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Collapse loads of reinforced concrete cylindrical water tanks using limit analysis approach
comprers
& s1nlctwes
Vol. 48, No. 2, pp. 20.5-211.
1993 0
F’rinted in Great Britain.
0045-7949~93 s6.00 + 0.00 1993 Pergamon Press Ltd
COLLAPSE LOADS OF REINFORCED CONCRETE CYLINDRICAL WATER TANKS USING LIMIT ANALYSIS APPROACH K. RAMANJANEYULU,S. GOPALAKR~SHNAN and T. V. S. R. APPA RAO Structural Engineering Research Centre, Madras-600 113, India (Received 30 April 1992) Abstract-Evaluation of the margin of safety against collapse of reinforced concrete cylindrical water tanks requires knowledge about the load carrying capacity of the tanks. The limit analysis procedure can be used to estimate the collapse loads of cylindrical tanks. In this paper, determination of the collapse loads of reinforced concrete circular cylindrical water tanks, having varying reinforcement along the height, by applying the limit analysis approach is discussed. The relevant mathematical formulations and the details of the computer program TANK developed for estimating the collapse loads of short, medium height and long cylindrical tanks are presented. The formulations presented are general in nature and different distributions of hoop reinforcement along the height of the tank wall can be considered. The results obtained using this program are found to be in good agreement with those reported in the literature. Numerical results for example problems with different hoop reinforcement distributions are given.
NOTATION distance from the origin total height of the tank radius of the tank thickness of the tank wall non-dimensional length parameter (=4 L’/Rt) compressive yield stress of concrete tensile yield stress of concrete yield stress of steel reinforcement per unit width in circumferential direction reinforcement per unit width in longitudinal direction percentage reinforcement in circumferential direction (= A&) percentage reinforcement in longitudinal direction (= Ax/t) non-dimensional length ( = x/L) reduced generalized stress resultants maximum positive moment capacity maximum negative moment capacity bending moment for ith region shear force for ith region
non-dimensional pressure
1. INTRODUCTION For the limit state design of reinforced concrete liquid retaining structures it is necessary to know load carrying capacities. The collapse load cannot be determined through elastic analysis. The use of nonlinear finite element analysis to model the post-cracking behaviour and to trace the complete load deflection response of the structure up to failure is computationally intensive. Hence, there is a need for the development of a simple and computationally efficient procedure to determine the collapse loads of reinforced concrete (RC) cylindrical tanks. The limit
analysis approach can be used for this purpose. In this paper a general formulation for limit analysis of cylindrical water tanks is presented. Assuming linearly varying hoop tension capacity distribution along the height, Sawczuk and Olszak [ 1] have derived equations to estimate the collapse loads of short and long water tanks. For the case of medium height tanks, in order to avoid the mathematical rigour involved in the derivations, Sawczuk and Olszak [1] assumed uniform hoop tension capacity distribution along the height of the tank wall. In practice, hoop reinforcement along the height of a circula.. cylindrical water tank is usually provided in a stepped pattern. The equations derived by Sawczuk and Olszak [l] cannot be directly applied to estimate the collapse loads of cylindrical tanks with stepped distribution of reinforcement. Depending on the type of reinforcement distribution, the governing equations need to be derived afresh. This problem can be circumvented if the governing equations for the determination of the collapse load are based on the generalized distribution of hoop reinforcement. In this paper, based on the limit analysis approach, a new set of general formulations for the estimation of collapse loads of cylindrical water tanks are presented. By using certain control parameters any type of reinforcement pattern along the tank wall can be considered without the need to change the basic formulation. The two cases, the uniform and linearly varying hoop tension capacity distributions, used by Sawczuk and Olszak are special cases of the general distribution for which the formulations are presented in this paper. Based on these formulations, a computer program named TANK has been developed. 205
206
K.
RAMANJANEYULU ef al.
Using TANK, the collapse pressure and the factor of safety against the collapse of a circular cylindrical tank of given con~guration can be obtained. Results
of parametric studies conducted to evaluate the coilapse pressure of cylindrical tanks of different sizes and hoop tension capacities are also presented in non-dimensional form. 2. METHODOLOGY FOR LIMIT ANALYSIS REINFORCED CONCRETE CIRCULAR CYLINDRICAL TANKS
OF
The procedure for estimating the collapse of a reinforced concrete cylindrical tank consists of the following steps:
1. Establishing the relationship between material strength and the critical combination of internal forces which the material has to withstand. 2. Arriving at the yield condition and the flow rule. 3. Solution of the governing equilibrium equations using the prescribed stress boundary conditions and the above relations. 2.1. G~uer~jng equilibria
eq~~~io~
As the geometry and the internal pressure are axisymmetric, the yield condition for a circular cylindrical tank can be expressed in terms of three stress resultants: M, (longitudinal bending moment), N, and N, (membrane normal stress resultants). The sign convention for these stress resultants (positive direction) is shown in Fig. 1. It may be noted that due to rotational symmetry, there is no circumferential displacement and the ci~umferential curvature rate (&) vanishes. Thus, the moment in the circumferential direction, M,,, is zero. The equilibrium equation of a cylindrical shell subjected to axisymmetric loading is given by dZM
N,
d.---p,=& dx= R
Fig. 2. Failure criterion for concrete in plane stress.
Equation (I) can be expressed in terms of nondimensional parameters as follows: 1 d2m -L-.n,-p=o, C2 dc2
where <=x/L P =pxRIN, m, = M, IMP no = N%lN~ nx = NxIN, Np =f,t MP =fct2j4
N L=
c==.__L-
4L2 = Rt = length parameter
Rjw, and p. m, and n, are the non-dimensional parameters for pressure, longitudinal bending moment and membrane normal stress resultant in the circumferential direction, respectively. N,, and M, are the compressive and bending resistance of the tank wall at collapse, respectively. Any distribution of m, and n, which satisfies this equilibrium equation and the values of which do not exceed the corresponding capacities at any section are statically admissible. 2.2. Failure criterion The failure criterion, given by Sawczuk and Olszak [I] for the circular cylindrical shell under axisymmetric loading, is adopted in this study. The failure condition for concrete in plane-stress (Fig. 2), assuming the compressive stresses as positive, is given by
Fig. 1. Coordinate system for cylindrical tank
Collapse loads of reinfonxd ccwrete water tanks
201
rectangle for shdts without
Fig. 3. Yield surface for reinforced concrete. cylindrical shells [3].
The yield condition given by
for the reinforcement
a=-fy,
a
is
‘-a6
?I, -
+/&=O
03)
1 = 0,
(9)
where
‘*
m,(l + a) + 2nI + 2n,(28cl, + a - 1) 4/&-2a
n,+a
and compression
00
Considering the longitudinal (x) and circumferential (0) directions as the principal directions, and assuming compressive forces and stresses as positive, the yield surface (failure criterion), as shown in Fig. 3, is formed by two parabolic cylinders [l-4] which are given below
+2/+lI-
the tangential forces in tension respectively, given by
(4)
The collapse of an element can occur only if both eqns (3) and (4) are satisfied. The instantaneous deformation mode is given by
ac.-Z
Fig. 4. Simplified failure criterion.
10,
for
m,> 0
(6)
From eqns (6-9) it can be seen that the yield locus is dependent on the percentages of reinforcement in the circumferential and longitudinal directions. Hence, the collapse load also depends on these parameters. If the longitudinal force, n, , is small in comparison with the other stress resultants, the analysis can be restricted to the m,-n, plane. The intersection of the surfaces given by eqns (6) and (7) with the plane n, = 0 results in the simplified yield criterion shown in Fig. 4.
2.3. Hoop tension capacity distribution -m,(l
+a)+2nt+2nX(a
- I)-2a for
-0, m,
(7)
..... b
and these two parabolic cylinders are limited by two parallel planes, represented by the critical values of
T
c
Cow (0
The common practice of curtailing the reinforcement along the height of the tank wall leads to the stepped distribution for hoop tension capacity. Six possible distributions for hoop tension capacity are shown in Fig. 5. The distribution shown in Fig. 5, case (i), is expressed as below.
car. (ii)
Fig. 5. Different hoop-tension capacity distributions.
