Engineering Analysis with Boundary Elements 71 (2016) 70–78

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Engineering Analysis with Boundary Elements

http://d0955-79

n CorrE-m

youssef

journal homepage: www.elsevier.com/locate/enganabound

Improved hybrid boundary solution for shell elements

Taha H.A. Naga a, Youssef F. Rashed b,n

a Department of engineering mathematics and physics, Faculty of engineering – Shoubra, Benha University, Egyptb Supereme Councile of Universities, Egypt, and Department of Structural Engineering, Cairo University, Giza, Egypt

a r t i c l e i n f o

Article history:Received 1 June 2016Received in revised form12 July 2016Accepted 19 July 2016

Keywords:Finite element methodBoundary element methodVariational formulationStiffness matrixShear-deformable platesShell structures

x.doi.org/10.1016/j.enganabound.2016.07.01197/& 2016 Published by Elsevier Ltd.

esponding author.ail addresses: thnaga@feng.bu.edu.eg (T.H.A. N@eng.cu.edu.eg (Y.F. Rashed).

a b s t r a c t

In this paper, a new finite element for shell structures is developed using an improved hybrid boundarysolution. First the variational boundary integral equation for shear-deformable plate bending problems isdeveloped based on quadratic boundary elements. Hence such a formulation is coupled to a similarformulation for 2D plane stress problems to produce the developed shell elements. Numerical examplesare presented to demonstrate the accuracy and validity of the proposed formulation.

& 2016 Published by Elsevier Ltd.

1. Introduction

Shell structures gain huge attention especially in structuralengineering [1]. Engineers, nowadays, use several commerciallybased finite element packages (such as, SAP [2], ANSYS [3], etc.) toanalyze such problems. Most of shell elements used in thesepackages are based on the hybrid integral equation formulation ofPian and Tong [4]. Alternatively, few packages were developedbased on alternative technology, the boundary element method,such as the PLPAK [5]. The mathematical base behind this package[5] is the direct boundary integral equation formulation based onthe well-known virtual work statement, in which the virtual stateis the fundamental problem, or the solution due to unit general-ized load in an infinite domain. This solution is called the funda-mental solution [6].

It has to be noted that Zienkiewicz [7] (page 646) in 1977, wasthe first textbook referred to use fundamental solutions as trialfunctions within finite elements to form the “Boundary Solutionprocedures”. However, the solution presented in Ref. [7] was basedon the traditional energy based functional (without the term thatforces the compatibility between the domain and boundary dis-placements along the boundary). This termwas introduced in Tong[4]. Zienkiewicz in Ref. [7] solved 2D problems and since that time,most of finite element formulations employed polynomial based

aga), yrashed@scu.eg,

trial functions.In 1989, Dumont [8] and Tania [9] developed boundary element

formulation based on the variational functional of Tong [4]. Theirformulation has several advantages, such as it leads to similarformulation to finite elements in terms of sparse symmetric stiff-ness matrices. They considered this formulation for potentialproblems, and 2D elasticity problems. One of the disadvantages oftheir formulation is the appearance of higher order singularities;which led to use equally divided boundary elements (constant orquadratic) to cancel such singularities naturally. Several papershave then been appeared in this context, such as the work ofLeung et al. [10] to couple boundary element method to finiteelement method, Dumont [11] for hybrid stress formulation, Nagaand Rashed [12] for shear-deformable plate bending problemswith constant elements. It has to be noted that all these publica-tions considered this variational formulation as new formulationinside the boundary element method.

Qin [13–17] developed new fundamental solution-based finiteelements; which employs the formulation of Tong [4]. For ex-ample, in Ref. [13], Qin developed a new type of hybrid finiteelement formulation for solving 2D heat conduction problems. InRef. [14], Qin considered the solution of two-dimensional ortho-tropic elasticity. Qin [15] developed a new type of finite elementswith special fundamental solutions for analyzing plane elasticproblems containing holes. In Ref. [16], Qin developed the sameelements for three-dimensional elastic problem. In Ref. [17], Qinclassified his developed fundamental solution-based finite ele-ments as a fourth type of the finite element method. He

T.H.A. Naga, Y.F. Rashed / Engineering Analysis with Boundary Elements 71 (2016) 70–78 71

demonstrated the main advantage of his new elements comparedto the traditional finite element methods.

It has to be noted that in all of the previous work [13–17], theso-called “regular collocation” or placing to collocation point out-side the element domain is considered to avoid computing sin-gular integrals. Such a technique has proven to be efficient.

