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Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2007, Article ID 78507, 19 pages doi:10.1155/2007/78507 Research Article A Finite Circular Arch Element Based on Trigonometric Shape Functions H. Saari and R. Tabatabaei Received 27 August 2006; Accepted 26 February 2007 Recommended by Jan Awrejcewicz The curved-beam ﬁnite element formulation by trigonometric function for curvature is presented. Instead of displacement function, trigonometric function is introduced for curvature to avoid the shear and membrane locking phenomena. Element formulation is carried out in polar coordinates. The element with three nodal parameters is chosen on curvature. Then, curvature ﬁeld in the element is interpolated as the conventional trigonometric functions. Shape functions are obtained as usual by matrix operations. To consider the boundary conditions, a transformation matrix between nodal curvature and nodal displacement vectors is introduced. The equilibrium equation is written by minimizing the total potential energy in terms of the displacement components. In such equilibrium equation, the locking phenomenon is eliminated. The interesting point in this method is that for most problems, it is sucient to use only one element to obtain the solution. Four examples are presented in order to verify the element formulation and to show the accuracy and eciency of the method. The results are compared with those of other concepts. Copyright © 2007 H. Saari and R. Tabatabaei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis- tribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Curved beams are more ecient in transfer of loads than straight beams because the transfer is aected by bending, shear, and membrane action. Some of the structures such as arches and arch bridges are modeled using curved beam elements. The ﬁnite element analysis of curved beam has been given signiﬁcant attention by researchers in recent years mostly because it is a versatile method for solving structural and other mechanical prob- lems.

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### Transcript of arches elements Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2007, Article ID 78507, 19 pagesdoi:10.1155/2007/78507

Research ArticleA Finite Circular Arch Element Based onTrigonometric Shape Functions

H. Saffari and R. Tabatabaei

Received 27 August 2006; Accepted 26 February 2007

Recommended by Jan Awrejcewicz

The curved-beam finite element formulation by trigonometric function for curvature ispresented. Instead of displacement function, trigonometric function is introduced forcurvature to avoid the shear and membrane locking phenomena. Element formulationis carried out in polar coordinates. The element with three nodal parameters is chosenon curvature. Then, curvature field in the element is interpolated as the conventionaltrigonometric functions. Shape functions are obtained as usual by matrix operations.To consider the boundary conditions, a transformation matrix between nodal curvatureand nodal displacement vectors is introduced. The equilibrium equation is written byminimizing the total potential energy in terms of the displacement components. In suchequilibrium equation, the locking phenomenon is eliminated. The interesting point inthis method is that for most problems, it is sufficient to use only one element to obtainthe solution. Four examples are presented in order to verify the element formulation andto show the accuracy and efficiency of the method. The results are compared with thoseof other concepts.

Copyright © 2007 H. Saffari and R. Tabatabaei. This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, dis-tribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Curved beams are more efficient in transfer of loads than straight beams because thetransfer is affected by bending, shear, and membrane action. Some of the structures suchas arches and arch bridges are modeled using curved beam elements. The finite elementanalysis of curved beam has been given significant attention by researchers in recent yearsmostly because it is a versatile method for solving structural and other mechanical prob-lems. 2 Mathematical Problems in Engineering

The analysis of curved beam is conventionally formulated based on displacementfields. Such formulation often leads to excessively stiff behavior in the thin regimes. In

such analyses, shear locking phenomenon occurs when lower-order elements are used inmodeling. This is because in such models, only flexural deformations are considered andshear deformations are neglected. Another phenomenon is called membrane locking. Itoccurs when other classical curved finite elements are used for modeling thin and thickcurved beams; because they exhibit excessive bending stiffness alone in approximatingthe extensional bending response and the lower-order element cannot bend without be-ing stretched. It means that elements are unable to represent the condition of zero radialshear strains. Therefore, these two phenomena are associated with highly undesirable sit-uation and numerical deficiency. Thus much attention has been focused to remedy thelocking phenomena.

