Title no. 83-22
Mo
C om om p
Re
me
Frank J.
io
Collins
An anal anal ti odel odel is pres presen ente te that that is apab apable le of pred predic icti ting ng th load load-d -def efor orma mati tion on resp respon onse se of rein reinfo forc rced ed conc concre rete te elem elemen ents ts subj subjec ecte te to in-p in-pla lane ne shea shea an norm normal al stre stress sses es In th odel odel crac cracke ke conc concre rete te treate te as ne mate materi rial al with with it ow stre stress ss-s -str trai ai char charac acte teri rist stic ics. s. is trea Equi Equili libr briu ium, m, comp compat atib ibil ilit ity, y, an stre stress ss-s -str trai ai rela relati tion onsh ship ip ar forformula mulate te in term term of aver averag ag stre stress sses es an aver averag ag stra strain ins. s. Cons Consid ider eraais
Th stre stress ss-s -str trai ai rela relati tion onsh ship ip fo th crac cracke ke conc concre rete te were were dete deterrmine mine by test testin in 30 rein reinfo forc rced ed conc concre retepan tepanel el unde unde vari variet et of well well defi define ne un fo ia al st ss nclu ncludi di ur shea shear. r. It wa ou that that crac cracke ke conc concre rete te subj subjec ecte te to high high tens tensil il stra strain in in th dire direct ctio io norm normal al to th ompr ompr ssio ssio is so te an ak in ompr ompr ss on than than conc concre rete te in stan standa dard rd cyli cylind nder er test test Addi Additi tion onal ally ly sign signif ific ican an tens tensil il efo high high valu values es of aver averag ag tens tensil il stra strain in
Keywor Keywords: ds: aggreg aggregate ate interl interlock ock axial axial loads; loads; biaxia biaxia loads; loads; cracki cracking ng (fract (fracturi uring) ng) rack rack widt widt an sp cing cing init init elem elemen en etho ethod; d; of shor shor truc tructu tu es rein rein forc forced ed conc concre rete te shea shea stre streng ngth th stif stiffn fnes ess; s; stre stress sses es stre stress ss-s -str trai ai rela relati tion onsh ship ips; s; struct structura ura analys analysis; is; tensio tension; n; tests. tests.
he safe safety ty of la ge scal scal st uctu uctu su ffsh ffsh re stru struct ctur ures es fo nucl nuclea ea powe powe nd lo g-sp g-sp ridg ridg depe depe
in this this pr di tion tion
th
olve olve th tw inte inte rela relate te
comp comple le ivil ivil ngin nginee ee in il latf latfor orms ms onta ontain inme ment nt plan plants ts high high-r -ris is buil buildi ding ngs, s, ds th esig esigne ne 's abil abilit it
esig esigne ne typi typi ally ally
ta ks
on pt l-
et mini mini
ly appl applie ie load load (ele (eleme ment nt anal analys ysis is). ). urin urin th last last ye rs te hniq hnique ue have have be th ir po
tu no
an
el ance ance
is rein reinfo forc rced ed conc concre rete te elem elemen ents ts subj subjec ecte te to in-p in-pla lane ne shea shea
of in-p in-pla lane ne stre stress sses es di ting ting th espo espo se
th
th simp simple le
info inforc rc
load loads, s, ne
ra ks ma fo m, prepre- xist xistin in
rein rein or in
ba s.
