Theory and Application of Naturally Curved and Twisted Beams with Closed Thin-Walled Cross Sections
Aimin Yu* - Rongqiang Yang - Ying Hao Tongji University, School of Aerospace Engineering and Applied Mechanics, China
A novel theory for analyzing naturally curved and twisted beams with closed thin-walled cross sections is presented based on the small displacement theory. By introducing the eigenwarping functions and expanding axial displacements or axial stress distribution in a series of eigenwarpings, the differential equation for determining generalized warping coordinate and the expression for eigenvalues can be derived from the principle of minimum potential energy. In the derivation procedure, the effects of the initial torsion and small initial curvature of the beams are accurately taken into account. The non-classical influences relevant to the beams are transverse shear and torsion-related warping deformations. Improved solutions can be obtained by adding a series expansion in terms of eigenwarpings to the uncorrected solution. The present theory is used to investigate the stresses and displacements of a cantilevered, rectangular box curved beam subjected to a uniformly distributed load. It is observed that the numerical results obtained agree well with the data from FEM. ©2009 Journal of Mechanical Engineering. All rights reserved.
Keywords: naturally curved and twisted beam, small initial curvature, small displacement theory, eigenwarping, generalized warping coordinate
0 INTRODUCTION
Static and dynamic analysis of naturally curved and twisted beams with closed thin-walled cross sections has many important applications in mechanical, civil and aeronautical engineering due to their outstanding engineering properties, such as streamlined modeling and favorable loaded characteristics. The structural behavior of the beams is no longer appropriately modeled with the classical beam theory ([1] to [3]), and a more advanced theory is much needed to overcome the demerits of the classical beam theory. Much research has been done in the theories for straight beams and curved beams ([4] to [14]), however, much less has been done for naturally curved and twisted beams. Bauchau and Bauchau et al. ([15] and [16]) provided a comprehensive treatment to the problem of warping using variational principles to model thin-walled straight beams made of anisotropic materials, however, their modes can not be used for naturally curved and twisted beams straightly. Based on small displacement theory, the main contribution of the present work is to derive a set of orthonormal eigenwarpings and equivalent constitutive equations that can be used for the analysis of naturally curved and twisted beams. In addition, the correction to transverse shear
deformations is also included in the present formulations.
1 GEOMETRY AND CONSTITUTIVE RELATIONS OF THE BEAM
Let the locus of the cross-sectional centroid of the beam be a continuum curve l in space, the tangential, normal and bi-normal unit vectors of the curve are t, n and b, respectively. The Frenet-Serret formula, for a smooth curve, is:
t' = k1n n' = -k1t + k2b b' = -k2n,
(1)
where ( )' means derivative with respect to s. s, k and k2 are arc coordinate, curvature and torsion of the curve, respectively.
We introduce % - and n - directions in coincidence with the principal axes through the centroid Ob as shown in Fig. 1. The angle between the % - axis and normal n is represented by 0, which is generally a function of s. If the unit vectors of Ox % and Ox n are represented by i^ and in, then:
= n cos0+ b sin0
in=-n sin0+ b cos0.	(2)
From Eq. (1) the following expressions are obtained:
Corr. Author's Address: School of Aerospace Engineering and Applied Mechanics,	733
100 Zhangwu Road, Shanghai, China, aimin.yu@163.com
t — k^iç kÇiq
h =-K*+ks n t
in — kçt - ks iç
(3)
which kç — k1sin^, kn — kposO, ks — k2 + 0'.
Fig. 1. Geometry of the beam
Fig. 2 depicts a cross section of the beam, where Z is the curvilinear coordinate describing the contour of the section, denoted C. In the present work, the basic assumption is that the contour does not deform in its own plane, but is free to warp out of plane. This means that the degree of freedom of the deformation is fully represented by six rigid body modes us(s), u^s), un(s), (s(s), ((s) and (n (s), respectively, the three translations and three rotations of the section, and any applied transverse load only induces membrane stresses in the structure, specifically an axial stress flow n, and a shear stress flow q. These two stress flows are acting in the plane of contour and are uniform across the thickness of the walls that form the cross-section. The constitutive relations are:
n = E • t • e
q = G •t Y	(4)
where t is the wall thickness, E is the modulus of elasticity and G is the shear modulus of the material, respectively, e and y are the membrane axial strain and (engineering) shear strain, respectively.
