Wind-loaded reinforced-concrete cooling towers: buckling or ultimate load?

Wind-loaded reinforced-concrete cooling towers: buckling or ultimate load? Herbert A. Mang, Helmut Floegl, F r i e d r i c h Trappel and Herbert Walter Institute of Structural Analysis and Strength of Materials, Technical University of Vienna, Vienna, Austria (Received August 1981; revised March 1982)

Wind Ioadings govern the design of most cooling towers. Until now, proof of sufficient safety against buckling under wind load has been a major concern for the designers of such shells. In this paper it is demonstrated that a typical cooling tower made of reinforced concrete would not buckle - at least not in the classical sense of the word. Failure would rather be initiated by rapid propagation of cracks in tensile zones followed by temporary stiffening and, finally, by yielding of the reinforcement. The theoretical part of this paper is restricted to a presentation of the constitutive model, discussion of the equation for incremental-iterative ultimate-load analysis and of the condition for instability. The numerical part contains a detailed study of a built hyperbolic cooling tower. It is shown that: (a), buckling loads resulting from linear and geometrically nonlinear prebuckling analyses are considerably larger than the ultimate load; and (b), results based on a certain form of 'equivalent axisymmetric pressure' are on the unsafe side of corresponding results from the 'actual' wind load. It is also demonstrated that the 'crack load', representing a lower bound to the ultimate load, can be estimated by means of a linear-elastic nonaxisymmetric analysis of the cooling tower. Key words: cooling tower, reinforced concrete; wind load, equivalent axisymmetric pressure; finite element analysis, layer model, smeared cracks; buckling, ultimate load

Introduction The collapse of large hyperbolic cooling towers such as the one in Ferrybridge, England, in 1965 (see Figure 1), 1 has stimulated interest in research on possible reasons for these spectacular structural failures. Pertinent research efforts in several industrialized countries have been aimed at a better understanding of the structural behaviour of cooling towers subjected to wind loads and earthquake excitations, respectively. Doctoral dissertations (e.g. Floegl2 and Chang3), proceedings of symposia,4 state-of-the-art papers (e.g. Abel and Gould s and Zerna et al.6), reports of professional committees 7 and many scientific papers, some of which are listed, for example, in reference 5, reflect the activities of those of the scientific community who are interested in the structural performance of cooling tower shells. So far, the demonstration of a sufficient degree of safety against buckling under wind load has been considered to be one of the most important factors in the analysis of hyperbolic cooling towers. In a recent paper, Abel et al. s discussed several methods for determination of buckling pressures. Together with other results, they presented 0141-0296/83/03163-18/$3.00 © 1983 Butterworth & Co. (Publishers) Ltd

buckling pressures for the Der and Fidler 9 cooling tower models, made of PVC and copper, respectively, and compared them to the test results. The investigation of Abel et al. is restricted to the assumption of a linear prebuckling path. Emphasis is laid on 'equivalent axisymmetric concepts' as being computationally economic. However, it is questionable whether axisymmetric buckling concepts can generally be applied to nonaxisymmetric situations. The problem is whether or not such concepts necessarily provide lower bounds to the buckling pressure obtained from a general (nonaxisymmetric) analysis. No less it is questionable whether the assumption of a linear prebuckling path, restricting investigations of instability to the determination of bifurcation points on the linearized load-displacement path, is justified. Strictly speaking, the prebuckling path will only be linear if the rotations of the tangents to the parameter lines of the shell vanish and if the constitutive equations are linear. If this is not the case, the assumption of a linear prebuckling path, at best, will be an approximation.

Eng. Struct., 1983, Vol. 5, July 163

Wind loaded RC cooling towers: hi. A. Many et aL

Figure I

Collapse of cooling tower 2A in Ferrybridge, UK, in

19651

If the prebuckling path is nonlinear, the buckling condition will also be nonlinear. Frequently, this condition is linearized which, at best, is an approximation. It must be emphasized that the linearized buckling condition differs from the buckling condition which results from the aforementioned assumption of a linear prebuclding path. Identical results will only be obtained if the prebuckling path is, indeed, linear. For the sake of completeness, it should be mentioned that a procedure termed 'accompanying linear eigenvalue analysis 'lz and characterized by solving the nonlinear buckling problem by means of a sequence of linear buckling problems permits a comparison of 'linear' and 'nonlinear' buckling pressures. When assessing the structural safety of a hyperbolic cooling tower made of reinforced concrete (RC) against collapse, it is necessary to consider the nonlinearity and fracture of concrete. If this is done while maintaining axisymmetry, fracture of concrete in biaxial compression will be the dominant failure mode. This has been demonstrated in a recent paper by Zerna et al., ~3 which reported on the ultimate (collapse) load analysis of a RC cooling tower for an 'equivalent axisymmetric wind pressure'. They employed a layered, axisymmetric finite element (FE) model, which allows consideration of the fracture of concrete. Their comment ~3that ' . . . in the investigated example mainly compressive forces are acting in the hoop as well as in the meridional direction...' characterizes the state of affairs. Zerna et al. compared bifurcation loads, based on linear elasticity, for several load intensity factors, with buckling loads, based on more realistic material properties. As expected, they found that, at larger load

164

Eng. Struct., 1983, Vol. 5, July

intensities, the latter become smaller than the former. This is a situation similar to that frequently encountered in the buckling of metallic structures. However, with regards to RC cooling towers subjected to wind loads, the actual situation is considerably different. It is characterized by several zones of tensile forces. Consequently, fracture of concrete in tension rather than its failure in compression should be expected to initiate the process of structural failure of the shell. This has motivated the writers to investigate whether or not buckling loads, based on a linear elastic material which can carry tensile forces equally as well as compressive forces, are a relevant parameter for assessing the structural safety of RC cooling towers. In order to be able to answer this question, a concept for ultimate load analysis of thin shells was developed)' 14 It is based on the FEM. Curved, triangular, thin-shell elements, as subdivided into a sufficiently large number of thin layers, such that, approximately, a plane state of stress may be assumed to exist in each layer, serve as the analysis tool. The concept considers: (a), nonlinearity of concrete in the compression-compression domain and in parts of the compression-tension domain; (b), fracture of concrete within the framework of a concept of 'smeared cracks'; (c), strain hardening of the reinforcement assumed to be 'smeared' to thin layers of steel; (d), tension stiffening, employing a concept developed by the first two authors, 16-18 (e), geometric nonlinearity and (f), follower loads in the form of hydrostatic pressure. For buckling analyses on the basis of a linear elastic material, a special computer program, restricted to geometric nonlinearity, is available. 19 It is characterized by sophisticated provisions for a reliable extraction of eigenvalues)° The present paper consists of a theoretical and a numerical part. The former is restricted to a presentation of the constitutive model, discussion of the basic equations for incremental-iterative ultimate load analysis and of the general condition for instability. This condition is linearized and compared with the buckling condition obtained for a linear prebuckling path. The numerical part consists of a detailed study of a hyperbolic cooling tower, built in Port Gibson, Miss., USA. For this cooling tower, results from the buckling analysis have been reported elsewhere.2' 21,22 The numerical investigation contains results from linear as well as geometrically nonlinear buckling analyses for the 'actual' wind load as well as for an 'equivalent axisymmetric wind pressure', s's It also contains results from a number of ultimate load analyses In one case, geometric nonlinearity is disregarded. In another, the tension-stiffening effect is neglected. Also, the percentage of reinforcement is varied. The two previously mentioned modes of loading are also taken into account. Subsequent to the numerical study, conclusions from the present research are presented.

Theoretical concept Model for ultimate load analysis Figure 2 shows the middle surface S of an undeformed RC shell with the boundary 1-'. The figure also contains the deformed shell with the middle surface S' and the boundary F'. The FE mesh consists of m C°-conforming, curved triangular RC finite elements. Details of the element em-

Wind loaded RC cooling towers: H. A. Mang et aL

[3, s V,W

z

I

/

/ S

~

_i:*'/,

/%vdS

Steel- layers, p

z

D(UN)

cl.,u

i

F

PsdS'

Concrete

D(CR)

--'--'~///

X

Figure 2

Reinforced concrete

i-' shell: middle surface of undeformed shell; layered finite element model of deformed, partially cracked shelP~

ployed may be found elsewhere. Is Each element is subdivided into n sufficiently thin layers such that, approximately, a plane state of stress may be assumed in each layer. The thickness of layer l is gO)h, where h is the thickness of the shell and g(t) is the weight coefficient for Gauss quadrature; x , y , z are global Cartesian coordinates; tz,/3, v are global surface coordinates; o~*,#* are local surface coordinates, referred to the common boundary b of two neighbouring elements, e and k; &, fi are parallel to an orthogonal net of reinforcement bars with diameters ~ba, ~ . The reinforcement is 'smeared' to steel layers of thickness ~ua(tlt~) h ;/~a(/a~) are the percentages of reinforcement. Dead load and hydrostatic pressure (undeformed configuration: Psv dS, deformed configuration: P s dS') are considered; ks* ds* is an unknown distributed bending moment for enforcement of CLcontinuity at b; u, v, w are displacement components parallel to a,/3, v; ~ ( t r ~ (~(cRj) denotes the entirety of uncracked (cracked) subregions of the shell. The defmition of such subregions is only meaningful for a concept of 'smeared cracks' as is applied in the present investigation.

