A Study on the Superposition Method to Estimate the

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A Study on the Superposition Method toEstimate the Ultimate Strength of SteelReinforced Concrete Column Subjected toAxial Thrust and Bending MomentSimultaneously

NAKAMURA, Takeshi; WAKABAYASHI, Minoru

NAKAMURA, Takeshi ...[et al]. A Study on the Superposition Method to Estimate the Ultimate Strength of SteelReinforced Concrete Column Subjected to Axial Thrust and Bending Moment Simultaneously. Bulletin of the DisasterPrevention Research Institute 1976, 26(3): 163-193

1976-09

http://hdl.handle.net/2433/124861

Rut Dins. Prey. Res. Inst., Kyoto Univ., VoL 26, Part 3, No. 242, Sept., 1976 163

A Study on the Superposition Method to Estimate the Ultimate Strength of Steel Reinforced Concrete Column Subjected

to Axial Thrust and Bending Moment Simultaneously

By Takeshi NAKAMURA and Minoru WAKABAYASHI

(Manuscript received October 3, 1976)

Abstract

In this paper, considered is the further applicability of the "Superposition Method" to estimate the load carrying capacity of steel reinforced concrete column subjected to axial thrust and bend-

ing moment. The accuracy of the superposition method is checked in the combination use of various grades of concrete and steel. It is confirmed that the unsafe-side error of superposition

method is not large even though steel portion with shallow depth is put in a cross section of a column and/or an asymmetrical steel shape is used, provided that the yield strength of steel is not too high. The unsafeside error of the superposition method is reasonably compensated by the

reduction of ultimate strength of concrete which was specified in the design standard of AIJ for SRC structures.

1. Introduction

Steel reinforced concrete structural system (concrete encased steel structural system) was imported from the West at the end of The Meiji Period. At present, it is one of the most frequently used structural systems for the design of building structures in Japan. Steel reinforced concrete structures (SRC-structures) have the engineering properties of two elemental structural systems; ordinary reinforced con-crete structural system and steel structural system. In this system, these two ele-mental structural systems compensate each other. The property of brittle failure caused by the material property of concrete is offset by the ductile property of steel. On the other hand, the weakness of steel against fire and high temperature is rein-forced by an outer concrete encasement.

In Japan, strong earthquakes frequently occur. Under such severe natural circum-stances, a steel reinforced concrete structural system has been developed as the most effective earthquake-resistant-structural system. Until quite lately, SRC system has been applied to design most high-rise building structures with more than six stories. Although a number of pure steel structures has been recently increasing since light-weight fire-proof material was developed and accepted by laws, the SRC system is still the most frequently used system for building structures about 45 in high and for the lower or underground portions of steel high-rise buildings.

The design formulas for beams and columns of SRC structures has been based on the so-called "Superposition Method" since the first design specification for steel reinforced concrete structures was completed by the Architectural Institute of Japan

(AIJ) in 1958. The "Superposition Method" is based on the consideration that the

164 T. NAKAMURA and M. WAKABAYASHI

ultimate load carrying capacity of SRC structural elements is given by the sum total of the ultimate load carrying capacity of the individual elements which compose the SRC section, i.e., ordinary reinforced concrete and steel portion. It has been verified under the plasticity theory that the load carrying capacity of a composite structure composed of two or more components with different load carrying capacity is not smaller than the sum total of the load carrying capacity of the individual com-

ponents3)4). However, this verification is valid only for the component materials which behave in perfectly plastic manner. It has been confirmed in experiments that the value of the ultimate strain of concrete for crushing in compression is about

0.30.35°/0. It has been also understood that the "Superposition Method" to the load carrying capacity of the components composed of such materials does not give always the safeside estimation of load carrying capacity of a composite structure.