2 INTERNAL FORCES, EQUILIBRIUM EQUATIONS AND KINEMATIC EQUATIONS
Simplifying stress vectors to the centroid O1 on the cross section A, the principal vector Q and principal moment M can be obtained, of which components are respectively denoted by
Qs, Q4, Qn and Ms, M4, Mn so:
Q = Qs t + + Qvtv,
M = Ms t + M4i4 + Mviv,
where Qs is axial force, Q4 and Qn are shear forces, Ms is torque, M4 and Mn are bending moments, as shown in Fig. 3. The external forces and moments per unit length along the axis of the beam are indicated by p and m as:
p = Pst + M+ Pnn m = mst + mj4 + m i .
Fig. 3. Stress resultants developed on a typical beam element
The equilibrium equations are:
d {Q}-[K ]{Q}+{p} = {°}'
d {M }-[K ]{M }-[// ]{Q} + {m} = {0},
(5)
where
{0} — [Os 0ç oj, {M} — [M M {p} — [Ps Pç pj , {m} = \ms mç ,
[K ] —
k -k.
-k
k.. - k
, [H ] —
"0 0 0 0 0 1 0 -10
Fig. 2. Closed cell thin-walled beam model
The general solutions are [171:
[Q] = [A]-({Q0H 0[a] -{p}ds), {M} = [A].{{M0} + £[A] ■([H].[A]. ({Qo } + {Q'})>}) ds},
(6)
where {Q0} and {M0} are integration constants,
[Q*}=-i: mt -{p)ds .
If the base vectors of special fixed right-handed rectangular coordinate system are ix, iy, iz,
then :
t ■ i„
t ■ i..
i, ■ i U ■ i
t ■ h L ■ i
[A] =
The kinematic equations are:
i ■ i i ■ i i ■ i
n x n y n z
(7)
s = u' - k u, + k,u ,
s s n ( in'
(8)
S = u'(+ kvus -ksuv-pr|,
sn = u'n — k.us + ksu.+P. ws = ps — knP.+ k((pv, w.=p.+ kp — ksPn,
w = p — kp + k p.,
where ss, s., sn, ws, w., wn are respectively generalized strains corresponding to the internal forces Qs , Q. , Qn , Ms , M., Mn and us, u., un, ps, 9, pn, are the displacement components corresponding to the loads ps, p., pn, ms, m., mn. The boundary conditions should be given by the following prescribed qualities:
Qs or us , Q. or u. , Qn or un, Ms or ps, M. or 9, Mn or pn.
(9)
Eqs. (8) can be rewritten as:
	-[K ]{P}-		>} = {0},		
d W where:	-[K ]-{u}-		~[H HP}-{s} =	{0},	(10)
{p}=		P,	Pnl , iu] = [us	u,	n,
{p}=	[P.	P,	PnJ , {u} = [u*	. u,	unl ,
a}=		a,	anl, {S} = [S	s.	S]" ,
so the general solutions to the kinematic equations are [17]:
{p}=[ 4'({Po} + {P*}),
[u} = [A].{[U0} + j;[A] .({s} + [77].[A]
•({P0 } + {p}) ds},	(11)
in which {p0} and {U0} are integration constants,
PH'[4 •{w}ds •
3 STRUCTURAL ANALYSIS BY THE EIGENWARPING APPROACH
Eigenwarping theory is in fact to transform the solution of out-of-plane torsional warping of the cross-section into a problem for finding the eigenvaules and eigenvectors. The eigenvalue problem can be solved using a finite element technique where the eigenwarping function is discretized over the section of the beams. The solution to the problem will be improved by adding a series expansion in terms of eigenwarpings to the uncorrected solution. In addition, the correction to transverse shear deformations is also included in the present formulations.
