Torsion-extension coupling in initially twisted beams by finite elements

Eur. J. Mech. A/Solids 20 (2001) 501–508

2001 Éditions scientifiques et médicales Elsevier SAS. All rights reservedS0997-7538(00)01131-1/FLA

Torsion-extension coupling in initially twisted beams by finite elements

Wen-Guang Jianga,1, J.L. Henshallb,∗

aSchool of Mechanical Engineering, Yanshan University, Qinhuangdao, Hebei, 066004 PR ChinabDepartment of Mechanical Engineering, Brunel University, Uxbridge, Middlesex, UB8 3PH, UK

(Received 22 May 2000; revised and accepted 9 October 2000)

Abstract – The analysis of initially twisted beams subjected to axial loads (extension and/or torsion) may be performed by finite element modelling ofonly a small slice of the beam cross-section. Accurate relationships between the related translational nodal degrees of freedom were formulated andusedto maintain the helically symmetric deformation features. Consequently, conventional three-dimensional solid elements can be used for the structuraldiscretization. Both linear and nonlinear examples are presented to show the validity of the method. Excellent agreement has been achieved comparedwith the previous analyses and experimental results. 2001 Éditions scientifiques et médicales Elsevier SAS

twisted beam / tension and torsion load / accurate boundary condition

1. Introduction

The present paper is concerned with the analysis of uniformly pre-twisted constant cross-sectional beamssubjected to torsion and/or extension loads (figure 1). Okubo (1951, 1954) formulated the appropriate three-dimensional elasticity equations in the form of a two-dimensional boundary value problem and solved forelliptical contours (both for helical springs and twisted beams). Shield (1982) extended the analysis for generalsections, and Krenk (1983-a, 1983-b) obtained an explicit integral representation of the exact asymptoticcoupling term. These analyses were based on exact analytical theory of essentially the same equations. Thesetwo-dimensional equations were solved by numerical techniques by Kosmatka (1992) in a non-linear settingfor non-trivial cross-sections. Aside from these there are several papers on approximate theories for slendersections (Knowles and Reissner, 1960; Reissner and Wan, 1968; Rosen, 1983).

In the present paper a more rigorous approach using the finite element method is presented to analyze thisproblem. Accurate boundary conditions have been developed and applied using conventional finite elementanalysis. Several applications of the method to the solution of typical mechanical components such as wirerope strand, which exhibit helically symmetric features, have been previously satisfactorily implemented byJiang and co-workers (Jiang, 1999; Jiang et al., 1999, 2000). In this paper the methodology has been extendedto derive appropriate boundary conditions to perform both linear and nonlinear analyses of initially twistedbeams. This approach has the advantages of being able to consider exact geometry, can analyze beams withnon-linear, anisotropic and non-uniform material properties, can analyze geometric non-linear situations, andprovides accurate results with minimum approximations and very limited computational resources.

∗ Correspondence and reprints: [email protected] Currently at: Institute of Computational Mathematics, Brunel University, Uxbridge, Middlesex, UB8 3PH, UK

502 W.-G. Jiang, J.L. Henshall

Figure 1. Example of a model cross-sectional slice taken from an axially loaded initially twisted beam (this example shows a rectangular cross-sectionbeam, but the model applies to any uniform cross-section).

2. Finite element model

When an initially twisted beam is subjected to a twisting momentM and/or an axial forceF, the stressdistributions over all cross sections, and the warping and distortion of all cross sections will be identical. Inother words, the problem may be reduced to an analysis of a typical small slice of the beam as shown infigure 2. The coordinate system chosen is such that thez-axis coincides with the initially twisted axis of thebeam analyzed, and without loss of generality, the origin is located on the bottom cross-section plane of theslice model (seefigure 1). Three-dimensional solid brick elements are used for the structural discretization.The nodes of the elements have three degrees of freedom on each, i.e. translations in thex, y, andz directions.Using the present approach only one element division is required in the beam axial direction. Precise boundaryconditions are maintained by using the constraint equations which relate the displacements of the correspondingnodes on the corresponding beam cross-sections of the FE model.

The initial twist angle,θs , between the top and bottom cross-section is:

θs = �0�h, (1)

where�0 is the initial twist angle per unit length of the beam, and�h is the thickness of the cross-sectionalslice.