on
ck ma
tr evel evel
he st ss
in th
ei forc forcin in
rs
Unfo Unfort rtun unat atel ely, y, th mode models ls
an lysi lysi proc proc du no th ompu omputa tati tion on av il le to th stru stru tu al engi engi ee
CI JO RNAL RNAL
ho
memb membra rane ne elem elemen ents ts
po
Rece Receiv ived ed July July 29 1985 1985 an revi review ewed ed unde unde Inst Instit itut ut publ public icat atio io poli polici cies es Copy Copyri righ gh 1986 1986 Amer Americ ican an Conc Concre rete te Inst Instit itut ute. e. Al righ rights ts rese reserv rved ed incl includ udin in th maki making ng of copi copies es unle unless ss perm permis issi sion on is obta obtain ined ed from from th copy copyri righ gh prop propri ri etor etors. s. Pert Pertin inen en disc discus ussi sion on will will be publ publis ishe he in th Janu Januar aryy-Fe Febr brua uary ry 1987ACI 1987ACI JOUR JOURNA NALf Lf rece receiv ived ed by Oct. Oct. I, 1986 1986
219
panels. It fo
AC memb Fran ecchio is an ssistant rofe so in th Departme Civi Engineerin at th University of Toronto. Prio to joinin th university he wa structural research ngin er with Ontari Hy ro wh re he wa in volved in research relate to th analysis an design of reinforced concrete nu learpowe plan st uctu s. He is memb of AC Committe 435. Deflections of Structures
fo re
Michae P. COllins.FACI is professo in th Department of Civi Engineer in at th University oronto He ischai ma of oint CI ASCE Commit tee 445. Shea an Torsion; chairman of th Canadian Standard Associatio (CSA Technica Committee Canadian 474. Concrete Offshore Structures delegate to Comite Euro-International du Beton; member of CS Technica Committe A23.3. Reinforced Concrete Design an member of AC Commit tee 358. Concrete Guideways. an E90l Scholarships an of Subcommittee 3/8E Shea nd To ion. He ha ct as on ultant on th sh ar design Condeep concrete offshore platforms
re
structures.
ff
re
re
rm fi fi
theory9.10 mi ting shea an co pres io
at th co tact locati ns r,
material with it ow stress-strai characteristics. Equi ibri m, compatibilit an tress-strain relationship
re fu
fo
re twee th crac s, an
cret
rc anel whic were te te in this in esti ation. Fo
cracke
rtio
Loading ratios PVl
1:0:0
PV4 PV5
1:0:0 1:0:0
Longitudinal steel
Transverse
o,
/y" MPa
0.0018
428
a;
1:0:0 1:0:0
PVI2
1:0:0
MPa
MPa
662
-0.0023 -0.0023
PV14
1:0:0
PV16 PV17 PVI8
1:0:0 0:-1:0 1:0:0
0.0074
1:0:0
PV25 PV27 PV28 PV29 PV30
1:0.32:0.32
concrete.
of
reinforced co cret structure. It
V('~'
MPa
v"
MPa
Failur f:>./f.y_~
strain
"/f" 1.04 0.43
1.79
-29.8
255
0.0179 0.0074
255
-0.0020
0.0074 0.0074
255 255
-0.0020 -0.0020
458
-20.4
420
0.0179
466
0.93 0.37
0.14 (>3.04
-18.6
Edge failur Stee brittl
fracture
Edge failur
8.56
299
0.0152
0.48 0.10 0.21
Comments
0.23
0.84
235 248
f/C_',.
1.38
-0.0027
0.0089 PV21 PV22
-26.6 - 2
0.0179
erifie
stee
0.0179 PV8 PV9 PVIO
empl ys ex erimentall
4.12
Edge failur
Explosiv failur Edge failur
0.46
4.26 -0.0018
-19.5
2.35 -0.05 0.13
463
2.00
5.41
483 324
1.66
5.80
-0.0019
3.59
0.81
-0.03 0.17
0.37 1.47 0.53
0.92
-19.1
'Precracke in biaxia tension. 'Value of Note: MPa 14 psi.
220
AC JOURNA
March-Apri 1986
loading
Fig.
Deformation
Membrane elemen
longitudinal id with th
an transverse (y) ei forc me dire tions.
(x)
oads ctin on 'Y
unifor stress
axia stresses fx and J; XY
strains
Ex
and
'Yxv'
fy
in-plane stresses fx and plan strain and 'YXY' followin additional assumption
of loadin
isto
io is signific nt
ve ge lu en ta ve enough to includ severa cracks
i. .,
il
ot
as
tr ted. istanc
larg
verall slip).
re ni orml
st ai
will be made
istribut
ve th el
il be take
ent. ll Fig.
negati e.
Compatibilit
conditions fo
If
OMPATI ILIT
cracke elemen fy,
and
'Yxy
are
CONDIT ON
th mation xperienc
qu strain
ha ge in st
th
on rete mu
at ed by
iv
conc et
om tr 'Yxy
st in
th surroundin
om it
in lu <,
lan8
(3
en (1)
and
and (2)
AC JOURNA
221
where f] ri cipa compressiv strain
111!
~~
....
CONDITIONS
Th forces ap lied to th reinforced co cret elemen ar resi te by stresses in th co cret an stresses in th
-~=~=~=~=F··""'····"--"""'''''' II
II
fx
EQUILIBRIU
:---.---1.....
