The Impact of Diameter, Number of Ribs, Percentage of Steel, Compressive Strength and Cover...

8/6/2019 The Impact of Diameter, Number of Ribs, Percentage of Steel, Compressive Strength and Cover Thickness on the Large Concrete Dome- Hani Aziz Ameen

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AMERICAN JOURNAL OF SCIENTIFIC AND INDUSTRIAL RESEARCH© 2010, Science Huβ, http://www.scihub.org/AJSIR

ISSN: 2153-649X doi:10.5251/ajsir.2010.1.3.472.495

The impact of diameter, number of ribs, percentage of steel, compressivestrength and cover thickness on the large concrete dome

Dr. Hani Aziz Ameen

Asst. Prof., Pumps Engineering Department - Technical College / Al-Musaib – IraqE-mail:[email protected]

ABSTRACT

This paper presents an investigation of the finite element analysis via ANSYS12 software for the large concrete dome. Very few studies have been devoted to the case of the analysis of large concrete dome. The prediction of the analysis of large diameter concrete dome is achallenging task. This paper have a novel discussion for the non linear analysis of reinforcedconcrete or composite hemispherical dome of monolithically or partially interconnected ribs,running in the meridional of the dome. A simplified three dimensional finite element model isdeveloped in ANSYS12, this analysis, which may be applied to three cases of hemisphericaldome: unribbed, monolithic ribs, and composite ones. The interfaces between the precast radialribs and the abutting cover in composite dome is modeled. Material non-linearity due to the

cracking and crashing of the concrete and yielding of the reinforcing steel bars are taken intoconsideration during the analysis. The current application included initially, the analysis of unribbed hemispherical reinforce concrete dome by the recent finite element model and by themembrane theory of shells, have shown very high agreement in the membrane meridional andhoop normal stresses values which were all within the 3.3% margin of difference. Application of the present numerical model to hemispherical ribbed dome of the two specified types(monolithic concrete and composite ones) has produced variations of the three orthogonalnormal stresses and the vertical displacement along meridional path from apex to base of dome, which are in perfect agreement with the logical structure behavior of such dome that isdrawn by high structural inspection. A wide parametric study the impact of diameter, number of ribs, percentage of Steel, compressive strength and cover thickness on the large dome for thethree cases to evaluate the quantities affect of the main parametric in the aspects of geometry,constitutions and material properties. Numerical percentages of the variation of ultimate loads

and vertical deflection at dome apex with variation of each of those parametric has beencomputed when given the fundamental guidelines for the structural design of such dome.

Keywords: finite element method, concrete, spherical shell, ANSYS, dome

INTRODUCTION

In the recent years, thin shell structures find wideapplications in many branches of technology such asspace vehicle, nuclear reactor, pressure vessels,roofs of industrial building and auditoriums(Chandrashekhra,1995 and Tony,1996). From thepoint of view of architecture, the development of shellstructure offers unexpected possibilities and

opportunities for the combined realization of functional, economic and aesthetic aspects(Chandrashekhra,1995). Chen,1979, studied andtested conical concrete-shell specimens with widelyvarying material properties and traced their load-deformation response, internal stresses and crackpropagation through the elastic, inelastic, andultimate stress ranges. Zweilfel,1997, presented a

small diameter concrete dome with alternate formingsystem, he built a dome of 16 ft in diameter and 8 ftin height. Manasrah, 1993, investigated theeffectiveness of connection elements of thesegmental cylindrical shell units subjected to knife-edge load at the crown of the shell. A total of eight fullscale shell unit models with four types of connectionswere constructed and tested. All the models were of 450 mm width, 35 mm thickness, 1000 mm head of the crown and 4000 mm covered span. Ford,2001,studied the stress-strain and fracture behavior for both reinforced and unreinforced concrete is tailoredto range from strong linear elastic but brittle to toughand ductile by various combinations of rubberymethyle methacrylate. Such composite specimensare found ideal for the purpose of comparison withvarious material models now available in the



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extended NONSAP program. Jackson, 2002,reported the finding of a structural evaluation of the 5meter diameter observatory dome structureconstructed by Observa-Dome laboratories, Inc.Regarding to the presentation of literature review, it

should be emphasized that no investigation related tothe analysis of large concrete thin shell dome isfound. So it can be represented this work as a firstone in the field of the study of concrete dome. Themain objectives of this study are conducting ananalytical study on the behavior of reinforcedconcrete ribbed dome with precast rib and cast-in-place cover and concrete slab under monotonicallyincreasing loads by using three dimensional finiteelement method of analysis with inclusion of materialnonlinearities due to cracking of concrete, crushing of concrete, yielding of steel bars, nonlinearity of thestress-strain response of concrete and steel, and theinterface changing status in analysis, modeling of theinterface used to simulate the interface by taking intoaccount both of the shear-friction mechanism(between precast reinforced concrete rib and cast-in-place cover) and the dowel action mechanism (of thereinforcing bars that are crossing the interface, which

contribute to the overall shear stiffness of the joint)and the study of the load–deflection behavior of suchcases as well as the maximum attained loads (loadcarrying capacity). The computer program ANSYS12has also been used to study the effects of some

material parameters used in the finite element modelwhich deal with the composite construction. Checkingand assessing the validity and accuracy of theadopted finite element modeling with availableexperimental test results are made.