When the depth of steel portion put in an SRC cross section is not very small in com-

parison with the depth of a resultant concrete section and ordinary low carbon steel with not very high yield strength (,ay=2.4,--3.3 t/cm2) is put in the concrete with ordinary ultimate strength (//0---150200 kg/cm2), the unsafeside error of the "Superposition Method" is not so large that the application of that method becomes

impossible. In such cases, the partial unsafeside error of the "Superpositions Method" is reasonably compensated by the use of the reduced ultimate strength of concrete. This reduction of concrete strength was taken into the specification to compensate for the imperfect casting of concrete due to the existence of steel portion and reinforc-ing bars which are complicatedly put in the SRC section and errors involved when

the gross area of concrete cross section is used in the analysis. Recently, high strength concrete and high strength steel have been developed to

be used in the design of buildings. When high strength steel is used in the design of SRC structures, it is easily understood that the supperposition method does not give the safe-side value of the ultimate load capacity under pure compression if the yield limit strain of steel is larger than the crushing strain of concrete. In the combina-tion use of concrete and high strength steel, a larger unsafe-side error of the super-

position method would be anticipated. Further, the order and amount of the error of the superposition method has not been checked sufficiently for the case that the steel section with considerably smaller depth in comparison with the depth of con-

crete section is used. In this paper, the applicability of the superposition method for using high strength steel and the steel portion with the small depth in the SRC cross

section are considered mainly.

2. General Description of the Superposition Method for the Ultimate Load Capacity of the SRC Column Section Subjected to Axial Thrust

and Bending Moment

As previously mentioned, the principle of the superposition method is based on the consideration that the ultimate load carrying capacity of SRC structural elements is gitien by summing the load carrying capacity of the individual components, i.e.

A Study on the Superposition Method to Estimate 165

reinforced concrete and steel. In order to help to understand, the fundamental

superposition method is graphically shown in Fig. 1. The ordinate is axial thrust and the abscissa is bending moment. In the figure, the ultimate strength curves

of plain concrete with rectangular cross section and pure steel portion with ideal

I-section subjected to axial thrust and bending moment simultaneously are shown

by chained line and dashed line, respectively. They are referred to as M-N interac-

tion curves of concrete and steel portions, respectively. M-N interaction curve of SRC cross section by the superposition method is given by the envelope curve (dotted

curve) of a group of curves which are obtained from making the origin of the M-N

interaction curve of steel move on the M-N interaction curve of concrete. The

equations to calculate the load capacity due to the superposition method is shown

in section 4.

N

St eel Reinforced .

, Concrete

Concrete--- , ,, ..„.., -..„,,. ---...

•-•:- • ..

.

^ 4.4:10' „to. • • • 41- • • •,,, ...

.. "17 . %. •

••t•• ... ••••O.•:", • •%

, , ,

7 .... . 4 eier- ,- 7

M

Steel

----

Fig. 1. Graphical Method of Superposition of Individual Component.


3. Calculation Method of the Ultimate Load Carrying Capacity of SRC Section Subjected to Axial Thrust and Bending Moment Simultaneously

The "Ultimate Strength Method" is well known as the most accurate method of

calculation to obtain the ultimate load carrying capacity of SRC column section under axial thrust and flexure. The complete description of the "Ultimate Strength Method" is given in Refs. 1) and 2). Only the stress-strain relationship of concrete is different from that in Refs. 1) and 2). In this paper, the stress-strain relationship of concrete is assumed as shown in Fig. 2. Before the attainment of the maximum stress ca„ the stress-strain relationship is parabolic. After the attainment of the

maximum stress ,o0, the maximum value of stress coo is maintained until the strain

GeB) reaches the value of (I +r) • ceo, i.e. the crushing strain of concrete. In the strain range larger than (1+ x) • ce 0, concrete can not sustain any load in com-

pression. In tension range, concrete can not sustain any load. The stress-strain relationship of steel is assumed to be elastic-perfectly plastic in both tension and compression. The other main assumptions taken in this method of calculation are as follows. C) A plane of transverse section remains plane after deformation. C) The ultimate load carrying capacity of SRC section is attained when the strain in the

extreme fiber of concrete in compression reaches the crushing value (1-“)-ceo. It was confirmed by a number of experimental investigations in the past that the ulti-mate strength based on the assumption ® gives a good approximation to the real ultimate strength of a column section. Under the assumptions ® and ®, the strain distribution in a cross section at the ultimate state is easily obtained by giving the location of the neutral axis in flexure. Corresponding to the ultimate state of strain, the ultimate state of stress is calculated using the stress-strain relationships of the element materials. The ultimate load carrying capacity is obtained as the generalized stress resultants being in equilibrium with the ultimate state of stress. The results

of the ultimate strength method are used to check the accuracy of the superposition method in this paper. The equations in the ultimate strength method are developed in the next section.