1,=1,=1,=1
f2
II
t""""+
~II= = ~ II =~=~=t=:o=3--"""'~"" II
....
1 == 1
1 = 4 ]= =
Ignoring th smal reductio
in concrete cross-sectiona
becomes
L.
Fig.
ree- dy di ra
part of el
(7)
en
(8) (9)
and (lO)
Assuming that Ve
Vc
CXy
re (a Av ra Co cr te Stresses
(b) Principa Stress in Concrete
fex>ey, and Fig.
cxy
ar known.
yields th followin useful relationship (11)
-
CY
(12)
and (13)
STRESS·STRAI
RELATIONSHIP
re
(c
Fig.
r'
Ci cl
fo Av
cr
Stresses in cracke concrete
tr ss
fe ro loca tres -l ca strain relati de er ined fr stan ar ateria test Furt er re th averag stre s-av ra fo th co cret will ot co letely ndepen en of each simplicity of th model.
AC JOURNA
March-Apri 1986
in th
reinforcement.
It to
re-
E,
f"
tr ss st ai
E,
el ti nshi
fo
ei forc
en
f, (16)
Vn
\X
(17)
..
EXPERIMENTAL
ROGRAM
1).
Fig.
[0
in.).
Fig.
ACI JOURNAL
223
6.--..----------.
8.-----------.---,
AV
AG NS
OF
AG ONCR
RA
O~~~~----+H~~
prin ip st in xe concrete coincide
nd th
ri ip
st ss xe fo th fe
Co cr te St (millistrains)
Fig. circlesfo
ircl
compressive strain E2 pa tensil strain EI' Thus cracke concrete subjecte to
St (MPe)
xperimentall determined trai nd stress Specimen 26 (1 MPa 145 psi) suggeste is
.....
~, .:.~
<::>
ui Ul
..
f0-
Ul
2,
(ISa)
where
., ....
a:
2, l_
ma
fc2max
ij
(I8b)
t/t/
Ul
a:
a. ::
increasing
a.
t; wil reduc
max
a. LL
show decreasing values of fe i=
-e
cracking (i.e.,
::::i
with increasing values of
is (19)
where 2f:
directio
gested afte cracking (i.e.,
with principa compressiv strain directio
fa .,/200
I" Jcl
th reinforced concrete Re re gram.
11 gi
(fx,
a)
is
EI
(20)
/y, and
ul detail of th
and xperimenta
'YXy).
ro
nd tr sv se di tion nd fr th su st ss st in hara te isti of th in or ment si th se
be us th ulti at ca ci nowi
th
ppli
he
st ss
ting on th
le ent,
tr smit tensio
os th cr ks
It
dete mine le to th
of
lationship linkin th conc te st ss ci nc et strain ircl AC
OURN
bi iall stress
le
t::~ .
-
;'
"~t
~ 1 " " " "
c:l
/. cf.df.~
(8
S tr en -S tr ai n
l at i n sh i
fo
C ra ck e
'o
Co cr te
in Compressio
1.4
10'
(d
T hr ee - i me ns io na l S ir en -S tr ai n
',/' (c
C or re l t io n
es
at
fo
A ep r. s. nt at io n
o f C o p re sa iv e
e la ti on sh i
1.0 C ra ck e
Co cr te
In Oompreeetcn
0.
\~':1:.: .••
0.
C1
" -
0.
1+/200(;,
:.~~-
....
~. .. .......
-,....___
0.2
°O~~L_~-~-~-~O-*-~,'~~,6· (.
(e
A .v er a
S tr .s s- St ra i
C r . ck e
Co cr te
l at i n sh i In
(f C or re l t io n
lo
o f T es t
(x 10
ta fo
Cr ck
Co cr te
in Te si
si
Fig. 11 it
2, Thes
stresses ci pressive stresses
shea Psx( xe
fe
Isx)
fe
across th crac and vx are
ro ,x
rc
sinO
in
re
Ps
(26)
and p,J,ycos()
!cICOSO p ,J ,
COs
(27)
d CO s
(2 ). In this case eq ilibri (23)
AC JOURNA
March-Apri 1986
will requir shea
tresse
Vet'
the
je
0.18
CD,'
nmax Velmax
where
cimax
(a
Stress
Applie
to Cr ck
0.31
24 w/(a
16)
(29)
El
It can sycr
S8;
(30) where (31)
-
(b) Calculate
Average
•••
(c) Loca Stresse
Stresses •••
Fig. 12 calculate averag stresses
••
·v···'· ••
•• ..