Concrete Models Adopted in the Present Study:In the current study, concrete material models thatdeal with the nonlinear three dimensional analysis of reinforced concrete members under static increasingload are considered. These models treat the concreteas being a linear elastic-perfectly plastic-brittle-fracture material as shown in Fig.(1). The concreteunder a triaxial stress state is assumed to crush or

crack completely once the fracture surface isreached. The complete stress-strain relationship for aperfectly plastic-brittle fracture model is developed inthree parts (Chatterjee,1998): (1) Before yielding, (2)During plastic flow, (3) After fracture.

Fig. (1) Uniaxial stress-strain relationship used for concrete.

This stress-strain relationship is expressed by a single value of Young’s modulus, E, and a constant poisson’sratio,υ . So this relation can be written in matrix form as:

{ } [ ] { }ε=σ cD …………………………………………………………….… (1)

The matrix [ ]cD for uncracked elastic concrete can be defined by (Chatterjee,1998):

E

1

ƒ

σ

εε ε



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[ ]( )( )

( )( )

( )( )

( )( )

⎥⎥⎥⎥

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎢⎢⎢⎢⎢⎢

⎣

⎡

υ−

υ−

υ−υ−υυ

υυ−υ

υυυ−

υ−υ+=

2

2100000

02210000

002

21000

0001

0001

0001

211

ED c

….…(2)

The equivalent uniaxial stress-strains in the various stages are given by:

1) For σ ≤ fć then σ = Eε

2) For E

fć≥ε then σ = fć

The incremental stress-strain relationship can be expressed as:

{ } [ ] { }ε=σ dDd c ………….………………………………………………..………………… (3)

Determination of the Model Parameters: A total of

five strength parameters are needed to define thefailure surface as well as an ambient hydrostatic

stress state ( cf ′ , f t, f cb, f 1, f 2 andahσ ), these are

shown in Fig.(2).

Fig.2 A profile of the failure surface as a function of five parameters

cf ′ and f t can be specified from two simple tests. The other three constants can be determined from :

f cb = 1.2 cf ′ ........................................................................................................................ (4)

f 1 = 1.45 cf ′ ....................................................................................................................... (5)

f 2 = 1.725 cf ′ ...................................................................................................................... (6)

However, these values are valid only for stress states where the following condition is satisfied:

3h ≤σ cf ′…………..……...……………………………………………………………….. (7)

where:

hσ = ( )zpypxp3

1σ+σ+σ ………………………………………………………………..…….… (8)

Condition(7) applies to stress situations with a lowhydrostatic stress component. In Fig.(2), the lower

curve represents all stress states such that ( °=θ 0 ),

while the upper curve represents stress states for

( °=θ 60 ). Here θ is defined as the angle of

symmetry. The axis (ξ) represents the hydrostaticlength. The materials properties of our research istaken as in table(1).

ατ

ξo

ξtf

1ξ

cbξcξ tξ2ξ

°=θ 0

1f cbf

2r °=θ 602f

1r

fć



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Table 1. Material properties used for reinforcing concerted ribbed dome

Concrete

Ec* Young’s modulus (MPa ) 25800

c f

' Compressive strength (MPa ) 30.36

t f ** Tensile strength (MPa ) 3.1

υ*** Poisson

’s ratio 0.2

Reinforcing steel

Es*** Young

’s modulus (MPa ) 200000

y f Yield stress (MPa ) 344.8

υ *** Poisson’s ratio 0.3

Interface properties

Kn Normal stiffness (kN/m) 1638.3

Ks Tangential shear stiffness (kN/m) 1638.3 x 10-2

Coefficient of friction 1

1α *** 60

2α ***

Tension stiffening parameters0.6

βο*** 0.3

βc***Shear transfer parameters

0.7

(*) Ec= ا

c f 4700 (**) t f = ا

c f 55.0 (***) Assumed value

Finite Element Model of Concrete: In the currentstudy, three dimensional 8-node solid elements(Solid65) are used to model the concrete (ANSYS12manuals help,2009). The element has eight corner nodes, and each node has three degrees of freedomu, v and w in the x, y and z directions respectively.