4. Method of Calculation

4.1 Assumed Stress-Strain Relationships

(1) Concrete (see Fig. 2(a)) In compression,

a = cao—(2— )for 0 <e < gio teatet1

( 1 ) —~Qa for tee <e < (1 ± x) cc,

= 0 for e> (1±,^)ce,

In tension,

a=0 ( 2 )


0

(Comp.)

e o I I

a . {2E,..(E)z)a crorro ca

i 1

rcg ^ (14400E0

Atan-ICE I

I

C

cCo ceg (Cont.)

(a) Concrete

a

ray-- 1 1

Cyi a

7

tan-I rE -rey

C

rey

^^-ray

(b) Reinforcing Bar

a

a ay

1 I

c I tan-IsE sy i i C

i BEY

I I

1 i -

say

(c) Steel

Fig. 2. Stress-Strain Relationships.


(2) Steel (see Figs. 2(b) and (c)) In both tension and compression

a= iEe for I el tie), a-=y fore I> ( 3 ) f

where, i=r, s, iv and j= 1, 2, 3, •

4.2 Strain Distribution in a Cross Section at the Ultimate State At the ultimate state, the strain in the extreme fiber of concrete in compression

is assumed to be ce3= (1+c)00, as shown in Fig. 3. Under the assumption (:), the strain in an arbitrary longitudinal fiber in a section is expressed in the next equa-tion.

ecr-(14%),co c o

kD_________ D

(1-k)D 4 Fig. 3. Strain Distribution and Stress in Concrete at Ultimate State.

k —yid, ijec OB k

1 —k— ijd (4) &its— c€.8 k

where k is the location of the neutral axis in dimensionless form.

,.De and;id,=

d =s, D

i r, 3-, w

j 1, 2, 3, • ••

r, s and w denote reinforcing bars, flange of the steel portion and the web of the steel

portion, respectively. 1, 2, 3, • • • are the number of layers of steel and reinforcements. Corresponding to the strain distribution, ilk state of stress in a cross section is deter-mined according to the stress-strain relationships. The definition of a sign of strain


and stress is that compressive strain and stress are positive in the fiber in thee.com

pression side of a section, and tensile strain and stress are positive in the fiber in the tension side.

4.3 M-N Interaction of Concrete Portion

M-N interaction curve of concrete portion at the ultimate state can be directly

calculated under the assumptions © and T. The state of stress and the stress result-

ants at the ultimate state are shown in Fig. 4, with main geometrical values in a

cross section. The resultant axial force Ne and bending moment Me at the ultimate

state are expressed as Eq. 5 expressed in a dimension of stress.

mEmmiÎ^ 1 K 1111•^ , Trickipk.a•BD k

D^WorangTricC0 1-1-K

1 K ^1=M • — 14kDiiÎ^mmmin -K a

c

kD 3 1 , mÊ^sv 2 . k .C1,50• BD n 3 1-tx 1

kD ÎMI^ 1-1-K```' ^•• 5

11 To ^^^• ^••^

(1-10D

Fig. 4. Stress in Concrete at Ultimate State.

N 2 1 r )k2+5xto kcao BD\3 1+x+1+x3(1+4 ( 5 )

BD2Me 2 1 ( 15 1IC =—k+ k)+ 1 kkcoof {3 1+k 2 81+x2(1+x)(1+x Equation 5 can be applied to the both cases of the superposition method and the

ultimate strength method.

170 7'. NAKAMURA and M. WAKABAYASHI

4.4 N and M of Reinforcing Bar and Steel Flange at the Ultimate State

Stress in a reinforcing bar or steel flange is exppressed as follows, corresponding

to a strain shown in Fig. 5.

eels

0 0 j€c L-- —1 -• slec 1 wjec lcD

D

wj et I-- sjet(1-k)D

0 0 tier

Fig. 5. Strain Distribution.

