 100 0 50 100

(c) k/D = 25.7, 8 = 22.5° (d) LID = 25 .7, e/D = 0.2

Fig. 12 Interaction diagrams for column section illustrating load-moment behavior up to failure (units: 1 t =9. 8 kN, 1 t• cm =0. 098 kN•m).

sponding analytical data. When the column is short, the additional eccentricity due

to lateral deflection at the critical section is negligible and the maximum moment can

be estimated by the end moment at all stages. When the N-M path with increasing

load has reached the interaction line denoted by "1/D=0" (where o-c=Fe and Eu=

2 Er), a material failure of the section occurs. If the column is long, the additional

eccentricity at the critical section increases more rapidly at high load levels, and the

N-M path is curved. The maximum load of the column eventually is reduced by the

amplified bending moment caused by additional eccentricity.

There are two types of long column behavior for the case of //D=15.7. When

the eccentricity at the column ends is relatively large, a material failure of the sec-

tion occurs causing the N-M path to reach the interaction line. However, when the

load is applied with small eccentricity, the column becomes unstable before reaching

the interaction line and even at the critical section the column does not demonstrate

its material strength capacity. This is called an instability failure. For all columns

of 1/D=25.7, the instability failure occurred, except in the one case of e/D=0.5 in

which material failure occurred. In the test results, the N-M path reversed to inside

of the interaction line after reaching the interaction line. This is because the re-


,r, 0 0 N(t)N(t)4NO)

(a) 9=0'60 f eo.'(6) 8 *22 5' (c) 8.45• 6060 -- -..,,. ,_-_,...i.„... ,„ ON .!"..0.-.°' 'Nriiii.,d.O.se‘-'1P.R.' 4.0s...e.,eta/II','`(: 4o 40 1640 40 /..-0*4.);.. ,\

e.., /4'..., • ' • , , et, /4.,..,

.Y40**,elo'°5%

^ I 20*k. i 200. 20

r ,

, . 1 ,

,

•.,,1...'-i M(t cm)s..../11111111111*.1 ' m(t•cm)'..t /hgt cm) 50 100 000050 10000 50 100

Fig. 13 Square long column interaction diagrams (units: 1 t =9. 8 kN, 1 t•cm =O. 098 kN•m).

Table 4 Summary of analytical results of square long columns (unit: 1 t =9. 8 kN).

Specimen trc* Pm.x u v 0, OY E.., name kg/cm2 t mm mm deg deg x 10-3

B000 52.8 2. 12 1.06 0. 105 0.212 2. 10 B 205 42. 1 7. 42 3. 39 0. 350 0. 773 3. 11

B210 34. 9 9. 19 4. 48 0. 462 0. 947 3. 32

B220 280 25.4 14.30 7.64 0.777 1.49 4.37 B250 13. 2 17. 88 9. 24 0. 976 1. 95 4. 30

B020 26. 1 13. 45 0.00 0.000 1. 43 2. 95

B420 24.9 10.93 10.93 1. 15 1. 15 4.27

A000 39. 1 12. 1 5. 0 0. 30 0. 72 2. 05

A 205 30. 5 16. 1 7. 6 0. 47 1. 00 2. 20

A210 24. 2 19. 2 9. 7 0. 60 1. 20 2. 30 A220 310 17. 3 28. 9 14. 7 0. 90 1. 80 2.90

A250 9. 8 38. 7 18. 4 1. 17 2. 52 3. 21

A020 18. 0 38. 1 0. 00 0. 00 2. 35 2. 52 A420 16. 9 22. 0 22. 0 1. 38 1. 38 2. 92

* ce: Assumed concrete strength

duction of load carrying capacity of the critical section occurs by the spalling of con-

crete and the buckling of main reinforcements.

Figure 13 is the family of square column interaction curves giving the combina-

tion of the axial load and the end moment which cause failure of the column. Solid

lines are obtained analytically, indicating the reduction in ultimate load due to slen-

derness for various loading cases. All the test data are plotted by the small circles.

Analytical results of the maximum loads P..x and the deformation at P=Prn,,,, for long

columns are summarized in Table 4. In the case of square long columns, it is re-

markable that there is not much difference of the ultimate loads of the columns in

spite of variation of the angle of eccentricity of the applied load. The analytical re-

sults agree quite well with the test results.