208
K.
RAMANJANEWLU
For region I
et al.
obtained from the distribution (Fig. 5) as follows: ng = n,, + k, t.
(10) a
For region II no =
shown for case (i)
(11)
n21 + k2(< -0
For region III
:b: (4
~11,
no = njl + kAt -(B +
(12) (d)
where (e)
kI = (n12- n,, )/B
when k, = 0 and n,, = n,2 = n2,, the distribution shown in case (ii) can be obtained; whenk,=O,k,=O,k,=Oandn,,=n,,=n,,= n,,; n22 # n,, , n,, = nj2, the distribution shown in case (iii) can be obtained; when k, = 0, n,, = q2, n,2 # n2,, k2 = 0, n2, = n22, n22Z n31 and k, = 0, n3, = n,,; the distribution shown in case (iv) can be obtained; when k, = k2 = k, = 0 and n,, = n,2 = n2, = nz2 = 4, = n32Tthe distribution shown in case (v) can be obtained; when k, = k,= k,,n,, =n,, and n,,=n,,, the distribution shown in case (vi) can be obtained.
k2 = (n22- n2,)/Y 2.4. Classification
of cylindrical corresponding collapse mechanisms
k3 = (n32- n3, Ml - (B + Y11.
, \ I T L=T \ / il-l
The hoop tension capacity distributions shown against other cases [Fig. 5, cases (ii)-( can be
tanks
Based on the mode of failure at collapse, the cylindrical water tanks are classified into three categories: shallow (short), medium height and long
Region-l
,
:
I
I\ \ i \ \
L
T
Zl
/z i c,
\
/i
g=1
(ii)
0)
(iii) Mu
(a)
’
/
J
t
z
\
?
/’
\
and
(iI
+J!??:’ 1 n, .. g.’ p f
IL ’ mbwlt
(ii)
v.. I
MU\
i;iu
Y
“22 . . . ,. .
\
3,
tq
t% (iii)
Mu
(C) Fig. 6. Collapse mdanim (i), hoop-tension capacity distribution (ii), and bending moment distribution (iii) for (a) short tanks, (b) medium height tanks and (c) long tanks.
Collapse loads of reinforced concrete water tanks
209
tanks. The mode of failure depends upon the value of the non-dimensional length parameter (C’). For shallow (short) cylindrical tanks fixed at the bottom and free at the top, only one plastic hinge circle will develop at the base (Fig. 6a). For medium height cylindrical tanks, the top portion of the tank will be in a state of hoop compression. Two hinge circles form in this type (Fig. 6b). For long tanks, partial failure will occur. This type of failure will be confined to the base region. In this type, two positive hinge circles and a negative hinge circle will be formed (Fig. 6c).
By integrating eqn (14) successively and using the boundary conditions, the equations for shear force and the bending moment are obtained as
2.5. Mathematical formulations for the tanks with general hoop tension capacity distribution [Fig. 5, case (i)]
F
$x,= -(n,,t +k,;)+p; -(n,,~+k,~)+p~.
&mxI=
Region II. Substituting eqn (11) into eqn (13)
1
the value of n, from
=
(17)
(t: -lv2
2
2.51. Short cylindrical tanks. For the case of short cylindrical tanks the collapse mode and the bending moment distribution are shown in Fig. 6(a). The origin of the coordinate system is at the top of the tank. The shell is clamped at r = 1 and free at r =o. The equation of equilibrium for the cylindrical shell subjected to hydrostatic pressure can be expressed as follows:
=()*
C2 dC2
The boundary as: (i) mxl =0
and continuity
1 1 =
342
(19) Using the conditions n&r= tix2 and mx, = mx2 at { = fi, the values of C, and C, are obtained as
P2 G = -(n2, -n,,)y+k,-.
conditions
are given
&, = 0
at
1 -@ix,=
--[n,,+k,{<
n3,t
1 =
-
n31
-0
and
(iii) m,, = m,,
and
(iv) m,,=m,
at
ti,, = ti,, +I,, = ti,,
+k, It -(B+Y)12 2
{e -(B+vW 6
C+k, 2
1 F %I = -h
(20)
1
+pc+C5 2 (21)
1
r =O, at
at
(22)
4 = (Y + B),
By using the conditions tirx2= &, and rn,, = m,, at t = (B + y), the values of C, and C, are obtained as
C5= (4,
-n21)(B+y)-k,~-kl~+(n2,
the value of n, from eqn (10)
G= -h, +k,O+pt.
+c,.
{ = /?,
c = 1.
Region I. Substituting into eqn (13)
3
+r)}l+pt
+p;+c,e: (ii) m,, = m,,
8’
Region III. Substituting the values of no from eqn (12) into (13)
3mx3
and
n2,~+k2~]+p~+C,C+C,.
-
L&3= c2
2
Id-nn,_P~
(18)
+pf+C,
2
Mathematical formulations for limit analysis of short, medium height and long cylindrical tanks, fixed at the base and free at the top, with the general hoop tension capacity distribution [Fig. 5, case (i)] along the height are presented in the following sections. Depending on the values of n,,, . . . , n32 this general hoop tension capacity distribution degenerates to particular cases (ii)shown in Fig. 5.
(16)
-In,, +k2(t -B)]+pt:
I
42
(15)
(14)
_n
jP+r)’ ; ,j+b 8’
,(B+v) - 2
-k,g--(n,,
-n,,)f.
k
21
2
K.
210
RAMANJANEYULU
Substitution of tn,, = m, (i.e. maximum positive moment capacity), at e = 1, in eqn (22) yields
[l - (B + r)13 6
(B+r)* +@3,
+k,~_k,Y’?!?d
-n2,)2
2 8*
+
@21-
4,)
The boundary Fig. 6a) gives
B’ -
k,
Ti-
condition
0.X:(--k,+p) +
y
(n2,
-
t,h21-
-
n,,)/.I
*
+
8’
k, ; + k, T
.
(23)
A,, = 0 at [ = 4, (see
et al.
When mX2 is greater than t&. (maximum negative moment capacity) at C:= e,, then the solution obtained using mechanism applicable for short tanks is not valid. The flow chart of the program TANK outlining the procedure to compute the collapse pressure for the case of short tank, is given in Fig. 7(a). 2.5.2. Medium height cyrindrical tanks. The details of collapse mechanism and bending moment distribution for medium height cylindrical tanks are shown in Fig. 6(b). The origin of the coordinate system is at the top of the tank. The shell is clamped at 5 = 1 and free at 6 = 0. Top portion: ne = + 1 (top portion is in hoop compression). Substituting the value of n, into eqn (13)
k2f-9
-
O.S/?*(k,+ k,) = 0.
Read
Height,
,unit
weight
Radius, of fluid
(24)
Thickness, in the
tank
‘“22’“31’“32’$,1/ 1
K
Compute
I
8 mu
Considering short tank collapse mode, compute non-dimensional pressure ( P 1 -Eqn. (23)
1
I
Compute negative
the location (ct) moment (rn821
-1
of maxImum -- Eqn. (24)
1 Compute
mx2 at r, Short not
tank
valid.
case
is
Hence
medium
height
tank
collapse
mode
is
examined.
Print and
eollap*e safety
pressure factor A
Fig. 7(a). Flow chart of the computer program TANK: for the case of a short tank.
Collapse loads of reinforced concrete water tanks By integrating eqn (25) successively and substituting the boundary conditions, the shear force and bending moment are obtained as
211
1 = -[n,,;+k,;]+P;+C,( F mxl
+c*.
(30)
By making use of conditions tifiOp= ti,, , mnop = m,, at [ = &,. the constants C, and C, are obtained
1 . 3 %oP
(26)
as
C, = &(l + n,,) + k, f
Region I: n, = nI, + k, r. Substituting n, into eqn (13)
(l+n,,) c,= -c+---
the value of
1 .. eqn (28) successively
~~~,=-[n,,e+k,~]+~~+c,
’ 3’
Region ZZ
(28) BY integrating
3
k
(29)
= -[n,,+k,(c
;:I=
-B)I+p5
-[nz,<+kzy]+pg+c3
Q A
Assume
Compute
go= 0
1 non -dimensional - -
pressure
(p)
Eqn.( 37) 1
Calculate
negative
hinge
&
-
Eqn.