Assuming that the deformations of the beam consist of stretching, bending and torsion, thus the displacement field can be written as follows:
u = Wt + Ui, + Vin,
(12) (12a)
in which:
W = Us (s) + nP, (s) - ,(pn (s),
U = u,is) -nps (s), V = Un (s) + p (s). (12b) The strain-displacement relations are [1]:
4gen =ss +n(o,-,(ov,
24se12 = S,-nas,
24gel3 = sn + ,as,	(13)
where ss, s,, Sn, as, a,, ®n are the same as Eq. (8). For simplicity, the initial curvature k is assumed small to assure that:
4s * !■
The assumption is realistic for most practical applications, hence it does not seriously
restrict the applicability of this model. In this development a set of orthonormal eigenwarpings that can be used for naturally curved and twisted beams is derived. An unloaded beam is now considered (i.e. {p}, {m},=0) and a solution of the following form is assumed ([1] and [15]):
W (Z, ^) = P(Z)a(s),	(14a)
S[,( s) = Ua(s), sv( s) = Va( s),
ms (s) = Ea(s),	(14b)
where p(Z) and a(s) are the eigenwarping modes of the cross-section and the generalized warping coordinates, respectively, and U, V and E are three unknown parameters. Substituting Eqs. (14) into the strain-displacement relations in [1], we obtain:
e11 = cpa'(s) + ks dÇ
Y = 2ei_
dZ
+ 2e,
( Pn-( P
dç dn dn 3 ~dÇ
a( s ),
= Ua(s) - n^oc(s)
dZ	dZ
( dP) + kp
dç n
( ) dÇ
dZ
+ Va( s ) dn + ÇSa(s )
dZ	dZ
tdP\ i
(T~)- kçP dn
a{ s ) n
dZ
,dp , dn	dE
= (— - k.(p—- + km— dZ E dZ n dZ
+ U — + V— + rE)a( s),
dZ dZ	(15)
where the variables are separated, and r is the distance from the centroid O\ to the tangent to the cross-sectional curve(see Fig. 2). The total strain energy in the beam is there:
n = 2 i0 ic (Eteh + GtY )dZds,
where the product of the generalized warping coordinate and its derivative can be eliminated according to small displacement theory, and the differential equation defining generalized warping coordinates and associated eigenvalues can be
derived by minimizing with respect to p, U , V and E. The derivation is similar to that in [15]. The associated eigenvalues ]] can also be written in the form of a Rayleigh quotient:
JG
ft=-
dPi
dn
(—^ - k.p. —-dZ Ç dZ
+kp u.—1
^ dZ 'dZ
+V.én + Br )2 + • dZ
Ek2
G s

dZ
JCEtpp dZ
(16)
Unlike the case of straight beams, Eq. (16) contains the terms related to the initial curvature and torsion of the beam, and a set of orthonormality relationships for the present problem is:
icEtpipjdZ = 8ij, icGtrridZ = M2Sj, (17) where:
r =
dP
dn
dÇ , n dÇ , tT dn

dZ
dZ
dZ dZ dZ
+sr )2 + Ek]
• G s
tJn-ipÇ
1
2 A 2
4 IMPROVED BEAM THEORY AND EQUIVALENT CONSTITUTIVE EQUATIONS
Improved solutions will be obtained by using Eqs. (12a), (8) and (14) accordingly:
W = Wor +gpiai,
(18a)
Si=eior +XUiai(s ),
Sn=Snor +ZV a( s),
co = co +V ¿z.a.( s),
s	sor / , • • V
(18b)
where Wor coincides with Eq. (12a), and s^or, enor, cosor coincide with e, en, cos in Eq. (8). Thus, the final strain components e and y are now:
e=en+ZV, (Z)a'( s)
+ks Z
(V )n-(V X
og dn
a (s)
= s +n®g +Zv (ZK(s)
+K Z
(V )n-(V g
dg dn
a (s),
y 2e'2ordJ+2e13ordZ+ZZ y
dg dn v ,d% dn = e„^- + e —- + rrn + Z (—— - kv—-
g dZ n dZ s T dZ g dZ
+KV^ + V,n+ rSM (s),
dZ i dZ i dZ i
(19)
where en, ei2or, ei3or coincides with Eq. (13). The strains corresponding to this displacement field can be used to evaluate the total potential energy
n
1 t'
n = - Jo Jc (Ete2 + Gty2 )dZds
-J0' (pgUg+ pvuv + msVs ^
(20)
and taking into account the orthonormality relationships (17) it reads:
n=n +Zj(
1(af + p2 a2) - diai
ds,
(21)
where nor is the total potential energy for the original problem, and:
dt = Qg(Ui + knV) + Qn(V -kgV) + MsEi. (22)
Eq. (21) shows the solutions to be decoupled from the corrective terms a and the different corrective terms decoupled from each other. Minimizing nor will render the equilibrium Eq. (5) described with the internal forces and minimizing the corrective terms with respect to ai gives:
a"-rfa =-d<.	(23)
This differential equation is solved readily and yields the solution of the problem as a series expansion of eigenwarpings:
W = wor+Zva,
i
s = s +zV.a.,
n nor / i i i > i
n = nor + Z EtVa'i
s, = s. +
ZUa
®s =®sor
+ ks Z Et
dp dyt
hrt )n- (~r~)g
dg dn
a
^ dp dn , dg
q = qor + z Gt(—^-k.v—+kncp,—
or V dZ g dZ n dZ
+Uidg + Vtdn + rsa (s),
dZ dZ	(24)
where nor and qor are the uncorrected solutions. When infinite series are used, Eq. (24) give the theoretical solution of the problem under the assumption of infinite in-plane rigidity of the section.