Geometric relationships Nonlinear strain-displacement equations, restricted to consideration of moderately large rotations, are applied within the framework of a total Lagrangian formulation. The selected geometric relationships, which are taken from Koiter, :a should be suitable for ultimate load analysis of RC cooling towers. For such structures the maximum displacements just before final collapse are expected to be small in comparison to the radii of curvature but certainly not small in comparison to the shell thickness. On the basis of Kirchhoff's normal hypothesis, the strains are obtained as:

e,. = e~°)-

~i/

" - C o n c r e t e layer, rn

(1)

where e~°) are the middle surface strains and Kij are the changes of curvature of the middle surface. For thin shells such as cooling towers, use of Kirchhoff's normal hypothesis is admissible; at least, so long as the concrete remains intact. For the present research it is assumed that also for the cracked subregions of the shell Kirchhoff's normal hypothesis is an acceptable approximation. The middle surface strains can be written as: (2)

6~O) = eli q- Ill/

where eii is the linear and rhi is the nonlinear (quadratic) part of the strain tensor. Details concerning eli and r/i] are given elsewhere./7

Constitutive equations Concrete. The assumption of a membrane state of stress in the concrete layers allows biaxial constitutive models to be used. For the present research, biaxial stress-strain equations, proposed by Liu et al., 24 are used. Let o~ = Ocl denote the algebraically larger principal stress. Then, these equations can be cast in the following form:: OCi

=-- 0 i - -

_

_

I -- VcC~i ei

E(O)

(3) \

eiu/ eiu

where: oti : - oi

i :/: j

(4)

Eng. Struct., 1983, Vol. 5, July

165

Wind loaded RC cooling towers: H. A. Mang et aL Table I Functionsg~(c%),g2(a 2) for equation (3) Ratio of principal stresses e2 =

a~/a=,a~ > a2

~2

1 ) 0

~ 0 i> - 0 . 1

< -0.1 > -0.325

~ -0.325 > -~

g1(%) g2(cq)

1 1

1 1 + 10cq

1 0

0

0

~

oo

1.2

~ ,E2

~1.0

0 0

--Elu ~_ _

?

~"

1~

Y

I|

~)-~"

fcu=-3240NlCm2

//

analytical 24

0 2 ~-i I

-

//

ot, is the ratio of principal stresses, ei are the principal strains, aiu and eiu are experimentally obtained values of maximum stress and corresponding strain and E(ff)/(1 -- vc ai) are the effective initial moduli; E(ff) and vc are the initial tangent material modulus and Poisson's ratio for uniaxial loading respectively. Table I contains the functions gi(a2). Figure 3 shows a comparison of typical stress-strain diagrams on the basis of equation (3) with diagrams from experiments conducted by Kupfer. 2s In Figure 3,feu is the prism strength of concrete. As a compressive stress,fcu < 0. The employed constitutive model is termed 'equivalent uniaxial model' because of the description of the biaxial stress-strain behaviour of concrete in the form of equivalent uniaxial stress-strain relations. Advantages and disadvantages of this model as well as of other models are discussed, for example, by Chen and Ting. 26 For thin shells, such as hyperbolic cooling towers, analysed for monotonically increasing, proportional loads, the chosen constitutive model is adequate. In this context it should be noted that, especially at higher load intensities, fracturing of concrete is the decisive parameter for the structural behaviour of cooling towers. Consequently, it would not be meaningful to select a very sophisticated constitutive model for the intact concrete without improving the analytical model for crack formation and propagation. With regard to crack propagation, resort to fracture mechanics would have to be made. Within the framework of the FEM, promising work in this direction has been done by Ba~ant and Cedolin 27 for the comparatively simple case of panels. For l eil>~leiul concrete falls by fracturing. For --oo< a2 ~< a2L with ~2z. = f(fcu/ftu), where ftu denotes the uniaxial tensile strength of concrete, and 1 ~< a2 ~<~, fracture occurs by cracking of concrete normal to the direction of e~. The alternative fracture mode of concrete crushing - occurs ifa2L < ct2 ~< 1. In this case, concrete loses its strength completely. When concrete cracks, it retains its capacity to carry internal forces in the direction parallel to the cracks so long a s / ~ < 02 < ftu. Because of tension stiffening, concrete also retains part of its tensile strength normal to the cracks. Figure 4 illustrates the fracture envelope of concrete, representing the geometric locus of biaxial states of fracture stress Oiu. Analytical expressions for description of the fracture envelope are given by Kupfer. 2s

~

E2u

----experi~nt°125

~_

I

1

~

I

-1

-2

-3

g 103 •

Figure 3 Typical stress-strain diagrams for plain concrete subjected to biaxial states of stress ~6

(5/02 : 0

// / ol/02=~uc,/4u) ....... T

~

/U

b~" 4u

_

l,~i.iil, ~-q........... '- ///!-/'7-;]?

| \

/.#"

"/'/'/"

Cra d .........

/"

/

.//

,' II I|/ Nonlinear

reg,oo -

.

/

I

al / n

\\

Ill \ .... I: w°,/o2=-o325

'o,,o,

Figure 4 Fracture envelope for plain concrete subjected to biaxial states of stress l~

Drrectlonsof prlncipol stresses 0.-_:1 ~3"- = ~

Sc~e,,u~ /

!

\

~

Cracks,n

Steel. Reinforcement steel is considered as an elastoplastic material. The stress-strain diagram is assumed to be bilinear and identical in tension and compression. Ineremental-iterative FE-analysis For tracing the nonlinear load-displacement path of RC shells, the load must be applied incrementally. In order to reduce the unbalance between external and internal forces,

166

Eng. Struct., 1983, V o l . 5, J u l y

ear

~ - ~ ~Vc~') parallel crackstn ~....~.~ /// one dtrectpon ~ C L • Integration point Figure 5 Middle surface of partially cracked concrete-layer I of finite element e ~

Wind loaded RC cooling towers: H. A. Mang et al.

equilibrium iterations must be performed. Such iterations are especially important when fracturing of concrete occurs, resulting in a sudden energy release. For the present research, a full Newton-Raphson iteration is carried out at each load increment which requires updating of matrices depending on the state of deformation. In the following, the mathematical relationship for the qth step of the equilibrium iteration at a certain load increment is given. This relationship is taken from work by Floegl and Mang, 17where it has been presented in a slightly different form. Indicial notation is used because it facilitates explanation of the individual terms. Einstein's summation convention is applied, requiring summation over the range of repeated indices. The mentioned relationship is given as follows:

S(ce, t; UN )

+ o('f'(q_l)6Arh,f(,z))h

f

dS +

-(a (a (Ao(31~(q)6eO}(q_l)

s(ce, t; cR )

-(a) (a), ] + a,,.. •/'~q_--l) 6 Art i'.i (q) ) h dS f

+~[

p=l

(Ao~(,z)Se~(q-l) + (ri(q-1)SArli(,z) h dS

s(se, ~ ; UN)

"t-

(AO~(q)c~et(q_l)

3L O~'(q_l)~A~'(q)) h dS

s(se, P ; c R )

- Y'

b--1

e=l

(AX~*(q)~01*(q_z) + A0~.(q)~X~.(~_z)) ds*

In equation (5) the first sum extends over the finite elements. The second sum stretches over the concrete layers of an element. The third sum extends over the steel layers of an element. The fourth sum extends over all element boundaries along which C t continuity must be enforced. The same sequence of sums occurs on the fighthand side of equation (5). The integrals in this equation represent virtual work terms, most of which are given in local coordinates. The local coordinate systems &,/~ and a*,/3* have been introduced in the section on the model for ultimate load analysis. The local coordinate system a",/3" refers to the principal directions of stress and strain in the intact concrete. The reason for this choice is that the constitutive equations for the intact concrete are given in principal directions. The local coordinate system a',/3' refers to the normal and the parallel to the crack band associated with the tributary domain of an integration point in concrete layer I of element e (see Figure 5). Prior to assemblage of elements, transformation from local to global quantities must be performed. The first integral in equation (5) represents an increment of the virtual work of the internal forces acting in the part of concrete layer l of element e belonging to ~(UN). Its middle surface is denoted as S (e't; uN3 (see Figure 5). Here and in the following, the symbol A denotes an increment of the subsequent quantity. Note that the subscript of the considered step in the equilibrium iteration, q, is attached to all incremental quantities in equation (5). The subscript q--1 is attached to the remaining quantities in equation (5) which should indicate that these quantities are known from the preceding step of the iteration. Note that AOl,,2,,(q ) 4= 0 because the directions of the principal axes are not the same before and after the qth step. The stress increment Aoi"]"(q) is obtained as: Aoi"] "(q ) = Di"] "k"t"(q - 1) Aek"l" (q )

l*(b)

where Di"j"k"l"(q_l) denotes the tangent material matrix, evaluated for ei,V,,(q_l) ==-el(q-l), and Aet~"t"(q) is the corresponding strain increment. For q = 1, ei,,],,(o) = ei, where ei are the principal strains at termination of the equilibrium iteration for the preceding load increment. For the constitutive model employed in this research, the tangent material moduli for fractured (cracked or crushed) concrete and the reinforcement steel are taken from a paper of Cedolin and Dei Poll. 28 Since the principal stresses in the intact concrete, Oi'T' = ai, depend on the unknown ratio of principal stresses, ai = ct//oi, the quantities Gi"j"(q_l) in the first integral of equation (5) must be determined iteratively. Details of this iteration are given in reference 28. The second integral in equation (5) extends over S(ce, t; cn) which belongs to ~(cR); A~,a..),are increments of average values of stresses between nei~bouring cracks which would act in the concrete if there was no tensionstiffening effect. For parallel cracks in one direction (see