The purpose of this paper is to derive “Hybrid Boundary solu-tion” procedures for shell elements (or fundamental solution-based shell elements). The formulation for the uncoupled 2Dmembrane and plate bending will be based on the extended Tonghybrid variational functional. Fundamental solutions are employedfor both 2D membrane and plate in bending. Quadratic boundaryelements are used to represent the boundaries of the new devel-oped shell elements. In this work, regular collocation is also usedto avoid singularities; especially in the 3D spatial space for gen-erally inclined shell element. The derived element is going to betested via numerical examples.

Fig. 2. The proposed shell finite element.

2.50m 2.50m

AA Thick. 500mm

Thick. 250mm

Simply Supported

Fig. 3. The Multi-thickness circular plate in Example (1).

2. The proposed new finite element for plate bending

Without losing the generality, the proposed element could beformed from any multi-line/curve closed polygon; however, in thispaper a traditional eight-node quadrilateral shape is considered.Consider the new plate finite element of domain Ω(y) with eightnodes and 24 degrees of freedom bounded by a closed boundarysurface Γ as shown in Fig. 1. A generalized form of the hybridenergy functional (Π ) of the type proposed by Tong [4] for Re-issner's plate bending could be written as follows:

( )( )

( )

∫

∫ ∫

∫

( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( )

( ) ( )

( )

Π ˜ ˜

= χ + ψ Ω

− Ω − ¯ ˜ Γ

− ˜ ˜ − Γ( )

Ωαβ αβ α α

Ω Γ

Γ

d

d d

d

y x x

y y y y y

y y y x x x

x x y x y

u , u , p

12

M Q

b u p u

p u u ,1

y

y x

x y

i i i

3 3

i i i i

,i i i

P

Where ( ) ( ) ( ) ( )αβ α uy y y yM , Q , b ,3 i i are the moment, shear stressresultants, body loads and generalized displacements respectively.The symbols χ ( ) ψ ( )αβ αy yand 3 denote bending and shear strain. ΓP isthe portion of the boundary where generalized tractions are pre-scribed. The over bars ( ( )xpi ) indicate prescribed boundary valuesfor tractions and the over tilde ( ( ) ( )x xu andpi i ) represents a dis-placement value or traction value which is defined only on the

1

8

76

5

4

3

2

Ω(y)

ξ

ξ 8

ξ 6 ξ 5

ξ 4

ξ 3ξ 2

1

Γ(x,y)

xξ 7

Fig. 1. The proposed finite element for plate bending. Fig. 4. Mesh (A) in Example (1), 80 elements with 8 nodes each.

Fig. 5. Mesh (B) in Example (1), 192 elements with 8 nodes eac.

T.H.A. Naga, Y.F. Rashed / Engineering Analysis with Boundary Elements 71 (2016) 70–7872

problem boundary. Integrating by parts the first domain integralon the right hand side of Eq. (1), it gives [18]:

( ) ∫ ∫

∫

∫ ∫

∫ ∫

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

( ) ( )

( )

( ) ( )

( ) ( )

Π = Γ −

Ω − Ω −

Ω −

Γ + Γ −

Γ ( )

Γ Ωαβ β

α

Ωα α

Ω Γ

Γ Γ

d

d d

d d

d

u d

y x x y y y y

y y y y y

y y y x x

x x x x

y x y

u , u , p12

u p12

M

u12

Q u

b u u p

p u p

, 2

y y

y

y x

x x y

i i i i i,

3 ,3

i i i i

i i

,

i

i

P

The main idea of the present formulation is to approximate thefield variables ( ( ) ( ) ( ) ( ))y y x xu , p , u , pi i i i before applying the var-iational operator to the functional in Eq. (2).

-0.005

-0.0045

-0.004

-0.0035

-0.003

-0.0025

-0.002

-0.0015

-0.001

-0.0005

0-5.5 -3.5 -1.5

Distance along t

Analytical [21]

FEM Mesh (A)

FEM Mesh (B)

Present Formulation Mesh (A)

Ref. [21] (441 division)

Fig. 6. Deflections along secti

2.1. Approximation of the domain variables

The first four integrals in Eq. (2) contain the unknown domainvariables ( ( ) ( ))y yu , pi i . which are approximated as the summation

of product of fundamental solution ( ( )ξ* yU ,ki n ) [18] and an un-known set of fictitious concentrated generalized tractions ( ( )ξγk n )located at a set of arbitrary source points ( ξ );n where n¼1 to 8(recall Fig. (1)) [19], as follows:

( ) ( )∑ ξ ξ( )= * γ( )=

=

y yu U ,3

in 1

n 8

ki n k n

Where ( )ξγk n is the set of fictitious generalized tractions appliedalong the direction (xk). In this paper, (n) is taken equal to 8 torepresent the same number of the boundary points and placedalong offset equal to the element length as shown in Fig. (1). In asimilar way, the generalized traction components can be written inmatrix form as follows:

( ) ( )∑ ξ ξ( )= * γ( )=

=

y yp P ,4

in 1

n 8

ki n k n

Where ( )( )* ξP y,ki n is the traction fundamental solution kernel [18].