(38)
&
(new
value
ofgo
known
p and
Compute
-
gf
--
circle
Eqn.
location
) for
the
(39)
Yes Compute
the
location
(ET)
of maximum
Fig. 7(b). Flow chart of the computer program TANK: for the case of a medium height tank.
ill:
212
K. 1
-
Fmx2=
[
RAMANJANEYULU et
n2,;+k2q]
al.
The boundary
condition
ti,, = 0 at 5 = 5, gives
0.55:(P -kd-t,(n2,
-kd)+(n~,
-n,,)B
- 0.5k,j2 - 0.5k2B2 + c,(l + nl,)
The continuity conditions tix, = fix2 and m,, = m,, at + 0.5k, 1;; = 0.
5 = /I gives
Using the boundary
Region III
+k
i&3 =C2
n3,5 + k,
-(B +r)Jl+~t {C -(B
+v)12
2
zmx3=
-
n3, ‘2+k, 2
It
6
using
mx2=mx3
continuity at
-
Kn2,-
nil
M - O.% 821t, (39)
By solving the non-linear eqns (37)-(39) iteratively as shown in Figs 7(b) and 7(c), the values of to, 5, and p are computed.
9 B
(36)
tix2 = tix3 and
t
I
I
5 =(B+Y)
C5=(n3,-n,,)(B
+y)--k2g-k,T
+tn,,
c,=
conditions
5 (35)
1
+P;+C,t+Cc+
By
6
-k,~+0.5(n,,-n,,)~2-k,~=0.
+$+, 2
r)>’
-
6
(34)
1
- (B +
1
condition mx2 = ni, at c = r,
G,-B13 PC 2~
&+h3= -[n3,+k3{t
(38)
-n,,M+C,U
+n,,)+k,+
-(n3,-n2,)~+k,~+k2y2~ 2
-k2$-(n2,-n,,)-i--_i--
P2
(l+n,,)
2
of m,, = mu at 5 = 1, in eqn
Substitution yields
I
r
I
I
k 5
’ 3’ (36),
I
t Compute
1 1, f’, f”
I
I
P - (B + r113 6 - (n3,- n2,W + Y) + 0.5k2y2 + 0.5k,p2 -
tn2,- h)B + O.%,, - nzl)tB + r12
+
k,;- 0.5k2y2(B + y) -
+C’tl+nll)+k 0-
2
G ‘3, 1
P
DIFF = &,,, - go
OSk, t;
Fig. 7(c). Flow chart for the solution of a cubic non-linear equation.
Collapse loads of reinforced concrete water tanks Boundary condition
1
ti,r = 0 at r = & gives
@x3
-nr,&
- 0.5M1;2 - I-V + 0.5& +
b2,
-
=
-n32t
=
-n32
+n,,)+0.5k,ti=0.
(40)
When the moment at < = r2 reaches m,, this mechanism is no longer valid, only partial failure occurs, which is the case for long tank. 2.5.3. Long cylindrical tanks. The collapse mode and the bending moment distribution for long cylindrical tanks are shown in Fig. 6(c). In this case, it is convenient to choose the origin of coordinate system at the bottom of the tank. The bending moment diagram between c = r2 and 5 = 1 is continued in a statically admissible manner. The hoop tension capacity distribution for this case is expressed as below. Region I ne =
42
-
W
-
(B,
+
(43)
+k,g+p
n,,)B 2
1 -0.5k,jP+r0(1
213
F
mx3
r+k$+p 2
Region II 1 @iX2 = -n,+k2(r
&litx2 =
-n,,t
-S,)+p(l
(45)
-r)
+ WI
1 F mx2 =
-n22
c+k 2 + c3c
~11. Using
Region II
the boundary
+ c,.
and continuity
(47)
conditions
specifiedin eqn (41), the constants are obtained as n0 = n22- k2(r - B, ).
G=&
Region III
- 0.5k2G - PIj2-P
no = n32- k, 5,
(using ti, = 0
where
Using continuity
condition
at
r = 5,).
rit,, = +I,~ at c = j3,
k, = (n,2 - n,,)l[l - (8, + r)l k2 = (n22- n2, )/Y k, = (n32- nsl )I&
-0.5k,(&-&)‘-p
B, = [l - (B + Y)I. The boundary and continuity conditions for this case are as follows:
C2 = $
(using mx3 = m,
Using continuity (i)
43
=m,
(ii)
42
=
m,
and and
(iii) mx3 = mx2 and
&,=O
at
ti,,=O
c=O
at
tiX3= tir,
(41b)
t = fl, (4lc)
(iv) m, = m, (v)
m2
=
m,,
and and
ti,, = 0
at
ti,, = ti,,
r = t2 at
r
(41d)
=Cf%+y). (414
CAS 48,2-c
-n32 + k& +p(l - 5)
mX3= mx2 at 5 = fi,
C.,=(n,,-n,,)T-k,q+$.
Using the boundary
(42)
condition
mx2 = tiiu at { = c,
p[2[; - 31;;] = -6
-
3n2,ti
- 3k25fB, + k2B: - 3(n,2 -n&j?: The condition yields
Region III 1 @ijix3=
5 = 0).
(41a)
r ={, at
condition
at
r2 = VP - %2
+
X2G
+ 2k&.
(48)
tiX2= 0 at c = e, and also at t = c2
- 2k2B, - pt,
+
M,I/(P
-k,).
(49)
K.
214
~MANJA~~LU
et al.
(a) Height (m), radius (m) and thickness (m) of the tank wall.
(b) Compressive ;;
t Compute Pr tssur*
non-dimensional ‘p’
--
Eqn.
(48)
W
i cokuwe
--
g,
Eqn. (69)
1
(0
strength of concrete cf,), yield strength of reinforcement U;). Unit weight of the fluid in the tank (kg/m3). Vertical reinforcement on the inner face in cm2/m, and on the outer face in cm*/m. Hoop reinfor~ment areas in cm’/m along the height of the wall (i.e. values for nll,n 12,n,, , n2*, n3,, t132 shown in Fig. 5). /I (height of the region I), y (height of the region II) (shown in Fig. 5).
i Compute
(&J - -
the new value Eqn. (SO)
of gl
1 OlFF *
gtn-
!&
I
Print
hfn$e
pr+ssurr
localfir, and
rcrf*ty
COltapse
factor
1
&
stop
Fig. 7(d). Flow chart of the computer program TANK: for
the case of a long tank.
Substitution
The slope parameters k,, k2 and k, (Fig. 5) of the reinforcement distribution pattern and length parameter (C2) for the cylindrical tank are computed first. Then the collapse pressure is calculated by assuming the tank to be a short tank; the applicability of the mechanism for the short tank is then checked. If this mechanism is found to be invalid, collapse pressure according to the mechanism for medium height tanks is computed and the applicability of the mechanism is checked. If the mechanism for medium height tanks is also found to be invalid, collapse pressure will he calculated according to the mechanism for long tanks. Thus, the computer program automati~lly selects the valid m~hanism for the given configuration of the tank and calculates the collapse pressure and factor of safety. The output results consist of an echo of input data, followed by the information regarding hinge positions, collapse pressure and factor of safety against collapse.