;jar= for —iie,<,/,�11e,

Vie for ( 6 ) for +Jet<

in compression side.

jai=ijEije,for —ijey�iie,�iity

tics = rjcy for rie:> i 16 ( 7 )

fiat = —iicy for tier < —They

in tension side. The resultant thrust iiN(;iNIBD) and moment i1M(i1MIBD2) are obtained from

these stresses, as follows.

tr 0 0 opc

wD I--.

D-wD

wDt 0 0 11-4apt Oh

Fig. 6. Dimension of Cross Section.

A Study on the Superposition Method to Estimate

kiN BD -= Jaey?[—oat iiPt

( 1 1 BD'=',fie\2—,ide)±,fitiiPt(2 116C) Equation 8 is applied to the ultimate strength method. In the superposition method, M-N interaction curve of reinforcing bars and steel with ideal I-section is obtained as the linear line elements passing through the points representing the ultimate load carrying capacity under pure thrust on the N-axis and the point representing the

ultimate strength in pure flexure on the M-axis.

4.5 N and M in a Steel Web at the Ultimate State

The ultimate states of stress in a steel web are composed of a total of ten cases corresponding to the location of the neutral axis. The states of strain and stress in each case are shown in Fig. 7. The equations for the resultants N and M in a steel web or a steel portion with a rectangular cross section are developed hereafter.

(1) When k< wide

Case 1

N 1 BD — 2 • RiE ;6 to/ c)(1 ejdt wide) toil

M 1 BD2 = 2 • siE(1— widt—logdc)[wiecCufdt—tel ( 9 ) e)

^.et*elee32){-2(1A) —I.4}1t 1 Case 2

,22/± _1 BD — 2 (wi° wiec) 17—wiGY (1 —G A-1°,4 x))

M

ITE71°.1 = key( i moide—widt— Y ) Y • • • • • • (10)

±-1 ,„2Y (siva,,- -..E •ut.ec)(-2Y+tv.4-12)•3i —

"E•,,jec• +wjd`2 2

where, Y---(k—..ide)(1+°16.) lute Case 3

BD ..N =---—„, Jo1 —,„14— ida ,, it, (1 1 )

BwiM 1D2 = 2 wia


STRAIN STRESS

ar Pa EIANG.PCONTR. TeNT.17471.11P. E a -weswac-.-sEvec CASE 1

tEIt

®E ?0wac = 'aEw€c vata sayCASE 2

4-0I Pm. wet

vat a sayCASE 3 P I

E

vac a sEwec CASE 4

E Ar

,

WEC P vac = say e—t CASE 5

wd, s R s 1 -,dt 7E E vat = sEvet I

mac=sEvec CASE 6

brat= say wet P

I POs a say

CASE 7 peat = say

E

fee1'PC ' stwee CASE 8

wat° "sEvEt

:t E

IPEwow . say J k> 1-sclsCASE 9

'Pt - 'sEvat

P "

brGt

.. di, say CASE 10

P

4 4 4 say say

Fig. 7. Ultimate State of Steel Web.

A Study on the Superposition Method to &climate 173

(2) When .idc<k< 1 muid,

Case 4

OJN BD = pint $02

..M 1 1 BD2 = id •—T (k— wide)} (12) 1 1

— id, —T (1—k— . id,)

where, win, = 2 „jec(k moid,)„iEwiti

win2 =—1„.id,)„iE • „it, 2

Case 5

„jAr .n,.n2.n BD —1°w—s"J3

1 1 wr BD22—wide2 (kwide)(]s „ee)} • (13) ••r in2W—••id—widc)(1—”ire(kd))

4-win.st-1.d,— (1—k— .id;)} 2—wide—

where, win, = wia y(k— . id 41 .6 ).JI,

J

1 wing =wiay(k--.14)wi it

wj C

w ins = luret• .E (1—k — widt)„itt

2 Case 6

wiN BD2 w in2—wjn3

11

BugD2 —14;74— to fd•---3- (k— wic 4)) (14) 2 e1 win2{k ±3— k — ..d,) ir„ie,2 }

1 1 +,,in,f(1 —k — „ id :)(1 — wie jet

174 T. NAKAMURA and M. WAKABAY ASH!

where, ,01 = '2. je t• „/E(k — wide)wit,

wine = 2,piay(1—k-01d,):iieeit •wit, is „in, = „;a,(1 —k —/4)(1 —Ivet

wits wj

Case 7

wiN BD = —wins

M 1 1

BD,= winI{T—„jdc—T(k —„jd)(1 lejec) wide)(1—a•)--1(k—.dc)"ie•(15)