Reinforcement Idealization: In developing a finiteelement model for reinforced concrete members theauthor suggested three alternative representations of reinforcement can usually be used, these are givenas follows:Distributed Representation: In this approach,

the reinforcement is assumed to be distributed in

a layer over the element in any specifieddirection. To construct the constitutive relation of a composite concrete-reinforcement, perfectbond is assumed (Fig.(3.a)).

Discrete Representation:One dimensional bar element may be used in this approach tosimulate the reinforcement. Discreterepresentation has been widely used due to itsversatility and capability to adequately accountfor the bond-slip and dowel action phenomena(Fig.(3.b)).



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Fig. 3 Reinforcement representation type

Embedded Representation: The embeddedrepresentation is often used with high order isoparametric elements. The bar elements areassumed to be built into the brick elements. Inthis approach perfect bond is assumed betweenthe reinforcing bars and the surrounding

concrete. The stiffness of steel bars is added tothat of the concrete to obtain the global stiffnessmatrix of the element. It is assumed that the barsare restricted to be parallel to the local coordinate

axes ξ , ηand ζ of the brick element

(Fig.(3.c)).In the present work, the reinforcement is includedwithin the properties of the 8-node brick solid65

elements (embedded representation) to include thereinforcement effect in the concrete structures,excluding the reinforcing bars that are crossing the

joint, which are represented by using “bar Likn8elements” (Discrete representation). In the twomanners, the reinforcement is assumed to be

capable of transmitting axial forces only, and perfectbond is assumed to exist between the concrete andthe reinforcing bars.

Model Generation: Two different methods used incurrent study to generate a model: Solid model anddirect generation. In solid modeling some one candescribe the boundaries of the model, establish



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controls over the size and desired shape elementsautomatically, by contrast. In the direct generationmethod, determine the location of every node andsize, shape and connectivity of every element prior todefining these entities in ANSYS model.

In this study the dome with large diameter (50,70 and100) meter is analysis with rib and without, firstly thedome is plotted and meshed using solid65 with

different thickness (5,7 and 10) cm as shown inFig.(4).Then the dome is reinforcement using steel elementLink8 as shown Fig.(5).

The study is concerned on the effect of the ribs onthe deflection of the dome. The section of the rib istaken as T- section as shown in Fig.(6).

Fig 4 Model of dom

Fig.5 Reinforcement in the dome

Fig 6 T-section of the rib , a = 0.6 cm , b= 0.3 cm , c= 0.6 cm , t= 0.1 cm

Three cases are studied, the first one is the domewith one rib, the second is the dome with two ribs and

third one is the dome with three ribs, in the case of one rib, the rib is located half on the dome and in the



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second case the two ribs were crossed and the thirdcase the arrangement of the ribs were two ribs crossand the third with 45 degree between the two ribs asshown in Fig.(7)

Several difficulties was attacked in the building of the

model as shown Fig.(8).

Fig 7 Dome model with one, two and three ribs

Fig.8 Interface elements modeling in ANSYS dome with pre-cats ribs and cast in place cover



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There are two major cases are studied, the first oneis the rib embedded in the dome and in this case therib is done by using direct generation using beam188with T-section while in the second case the T-sectionas an area and then dragging with path to form the

rib, this steps is done with one and two ribs while in

three ribs the transformation coordinates had beendone to get the rib at 45 degree as shown in Fig.(9)(Hani, 2010). To connect the rib with the dome, thedowel is used which represented by link8 as shown inFig.(10).

Fig 9Transformation coordinates (Hani,2010) Fig.10 Dowel caseFrom Fig.(9) it can be deduced that

1 x x x −= …………………………………………………………………..……(9)

1 y y y −= …………………………………………………..…………....……(10)

θ θ sincos1 y x x xn −+= …………………………………..…..…………..(11)

θ θ cossin1 y x y yn −+= ..……………………….……….…..…….…. (12)

The present work covers the finite element analysisof reinforced concrete ribbed dome, three cases areused to calculate values of the stress and deflection.

The first case is the flat dome (without ribs), thesecond case is the monolithic ribbed dome and thethird case is the composite concrete ribbed domewith interface between ribs and cover. All models areconstructed with the same radius and cover thickness. Nodes of the dome base are consideredfixed against translations and rotations in alldirections (fixed ring beam). Vertical downward loadis applied at the crown which is suggested to act theouter surface of the spherical dome. ANSYS12 finiteelement program is used to analyze the study casesmentioned above to determine the stress and

deflection values within the reinforced concretedome. ANSYS results are then compared with thetheoretical ones (David Billington,1990) “shell theory

equations” which achieved good agreement.Case One: Plain Dome (without ribs)Description of Structure and Finite Element Modeling: A 50 m reinforced concrete dome without ribs of 0.05 m cover thickness and subjected to a verticaldownward point load, P=1150 N/m

2applied at

crown, is modeled by (101210) 8-node brickelements Solid65 to represent the concrete of thedome. The finite element mesh, boundary conditionsand loading are all modeled by ANSYS12 systemand shown in Figs.(11) and (12).