154 S. IW AI, K. MIN AMI and M. W AKABAY ASHI

7. Ultimate Loads of Rectangular Columns

Figure 14 shows how the maximum load P„,,,„ varies with changing the angle 8

of eccentricity for rectangular columns subjected to uniaxially or biaxially eccentric

loads. Analytical results are listed in Table 5. Because of the difference of load car-

rying capacity and bending stiffness about each of two principal axes, i.e. the strong

axis and the weak axis, the maximum load of column decreases with increasing 0.

For all short rectangular columns of 1=120 cm (1/b=10), the material failure occured

at the critical section. When the load is applied so as to cause bending about the strong

axis, the effect of the additional bending moment caused by deflection on the ultimate

o Experiment Pmax (t) • Analysis

60 - II I ..120cm

50 -N. e = 3 cm .

,

.

40 -

aN l - 300cm 30 --., e 3 cm `‘ ..,

20 - "---e-- -a Fig . 14 P...x-0 relationship of rectangular columns (unit: 1 t =9. 8 kN). 10 - I = 300 cm

e = 6 cm

0 - ' ' ' " 9 (deg) 0 22.5 45 67.5 90

Table 5 Summary of analytical results of rectangular columns (unit: 1 t =9.8 kN).

Specimen 6c* Prnax U v Ox By Eniax name kg/cm' t mm mm deg deg x 10-3

RS300 58. 7 5. 35 0 0 0. 92 4. 88

RS322 53. 0 3. 83 5. 12 0. 85 0. 66 5. 52 RS345 340 46. 4 2. 68 6. 58 1. 12 0. 46 5. 12

RS367 42. 7 1. 53 8. 23 1. 38 0. 26 5. 09

RS390 41. 4 0 9. 87 1. 60 0 5. 00

RL300 230 34. 2 18. 77 0 0 1. 26 2. 69 RL322 280 27. 2 10. 98 20. 43 1. 30 0. 74 2. 78

RL345 320 23. 9 7. 95 30. 15 1. 91 0. 53 2. 85 RL367 330 21. 6 3.92 35. 00 2. 23 0. 26 2. 57

RL390 340 21. 1 0 37. 72 2. 41 0 2. 30

RL622 270 17. 1 15. 83 26. 22 1. 71 1. 09 3. 19 RL645 280 13. 8 10. 96 37. 22 2. 44 0. 74 3. 17

RL667 340 13.1 5. 02 38. 41 2. 58 0. 34 2. 42

*6, : Assumed concrete strength


load is negligible. However, when the load is applied to cause bending about the

near-weak axis, there is some amount of reduction of ultimate load due to the addi-

tional moment. For the long columns of 1=-300 cm (l/b =25), the ultimate load was

affected largely by increasing deflection with increasing load and the instability fail-

ure occurred. When the columns are subjected to a biaxially eccentric load causing

bending about the near-weak axis, that is, the load is applied when angle 0 ranges

from about 45° to 90°, the maximum load and the deformation of the columns closely

resemble the behavior under uniaxially eccentric loading conditions about the weak

axis. Therefore, those columns can be approximated by the columns under only uni-

axial bending about the weak axis. However, when the columns are subjected to

biaxially eccentric load to cause bending about the near-strong axis, with 0 ranging

from 0° to 45°, the ultimate strengths seriously depend on the angle of eccentricity of

the applied load, as shown in Fig. 14, and consequently the column must be analyzed

exactly by taking into account the biaxial loading effect.

8. Conclusions

The following conclusions can be drawn for the ultimate load and the deforma-

tion behavior of slender reinforced concrete columns subjected to a biaxially eccen-

tric load.

(1) On the basis of the test results, it is recognized that the concrete and the re-inforcing bar strains conform closely to the plane strain distribution even in the case

of biaxial bending.

(2) The numerical analysis adopted herein to solve the load-deformation response of the column very well predicts the test behavior.

(3) The ultimate load carrying capacity of a slender column is reduced by the ad-ditional eccentricity due to lateral deflections, even in a column having a length to

depth ratio of 15.

(4) In the case of square long columns, there is not much difference on ultimate loads of the columns in spite of variation of the angle of eccentricity of the applied

load.

(5) In the case of rectangular long columns, when the columns are subjected to biaxially eccentric load to cause bending about the near-strong axis, the ultimate loads

and the deformation behavior seriously depend on the angle of eccentricity of the ap-

plied load. Consequently those columns must be analyzed exactly by taking into ac-count the biaxial loading effect. However, the columns subjected to biaxial bending

about the near-weak axis can be approximated by that under only uniaxial bending

about the weak axis.