of m,. = m, at < = c2 yields 4. NUMERICAL
RESULTS AND DISCUSSION
4.1. C~~~~r~cuI tank with uniform hoop tension capacity distribution
By solving the non-linear equations (48)-(50) iteratively as shown in Fig. 7(d), the values of {,, t$ and p are computed. 3. SALIENT
FEATURES OF THE COMPUTER PROGRAM TANK
Based on the mathematical formulations presented in the previous sections, a computer program, TANK, has been developed to find the collapse pressure as well as the factor of safety against collapse of cylindrical water tanks. Flow charts for this program are shown in Figs 7(a)-(d). The input data for the program are as follows:
The mathematical formulations presented in this paper are applicable for a general variation of hoop tension capacity along the height of the tank wall. Particular cases, such as constant (uniform) hoop tension capacity and linearly varying hoop tension capacity along the height of the tank wall, can be analysed using the computer program TANK by approp~ately inputting the hoop reinfor~ent details (i.e., n,, , . . . , n3z). Sawczuk and Olszak [l] have analysed a cylindrical tank with uniform hoop tension capacity distribution along the height of the wall and have given charts for non-dimensional collapse pressures for different length parameters. Toi and Kawai [S] have analysed this problem using the discrete limit analysis and have given collapse pressures for different length parameters. For the same problem, collapse pressures for different length parameters (C*) are computed using TANK in this study. The comparison of the results obtained in the present study with those reported by
215
Collapse loads of reinforced concrete water tanks Table 1. Comparison of collapse pressures of cylindrical tank with uniform hoop tension capacity distribution rig. 5, cae (v)] with n, = m, = L@” = 1 for different length pammeWs Nondimensional collapse pressure (p) Length parameter (C3
Sawcxuk and Olszak 111
16 40
Preacnt study
3.370
3.375 2.457
2.170
2.212 2.180 2.081 2.014 1.821
2.45
z 80 100 144
2.20 2.08 2.00
1.770
pressure which would be closer to the true collapse load. The same cylindrical tank was analyzed using the computer program developed in this study. It is found that this geometry has the medium height tank collapse mode and the collapse pressure is found to be 0.218. Thus it is noted that the collapse pressure given by Sawcxuk and Olsxak [l] is on the higher side and is not the lower bound value. However, for the purpose of comparison, the same tank was also analysed in the present study by assuming it to be a long tank (partial collapse mode) and p = 0.221 is obtained. The collapse pressures obtained in the present study are compared with that of Sawcxuk and Olsxak in Table 3. For the tank considered in this section, the valid mechanism is that of the medium height tank and the collapse pressure is 0.218.
Sawcxuk and Olsxak [I] and Toi and Kawai [5] is shown in Table 1 and Figs 8 and 9. It may be noted that Sawcxuk and Olsxak have also given hinge locations for medium height tanks and these values are also compared with those obtained in the present study. From Fig. 8, it can be seen that there is good agreement between the results. 4.2. Cylindrical reinforcement
Toi and Kawai 151
tank with linearly varying hoop
Sawcxuk and Olsxak have analysed a reinforced cylindrical water tank with linearly varying reinforcement along the height of the tank wall. The geometry, material and reinforcement details of this tank are given in Table 2. Since the length parameter (C*) for this geometry was such that it did not fall within the short tank collapse mode, they considered it to have the long tank collapse mode and computed the collapse pressure as p = 0.23. It may be noted that they did not examine for the case of a medium height tank collapse mode. It is observed that only a correct collapse mode can give lower bound to the collapse
4.3. Cylindrical tank with stepped dktribution of hoop reinforcement A cylindrical tank with stepped distribution of reinforcement is analysed using the computer program TANK. Details are given below.
x
sawuuk
A
aal H
1
2s
1
I
I
35
45
IS
Length
parameter
nawai
I
I
gs Cc?= 4&Rt
and oIuak[I]
TO; and
75
[s]
- r5
I
I
85
SP
)
Fig. 8. Comparison of hinge locations and collapse pressuns for medium height hoop-tension capacity distribution.
tanks with
uniform
K.
216 0.8
et al.
RAMANJANEYLJLU
I: I
a6
El,&
I
0.4
0.2
0
b]; i lo5
115
125
135
145
155
Length ~ammrkr
165
175
165
155
205
C2 * 4L%?t
Fig. 9. Variation of hinge locations and collapse pressure for long tank with uniform hoop-tension capacity distribution.
A cylindrical tank of 500,000 litre capacity was designed according to 13:3370-1967 [6]. The dimensions and the reinforcement details are given in Table 2. The collapse pressure and the factor of safety of this tank are obtained using the computer program. The tank is found to have medium height tank collapse mode and the non-dimensional collapse pressure (p) is found to be 0.299. For a correct collapse mechanism, the collapse pressure has to be minimum. This aspect is examined by computing the collapse pressure for other two collapse modes, namely, short as well as long tank collapse modes as well. These values are given in Table 4. It may be seen
from this table that only for the case of medium height tank collapse mode, the p value is minimum and this gives the correct collapse pressure. The collapse pressure (P,) is calculated from the nondimensional collapse pressure, p, as follows: Pf=‘G
= 13.8 t/ml.
The safety factor against collapse is obtained as the ratio of the collapse pressure ;p/) to the actual pressure (height x unit weight of the fluid in the tank). In the present case, the actual pressure
Table 2. Details of the cylindrical tanks with two different patterns of reinforcement
Particulars Height Radius Thickness
Tank with linearly varying reinforcement [Fig. 5, case (vi)] 5.0 m 5.0m lO.Ocm
Tank with stepped distribution of reinforcement [Fig. 5, case (iv)] 4.0 m 6.5 m 15.ocm
f,
200 kg/cm2
200 kg/cm2
f, Hoop reinforcement
2500 kg/cm*
2500 kg/cm’
linearly varying from 3.0cm2/m at top to 14.1 cm2/m at bottom of the tank
@ 11.31 cm’/m for the top 1.2 m @ 16.16cmz/m for the next 2.0m and @ 11.31 cm*/m for bottom 0.8 m
Vertical reinforcement (outer face)
1.01 c&/m
2.513 cm*/m
Vertical reinforcement (inner face)
3.52 cm*/m
10.27 cm*/m
Collapse loads of reinforced concrete water tanks Table 3. Comparison
217
of collapse pressure of cylindrical tank with linearly varying reinforcement [Fig. 5, case (vi)] Non-dimensional
collapse pressure (p) Present study
Length parameter (C2)
Sawczuk and Olszak [l] using long tank solution
As medium height tank
As long tank
200
0.23
0.218t
0.221
t This corresponds to the correct collapse mechanism.
Table 4. Collapse pressures of cylindrical tank with stepped distribution of reinforcement
[Fig. 5, case (iv)] Nondimensional collapse pressure(p) Length parameter (C2)
As short tank case
As medium height tank case
As long tank case
65.641
0.367
0.299t
0.33
t This corresponds to the correct collapse mechanism.
out to be 4.0 t/m2. The factor of safety against collapse of this tank is calculated as F.S. = 13.814.0 = 3.45.
Acknowledgements-The authors wish to thank Shri N. V. Raman, Director, Structural Engineering Research Centre, Madras, for his constant encouragement and support in carrying out this study. This paper is published with the kind permission of the Director, S.E.R.C., Madras.
5. CONCLUDING REMARKS
REFERENCES
works
The mathematical formulations presented in this paper for the estimation of collapse loads of short, medium height and long tanks are for the general distribution of hoop reinforcement along the height of the tank wall. Different patterns of reinforcement distribution: uniform, linearly varying and stepped distributions, can be handled by suitably choosing the parameters nu, . . . , nj2. Thus, the formulations are versatile. The results obtained using the program TANK are found to be in good agreement with the results reported by Sawczuk and Olszak [l]. The computer program TANK runs on an IBM compatible PC-XT.
1. A. Sawczuk and W. Olszak, A method of limit analysis of reinforced concrete tanks. Simplified Calculation Methoa!vof Shell Structures, Proceedings of Colloquium, Brussels,pp. 416437 (1961). 2. W. Olszak and A. Sawczuk, Inelastic Behaviour in Shells. Noordhoof (1967). 3. M. Save, Limit analysis and design of containment vessels. Nuclear Engng Design 79, 343-361 (1984). 4. M. Save and C. E. Massonnet, Plastic Analysis and Design of Plates, Shells and Disks. North-Holland, Amsterdam (1972). 5. Y. Toi and T. Kawai, Discrete limit analysis of plate and shell structures. Compur. Srrucr. 19, 251-261 (1984). 6. Indian Standard code of practice for concrete structures for the storage of liquids (13:3370--1967). Indian Standards Institution, New Delhi (1967).
