geometrically nonlinear finite element model of spatial thin-walled beams with general open cross...

Published by AMSS Press, Wuhan, China.Acta Mechanica Solida Sinica, Vol. 22, No. 1, February, 2009 ISSN 0894-9166

GEOMETRICALLY NONLINEAR FINITE ELEMENTMODEL OF SPATIAL THIN-WALLED BEAMS WITH

GENERAL OPEN CROSS SECTION�

Xiaofeng Wang Qingshan Yang

(School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China)

Received 26 November 2007, revision received 11 September 2008

ABSTRACT Based on the theory of Timoshenko and thin-walled beams, a new finite elementmodel of spatial thin-walled beams with general open cross sections is presented in the paper, inwhich several factors are included such as lateral shear deformation, warp generated by nonuni-form torsion and second-order shear stress, coupling of flexure and torsion, and large displacementwith small strain. With an additional internal node in the element, the element stiffness matrix isdeduced by incremental virtual work in updated Lagrangian (UL) formulation. Numerical exam-ples demonstrate that the presented model well describes the geometrically nonlinear propertyof spatial thin-walled beams.

KEY WORDS spatial beams, thin-walled structures, geometrically nonlinear, finite element, stiff-ness matrix

I. INTRODUCTIONGeometrical nonlinearity of spatial thin-walled beams is invariably one of main concerns in the study

of spatial structures consisting of bars, in which domestic and foreign researchers have done a lot ofwork.

Reference [1] derived stiffness matrix of thin walled bars with open cross-section under axial, flexuraland torsional displacements, but neglecting the shear deformation due to nonuniform bending. Basedon the theory of finite displacements, Ref.[2] developed a nonlinear model of thin-walled beams withopen cross-section under non-uniform torsion and presented a static stability criterion which includedthe effects of changes in beam geometry such as initial bending curvature prior to instability whensubjected to conservative loads. Reference [3] studied the mechanical behavior of thin-walled barswith open cross-section when subjected to axial force, flexural and torsional moments and derived thenonlinear expressions in accordance with the beam theory. However, the influence of shear deformationdue to non-uniform bending and torsion was neglected. Reference [4] presented a consistent co-rotational(CR) total Lagrangian (TL) finite element formulation for the geometric nonlinear buckling and post-buckling analysis of thin-walled beams with generic open section and considered all coupling amongbending, twisting, and stretching deformations of the beam element. A stability matrix was introducedto the element tangent stiffness matrix. According to the theory of thin-walled beams with open crosssection, Ref.[5] took into account geometric nonlinearities and longitudinal deformations caused bylarge cross-sectional rotation of the beam. Reference [6] obtained a doubly symmetric thin-walled beamelement with open section using co-rotational total Lagrangian formulation. The effects of deformation-

� Project partly supported by the National Science Fund for Distinguished Young Scholars (No. 50725826).

Vol. 22, No. 1 Xiaofeng Wang et al.: Finite Element Model of Spatial Thin-walled Beams · 65 ·

dependent third-order terms of element nodal forces on the buckling load and post-buckling behaviorwere investigated. All coupling among bending, twisting, and stretching deformations for the beamelement was considered by consistent second-order linearization of geometrically nonlinear beam theory.Reference [7] included bending-torsion coupled terms and shear deformation effects in the analysisand presented an improved co-rotational formulation, separating incremental displacements by rigidbody motions and pure deformations. Reference [8] accounted for restrained warping as well as thesecond-order displacement terms due to large rotations and ensured the joint moment equilibriumconditions of adjacent non-collinear elements by the introduction of semi-tangential moment into thegeometric potential. Reference [9] considered large deflection, buckling, the cross-sectional warping andnonuniform torsion. The correction matrix was adopted for accurate consideration of finite rotation atframe joints. Reference [10] deduced a new stiffness matrix for geometrically non-linear analysis of three-dimensional beam-columns with bisymmetrical, thin-walled, I-type cross-sections and made correctionsin the element matrix to properly consider the behavior under finite rotation. Reference [11] establishedan asymmetric thin-walled beam-column element, in which the coupling among axial stretching, thelateral deflection and torsional deformations was incorporated with the influence of sectorial warpingin the section neglected. In accordance with the space beam element model, Ref.[12] developed a thin-walled beam element model considering the restrained torsion. The effects of axial force, shearing force,biaxial bending moment and bimoment were included in the geometrical stiffness matrix of the element.Reference [13] studied the influence of member-distributed loading on the geometric nonlinearity andof warping deformation and solved the problem of rotational discontinuities at the joints of deformedspace-frame structures by the adoption of Rodriguez’s modified rotation vector to represent angulardeformations.

