Engineering Structures 30 (2008) 653–663www.elsevier.com/locate/engstruct

Shear deformation effect in nonlinear analysis of spatial beams

E.J. Sapountzakis∗, V.G. Mokos

School of Civil Engineering, National Technical University, Zografou Campus, GR-157 80 Athens, Greece

Received 31 August 2006; received in revised form 2 May 2007; accepted 7 May 2007Available online 18 June 2007

Abstract

In this paper a boundary element method is developed for the nonlinear analysis of beams of arbitrary doubly symmetrical simply or multiplyconnected constant cross-section, taking into account shear deformation effect. The beam is subjected in an arbitrarily concentrated or distributedvariable axial loading, while the shear loading is applied at the shear centre of the cross-section, avoiding in this way the induction of a twistingmoment. To account for shear deformations, the concept of shear deformation coefficients is used. Five boundary value problems are formulatedwith respect to the transverse displacements, the axial displacement and to two stress functions and solved using the analogue equation method, aBEM-based method. Application of the boundary element technique yields a system of nonlinear equations from which the transverse and axialdisplacements are computed by an iterative process. The evaluation of the shear deformation coefficients is accomplished from the aforementionedstress functions using only boundary integration. Numerical examples are worked out to illustrate the efficiency, the accuracy and the range ofapplications of the developed method. The influence of both the shear deformation effect and the variability of the axial loading are remarkable.c© 2007 Elsevier Ltd. All rights reserved.

Keywords: Transverse shear stresses; Shear centre; Shear deformation coefficients; Beam; Second order analysis; Nonlinear analysis; Boundary element method

1. Introduction

An important consideration in the analysis of thecomponents of plane and space frames or grid systems isthe influence of the action of axial, lateral forces and endmoments on the deformed shape of a beam. Lateral loadsand end moments generate deflection that is further amplifiedby axial compression loading. The aforementioned analysisbecomes much more accurate and complex taking into accountthat the axial force is nonlinearly coupled with the transversedeflections, avoiding in this way the inaccuracies arising froma linearized second-order analysis.

Over the past twenty years, many researchers havedeveloped and validated various methods of performing alinearized second-order analysis on structures. Early efforts ledto methods based on accounting for the aforementioned effectby using magnification factors applied to the results obtainedfrom first-order analyses [1–3]. An example of such a methodis the “B1 and B2 factor approach” provided in the AISC-LRFD specification [4]. Since the modifications used in this

∗ Corresponding author. Tel.: +30 2107721718.E-mail addresses: cvsapoun@central.ntua.gr (E.J. Sapountzakis),

vgmokos@central.ntua.gr (V.G. Mokos).

0141-0296/$ - see front matter c© 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2007.05.004

method are only applied to the moments of the columns andnot of the beams, the results obtained from this method areoften unsatisfactory especially for cases involving moderate tolarge deformations [2]. Consequently, due to the demand ofmore rigorous and accurate second-order analysis of structuralcomponents several research papers have been publishedincluding a nonlinear incremental stiffness method [5], closed-form stiffness methods [6,7], the analysis of nonlinear effectsby treating every element as a “beam–column” one [8], a designmethod for space frames using stability functions to capturesecond-order effects associated with P–δ (for one element)and P–∆ (for multiple elements) effects [9], uniform formulaerestricted to a single bar of a skeletal structure and to only afew loadings [10], the finite element method using linear andcubic shape functions (approximate solutions) [11] and a 3-Dsecond-order plastic-hinge analysis accounting for material andgeometrical nonlinearities [12,13]. Recently, Katsikadelis andTsiatas [14] presented a BEM-based method for the nonlinearanalysis of beams with variable stiffness. In all these studiesshear deformation effect is ignored.

Moreover, Kim et al. presented a practical second-orderinelastic static [15] and dynamic [16] analysis for 3-D steelframes, Machado and Cortinez [17] a geometrically nonlinear

654 E.J. Sapountzakis, V.G. Mokos / Engineering Structures 30 (2008) 653–663

beam theory for the lateral buckling problem of bisymmetricalthin-walled composite simply supported or cantilever beams,taking into account shear deformation effect. Nevertheless, inall of the aforementioned research efforts the axial loadingof the structural components is assumed to be constant.Finally, the boundary element method taking into accountshear deformation effect has been employed only for first-orderanalyses [18–20].