wie, 3wjec

±„in2[2—„,/,—{1 wwi ate: (k — wide)} 3 (k—d) wje7 w"Wjec — 1 1

+wilts 2Thd#—F2-{1 —kmdt— wie(kmdc)}] wiec

where.n=.a(k —sejdc)(I — Met,e) 1WI,,.t, j

1 *;E9 #1.A\ wine =2.55reiswi

c

wins = .15( 1—k — „id t— (k—,I41,)}„it, IOC

(3) When k�1 —„id,

Case 8

N 1 = 2 Cie,— wie,)(1„id,),iE • wit, BD

1 1

_

BB, — —„ie,• wiE(1-534-51d,){T—wide—T (1— widemid,)},it, (16) 1111 ±—(..e)...d){——d——(—d) 2aelwl*c)E(1—dc—23de—}art

Case 9

BD — .02

} (17) 404 f1111ee BY2—2(1 — w)de—l'idr)}±,02{2—1°,5—3wl1(k—eicia)

A Stun)on the Superposition Method to Estimate 175

where, win, = 1 sq6y+j I aj4 mina =2Gsva.1+soje t• wincstiehell

Care 10

wiN BD = wicr 3,(1— id,— wid,)„,t,

N 1 (18) es_ Bpz — wad t)(wielt—wide),qt,

Equations 9 to 18 are to be used in the ultimate strength method.

4.6 M-N Interaction of Steel Shape at the Ultimate State

The M-N interaction of steel portion which is used in the superposition method is as follows.

(1) when k< „de

BD — 'GA+ 3;Pr+ wiP) (19)

= o BD2

(2) when side�k� 1 mid,

BD = 3,a,(31Pc-3;Pt)

+ Witt(widg—„jd,+2k— 1) .M 1 1 jgDz = sji IjPe(2—side)-1-sitt(2—,3€11)} (20)

-F—21wa"•ti{(k— wide) (1 —k — wide)

+(k—wid.)(1—k-0/11)}

(3) when k> ,Id, st

BD = sic,LiPc±siPs+NIP) (21) =---0 BD2

S. Results of Computation

The M-N interaction curve based on the superposition method is obtained from

the vector sum of the M-N interaction curves of each component put in an SRC cross


600

0.5%

0.1 D D It N/RD • %

1% (kg/cm2)

co, = 300 kg/cm'

400 ray - 3000 kg/cm'

%

5 \

200 % say - 4100 kg/cm2

\5

sirsoy = 3300 kg/cm' say a 2400 kg/cm'

100

0 r a M/I3D2 (kg/ cm2 )

a

to.

-200 0.22 0. 352 E

Fig. 8. Example of Computed Results.

section which is computed using the equations shown in the previous sections 4.3,

4.4 and 4.6. On the other hand, the M-N interaction curve based on the ultimate

strength method is directly computed using the equations in sections 4.3, 4.4 and

4:3. An example of computed results is shown in Fig. 8. Dashed lines are the

results due to the ultimate strength method and solid lines are the ones due to the

superposition method. In the figure, the ordinate and abscissa represent magnitudes

of axial thrust and bending moment expressed in a dimension of stress, respectively.

5.1 Combination Use of Various Grades of Steel and Concrete

Figures 9 to 11 shows the effects of a variation of the yield stress of steel put in

A Study nn the Superposition Methodic- rEstimate 177

a SRC section. The discrepancy between the results of two methods are very small for SS41 steel (scrI=2400 kg/cm2) and SM50 steel (,ay=3300 kg/cm2), for the

case of SM58 steel- (;ay=4100 kg/emz), it is a bit larger. Maximum error of the superposition method in that case is about 10%.