Introducing the internal forces defined by:
Qs =jcndZ, Ms =lcqrdZ,
Qg = |CqdzdZ, Mg=jcnndZ, dn
Qn = jcq-ZdZ, Mn=-j ngdZ.
Jc dZ	Jc	(25)
In view of Eqs. (4), (19) and (25), noting
that:
jc EtgdZ = jc EtndZ = jc EtgndZ = 0,
we obtain the equivalent constitutive equations described with the generalized strains and generalized warping coordinates:
Qs = Sss + Z jc EtpdZa'
+K ZjcEt
i
Qg = GgAggsgor + GnAgnS
dg
dp fiV, g
dg on
dZat,
+ f Gtr^dZ® +Z i Gtr ^dZa., Jc dZ sor ¿flc 1 dZ "
Q = G.A. s. + G A s
¿¿n g gn gor l nn l
d
+ j Gtr ^L dZ(0 +Zi Gtr ^L dZa.,
Jc dZ sor ¿fJc	1 dZ "
M — IPrn + J Gtr—dÇe.
s P sor Je dÇ
+ \cGtrjZ dfrn +Y\cGtrirdÇai,
Mç — IçCOç + £| EtricpdÇa'i
P—-
kn — k1 —
+ ks XJ>
i
Mr — Irra)r -XlCEtÇPdZa'
(dp )r-(p)Ç
dç d,
- ks ZlcEt
(p)r-(p )Ç
dÇ d,
rdZ,ai,
ÇdÇar
(26)
where Gç and Gr are the shear coefficients in Ç -and , - directions for thin-walled beams [18], S — JcEtdZ is the axial stiffness,
Içç — JcEtr2dÇ is the bending stiffness (similar
definition for I ),
- nn ■
A44=\c Gt(42 dZ is the shear stiffness
(similar definitions for Ann and A4n), and IP =jcGtr2 dZ is the torsional stiffness.
It is observed that the internal forces of the beam depend on not only the generalized strains but also the eigenwarping functions, generalized warping coordinates and their derivatives.
5 EXAMPLE - A CURVED BEAM UNDER A UNIFORMLY DISTRIBUTED LOAD
Some numerical results are given to demonstrate the theoretical formulations derived in previous sections, which will be directly applied to compute the stresses and displacements of a curved, thin-walled rectangular beam (see Fig. 4). In this case 0, ks and k; in Eq. (3) are zero and kn is 1/R. Fix the origin of the rectangular coordinate system at the end of the beam ( s = 0), the axis of the beam being on the plane Oxy. The load acting is
{m} = [0 0 0], {p} = [0 0 Pn]T.
If the axis of the beam is a circle with radius a, one has
a	a
x = asinp, y = a(1 - cosp).
Using Eqs. (6) and (11), we have:
Ms = M0s cos p + M0; sin p
+ Q0na(1 - cos P) + Pna2 (sin p - p, M; = -M0s sin p + M0; cos p + Q0na sin p - Pna2(1 - cos p),
Qn= Q0n- Pns,
( =(s cosp + (0;sinp
+ acosp j ^ (ws cosp - w4sinp)d p +asinpj^(ws sinp + ®4cosp)dp,
(4 =-( ssinp + (0;cosp
- asinp j ^ (wscosp - w^ sinp)dp +acospj J(®isinp + ^cosp)d p,
un = U0n +(0sy - (04x + a j0" sndp
+aj^ asinpj^(®scosp-^sinp) dp
-a cosp j" (wssinp +w4 cos p) dp] dp, ^7)
where M0s, M04,, Q0n are the values of Ms, M4, Qn, at the end s = 0, and (0s, (04, U0n are the values of (s, (, un at s = 0, ws, w^, and sn are described with Ms, M4, Qn, a and a' by Eqs. (26).
The curved beam with radius a = 400 mm is assumed to be fixed at one end (s = 0) and free at the other (s = l). Fig. 4b illustrates the cross-section at the free end of the beam. The following properties of this material are:
E = 2.106-105 MPa, G = 81.6-105 MPa. The first step is to calculate the eigenwarpings ( and the associated eigenvalues J«i. The eigenvalue problem (16) can be solved using a finite element technique. For this example, 43 nodal points are used to model the section and 24 eigenwarpings are extracted.