V(e)

4- f (Pse(q-1) -4- Aps#(q) ) ~Ug(q_l ) dS S(e)

f

o(,].(q_l)Se(,i.(q_l)h dS sCce, z; UN) ei'j'(q--1) u='dS) + f ol'j'(q--1)O =
S~' l; cR)

f

--p~=rl (

(6)

Ot(q-1)6ei(q-1)hdS

S(se,p; UN)

Figure 6a )

#~a) i(q-1) 6(a) ei'(q-1) n-dS)]

+

s(se, p; CR ) + ~

b=l

=

(Xl*(e-x)601*(q_l) + 0z*(~-z)~Xl*(q_l)) ds*

l*(b) g=l,2,3;i,]=l,2;l==oq2=l~,3--v;q=l,

2...

(5)

,;(a) / "'~'t37

=0

(7)

~ 2 Ge~a)#,

where G, taken from reference 28, enables a rough estimate of the effect of aggregate interlock. For parallel cracks in two directions (see Figure 6b), also 0t3, _(a)~, = O. Only two

Eng. Struct., 1983, Vol. 5, July

167

Wind loaded RC cooling

towers: H. A.

Mang et aL

ii

(1'

; Reinforcement Imrs :eme~ 04

--I

b~.'

/~../~ Average steel lengths bo~.b~' Crack spacings

a Figure 6 Lengths of reinforcement bars between neighbouring parallel cracks ~7

crack bands are taken into account. Strictly speaking, the simplifying assumption that the second crack band is normal (a) to the first one would require ca,o, to be zero. The motivation for this simplifying assumption is that: (a), in general, accurate modelling of formation of additional crack bands is less important than that of the first crack band; and (b), increased sophistication concerning the modelling of formation of additional crack bands should be accompanied with a fracture-mechanics based model for crack propagation which, at this point in time, is not available. The third and the fourth integral in equation (5) refer to the reinforcement in steel-layer p of element e;S(se'pi UN) denotes the part of the respective middle surface which is , ~UN)( ~ (cR) );6~ is given as: located in d~ O1 = # i o i

(8a)

6~ = U~o~

(8b)

tensile stresses transferred by means of bond from the reinforcement to the surrounding concrete. As indicated by the equations (9), derived by Floegl and Mang, ~7 in the analysis the tension-stiffening effect is lumped to the steel. On the basis of equations (9), the tangent material matrix for the reinforcement in ~ ( c n ) is obtained as: 1 S0) 1 -tJ=fiE

G °) :

and (a)

(a)

o~ = #~f~a~

(9b)

where o~a) are average stresses in the reinforcement bars between neighbouring cracks and fl ~> 1 are so-called 'tension stiffening factors', accounting for the tensionstiffening effect) 7 This effect represents the capacity of the intact concrete between neighbouring cracks to carry

168

Eng. Struct., 1983, Vol. 5, July

i] (lO)

0

0

where E(s°) is the modulus of elasticity of reinforcement bars ~i. Since the tension-stiffening factors are not constants in fact, they depend on a number of variables: 17

and

where o[ are the stresses in the reinforcement bars and /~i are the corresponding percentages of reinforcement. O}a) is obtained as: (a) (a) °i = #i f i o i (9a)

0

o

A6~a) = #i(fiAol a) + Afiol a))

(1 la)

(a),

(llb)

and AAa)

(a)

_,.

In the actual numerical analysis the terms/a i Afi ola) and is admissible within the framework of equilibrium iterations. Disregarding these terms results in a symmetric contribution to the tangent stiffness matrix which otherwise would not be the case. The fifth integral represents the variation of an increment of the load potential Xx,0t, with x~b,) = 3.Z, as the unknown load - a Lagran~ian multiplier in the mathe~b) matical sense -- and with 01, - 01, as the corresponding 'displacement' in the form of the difference of normal slopes of elements e and k at b. The integral extends over the length of b, l *~b).

#~Af$o~'0 are disregarded which

Wind loaded RC cooling towers: H. A. Mang et aL

The first integral on the right-hand-side of equation (5) represents the virtual work of the volume forces Pvg. The integral extends over the volume of f'mite element e, IAe). The second integral on the right-hand-side of equation (5) expresses the virtual work of the total hydrostatic pressure Psgtq-~) + Apsg(q), where PSg(q-1) is the total hydrostatic pressure after step q -- 1 of the equilibrium iteration, referred to global (fixed) curvilinear coordinates a,/3, v, and where Apsg(q) is an increment of Psg(q-~). The reason for attribution of the subscript q to the pressure load is the dependence of hydrostatic pressure on the state of deformation. The latter changes from step to step of the equilibrium iteration. Psg(q-x) can formally be written as: Psg(q-1)

f(P8,

=

(12)

Ug(q-1))

where Ps = Ps( a,/3, v) is a pressure function which does not depend on the state of deformation. Apsg~q) can formally be written as: APsg(q) = f( APs, Ugtq-X)) + g(Ps, Ug~a-x), AUg¢a))

-- ~ f b=l

(dX,.60,. + d0x.6X,.) ds*

l*(b)

Lf=

s(e)

g = l , 2 , 3 ; i , ] = l,2; l=a, 2-{J, 3=-v

(15)

It is easy to show that the condition for a bifurcation point on the load-displacement path is obtained by replacing the differential symbol d in equation (15) by the variation symbol 6. Expressing displacements, strains and stresses referred to local coordinate systems in terms of displacements, strains and stresses referred to the global coordinate system, performing summation over the layers and the elements and introducing matrix notation, equation (15) can be written as:

(13) where Aps represents an increment of Ps, applied after termination of the preceding equilibrium iteration, and /Xu~q) is the increment of displacements resulting from the qth iteration step. The respective integral in equation (5) stretches over the loaded part of the surface of element e, S~ne). The remaining terms on the right-hand side of equation (5) represent the virtual work of internal forces obtained after the preceding step of the equilibrium iteration. The iteration is terminated if the unbalance between external and internal forces becomes smaller than a prespecified tolerance. Instability Loss of stability of a shell is usually associated with an extreme value (limit point) on the load-displacement path. Occasionally it may be associated with a bifurcation point on this path. A relative extreme value indicates snapthrough. The absolute extreme value signals the ultimate load. The mathematical condition for an extreme value on the load-displacement path may be obtained from equation (5) by first reducing this relationship to a purely incremental one and then performing the limit value process, that is, replacing the symbol A by the differential symbol d. For a limit point, dPs = 0. According to equation (13):

dPsg = g(Ps , Ug, dug)

(14)

resulting in: Det

+

~,(lOi,j, j, ~i,f)h ,--(a)~Oei,/, (a) + Oi, -(a)~d (a), dS ]

f

S~" LCR) + ~ p=l

+

f

(d6i6e[+6";6drl~)hdS

S(Seop; UN) ^
S(se, p; OR)

SIt JJ

(18)

where K is the small-displacement stiffness matrix and K is a matrix with constant coefficients. The coefficients of NI (N2) depend linearly (quadratically) on the nodal displacements q. The tangent matrix may also be written as: 11

K T = K + Ko(o(q2)) + KL(q 2)

(19)

where K o is the initial-stress stiffness matrix and K L is the initial-displacement stiffness matrix. The matrix K ~ ) in equation (1 7) is the tangent pressurestiffness matrix which can be written as: 29

K~P)(Ps) + K(2P)(Ps,q)

(20)

(P)

where K1 ( K 2 ) is the first (second) pressure-stiffness matrix. The matrices K~P) and K~P) are unsymmetric matrices. C is the constraint matrix. Multiplication of this matrix by the vector of nodal displacements gives a set of linear constraint conditions:

Cq = 0

[

(17)

N1(N2) is the first (second) geometric stiffness matrix.