& s1nlctwes
Vol. 48, No. 2, pp. 20.5-211.
1993 0
F’rinted in Great Britain.
0045-7949~93 s6.00 + 0.00 1993 Pergamon Press Ltd
COLLAPSE LOADS OF REINFORCED CONCRETE CYLINDRICAL WATER TANKS USING LIMIT ANALYSIS APPROACH K. RAMANJANEYULU,S. GOPALAKR~SHNAN and T. V. S. R. APPA RAO Structural Engineering Research Centre, Madras-600 113, India (Received 30 April 1992) Abstract-Evaluation of the margin of safety against collapse of reinforced concrete cylindrical water tanks requires knowledge about the load carrying capacity of the tanks. The limit analysis procedure can be used to estimate the collapse loads of cylindrical tanks. In this paper, determination of the collapse loads of reinforced concrete circular cylindrical water tanks, having varying reinforcement along the height, by applying the limit analysis approach is discussed. The relevant mathematical formulations and the details of the computer program TANK developed for estimating the collapse loads of short, medium height and long cylindrical tanks are presented. The formulations presented are general in nature and different distributions of hoop reinforcement along the height of the tank wall can be considered. The results obtained using this program are found to be in good agreement with those reported in the literature. Numerical results for example problems with different hoop reinforcement distributions are given.
NOTATION distance from the origin total height of the tank radius of the tank thickness of the tank wall non-dimensional length parameter (=4 L’/Rt) compressive yield stress of concrete tensile yield stress of concrete yield stress of steel reinforcement per unit width in circumferential direction reinforcement per unit width in longitudinal direction percentage reinforcement in circumferential direction (= A&) percentage reinforcement in longitudinal direction (= Ax/t) non-dimensional length ( = x/L) reduced generalized stress resultants maximum positive moment capacity maximum negative moment capacity bending moment for ith region shear force for ith region
non-dimensional pressure
1. INTRODUCTION For the limit state design of reinforced concrete liquid retaining structures it is necessary to know load carrying capacities. The collapse load cannot be determined through elastic analysis. The use of nonlinear finite element analysis to model the post-cracking behaviour and to trace the complete load deflection response of the structure up to failure is computationally intensive. Hence, there is a need for the development of a simple and computationally efficient procedure to determine the collapse loads of reinforced concrete (RC) cylindrical tanks. The limit
analysis approach can be used for this purpose. In this paper a general formulation for limit analysis of cylindrical water tanks is presented. Assuming linearly varying hoop tension capacity distribution along the height, Sawczuk and Olszak [ 1] have derived equations to estimate the collapse loads of short and long water tanks. For the case of medium height tanks, in order to avoid the mathematical rigour involved in the derivations, Sawczuk and Olszak [1] assumed uniform hoop tension capacity distribution along the height of the tank wall. In practice, hoop reinforcement along the height of a circula.. cylindrical water tank is usually provided in a stepped pattern. The equations derived by Sawczuk and Olszak [l] cannot be directly applied to estimate the collapse loads of cylindrical tanks with stepped distribution of reinforcement. Depending on the type of reinforcement distribution, the governing equations need to be derived afresh. This problem can be circumvented if the governing equations for the determination of the collapse load are based on the generalized distribution of hoop reinforcement. In this paper, based on the limit analysis approach, a new set of general formulations for the estimation of collapse loads of cylindrical water tanks are presented. By using certain control parameters any type of reinforcement pattern along the tank wall can be considered without the need to change the basic formulation. The two cases, the uniform and linearly varying hoop tension capacity distributions, used by Sawczuk and Olszak are special cases of the general distribution for which the formulations are presented in this paper. Based on these formulations, a computer program named TANK has been developed. 205
206
K.
RAMANJANEYULU ef al.
Using TANK, the collapse pressure and the factor of safety against the collapse of a circular cylindrical tank of given con~guration can be obtained. Results
of parametric studies conducted to evaluate the coilapse pressure of cylindrical tanks of different sizes and hoop tension capacities are also presented in non-dimensional form. 2. METHODOLOGY FOR LIMIT ANALYSIS REINFORCED CONCRETE CIRCULAR CYLINDRICAL TANKS
OF
The procedure for estimating the collapse of a reinforced concrete cylindrical tank consists of the following steps:
1. Establishing the relationship between material strength and the critical combination of internal forces which the material has to withstand. 2. Arriving at the yield condition and the flow rule. 3. Solution of the governing equilibrium equations using the prescribed stress boundary conditions and the above relations. 2.1. G~uer~jng equilibria
eq~~~io~
As the geometry and the internal pressure are axisymmetric, the yield condition for a circular cylindrical tank can be expressed in terms of three stress resultants: M, (longitudinal bending moment), N, and N, (membrane normal stress resultants). The sign convention for these stress resultants (positive direction) is shown in Fig. 1. It may be noted that due to rotational symmetry, there is no circumferential displacement and the ci~umferential curvature rate (&) vanishes. Thus, the moment in the circumferential direction, M,,, is zero. The equilibrium equation of a cylindrical shell subjected to axisymmetric loading is given by dZM
N,
d.---p,=& dx= R
Fig. 2. Failure criterion for concrete in plane stress.
Equation (I) can be expressed in terms of nondimensional parameters as follows: 1 d2m -L-.n,-p=o, C2 dc2
where <=x/L P =pxRIN, m, = M, IMP no = N%lN~ nx = NxIN, Np =f,t MP =fct2j4
N L=
c==.__L-
4L2 = Rt = length parameter
Rjw, and p. m, and n, are the non-dimensional parameters for pressure, longitudinal bending moment and membrane normal stress resultant in the circumferential direction, respectively. N,, and M, are the compressive and bending resistance of the tank wall at collapse, respectively. Any distribution of m, and n, which satisfies this equilibrium equation and the values of which do not exceed the corresponding capacities at any section are statically admissible. 2.2. Failure criterion The failure criterion, given by Sawczuk and Olszak [I] for the circular cylindrical shell under axisymmetric loading, is adopted in this study. The failure condition for concrete in plane-stress (Fig. 2), assuming the compressive stresses as positive, is given by
Fig. 1. Coordinate system for cylindrical tank
Collapse loads of reinfonxd ccwrete water tanks
201
rectangle for shdts without
Fig. 3. Yield surface for reinforced concrete. cylindrical shells [3].
The yield condition given by
for the reinforcement
a=-fy,
a
is
‘-a6
?I, -
+/&=O
03)
1 = 0,
(9)
where
‘*
m,(l + a) + 2nI + 2n,(28cl, + a - 1) 4/&-2a
n,+a
and compression
00
Considering the longitudinal (x) and circumferential (0) directions as the principal directions, and assuming compressive forces and stresses as positive, the yield surface (failure criterion), as shown in Fig. 3, is formed by two parabolic cylinders [l-4] which are given below
+2/+lI-
the tangential forces in tension respectively, given by
(4)
The collapse of an element can occur only if both eqns (3) and (4) are satisfied. The instantaneous deformation mode is given by
ac.-Z
Fig. 4. Simplified failure criterion.
10,
for
m,> 0
(6)
From eqns (6-9) it can be seen that the yield locus is dependent on the percentages of reinforcement in the circumferential and longitudinal directions. Hence, the collapse load also depends on these parameters. If the longitudinal force, n, , is small in comparison with the other stress resultants, the analysis can be restricted to the m,-n, plane. The intersection of the surfaces given by eqns (6) and (7) with the plane n, = 0 results in the simplified yield criterion shown in Fig. 4.
2.3. Hoop tension capacity distribution -m,(l
+a)+2nt+2nX(a
- I)-2a for
-0, m,
(7)
..... b
and these two parabolic cylinders are limited by two parallel planes, represented by the critical values of
T
c
Cow (0
The common practice of curtailing the reinforcement along the height of the tank wall leads to the stepped distribution for hoop tension capacity. Six possible distributions for hoop tension capacity are shown in Fig. 5. The distribution shown in Fig. 5, case (i), is expressed as below.
car. (ii)
Fig. 5. Different hoop-tension capacity distributions.
