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Bulletin of Applied Mechanics 3(12), 105–114 (2006)105

A Calculation Method of Hollow Circular Com-

posite Beam Under General Loadings

Autor: Jiang Sun1, Milan Růžička2

1Jiaxing University, Department of Mechanical and Electronical Engineering, Jiaxing Zhejiang, China

2Ústav mechaniky, biomechaniky a mechatroniky, Fakulta strojní, ČVUT v Praze

<sunjiang@mail.zjxu.edu.cn>

Keywords: composite beam, general loading, laminate, circular beam

Abstract

A calculation is presented for hollow circular composite beam. This method is applied to slender

cantilever composite beams subjected to torsion, tension, shear, and combined loadings. The

numerical results from the present approach are compared with available experimental data,

other analytical results, and 3D finite element analysis. Good correlation between the present

method and other results is achieved for all test cases, regardless of wall thickness.

1 Introduction

Composite materials are finding increasing applications for thick or thin structures, owing to their

innovative and cost effective manufacturing technology. Many primary or secondary elements of

composite structures such as aircraft wings, helicopter rotor blades, robots arms, bridges and

structural elements in civil engineering constructions can be idealized as thin- or thick-walled

beams, which can be studied by considering simpler governing equations.

Since non-classical phenomena, such as large torsional warping and coupled deformations,

arise inherently from the orthotropy or fiber orientation in composites, detailed structural models

of composite thin- and thick-walled beams are essential for a full exploitation of special non-

classical effects in the design of macroscale structures.

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106

Figure 1: Circular beam and beam element

Elastic coupling in orthotropic laminae arises from the material structure resulting to cou-

pling between normal stress and shear strain as well as between shear stress and normal strain.

The classical lamination theory based on Kirchoff hypothesis was presented and discussed by

Reissner and Stravsky [1], Dong et al. and Ashton and Whitney [2] for determining adequately

the stresses and deflections only in thin laminates. An adequate prediction of elastic coupling

must be based on a detailed modeling of the distortion that occurred at planes perpendicular to

the beam axis before deformation. This distribution of shear deformation will be further referred

to as ”out-of-plane” warping”. Mansfield et al., Chang and Libove and Libove [3] have developed

various theories for composite one- or two-cell thin-walled cylindrical tubes without taking into

account the warping effect. Furthermore, Kim et al. [4] investigated the bending slope as well

as the twist angle variation of thin-and thick-walled composite beams by considering torsional

warping and transverse shear effects.

In this paper, a calculation method is presented for calculating a hollow composite beam un-

der general loadings. Calculation method is then applied to a thin-walled circular beam and a

thick-walled beam under general loadings. The numerical results from the present approach are

compared with available experimental data, other analytical results.

2 Analysis

2.1General descriptions of hollow beam and basis assumptions

The problem considered consists of a long hollow shell of length L, thickness h, and radius r, as

show in Fig.1. It is assumed that 2r≪L and no dimensional restrictions on the wall thickness

h are imposed. Here, a thick-walled beam is defined as a beam which satisfies h/2r≥0.1. The

Cartesian coordinate system (x, y, z) and the local curvilinear system (ξ, η, z) and local Cartesian

coordinate system (xj, yj, z) are used in the present analysis. The circumferential coordinate ξ is

measured along the tangent to the middle surface in the clockwise direction, whereas η is measured

along the normal to the middle surface. The material is anisotropic and its elastic properties can

vary circumferentially and in the direction normal to the middle surface as well. In the following

analysis, first, the hollow beam is divided to n beam elements, then the displacements, stress

and strain are analyzed on the local curvilinear system (ξ, η, z), then transfer them to the local

Cartesian coordinates (xj, yj, z), after that, transfer and integrate to global Cartesian system

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Bulletin of Applied Mechanics 3(12), 105–114 (2006)107

Figure 2: Displacement of beam element

(x, y, z).

The basis assumptions are hold in developing kinematic relations.

i The contours of the original beam cross-sections do not deform in their own planes. This

statement implies that the inplane deformation of the beam cross-section is neglected. This

assumption is particularly valid as the wall thickness of the beam increases.

ii The out-of-plane displacement of the beam cross-section due to bending and shear is as-

sumed to be described by a cubic function of the cross-sectional coordinates y and z (Levin-

son, 1981; Reddy, 1984). This means that transverse shear effects of the beam are considered

and transverse shear strains vary parabolically across the thickness.

iii Any general beam wall segment behaves as a thick shell. This implies that the transverse

effects of the wall segments are also taken into account.

iv Warping effects are taken into account.