Most of researches have proposed various schemes to alleviate locking such as re-duced integration, (Zienkiewicz et al. ; Stolarski and Belytschko [2, 3]; Pugh et al. ),discrete Kirchhoff ’s theory (Batoz et al. ), penalty relaxation method, (Tessler andSpiridigliozzi ), hybrid/mixed concept (Reddy and Volpi ), isoparametric inter-polations, (Ashwell et al. ), free formulation, (Dawe ; Ashwell and Sabir ),and assumed displacement field (Raveendranath et al. ). However, most of these ap-proaches have been used for thin regimes, however the given results have shown thatthese methods cannot represent the behavior of the curved beam quite correctly in thethick regimes. To overcome this problem, a three-nodded curved beam element is for-mulated by Bathe , considering shear and tangential rigidity. Curved beam elementwith straight beam elements is modeled Mc Neal and Harder ; however this resultsin a large degree of freedom in a modeling. A three-nodded locking-free curved beamelements based on curvature is formulated by Lee and Sin . The latest attempt bySinaie et al.  considers eliminating shear-locking phenomenon and involves six nodalcurvatures. A new two-nodded shear flexibility curved beam element by assuming poly-nomial radial displacement field is derived by Raveendranath . Shear force effects oncurved beam element behavior were considered by Sheikh . Nonlinear formulationof curved beam element by using curvilinear system and neglecting shear force effectsis investigated by Wen and Suhendro  and Wen and Lange  and Calhoun andDaDeppo .

In this paper, a new curved beam finite element formulation by trigonometric func-tion for curvature is presented. Unlike other investigators, trigonometric function is usedfor curvature and this enables efficient elimination of the shear and membrane-lockingphenomenon. This is achieved by introducing strain-displacement relationships in polarcoordinate system using a three-nodded element. Relationship between nodal curvatureand nodal displacement is obtained using transformation matrix. The total potential en-ergy equation is written and minimized; then, force-displacement relation is derived andan algorithm for analysis is proposed. Finally, four numerical examples are solved andthe results are compared with solutions given by other methods. The results show thatperformance of the new-presented element without locking phenomena is preferred overthe other types of elements. H. Saffari and R. Tabatabaei 3

LPr

Pt

S

χ1

χ3

χ2

m

R

Figure 2.1. Nodal curvatures and applied loads in a 3-node circular arch element.

2. The curvature field and matrix operations

The matrix operation for the curvature χ of circular arch element is being presented inthis section (see Figure 2.1). Prior to present the formulation, it is worthy to note herebefore the formulation that curvature can be interpolated with conventional trigonomet-ric functions. In present study, a three-node element for the curvature is chosen. Theshape function for this type of element is the simplest one, which express the behavior ofa curved beam. A trigonometric function representation for curvature field χ is assumedas

χ = a1 + a2 Cos(Φ) + a3 Sin(Φ) (2.1)

or

χ = [ f ] · [a], (2.2)

where [a] and [ f ] are two vectors defined by

[a]T = [a1 a2 a3],

[ f ]= [1 Cos(Φ) Sin(Φ)] (2.3)

in which Φ is obtained by

Φ= S

R. (2.4)

The relation nodal for curvature vector [κ] can be written as

[κ]= [ω] · [a], (2.5) 4 Mathematical Problems in Engineering

L U

θ

S W

R

Figure 3.1. Components of displacement in a typical circular arch beam.

where

[κ]T =[χ1 χ2 χ3

],

[ω]=

⎢⎢⎢⎢⎢⎢⎣

1 1 0

1 CosL

RSin

L

R

1 CosL

2RSin

L

2R

⎥⎥⎥⎥⎥⎥⎦

.(2.6)

Substituting (2.5) in (2.2) results in

χ = [ f ] · [ω]−1 · [κ] (2.7)

as depicted in Figure 2.1. If the curvature in the element is interpolated the followingequations are obtained:

[fκ]= [ f ] · [ω]−1, (2.8)

χ = [ fκ] · [κ], (2.9)