However, studies in all these geometrically nonlinear literatures just concentrated partially on theshear deformation, warp and coupling between flexure and torsion. Literatures which dabbled in allthese aspects have not yet been found.

On the basis of Ref.[14] which synthetically considered the elastic effects of shear deformation,nonuniform torsion, coupling of flexure and torsion, and warp induced by second-order shear stress, thispaper develops a new geometrically nonlinear model of spatial thin-walled beams with general opencross section.

Some hypotheses for spatial thin-walled beams are adopted as follows: (a) Thin-walled beams withgeneral open and consistent cross section are studied in the paper. (b) Although large deformationis assumed, beam elements are elastic. (c) Beam elements comply with the theory of Bernoulli-Eulerbeam when subjected to lateral deflection and with that of Vlasov thin-walled beam when subjectedto nonuniform torsion. (d) Characteristics of the beam section are invariable in the deformation.

II. SHAPE FUNCTIONLocal coordinates of the spatial thin-walled beam

element are specified in Fig.1. Axis x is located atcentroid. y and z are principal centroidal axes of iner-tia. o is origin of element coordinates. i and j denotenodes at both ends of the element, respectively. m

is internal node of the element, which is located atthe midpoint of the element as showed in Fig.1. Fig. 1 Local coordinates of element.

According to Ref.[14], displacements of the element are derived:

u0 = N1L2u0i + N2

L2u0j (1)

vs = N1Hvsi + N2

Hvsj + N3H

dvs

dx

∣∣∣∣i

+ N4H

dvs

dx

∣∣∣∣j

(2)

ws = N1Hwsi + N2

Hwsj + N3H

dws

dx

∣∣∣∣i

+ N4H

dws

dx

∣∣∣∣j

(3)

· 66 · ACTA MECHANICA SOLIDA SINICA 2009

θy = N1L3θyi + N2

L3θyj + N3L3θym (4)

θz = N1L3θzi + N2

L3θzj + N3L3θzm (5)

θx = N1Hθxi + N2

Hθxj + N3H

dθx

dx

∣∣∣∣i

+ N4H

dθx

dx

∣∣∣∣j

(6)

θ = N1L3θi + N2

L3θj + N3L3θm (7)

where u0 is the displacement of centroid along axis x. vs and ws are displacements of the shear centrealong axis y and z, respectively. θx, θy and θz are rotations of cross section around axis x, y and z. θ

is the warp angle of the cross section.

ζ =x

L(8)

N1L2 = 1− ζ, N2

L2 = ζ (9)

N1L3 = 1− 3ζ + 2ζ2, N2

L3 = −ζ + 2ζ2, N3L3 = 4ζ − 4ζ2 (10)

N1H = 1− 3ζ2 + 2ζ3, N2

H = 3ζ2 − 2ζ3, N3H = (ζ − 2ζ2 + ζ3)L, N4

H = (−ζ2 + ζ3)L (11)

Formulae above can be described in matrix as follows:

{δ} = [N ] {δ0} (12)

where

{δ} ={

u0 vs ws θx θy θz θ}T

(13)

{δ0} ={

δeT δiT}T

(14)

{δe} ={

u0i vsi wsi θxi θyi θzi θi u0j vsj wsj θxj θyj θzj θj

}T(15)

which is external node displacements involved in the C0 continuity.

{δi}

=

{dvs

dx

∣∣∣∣i

dws

dx

∣∣∣∣i

dvs

dx

∣∣∣∣j

dws

dx

∣∣∣∣j

dθx

dx

∣∣∣∣i

dθx

dx

∣∣∣∣j

θym θzm θm

}T

(16)

which is internal node displacements irrelevant to harmony of displacements between elements.

[N ] =[HT

u HTv HT

w HTθx HT

θy HTθz HT

θ

]T(17)

Hu ={N1

L2 0 0 0 0 0 0 N2L2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

}(18)

Hv ={0 N1

H 0 0 0 0 0 0 N2H 0 0 0 0 0 N3

H 0 N4H 0 0 0 0 0 0 0

}(19)

Hw ={0 0 N1

H 0 0 0 0 0 0 N2H 0 0 0 0 0 N3

H 0 N4H 0 0 0 0 0 0

}(20)

Hθx ={0 0 0 N1

H 0 0 0 0 0 0 N2H 0 0 0 0 0 0 0 N3

H 0 N4H 0 0 0

}(21)

Hθy ={0 0 0 0 N1

L3 0 0 0 0 0 0 N2L3 0 0 0 0 0 0 0 0 0 0 N3

L3 0}

(22)