In this paper a boundary element method is developedfor the nonlinear analysis of beams of arbitrary doublysymmetrical simply or multiply connected constant crosssection, taking into account shear deformation effect. Thebeam is subjected in an arbitrarily concentrated or distributedvariable axial loading, while the shear loading is applied atthe shear centre of the cross-section, avoiding in this way theinduction of a twisting moment (the torsional loading of thebeam can be independently analysed). To account for sheardeformations, the concept of shear deformation coefficientsis used. Five boundary value problems are formulated withrespect to the transverse displacements, the axial displacementand to two stress functions and solved using the analogueequation method [21], a BEM-based method. Application ofthe boundary element technique yields a system of nonlinearequations from which the transverse and axial displacementsare computed by an iterative process. The evaluation ofthe shear deformation coefficients is accomplished fromthe aforementioned stress functions using only boundaryintegration. The essential features and novel aspects ofthe present formulation compared with previous ones aresummarized as follows:

i. The beam is subjected in an arbitrarily concentrated ordistributed variable axial loading.

ii. The beam is supported by the most general boundaryconditions including elastic support or restraint.

iii. The performed analysis is a nonlinear one, arising from thefact that the variable axial force is nonlinearly coupled withthe transverse deflections, taking into account additionalterms and avoiding the restrictions of a linearized second-order analysis.

iv. Shear deformation effect is taken into account.v. The shear deformation coefficients are evaluated using

an energy approach, instead of Timoshenko’s [22] andCowper’s [23] definitions, for which several authors [24,25] have pointed out that one obtains unsatisfactory resultsor definitions given by other researchers [26,27], for whichthese factors take negative values.

vi. The effect of the material’s Poisson ratio ν is taken intoaccount.

vii. The proposed method employs a pure BEM approach(requiring only boundary discretization) resulting in lineor parabolic elements instead of area elements of theFEM solutions (requiring the whole cross-section to bediscretized into triangular or quadrilateral area elements),while a small number of line elements are required toachieve high accuracy.

Fig. 1. Prismatic beam in torsionless bending (a) with an arbitrary doublysymmetrical cross-section occupying the two-dimensional region Ω (b).

Numerical examples are worked out to illustrate theefficiency, the accuracy and the range of applications of thedeveloped method. The influence of both the shear deformationeffect and the variability of the axial loading are remarkable.

2. Statement of the problem

Consider a prismatic beam of length l with a doublysymmetrical cross section of arbitrary shape, occupying thetwo dimensional multiply connected region Ω of the y, zplane bounded by the K + 1 curves Γ1,Γ2, . . . ,ΓK ,ΓK+1,as shown in Fig. 1. These boundary curves are piecewisesmooth, i.e. they may have a finite number of corners. Thematerial of the beam, with shear modulus G and Poisson’sratio v is assumed homogeneous, isotropic and linearly elastic.Without loss of generality, it may be assumed that the x-axisof the beam principal coordinate system is the line joiningthe centroids of the cross-sections. The beam is subjected toan arbitrarily distributed axial loading px and to torsionlessbending arising from arbitrarily distributed transverse loadingpy , pz and bending moments m y , mz along y and z axes,respectively (Fig. 1(a)).

According to the nonlinear theory of beams for moderatelarge deflections ((∂u/∂x)2

∂u/∂x, (∂u/∂x) (∂u/∂y)

(∂v/∂x)+ (∂u/∂y) , (∂u/∂x) (∂u/∂z) (∂w/∂x)+ (∂u/∂z))and assuming small rotations [28,29], the angles of rotationof the cross-section in the x–z and x–y planes of the beamsubjected to the aforementioned loading and taking into accountshear deformation effect satisfy the following relations:

cos ωy ≈ 1 (1a)

cos ωz ≈ 1 (1b)

sin ωy ≈ ωy = −dw

dx= θy − γz (1c)

sin ωz ≈ ωz = −dv

dx= −θz − γy (1d)

E.J. Sapountzakis, V.G. Mokos / Engineering Structures 30 (2008) 653–663 655

F

E

ar

θ

a

a

α

where w = w (x), v = v (x) are the beam transversedisplacements with respect to z, y axes (Fig. 2(a)) [28,29],respectively, while the corresponding curvatures are given as

ky =dθy

dx= −

d2w

dx2 +dγz

dx= −

d2w

dx2 −pz

G Az(2a)

kz =dθz

dx=

d2v

dx2 −dγy

dx=

d2v

dx2 +py

G Ay(2b)

where γy , γz are the additional angles of rotation of the cross-section due to shear deformation and G Ay , G Az are its shearrigidities of the Timoshenko’s beam theory, where

Az = κz A =1az

A (3a)

Ay = κy A =1ay

A (3b)

are the shear areas with respect to y, z axes, respectivelywith κy , κz the shear correction factors, ay , az the sheardeformation coefficients and A the cross section area. Referringto Fig. 2(b) [28,29], the stress resultants Rx , Rz acting in the x ,z directions, respectively, are related to the axial N and shearQz forces as

Rx = N cos ωy + Qz sin ωy (4a)

Rz = Qz cos ωy − N sin ωy (4b)

which by virtue of Eq. (1) become

Rx = N − Qzdw

dx(5a)

Rz = Qz + Ndw

dx. (5b)