(P)

S~' l; UN)

= 0

K T =K +N,(q) +N2(q 2)

K~ ) = /=1

i ,

as the condition for an extreme value on the load-displacement path. Replacing the differential symbol in equation (16) by the variation symbol, it is seen that equation (17) also represents the condition for a bifurcation point on the load-displacement path. In equation (17), KT is the tangent stiffness matrix which can be written as: 22

The mentioned condition is then obtained as follows:

e=l

C

(21)

for enforcement of CLinterelement continuity. Frequently, the tangent stiffness matrix is linearized which means that N2 and K2~) a r e disregarded and that N1 and K~P) are set: N, = N, (q*) + xN,(q(R)) K~P) =xK,(P)(Ps) - (R)

(22)

(23)

Eng. Struct., 1983, Vol. 5, July 169

W i n d l o a d e d R C c o o l i n g towers: H. A. Mang et aL

where:

q* = q (Pv)

(24) (25)

q(m = q (p(n))

are the displacements resulting from volume forces and a reference pressure distribution, respectively; X is the load intensity factor. The condition for a bifurcation point in case of linearization of the tangent stiffness matrix is obtained as: Det

[ K + N , ( q * ) + N×[ ,(q (n) ) - K , ( P )(Ps)]', -(R) CT ] = 0 c i (26)

In general, symmetrization of K~P) is admissible. 29 Equation (26) yields the so- called 'initial linear buckling pressure'. 12 Stricklin and Martinez s° have shown that initial linear buckling loads based on equation (26) are smaller than buckling pressures resulting from a nonlinear prebuckling analysis. This state of affairs was confirmed by Brendel et alJ 2 who analysed cylindrical tanks for wind loads. Within the framework of the previously mentioned accompanying linear eigenvalue analysis, the authors of reference 12 also applied another form of linear buckling matrix. This matrix follows from linearizing KT according to equation (19). However, only Ka(oz(qCn))) is then amplified by the load intensity factor ×. For the given class of problems this would result in:

-K + KL(q* +q(R)) + Ko(OL(q*)) ] Det + x[K°(OL(q(R)))--KIP)(P~R))] CT ] = 0 C (27) where eL are stresses depending linearly on the displacements. For the problems solved by the authors of reference 12, equation (27) results in initial linear buckling pressures which are significantly larger than the buckling pressures obtained from a nonlinear prebuckling analysis, the latter being larger than initial linear buckling pressures resulting from equation (26). Although, strictly speaking, equation (27) is not the linearized form of equation (18), equation (27) is, nevertheless, perfectly suitable for a nonlinear buckling analysis by means of an accompanying linear eigenvalue analysis. If the prebuckling path is linear: lo

KL = 0 N 1"= K~ (OL)

(28)

: 0

(30)

are the constitutive equations linear. Consequently, one cannot be absolutely sure not even if material nonlinearity was disregarded - that bifurcation points obtained from any one of the three linear buckling conditions are good approximations to corresponding instability points resulting from the nonlinear instability criterion (equation (17)). Numerical

investigation

This section contains a detailed numerical study of a hyperbolic cooling tower of approximately 150 m height, erected in Port Gibson, Miss., USA. The shell is analysed for wind load. The purpose of this investigation is to (a), compare instability points on the load-displacement path based on simplifying assumptions such as a linear-elastic material and/or an equivalent axisymmetric pressure with the ultimate load and (b), study the behaviour of the structure from the uncracked state until final collapse. At first, the analysis model is presented. Geometric, material and load data are given. The reasons for the selec tion of the FE mesh are explained. Then, buckling analyses are performed, assuming a linear-elastic material. Finally, ultimate load analyses are carried out, employing the constitutive model described in the preceding section.

Analysis model Figure 7 contains characteristic dimensions of the shell, given in metres. The part of the generatrix of the shell above (below) the throat is part of a hyperbola (ellipse). The equations of these two parts of the generatrix are obtained by substituting the coefficients A, B . . . . . F in Table 2 into the equation for a general curve of second order, given as: Az 2

+ 2Brz + Cr2 + 2Dz + 2Er + F = 0

where r = r(z) is the radius of the middle surface of the shell at z -=/3 with/3 > 0 above the throat. At the throat, continuity of r, dr/dz and d2r/dz 2 is given. Note that the

_

30.50

203

[K+Ko(~L(q*))+x[Ko(OL(q(R)) cT 1 -- I((P)[~(R)~] I " 1 ~vs JJ ~ = 0 c

It is seen that for the special case of a linear prebuckling path identical eigenvalues are obtained from equations (26), (27) and (31). For cooling towers subjected to wind loads, neither the rotations of the tangents to the parameter lines vanish nor

Eng. Struct., 1983, V o l . 5, J u l y

1.017 )203

=

0.258

(31)

170

°

36.33

(29)

In this case, the condition for a bifurcation point on the load-displacement path is obtained as:

Det

(32)

_ Q342 - 0.762 59.68

Figure 7 Cooling tower at Port Gibson, Miss., USA. Characteristic dimensions

Wind loaded RC cooling towers: H. A. Mang et al.

contains material numbers, referring to smeared-steel layers. In the second column the boundaries/~ = constant of these steel layers are listed. The third column contains average distances +-vs of the middle surfaces of the steel layers from the middle surface of the shell divided by the thickness h. The concrete cover on both faces of the shell is relatively large, resulting in relatively small values of vs/h. The fourth (sixth) column contains the diameters of the reinforcement in the hoop (meridional) direction. The fifth (seventh) column contains the thicknesses of the respective smearedsteel layers on one of the two faces of the shell. The reference pressure function, that is, the part of the reference wind load which does not depend on the displacements, is given as: 2' 7

Table 2 Coefficients for determination of generatrix of shell z-~#;~0 Hyperbola A B

[1 ] [1 ]

C

[1]

1

D

[m]

1.417 4

z------#<0 Ellipse

--0.002 601 0 --0.038 937

0.860 2 0.710 4

1

--25.859 8

E

[m]

--35.764 2

--247.231 9

F

[m 2]

1278.667 8

16674.190 4

Table 3 Material properties of concrete and steel

Concrete E~ ) vC 3"

2.826 8 0.175 24.25 -3.447 0.319 1.0 --0.043 47

fcu ftu ZBu

C=2L

MN/cm 2 kN/m 3 kN/cm 2 kN/cm 2 kN/cm 2

p(sR) = 413.013Ha(3.281/~ + 393.696) 2:7 where: 12 H a = - - 0 . 5 + ~ A, cos(ne) n=O

-

Steel

E~

) aSy eSu

Table 4

20.594

MN/cm2

1.030 41.37 62.05

MN/cm ~ kN/cm 2 kN/cm 2

Location and thickness of 'smeared-steel layers'

Mat. no.

From/to (m)

~s/h

q~

~h

¢~

I.~h

(1)

(cm)

(era)

(cm)

(cm)

1 2 3 4 5 6 7 8 9

30.50 23.52 0.00 --18.94 --37.88 --56.82 --75,76 --94.70 --113.64 --120.00

+0.410 -+0.266 +-0.266 +0.250 -+0.251 -+0.278 -+0.312 -+0,339 -+0,406

2.38 0.96 0.96 0.96 1.27 1.27 1.27 1.27 1.60

0.203 0.035 0.035 0.035 0.049 0.049 0.050 0.057 0.142

1.27 0.96 1.06 1.59 1.75 1.91 1.43 1.43 1.43

0.063 0.036 0.051 0.098 0.117 0.124 0.119 0.116 0.108

1 6 6 6 9 9 4 3 0

(33)

0 9 3 6 0 7 3 0 3

numbers in Figure 7 and Table 2 were obtained converting data given in American standard units to metric units. Restricting the accuracy of consideration of the height and the radius of the shell to cm and of its thickness to mm obviously results in a loss of accuracy which, for practical purposes, is irrelevant. The simplifying assumption of a rigid, hinged base was made. Recently, Meh131 has demonstrated that the influence of the flexibility of the supports on the buckling pressure of the given shell is relatively small. Material data are compiled in Tables 3 and 4. Table 3 contains material properties of concrete and steel. Quantities in this table which were not def'med previously are 3'

~Sl;:tc~t~ Wo~,sg~t:)~):~h~b ~l~d ~: ~ n ggthm)odE(us~:m?deUle~;, of asy (yield stress of steel), and osu (ultimate stress of steel). The quantities Ets°), Etsh), (ray, and aSu defme the bilinear stress-strain diagram for steel, mentioned earlier in the paper. Table 4 contains information on two orthogonal reinforcement nets (one near the outer and one near the inner face) varying with the height. The first column

(34)

The dimensions of/~ = z and p(sR) are m and N/m 2, respectively. The coefficients A, are given in Table 5. Figure 8 shows the distribution of the wind load for the luff and the lee meridian. The wind load is acting on the deformed shell shown by full lines. The dashed lines refer to the undeformed shell. Because of symmetry of the wind load, only one half of the cooling tower needs to be analysed. Figure 9 shows the FE mesh for one half of the shell. It consists of 12 x 12 x 2 = 288 curved triangular finite elements (12 tings of elements in the hoop direction, 12 stripes of elements in the meridional direction). Before the f'mal decision on the form of the mesh was made, detailed numerical studies were performed. They were restricted to linear-elastic, static analysis. In the course of these studies, a 24 x 24 x 2 mesh resulting in 1152 elements was also used. For the 12 x 12 x 2 mesh, different divisions in the vertical direction were taken into account. The motivations for selection of the 12 x 12 x 2 mesh shown in Figure 9 were (a), good agreement of stresses and of the determinant of the stiffness matrix with corresponding quantities from the 24 x 24 x 2 mesh and (b), coincidence of element boundaries and of boundaries of zones of different degree of reinforcement. Table 6 contains coordinates/~ = z of circumferential boundaries of elements as well as corresponding thicknesses of the shell. Between specific values of the shell thickness linear interpolation is used in the meridional direction. The same mesh was used for all numerical analyses.