U0 s = U0,= U 0n= 0,
0.16m
axis of the beam {a)
W
		
		
A		| 0.04m
		C
P, (b)
Fig. 4. Axis and cross-section at the free end of a plane curved beam
The bending and twisting behavior of the beam subjected to a uniformly distributed load pn in the n direction will be described. In this case the differential Eq. (23) to be solved for improved solution is:
a	= -PnaVi(— -p)
J2 ~ ( n
" 2 (28)
- Pna *Si -P- Gos^).
The solution to Eq. (28) must be:
a = Cj ch /ui s + C2 sh /ui s + ai *,	(29)
in which a* is a particular solution to Eq. (28). The complete solution of Eq. (28) becomes:
Mi
M
1	-,n
2	Pn . 2
ai = Ce + Cf»' + pvaV% (^-P)
+ M Pna ( 2-P)
(t+mM )
-p a2Stcosp.
(30)
s = 0(P = 0),
P0 s = P0, = Pn = 0 ai = 0, s = ip = p ), Ms = M,= Qv= 0,
a' = 0,
where I = na/2, the integration constants determined by the aforementioned conditions are:
M 0 s = Pva 2(f -1),
M 0, =- Pna ^
0,
n 2
Q0V =~ Pna,
C =
1	a Mi n + Mina - 2e2 Mi a - 2e2 v
2	mM (1 + eMna)(1 + MMa2) i
C =
1 a Min + a Mn - 2a Mi - 2e2 ' a —
"2 mM (1 + eMa )(1 + M)a2 ) ' 1
—e2 x 2
2 + e2 aMi^ + e2 aM,n +
77 1
V i--e2
2
The boundary conditions are:
rf (l + erf'a )(1 + rfa2)
4 3	2	,-s	4 3
2a + e2 a rf n + e2 a rfn — 2e2 a rf
(31)
So far, the solutions to this problem have been obtained.
The determination of stresses and displacements is a significant problem in the static analysis of thin-walled beams. Let VA present the displacement in the n - direction of point A on the cross section (fi = n/2) of the beam due to the load pn shown in Fig. 4b, and ps presents the tip twist angle of the beam. Theoretical results for VA and ps are obtained and compared with a 2-D finite element analysis (referred as the FEM results), according to the ANSYS program. To analyze the beam shown in Fig. 4 by the finite element method, we partition it into 3600 shell elements (SHELL 92), and the total number of nodal points is 10880. These cases for pn (0 ^ 1000 N/m) are shown in Fig. 5 a and b. It is evident that the theoretical results are close to the data from FEM.
1
4
2
1
0,
Fig. 5a. Vertical displacement VA of point A at the free end of the beam under a uniformly distributed loadpn (0 1000 N/m)
a -0.5 1
-1.5
l "2 sT "2 5
•3 -3.5 -4 -4.5
0 100 200 300 400 500 600 700 800 900 1000
Pn [N/m]
Fig. 5b. Tip twist angle ps of the beam under a uniformly distributed loadpn (0 — 1000 N/m)
It is also interesting to compute the stress distributions of the beam. Fig. 6 shows the distributions of the axial stress flow n in the upper face at the root (P = 0) and the shear stress flow q in the upper face at cross section (P = 45°) of the beam under a uniformly distributed load pn = 1000N/m.
6 CONCLUSIONS
An analysis method for determining the stresses and displacements of naturally curved and twisted beams with closed thin-walled cross sections is developed based on small displacement theory. The effects of torsion-related warping, transverse shear deformations and extension-shearing coupling are included in the proposed model. The above results clearly indicate that the key factor to improving the stress and displacement predictive capability of a theory
Fig.6a. Distribution of the axial stress flow n in the upper face at the root of the beam under a uniformly distributed loadp= 1000 N/m
o.a 0.6 0.4
n 0.2
s
2, 0
^ -0.2
-0.4 -0.6 -o.a -1
-0.08 -006 -0.04 -0.02 0 0.02 0 04 0.06 0 08
i[m]
Fig. 6b. Distribution of the shear stress flow q in the upper face at cross section (P = 45°) of the beam under a uniformly distributed load pn = 1000 N/m
is to account for these non-classical effects correctly and find the eigenwarping modes of the cross-sections. The eigenwarpings depend on not only the material properties and geometry of the sections, but also the initial curvature and torsion of the beams. The theory suggested in this paper is not limited to thin-walled box beams. In the case of solid cross sections, the concept of eigenwarpings can be extended as long as the basic assumption remains valid, i. e., as long as the section can be assumed infinitely rigid in its own plane.