Hθz ={0 0 0 0 0 N1

L3 0 0 0 0 0 0 N2L3 0 0 0 0 0 0 0 0 0 0 N3

L3

}(23)

Hθ ={0 0 0 0 0 0 N1

H 0 0 0 0 0 0 N2H 0 0 0 0 0 N3

H 0 N4H 0 0

}(24)


III. DISPLACEMENT AND STRAINNatural coordinates of the cross-section are showed

in Fig.2, where y and z are principal centroidal axesof inertia, o is centroid, s is natural coordinate at themid-surface of the cross section, S is shear centre, ys

and zs are Descartes’ coordinates of shear centre, P

is a discretionary point on the mid-surface, naturalcoordinate of which is s, ξ and η are tangential andnormal displacements at point P , respectively, andh is the distance from shear centre to tangential di-rection of point P . Then, displacements at the pointP can be expressed in the following form[15,16]:

Fig. 2 Natural coordinat.

⎧⎪⎪⎨⎪⎪⎩

u

ξ

η

⎫⎪⎪⎬⎪⎪⎭ =

⎡⎢⎢⎢⎢⎣

1 0 0 0 z −y ω

0∂y

∂s

∂z

∂sh 0 0 0

0 −∂z

∂s

∂y

∂srn 0 0 0

⎤⎥⎥⎥⎥⎦ {δ} =

⎡⎢⎢⎣

Φu

Φξ

Φη

⎤⎥⎥⎦ {δ} = [Φ] {δ} (25)

where u is the axial displacement, ω is the sectorial coordinate and ω =∫ s

0hds for the open cross

section,

rn = (y − ys)∂y

∂s+ (z − zs)

∂z

∂s(26)

According to Eqs.(25) and (12), incremental Green strain in update Lagrange (UL) formulation canbe expressed as

Δεx = ΔεLx + ΔεN

x (27)

Δγsx = ΔγLsx + ΔγN

sx (28)

where

ΔεLx =

∂(Δu)

∂x= [Φu]

∂(Δ {δ})

∂x= [Φu]

∂ [N ]

∂x(Δ {δ0}) (29)

ΔεNx =

1

2

[(∂(Δu)

∂x

)2

+

(∂(Δξ)

∂x

)2

+

(∂ (Δη)

∂x

)2]

=1

2(Δ {δ0})

T ∂ [N ]T

∂x([Φu]

T[Φu] + [Φξ]

T[Φξ] + [Φη]

T[Φη])

∂ [N ]

∂x(Δ {δ0}) (30)

ΔγLsx =

∂(Δu)

∂s+

∂(Δξ)

∂x=

(∂[Φu]

∂s[N ] + [Φξ]

∂[N ]

∂x

)(Δ {δ0}) (31)

ΔγLsx =

∂(Δu)

∂s

∂(Δu)

∂x+

∂(Δξ)

∂s

∂(Δξ)

∂x+

∂(Δη)

∂s

∂(Δη)

∂x

= (Δ {δ0})T[N ]T

(∂ [Φu]

T

∂s[Φu] +

∂ [Φξ]T

∂s[Φξ] +

∂ [Φη]T

∂s[Φη]

)∂[N ]

∂x(Δ {δ0}) (32)

IV. GEOMETRICAL STIFFNESS MATRIXAccording to the linearized incremental virtual work equation in UL formulation:∫

V

δ(tΔεLij)(

tΔSij)dV +

∫V

δ(tΔεNij )(

tτij)dV = tΔW −

∫V

δ(tΔεLij)(

tτij)dV (33)

where tΔSij is the increment of Piola-Kirchhoff stress tensor from t to t + Δt, tτij is Euler’s tensor atthe moment of t, tΔεij is the increment of Green strain tensor from t to t + Δt, tΔW is incremental


work done by external loads from t to t + Δt, and the superscripts L and N denote linear part andnon-linear part of Green strain tensor respectively, we conclude:∫

V

δ(tΔεLx )E(tΔεL

x )dV +

∫V

δ(tΔγLsx)G(tΔγL

sx)dV +

∫V

δ(tΔεNx )(tτx)dV

+

∫V

δ(tΔγNsx)(tτsx)dV = tΔW −

[∫V

δ(tΔεLx )(tτx)dV +

∫V

δ(tΔγLsx)(tτsx)dV

](34)

Considering the theory that virtual displacements are arbitrary at the admission of constraintconditions and Eqs.(27)-(32), Eq.(34) can be simplified:

t [K] (tΔ {δ0}) = (t [KL] + t [KG])(tΔ {δ0}) = t {Q} − t {fσ} (35)

where t {Q} is the vector of nodal forces at the moment of t,

t [KL] =

∫V

E

(∂ [N ]