600

0.52 .411pmk,„ 12 0 o

1% 0.10 D N/BD

(kg/cm2) 12 o 0 AF

0.52

400 sus - 225 kg/ore

ray = 3000 Icg/cm2

\ 200 \ \ 4100 kg/cm2

\ \ \ \

\ \ \ \

is say 3300 kg/cm2

say =2400 kg/cm2

0 100

WED' (kg/co2)

a

cas-71-7 -200

0.21 0.352 c

Fig. 9. Effects of Steel Strength (Strong Axis Bending).


Figure 12 shows the effects of variation of concrete strength in compression. The

discrepancy between the two methods is also small. Such an amount of unsafe

side error of the superposition method is sufficiently compensated for by the use of

the reduced strength of concrete in the actual design procedure as shown in Fig. 13.

600

0.5% 0 (ET N/BDn0.1D D

(kg/cm2) 0.52 0 a IF I

400 sus = 225 kg/cm'

ray- 3000 kg/®'

say 4100 kg/cm2 kililli\iNsrsaykg/cm kg/c

It I say2400 kg/cm' 200

50 100 M/BD2

(kg/cm2)

sus-7n

0.2S 0.35%E

-200

Fig. 10. Effects of Steel Strength (Weak Axis Bending).

A Stun) on the Superposition Method to Estimate 179

5.2 Effects of the Depth of Steel Portion

Figure 14 shows the effects of the depth of steel portion in an SRC section. From

the figure, it is understood that the error of the superposition method is pretty small

even if a steel section with a small depth is put in a column section. Maximum

600

0.5%

NISD 1% (kg/cm2)0.1D D 1%

0.52

cao. 150 kg/cm2 400 ray 3000 kg/cm2

200

•s . say•3300 kg/cm2 • s•

say T. 2400 kg/cm2

0 7p.1 50 100

M/BDI

(kg/cm2)

a

ear 771:

0.2%0.35% E

-200

Fig. 11. Effect of Steel Strength (roe)=150 Icgicmz).


unsafeside error of the superposition method is about 15% in comparison with the results of the ultimate strength method for the case where the depth ratio of the steel

portion alD=0.2.

600

0.52 4r" (kg/cmt) /X 0. / D

0.5X I ray 2400 kg/cm2

400 say - 2400 kg/cm7

•••

• coo = 270 kg/cm7 200 •

S. = 225 kg/cm7

S. ••• • N. ‘111 cao •. 180 kg/cm2

%

/7 coo = 150 kg/cm, 0 pry

50 100 M/107

(kg/cm')

cer 0 - -

0.2X 0.35X E

-200

Fig. 12. Effects of Concrete Strength.


5.3 Effects of Crushing Strain of Concrete

Figures 1521 show the effects of the crushing strain of concrete. If concrete crushes in too small range of strain in compression, the discrepancy between the two methods is pretty large (Figs. 17 and 18). However, when concrete performs in an

600

0.5% 4_ .0%

N(BD 1.010.1 D 0 (kg(cms) 1.0%

0.5%

ioY = 3000 kg/cm' 400

soy= 3300 kg/cm2

200 `...‘ coo = 225 kg/cm'' - ( 1 - 15spc )con

191.25 kg/cm'

co; : Reduced Strength of Concrete

0 50 100

M/BD2 (kg/cm')

CC° ITT 0.210.35% E

-200

Fig. IS. Compensation of Unsafe-side Error of Superposition Method by

Reduction of Concrete Strength.

182 7'. NAKAMURA and M. WAKABAYASHI

ordinary manner in compression according to the reasonable stress-strain relation-ship (Figs. 9 and 15), the discrepancy is so small as to be able to be offset by the use of reduced strength of concrete. The reduced strength of concrete gives a quite conservative M-N interaction for an SRC cross section (see Fig. 13).

600

2% '7= tq,B. an1)

(cticae)=Ira4-1

cao^225 kg/®2400

2

aY- 2400 kg/cm4

a 0.8

• a 0.6 a 0.4

a - 0.2 o For 50 100 WW2

(42/m2)

a

cap --

1

-200 0.2% 0.151 C

Fig. 14. Effects of Depth of Steel Portion.


5.4 Effects of the Amount of Steel Reinforcement

Figure 22 shows the effects of the amount of steel reinforcement in an SRC Section.

The discrepancy is also small.