Fig.11 Plain concrete flat dom (case one); Fig.12 FE model of unribbedgeometry , constituents & B.C. reinf. concrete dome



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0.00 20.00 40.00 60.00 80.00 100.00

Meridion Angle (Deg.)

-0.80

-0.40

0.00

0.40

0.80

1.20

H o o p S t r e s s , M

P a

Stress,FEA

Stress,Theory

B- Normal Membrane Stress in the Hoop

Direction ; θσ

Those stresses as computed by the present FE-model and the membrane theory of shell for values of

the angle φ ranging between (0°- 90°) are all shown

in table(2) and Fig.(14) .

Fig.14 Comparison of FEA and shell theory with angle vs. hoop stress

Inspection of Fig.(14) clarifies the excellentagreement between the FE-model and themembrane shell theory computation for whole range

of angle φ values. The average difference in value

of those stress is 3.3% . This excellent is fact reflectthe high efficiency of the membrane theory in

computing normal hoop stresses in axsysmmetricdouble curvature shells, since no bending stress inthe hoop direction of such shell exists.Case Tow: Monolithic Dome1- Description of Structure and Finite ElementModeling

A 50 m reinforced concrete ribbed dome of 0.05 mcover thickness and content four ribs distributioneach 45 degree, are subjected to a vertical downwardpoint load, P=0.728 MPa applied at crown, ismodeled by (101210) 8-node brick elements Solid65to represent the concrete of the dome and (330) 2-

node Beam188 element to represent the ribs of thedome. The finite element mesh, boundary conditionsand loading are all modeled by ANSYS12 systemand shown in Fig.(15).

-a- - b -

Fig.15 FE-Modeling of monlthich rib reinf. concrete dome (case(2))

The longitudinal reinforcing bars are embedded intothe brick elements at their correct positions andmodeled by Link8 finite element of ANSYS. Full

Newton-Raphson method to carry out the nonlinear analysis is adopted. A displacement convergencecriterion is used with tolerance of (0.5%).



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0.00 20.00 40.00 60.00 80.00 100.00

Meridion angle (deg.)

-12.00

-8.00

-4.00

0.00

4.00

S t r e s s M P a

stress in the x direction

stress in the y direction

stress in the z direction

2- Presentation and Discussions of ResultsA) Normal stresses in the three orthogonal direction

zyx ,, σσσ versus of angle φ , table(3) show the

values of normal stress in the three orthogonal

direction of the cartezian coordinate x, y and z for seven locations in the dome at intersection, those

value of stresses are obtained from the present finiteelement model with x-y plane, those locationspecified seven point given in table(3) are those

making φ angle of values (0,15,30,45,60,75,90)

degree.

Table 3 Stress and deflection from crown to the bottom

Values of stresses in the three orthogonal directions (i.e.

zyx ,, σσσ) which are recorded in table(3) above are all graphically plotted

versus the angle φ and shown in Fig.(16) .

Fig.16 Stress in dome with different meridian angles

From the inspection of table(3) and Fig.(16) it seenthat the values the normal stress( xσ ) has a

maximum compressive value at the crown (under theconcentrated load) with moving along the meridionalsection in the x-y plane ( i.e. with increasing the value

of angle φ from zero) . It seen that the compressive

value of xσ decreases gradually until became zero

at φ angle equal 18° beyond this location ( i.e. with

increasing the φ value beyond 18°) the xσ became

tensile and increases rapidly until it reached its max.

tensile value at φ angle equal 31°. Beyond the angle

φ value 31° the xσ value reduced slowly until its

vanishes at angle φ value equal to 90° ( i.e. at the

dome base) .

φ , degreesxσ ,MPa yσ ,Mpa zσ ,Mpa

Deflection , mm

0 -6.29 0.256 -8.68 -4.3

15 -0.367 -0.607 -2.68 -1.1

30 0.855 -0.0964 -0.423 -0.13

45 0.458 -0.169 0.132 -0.1

60 0.285 -0.145 0.0515 -0.06

75 0.16 -0.146 0.0011 -0.034

90 0.0054 -0.134 0.035 -0.021



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The variation of the normal stressyσ with angle φ (

at dome section in the x-y plane) is quiet different

from that of xσ . The variation of yσ with angle φ is

seemed uniform (i.e. extremely small variation of yσ

with angle φ ) this trend is quiet logical as the dome

meridional slice within x-y plane, act as an arch withtwo clamped edge and subjected to verticaldownward point load at the crown. The curve

representing the relation between the normal zσ

and angle φ is clearly closed in its trend to the

corresponding curve of xσ , with the zσ values

smaller than their corresponding once. This fact maybe attributed to the asysmmetric properties of theanalysis shell. Figs.(17),(18) and (19) are shown thevariation of each of the three normal membrane

stress zyx ,, σσσ over the whole outer surface of

the analysis dome respectively. From these figures

the similarity of the distribution xσ and zσ stresses

is quiet clear (with 90° orientation about y- axis) thisfact reflect the symmetric properties of the analysisdome.