The deformation relationship between the corresponding nodesn(r, θ,0) and n′(r, θ + θs,�h) on thecorresponding cross-sections can be expressed as (seefigure 2):

�r ′ = �r, (2a)

�θ ′ = �θ + δθ, (2b)

u′z = uz + δh, (2c)

Torsion-extension coupling in initially twisted beams by finite elements 503

(a) (b)

Figure 2. The deformation relationship between the corresponding nodes on the corresponding cross-sections of the model: (a) 3D view; (b) top view.

where�r , �θ , uz and�r ′, �θ ′, u′z are the displacement components of noden and its corresponding noden′

in the cylindrical coordinate system respectively.δθ andδh are the changes in angle and axial distance betweenthe two corresponding nodes respectively due to the loading.

Transforming from cylindrical coordinates into Cartesian coordinates, the constraint equations relating thedisplacements between the corresponding nodes can be expressed as

u′ = Ru + u� + uε (3)

in which,

u = (ux, uy, uz)T (4)

is the displacement vector of noden,

u′ = (u′x, u

′y, u

′z)

T (5)

is the displacement vector of the corresponding noden′,

u� = (−2r sin(δθ/2)sin(θ + θs + (δθ/2)

),2r sin(δθ/2)cos

(θ + θs + (δθ/2)

),0)T

(6)

is the relative rotational displacement vector,

uε = (0,0, δh)T (7)

is the relative extension displacement vector and

R =

cos(θs + δθ) −sin(θs + δθ) 0

sin(θs + δθ) cos(θs + δθ) 0

0 0 1

(8)

is the rotational matrix.


δh andδθ can be expressed in terms of the applied axial strainε and the applied further twist angle per unitinitial length�, respectively, as

δh = ε�h, (9)

δθ = ��h. (10)

The implementation of equations (2a–c) or (3) ensures that the slice model can only move along, or rotate about,thez-axis of the coordinate system. To avoid singularity of the system of equations, more constraints should beintroduced to prevent this rigid body movement. This may be performed by constraining an arbitrarily selectednode away from thez-axis,n0(r0, θ0,0), from moving along axialz andθ directions. This can be realized bythe constraint equations:

uy = ux tanθ0, (11a)

uz = 0 (11b)

and the constraints on its corresponding node,n′0(r0, θ0 + θs,�h), can be derived by substituting equations

(11a, b) into equation (3), i.e.:

(u′

x

u′y

)= cos(θs + δθ) −sin(θs + δθ)

sin(θs + δθ) cos(θs + δθ)

(

ux

ux tan(θ0)

)+(−2r0 sin(δθ/2)sin(θ0 + θs + (δθ/2))

2r0 sin(δθ/2)cos(θ0 + θs + (δθ/2))

), (12a)

u′z = δh. (12b)

In the present method, the loading process has to be realized by applying an axial strainε and twist angle perunit length�, and the resultant axial forceF and twisting momentM can be derived by the summation of thereaction nodal forces on the cross-section of the FE model of the beam. When the axial strainε and twist angleper unit length� are given, the formulated constraint equations become linear relationships between nodaltranslational displacements, and can be conveniently incorporated into the system equations. The degrees offreedom on the left-hand side in equations (3), (11) and (12) can be expressed in terms of the degrees offreedom on the right-hand side, and therefore eliminated during the solution of the equations. After solution,the eliminated degrees of freedom can be recovered from the equations.

3. Numerical examples

3.1. Linear elastic analysis

Being an elastic body, the constitutive relationship relating the axial extension strainε and twist angle perunit length� to the resultant axial forceF and twisting momentM may be expressed in the general form

[k11 k12

k21 k22

](ε

�

)=(

F

M

), (13)

wherek11, k12, k21, k22 are the stiffness constants, andk12 = k21 based upon reciprocity.

For the present finite element analysis, any two sets of solutions, for example, extension with no-torsion(finite ε, � = 0) and torsion with no-extension (finite�, ε = 0) may be performed to derive all of the stiffness


constants by the following equations respectively

k11 = F/ε, k21 = M/ε, (14a, b)

k12 = F/�, k22 = M/�. (15a, b)

The accuracy of the proposed FEM is first examined in the case of linear elastic analysis. An elliptical cross-section beam with minor to major aspect ratio ofb/a = 0.1, material properties given byE = 1.0 andν = .30was analyzed. The thickness of the slice was chosen asb/2. Eight-noded linear brick elements were used forthe structural discretisation. Only one element division is needed along the thickness (beam axial direction).The cross-sectional meshes are shown infigure 3.

The analysis results show that the relationshipk12 = k21 is very precisely satisfied. The variation of extension-torsion coupling, 2a�/ε (= −2ak21/k22), as a function of pretwist level(2a�0) for centered beams(x0 = y0 =0) is presented infigure 4. The variation in the extension-torsion coupling of helical beams(2a�0 = 0.1) as afunction of the section offset from the initial twist axis is shown infigure 5. The present results are in goodagreement with those of Kosmatka (1992). Movement of the initial twist axis along the minor axis results in agradual decrease in coupling that approaches zero fory0 � b. As the initial twist axis is offset from the section

Figure 3. Cross-sectional finite element mesh.