[fκ]=

[f κ1 f κ2 f κ3

], (2.10)

f κ1 ={

14

(Cos

(L

4R

)−Cos

(3L− 4S

4R

)/(Sin(L

4R

)2

Cos(L

4R

)))}, (2.11)

f κ2 ={− 1

2

(Sin(L− 2S

4R

)Sin(S

2R

)/(Sin(L

4R

)2

Cos(L

4R

)))}, (2.12)

f κ3 =(

Sin(L− S

2R

)Sin(S

2R

)/Sin(L

4R

)2). (2.13)

3. The sectional rotation field

The geometry of a circular arch element with radius R and centroidal arc length L loadedin plane is shown in Figure 3.1. H. Saffari and R. Tabatabaei 5

Considering the figure, the sectional rotation field θ for a typical circular arch elementin polar coordinates is derived using the curvature χ as

χ = θ,S, (3.1)

where subscript S denotes differentiation with respect to arc length. Substituting (2.9) in(3.1) and integrating from (3.1) results in

θ = [ fθ] · [κ],

[fθ]=

∫ S

0

[fκ] ·ds+C1,

(3.2)

where C1 is the integration constant and [ fθ] is determined as detailed in Appendix A.

4. The radial and tangential displacement fields

The radial displacement W and the tangential displacement U for a typical circular archelement are illustrated in Figure 3.1. The relationships between these components of dis-placement and the shearing strain γ as well as the tangential strain ε are as follows:

ε =U,S−W

R,

γ =W,S− θ +U

R.

(4.1)

The membrane force N , the bending moment Mb, and the shear force V , respectively aregiven by

N = EAε, Mb = EIχ, V = kGAγ, (4.2)

where A is the cross sectional area, I is the moment of inertia, E is the modulus of elas-ticity, G is shear modulus and k is the shear coefficient. The equilibrium equations of acircular arch element, take the following form (Timoshenko and Gere ):

Mb,S +V = 0,

V,S +N

R= 0,

N,S +Mb,S

R= 0.

(4.3)

Substituting (4.1) in (4.2) and then in (4.3) yields

W,SS +W

R2= χ−

(EI

GAk+I

A

)· χ,SS,

U = R ·(θ−W,s− EI

GAk· χ,s

).

(4.4) 6 Mathematical Problems in Engineering

Solving (4.4) for the radial and tangential displacements can be calculated in terms ofcurvature as

W = fW · κ+C2 · cos(Φ) +C3 · sin(Φ),

U = fU · κ+R ·C1 +C2 · sin(Φ)−C3 · cos(Φ),(4.5)

where constants, C1, C2, and C3, are the components of rigid-body displacements whilefθ , fW , and fU are detailed in Appendix A.

5. Nodal curvatures and nodal displacements relation

Consider a circular arch element with six specified boundary conditions as shown inFigure 3.1. By applying boundary conditions to nodes 1 and 2, the deformation-curvaturerelations are obtained as

W1 = fW∣∣S=0 · κ+C2, (5.1)

U1 = fU∣∣S=0 · κ+R ·C1−C3, (5.2)

θ1 = fθ∣∣S=0 · κ+C1 = C1, (5.3)

W2 = fW∣∣S=L · κ+C2 · cos

L

R+C3 · sin

L

R, (5.4)

U2 = fU∣∣S=L · κ+R ·C1 +C2 · sin

L

R−C3 · cos

L

R, (5.5)

θ2 = fθ∣∣S=L · κ+C1, (5.6)

where fi|S=a is the value of fi at S = a. Substituting (5.1)–(5.3) in (5.4)–(5.6), respec-tively, and eliminating the constants C1, C2, and C3, the nodal displacements and nodalcurvature obtained as