208
K.
RAMANJANEWLU
For region I
et al.
obtained from the distribution (Fig. 5) as follows: ng = n,, + k, t.
(10) a
For region II no =
shown for case (i)
(11)
n21 + k2(< -0
For region III
:b: (4
~11,
no = njl + kAt -(B +
(12) (d)
where (e)
kI = (n12- n,, )/B
when k, = 0 and n,, = n,2 = n2,, the distribution shown in case (ii) can be obtained; whenk,=O,k,=O,k,=Oandn,,=n,,=n,,= n,,; n22 # n,, , n,, = nj2, the distribution shown in case (iii) can be obtained; when k, = 0, n,, = q2, n,2 # n2,, k2 = 0, n2, = n22, n22Z n31 and k, = 0, n3, = n,,; the distribution shown in case (iv) can be obtained; when k, = k2 = k, = 0 and n,, = n,2 = n2, = nz2 = 4, = n32Tthe distribution shown in case (v) can be obtained; when k, = k,= k,,n,, =n,, and n,,=n,,, the distribution shown in case (vi) can be obtained.
k2 = (n22- n2,)/Y 2.4. Classification
of cylindrical corresponding collapse mechanisms
k3 = (n32- n3, Ml - (B + Y11.
, \ I T L=T \ / il-l
The hoop tension capacity distributions shown against other cases [Fig. 5, cases (ii)-( can be
tanks
Based on the mode of failure at collapse, the cylindrical water tanks are classified into three categories: shallow (short), medium height and long
Region-l
,
:
I
I\ \ i \ \
L
T
Zl
/z i c,
\
/i
g=1
(ii)
0)
(iii) Mu
(a)
’
/
J
t
z
\
?
/’
\
and
(iI
+J!??:’ 1 n, .. g.’ p f
IL ’ mbwlt
(ii)
v.. I
MU\
i;iu
Y
“22 . . . ,. .
\
3,
tq
t% (iii)
Mu
(C) Fig. 6. Collapse mdanim (i), hoop-tension capacity distribution (ii), and bending moment distribution (iii) for (a) short tanks, (b) medium height tanks and (c) long tanks.
Collapse loads of reinforced concrete water tanks
209
tanks. The mode of failure depends upon the value of the non-dimensional length parameter (C’). For shallow (short) cylindrical tanks fixed at the bottom and free at the top, only one plastic hinge circle will develop at the base (Fig. 6a). For medium height cylindrical tanks, the top portion of the tank will be in a state of hoop compression. Two hinge circles form in this type (Fig. 6b). For long tanks, partial failure will occur. This type of failure will be confined to the base region. In this type, two positive hinge circles and a negative hinge circle will be formed (Fig. 6c).
By integrating eqn (14) successively and using the boundary conditions, the equations for shear force and the bending moment are obtained as
2.5. Mathematical formulations for the tanks with general hoop tension capacity distribution [Fig. 5, case (i)]
F
$x,= -(n,,t +k,;)+p; -(n,,~+k,~)+p~.
&mxI=
Region II. Substituting eqn (11) into eqn (13)
1
the value of n, from
=
(17)
(t: -lv2
2
2.51. Short cylindrical tanks. For the case of short cylindrical tanks the collapse mode and the bending moment distribution are shown in Fig. 6(a). The origin of the coordinate system is at the top of the tank. The shell is clamped at r = 1 and free at r =o. The equation of equilibrium for the cylindrical shell subjected to hydrostatic pressure can be expressed as follows:
=()*
C2 dC2
The boundary as: (i) mxl =0
and continuity
1 1 =
342
(19) Using the conditions n&r= tix2 and mx, = mx2 at { = fi, the values of C, and C, are obtained as
P2 G = -(n2, -n,,)y+k,-.
conditions
are given
&, = 0
at
1 -@ix,=
--[n,,+k,{<
n3,t
1 =
-
n31
-0
and
(iii) m,, = m,,
and
(iv) m,,=m,
at
ti,, = ti,, +I,, = ti,,
+k, It -(B+Y)12 2
{e -(B+vW 6
C+k, 2
1 F %I = -h
(20)
1
+pc+C5 2 (21)
1
r =O, at
at
(22)
4 = (Y + B),
By using the conditions tirx2= &, and rn,, = m,, at t = (B + y), the values of C, and C, are obtained as
C5= (4,
-n21)(B+y)-k,~-kl~+(n2,
the value of n, from eqn (10)
G= -h, +k,O+pt.
+c,.
{ = /?,
c = 1.
Region I. Substituting into eqn (13)
3
+r)}l+pt
+p;+c,e: (ii) m,, = m,,
8’
Region III. Substituting the values of no from eqn (12) into (13)
3mx3
and
n2,~+k2~]+p~+C,C+C,.
-
L&3= c2
2
Id-nn,_P~
(18)
+pf+C,
2
Mathematical formulations for limit analysis of short, medium height and long cylindrical tanks, fixed at the base and free at the top, with the general hoop tension capacity distribution [Fig. 5, case (i)] along the height are presented in the following sections. Depending on the values of n,,, . . . , n32 this general hoop tension capacity distribution degenerates to particular cases (ii)shown in Fig. 5.
(16)
-In,, +k2(t -B)]+pt:
I
42
(15)
(14)
_n
jP+r)’ ; ,j+b 8’
,(B+v) - 2
-k,g--(n,,
-n,,)f.
k
21
2
K.
210
RAMANJANEYULU
Substitution of tn,, = m, (i.e. maximum positive moment capacity), at e = 1, in eqn (22) yields
[l - (B + r)13 6
(B+r)* +@3,
+k,~_k,Y’?!?d
-n2,)2
2 8*
+
@21-
4,)
The boundary Fig. 6a) gives
B’ -
k,
Ti-
condition
0.X:(--k,+p) +
y
(n2,
-
t,h21-
-
n,,)/.I
*
+
8’
k, ; + k, T
.
(23)
A,, = 0 at [ = 4, (see
et al.
When mX2 is greater than t&. (maximum negative moment capacity) at C:= e,, then the solution obtained using mechanism applicable for short tanks is not valid. The flow chart of the program TANK outlining the procedure to compute the collapse pressure for the case of short tank, is given in Fig. 7(a). 2.5.2. Medium height cyrindrical tanks. The details of collapse mechanism and bending moment distribution for medium height cylindrical tanks are shown in Fig. 6(b). The origin of the coordinate system is at the top of the tank. The shell is clamped at 5 = 1 and free at 6 = 0. Top portion: ne = + 1 (top portion is in hoop compression). Substituting the value of n, into eqn (13)
k2f-9
-
O.S/?*(k,+ k,) = 0.
Read
Height,
,unit
weight
Radius, of fluid
(24)
Thickness, in the
tank
‘“22’“31’“32’$,1/ 1
K
Compute
I
8 mu
Considering short tank collapse mode, compute non-dimensional pressure ( P 1 -Eqn. (23)
1
I
Compute negative
the location (ct) moment (rn821
-1
of maxImum -- Eqn. (24)
1 Compute
mx2 at r, Short not
tank
valid.
case
is
Hence
medium
height
tank
collapse
mode
is
examined.
Print and
eollap*e safety
pressure factor A
Fig. 7(a). Flow chart of the computer program TANK: for the case of a short tank.
Collapse loads of reinforced concrete water tanks By integrating eqn (25) successively and substituting the boundary conditions, the shear force and bending moment are obtained as
211
1 = -[n,,;+k,;]+P;+C,( F mxl
+c*.
(30)
By making use of conditions tifiOp= ti,, , mnop = m,, at [ = &,. the constants C, and C, are obtained
1 . 3 %oP
(26)
as
C, = &(l + n,,) + k, f
Region I: n, = nI, + k, r. Substituting n, into eqn (13)
(l+n,,) c,= -c+---
the value of
1 .. eqn (28) successively
~~~,=-[n,,e+k,~]+~~+c,
’ 3’
Region ZZ
(28) BY integrating
3
k
(29)
= -[n,,+k,(c
;:I=
-B)I+p5
-[nz,<+kzy]+pg+c3
Q A
Assume
Compute
go= 0
1 non -dimensional - -
pressure
(p)
Eqn.( 37) 1
Calculate
negative
hinge
&
-
Eqn.