Fig.17 Stress in the x- direction Fig.18 Stress in the y- direction

Fig.19 Stress in the z- direction

B) Vertical Deflection Values of vertical displacement at specified locationon dome surface at x-y plane which are shown intable(3) for the specified values of the meridional

angle φ are all plotted in Fig.(20) versus the angle

φ .



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0.00 20.00 40.00 60.00 80.00 100.00

Meridion angle (deg.)

-40.00

-30.00

-20.00

-10.00

0.00

D i s p l a c e m e n t ( m m )

Displacement (mm)

Fig. 20 Deflection in dome with different meridian angles

From this figure it easily observed that the verticaldisplacement has max. value at crown ( where the

angle φ is zero) then the displacement reduce

rapidly with the increase of value until it vanish for

angle φ value from equal to 16° after that angle φ

value the vertical displacement remain zero till thedome base.C) Fracture Pattern

Fig.(21) shows the fracture pattern of the analysisdome of the case two (concrete monolithic dome)subjected to the concentrated downward point load atcrown. This fracture pattern is as failure stage of thedome. The inspection of that figure demonstrated thecrashes concrete and compression within around thedome apex and the tensile failure cracking at rear theedge.

Fig. 21 Fracture pattern of dome case(2) at ultimate stage



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0.00 20.00 40.00 60.00 80.00 100.00


-0.80

-0.40

0.00

0.40

0.80

1.20

S t r e s s M P a

Stress in the X direction

Stress in the Y direction

Stress in the Z direction

Case Three: Dome With Precast Rib and Cast inPlace Cover 1- Description of Structure and Finite ElementModelingA 50 m reinforced concrete ribbed dome of 0.05 m

cover thickness and four ribs are subjected to avertical downward point load, P=0.61 MPa applied atcrown, is modeled by (102970) 8-node brick

elements solid65 to represent the concrete of thedome and (300) the effect of dowels which arepassing through the interface between web andflange for rib is modeled by 300 nonlinear springelements (combin14). The finite element mesh,

boundary conditions and loading are all modeled byANSYS system and shown in Fig.(22).

-a - - b -

Fig. 22 Distribution of Dome ribs model (case(3))

2- Presentation and Discussions of ResultsA) Normal stress in the three orthogonal direction

zyx ,, σσσ versus of angle φ , table(4) show the

values of normal stress in the three orthogonaldirection of the cartezian coordinate x, y and z for seven locations in the dome at intersection, thosevalue of stresses are obtained from the present finiteelement model with x-y plane, those location

specified seven point given in table(4) are those

making φ angle of values (0,15,30,45,60,75,90)

degree. Values of stresses in the three orthogonal directions

(i.e. zyx ,, σσσ ) which are recorded in table(4)

above are all graphically plotted versus the angle φ

and shown in Fig.(23).

Fig.23 Stress in dome with different meridian angles



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Table (4): Stress and deflection from crown to the bottom

From the inspection of table(4) and Fig.(23), it seenthat the values the normal stress has it maximumcompressive value at the crown (under theconcentrated load ) with moving along the meridionalsection in the x-y plane (i.e. with increasing value of

angle φ from zero). It seen that the compressive

value of

x

σ decreases gradually until became zero

at φ angle equal 20° beyond this location (i.e. with

increasing φ value beyond 20°) the xσ became

tensile and increases rapidly until it reached its max.

tensile value at φ angle equal 30°. Beyond angle φ

value 30° the xσ value reduced slowly until its

vanishes at angle φ value equal to 90° ( i.e. at the

dome base) .

The variation of the normal stress yσ with angle φ

( at dome section in the x-y plane) is quiet different

from that of xσ . The variation of yσ with angle φ

is seemed uniform (i.e. extremely small variation of

yσ with angle φ ) this trend is quiet logical as the

dome meridional slice within x-y plane, act as an archwith two clamped edge and subjected to verticaldownward point load at the crown . The curve

representing the relation between the normal zσ

and angle φ is clearly closed in its trend to the

corresponding curve of xσ , with the zσ values

smaller than their corresponding once. This fact maybe attributed to the asysmmetric properties of theanalysis shell.Figs.(24),(25) and (26) show the variation of each of

the three normal membrane stress zyx ,, σσσ

over the whole outer surface of the analysis domerespectively. From these figures the similarity of the

distribution xσ and zσ stresses is quiet clear ( with

90° orientation about y- axis) this fact reflect thesymmetric properties of the analysis dome.