Buckling without consideration of material nonlinearity This subsection contains results of buckling analyses of the mentioned cooling tower for the simplifying assumption Table 5 Fourier-coefficients for determination of distribution in hoop direction

wind-load

n

An

.

An

0 1 2 3 4 5 6

0.383 30 --0.279 18 --0.619 78 --0.509 27 --0.091 67 0.11794 O.033 33

7 ,8 9 10 11 12

--0.044 74 --0.008 --0.009 --0.013 --0.005 --0.016

33 72 56 97 67

Eng. Struct., 1983, Vol. 5, July

171

Wind loaded RC cooling towers: H. A. Mang et aL i_

7704

I-

l

_1

-r

! 30.50

Ps(~=n)

Ps ( a = o )

120. O0

i= F'-

-t

11936

Figure 8 Wind profile for luff and lee meridian

Figure 9 Finite element mesh

of a linear-elastic material. Material parameters (u C and 3') are given in Table 3. Considering the stiffness of the reinforcement, results in a modified initial tangem material modulus, given as if(o) = 2.9569 MN/cm 2. Linear buckling analyses on the basis of equation (26) as well as nonlinear instability analyses (equation (l 7)) were performed. Reasons for choosing equation (26) for computation of initial linear buckling pressures instead of either equations (27) or (31) were that: (a), from a mathematical viewpoint, equation (26) represents a consistent linearization of equation (17) and (b), equation (26) was expected to result in lower buckling pressures than equation (17). 12'30 It should be noted that the consistency of equation (26) with equation (17) does not imply at all that initial linear buckling pressures obtained from equation (26) are better approximations to the instability load resulting from the nonlinear buckling condition than are initial linear buckling pressures obtained from the two alternative modes of linear buckling analyses. In addition to the actual pressure function, given by equation (33), an equivalent axisymmetric pressure function was considered. It is defined as the pressure function for the luff meridian (a = 0). This was done to check whether or not the concept of equivalent axisymmetric pressure would yield results on the safe side. In passing, it is noted that the axisymmetric loading used by Zerna et al. ~3 is different from the equivalent axisymmetric pressure function employed by the present writers. Two linear buckling analyses were performed disregarding the influence of dead load. Table 7 contains the results in the form of 'critical load intensities' ?(ca. Results in parentheses were communicated to the first author by Abel. 32 These results were obtained using triangular isoparametric elements with 36 degrees of freedom? It is seen that (a), initial linear buckling pressures computed on the basis of equation (26) are considerably smaller than corresponding buckling loads from nonlinear buckling analyses; (b), results agree well with results communicated by Abel; (c), buckling loads based on the equivalent axisymmetric concept are considerably larger than corresponding buckling pressures from the actual pressure function; and (d), the influence of dead load on initial linear buckling pressures is negligibly small in case of using the actual pressure function; it is greater in the case of applying the equivalent axisymmetric pressure function. Abel also communicated a buckling pressure based on the condition for a linear prebuckling path (equation (31)). The critical load intensity, ?(cn, was obtained as 18.7. This value differs relatively little from the instability pressure resulting from a nonlinear prebuckling analysis. Quoting from Abel, 32 'thus it appears, for these problems, that the Table 7 Critical load intensities ×CR, disregarding material nonlinearity

Table 6 Coordinates/3 ~ z of circumferential boundaries of finite elements and corresponding shell thicknesses /3--=z (m)

h (m)

~=--z (m)

h (m)

30.500 23:523 17.642 11.762 5.881 0.000 --18.940

1.017 0.203 0.203 0.203 0.203 0.203 0.203

--37.879 --56.818 --75.758 --94.697 - - 113.637 - - 119.999

0.203 0.203 0.258 0.300 0.342 0.762

172

Eng. Struct., 1983, V o l . 5, J u l y

Critical load intensity ×CR

Load

XCR X wind load Dead load + (×CR X wind load) ×CR X equivalent axisymmetric pressure Dead load+ (×CR X equivalent axisymmetric pressure)

Linear buckling Nonlinear using eq. (26) instability 4.80 4.80 (4.9732 ) - - 2 0 * (19.532 ) 13.67 12.85

~ 33 t

* Load increments ,~× = 2. t Load increments A× = 3.

Wind loaded RC cooling towers: H. A. Mang et aL

linear and quadratic terms of the initial-displacement stiffness matrix tend to offset each other and that the linear-initial displacement buckling remits are misleadingly low'. Nevertheless, even these 'misleadingly low' results will be found to be considerably higher than corresponding results obtained from ultimate load analyses. Consequently, in the given ease, an assessment of results from different modes of linear buckling analyses should not ignore the result from ultimate load analysis. Apart from this fact, the question for the 'best mode' of linear buckling analyses of hyperbolic shells subjected to nonaxisymmetric pressure loads is an important topic for pertinent research. With regards to (c), it is interesting that Chang 3 holds the opinion that the concept of equivalent axisymmetric pressure ' . . . always gives lower bounds to the solutions of nonaxisymmetric linear buckling and full nonlinear analysis when the internal suction is included in the equivalent axisymmetric pressure approach'. As was shown by the writers, this is not true. In this context it should be emphasized that the load-carrying mechanism of the wind-loaded shell is strikingly different from the one of the axisymmetrically loaded shell. The former is characterized by alternating bands of meridional tensile and compressive stresses which approximately coincide with regions of compression and suction of the wind load. This type of Table 8 Dependence of initial linear buckling pressures on factor p in/' = p .r(z) f o r wind load and equivalent axisymmetric pressure Critical load intensity XCR

XCR X equivalent p

XCR X Wind load

axisymmetrical pressure

1.0 1.5 2.5

4.89 9.02 9.40

13.67 11.25 8.71

load produces meridional stresses which are considerably larger than the ones resulting from the equivalent axisymmetric pressure. The latter primarily activates the loadcarrying capacity of the shell as a ring which, depending on the dimensions of the shell, may be considerably larger than its capacity to carry the stresses from the 'actual' wind load. By variation of characteristic shell parameters such as, for example, a proportional change of the shell radius r(z), resulting in f = pr(z) where p is a proportionality factor, it is easy to show (see Table 8) that the concept of equivalent axisymmetric pressure may yield results on either side of the corresponding buckling pressure for the nonaxisymmetric wind load. In order to ensure that the results are on the safe side, the equivalent axisymmetric pressure would have to be made dependent on the dimensions of the shell. Because of the large number of possibilities for variation of parameters, this might prove to be unfeasible. Figure 10 gives an idea of the eigenform of the shell obtained from a linear buckling analysis considering dead load and wind. Transverse displacements are plotted along 13 meridians of one half of the shell. Figure 11 shows a plot of the transverse displacement w at point a = 0, ~ = 0 versus the load intensity factor X. Note that the latter refers only to wind load. It is seen that approximately up to X = 16.0 the behaviour of the shell is nearly linear. After the second step of the iteration at X = 20, a change of sign of the buckling determinant was noticed. Because it was expected that the ultimate load of the RC shell would be considerably smaller than × = 20, no attempt was made to refme the geometrically nonlinear analysis in the vicinity of this load intensity.