••
0.4
I.Ii
Walraven's
0.
13
aggregate interlock 226
Fig. 14 ip et cros cr ck an co pressive stress on crac
perl enta
xy
.•
POints
(MPa) Calculate
Fig. 15
is
respo se
ca
145 psi)
erve
e-
and Smy contro characteristic of th x-reinforcemen an th yreinforcement, respectively. Smx
stresses
Vci
andfei
in
Fig. 16
'.10'
8, deg
Specimen PV20 afte failur
r:
I,~
0.50
SOLUTION TECHNIQU
ir
1.00
42.0
73
1.50
41.6
107
roce ur to
lcul te th st ss
0.89
3.00 5.00
ncrete
ross th
10'
Remarks
7.50
ra ks
296.9
3.03 3.95
0.37
4.55
0.45
189
2.29
407
8.80
37.9
is th in th
v.,
Veil
I:
1.65
198
ason ly
r:
1.33
principa tensil stress fe tion is possible
is
xy
MPa
305
36.3 ee
Concrete crushing
14 psi.
INTERACT ON
DIAGRAMS
ally loaded element.
Smx
singth soluti
Smy
verage f:
at ia properti
J;,. The of th four pe im ns re fyx fyy 477
1.47 MP (210 psi) proc dure outlined in th pp dix, th x- nd -r in or em nt
lo ds or th sile tr in
high
.007 th
ou sp im
th
test
ve ge te sile tr ss in (1)
shea failure.
AC JOURNA
nd havi
March-Apri 1986
ig
ia ia te sion
ieldin
XYMPa)
10
0.
0.1
Cr ckin
Fig. 17 Shea strength-axial strength interactio gram MPa 145 psi)
0.0179
sx
lo
diasy
Fig. 19 forcemen
0.
yy If~
Shea strength variatio only is increased
as transver
rein
Stee yields 0. Vu
0.1
to th
Fig. 18 Shea strength variatio as both longitudinal an transverse reinforcemen ar increase it
oncr te ailing om ressiv st esse :, levels failur is cont olle by reaching
acki
lo
ould be un ns rvativ
ic in material properties andfyx
(-
on id rtr in easing hear stre gth.
ieldin
of th longitudin fe
SHEA
Approximatel
STRENGTH
two-thirds of th specimen describe x- and y-
ment ratios influenc shea strength nd
ent.
27 th
moun of tr ns rs rein orcement wa
he predicte
strength we
base
th
ollo
ment.
Th odifie ompr ssio -field th or is pabl predicting th response of reinforced concrete elements to in-plane sh nd xi stress by on id ring equi stress st ai relationships, ll xpress in term of av er ge stress nd er ge st ins. Consider tion is ls lo io
corpor te reinforcement
(Psx
om ressiv
stress prin ip
fe/fyx), schematicall
228
or prin ip
summarized in Fig. 20
AC JOURNA
March-Apri 1986
Reinforcement
Cracked
Reinforced
Concrete
Concrete
'"
.2 'C
c:
ACKNOWLEDGMENTS
'C
c:
'"
'"
Ultimate Yield
'-Cracking
NOTATION E, E,
/:.'
modu us
of elastici
(negativ
quantity
of einforce en
membrane elements
.,
L. stress in conc et
x-direct on
averag
stress in x-reinforcemen
averag
stress in y-reinforcemen
juu j.,
1,", j. j..
yiel
stress of x-reinforcemen
j"
yiel
stress of y-reinforcemen IJ
s.;
x-rein forcement
s.:
aver ge
pacing of cracks pe pendicular
to
the y-reinforce-
ment V"m~
v" axes
to
'Y
shea
stress on x-reinforcemen
shea
stress on y-reinforcemen
crac
widt
v.