156 S. IWAI, K. MINAMI and M. WAKABAYASHI

Acknowledgment

The authors wish to express the gratitude to Dr. Takeshi Nakamura of the Dis-

aster Prevention Research Institute, Kyoto University, for valuable discussions and

critical reading of the mamuscript.

References

1) Architectural Institute of Japan: AIJ Standard for Structural Calculation of Reinforced Con- crete Structures (revised 1985), Chap. 4.

2) Iwai, S.: Studies on the Elastic-Plastic Behavior of Reinforced Concrete Long Columns Sub- jected to Static and Dynamic Loads, Thesis submitted in partial fulfillment of the require-

ments for the degree of Doctor of Engineering at Kyoto University, Kyoto, Japan, Dec. 1984 (in Japanese).

3) Iwai, S., K. Minami and M. Wakabayashi: Ultimate Strength of Slender Reinforced Concrete Columns Subjected to Biaxially Eccentric Loads (Part 1- Loading Tests of Square Cross Sec-

tion Columns), Trans. Architectural Inst. of Japan, No. 367, Sep. 1986, pp. 59-68 (in Japa- nese).

4) Iwai, S., K. Minami and M. Wakabayashi: Ultimate Strength of Slender Reinforced Concrete Columns Subjected to Biaxially Eccentric Loads (Part 2 - Comparison of Experiment and Anal-

ysis), to be submitted to Trans. Architectural Inst. of Japan (in Japanese). 5) Bresler, B.: Design Criteria for Reinforced Columns under Axial Load and Biaxial Bending,

Journal of ACI, Vol. 57, No. 5, Nov. 1960, pp. 481-490. 6) Ramamurthy, L.N.: Investigation of the Ultimate Strength of Square and Rectangular Columns

under Biaxially Eccentric Loads, ACI Publication SP-13, 1966, pp. 263-298. 7) ACI Committee 318: Building Code Requirements for Reinforced Concrete (ACI 318-83),

Chap. 10, 1983.

Appendix-Notation

The following symbols are used in this paper.

b=width of column section

D=depth of column section

E3 =Young's modulus of reinforcing bar

e= Vex2+ ey2, resultant eccentricity of load

ex =componentof eccentricity of load in the x direction ey= component of eccentricity of load in the y direction

Fe=concrete cylinder strength

/=length of column

M=biaxial bending moment

Mu =ultimate biaxial bending moment (Eq. (7))

Muyo=ultimate uniaxial bending moment about the y axis (Eq. (7))

Mx=bending moment about the x axis MM =bending moment about the y axis

N= axial force


P=compressive load applied at the ends of column

Peal = maximum load calculated by the numerical analysis

P.= maximum load

Po=ultimate load under pure axial compression (Eq. (6))

Pte.t=maximum load obtained by test

Pu=ultimate load under biaxial compression (Eq. (6))

Px=ultimate load under compression with uniaxial eccentricity ex (Eq. (6))

Pv=ultimate load under compression with uniaxial eccentricity ey (Eq. (6))

u=component of lateral deflection in the x direction

v=component of lateral deflection in the y direction

w=total axial deformation of column

x=principal axis of the cross section

y=principal axis of the cross section z=coordinate along original column axis

a=torsional rotation angle at the mid-height of column

E = strain

E max= compr ession corner strain at maximum load

DE.=compression corner strain at maximum load measured by potentiometers

Gemax=compression corner strain at maximum load measured by strain gages Eo=strain at the centroid of cross section

Ep=strain at maximum stress of concrete

E u =failure limit strain of concrete

Ey=yield strain of reinforcing bar

8=inclination of plane including the loading point and the centroid of the end sec-

tion to the x axis Ox=slope of deflection curve of column about the x axis

Ost=rotation angle of the lower end of column about the x axis

Ozo=deflection slope about the x axis at origin of the z axis (see Fig. 5)

Ox2,=rotation angle of the upper end of column about the x axis

ey=slope of deflection curve of column about the y axis

tipz=rotation angle of the lower end of column about the y axis

00=deflection slope about the y axis at origin of the z axis (see Fig. 5)

8,.=rotation angle of the upper end of column about the y axis

6 = stress

6 c= ultimate strength of concrete used in the analysis

6v =yield strength of reinforcing bar

0,v=curvature about the x axis

Øy =curvature about the y axis

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