2.2. Approximation of the boundary variables

The unknown generalized boundary displacement and tractionvectors denoted by ( ) ( )x xu andpi i are approximated using quad-ratic shape function ηϕ ( )m , as follows:

∑ η ( )= ϕ ( ) ( ) ∀ Γ( )=

=

ux x xu in5

eim 1

m 3m

im

e

∑ η ( )= ϕ ( ) ( ) ∀ Γ( )=

=

px x xp in6

eim 1

m 3m

im

e

Where, ( ) ( )u px and xim

e im

e are vectors of the nodal values of theelement ( )xe for boundary displacements and boundary tractions.

2.3. The proposed stiffness equation

Using the representation given in Eqs. (3)–(6), Eq. (2) could bere-written as follows:

0.5 2.5 4.5

Def

lect

ion

(m)

he strip (m)

on (A-A) in Example (1).

-16

-14

-12

-10

-8

-6

-4

-2

0

2

-5.5 -4.5 -3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.5 4.5 5.5

Ben

ding

mom

ent(t

.m)

Distance along the strip (m)

Analytical [21]

FEM Mesh (A)

FEM Mesh (B)

Present Formulation Mesh (A)

Ref. [21] (441 division)

Fig. 7. Bending moment along section (A-A) in Example (1).

Fig. 8. The multi–thickness cantilever plate in Example (2).

5678910

Distance along the strip

Fig. 9. Deflection of the multi-thickness plat

T.H.A. Naga, Y.F. Rashed / Engineering Analysis with Boundary Elements 71 (2016) 70–78 73

{ }{ } { } { }

{ } { } { } { }{ } { }

Π γ γ

γ

γ

= – ¯

+ −

− ( )

× × × × ×

× × × × × ×

× ×

⎡⎣ ⎤⎦⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦

F u P

p L u p G

B

12

7

T T

T T T

T

3 24 1 24 24 24 1 24 1 24 1

24 1 24 24 24 1 24 1 24 24 24 1

24 1 24 1

Where

∫ ( ) ( )ξ ξ= * * Γ( )( )× Γ( )

⎡⎣ ⎤⎦ dF y y yU , P ,8y24 24 ki n mi n

∫ ( )η ξ= ϕ ( ) * Γ( )( )× Γ ( )

⎡⎣ ⎤⎦ dG x xU ,9x

e e24 24

Tj

ki nee

∫{ } ϕ η¯ = ( ) ¯ ( ) Γ( )( )

×Γ ( )

dP x xp10x

e e24 1j

i

ee

∫ ϕ ϕη η= ( ) ( ) Γ( )( )× Γ ( )

⎡⎣ ⎤⎦ dL x11x

e24 24j j T

ee

∫{ } ( )= * ξ ( ) Ω( )( )× Ω( )

dB y y yU , b12y24 1 ki n i

At equilibrium conditions, the functional Π3 is stationary i.e. its

-2.5

-2

-1.5

-1

-0.5

001234

Def

lect

ion

(m)

(m)

Analytical [21]

FEM Mesh (A)

FEM Mesh (B)

Ref. [21] (5x20 Divisions)

Ref. [21] (5x25 Divisions)

Ref. [22]

Proposed Formulation Mesh (A)

e along the center line, in Example (2).

0

20

40

60

80

100

120

012345678910

Ben

ding

mom

ent(t

.m)

Distance along the strip (m)

Analytical [21]

FEM Mesh (A)

FEM Mesh (B)

Ref. [21] (5x20 Divisions)

Ref. [21] (5x25 Divisions)

Proposed Formulation Mesh (A)

Fig. 10. Bending moment of the multi-thickness slab along the center line, in Example (2).

A

A

q

Fixed support

Fig. 11. The L-shaped plate structure in Example (3).