T

∂x[Φu]

T[Φu]

∂ [N ]

∂x

)dV +

∫V

G

([N ]

T ∂ [Φu]T

∂s

∂ [Φu]

∂s[N ]

)dV

+

∫V

G

([N ]

T ∂ [Φu]T

∂s[Φξ]

∂ [N ]

∂x

)dV +

∫V

G

(∂ [N ]

T

∂x[Φξ]

T ∂ [Φu]

∂s[N ]

)dV

+

∫V

G

(∂ [N ]T

∂x[Φξ]

T [Φξ]∂ [N ]

∂x

)dV (36)

is the elastic stiffness matrix at the moment of t,

t [KG] =

∫V

∂ [N ]T

∂x

([Φu]

T[Φu] + [Φξ]

T[Φξ] + [Φη]

T[Φη]

) ∂ [N ]

∂x(tτx)dV

+

∫V

[N ]T

(∂ [Φu]

T

∂s[Φu] +

∂ [Φξ]T

∂s[Φξ] +

∂ [Φη]T

∂s[Φη]

)∂ [N ]

∂x(tτsx)dV

+

∫V

∂ [N ]

∂x

([Φu]

T ∂ [Φu]

∂s+ [Φξ]

T ∂ [Φξ]

∂s+ [Φη]

T ∂ [Φη]

∂s

)[N ] (tτsx)dV (37)

is the geometrical stiffness matrix at the moment of t,

tτx =N

A+

Myz

Iy

−Mzy

Iz

−Bω

Iω

(38)

is Euler’s normal stress at the moment of t,

tτsx = −QySz

Izt−

QzSy

Iyt−

MωSω

Iωt(39)

is Euler’s shear stress at the moment of t, and

t {fσ} =

∫V

∂ [N ]T

∂x[Φu]

T(tτx)dV +

∫V

([N ]

T ∂ [Φu]T

∂s+

∂ [N ]T

∂x[Φξ]

T

)(tτsx)dV (40)

is the vector of equivalent nodal forces corresponding to the stresses.

V. FINITE ELEMENT SOLUTIONThe nonlinear equilibrium equation as showed by Eq.(35) can be solved by the N-R method combined

with spherical explicit arc-length. The vector of incremental displacements at the iteration of j at theload step i can be divided into two parts[17]:

δu = δuI + δuII (41)


where δuI = K−1Q, K is the stiffness matrix updated at the iteration of j − 1 at the load step i, Q isthe vector of loads, δuII = −K−1R, R is the residual vector of nodal forces after the iteration of j− 1at the load step i.

The incremental load factor at the iteration of j at the load step of i can be expressed in the followingform:

ijδλ =

ijR−

ijΔuTi

jδuII

ijΔλ + i

jΔuTijδu

I(42)

where ijR = −

s2∥∥ij+1t

∥∥ (∥∥i

j+1t∥∥ − s), i

j+1t =

√∥∥ijt∥∥2

+ ijδu

Tijδu + i

jδλ2, ‖·‖ is the norm of matrix, i

jt

is the iterative vector at the iteration j at the load step of i, s is incremental arc length, ijδλ =

−ijΔuTi

jδuII

ijΔλ + i

jΔuTijδu

I, i

jΔu =

j−1∑k=1

ikδu and i

jΔλ =

j−1∑k=1

ikδλ.

VI. EXEMPLIFICATIONIn terms of the geometrical stiffness matrix deduced by this paper, a finite element program is

developed using C#.Net. There are five examples given in the following sections. Example 1 validatesthe geometrically nonlinear algorithm, example 2 testifies to the validity of the geometrically nonlinearmodel and the other three examples analyze effects of three factors including traverse shear deformation,coupling of flexure and torsion and warp.6.1. Example 1

The William’s plane frame showed in Fig.3 is a classical example widely used in the geometricallynonlinear analysis for structures. The length unit is millimeter. The basic parameters are those in whichelastic module E = 71.0×103 MPa, cross sectional area A = 1.181 cm2 and inertia moment I = 0.0375cm4. A vertical load P is applied at the top.

The relationship between the load and the displacement at the top calculated by both finite elementprogram and the pertinent theory is showed in Fig.4.

It is easy to find from Fig.4 that the the present result is very close to that obtained from the theory.Therefore, the finite element program is valid on the aspect of the geometrically nonlinear algorithm.

Fig. 3. Plane frame of William.

Fig. 4. Comparison of results.