(38)
&
(new
value
ofgo
known
p and
Compute
-
gf
--
circle
Eqn.
location
) for
the
(39)
Yes Compute
the
location
(ET)
of maximum
Fig. 7(b). Flow chart of the computer program TANK: for the case of a medium height tank.
ill:
212
K. 1
-
Fmx2=
[
RAMANJANEYULU et
n2,;+k2q]
al.
The boundary
condition
ti,, = 0 at 5 = 5, gives
0.55:(P -kd-t,(n2,
-kd)+(n~,
-n,,)B
- 0.5k,j2 - 0.5k2B2 + c,(l + nl,)
The continuity conditions tix, = fix2 and m,, = m,, at + 0.5k, 1;; = 0.
5 = /I gives
Using the boundary
Region III
+k
i&3 =C2
n3,5 + k,
-(B +r)Jl+~t {C -(B
+v)12
2
zmx3=
-
n3, ‘2+k, 2
It
6
using
mx2=mx3
continuity at
-
Kn2,-
nil
M - O.% 821t, (39)
By solving the non-linear eqns (37)-(39) iteratively as shown in Figs 7(b) and 7(c), the values of to, 5, and p are computed.
9 B
(36)
tix2 = tix3 and
t
I
I
5 =(B+Y)
C5=(n3,-n,,)(B
+y)--k2g-k,T
+tn,,
c,=
conditions
5 (35)
1
+P;+C,t+Cc+
By
6
-k,~+0.5(n,,-n,,)~2-k,~=0.
+$+, 2
r)>’
-
6
(34)
1
- (B +
1
condition mx2 = ni, at c = r,
G,-B13 PC 2~
&+h3= -[n3,+k3{t
(38)
-n,,M+C,U
+n,,)+k,+
-(n3,-n2,)~+k,~+k2y2~ 2
-k2$-(n2,-n,,)-i--_i--
P2
(l+n,,)
2
of m,, = mu at 5 = 1, in eqn
Substitution yields
I
r
I
I
k 5
’ 3’ (36),
I
t Compute
1 1, f’, f”
I
I
P - (B + r113 6 - (n3,- n2,W + Y) + 0.5k2y2 + 0.5k,p2 -
tn2,- h)B + O.%,, - nzl)tB + r12
+
k,;- 0.5k2y2(B + y) -
+C’tl+nll)+k 0-
2
G ‘3, 1
P
DIFF = &,,, - go
OSk, t;
Fig. 7(c). Flow chart for the solution of a cubic non-linear equation.
Collapse loads of reinforced concrete water tanks Boundary condition
1
ti,r = 0 at r = & gives
@x3
-nr,&
- 0.5M1;2 - I-V + 0.5& +
b2,
-
=
-n32t
=
-n32
+n,,)+0.5k,ti=0.
(40)
When the moment at < = r2 reaches m,, this mechanism is no longer valid, only partial failure occurs, which is the case for long tank. 2.5.3. Long cylindrical tanks. The collapse mode and the bending moment distribution for long cylindrical tanks are shown in Fig. 6(c). In this case, it is convenient to choose the origin of coordinate system at the bottom of the tank. The bending moment diagram between c = r2 and 5 = 1 is continued in a statically admissible manner. The hoop tension capacity distribution for this case is expressed as below. Region I ne =
42
-
W
-
(B,
+
(43)
+k,g+p
n,,)B 2
1 -0.5k,jP+r0(1
213
F
mx3
r+k$+p 2
Region II 1 @iX2 = -n,+k2(r
&litx2 =
-n,,t
-S,)+p(l
(45)
-r)
+ WI
1 F mx2 =
-n22
c+k 2 + c3c
~11. Using
Region II
the boundary
+ c,.
and continuity
(47)
conditions
specifiedin eqn (41), the constants are obtained as n0 = n22- k2(r - B, ).
G=&
Region III
- 0.5k2G - PIj2-P
no = n32- k, 5,
(using ti, = 0
where
Using continuity
condition
at
r = 5,).
rit,, = +I,~ at c = j3,
k, = (n,2 - n,,)l[l - (8, + r)l k2 = (n22- n2, )/Y k, = (n32- nsl )I&
-0.5k,(&-&)‘-p
B, = [l - (B + Y)I. The boundary and continuity conditions for this case are as follows:
C2 = $
(using mx3 = m,
Using continuity (i)
43
=m,
(ii)
42
=
m,
and and
(iii) mx3 = mx2 and
&,=O
at
ti,,=O
c=O
at
tiX3= tir,
(41b)
t = fl, (4lc)
(iv) m, = m, (v)
m2
=
m,,
and and
ti,, = 0
at
ti,, = ti,,
r = t2 at
r
(41d)
=Cf%+y). (414
CAS 48,2-c
-n32 + k& +p(l - 5)
mX3= mx2 at 5 = fi,
C.,=(n,,-n,,)T-k,q+$.
Using the boundary
(42)
condition
mx2 = tiiu at { = c,
p[2[; - 31;;] = -6
-
3n2,ti
- 3k25fB, + k2B: - 3(n,2 -n&j?: The condition yields
Region III 1 @ijix3=
5 = 0).
(41a)
r ={, at
condition
at
r2 = VP - %2
+
X2G
+ 2k&.
(48)
tiX2= 0 at c = e, and also at t = c2
- 2k2B, - pt,
+
M,I/(P
-k,).
(49)
K.
214
~MANJA~~LU
et al.
(a) Height (m), radius (m) and thickness (m) of the tank wall.
(b) Compressive ;;
t Compute Pr tssur*
non-dimensional ‘p’
--
Eqn.
(48)
W
i cokuwe
--
g,
Eqn. (69)
1
(0
strength of concrete cf,), yield strength of reinforcement U;). Unit weight of the fluid in the tank (kg/m3). Vertical reinforcement on the inner face in cm2/m, and on the outer face in cm*/m. Hoop reinfor~ment areas in cm’/m along the height of the wall (i.e. values for nll,n 12,n,, , n2*, n3,, t132 shown in Fig. 5). /I (height of the region I), y (height of the region II) (shown in Fig. 5).
i Compute
(&J - -
the new value Eqn. (SO)
of gl
1 OlFF *
gtn-
!&
I
hfn$e
pr+ssurr
localfir, and
rcrf*ty
COltapse
factor
1
&
stop
Fig. 7(d). Flow chart of the computer program TANK: for
the case of a long tank.
Substitution
The slope parameters k,, k2 and k, (Fig. 5) of the reinforcement distribution pattern and length parameter (C2) for the cylindrical tank are computed first. Then the collapse pressure is calculated by assuming the tank to be a short tank; the applicability of the mechanism for the short tank is then checked. If this mechanism is found to be invalid, collapse pressure according to the mechanism for medium height tanks is computed and the applicability of the mechanism is checked. If the mechanism for medium height tanks is also found to be invalid, collapse pressure will he calculated according to the mechanism for long tanks. Thus, the computer program automati~lly selects the valid m~hanism for the given configuration of the tank and calculates the collapse pressure and factor of safety. The output results consist of an echo of input data, followed by the information regarding hinge positions, collapse pressure and factor of safety against collapse.