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

1.10

1.20

-0.10 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10

Distance along the strip (m)

Analytical [22]FEM Mesh (A)FEM Mesh (B)Proposed Formulation Mesh (A)Undeformed shapeRef. [22]

Fig. 12. Deformed shape of L-shaped plate structure along section (A-A), in Ex-ample (3).

Fig. 13. Bending moment of L-shaped plate structure along section (A-A), in Ex-ample (3).

Fixed support

P1

P2

Fig. 14. The hollow cantilever box considered in Example (4).

T.H.A. Naga, Y.F. Rashed / Engineering Analysis with Boundary Elements 71 (2016) 70–7874

first variation δΠ3 vanishes for any arbitrary values of(δγ δ δ )u and, p .Therefore the corresponding generalized Euler-Lagrange equationsare:

{ }γ{ } − { } − = ( )× × × ×⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦F G p B 0 1324 24 24 1 24 24 24 1

{ }{ } − ¯ = ( )× × ×⎡⎣ ⎤⎦L p P 0 14

T

24 24 24 1 24 1

Fig. 15. Vertical displacements of the box in Example (4) due to bending loading.

-0.09

-0.07

-0.05

-0.03

-0.01

0.01

0.03

0.05

0.07

0.09

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25z/L2

x/L2

FEM mesh (A)FEM mesh (B)FEM mesh (C)Proposed Formula�on mesh (A)Proposed Formula�on mesh (B)Ref. [22] mesh (D)

Fig. 16. Deformed shape of cross section at y¼2 in the box of Example (4) due to torsional loading.

T.H.A. Naga, Y.F. Rashed / Engineering Analysis with Boundary Elements 71 (2016) 70–78 75

γ{ } − { } = ( )× × × ×⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦L u G 0 15T

24 24 24 1 24 24 24 1

Eqs. (13)–(15) can be written in terms of the boundary dis-placements only as follows:

{ }{ } = ( )× × ×⎡⎣ ⎤⎦K u Q 1624 24 24 1 24 1

Where:

= ( )× × × ×⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦K R F R 1724 24 24 24

T

24 24 24 24

=( )× ×

−

×⎡⎣ ⎤⎦ ⎡

⎣⎢ ⎡⎣ ⎤⎦ ⎤⎦⎥ ⎡⎣ ⎤⎦R G L 1824 24 24 24

T 1

24 24

In which, { } ×u 24 1 is the generalized displacement vector, andgiven by:

{ }= { } { } { } ⋯⋯⋯⋯⋯{ } ( )× ×⎡⎣ ⎤⎦u u u u u 1924 1

Ti1 i2 i3 i8 1 8

Where{ } =ju , 1to8ij is the component of the generalized displace-ment, postulated at each degree of freedom, for example:

{ }θ θ{ } = ( )u u 20i4 14 24 34

And { } ×Q24 1

is the force vector, and could be written as:

{ }{ } { } { } { }= ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ( )×⎡⎣ ⎤⎦Q Q Q Q Q 2124 1

T

i1 i2 i3 i8

Where { } =jQ 1 to 8ij

is the component of the force, postulated at

each degree of freedom, for example:

{ } { }= ( )Q M M Q 22i4 14 24 34

It has been noted that the obtained [K] or the stiffness matrix issymmetric, positive definite and similar to the one obtained fromthe finite element method [7].

As previously mentioned the source points ( )ξn are locatedoutside the proposed element domain; therefore the integralsinvolved in computing values of matrices [F], [G], [P], [L] and [B]are regular, and could be computed numerically using the stan-dard Gauss quadrature scheme. It has to be noted that the domainintegral { }B is evaluated using equivalent boundary integrals asgiven in Ref. [18].

Fig. 17. The cylindrical barrel roof considered in Example (5).

T.H.A. Naga, Y.F. Rashed / Engineering Analysis with Boundary Elements 71 (2016) 70–7876

3. The proposed shell finite element

The previously derived stiffness matrix for the plate bending inSection (2) is coupled to the previous work for 2D membraneformulation in Ref. [9] to produce shell element. Formulation ofgenerally inclined element is presented in this section. Fig. (2)demonstrates general configuration of the proposed shell finiteelement.