6.2. Example 2

A thin-walled cantilever and its geometrical parameters are showed in Fig.5. The length unit iscentimeter. Axis x is located at centroid. y and z are principal centroidal axes of inertia. Load P isexerted at the centroid on the cross section of the free end and P = 100000 N, the disturbing loadQ is applied at the shear center and Q = 0.001P = 100 N, elastic module E = 2.1 × 107 N/cm2 andPoisson’s ratio μ = 0.25.

Comparison is conducted on the results calculated by the finite element model developed in thispaper and by beam189 and Shell63 element of ANSYS.

Figure 6 illustrates that the result calculated by this paper is consistent with that by Shell63 while theresult calculated by beam189 element deviates from both when geometrically nonlinear characteristics


Fig. 5. Cantilever and parameters of its sections.

Fig. 6. Comparison with results calculated by shell63.

of the element begins to be predominant. Apparently, beam189 element erroneously underestimatesthe beam rigidity at its geometrically nonlinear phase, thus is deficient at least in this aspect. Figure6 shows that the present model has a distinct advantage over beam189 element and can be used inengineering applications.

6.3. Example 3

Still take the same cantilever as showed in Fig.5 for example, except for the varied beam length tostudy the effect of shear deformation and the load case that only traverse force Q is applied at the shearcenter on the cross section of the free end with Q = 1000 N.

The quotient of displacements with and withoutshear deformation varying with the ratio of elementlength to section height is showed in Fig.7, whichmanifests that the shear deformation effect is nega-tively relevant to the ratio, in accordance with thefamiliar theory of material mechanics. The quotientbecomes smaller with the ratio increasing and largerwith its decreasing. The error due to the absence ofconsideration of shear deformation would be reducedto 4.5%, smaller than 5% (upper limit of tolerance)and could be accepted in engineering applicationswhen the ratio mount up to 10. Beyond that limit,beams are regarded as slim enough for shear deforma-tion to be neglected with acceptable error. However,

Fig. 7 Dependence of shear deformation on the ratio ofthe beam length to its cross section height.

it must be considered in the case of so-called stout beams whose ratio is below that limit, for that theshear deformation is dominant in the traverse displacement as easily seen in the left part of Fig.7.

6.4. Example 4

The example is also the same as showed in Fig.5. However, eccentricity of shear centre to centroid isregarded as a variable and only traverse force Q is assumed to be applied at the centroid on the crosssection of the free end and Q = 1000 N.


Fig. 8. Variation of the coupling effect with the ratio of theeccentricity to flank width.

Fig. 9. Variation of warp effect with the ratio of length andheight.

The relationship of the quotient of displacements considering and leaving out the coupling of flexureand torsion and the ratio of the eccentricity to flank width is showed in Fig.8.

From the figure, it is found that the quotient is positively relevant to the ratio. When the ratio getslarger, the quotient rises. It reflects that the coupling is being intensified, making the rigidity of thebeam weakened. When the ratio comes up to about 0.4, the quotient ascends to 104.9%. Over thatpoint the coupling should be taken into account to abstain from error larger than 5%.6.5. Example 5

Assume that twist moment T is applied at the free end of the thin-walled cantilever shown in Fig.5with varied beam length to study the warp effect and T = 1000 N·cm.

The quotient of displacements including and excluding warp varying with the ratio of the beamlength to the section height is showed in Fig.9.

In the figure, it is seen that warp effect makes the beam to be more rigid considering that thequotient is less than 1.0. With the ratio increasing and the effect of the warp constraint at the fixed enddeclining, the quotient gets larger. This means that warp effect is abated. However, the curve becomesflat with augment of the ratio, which means the discrepancy resulting from the warp effect diminishesmore slowly, with error of more than about 25% corresponding to the ratio greater than 10. Namely,even for the slim thin-walled beam, the warp effect is all the same notable and cannot be neglected.Therefore, the warp effect should be considered for thin-walled beams, be slim or stout.

VII. SUMMARYThis paper principally studies spatial thin-walled beams with general open cross section. Resorting

to the way that flexure angle and warp angle are independently interpolated, the model developedin this paper considers such factors as shear deformation, warp caused by non-uniform torsion andsecond-order shear stress, and the coupling between flexure and torsion. The first two examples verifythe satisfying accuracy of the present model, its advantages over beam189 element of ANSYS and itsavailability in the engineering. The other three examples give quantitative analyses to the extent of thesefactors’ influences and explicitly illuminate indispensability of inclusion of them under some conditions.

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geometrically nonlinear finite element model of spatial thin-walled beams with general open cross...

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