The second term in the right-hand side of Eq. (5a), expressesthe influence of the shear force Qz on the horizontal stressresultant Rx . However, this term can be neglected since Qzw

′

is much smaller than N [28,29] and thus Eq. (5a) can be writtenas

Rx ≈ N . (6)

Similarly, the stress resultant Ry acting in the y direction isrelated to the axial N and shear Q y forces as

Ry = Q y + Ndv

dx. (7)

The governing equation of the beam transverse displacementw = w (x) will be derived by considering the equilibrium of thedeformed element in the x–z plane. Thus, referring to Fig. 2(b)we obtain

dRx

dx+ px = 0 (8a)

dRz

dx+ pz = 0 (8b)

dMy

dx− Qz + m y = 0. (8c)

ig. 2. Displacements (a) and forces (b) acting on the deformed element in thexz plane.

Substituting Eqs. (6) and (5b) into Eqs. (8a) and (8b), usingq. (8c) to eliminate Qz , employing the well-known relation

My = E Iyky (9)

nd utilizing Eq. (2a) we obtain the expressions of the angle ofotation due to bending θy and the stress resultants My , Rz as

y = −dw

dx+

1G Az

(−E Iy

d3w

dx3 −E Iy

G Az

(dpz

dx+ N

d3w

dx3

− 2pxd2w

dx2 −dpx

dx

dw

dx

)+ m y

)(10)

My = −E Iyd2w

dx2 −E Iy

G Az

(pz +

dN

dx

dw

dx+ N

d2w

dx2

)(11a)

Rz = −E Iyd3w

dx3 −E Iy

G Az

(dpz

dx+ N

d3w

dx3 − 2pxd2w

dx2

−dpx

dx

dw

dx

)+ m y + N

dw

dx(11b)

nd the governing differential equation as

E Iy

(1 +

N

G Az

)d4w

dx4 = pz − pxdw

dx+ N

d2w

dx2 +dm y

dx

−E Iy

G Az

(d2 pz

dx2 − 3pxd3w

dx3

− 3dpx

dx

d2w

dx2 −d2 px

dx2

dw

dx

). (12)

Moreover, the pertinent boundary conditions of the problemre given asz1w (x) + αz

2 Rz(x) = αz3 (13a)

656 E.J. Sapountzakis, V.G. Mokos / Engineering Structures 30 (2008) 653–663

βz1θy (x) + βz

2 My(x) = βz3 at the beam ends x = 0, l (13b)

where αzi , β

zi (i = 1, 2, 3) are given constants, while the angle

of rotation θy and the stress resultants My , Rz are given as

θy = −E Iy

G Az

(1 +

N

G Az

)d3w

dx3 −dw

dx(14a)

My = −E Iy

(1 +

N

G Az

)d2w

dx2 (14b)

Rz = −E Iy

(1 +

N

G Az

)d3w

dx3 + Ndw

dx

at the beam ends x = 0, l. (14c)

Eq. (13) describe the most general boundary conditionsassociated with the problem at hand and can include elasticsupport or restrain. It is apparent that all types of theconventional boundary conditions (clamped, simply supported,free or guided edge) can be derived form these equations byspecifying appropriately the functions αz

i and βzi (e.g. for a

clamped edge it is αz1 = βz

1 = 1, αz2 = αz

3 = βz2 = βz

3 = 0).Similarly, considering the beam in the x–y plane we

obtain the boundary value problem of the beam transversedisplacement v = v (x) as

E Iz

(1 +

N

G Ay

)d4v

dx4 = py − pxdv

dx+ N

d2v

dx2 −dmz

dx

−E Iz

G Ay

(d2 py

dx2 − 3pxd3v

dx3 − 3dpx

dx

d2v

dx2 −d2 px

dx2

dv

dx

)inside the beam (15)

αy1 v (x) + α

y2 Ry(x) = α

y3 (16a)

βy1 θz (x) + β

y2 Mz(x) = β

y3 at the beam ends x = 0, l (16b)

where αyi , β

yi (i = 1, 2, 3) are given constants and the

expressions of the angle of rotation θz and the stress resultantsMz , Ry inside the beam are given as

θz =dv

dx−

1G Ay

(−E Iz

d3v

dx3 −E Iz

G Ay

(dpy

dx+ N

d3v

dx3

− 2pxd2v

dx2 −dpx

dx

dv

dx

)− m y

)(17a)

Mz = E Izd2v

dx2 +E Iz

G Ay

(py +

dN

dx

dv

dx+ N

d2v

dx2

)(17b)

Ry = −E Izd3v

dx3 −E Iz

G Ay

(dpy

dx+ N

d3v

dx3

− 2pxd2v

dx2 −dpx

dx

dv

dx

)− m y + N

dv

dx. (17c)

In both of the aforementioned boundary value problems theaxial force N inside the beam or at its boundary is given fromthe following relation [28,29]:

N = E A

[du

dx+

12

(dw

dx

)2

+12

(dv

dx

)2]

(18)

where u = u (x) is the bar axial displacement, which can beevaluated from the solution of the following boundary valueproblem:

E A

[d2u

dx2 +d2w

dx2

dw

dx+

d2v

dx2

dv

dx

]= −px

inside the beam (19)

c1u (x) + c2 N (x) = c3 at the beam ends x = 0, l (20)

where ci (i = 1, 2, 3) are given constants.The solution of the boundary value problems prescribed

from Eqs. (12), (13a), (13b), (15), (16a) and (16b) presumesthe evaluation of the shear deformation coefficients az , aycorresponding to the principal centroidal system of axes Cyz.These coefficients are established equating the approximateformula of the shear strain energy per unit length [25]

Uappr. =ay Q2

y

2AG+

az Q2z

2AG(21)

with the exact one given from

Uexact =

∫Ω

(τxz)2+(τxy)2

2GdΩ (22)

and are obtained as [30].

ay =1κy

=A

∆2

∫Ω

[(∇Θ) − e] · [(∇Θ) − e] dΩ (23a)

az =1κz

=A

∆2

∫Ω

[(∇Φ) − d] · [(∇Φ) − d] dΩ (23b)

where (τxz) ,(τxy)

are the transverse (direct) shear stresscomponents, (∇) ≡ iy (∂/∂y) + iz (∂/∂z) is a symbolic vectorwith iy, iz the unit vectors along y and z axes, respectively, ∆is given from

∆ = 2 (1 + ν) Iy Iz (24)

ν is the Poisson ratio of the cross-section material, e and d arevectors defined as

e =

(ν Iy

y2− z2

2

)iy +

(ν Iy yz

)iz (25a)

d = (ν Iz yz) iy +

(ν Iz

z2− y2

2

)iz (25b)

and Θ (y, z), Φ (y, z) are stress functions, which are evaluatedfrom the solution of the following Neumann type boundaryvalue problems [30].

∇2Θ = −2Iy y in Ω (26a)

∂Θ∂n

= n · e on Γ =

K+1⋃j=1

Γ j (26b)

∇2Φ = −2Izz in Ω (27a)

∂Φ∂n

= n · d on Γ =

K+1⋃j=1

Γ j (27b)

E.J. Sapountzakis, V.G. Mokos / Engineering Structures 30 (2008) 653–663 657

where n is the outward normal vector to the boundary Γ . Inthe case of negligible shear deformations az = ay = 0. It isalso worth here noting that the boundary conditions (26b) and(27b) have been derived from the physical consideration that thetraction vector in the direction of the normal vector n vanisheson the free surface of the beam.

3. Integral representations — numerical solution

According to the precedent statement, the nonlinear analysisof a beam including shear deformation reduces in establishingthe transverse displacements w = w (x), v = v (x), the axialdisplacement u = u (x) and the stress functions Φ (y, z),Θ (y, z). The transverse and the axial displacements musthave continuous derivatives up to the fourth and second order,respectively, with respect to x , while the stress functions musthave continuous partial derivatives up to the second order withrespect to y, z.

3.1. For the transverse displacements w, v

The numerical solution of the boundary value problemsdescribed by Eqs. (12), (13a), (13b), (15), (16a) and (16b) issimilar. For this reason, in the following we will analysethe solution of the problem of Eqs. (12), (13a) and (13b)noting any alteration or addition for the problem of Eqs.(15), (16a) and (16b). Eq. (12) is solved using the analogueequation method [21]. This method has been developed forthe beam equation including axial forces by Katsikadelis andTsiatas [31]. However, another formulation is presented in thisinvestigation.

Let w be the sought solution of the boundary value problemdescribed by Eqs. (12), (13a) and (13b). Differentiating thisfunction four times yields

d4w

dx4 = qz (x) . (28)

Eq. (28) indicates that the solution of the original problemcan be obtained as the deflection of a beam with unit flexuralrigidity and infinite shear rigidity subjected to a flexuralfictitious load qz (x) under the same boundary conditions. Thefictitious load is unknown. However, it can be established usingBEM as follows:

The solution of Eq. (28) is given in integral form as

w (x) =

∫ l

0qzw

∗dx −

[w∗

d3w

dx3 −dw∗

dx

d2w

dx2

+d2w∗

dx2

dw

dx−

d3w∗

dx3 w

]l

0(29)

where w∗ is the fundamental solution, which is given as

w∗=

112

l3(

2 + |ρ|3− 3 |ρ|

2)

(30)

with ρ = r/ l, r = x − ζ , x , ζ points of the beam, which is aparticular singular solution of the equation

d4w∗

dx4 = δ(x, ξ). (31)