[Tκ] · [κ]= [Tu

] · [δ], (5.7)

where [δ] is the boundary condition vector and is defined as

[δ]=[U1 W1 θ1 U2 W2 θ2

]T,

[Tκ]=

⎢⎢⎢⎢⎢⎢⎣

fW∣∣S=Lκ− fW

∣∣S=L cos

L

R+ fU

∣∣S=0 sin

L

R

fU∣∣S=Lκ−FW

∣∣S=0 sin

L

R− fU

∣∣S=0 cos

L

R

fθ∣∣S=L

⎥⎥⎥⎥⎥⎥⎦

,

[Tu]=

⎢⎢⎢⎢⎢⎣

−cosL

Rsin

L

R−Rsin

L

R1 0 0

−sinL

R−cos

L

R−R(

1− cosL

R

)0 1 0

0 0 −1 0 0 1

⎥⎥⎥⎥⎥⎦.

(5.8) H. Saffari and R. Tabatabaei 7

The transformation matrix [T], which relates the nodal curvature vector and the nodaldisplacement vector, may be expressed as follow:

[T]= [T−1κ

] · [TU], [κ]= [T] · [δ]. (5.9)

6. Equilibrium equation of the element

The finial finite element equilibrium equation is written in terms of the displacementcomponents of the two nodes. The shear and tangential strains are incorporated into thetotal potential energy by the force equilibrium equation. Therefore, the presented analysisformulation of the circular arch element is ensured to be free of the locking phenomena.The total potential energy in a circular arch element shown in Figure 2.1 is written as

π = 12EI∫ L

0χ2ds+

12GAk

∫ L

0γ2ds+

12EA∫ L

0ε2ds− [Pe

]. (6.1)

In the above equation, Pe is the equivalent nodal load vector. Considering the expressionfor each strain and displacement field and invoking the stationary condition of the givensystem, that is, δπ = 0 the relation between force and displacement is obtained in thefollowing steps:

[K] · [δ]= [Pe]. (6.2)

The stiffness matrix [K] is given by

[K]= [T]T ·[EI(∫ L

0f Tκ · fκ dS+α

∫ L

0f Tκ,s · fκ,s dS+β

∫ L

0f Tκ,ss · fκ,ss dS

)]· [T] (6.3)

and the equivalent nodal load vector [Pe] is obtained by:

[Pe]=

∫ L

0VTW ·Pr ·dS+

∫ L

0VTU ·Pt ·dS+

∫ L

0VTθ ·m ·dS, (6.4)

where Pr , Pt and m are the distributed radial, tangential, and moment loads applied,respectively, as shown Figure 2.1. The VW , VU and Vθ are defined later in Appendix B.

7. Numerical studies

Four sample problems of a curved beam are solved in this section. (1) An arc for varioussubtended angles supported on a fixed and a roller at the other end. (2) A quarter cir-cular cantilever circular arch. (3) A pinched ring and (4) A quarter circular arch with amoment load applied. These problems are elaborately selected because they are of typicaltheoretical studies to investigate shear and membrane-locking phenomena. Comparisonof the results obtained in this study is shown with those obtained by other investigators.

7.1. Example 1: an arc for various subtended angles. An arc of a ring with angle α andwith one end fixed and a concentrated load P at the other roller end, as in Figure 7.1, ischaracterized by slenderness, R/h = 50. The vertical displacements obtained under load 8 Mathematical Problems in Engineering

1

h

P

2

W2

R= 100P = 100E = 10.5e6G= 4e6h= 2

Figure 7.1. An arc for various subtended angles α.

Table 7.1. Comparison of the finite element solution (radial displacement under load) of one-quarterring with other methods.

Angle α(degree)

Results byLee and Sin 

Results byRaveendranath et al. 

Results bySinaie et al. 