ACKNOWLEDGEMENTS
The work is supported by the National Natural Science Foundation of P. R. China (Grant NO. 10572105) and the Shanghai Leading Academic Discipline Project, Project Number: B302.
7 REFERENCES
[1]	Washizu, K. Some considerations on a naturally curved and twisted slender beam, Journal of Mathematics and Physics, 1964, vol. 43, no. 2, p. 111-116.
[2]	Yu, A.M., Fang, M., Ma, X. Theoretical research on naturally curved and twisted beams under complicated loads, Computers and Structures, 2002, vol. 80, no. 32, p. 2529-2536.
[3]	Yu A.M., Yang X.G., Nie G.H. Generalized coordinate for warping of naturally curved and twisted beams with general cross-sectional shapes, International Journal of Solids and Structures, 2006, vol. 43, no. 10, p. 2853-2867.
[4]	Katori, H. Consideration of the problem of shearing and torsion of thin-walled beams with arbitrary cross-section, Thin Walled Structures, 2001, vol. 39, no. 8, p. 671-684.
[5]	Jonsson, J. Distortional warping functions and shear distributions in thin-walled beams, Thin Walled Structures, 1999, vol. 33, no. 4, p. 245-268.
[6]	Jonsson, J. Distortional theory of thin-walled beams, Thin Walled Structures, 1999, vol. 33, no. 4, p. 269-303.
[7]	Spatial stability of shear deformable curved beams with non-symmetric thin-walled sections, I: Stability formulation and closed-form solutions. Computers & Structures, 2005, vol. 83, no. 31-32, p. 2525-2541.
[8]	Kim, M.K., Kim, S.B., Kim, N.I. Spatial stability of shear deformable curved beams with non-symmetric thin-walled sections, II: F.E. solutions and parametric study. Computers & Structures, 2005, vol. 83, no. 31-32, p. 2542-2558.
[9]	Kim, Y., Kim, Y.Y. Analysis of thin-walled curved box beam under in-plane flexure, International Journal of Solids and Structures, 2003, vol. 40, no. 22, p. 61116123.
[10]	Saade, K., Espion, B., Warzee, G. Nonuniform torsional behavior and stability of thin-walled elastic beams with arbitrary
cross sections, Thin Walled Structures, 2004, vol. 42, no. 6, p. 857-881.
[11]	Li, J., Shen, R., Hua, H., Jin, X. Coupled bending and torsional vibration of axially loaded thin-walled Timoshenko beams, International Journal of Mechanical Sciences, 2004, vol. 46, no. 2, p. 299-320.
[12]	Li, J., Li, W., Shen R., Hua, H. Coupled bending and torsional vibration of nonsymmetrical axially loaded thin-walled Bernoulli-Euler beams, Mechanics Research Communications, 2004, vol. 31, no. 6, p. 697-711.
[13]	Li J., Shen R., Hua H., Jin X. Response of monosymmetric thin-walled Timoshenko beams to random excitations, International Journal of Solids and Structures, 2004, vol. 41, no. 22-23, p. 6023-6040.
[14]	Halilovic, M., Štok, B. Analytical tracing of the evolution of the elasto-plastic state during the bending of beams with a rectangular cross-section, Strojniški vestnik -Journal of Mechanical Engineering, 2007, vol. 53, no. 12, p. 806-818.
[15]	Bauchau, O.A. A beam theory for anisotropic material, Journal of Applied Mechanics, 1985, vol. 107, no. 6, p. 416422.
[16]	Bauchau, O.A., Coffenberry, B.B., Rehfifld, L.W. Composite box beam analysis: Theory and Experiments, Journal of Reinforced Plastics and Composite, 1987, vol. 6, no. 1, p. 25-35.
[17]	Yuchun, Z., Peiyuan, Z., Bo, Y. Doublemoment of spatial curved bars with closed thin-wall cross-section, Applied Mathematics and Mechanics, 1999, vol. 20, no. 12, p. 1350-1357.
[18]	Bank, L.C., Bednarczyk, P.J. A Beam Theory for Thin-walled Composite Beams, Composites Science and Technology, 1988, vol. 32, no. 4, p. 265-277.