Employing Eq. (30) the integral representation (29) can bewritten as

w(x) =

∫ l

0qzΛ4(r)dx −

[Λ4(r)

d3w

dx3 + Λ3(r)d2w

dx2

+ Λ2(r)dw

dx+ Λ1(r)w

]l

0(32)

where the kernels Λi (r), (i = 1, 2, 3, 4) are given as

Λ1(r) = −12

sgn ρ (33a)

Λ2(r) = −12

l(1 − |ρ|) (33b)

Λ3(r) = −14

l2|ρ| (|ρ| − 2)sgn ρ (33c)

Λ4(r) =1

12l3(

2 + |ρ|3− 3 |ρ|

2)

. (33d)

Notice that in Eq. (32) for the line integral it is r = x−ξ , x, ξ

points inside the beam, whereas for the rest terms r = x − ζ , xinside the beam, ζ at the beam ends 0, l.

Differentiating Eq. (32) results in the integral representationsof the derivatives of the deflection w as

dw(x)

dx=

∫ l

0qzΛ3(r)dx −

[Λ3(r)

d3w

dx3

+ Λ2(r)d2w

dx2 + Λ1(r)dw

dx

]l

0(34a)

d2w(x)

dx2 =

∫ l

0qzΛ2(r)dx −

[Λ2(r)

d3w

dx3 + Λ1(r)d2w

dx2

]l

0(34b)

d3w(x)

dx3 =

∫ l

0qzΛ1(r)dx −

[Λ1(r)

d3w

dx3

]l

0. (34c)

The integral representations (32) and (34a) written forthe beam ends 0, l together with the boundary conditions(13a) and (13b) can be employed to express the unknownboundary quantities w, w′, w′′ and w′′′ in terms of qz . Thisis accomplished numerically. If L is the number of the nodalpoints along the beam axis, this procedure yields the followingset of linear equations:

[E11] [E12] [E13] [E14][0] [E22] [E23] [0]

[E31] [E32] [E33] [E34][0] [E42] [E43] [E44]

ww′

w′′

w′′′

=

az

3

βz

3

0

0

+

[0][0]

[F3][F4]

qz (35)

in which [E22], [E23], [E1i ], (i = 1, 2, 3, 4) are 2 × 2 matricesincluding the nodal values of the functions az

1, az2, β

z1, βz

2 ofEqs. (13a) and (13b) and

[Ei j], (i = 3, 4, j = 1, 2, 3, 4)

are square 2 × 2 known coefficient matrices resulting from thevalues of the kernels Λi at the beam ends;

az

3

,βz

3

are 2 × 1

658 E.J. Sapountzakis, V.G. Mokos / Engineering Structures 30 (2008) 653–663

known column matrices including the boundary values of thefunctions az

3, βz3 in Eqs. (13a) and (13b) and [Fi ] (i = 3, 4)

are 2 × L rectangular known matrices originating from theintegration of the kernels on the axis of the beam. Finally, w,w′,w′′

andw′′′

are vectors including the two unknown

nodal values of the respective boundary quantities and qz isa vector including the L unknown nodal values of the fictitiousload.

The discretized counterpart of Eq. (32) when applied to allnodal points in the interior of the beam yields

W = [F] qz −([E1] w + [E2]

w′

+ [E3]w′′

+ [E4]w′′′

)(36)

where [F] is an L × L known matrix and [Ei ], (i = 1, 2, 3, 4)

are L × 2 also known matrices. Elimination of the boundaryquantities from Eq. (36) using Eq. (35) for homogeneousboundary conditions Eqs. (13a) and (13b) (az

3 = βz3 = 0) yields

W =[Bz]qz (37)

where[Bz]

is an L × L matrix.Moreover, the discretized counterpart of Eqs. (34a)–(34c)

when applied to all nodal points in the interior of the beam, afterelimination of the boundary quantities using Eq. (35) yieldsW ′

=[B ′

z

]qz (38a)

W ′′

=[B ′′

z

]qz (38b)

W ′′′

=[B ′′′

z

]qz (38c)

where[B ′

z

],[B ′′

z

],[B ′′′

z

]are known L × L coefficient matrices.

Note that Eqs. (37) and (38a)–(38c) are valid for homogeneousboundary conditions (az

3 = βz3 = 0). For nonhomogeneous

boundary conditions, an additive constant vector will appear inthe right-hand side of these equations.