One-elementpresent study

10 0.004 19 0.004 19 0.004 19 0.004 19

20 0.007 7 0.007 7 0.007 7 0.007 7

30 0.014 2 0.014 1 0.014 2 0.014 23

40 0.028 7 0.028 6 0.028 7 0.028 76

50 0.054 0.054 0.054 0.054 00

60 0.093 1 0.093 0 0.093 1 0.093 16

70 0.149 9 0.148 5 0.149 9 0.149 95

80 0.228 8 0.227 5 0.228 8 0.228 8

90 0.335 0 0.334 5 0.335 0 0.335 1

100 0.475 0 0.474 5 0.475 0 0.475 2

120 0.889 2 0.889 2 0.889 2 0.889 8

140 1.557 0 1.556 5 1.557 0 1.558 9

160 2.608 9 2.607 8 2.608 9 2.613 8

180 4.240 3 4.240 0 4.240 3 4.253 9

P are compared with those given by the other methods (Raveendranath et al. ; Leeand Sin  and Sinaie et al. ), and are summarized in Table 7.1.

It can be seen, from Table 7.1 that the results of the present analysis agree well withthose obtained by other methods. Also, just one element can be calculating accuracy H. Saffari and R. Tabatabaei 9

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

iald

ispl

acem

ent,W

0 20 40 60 80 100 120 140 160 180

Angle, α (degree)

2 elements (Lee and Sin )4 elements (Raveendranath et al. )1-element present method

results by presented method. Figure 7.2 is the graphical representation of the results byvarious concepts mentioned above. While the results of present study are superposed. Inthis figure, the radial displacement of joint 2 of thin arch for various subtended angleα is shown. It can be seen from Figure 7.2 that the solutions given by Raveendranath etal.  and Lee and Sin  in good agreement with the results obtained by new formu-lation presented here. It should be remembered that in present study only one elementhas been employed whereas in other methods, the number of elements are large.

7.2. Example 2: a quarter circular cantilever ring. The details of a quarter circular can-tilever ring are shown in Figure 7.3. The finite element results for radial displacement(W), tangential displacement (U), and sectional rotation (θ) at the ring tip and for awide rang of slenderness, (R/h = 4 to 1000, thick to thin) and those obtained by othermethods are all summarized in Table 7.2. The free-locking exact solution can be derivedanalytically using Castigliano’s theorem (considering all bending, shear and membraneare strain energy components) and is shown as follows: (Lee and Sin )

Wc = πPR3

4EI+πPR

4GAk+πPR

4EA, (7.1a)

Uc = PR3

2EI− PR

2GAk− PR

2EA, (7.1b)

θc = PR2

EI. (7.1c) 10 Mathematical Problems in Engineering

1

h

P

2

W2

U2

R

θ2

R= 10Unit widthE = 10.5e6G= 4e6P = 1

Figure 7.3. A quarter circular cantilever circular arch.

Table 7.2. Comparison of the finite element solution ratio of a quarter circular cantilever circulararch.

Slendernessratio R/h

Two-element model byRaveendranath et al. 

One-elementpresent study

Wf

Wc

θ fθc

U f

Uc

Wf

Wc

θ fθc

U f

Uc

4 0.997 61 0.999 32 1.034 06 1.000 0 1.000 0 1.033 50

10 0.998 28 0.999 32 1.000 53 1.000 0 1.000 0 1.005 27

20 0.998 37 0.999 32 1.001 28 1.000 0 1.000 0 1.001 31

50 0.998 40 0.999 32 1.000 17 1.000 0 1.000 0 1.000 21

100 0.998 40 0.999 32 1.000 01 1.000 0 1.000 0 1.000 05

200 0.998 40 0.999 32 0.999 96 1.000 0 1.000 0 1.000 01

500 0.998 40 0.999 32 0.999 95 1.000 0 1.000 0 1.000 0

1000 0.998 40 0.999 32 0.999 95 1.000 0 1.000 0 1.000 0

Subscripts f and c denote the finite element solution and analytic solution by Castigliano’s theorem,

respectively.

Table 7.2 shows the accuracy of the present model and indicates that the results are betterthan those obtained by other methods.