Figure 4. The variation of extension-torsion coupling as a function of pretwist level(2a�0) for centered beams(x0 = y0 = 0).


Figure 5. The variation in the extension-torsion coupling of helical beams(2a�0 = 0.1) as a function of the section offset from the initial twist axis.

Figure 6. The variation of critical location for zero coupling(x0/a), with aspect ratios(b/a) for different Poisson’s ratio values.

centroid along the major axis, the coupling changes sign from negative to positive, thus the beam undergoesfurther twisting instead of untwisting for an applied extension load.Figure 6 shows the variation of criticallocation for zero coupling(x0/a), with aspect ratios(b/a) for different Poisson’s ratio values. To achieve avery precise result a finer mesh was used. It can be seen from this figure that, for the case of zero Poisson’s


Figure 7. The elastic twist(2a�) as a function of the nondimensionalized torsional moment 3M/(8Gb3), with constant forceF = 28.6 N for a thinrectangular steel strip with dimensions 10mm× 0.5mm.

ratio, the present prediction is in excellent agreement with theories of Shield (1982) and Krenk (1983-b) whichgive the equationx0/a = 1

2

√1− (b/a)2.

3.2. Geometrically nonlinear analysis

Rosen (1983) has presented experimental results for the elastic deformation of an initially twisted thinrectangular steel strip with dimensions of 10mm× 0.5 mm under the action of an external tension force andtorsional moment. The material properties wereE = 1.92 × 105 N/mm2 and ν = 0.2886. The initial twistwas 2a�0 = 0.224. A minimal constant tension force of 28.6 N was applied throughout. This geometry wasanalysed using the present method with a uniform mesh of 80×4×1 eight-noded brick elements. A twist angleper unit length� = 0.017 rad/mm was applied in 17 uniform increments in the analysis. In each loading step,a search for the axial strainε which produced an axial force of 28.6 N was performed by a differential method.The elastic twist(2a�) as a function of the nondimensionalized torsional moment 3M/(8Gb3), relative tothe caseM = 0, F = 28.6 N, is plotted infigure 7. The agreement between the present study, nonlinear theory(Rosen, 1983), and experimental results (Rosen, 1983) is good, but the linear theory begins to show a significantdiscrepancy for an elastic twist(2a�) as low as 0.05.

4. Conclusions

The analysis of initially twisted beams under axial and torsional load has been realized by finite elementmodeling of just a slice of the cross-section of the beam. Conventional three-dimensional solid elementswere used and exact boundary conditions were applied. Both linear and nonlinear numerical examples wereimplemented. Good correlation has been found with both previous analyses and experimental results.


References

Jiang, W.G., 1999. The development of the helically symmetric boundary condition in finite element analysis and its applications to spiral strands.PhD thesis. Brunel University, UK.

Jiang, W.G., Yao, M.S., Walton, J.M., 1999. A concise finite element model for simple wire rope strand. Int. J. Mech. Sci. 41 (2), 143–161.Jiang, W.G., Henshall, J.L., Walton, J.M., 2000. A concise finite element model for 3-layered straight wire rope strand. Int. J. Mech. Sci. 42 (1),

63–86.Knowles, J.K., Reissner, E., 1960. Torsion and extension of helicoidal shell. Quart. Appl. Math. 27 (4), 409–422.Kosmatka, J.B., 1992. On the behavior of pretwisted beams with irregular cross-sections. J. Appl. Mech. 59, 146–152.Krenk, S., 1983-a. The torsion-extension coupling in pretwisted elastic beams. Int. J. Solids Structures 19 (1), 67–72.Krenk, S., 1983-b. A linear theory for pretwisted elastic beams. J. Appl. Mech. 50, 137–142.Okubo, H., 1951. The torsion and stretching of spiral rods I. Quart. Appl. Math. 9, 263–272.Okubo, H., 1954. The torsion and stretching of spiral rods II. Quart. Appl. Math. 9, 488–495.Reissner, E., Wan, F.Y.M., 1968. On axial extension and torsion of helicoidal shell. Journal of Mathematics and Physics 47, 1–31.Rosen, A., 1983. Theoretical and experimental investigation of the nonlinear torsion and extension of initially twisted bars. J. Appl. Mech. 50,

321–326.Shield, R.T., 1982. Extension and torsion of elastic bars with initial twist. J. Appl. Mech. 49, 779–786.

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Torsion-extension coupling in initially twisted beams by finite elements