Ultimate load analysis This subsection contains results of ultimate load analysis of the considered cooling tower. Results are presented from

Crown

cO "o t

_E__

1 BQse ~=O

Ot =-~-

~=~

Figure 10 Eigenform (transversedisplacement)associatedwith initial linear buckling pressurefor dead load and wind according toequation (26)

Eng. Struct., 1983, Vol. 5, July

173

Wind loaded RC cooling towers: H. A. Mang et aL

20 Determinant changes sign a f t e r second step of equilibrium iterat=on

15

x 10

I -1.0

I -20

I I -3.0 -4.0 w,(cm)

I -5.0

I -6.0

Figure 11 Transverse displacement w at p o i n t a = O,/3 = 0 (intersection of l u f f meridian and throat) versus load intensity factor × (geometric nonlinearity only)

Yield plateau S t a t e "IT

Hardening

1.5 Crack plateau o

o

1.0 x

Figure 12 shows a plot of the transverse displacement w at point a = 0,/3 = -37.9 versus the load intensity factor ×, obtained from the aforementioned standard analysis. At each load increment, a full Newton-Raphson equilibrium iteration was performed. The maximum number of iteration steps was chosen as 15. In order to study the influence of this parameter on the solution, one run was also made with a maximum number of 10 iteration steps per load increment. With the exception of the right end of the flat part of the load-displacement curve, the two results agree very well (see Figure 13). As shown in Figure 12, the load-displacement path consists of two main parts, characterized by intact (state I) and fractured concrete (state II), respectivel_y: The latter can be divided into three sections. Section AB is termed the 'crack plateau'. It is characterized by rapid crack propagation associated with a rapid increase of displacements with practically no increase of the external load. As will be seen later, the length of the crack plateau primarily depends on the percentage of reinforcement. Section BC is characterized by hardening of the partially fractured structure. The last section, beginning at point C, is termed the 'yield plateau'. Points A and C determine the 'crack load' and the 'yield load', respectively. The crack load is controlled by the tensile strength of concrete. Because of hardening of the steel, the yield load is smaller than the ultimate load. For the analysis, however, the yield load is considered as a sufficiently accurate (lower bound) approximation to the ultimate load. In this context it should be mentioned that already at the end of the crack plateau (point B in Figure 12), the displacements are inadmissibly large. This is also illustrated in Figure 14, which contains a comparison of the transverse displacement w for a load intensity just below

,tate Z 0.5

M a x t m u m number of steps f o r equilibrium iteration at each load increment

1.50

0

I -0.1

I -0.2

t -0.3

I -0.4

I -05

[ -06

I -07

1.45

w(m)

Figure 12 Transverse displacement versus load intensity factor X

w at p o i n t ~ = O,/3= --37.9 x 140

(a), the 'standard analysis', characterized by consideration of the given reinforcement percentage, the 'actual' wind load, geometric nonlinearity and tension stiffening; and (b), a parameter study. Within the framework of the latter, the shell was analysed, for example, for the equivalent axisymmetric pressure. Moreover, the influence of the reinforcement percentage on the result was investigated by means of a variation of this parameter. Finally, one run each was made disregarding geometric nonlinearity and tension stiffening, respectively. For the analysis, all finite elements were subdivided into five concrete layers. Preliminary computer runs with seven as well as with three layers gave almost identical results. This is not astonishing because the shell carries the external load mainly through membrane action.

174

Eng. Struct., 1983, Vol. 5, July

1.35

1,30

0

-01

-0.2

-03

-0.4

-0.5

-06

-0.7

w(m)

Figure 13 Transverse displacement w at p o i n t e = O,/3= --37.9 versus load intensity factor X: influence of m a x i m u m number of steps for equilibrium iterations at each load increment

Wind loaded RC cooling towers: H. A. Manget al. 0 - O: I - 0;2 - O: 3 - 0;4-0:5

J~ ,(m)

..

w,(m)

! ! !

I I

-18.9

I I I

I I !

, /

30.5 23.5 17.6 11.8 5.9 +0.0

-37.9 |

I i !

- 56.8

I !

-75.8

---a--

X

=

1. 3 4 5

X

:

1.340

-g4.7

-113.6 -119.9 T~ 0~=~-

(z=O

~:

0.: ~__. 2

CI.: 1~

311:

4

Figure 14 Transverse displacement w at selected meridians f o r X = 1.330 and × = 1.345

l, II/',/42

! : ~ - - ~ . -. \.\ ,~ ,l !l

II

~V~

\ \\

II

\ \ ;-- -~.-~ \ ~ , \ \

./

I

t

I.

/

t

/

/

.

I

,

\ 1

4

,

,

# I

"Smeared crocks"

a

Layer 4

!

i

F

(outer face)

Figure 15 'Smeared cracks' and principal stresses in three d i f f e r e n t concrete layers f o r load intensity factor corresponding to p o i n t B (end o f crack plateau) in Figure 12

Eng. Struct., 1983, Vol. 5, July

175

Wind loaded RC cooling towers: 1"t. A. M a n g et al. Figure 15 - continued //i "i1"/ ,.,/,,. ii

l / ii ~/~,. '. 6" i

~,/~I /

,

% t t

I

X x \ x

t

,,,,if:

t

-\/

/

/

-

\ #

t

,

'

ttt ,, ttt, ttt,

•

]

•

I

,

t

]

-

\

"

~

"

t

•

t

t

t ,'r

Layer

b f~, / l i f t / •I z//II~'/1/11 II I

t

t

(centre) 1

~,.",! 4' / Ill) I

X

!

*

t t

1

, ,111't

4

4

I~f I

/

=~

F

,/ ;,, '.K... \ . .

F

it//

\

t

/

*

t

F /

L/

U

#

I

/

L

C

Layer 5

( inner fece ) I

0

the crack load with w for the load intensity corresponding to point B of the crack plateau. Figure 15 gives a good account of the state o f affairs. It contains plots o f 'smeared cracks' as well as o f principal stresses for three concrete layers for a load intensity factor corresponding to point B of the crack plateau. The directions o f the dashes in the illustrations o f the fractured shell indicate the directions o f the cracks. The lengths o f the

176

Eng. Struct., 1983, Vol. 5, July

,

I 1

,

J

I

f

!

Principal stressess ', ~ =tension 2 kN/cm 2 -compression

dashes are proportional to tributary areas of integration points located in fractured parts of the shell. Figure 15 illustrates that the load-carrying mechanism of the wind-loaded shell has little in common with the one resulting from axisymmetric pressure. In the latter case the shell is carried primarily by ring action. The wind-loaded shell, however - with the exception of the part above the throat - carries the load mainly through a band of meri-

Wind loaded RC cooling towers: 14. A. Mang et aL 2.4

Table 9 Comparison of crack and ultimate load intensities with critical load intensities disregarding material nonlinearity

Percentages of reinforcement 2.3

Load intensity factor × ( X C R , x U)

Mode of analysis Linear buckling Nonlinear instability

Dead load + (× X wind load) 4.80

Dead load + (X X equivalent axisymm, pressure)

Crack load

1.34

12.85 -33 b 12.75 c

Ultimate load

1.49

12.75 c

-20 a

2.4

a Load increments &X = 2 b Load increments ZXX= 3 c Load increments 4 × = 0.25

- - . ~ - - . - No reinforcement -~O

Regular (see Tables 4 and 6 )

- - ~ l

Doubled

/

/ []I I

/

2.0

l 1

/

1.9

I

1.8

!

X

d

I

I

1.6

Approximate determination of the crack load The concept presented for ultimate load analysis of cooling towers made of reinforced concrete is computationally expensive. However, the majority of the computer time

i

2.1

1.7

dional tensile forces, with the luff meridian as the axis of symmetry, and through two adjacent bands of relatively large meridional compressive stresses. Different loadcarrying mechanisms result in different failure modes and, in general, in different critical and failure loads, respectively. Table 9 contains a comparison of crack and ultimate load intensities with critical load intensities disregarding material nonlinearity. It is seen that the linear as well as the geometrically nonlinear buckling analysis of the given cooling tower, which may be considered a typical representative of this class of shells, yields results which are on the unsafe side of the crack load and the ultimate load, respectively. Furthermore, it is seen that for each of the three modes of analysis considered, the concept of equivalent axisymmetric pressure gives a result which is on the unsafe side of the corresponding result for wind load. Table 9 shows that the ultimate load is only 11.7% of the corresponding value resulting from the concept of equivalent axisymmetric pressure. Consequently, it is problematic to advocate consideration of material nonlinearity while maintaining axisymmetry. With regards to equivalent axisymmetric pressure, in Table 9 the same result is given for the crack load and the ultimate load. This is a consequence of having chosen relatively large load increments. Realizing that the concept of equivalent axisymmetric pressure is problematic, no attempt was made to 'improve' determination of the 'failure load' for this fictitious load case. Figures 16, 17 and 18 contain results from a study of the influence of the reinforcement percentage. It is seen that the ratio of the crack loads for the three considered reinforcement percentages is approximately equal to the ratio of the corresponding membrane stiffnesses. The length of the crack plateau decreases with increasing reinforcement percentage. Finally, the difference between the ultimate load and the crack load becomes larger with increasing reinforcement percentage. Figure 19 refers to a study of the influence of geometric nonlinearity and of the tension stiffening effect, respectively, on the transverse displacement w at point a = O, /3=--37.9. It is seen that disregard of geometric nonlinearity results in a 'stiffer' solution whereas disregard of tension stiffening yields a 'softer' solution.