It is concluded, from the results given in preceding sections, that trigonometric func-tion for curvature concept provides sufficient accuracy while using the least number ofelements employed. Moreover, the shear and membrane-locking phenomenon is com-pletely eliminated in this regime. H. Saffari and R. Tabatabaei 11

h2P

2P

R

R= 4.953h= 0.094Unit widthE = 10.5e6G= 4e6P = 100

(a) A pinched ring

1

2

P

θ1

θ1

U1

W1

U2

W2

U1 =U2 = θ1 = θ2 = 0

R= 4.953h= 0.094Unit widthE = 10.5e6G= 4e6P = 100

(b) 1/4 circular arch with nodal displacement

Figure 7.4. A pinched ring with a load 2P and model for 1/4 circular arch.

7.3. Example 3: a pinched ring. Another example, which is studied here, is a pinchedring typically shown in Figure 7.4. The same radial loads are applied at top and bottomof such ring. The ring is modeled by a 1/4 section with the boundary conditions as shownin Figures 7.4(a), 7.4(b) and then analyzed using only one element.

Using Castigliano’s theorem, the radial displacement under point load can be obtainedas (Lee and Sin )

W1 =−[PR3

EI

4−π2π

+PR

2GAk− PR

2EA

], (7.2a)

W2 = PR3

EI

π2− 84π

+πPR

4GAk+πPR

4EA. (7.2b)

Figure 7.5 shows the comparison of the finite element results with those of the theoreti-cal solution. In order to better represent the comparison, nondimensional axis is chosen.The ratios of the radial deflection obtained by all methods studied to the results of Cas-tigliano’s theorem are shown on abscise. It is mentioned here that again only one elementhas been employed in present finite element analysis.

The present model yields better result Wf /Wc = 1.0 as shown in Figure 7.5. However,results of element by (Raveendranath et al. ) are found to yield Wf /Wc = 0.955 andformulation of (Sinaie et al. ) is reported to yield result, Wf /Wc = 0.999. However,the results the other methods for the single element model are not reported here. 12 Mathematical Problems in Engineering

0.75

0.8

0.85

0.9

0.95

1

Nor

mal

ized

alde

flec

tion

1 2 3 4 5 6 7 8 9 10

(Babu and Prathap )(Shi and Voyiadjis )(Prathap )(Tessler and Spiridigliozzi )(Raveendranath et al. )(Sinaie et al. )Present method

Figure 7.5. Convergence of normalized radial displacement under load 2P and model for 1/4.

Mo

h

Ψ R β = 90Æ

R= 10h= 0.001Unit widthE = 1.2e10G= 4.615e9Mo = 1β= 90Æ

Figure 7.6. A half-circular arch with hinged supports, under the action of a concentrated moment.

7.4. Example 4: a quarter-thin circular arch with hinged supports. Figure 7.6 Showsthe details of a half-circular arch with hinged supports, subjected to a concentrated mo-mentMo at the middle of the span. This means that at the middle point, the bending mo-ment is expected to be discontinuous. The analytical solutions for central displacement H. Saffari and R. Tabatabaei 13

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Ben

din

gm

omen

t,M

0 10 20 30 40 50 60 70 80 90

Angle, Ψ (degree)

Exact4 elements (Raveendranath et al. )2-element present method

Figure 7.7. Internal bending moment distributions for hinge curved beam.

based on Euler theory of thin beams are reproduced here as follows (Raveendranath etal. ):

Wc = 0, (7.3a)

Uc =−0.0100489Mo ·R2

EI, (7.3b)

θc = 0.1211846Mo ·REI

. (7.3c)

In this problem, two elements are used to model this curved beam. Shear effects are ne-glected in the analysis of this sample problem for the sake of simplicity, as otherwise theexplicit calculation of the central deflections calculated via Castigliano’s theorem becomesvery complicated (Lee and Sin ):

M =

⎧⎪⎪⎨

⎪⎪⎩

(1−Cosψ + Sinψ)Mo

2, 0≤ ψ ≤ π

4,

−(1 + Cosψ− Sinψ)Mo

2,

π

4≤ ψ ≤ π

2,

(7.4a)

N =− Mo√2R

Sin(ψ− π

4

), (7.4b)

V = Mo√2R

Cos(ψ− π

4

). (7.4c)