Finally, applying Eq. (12) to the L nodal points in the interiorof the beam we obtain the following linear system of equationswith respect to qz[[

D′′ ′′z]−[D′′′

z

] [B ′′′

z

]−[D′′

z

] [B ′′

z

]−[D′

z

] [B ′

z

]]qz

= pz +

m′

y

−[Dz]

p′′z

(39)

where[D′′ ′′

z],[D′′′

z

],[D′′

z

],[D′

z

],[Dz]

are diagonal L × Lmatrices whose elements are given from

(D′′ ′′

z)

i i = E Iy

(1 +

Ni

G Az

)(40a)

(D′′′

z

)i i =

3E Iy

G Az(px )i (40b)

(D′′

z

)i i =

3E Iy

G Az

(p′

x

)i + Ni (40c)

(D′

z

)i i =

E Iy

G Az

(p′′

x

)i − (px )i (40d)

(Dz)i i =E Iy

G Az(40e)

at the L nodal points in the interior of the beam; qz, pz, m′y

and

p′′z

are vectors with L elements including the values of

the fictitious loading, the transverse loading, the first derivativeof the bending moment distributed loading and the secondderivative of the transverse loading at the L nodal points in theinterior of the beam. The values of the

(m y)′, (pz)

′′, (px )′ and

(px )′′ quantities result after approximating the corresponding

derivatives with appropriate central, forward or backward finitedifferences.

Similarly for the transverse deflection v = v (x), applicationof the Eq. (15) to the L nodal points in the interior of the beamresults the following linear system of equations with respect toqy[[

D′′ ′′y]−

[D′′′

y

] [B ′′′

y

]−

[D′′

y

] [B ′′

y

]−

[D′

y

] [B ′

y

]] qy

=

py

−m′

z

−[Dy]

p′′y

(41)

where[

B ′y

],[

B ′′y

],[

B ′′′y

]are known L × L coefficient matrices

similar with those mentioned before for the deflection w;[D′′ ′′

y],[

D′′′y

],[

D′′y

],[

D′y

],[Dy]

are diagonal L × L matrices

whose elements are given from

(D′′ ′′

y)

i i = E Iz

(1 +

Ni

G Ay

)(42a)

(D′′′

y

)i i

=3E Iz

G Ay(px )i (42b)

(D′′

y

)i i

=3E Iz

G Ay

(p′

x

)i + Ni (42c)

(D′

y

)i i

=E Iz

G Ay

(p′′

x

)i − (px )i (42d)

(Dy)

i i =E Iz

G Ay(42e)

at the L nodal points in the interior of the beam;qy,

py,

m′z

and p′′

y are vectors with L elements, similar with thosementioned before for the deflection w.

3.2. For the axial displacement u

Let u be the sought solution of the boundary value problemdescribed by Eqs. (19) and (20). Differentiating this functiontwo times yields

d2u

dx2 = qx (x) . (43)

Eq. (43) indicates that the solution of the original problemcan be obtained as the axial displacement of a beam with unitaxial rigidity subjected to a flexural fictitious load qx (x) underthe same boundary conditions. The fictitious load is unknown.

The solution of Eq. (43) is given in integral form as

u(x) =

∫ l

ou∗qx ds +

[u∗

du

dx−

du∗

dxu

]l

0(44)

E.J. Sapountzakis, V.G. Mokos / Engineering Structures 30 (2008) 653–663 659

where u∗ is the fundamental solution, which is given as

u∗=

12|r | (45)

with r = x − ζ , x , ζ points of the beam.Following the same procedure as in 3.1, the discretized

counterpart of Eq. (44) and its first derivative with respect tox , when applied to all nodal points in the interior of the beamyields

U = [Bx ] qx (46a)U ′

=[B ′

x

]qx (46b)

where [Bx ],[B ′

x

]are known matrices with dimensions L × L ,

similar with those mentioned before for the deflection w andthe following system of equations with respect to qx , qy and qzis obtained[

D′′x

]qx +

[D′′

x

] [[B ′′

z

]qz

]dg.

[B ′

z

]qz

+[D′′

x

] [[B ′′

y

] qy]

dg.

[B ′

y

] qy

= −px (47)

where the symbol []dg. indicates a diagonal matrix with theelements of the included column matrix, qx , px are vectorswith L elements, similar with those mentioned before for thedeflection w, while the axial force at the neutral axis of thebeam can be expressed as follows

N =[D′

x

] [B ′

x

]qx +

[D′

x

] [[B ′

z

]qz

]dg.

[B ′

z

]qz

+[D′

x

] [[B ′

z

]qz

]dg.

[B ′

z

]qz . (48)

In Eqs. (47) and (48)[D′′

x

],[D′

x

]are diagonal L×L matrices

whose elements are given from(D′′

x

)i i = E A (49a)(

D′x

)i i =

E A

2(49b)

at the L nodal points in the interior of the beam. Eqs. (39),(41), (47) and (48) constitute a nonlinear coupled systemof equations with respect to qx , qy , qz and N quantities.The solution of this system is accomplished iteratively byemploying the two term acceleration method [32,33].

3.3. For the stress functions Θ(y, z) and Φ(y, z)

The evaluation of the stress functions Θ (y, z) and Φ (y, z)is accomplished using BEM as this is presented in Sapountzakisand Mokos [30].