D------

ii

/

1.5

/

/

/

/

g

1.4 1.3 I I I I I 1 I I I 1.2, " f l -0.1 -0.2 -0.3 -0.4 -0.5-0.6 - 0 . 7 - 0 . 8 - 0 . 9 -1.0 w,(m)

Figure 16 Transverse displacement w at point c=: 0,/3 = --37.9

versus load intensity factor X for three different percentages of reinforcement

2.4 2.3

Per~ent~Jes of reinforcement o

Regular

/o

24

/ -- --a----

Doubled

2,4

/

2O

I

/

I

l

I

I 1.9

I

/ x

1.8

l

/ 1.7

/

1.6

l

l

1

// /

1.5

/

1.4 - ~ - - - - - - O - - J

1.3

I

1.2

y

o I

I

I

I

10

20

30

40

(Tsy/~s~ , ( k N / c m ) Figure 17 Meridional stress stress aS/~ at p o i n t e = O,/~----100 versus load intensity factor X for three different percentages of reinforcement

Eng. S t r u c t . , 1 9 8 3 , V o l . 5, J u l y

177

Wind loaded RC cooling towers: H. A. Mang et aL 24 23 I

Percentages of reinforcement 2.2 2.1

20

--'~-'--

No reinforcement

~

Regular

---o---

Doubled

I

¢

n = number of crocked subregions

! 19

x

I

I I

I

I

17

,i 16

J

/

/

/

/

,,

/ s*

/

/

f

1.3 I 1000

I 1500 n

I 2000

I 2500

Figure 18 Number of cracked tributary regions of integration points versus load intensity factor × for two different percentages of reinforcement

1.8

----o----

G e o m nonhnear, tension stiffening

--o----"~---

Geom. linear No tension stiffening

1.7

1.6

brane forces in the concrete along the luff meridian, resulting from dead load and the reference pressure function, respectively, and A s is the area of the existing meridional reinforcement provided per unit length. For the cooling tower investigated, Meb131 reports the smallest value of Xu as 1.53. This value agrees well with Xu = 1.49, given in Table 9. From an independent analysis Guedelhoefer 33 recently obtained Xu = 1.46.

¢ I

,~,

.a

A/ 1.4

(2,/~'~'"

~

Ultimate load analyses of the cooling tower investigated have shown that the load-carrying capacity of the shell is strongly tied to the yield limit of steel, asy. Beyond this limit, that is, in the strain-hardening regime of steel, the deformations increase rapidly without a significant increase of the load. Equilibrium iterations fail to converge. The indicated situation justifies the concept of approximate 'ultimate strength design' currently in use in the USA. 33 Assuming that the redistribution of stresses after cracking of concrete does not affect the integral, that is, the internal force, the load intensity factor for the ultimate load, ×,, can be estimated as follows:

n~L(PV ) + Xun~flL(p~R)) = Asosy (36) where ~##L(PV) and ~NL(p(SR)) are the meridional mem-

[2 I E~ I I I

×

(35)

Approximate determination of the ultimate load

0

I 500

] -- Ytu

O~g[3L(PV) and O(cO~¢L(p(s n)) are the meridional (membrane) stresses in the concrete along the luff meridian, resulting from dead load and the reference pressure function, respectively; ×cn is the load intensity factor for the crack load. The uniaxJal tensile strength of concrete, ftu, is given as 0.31 kN/cm 2 (see Table 3). Table 10 contains stresses O~aL(Pv) a n d O(~L(p~ R)) as well as load intensity factors Xcn for crack loads at different heights of the shell. At 13= --63.13 m, XCR takes on a minimum. It was obtained as 1.28. This value agrees well with XCR = 1.34, given in Table 9.

d"

1.2~

., ~(o) t~(R)~ ACRUC~L\VS

where

1.41 ---- --0-- --n-..C~ ,

-

OC(g3L(Pv) +

,/

1.5

(o)

I

t

1.8

to this meridian. It follows that the luff meridian represents a trajectory of principal stress. If the principal compressive stresses in the hoop direction are much smaller than the prism strength of concrete, as is the case for the cooling tower analysed, the load intensity factor for the crack load can be estimated as follows:

Conclusions

120J

i

J 01

"J

i I I l I 0.2 0.3 0.4 0.5 0 6

I I 07 08

I 09

I 10

w(m) Figure 19 Transverse displacement w at point a = O, ~ = --37.9 versus load intensity factor X: study of influence of geometric nonlinearity and of tension stiffening effect respectively

is required for tracing the load-displacement path from the crack load (point A in Figure 12) to the ultimate load (point C in Figure 12). It was shown that because of the small percentage of reinforcement the difference between the crack load and the ultimate load is relatively small. For design purposes it seems to be adequate to predict the crack load. Figure 12 shows that up to this load the load-displacement path is practically linear. Figure 15 illustrates that the cracks on the luff meridian are normal

178 Eng. Struct., 1983, Vol. 5, July

The following conclusions can be drawn from the present research:

(1) For the cooling tower analysed, buckling loads from linear as well as geometrically nonlinear analyses are con-

Table 10 Approximate determination of load intensity factors for crack loads, referred to the luff meridian (0)

•

.

(0)

(p(R)s

Height (m)

OCtaL (Pv)

OCfl~L

(N/cm 2)

(N/cm =)

Xcr

--25.25 --44.19 --63.13 -82.07 --101.01 -115.76

--159 -184 -193 --200 --203 --153

252 364 401 388 339 220

1.78 1.36 1.28 1.33 1.52 1.96

Wind loaded RC cooling towers: H. A. Mang et aL

siderably larger than the ultimate load. Because this cooling tower may be regarded as a typical representative of the given class of shells, it seems reasonable to extrapolate from the results obtained that the failure of wind-loaded cooling towers made of reinforced concrete is initiated by rapid propagation of cracks in the tensile zones of such shells and not by buckling. Consequently, analysis concepts which are based on buckling - be it in the conventional, global sense or in the sense that a certain state of biaxial compression is considered to be responsible for initiation of bucklinga4, as _ are inadequate for even an approximate assessment of failure loads of reinforced-concrete coolingtower shells. In this context it should be noted that the computer program would have signalled a bifurcation point or a limit point prior to cracking of the concrete and yielding of the reinforcement if compression had been relevant for the loss of stability of the shell. After submittinlg.the original manuscript of this paper, the third writer ~ analysed a cooling tower recently built in Austria. The trends he found from the analysis were similar to those reported in this paper. Thus, on the basis of the results obtained, in the design of cooling towers, there seems to be no need for a proof of sufficient safety against modes of loss of stability - bifurcation or limit-point buckling - which have little to do with the actual reason for failure of a reinforced-concrete cooling tower: cracking of concrete and yield of steel. (2) For the cooling tower which was analysed, linearization of the buckling condition for geometrically nonlinear instability analysis gives a considerably lower eigenvalue than has been communicated by Abel 32 on the basis of the buckling condition for the special case of a linear prebuckling path. The latter eigenvalue differs relatively little from the instability pressure resulting from a nonlinear prebuckling analysis. 'Thus, it appears, for these problems, that the linear and quadratic terms of the initial-displacement stiffness matrix tend to offset each other and that the linear-initial displacement buckling results are misleadingly low.'32 Nevertheless, they are considerably higher than corresponding results obtained from ultimate load analyses. Buckling results communicated by Abel 32 agree well with corresponding results reported in this paper. (3) For the cooling tower, results based on an equivalent axisymmetric pressure are on the unsafe side of corresponding results for the 'actual' wind load. By a proportional change of the shell radius without changing any other dimensions it was shown that the concept of equivalent axisymmetric pressure may yield results on either side of the corresponding buckling pressures for the 'actual' wind load. This is not surprising because the load-carrying mechanism of a cooling tower subjected to wind load is characterized by a combination of cantilever-beam and ring action. Depending on the dimensions of the shell, one of these two component actions prevails. The load-carrying mechanism of a cooling tower subjected to axisymmetric pressure, on the other hand, is dominated by ring action. If cantilever-beam action prevails, as is the case for the cooling tower considered, analysis concepts based on axisymmetry will be inadequate, making it almost impossible to assess the load-carrying capacity of reinforced-concrete cooling towers. In order to ensure that results from axisymmetric pressure are on the safe side, this type of load would have to be made dependent on the dimensions of the shell. Because of the large number of possibilities for variation of parameters, this might prove to be unfeasible. (4) By means of ultimate load analysis based on a concept