Fig 24 Stress in the x direction Fig.25 Stress in the y direction

φ , degrees xσ ,MPa yσ ,MPa zσ ,Mpa Deflection , mm

0 -.349 0.0965 -0.14 -4.3

15 -0.00604 0.807 -313 -0.78

30 0.696 -0.242 -0.486 -0.4845 0.401 -0.208 -0.182 -0.27

60 0.272 -0.208 -0.064 -0.016

75 0.214 -0.226 -0.0031 -0.0056

90 0.0199 -0..206 0.0015 -0.0013



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0.00 20.00 40.00 60.00 80.00 100.00


-5.00

-4.00

-3.00

-2.00

-1.00

0.00

D i s p l a c e m e n t m m

Displacement Uy

Fig. 26 Stress in the z direction

B) Vertical Deflection Values of vertical displacement at specified locationon dome surface at x-y plane which are shown intable(4) for the specified values of the meridional

angle φ are all plotted in Fig.(27).versus the angle

φ .

Fig. 27 Deflection in dome with different meridian angles

From this figure it easily observed that the verticaldisplacement has max. value at crown (where the

angle φ is zero) then the displacement reduce

rapidly with the increase of value until it vanish for

angle φ value from equal to 90° the dome base.

C) Fracture Pattern

Fig.(28) shows the fracture pattern of the analysisdome of the case three, subjected to theconcentrated downward point load at crown, thisfracture pattern is as failure stage of the dome. The

inspection of that figure demonstrated the crashesconcrete and compression within around the domeapex and the tensile failure cracking at rear the edge.



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0.00 1.00 2.00 3.00 4.00

Deflection (mm)

0.00

200.00

400.00

600.00

800.00

L o a d K n

Diameter = 50m

Diameter = 100m

Diameter = 150m

Fig.28 Fracture pattern of dome case(3) at ultimate stage

Parametric Study

Based on the present finite element model byANSYS12, a parametric study was performed toinvestigate the influence of several importantparameters on the behavior of concrete dome. Theparameters studied can be summarized as follows:1. Diameter of dome2. Number of ribs .3. Percentage of steel across the interface.4. Compressive strength of concrete.5. Cover thickness.The effect of each of the above-mentionedparameters on the behavior of the concrete dome isdiscussed below.

R.C Dome without Ribs:

1 -Effect of Diameter of DomeDome of different diameters have been considered tostudy this effect on the behavior of the load-deflectioncurve. Fig.(29) shows the load-deflection curve atthe crown of the dome with different diameter asdetermined by the present ANSYS model .Theselected values of dome diameter are 50m and150m, in addition to the original diameter value;100m. With respect to the originaldiameter;100m, the 50m diameter, caused adecrease in the ultimate load by about (45.6%).Similarly, the 150m diameter, caused an increase inthe ultimate load by about (54%) relative to the

original case (of 100m diameter).

Fig. 29 Load-deflection relationships at crown three un ribbed dome (case(1)) showing effect of diameter value.

2- Effect of Compressive Strength of Concrete:Fig.(30) shows the load-deflection curves at thecrown for un ribbed dome (case(1)) of various

concrete compressive strength values. Based on theresults of that analysis, the following effects have



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0.00 0.40 0.80 1.20 1.60

Deflection (mm)

0.00

100.00

200.00

300.00

400.00

500.00

L o a d k N

Fc'=25

Fc'=30

Fc'=35

0.00 0.40 0.80 1.20 1.60 2.00

Deflection (mm)

0.00

100.00

200.00

300.00

400.00

500.00

L o a d k N

Thickness cover=50 mm



occurred relative to the load–deflection relationshipsfor the original c f

' value (30 MPa):

i) Using c f ' =25 MPa for dome concrete, caused a

decrease in the ultimate load by (9%).

ii) The increase of c f ' value to 35 MPa, caused an

increase in the ultimate load by (10.7%).

Fig. 30 Load-deflection relationships at crown for three un ribbed dome (case(1)) showing effect of compressivestrength of concrete value.

3-Effect of the cover thickness:

Fig.(31) shows the response of deflection for variousthicknesses of cover for concrete dome. Based onthe results of the analysis, the following effects haveresulted relative to the original cover thicknesses;70mm.

i) Using thickness value t=50 mm, caused a 6.5%decrease in the ultimate load.

ii) On the other hand, increasing cover thicknessesto 100mm cased a 17.8% increase in the ultimateload value.

Fig. 31 Load-deflection relationships at crown for three un ribbed dome (case(1)) showing effect of the cover thickness value.

Monolithic R.C. Ribbed Dome:1- Effect of Diameter of dome

Three dome with different dome diameter value eachcontaining two ribs-have been considered to studythis effect on the load-deflection relationships.