of m,. = m, at < = c2 yields 4. NUMERICAL
RESULTS AND DISCUSSION
4.1. C~~~~r~cuI tank with uniform hoop tension capacity distribution
By solving the non-linear equations (48)-(50) iteratively as shown in Fig. 7(d), the values of {,, t$ and p are computed. 3. SALIENT
FEATURES OF THE COMPUTER PROGRAM TANK
Based on the mathematical formulations presented in the previous sections, a computer program, TANK, has been developed to find the collapse pressure as well as the factor of safety against collapse of cylindrical water tanks. Flow charts for this program are shown in Figs 7(a)-(d). The input data for the program are as follows:
The mathematical formulations presented in this paper are applicable for a general variation of hoop tension capacity along the height of the tank wall. Particular cases, such as constant (uniform) hoop tension capacity and linearly varying hoop tension capacity along the height of the tank wall, can be analysed using the computer program TANK by approp~ately inputting the hoop reinfor~ent details (i.e., n,, , . . . , n3z). Sawczuk and Olszak [l] have analysed a cylindrical tank with uniform hoop tension capacity distribution along the height of the wall and have given charts for non-dimensional collapse pressures for different length parameters. Toi and Kawai [S] have analysed this problem using the discrete limit analysis and have given collapse pressures for different length parameters. For the same problem, collapse pressures for different length parameters (C*) are computed using TANK in this study. The comparison of the results obtained in the present study with those reported by
215
Collapse loads of reinforced concrete water tanks Table 1. Comparison of collapse pressures of cylindrical tank with uniform hoop tension capacity distribution rig. 5, cae (v)] with n, = m, = L@” = 1 for different length pammeWs Nondimensional collapse pressure (p) Length parameter (C3
Sawcxuk and Olszak 111
16 40
Preacnt study
3.370
3.375 2.457
2.170
2.212 2.180 2.081 2.014 1.821
2.45
z 80 100 144
2.20 2.08 2.00
1.770
pressure which would be closer to the true collapse load. The same cylindrical tank was analyzed using the computer program developed in this study. It is found that this geometry has the medium height tank collapse mode and the collapse pressure is found to be 0.218. Thus it is noted that the collapse pressure given by Sawcxuk and Olsxak [l] is on the higher side and is not the lower bound value. However, for the purpose of comparison, the same tank was also analysed in the present study by assuming it to be a long tank (partial collapse mode) and p = 0.221 is obtained. The collapse pressures obtained in the present study are compared with that of Sawcxuk and Olsxak in Table 3. For the tank considered in this section, the valid mechanism is that of the medium height tank and the collapse pressure is 0.218.
Sawcxuk and Olsxak [I] and Toi and Kawai [5] is shown in Table 1 and Figs 8 and 9. It may be noted that Sawcxuk and Olsxak have also given hinge locations for medium height tanks and these values are also compared with those obtained in the present study. From Fig. 8, it can be seen that there is good agreement between the results. 4.2. Cylindrical reinforcement
Toi and Kawai 151
tank with linearly varying hoop
Sawcxuk and Olsxak have analysed a reinforced cylindrical water tank with linearly varying reinforcement along the height of the tank wall. The geometry, material and reinforcement details of this tank are given in Table 2. Since the length parameter (C*) for this geometry was such that it did not fall within the short tank collapse mode, they considered it to have the long tank collapse mode and computed the collapse pressure as p = 0.23. It may be noted that they did not examine for the case of a medium height tank collapse mode. It is observed that only a correct collapse mode can give lower bound to the collapse
4.3. Cylindrical tank with stepped dktribution of hoop reinforcement A cylindrical tank with stepped distribution of reinforcement is analysed using the computer program TANK. Details are given below.
x
sawuuk
A
aal H
1
2s
1
I
I
35
45
IS
Length
parameter
nawai
I
I
gs Cc?= 4&Rt
and oIuak[I]
TO; and
75
[s]
- r5
I
I
85
SP
)
Fig. 8. Comparison of hinge locations and collapse pressuns for medium height hoop-tension capacity distribution.
tanks with
uniform
K.
216 0.8
et al.
RAMANJANEYLJLU
I: I
a6
El,&
I
0.4
0.2
0
b]; i lo5
115
125
135
145
155
Length ~ammrkr
165
175
165
155
205
C2 * 4L%?t
Fig. 9. Variation of hinge locations and collapse pressure for long tank with uniform hoop-tension capacity distribution.
A cylindrical tank of 500,000 litre capacity was designed according to 13:3370-1967 [6]. The dimensions and the reinforcement details are given in Table 2. The collapse pressure and the factor of safety of this tank are obtained using the computer program. The tank is found to have medium height tank collapse mode and the non-dimensional collapse pressure (p) is found to be 0.299. For a correct collapse mechanism, the collapse pressure has to be minimum. This aspect is examined by computing the collapse pressure for other two collapse modes, namely, short as well as long tank collapse modes as well. These values are given in Table 4. It may be seen
from this table that only for the case of medium height tank collapse mode, the p value is minimum and this gives the correct collapse pressure. The collapse pressure (P,) is calculated from the nondimensional collapse pressure, p, as follows: Pf=‘G
= 13.8 t/ml.
The safety factor against collapse is obtained as the ratio of the collapse pressure ;p/) to the actual pressure (height x unit weight of the fluid in the tank). In the present case, the actual pressure
Table 2. Details of the cylindrical tanks with two different patterns of reinforcement
Particulars Height Radius Thickness
Tank with linearly varying reinforcement [Fig. 5, case (vi)] 5.0 m 5.0m lO.Ocm
Tank with stepped distribution of reinforcement [Fig. 5, case (iv)] 4.0 m 6.5 m 15.ocm
f,
200 kg/cm2
200 kg/cm2
f, Hoop reinforcement
2500 kg/cm*
2500 kg/cm’
linearly varying from 3.0cm2/m at top to 14.1 cm2/m at bottom of the tank
@ 11.31 cm’/m for the top 1.2 m @ 16.16cmz/m for the next 2.0m and @ 11.31 cm*/m for bottom 0.8 m
Vertical reinforcement (outer face)
1.01 c&/m
2.513 cm*/m
Vertical reinforcement (inner face)
3.52 cm*/m
10.27 cm*/m
Collapse loads of reinforced concrete water tanks Table 3. Comparison
217
of collapse pressure of cylindrical tank with linearly varying reinforcement [Fig. 5, case (vi)] Non-dimensional
collapse pressure (p) Present study
Length parameter (C2)
Sawczuk and Olszak [l] using long tank solution
As medium height tank
As long tank
200
0.23
0.218t
0.221
t This corresponds to the correct collapse mechanism.
Table 4. Collapse pressures of cylindrical tank with stepped distribution of reinforcement
[Fig. 5, case (iv)] Nondimensional collapse pressure(p) Length parameter (C2)
As short tank case
As medium height tank case
As long tank case
65.641
0.367
0.299t
0.33
t This corresponds to the correct collapse mechanism.
out to be 4.0 t/m2. The factor of safety against collapse of this tank is calculated as F.S. = 13.814.0 = 3.45.
Acknowledgements-The authors wish to thank Shri N. V. Raman, Director, Structural Engineering Research Centre, Madras, for his constant encouragement and support in carrying out this study. This paper is published with the kind permission of the Director, S.E.R.C., Madras.
5. CONCLUDING REMARKS
REFERENCES
works
The mathematical formulations presented in this paper for the estimation of collapse loads of short, medium height and long tanks are for the general distribution of hoop reinforcement along the height of the tank wall. Different patterns of reinforcement distribution: uniform, linearly varying and stepped distributions, can be handled by suitably choosing the parameters nu, . . . , nj2. Thus, the formulations are versatile. The results obtained using the program TANK are found to be in good agreement with the results reported by Sawczuk and Olszak [l]. The computer program TANK runs on an IBM compatible PC-XT.
1. A. Sawczuk and W. Olszak, A method of limit analysis of reinforced concrete tanks. Simplified Calculation Methoa!vof Shell Structures, Proceedings of Colloquium, Brussels,pp. 416437 (1961). 2. W. Olszak and A. Sawczuk, Inelastic Behaviour in Shells. Noordhoof (1967). 3. M. Save, Limit analysis and design of containment vessels. Nuclear Engng Design 79, 343-361 (1984). 4. M. Save and C. E. Massonnet, Plastic Analysis and Design of Plates, Shells and Disks. North-Holland, Amsterdam (1972). 5. Y. Toi and T. Kawai, Discrete limit analysis of plate and shell structures. Compur. Srrucr. 19, 251-261 (1984). 6. Indian Standard code of practice for concrete structures for the storage of liquids (13:3370--1967). Indian Standards Institution, New Delhi (1967).