600

0.5% 1. 0%It

11/BD1.0% 0.1 D (kg/ cm') 1.0%

0.5%

coo 225 kg/cm2

400 ray = 3000 kg / cmz

\ \ \\\\

• • • •

200\5 •• say 4 4100 kg / cm2 5

say = 3300 kg/cmi

I say = 2400 kg/ cm2

/

100 50 /1/1302

(kg/cm')

coo71-1

0.15% 0 . 3% E -200

Fig. 15. Effects of Stress-Strain Relationship of Concrete—(I).


5.5 Effects of Asymmetrical Arrangement of the Steel Portion in Cross

Section

An example of the effect of an ansymmetrical arrangement of the steel portion

is shown in Fig. 23. The discrepancy between the two methods is not so large as

600

0.5% ATI it---1 M/BD I%

(kg/Cm1)10.10D

0.15

225 kg/cm2

400raY3000 kg/cm2

.4.•••• •%

200 % say 4100 kg/cm2 • •••

% % say - 3300 kg/cm2

\

la soy 2400 kg/cm2

0 100

M/Be 05g/cef

0.2% 0.5%

-200

Fig. 16. Effects of,Stress-Strain Relationship of Concrete—(//).


to be worried about the asymmetrical steel portion.

6. Concluding Remarks

It is concluded based on the computed results that the superposition method gives

600

0.5% 0 N,.1 .0%I0 .,DD (kg/cm2) 1.0%

0% o oI 0.5%

coo - 225 kg/cm2

400 ray =3000 kg/cm2

Illi1/411Na say4100kg/cm2 say-3300kg/cm2

Nsoy = 2400 kg/cm2 200 (A):: ° rer 50/00

11/1302

(kg/cm2)

0

c a -71

-200I 0.2% E

Fig. 17. Effects of Stress.Strain Relationship of Concrete—(III).


a good approximation of the ultimate load carrying capacity of SRC column section

under the combination use of various grades of concrete and ordinary grades of steel.

The arrangement of the steel portion in a cross section and amount of steel reinforce-

ment do not affect very much the accuracy of the superposition method to estimate

600 0.554112ft ‘ r

N/ BD 1%'al10.1D4ID (kgicic2)

0.55

400 coo = 225 kg/cm'

ray = 3000 kg/cm2

iii%jer. soy = 4100 kg/cm2 1111*e- soy = 3300 kg/®' •

\41N4"1. so_- 2400 kg/cm2 200 ;‘11'

) 1

1 1 I a

)

50 100

M/BD2 0 (kg/CO2)

cos ___ .....,

0.2% c -200

Fig. 18. Effects of Stress-Strain Relationship of Concrete—(IV).

A Study on the Superposition Method So Estimate 187

the ultimate strength of an SRC column section. The "Superposition Method" is an appropriate and simple method to estimate the

ultimate load carrying capacity of the steel reinforced concrete column section sub-

jected to axial thrust and bending moment simultaneously, unless the steel with too

600

0.5% . ph N/BDIS Er

1% al0.1D (kg/cm2) 1%Oa_kzi

0.5%

400 co. = 225 kg/cm'

ray = 3000 kg/cm2

say = 3300 kg/cm2

•

• •

• ccs = 1.5% K 3.0% •

200 scs 2.0% K 3.5%

\ se° - 1.5% K = 3.0%

C c0 = 2.0% K = 3.5%

a vv. :

50 100

M/BD2 (kg/cm2)

"

coec (1+10s% C -200

Fig. 19. Effects of Stress-Strain Relationship of Concrete—(V) .


high yield strength is used in a design. The unsafeside error in the superposition

method is able to be sufficiently compensated for by using a reduced strength of

concrete.

600

1.02 C a --- 1.02 ----a

M' 1.0%---.- N0.10D (kg/cm2) 4 I

„.0, .

0.52„-----°,--

—

coo a 225 kg/0.2

400 yoy - 3000 kg/cm'

soy = 3300 kg/cm'

\ .

\ `s. \ ... .

\

.\\\ 200 \K= 1.5

\\K . 0.75 \\N

\V K = 0.75 \

% K = 0

/ / 0

coo_ rpr 50 100

M/002

(kg/cm') 0

71-1

1 j CEO (1+=)C£0 (

-200 sco = 0.2 2

Fig. 20. Effects of Stress-Strain Relationship of Concrete—(VI).