3.1. Local stiffness matrix of the proposed shell finite element

The local stiffness matrix ⎡⎣ ⎤⎦Kb and ⎡⎣ ⎤⎦Kp of the plate bendingand 2D membrane respectively are initially constructed from thepreviously stated formulation and from Ref. [9], relating the nodalboundary generalized displacement vector to the nodal boundarygeneralized force vector by the following relationships:

{ }{ } = ( )× × ×⎡⎣ ⎤⎦K u Q 23p 16 16 p 16 1 p 16 1

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0 5 10 15

Def

lect

ion

(ft)

Angle a

Analytical [1]

OFFSET =0.40

OFFSET =0.50 to 8.50

OFFSET =8.75

Fig. 18. Effect of source po

{ }{ } = ( )× × ×⎡⎣ ⎤⎦K u Q 24b 24 24 b 24 1 b 24 1

Where, the subscripts “p” and “b” denote quantities for 2D mem-brane and for plate bending, respectively.

Combining { } { } u uandp b to give:

{ } { } = θ θ θ ( )u u u u 251 2 3 1 2 3

And combining { } { } Q Qandp b to give:

{ } { } = ( )Q Q Q Q M M M 261 2 3 1 2 3

The relationship between the nodal local boundary generalizeddisplacement vector to the nodal boundary generalized forcevector of the shell can be written as follows:

{ }{ } = ( )× × ×⎡⎣ ⎤⎦K u Q 2748 48 48 1 48 1

Where ×

⎡⎣ ⎤⎦K48 48

is the shell stiffness in the local coordinate sys-tem; which can be constructed as follows:

=

( )θ

×

×

⎡⎣ ⎤⎦

⎡

⎣

⎢⎢⎢⎢

⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦

⎤

⎦

⎥⎥⎥⎥

K

K

K

K

0 0

0 0

0 0 28

p

b48 48

348 48

It can be seen that ×

⎡⎣ ⎤⎦K48 48

has zero in-plane rotational stiff-

ness. The value of θK 3is set equal to 10�7 times the smallestbending stiffness [20]. This value is chosen to remove the in-planerotational singularity from the element stiffness matrix when thelocal x1, x2;x3 axes coincide with the global X-Y-Z axes [20].

3.2. Transformation of the stiffness matrix

The stiffness matrix ⎡⎣ ⎤⎦K obtained in Eq. (28) represents thestiffness in the local coordinate system x1, x2, x3. Before the as-sembly of the over-all stiffness matrix of the structure, the shellstiffness matrix should be transformed to the global coordinatesystem X, Y, Z. The relationship between generalized displacementand generalized force vector in the global and local coordinatessystem could be written as follows

{ } = { } ( )× × ×⎡⎣ ⎤⎦u T u 2948 1 48 48 48 1

{ } { } = ( )× × ×⎡⎣ ⎤⎦Q T Q 3048 1 48 48 48 1

20 25 30 35 40

long line (A-B)

int location on results.

T.H.A. Naga, Y.F. Rashed / Engineering Analysis with Boundary Elements 71 (2016) 70–78 77

Where ×⎡⎣ ⎤⎦T

48 48is the transformation matrix (also known as the

direction cosines matrix) [20]. Finally, the stiffness matrix ×⎡⎣ ⎤⎦K

48 48of the shell element with reference to the global coordinate sys-tem can be obtained by substituting Eqs. (29) and (30) into Eq.(27), to give:

{ }{ } = ( )× × ×⎡⎣ ⎤⎦K u Q 3148 48 48 1 48 1

where

= ( )× × × ×⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦K T K T 3248 48 48 48

T

48 48 48 48

4. Numerical examples

In this section, several numerical examples are presented todemonstrate the accuracy and the validity of the proposed finiteelement for the plate bending problems and for shell problems.

4.1. Example 1: multi- thickness simply supported circular plate

A multi-thickness circular plate are simply supported along itsexternal boundary, as shown in Fig. (3), is considered in this ex-ample. The plate is subjected to a uniformly distributed load withintensity �1 t/m2. The Young's modulus for the plate material isE¼3�106 t/m2 and Poisson's ratioν is 0.20. Two meshes (A and B)are used, as shown in (Figs. (4) and 5). (Figs. (6) and 7) demon-strate the deflection and bending moment along section (A-A) ofplate using meshes (A) and (B). Results are plotted from the pro-posed new finite element together with analytical values; resultsobtained from the traditional finite element method [2] and thestiffness boundary element formulation in Ref. [21]. It can be seenthat, the present formulation is accurate compared to analyticalsolution even when using the coarse mesh (A).