The finite element results for the components of displacements are in very good agree-ment with the above analytical ones. Figures 7.7, 7.8, and 7.9 show, respectively, bendingmoment, membrane force, and shear force distribution along the arc length calculated at 14 Mathematical Problems in Engineering

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

Mem

bran

efo

rce,N

0 10 20 30 40 50 60 70 80 90

Angle, Ψ (degree)

Exact4 elements (Raveendranath et al. )2-element present method

Figure 7.8. Internal memberane force distributions for hinge curved beam.

the nodes and element centered for four elements of the arch using other method. Thesolutions for two elements model are in very good agreement with the exact solutiongiven by Castigliano’s theorem over the entire arc length.

8. Conclusions

A new finite element formulation of the circular arch element was presented in this paperusing Trigonometric function. Despite other conventional methods which use displace-ment functions, the element curvature in current study is defined by three nodal curva-tures. First the curvature field is defined by trigonometric function while the fundamentalrelations are derived in polar coordinate for the typical circular arch element. Second, thecurvature field in the circular arch is related properly to the three nodal curvatures. Thisentails that the shape functions are of trigonometric ones. Integration over the curvatureenables calculation of the tangential and the radial displacements as well as the sectionalrotation between the nodal curvature and displacement is derived by eliminating the rigiddisplacement components at a typical element node and by using a transformation ma-trix. Third, minimization of potential energy equation based on internal forces results inforce-deformation relation. Last, an algorithm for finite element analysis was presentedfollowed by numerical investigations on typical examples (problems) on curved beams.Four examples were studied in order to verify the validity of the present concept. Theresults obtained on these typical problems showed that the accuracy of the concept pre-sented is more than those of the other methods. Using the trigonometric functions, the H. Saffari and R. Tabatabaei 15

0.05

0.055

0.06

0.065

0.07

0.075

Shea

rfo

rce,V

0 10 20 30 40 50 60 70 80 90

Angle, Ψ (degree)

Exact4 elements (Raveendranath et al. )2 elements (Raveendranath et al. )2-element present method

Figure 7.9. Internal shear force distributions for hinge curved beam.

element formulation is largely developed such that only one element can be used to modela curved beam while exact results can be reached. This is because the trigonometric func-tion defines, from the geometrical point of view, the element curvature more accurately.Moreover, since the total potential energy due to axial, shear and bending forces haveall been considered in the new formulation, the membrane and shear-locking phenom-ena have been eliminated. These result in more accurate results compared to those of theother methods.

Symbols

W : Radial displacement χ: Curvature of elementU : Tangential displacement θ: Sectional rotationMb: Bending moment γ: Shearing strainV : Shearing force ε: Tangential strainN : Axial force k: Shear coefficientκ: Nodal curvature δ: Nodal displacementPe: Equivalent nodal load vectorR: Radius of arcA: Cross-sectionI : Moment of inertiaE: Young’s modulusG: Shear modulus 16 Mathematical Problems in Engineering

Appendices

A. Appendix

f θ1 =−14

{RCsc

(L

4R

)2

Sec(L

4R

)Sin(

3L4R

)+ Csc

(L

4R

)2

Sec(L

4R

)(SCos

(L

4R

)

+RSin(

3L4R

)− S

R

)},

f θ2 =−14

{RCsc

(L

4R

)Sec(L

4R

)+ Csc

(L

4R

)2

Sec(L

4R

)(SCos

(L

4R

)

+RSin(L

4R

)− S

R

)},

f θ3 =12

{RCsc

(L

4R

)2

Sin(L

2R

)−Csc

(L

4R

)2(SCos

(L

2R

)+RSin

(L

2R

)− S

R

)},

f W1 = 116R3

{(Csc

(L

4R

)2

Sec(L

4R

))(4R5 Cos

(L

4R

)−βRCos

(3L− 4S

4R

)

−αR3 Cos(

3L− 4S4R

))−(R5 Cos

(3L− 4S

4R

)