600

0.5% C.:. 0 I N/BD 1 % — ---11 0.1 n D (kg/cm2)

0.5% 0

cao ̂ 225 kg/em' 400 ray 3000 kg/cm1 3300 kg/cm'

\ \ \ \

\ \ 200 eVK a el."

K = 0.75

% K 0

/

0rpr. 50 100

N./AD'

(kg/cm')

car, (-7.7:

re0(1+00D0 E -200 re° 0.2 %

Fig. 21. Effects of Stress-Strain Relationship of Concrete—(VII).

190 T. NAKAMURA end -M. WAKARA ASH'

600

ppIIII OID N/BD P

w (kg/cm2)

OT{

.• 300 kg/cm2

400 soy = 2400 kg/cm2

\\ \\ \ \

200\ R- 10%

/1; = 6% p.,,= 3%

/ C

so/ 100

M/BD2 (kg/cm2)

cc° -77 -200

0.2% 0.35% E

Fig. 22. Effects of Amount of Steel.


600

a0.5%e

1 N/BD 2% ler coo r(kg/cm2) on 1S MllD 1110 1% --kliDI 0.2% 0.352 E 0.5%.11111.._e0.1

670 225 kg/cm2 400 ray = 3000 kg/cmz

/ S. / S. ^ • .

• • • / N.

/sN. ••\ /

e•• / •.• • / • / • • ‘

/ / • • • \ /•• /

/200••• / ••‘

/ / • • • / / \ • • /

/ 1 i i /1 1 / / / /

/ / //

i/ \ -50 m/BD2

(kg/cto2)

‘0101irdr50 • say = 4100 kg/sm2

say = 3300 kg/cm' say - 2400 kg/cm`

-200

Fig. 23. Results for Ansymmetrical Steel Portion.


Nomenclature

ija, : cross sectional area of main compressive reinforcement in i-component in j-th layer

: cross sectional area of main tensile reinforcement in i-component in j-th layer

B : width of cross section of column D : depth of cross section of column

sE : Young's modulus of steel iiE : Young's modulus of i-component in j-th layer

F, : ultimate stress of concrete : prefix denotes section component, i=r, s, w

: prefix denotes layer number k : non-dimensionalized location of neutral axis

M : bending moment Mc : bending moment in concrete

liM : bending moment in i-component in j-th layer of steel N : axial thrust

N, : axial thrust in concrete

iiN : axial thrust in i-component in j-th layer of steel wink : sub-variables

iip, : reinforcement ratio in compression side

liPe= BD

: reinforcement ratio in tension side

;Jai liP'= BD

wip : web reinforcement ratio SRC : steel reinforced concrete

„it : thickness of web in j-th layer wit, : non-dimensionalized web thickness in j-th layer

wit t = niB

: fiber strain

ceB : crushing strain of concrete

cen=(1±r)cen

Leo : yield strain of concrete

wie„ kiet: fiber strain of i-component in j-th layer ;fey : yield strain of i-component in j-th layer

se, : yield strain of steel


: variable which denotes the length of plastic range of concrete o : longitudinal stress

,e, : ultimate stress of concrete ijac, ;fa,: stress in i-component in j-th layer

;Jo, : yield stress of i-component in j-th layer

References

I) Wakabayashi, M., S. Takada and H. Saito: Steel Reinforced Concrete Structures, Structural Engineering Series (Kenchiku Kozogaku Taikei), Vol. 19, 1967, Shokokusha (Tokyo), pp. 32-72

(in Japanese). 2) Wakabayashi, M., et al.: Steel Reinforced Concrete Structures, Architectural Engineering

Series (Kenchikugaku Taikei), Vol. 18, 1970, Shokokusha (Tokyo), pp. 242-273 (in Japanese). 3) Tanaka, H.: Study on Additional Strength Theory, Transactions of A13, No. 57, July 1957,

pp. 261-264 (in Japanese). 4) Hirano, M.: Additional Strength of Cross Sections and that of Structures, Transactions of AU,

No. 63, October 1959, pp. 397-400 (in Japanese).

Documents

A Study on the Superposition Method to Estimate the