Am. J. Sci. Ind. Res., 2010, 1(3): 472-495

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0.00 0.40 0.80 1.20 1.60 2.00

deflection (mm)

0.00

200.00

400.00

600.00

L o a d k N

Legend Title

Diameter =40 m

Diameter =70 m

Diameter =100 m

0.00 1.00 2.00 3.00 4.00 5.00

Deflection (mm)

0.00

200.00

400.00

600.00

800.00

L o a d k N

Without Ribs

Two Ribs

Four Ribs

Fig.(32) shows the load-deflection curves at thecrown of the two ribbed dome with different diameter .The selected value diameter values were 40,70 and100 m .

With respect to the original diameter; 70m , using a50m diameter value, caused a decrease in theultimate load by (44.4%). Similarly, the increasediameter value to 150m, caused an increase in theultimate load by (16.9%).

Fig. 32 Load-deflection relationships at crown for three ribbed dome (case(2)) showing effect of diameter value

2- Effect of Number of Ribs Three ribbed monolithic domes with differentnumbers of ribs have been considered to study thiseffect on the load-deflection relationships.Fig.(33) shows the load-deflection curves at thecrowns of three ribbed monolithic domes of different

numbers of ribs. The selected numbers are 0 (withoutribs), 2 and 4 ribs. With respect to the originalnumber of ribs; two; the removal of ribs, caused a24.4% decrease in the ultimate load. On the other hand, increasing the number of ribs to four ,cased a25.5% increase in the ultimate load.

Fig. 33 Load-deflection relationships at crown for three ribbed dome (case(2)) showing effect of number of ribs .

3 -Effect of Compressive Strength of ConcreteFig.(34) shows the load-deflection curves for variousconcrete compressive strength values of ribbedmonolithic dome each having two ribs.

Using concrete of 25 MPa, c f '

value relative to the

original c f ' value; 30 MPa, caused a 11% decrease

in the ultimate load .



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0.00 0.40 0.80 1.20 1.60 2.00

Deflection (mm)

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200.00

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L o a d

k N

Thickness of Cover=50 mm



0.00 0.50 1.00 1.50 2.00 2.50

Deflection (mm)

0.00

200.00

400.00

600.00

800.00

L o a d k N

Fc'=25

Fc'=30

Fc'=35

On the contrary, the increase of c f ' value to 35

MPa, cause a 7.5% increase in the ultimate load

value relative to the original c f ' value; 30 MPa,

caused a 11% decrease in the ultimate load.

Fig.34 Load-deflection relationships at crown for three ribbed dome (case(2)) showing effect of compressivestrength of concrete value

4 -Effect of the Cover Thickness

Fig.(35) shows the response of deflection for variousthicknesses of cover for ribbed monolithic concretedome each having two ribs. Relative to the originalcover thickness (70mm), when the thickness (t) equal

to 50mm is used, the ultimate load is decrease by(11.7%). However the increase of cover thickness to100mm cases a 17.9% increase in the ultimate loadcapacity, relative to the original cover thickness(70mm).

Fig.35 Load-deflection relationships at crown for three ribbed dome (case(2)) showing effect of the cover thickness.

Dome with Pre-cast Concrete Ribs and Cast inPlace Concrete Cover 1 - Effect of Diameter of domeDome (case(3)) with different diameter values eachcontaining four ribs have been considered to studythis effect on the load-deflection relationships.

Fig.(36) shows the load-deflection curves at thecrowns three dome of this type varying in their diameters .The selected diameters values were 50,100 and150 m . Relative to the original diameter value; 100m, using 50m diameter value, caused a12.9% decrease in the ultimate load capacity. On theother side, the increase of diameter to 150m, caused



Am. J. Sci. Ind. Res., 2010, 1(3): 472-495

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0.00 2.00 4.00 6.00

Deflection (mm)

0.00

200.00

400.00

600.00

800.00

L o a d k N

Diameter=50

Diameter=100

Diameter=150

0.00 2.00 4.00 6.00

Deflection (mm)

0.00

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400.00

600.00

800.00

L o a d k N

Legend Title

Without ribs

Tow ribs

Four ribs

a 17.6% increase in the ultimate load capacity of the dome of 100m diameter .

Fig.36 Load-deflection relationships at crown for three ribbed dome (case(3)) showing effect of Diameter

2- Effect of number of ribsDome (case(3)) with different numbers of ribs havebeen considered to study this effect on the load-deflection relationships. Fig.(37) shows the load-deflection curves at the crown of the pre-castconcrete ribs and cast in place concrete cover of different numbers of ribs. The selected numbers are:

0 (without ribs), 2 and 4 ribs. With respect to theoriginal number of ribs; two; the removal of ribscaused a 17.8% decrease in the ultimate loadcapacity. On the other hand, increasing the number of ribs to four, caused a 18% increase in the ultimateload.

Fig. 37 Load-deflection relationships at crown for three ribbed dome (case(3)) showing effect of number of ribs.