v"
x,
t,
AC JOURNA
axes
Fig. 21
March-Apri 1986
Th
hell elemen te te
')'x,
(J
0,
»;
strain in concrete cylinder at peak stress j: (negative quantity) strain in concrete at cracking strain in oncret in x-direction st in in oncret in y-direction strain in reinforcin stee in x-directio strain in reinforcin stee in y-directio strain in x-direction strain in y-direction yield strain of x-reinforcemen yield strain of y-reinforcemen shea strain relative to x, axes angl of inclinatio of principa strain to x-axis ngle of nclination of prin ip stresses oncr te to xaxis reinforcemen rati fo reinforcin stee in x-directio reinforcemen rati fo reinforcin stee in y-directio
STRESSED ELEMENTS to find th relationship betwee shea stress v. an th resultin shea strain )'", Fo simplicity assume no prestresse reinforcement. Step 1- Determin th crac contro characteristic of th x-rein equations, or s.; nc s.; 1.5 maximu distance from y-bars Step f, tions. Step Estimate principa compresssive stress directio 0. Step Calculat averag crac widt usin q. 31 nd 30). Step Estimate averag stress in weaker reinforcement; assume that this is th y-reinforcement. Hence, estimate f,. Step Calculat averag tensio in th concrete j., usin Eq (19) an 20 subj to th ondition ha
(O.IS
"m"
REFERENCES
ethod, 3r Zienki ic O. ., he Finite Elem nt Gr w- il Book o. ew ork, 1977,787 pp El Edition, CM Publications Sout hampton, 1983 50 pp og
Edition,
where (29). Step
Conc te St uctures,
P"
0; an
U;
fJ
here v"',,,. is give by q.
Calculat sh ar st es v. ro
quilib iu
2nd
et
orsk
Step Step Step 10
j. from equilibriu usin Eq (13) ul f, usin Eq (18) Chec that .f/f.,,,,, ,;;; 1.0. If )c
Step
Calculate
de sels an Containments," ur Sectio 111,Divisio 2, American Societ of Mechanical Engineers, ew Yo k, 1983,376 pp "B or or ed Concrete CI 318-83 ," Americ Conc te nstitute e7. upta ., "Membran Rein or em nt in Conc te hells: Review," Nuclea Engineerin an Design Elsevier Scienc Publishers, Amsterdam, 1984 pp 63-75.
fYtan()
V".
sis," Report No 68-91, Department of Civi Enginering Massachu sett Institut of Technology Cambridge, Nov. 1968,226 pp Structures 1977 Appendix Veritas, Oslo 1980 22 pp
lanO
0.3k')
,.
I-
usin Eq (ISb)
Step
from geom tr usin
J.;
tan'()
f,
of crete Panels," Canadian Journa oj Civi Engineerin (Ottawa) V. 19 9. itch ll enis nd Collins, ichael ., "D agonal Co pr ssion Fi ld heor -A Ration od or St uctura oncret in Pu
q. 5)
lan'O Step Step I4
j.
he
usin q. (15) calculated agrees with estimate f,. If not,
JOURNAL,
396-408. Member in Shear, pp 649-666.
Step Step Step
Proceedings,
from geom tr usin
q, 4)
qu
J.,
oncret to In Pl ne Sh ar nd orma St sses," Publication No. 82-03, Department of Civi Engineering, University of Toronto, Mar. 1982 33 pp 12 al ven, oo ., "Funda enta nalysi of gg eg te In terlock," Proceedings, 2270. onally Reinforced Wall Subjecte to In-Plane Shea Force-Effecve Proceedings, nnua ting rchitectur nstitute of Japan, okoh ma Oct. 1984, pp 1807-1809. hD th sis, ep tmen of Civi bury Christchurch 1985
ng ne ring Univ sity of
Step
f/, (J
Step
j", If not, retu Increasing ()increases /,. v" and
j.,)
fj,f,
anterIf
0, th
v,
AC JOURNA
.Go
March-Apri
1986
If Il/"
tan8
0.18
Step
V"m.x
V,.,
If
;0
fw
Step
and
0.82/v"m~
b-
"8
'Y..
v,
f,
shea
Il /,,)/tanO
(f..,
I.
(f..,
If at ai ure: i. j,
AC JOURNA
+ [
increasing
f>
nd };'" ii.
v"tanli) o;
less th
is obtained
concrete
f,m,,,
governs.
iii. L, (f.,
from geometry
4AC)/2A
f",.
In
vv-
f,)/ranO
..
1.64
an
fm'
ay ex ee
v;
Otherwise
f",
and f" were calculated
f..
exceed
.. then yielding
v,/tanO/p"
March-April 1986
231