which considers (a), nonlinearity of concrete in the compression-compression domain and in parts of the compression-tension domain; (b), fracture of concrete within the framework of a concept of 'smeared cracks'; (c), strain-hardening of the reinforcement assumed to be 'smeared' to thin layers of steel; (d), tension stiffening; (e), geometric nonlinearity;and (f), follower loads of the form of hydrostatic pressure, the failure-mechanism of the wind-loaded cooling tower was explained. It was shown that a typical load-displacement path consists of two main parts, characterized by intact and fractured concrete, respectively. The second part can be divided into three sections, namely, the crack plateau, the hardening section and the yield plateau. (5) Through variation of the reinforcement percentages it was demonstrated that the ratio of the crack loads is approximately equal to the ratio of the corresponding membrane stiffnesses. It was also shown that the length of the crack plateau decreases with increasing reinforcement percentage and that the difference between the ultimate load and the crack load increases with increasing reinforcement percentage. (6) It was demonstrated that disregard of geometric nonlinearity results in a 'stiffer' solution. Disregard of tension stiffening, on the other hand, gives a 'softer' solution. (7) It was shown that the crack load as well as the ultimate load of the cooling tower analysed can be estimated by means of a linear-elastic nonaxisymmetric analysis. Since there is no reason to believe that other cooling towers made of reinforced concrete would not exhibit a predominantly linear behaviour up to the crack load, it seems to be justified to make the generalizing statement that linear-elastic nonaxisymmetric analysis is an adequate tool for the design of wind-loaded cooling towers. Concerning estimates of the ultimate load, it seems to be premature to make generalizations on the basis of the available numerical evidence. (8) Because failure of concrete in tension would initiate the collapse of cooling towers, the tensile strength of concrete is of crucial importance for the safety of these shells. It is well known that this quantity may vary considerably. Therefore, care should be exercised when specifying minimum values for the tensile strength of concrete in codes of practice. (9) Experimental investigations should consider cracking of the material and nonaxisymmetry of the load.

Acknowledgements The first writer is indebted to P. L. Gould, J. F. Abel and O. C. Guedelhoefer for stimulating discussions and helpful suggestions and, in the case of J. F. Abel, also for providing him with a llst of errors in the original manuscript. He also thanks J. K. Arnston of the Cooling Tower Division of ZURN Industries, Tampa, Fla., who made the necessary material data available. The help of P. Torzicky concerning computational aspects is gratefully acknowledged. The excellent co-operation of the Computing Center of Technical University of Vienna, Austria, in providing the writers with large amounts of computer time, was much appreciated. Financial support from the Foundation for Promotion of Scientific Research of Austria is also acknowledged.

Eng. Struct., 1983, Vol. 5, July 179

Wind loaded RC cooling towers: 1-t. A. Mang et aL

References 1 2

3

4 5 6 7 8 9 10

11 12 l3

14

15 16 17

Committee of inquiry into collapse of cooling towers at Fcrrybridge, Monday, 1 November 1965, Report, Central Electricity Generating Board, London, 1965 Floegl, H. 'Traglastermittlung donner Stahlbetonschalen mittels der Methode der Finiten Elemente unter Beriicksichtigung wirklichkeitsnahen Werkstoffverhaltens sowie geometrischer Nichtlinearit~it', Doctoral Dissertation, Technical University of Vienna, Vienna, Austria, May 1981 Chang, S. C. 'An integrated finite element nonlinear shell analysis system with interactive computer graphics', Department of Structural Engineering Report No. 81-4, Cornell University, Ithaca, N.Y., USA, February 1981 Zerna, W. (ed.). 'Kiihlturm Symposium 1977', Heft 29/30 der Berichte aus dem Institut fur Konstruktiven Ingenieurbau, Ruhr-Universit~it Bochum, Vulkan Verlag, Essen, 1977 Abel, J. F. and Gould, P. L. 'Buckling of concrete coolingtower shells', in Concrete Shell Buckling, ACI Special Publication SP-C7 (Medwadowski, S. and Popov, E. P., eds.), 1981 Zerna, W. et al. 'Cooling tower practice in Germany: state of the art', Preprint 81-005, ASCE Int. Cony. and Exp. 1981, New York, May 1981 ACI-ASCE Committee 334. 'Reinforced concrete cooling tower shells - practice and commentary', J. ACI, 1977, 74, 22 Abel, J. F. et al. 'Buckling of cooling towers', J. Struct. Div., ASCE, 1982, 108 (STI0) Der, T. J. and Fidler, R. 'A model study of the buckling behaviour of hyperbolic shells',Proc. Inst. Or. Engrg., 1968, 41,105 Floegl, H. and Mang, H. 'Zum Einfluss der Verschiebungsabh/ingigkeit ungleichf'6rmigen hydrostatischen Druckes auf das Ausbeulen diinner Schalen allgemeiner Form', Ing.-Archiv., 1981, 50, 15 Zienkiewicz, O. C. 'The f'mite element method' (3rd edn), McGraw-Hill, London, 1977 Brendel, B. et al. 'Linear and nonlinear stability analysis of thin cylindrical shells under wind load',Z Struct. Mech., 1981, 9, 91,1981 Zerna, W. et al. 'Bestirnmung der Beulsicherheit von Schalen aus Stahlbeton unter Beriicksichtigung der physikalisch-nichtlinearen Materialeigenschaften', Deutscher Ausschuss ftir Stahlbeton, Heft 315, Wilhelm Ernst, Berlin, 1980 Mang, H. A. and Floegl, H. 'Analytical prediction of short-term behavior of reinforced concrete panels, slabs and shells', Preprint 81-103, ASCE Int. Conv. and Exp. 1981, New York, May 1981 Thomas, G. R. and Gallagher, R. H. 'A triangular thin shell finite element: linear analysis', NASA Contractor Rep. 2483, Washington, DC, USA, 1975 Floegl, H. and Mang, H. A. 'Tension stiffening concept for rc panels based on bond slip', J. Struct. Div., ASCE (in press) Floegl, H. and Mang, H. A. 'On tension stiffening in cracked

181) Eng. Struct., 1983, Vol. 5, July

reinforced concrete slabs and shells considering geometric and physical nonlinearity', lng.-Archiv. 1981, 51, 215 18 Mang, H. A. and Floegl, H. 'Tension stiffening concept for reinforced concrete surface structures', 1ABSE- Colloquium on Advanced Mech. Reinforced Concrete, Final Report, Delft, The Netherlands, June 1981, 351 19 Mang, H. A. etal. 'Finite element shell instability analysis (FESIA), computer program manual', Report, Department of Structural Engineering, Cornell University, Ithaca, USA, June 1976 20 Walter, H. 'Zur numerischen Bestimmung yon Beuldriicken donner Platten und Schalen bei Beriicksichtigung einer Eigenschaft der Sturm'schen Folgen', Engineer's Thesis, Technical University of Vienna, Vienna, Austria, November 1980 21 Mang, H. A. et al. 'Finite element instability analysis of hyperboric cooling towers', Advances in civil engineering through structural mechanics, ASCE, New York, NY, USA, 1977, 246 22 Mang, H. et al. 'Deformation und Stabilit~it windbeanspruchter Kiihlturmschalen', Ing.-Archiv. 1978, 47, 391 23 Koiter, W. T. 'General equations of elastic stability for thin shells', Proc. Symp. on the theory of shells to honor Lloyd Hamilton Donell, University of Houston, Houston, Tx., USA, 1967 24 Liu et al. 'Biaxial sfress-strain relations for concrete'. J. Struct. Div., ASCE, 1972, 98 (ST5), 1025 25 Kupfer, H. 'Das Verhalten des Betons unter mehrachsiger Kurzzeitbetastung unter Beriicksichtigung der zweiachsigen Beanspruchung', Deutscher Ausschuss fiir Stahlbeton, Heft 229, Wilhelm Ernst, Berlin, 1973 26 Chen, W. F. and Ting, E. C. 'Constitutive models for concrete structures', J. Eng. Mech. Div., ASCE, 1980, 106 (EMI), 1 27 Ba~ant, Z. P. and Cedolin, L. 'Fracture mechanics of reinforced concrete',J. Eng. Mech. Div., ASCE, 1980, 106 (EM6), 1287 28 Cedolin, L. and Dei Poli, S. 'Finite element nonlinear plane stress analysis of reinforced concrete', Costruzioni in Cemento Armato, Studi e Rendiconti, Politecnico di Milano, 1976, 13, 1 29 Mang, H. A. and Gallagher, R. H. 'On the unsymmetric eigenproblem for the buckling of shells under pressure loading', J. Appl. Mech. ASME (in press) 30 Strickrin, J. A. and Martinez, J. E. 'Dynamic buckling of clamped spherical caps under step pressure loadings', A I A A J., 1969, 7, 1212 31 Mehl, M. 'Lrber das Tragverhalten yon Naturzugkiihltiirmen aus Stahlbeton', Doctoral Dissertation, Technical University of Vienna, Vienna, Austria, June 1982 32 Abel, J. F. (private communication), August 1981 33 Guedelhoefer, O. C. (private communication), July 1981 34 Mungan, I. 'Buckling stress states of hyperboloidal shells', J. Struct. Div., ASCE, 1976, 102 (ST10), 2005 35 Paduart, A. et al. 'Recommendations for the design of hyperboric or other similarly shaped cooling towers'. Report IASS Working Group No. 3, Brussels, 1977 36 Trappel, F. Engineer's Thesis, Technical University of Vienna, Austria (in preparation)