2.2 Kinematic equations of beam element

The displacements of j-th beam element may be expressed as (see Fig. 2)

w(ξ,η,z) = w0(z) − ξϕξ(z) − ηϕη(z) ,

uξ(η,z) = uξ0(z) − ηθ(z) ,

vη(ξ,z) = vη0(z) − ξθ(z) ,

(1a)

(1b)

(1c)

where the functions uξ, vη, and w are ξ-, η-, and z-directional displacements, respectively.

The functions uξ0(z), vη0(z) and w0(z) represent the rigid body translations of beam element

along the ξ , η and z axes. While the variables ϕξ(z) and ϕη(z) denote the rotations about the

η and ξ axes and θ is the twist angle of beam element.

The strain fields of beam element associated with the displacements are

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108

εzz=∂w

∂z= w

′

0(z) − ξϕ

′

ξ(z) − ηϕ

′

η(z) ,

(2a)

γzξ=∂w

∂ξ+∂uξ

∂η+∂uη

∂z

=

?

u

′

ξ0(z) − ϕξ(z)

?

− ηθ

′(z) ,

(2b)

γzn=∂w

∂z

=

?

v

′

η0(z) − ϕη(z)

?

+ ξθ

′(z)(2c)

where the prime denotes the differentiation with respect to z (i.e.,’=d/dz), hereafter. From the

assumed displacement fields in eqn (1) it can be readily seen that assumption (i) of cross-sectional

nondeformability (implying εξξ= εηη= γξη=0) is satisfied.

2.3 Reduced constitutive equations

The constitutive equations for the generally orthotropic ith layer can be written as

σzz

σξξ

σηη

τξη

τηz

τξz

i

=

Q11

Q12

Q13

0

0

Q16

Q12

Q22

Q23

0

0

Q26

Q13

Q23

Q33

0

0

Q36

0

0

0

0

0

0

Q16

Q26

Q36

0

0

Q66

Q44

Q45

0

Q45

Q55

0

i

εzz

εξξ

εηη

γξη

γηz

γξz

i

(3)

where Qijis the off-axis stiffness.

The anisotropic elastic characteristics of composite beam walls can result in highly three-

dimensional elastic behavior. The specific manner in which the three-dimensional dependence in

eqn (3) is accounted for in an equivalent one-dimensional beam theory is particularly important.

It is quite reasonable in a one-dimensional beam theory that the off-cross-sectional stress compo-

nents σξξ, σηηand τξηin eqn (3) are assumed to be negligibly small compared to the remaining

stress components in the absence of internal or external pressure fields. However, the correspond-

ing strains εξξ, εηη and γξη may not negligibly small and are included in this formulation. For

example, a significant through-the-thickness normal strain εηη can be generated in composite

cylindrical shells for certain types of lay-ups due to relatively large Poisson’s ratios, νzξand νzη

(Vison, 1993).

By neglecting σξξ, ηηηand τξη, the off-cross-sectional strain components can be expressed in

terms of cross-sectional ones as,

ǫξξ= B1ǫzz+ B2γzξ,

ǫηη= B3ǫzz+ B4γzη,

γξη= B5γzη,

(4a)

(4b)

(4c)

where

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Bulletin of Applied Mechanics 3(12), 105–114 (2006)109

B1=Q13Q23− Q13Q33

Q22Q33− Q

B2=Q23Q36− Q26Q33

Q22Q33− Q

B3=Q12Q23− Q22Q13

Q22Q33− Q

B4=Q26Q23− Q22Q36

Q22Q33− Q

2

23

,

(5a)

2

23

,

(5b)

2

23

,

(5c)

2

23

,

(5d)

B5= −Q45

Q44

,

(5e)

Substituting eqns (4a-c) into the matrix expression (3) and keeping only the cross-sectional

components, the reduced constitutive equations with three-dimensional elastic effects for the i-th

layer in the beam element are expressed as

σzz

τzξ

τzη

i

=

C11 C12

C12 C22

0

0

0

0

C33

i

εzz

γzξ

γzη

i

(6)

where

C11= Q11+2Q12Q13Q23− Q

2

12Q33− Q

2

23

2

13Q22

Q22Q33− Q

,

(7a)

C12= Q16+Q12Q23Q36− Q12Q26Q33+ Q13Q23Q26− Q13Q22Q36

Q22Q33− Q

C22= Q66+2Q23Q23Q36− Q

Q22Q33− Q

2

45

Q44

2

23

,

(7b)

2

26Q33− Q

2

23

2

36Q22

,

(7c)

C33= Q55−Q

,

(7d)

2.4Equilibrium equations for the beam element

The resultant forces and moments acting over the cross-section of beam element are related to

the stresses in the beam element by equilibrium in the local curvilinear coordinate system as