4.2. Example 2: multi-thickness cantilever plate

The cantilever multi-thickness plate, as shown in Fig. (8), isconsidered in this example. The plate is subjected to a uniformdistributed load with intensity �1 t/m2. The Young's modulus forthe plate material is E¼1�105 t/m2 and Poisson's ratio ν is set tozero. (Figs. (9) and 10) demonstrate the deflection and bendingmoment along the plate center line. Results are plotted from theproposed new finite element together with the traditional finiteelement method [2] and those of two boundary element strate-gies: the first is the traditional boundary element method with subregions [22] and the second is based on stiffness formulation of inRef. [21]. The results are also compared with the analytical values.Three meshes are considered:

� Mesh (A): consist of two elements with 8 nodes.� Mesh (B): consist of 64, four-node quadrilateral elements.� Mesh (C): consist of two sub regions with 12 boundary element.

(Figs. (9) and 10) demonstrate the superiority of the presentelement in term of accuracy; even with two elements.

4.3. Example 3: L-shaped shell structure

The previous two examples verified the present formulation forplate bending problems. This example and the next will test thepresent element for solution of shell structures.

The L-shaped plate shown in Fig. (11) with thickness 0.1 m,Young's modulus is E¼1�105 t/m2 and Poisson's ratio ν is 0.0 isconsidered in this example. The structure is subjected to

distributed load (q) of intensity 1t/m as shown in Fig. (11).(Figs. (12) and 13) demonstrate the deformed shape and thebending moment along section (A-A). Results are plotted from theproposed new finite element together with those obtained fromthe traditional finite element method [2] and the conventionaldirect boundary element method [22]. The results are comparedalso to the analytical values. Three meshes are considered:

� Mesh (A): consist of one element with 8 nodes for each surface.� Mesh (B): consist of 16 elements with 4 nodes.� Mesh (C): consist of one region with 12 boundary elements for

each surface.

(Figs. (12) and 13) demonstrate the accuracy of the presentformulation.

4.4. Example 4: hollow cantilever box

The hollow cantilever box of span (L2¼2000 mm), shown inFig. (14), and of thickness of 2mm is considered in this example.The box is subjected to different load cases:

1. Case (1): bending loading, in which P1¼P2¼ 5000N2. Case (2): torsion loading, in which P1¼ – P2¼ 5000N

The Young's modulus for the plate material is E¼7.0�104 MPaand Poisson's ratio ν is 0.30. (Figs. (15) and 16) demonstrate thevertical displacement and the deformed shape at y¼2 m for thebending and torsion cases of loading, respectively. Results areplotted from the proposed new finite element together with re-sults of the traditional finite element method [2] and the con-ventional direct boundary element method with sub regions [22].The results are also compared to the analytical values. Five meshesare used:

� Mesh (A): consist of 80 elements with 8 nodes.� Mesh (B): consist of 160 elements with 8 nodes.� Mesh (C): consist of 320, four-node quadrilateral elements.� Mesh (D): consist of one sub-region with 16 boundary element

for each region.� Mesh (E): consist of 4608, four-node quadrilateral elements.

Fig. (15) demonstrates that the results of the proposed for-mulation for mesh (A) and mesh (B) are close to the analyticalsolution. From Fig. (16), it can be seen that, the result of the pro-posed formulation for both meshes (A) and (B) are close to resultsof the finest finite element mesh (C).

4.5. Example 5: Effect of source point location

The previous examples verified the present formulation forplate bending and shell problems. This example demonstrates theeffect of source point location on the accuracy of the results. Thecylindrical barrel roof shown in Fig. (17) is considered. It has athickness of 0.25 ft, Young's modulus of E¼4.3�108 Ib/ft2 andPoisson's ratioν is set to zero. The roof is loaded by a distributedload of intensity 90 Ib/ft2. Only one quadrant is modeled with a6�12 shell elements. Fig. (18) demonstrates the variation of thedeflection along line A-B for different source point offset from(0.40 to 8.75) times the used element side length. It can be seenthat the result are accurate when the source point offset is withinthe range (0. 5 to 8.5) times the used element side length.

5. Conclusions

In this paper, a new finite element for shell structures was

T.H.A. Naga, Y.F. Rashed / Engineering Analysis with Boundary Elements 71 (2016) 70–7878

developed using an improved hybrid boundary solution. The for-mulation uses the fundamental solutions for 2D plane and platebending problems as trial functions. No singular integrals wasinvolved as the place of source points were chosen outside theelement domain. It was demonstrated that the choice of sourcepoint locations has almost no effect on the results within a re-commended range (0. 5 to 8.5) times the used element side length.Several examples with different boundary conditions and geo-metries were tested. It was demonstrated that the present for-mulation results were very accurate even with small number ofelements. The present formulation combines both advantages ofboundary and finite element methods.

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