3- Effect of Compressive Strength of Concrete:Fig.(38) shows the load-deflection curves for various

concrete compressive strength values of dome withpre-cast concrete ribs and cast in place concretecover having four ribs .Using concrete of 25 and 20

MPa, c f '

values relative to the original c f '

value;

30MPa, caused a 4% and 7% respectively

,decrease in the ultimate load capacity. On the

contrary, the increase of c f '

value to 35 and 40

MPa, cause a 5% and 10% respectively increase inthe ultimate load capacity value relative to the original

c f '

value; 30 MPa.



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0.00 1.00 2.00 3.00 4.00 5.00

Deflection (mm)

0.00

200.00

400.00

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Fc'=20

Fc'=25

Fc'=30

Fc'=35

Fc'=40

0.00 2.00 4.00 6.00

Deflection (mm)

0.00

200.00

400.00

600.00

800.00

1000.00

L o a d k N

Thickness =50 mm

Thickness =70 mm

Thickness =100 mm

Fig. 38 Load-deflection relationships at crown for ribbed dome (case(3)) showing effect of concrete compressivestrength values.

4 - Effect of the Cover ThicknessFig. (39) shows the response of deflection for various thicknesses of cover for dome(case 3) withpre-cast concrete ribs and cast in place concretecover having four ribs . Relative to the original cover thickness (70mm), when the thickness equal to

50mm is used, the ultimate load is decrease by(13.5%). However the increase of cover thickness to100mm cases a 13.9% increase in the ultimate loadcapacity, relative to the original cover thickness(70mm).

Fig. 39 Load-deflection relationships at crown for ribbed dome (case(3)) showing effect of the cover thickness.

5- Effect Percentage of steel across the interfaceTo study the effect steel percentage across the jointof dome (case(3)) with pre-cast concrete ribs and

cast in place concrete cover having four ribs withdifferent percentages ( ρ ) 0,0.18,0.32 and 0.46

(where ρ = steel percentage across the joint (area of

steel crossing the joint/ area of joint)). The resultsfrom Fig.(40) indicate that a large difference in failureload is noticed by using different steel percentage

across the joint. A 30%,60%, and 100% decrease of steel ratio caused a decrease of 15.9%, 29% and38% respectively in the ultimate load capacity.



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Deflection (mm)

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200.00

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L o a d k N

ρ=0%

ρ=0.18%

ρ=0.32%

ρ=0.46%

Fig.40 Load-deflection relationships at crown for ribbed dome (case(3)) showing effect of Percentage of steel

across the interface.

CONCLUSIONS 1- The present finite element model has given

very good accuracy in computing themembrane normal stresses σϕ in the

meridional direction and this conclusion hasbeen drawn from the comparison of σϕ

values obtained from the ANSYS model andthe classical membrane theory of shells. Thepercentage different average between thetwo approaches is 2.2%.

2- The present ANSYS12 model has produced

the same level of accuracy in computing themembrane stresses in the hoop direction σθ

for the whole range of the angle φ the

percentage different average for σθ values

between the ANSYS model and the classicalmembrane theory of shells is 3.3%.

3- Regarding the ribbed monolithic reinforceconcrete dome under the effect downwardpoint load at crown the present finite elementmodel has shown high efficiency indescribing the variation of membrane normalstresses in the three orthogonal direction x,y

and z with the meridional angle φ . Thenumerical analysis result followed strictly thebehavior given by the elementary membraneshell analysis. In the variation of membranenormal stresses in the x,y and z direction, the xσ and zσ increased from compression at

crown to max. tension at the dome base.

4- In the case of ribbed monolithic concretedome, the analysis by ANSYS12 model hasshown the monotontoic increases of xσ and

zσ values from the dome crown, where this

two stresses at angle φ values of 30o

and

46o, respectively given max. tensile stresses

value after with those stress decreasesgradually till they vanish at the dome base.

5- For the same dome case above (ribbedmonolithic type) the normal stresses yσ ,

varies slightly from crown (φ =0) to φ =31o

after which this stresses vanish continuallyalong the meridional path till the dome base

(φ =90o). Variation of vertical displacement

along the meridional path toward the domebase has similar trend of yσ . This behavior

consist of the primary inspection of theelementary membrane theory of shells.

6- Variation of the three principal normalstresses xσ , yσ and zσ , the vertical

displacement along the meridional path fromapex of ribbed composite concrete dome of partial interaction at interface (case(3) of the

present analysis ) to its base, predicted bythe present finite element model, has almostthe same trends as those of case(2) dome(ribbed monolithic concrete dome) but withlarger variations. This behavior is quiterealistic as it reflects the effect of the partialinteraction at interfaces.

The Impact of Diameter, Number of Ribs, Percentage of Steel, Compressive Strength and Cover...

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