+ 2βSSin(

3L− 4S4R

)+ 2αSR2 Sin

(3L− 4S

4R

)+ 2SR4 Sin

(3L− 4S

4R

))},

f W2 = 116R3

{(Csc

(L

4R

)2

Sec(L

4R

))(4R5 Cos

(L

4R

)−βRCos

(L− 4S

4R

)

−αR3 Cos(L− 4S

4R

))−(R5 Cos

(L− 4S

4R

)

+ 2βSSin(L− 4S

4R

)+ 2αSR2 Sin

(L− 4S

4R

)+ 2SR4 Sin

(L− 4S

4R

))},

f W3 = 18R3

{(Csc

(L

4R

)2)(−4R5 Cos

(L

2R

)+βRCos

(L− 2S

2R

)+αR3 Cos

(L− 2S

2R

))

+(R5 Cos

(L− 2S

2R

)− 2βSSin

(L− 2S

2R

)

− 2αSR2 Sin(L− 2S

2R

)− 2SR4 Sin

(L− 2S

2R

))}, H. Saffari and R. Tabatabaei 17

f U1 = 116R2

{(Csc

(L

4R

)2

Sec(L

4R

))(−4R5 Cos

(L

4R

)+ 4R3 Cos

(L

4R

)

+βRCos(

3L− 4S4R

)+αR3 Cos

(3L− 4S

4R

))

+(R5 Cos

(3L− 4S

4R

)− 4R4 Sin

(3L4R

)

+ 4αR2 Sin(

3L− 4S4R

)+ 4R4 Sin

(3L− 4S

4R

)− 2SβSin

(3L− 4S

4R

)

− 2αSR2 Sin(

3L− 4S4R

)− 2SR4 Sin

(3L− 4S

4R

))},

f U2 = 116R2

{(Csc

(L

4R

)2

Sec(L

4R

))(−4R5 Cos

(L

4R

)+ 4R3 Cos

(L

4R

)

+βRCos(L− 4S

4R

)+αR3 Cos

(L− 4S

4R

))

+(R5 Cos

(L− 4S

4R

)− 4R4 Sin

(L

4R

)

+ 4αR2 Sin(L− 4S

4R

)+ 4R4 Sin

(L− 4S

4R

)− 2SβSin

(L− 4S

4R

)

− 2αSR2 Sin(L− 4S

4R

)− 2SR4 Sin

(L− 4S

4R

))},

f U3 = 18R2

{(Csc

(L

4R

)2)(4R5 Cos

(L

2R

)− 4SR3 Cos

(L

2R

)

−βRCos(L− 2S

2R

)−αR3 Cos

(L− 2S

2R

))

−(R5 Cos

(L− 2S

2R

)+ 4R4 Sin

(L

2R

)− 4αR2 Sin

(L− 2S

2R

)

− 4R4 Sin(L− 2S

2R

)+ 2SβSin

(L− 2S

2R

)

+ 2αSR2 Sin(L− 2S

2R

)+ 2SR4 Sin

(L− 2S

2R

))},

(A.1)

where

β = IR2

Aα= EI

GAk. (A.2) 18 Mathematical Problems in Engineering

B. Appendix

Vθ = fθ ·T + fθ0,

VW =(fW − fW

∣∣

0 CosS

R+ fU

∣∣

0 SinS

R

)·T + fW0,

VU =(fU − fW

∣∣

0 SinS

R+ fU

∣∣

0 CosS

R

)·T + fU0,

fθ0 =[

0 0 1 0 0 0]

,

fW0 =[

CosS

R−Sin

S

RRSin

S

R0 0 0

],

fU0 =[

SinS

RCos

S

RR(

1−CosS

R

)0 0 0

].

(B.1)

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H. Saffari: Department of Civil Engineering, Shahid Bahonar University of Kerman, P.O. Box 133,Kerman 76169, IranEmail address: [email protected]

R. Tabatabaei: Department of Civil Engineering, Islamic Azad University of Kerman,P.O. Box 7635131167, Kerman 76175-6114, IranEmail address: [email protected]