Moreover, since the torsionless bending problem of beamsis solved by the BEM, the domain integrals for the evaluationof the area, the bending moments of inertia and the sheardeformation coefficients (Eqs. (23a) and (23b)) have to beconverted to boundary line integrals, in order to maintain thepure boundary character of the method. This can be achievedusing integration by parts, the Gauss theorem and the Greenidentity. Thus, the moments of inertia and the cross section area

can be written as

Iy =

∫Γ

(yz2 cos β

)ds (50a)

Iz =

∫Γ

(zy2 sin β

)ds (50b)

A =12

∫Γ

(y cos β + z sin β) ds (50c)

while the shear deformation coefficients ay and az are obtainedfrom the relations

ay =A

∆2

((4v + 2) Iy IΘ y +

14v2 I 2

y Ied − IΘe

)(51a)

az =A

∆2

((4v + 2) Iz IΦz +

14v2 I 2

z Ied − IΦd

)(51b)

where

IΘe =

∫Γ

Θ (n · e) ds (52a)

IΦd =

∫Γ

Φ (n · d) ds (52b)

Ied =

∫Γ

(y4z sin β + z4 y cos β +

23

y2z3 sin β

)ds (52c)

IΘ y =16

∫Γ

[−2Iy y4z sin β + (3Θ cos β

− y (n · e)) y2]

ds (52d)

IΦz =16

∫Γ

[−2Izz4 y cos β + (3Φ sin β

− z (n · d)) z2]

ds. (52e)

4. Numerical examples

On the basis of the analytical and numerical procedurespresented in the previous sections, a FORTRAN program hasbeen written and representative examples have been studiedto demonstrate the efficiency, wherever possible the accuracyand the range of applications of the developed method. Inall the examples treated each cross section has been analysedemploying NC S = 300 constant boundary elements alongthe boundary of the cross section and NBeam = 29 constantelements along the axis of the beam, which are enough to ensurethe convergence of the solution procedure. Moreover, the CPUtime at a Personal Computer Intel(R) 2.00 GHz for the analysisof each example is less than 6 s.

Example 1. A cantilever beam of a hollow rectangularcross section b × h loaded axially (either tensionally orcompressively) and transversely in both directions, as shownin Fig. 3 has been studied. In Figs. 4 and 5 the transversedeflections w, v and in Fig. 6 the axial displacement u alongthe beam axis are presented for both cases of tensile orcompressive axial loading as compared with those obtainedfrom a linear analysis taking into account or ignoring shear

660 E.J. Sapountzakis, V.G. Mokos / Engineering Structures 30 (2008) 653–663

Fig. 3. Cantilever beam of Example 1.

Fig. 4. Deflections w of the beam of Example 1.

deformation effect. As expected, from the first two figuresit is easily verified that tensile axial loading decreases whilecompressive increases the resulting deflections. From the lastfigure it comes up that even for tensile axial loading the resultsof the nonlinear analysis show a remarkable beam shorteningcoming from the intense transverse load. Moreover, from theaforementioned figures the increment of all the deflectionsand the axial displacement due to the influence of the sheardeformation effect is remarkable and should not be ignored innonlinear analysis. Furthermore, in order to demonstrate theaccuracy of the proposed method, in Fig. 7 the obtained beamtransverse deflections v at the free end of the cantilever beamfor tensile axial loading are presented as compared with thoseobtained from a FEM solution [34] using 60 beam elements andapplying the Full Newton–Raphson Method [28]. The resultsare in excellent agreement. Finally, in Table 1 the transversedeflections w, v and the axial displacement u at the free end ofthe cantilever beam for compressive axial loading are presentedfor various discretization schemes of NBeam constant elements,demonstrating the convergence and stability of the proposedmethod. It is apparent that 29 elements give good results.

Example 2. For comparison reasons, a clamped beam of lengthl = 0.508 m, of a rectangular cross section (E = 2.0688 ×

108 kPa, v = 0.3) by ×hz = 0.0254 m×0.003175 m subjected

Fig. 5. Deflections v of the beam of Example 1.

Fig. 6. Axial displacement u of the beam of Example 1.

Fig. 7. Deflections v at the free end of the cantilever beam of Example 1.

E.J. Sapountzakis, V.G. Mokos / Engineering Structures 30 (2008) 653–663 661

Table 1Transverse deflections w (cm), v (cm) and axial displacement u (cm) at the freeend of the cantilever beam of Example 1 for compressive axial loading

NBeam Nonlinear analysis with shear deformationw (cm) v (cm) u (cm)

5 29.9960 48.7650 −9.62679 29.5840 45.6095 −8.7167

13 29.5005 45.0107 −8.550517 29.4696 44.7923 −8.490421 29.4548 44.6882 −8.461925 29.4466 44.6305 −8.446129 29.4416 44.5952 −8.436450 29.4320 44.5277 −8.4179

100 29.4283 44.5022 −8.4110

Fig. 8. Deflections w (cm) at the mid-span of the beam of Example 2.

to a concentrated transverse load Pz ( kN) at its mid span hasbeen studied. In Table 2 the deflections w at the loading pointfor various values of the concentrated load Pz are presented ascompared with those obtained from a linear analysis taking intoaccount or ignoring shear deformation effect. From this tablethe significant influence of the nonlinear analysis effect is easilyverified especially in the case of intense loading, while as it wasexpected (since the cross section’s dimensions are very smallcompared with the beam’s length) the shear deformation effectin this case could be ignored. Moreover, in Fig. 8 the deflectionsw at the mid span for various values of the concentrated loadPz are presented as compared with those obtained from a FEMsolution using 8-noded plane stress rectangular elements [35].The accuracy of the results is remarkable.

In order to establish the influence of the shear deformationeffect the aforementioned clamped beam of l/4 lengthsubjected to a concentrated transverse load Py at its mid spanhas been studied. In Table 3 and in Fig. 9 the deflections v atthe loading point for various values of the concentrated load Pyare presented as compared with those obtained from a linearanalysis taking into account or ignoring shear deformationeffect. The increment of all the deflections due to the influenceof the shear deformation effect is noteworthy.

Table 2Transverse deflections w (cm) at the mid-span of the clamped beam ofExample 2

Pz (k N ) Linear analysis Nonlinear analysisIgnoringsheardeformation(az = 0)

With sheardeformation(az = 3.8741)

Ignoringsheardeformation(az = 0)

With sheardeformation(az = 3.8741)

0.044 0.2166 0.2168 0.1772 0.17730.089 0.4332 0.4336 0.2799 0.28010.222 1.0832 1.0841 0.4522 0.45230.356 1.7332 1.7346 0.5594 0.55950.445 2.1665 2.1682 0.6157 0.61580.890 4.3331 4.3365 0.8175 0.81771.334 6.4997 6.5048 0.9581 0.95831.779 8.6662 8.6731 1.0696 1.06992.224 10.832 10.8413 1.1638 1.16422.669 12.9994 13.0096 1.2460 1.24653.114 15.1660 15.1779 1.3197 1.3202

Table 3Transverse deflections v (cm) at the mid span of the clamped beam of l/4 lengthof Example 2

Py (k N ) Linear analysis Nonlinear analysisIgnoringsheardeformation(az = 0)

With sheardeformation(az = 3.8741)

Ignoringsheardeformation(az = 0)

With sheardeformation(az = 3.8741)

50 0.05944 0.07428 0.05942 0.07422100 0.11888 0.14856 0.11869 0.14809200 0.23776 0.29587 0.23629 0.29343400 0.47552 0.59424 0.46439 0.56759600 0.71328 0.89137 0.67859 0.81321800 0.95104 1.18849 0.87640 1.03012

1000 1.18879 1.48561 1.05776 1.222041250 1.48599 1.85701 1.26321 1.433331500 1.78319 2.22841 1.44816 1.619472000 2.37758 2.97122 1.76910 1.936542500 2.97199 3.71402 2.04101 2.20166

5. Concluding remarks

In this paper a boundary element method is developedfor the nonlinear analysis of beams of arbitrary doublysymmetric simply or multiply connected constant crosssection, taking into account shear deformation effect. Fiveboundary value problems are formulated with respect tothe transverse displacements, the axial displacement andto two stress functions and solved using the AnalogEquation Method, a BEM based method. The evaluationof the shear deformation coefficients is accomplished fromthe aforementioned stress functions using only boundaryintegration. The main conclusions that can be drawn from thisinvestigation are

a. The numerical technique presented in this investigationis well suited for computer aided analysis for beams ofarbitrary doubly symmetric simply or multiply connectedcross section.

662 E.J. Sapountzakis, V.G. Mokos / Engineering Structures 30 (2008) 653–663

Fig. 9. Deflections v (cm) at the mid-span of the clamped beam of l/4 lengthof Example 2.

b. The significant influence of geometrical nonlinear analysisin beam elements subjected in intense transverse loading isverified.

c. The discrepancy between the results of the linear and thenonlinear analysis demonstrates the significant influence ofthe axial loading.

d. The accuracy of the proposed method is noteworthy even forfew boundary elements.

e. The remarkable increment of all the deflections and the axialdisplacements due to the influence of the shear deformationeffect demonstrates its significant influence in nonlinearanalysis.

f. The developed procedure retains the advantages of a BEMsolution over a pure domain discretization method since itrequires only boundary discretization.

Acknowledgments

Financial support provided by the “HRAKLEITOS ResearchFellowships with Priority to Basic Research”, an EU fundedproject in the special managing authority of the OperationalProgram in Education and Initial Vocational Training. TheProject “HRAKLEITOS” is co-funded by the European SocialFund (75%) and National Resources (25%).

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