Analysis of thin-walled beams with variable monosymmetric cross section by means of Legendre polynomials

The problem of the free vibrations of nonprismatic thin-walled beam systems is interesting for two reasons. The first reason is the need to describe more precisely and solve this mathematically difficult problem that, except for special cases, has no closed analytical solutions. The second reason is practical and stems from the necessity to rationally shape and economically design contemporary civil engineering structures built from thin-walled beams with variable geometrical and material parameters.

The problems relating to thin-walled beam systems have been studied by many authors. As the literature on this subject is very extensive, only the research works that have been published in the recent years are presented here. Eisenberger (1997). Torsional vibrations of open and variable cross-section bars. Thin-Walled Structures. 28(3-4), 269-278. analysed the torsional free vibrations of a nonprismatic I-beam. Using the classical power series to approximate the sought functions, he determined the free vibration frequencies with and without the warping effect taken into account. On the basis of Vlasov’s theory of thin-walled systems, Ambrosini et al. (2000). A modified Vlasov theory for dynamic analysis of thin-walled and variable open section beams. Engineering Structures. 22(8), 890-900. DOI: 0.1016/S0141-0296(99)00043-7. presented equations describing the vibrations of a nonprismatic thin-walled beam with an open cross section, which take into account the effects of shear flexibility and rotatory inertia in the stress resultants. The same beam model was used by F. de Borbón and Ambrosini (2010). On free vibration analysis of thin-walled beams axially loaded. Thin-Walled Structures. 48(12), 915-920. DOI: 10.1016/j.tws.2010.06.002. to solve the free vibration problem with the influence of the static axial forces taken into account. The derived equations were solved using the method referred to as the state variables approach. The eigenproblem was also analysed by Arpaci and Bozdag (2002). On free vibration analysis of thin-walled beams with nonsymmetrical open cross-sections. Computers and Structures. 80(7-8), 691-695. The subject of the considerations was the vibrations of a prismatic beam with any cross section. The problem was solved analytically. The problem of the free vibrations of I-girders with intermediate bracings was analysed by Nguyen et al. (2011). Natural frequency for torsional vibration of simply supported steel I-girders with intermediate bracings. Thin-Walled Structures. 49(4), 534-542. DOI: 10.1016/j.tws.2010.12.001. Using trigonometric functions for approximation, they obtained a closed solution. Heyligerin (2015). Elasticity-based free vibration of anisotropic thin-walled beams. Thin-Walled Structures. 95:73-87. DOI: 10.1016/j.tws.2015.06.014. analysed the eigenproblem of prismatic thin-walled beams with, respectively, an open and a closed cross sections, taking into account anisotropy features. The derived equations were solved using the Ritz method, whilst polynomials and trigonometric functions were used to approximate the sought functions. Dynamic equations of the equilibrium of nonprismatic thin-walled beams with, respectively, an open and a closed cross sections were derived using Hamilton’s principle by Chang-New Chen (1998). Variational derivation of the dynamic equilibrium equations of nonprismatic thin-walled beams defined on an arbitrary coordinate system. Mechanics of Structures and Machines. 26(2),219-237. The equations were derived relative to an arbitrarily adopted axis, assuming the rotation of the cross sections relative to this axis. The obtained equilibrium equations and the boundary conditions are strongly conjugated. The problem of the warping of I-bars nonsymmetric in their cross section was solved by Mohri et al. in Mohri, F., Brouki, A. & Roth, J. C. (2003). Theoretical and numerical stability analyses of unrestrained, mono-symmetric thin-walled beams. Journal of Constructional Steel Research. 59(1), 63-90., where the effect of the load application point on the critical load value was investigated. The derived equations were solved using the Ritz method and the Galerkin method and trigonometric functions for approximation. The warping problem was also considered by Asgarian et al. (2013). Lateral-torsional buckling of tapered thin-walled beams with arbitrary cross-sections. Thin-Walled Structures. 62,96108. DOI: 10.1016/j. tws.2012.06.007. who used Hamilton’s principle to derive general equations of equilibrium describing a nonprismatic beam under the assumption that the beam cross section rotates relative to the centre of shear. The obtained differential equations were solved using the classical power series for approximation. Warping and free vibrations were also the subject of work [20] by Soltani et al. The same authors (2014b). Finite element method for stability and free vibration analyses of non-prismatic thin-walled beams. Thin-Walled Structures. 82, 245- 261. DOI: 10.1016/j.tws.2014.04.012. formulated a nonprismatic thin-walled finite element. In order to determine the exact shape functions, they used the method described previously in Asgarian, B., Soltani, M. & Mohri, F. (2013). Lateral-torsional buckling of tapered thin-walled beams with arbitrary cross-sections. Thin-Walled Structures. 62,96-108. and [20]. The formulated finite elements were used to analyse the eigenproblem and the stability problem for nonprismatic beams with any cross section. Similar to the previous works, the results were compared with the ones obtained using finite elements method (FEM). The works by Machado and Cortínez Mohri (2005). Non-linear model for stability of thin-walled composite beams with shear deformation. Thin-Walled Structures. 43(10),1615-1645. DOI: 10.1016/j.tws.2005.06.008. and Mohri et al. Mohri, F., Damil, N. & Ferry, M. P. (2008). Large torsion finite element model for thin-walled beams. Computers and Structures. 86(7-8), 671-683. DOI: 10.1016/j.compstruc.2007.07.007.are examples of the application of nonlinear analysis to thin-walled beam stability problems. In Machado, S. P. & Cortínez, V. H. (2005). Non-linear model for stability of thin-walled composite beams with shear deformation. Thin-Walled Structures. 43(10),1615-1645., the stability problem, including the behaviour after stability loss (buckling), was analysed for composite thin-walled beams with, respectively, an open and a closed cross sections. The nonlinear system of equations was solved by means of the incremental Newton-Raphson method. In Mohri, F., Damil, N. & Ferry, M. P. (2008). Large torsion finite element model for thin-walled beams. Computers and Structures. 86(7-8), 671-683., a nonlinear finite element with two joints and seven degrees of freedom was formulated and used to analyse the post-buckling behaviour of a thin-walled beam. The nonlinear equations stemming from this problem were solved using the iterative-incremental Newton-Raphson method. Extensive literature on the vibrations and stability of thin-walled beams can be found in, amongst others, the already cited work Asgarian, B., Soltani, M. & Mohri, F. (2013). Lateral-torsional buckling of tapered thin-walled beams with arbitrary cross-sections. Thin-Walled Structures. 62,96-108..

The subject of this article is the problem of the free vibrations of a nonprismatic monosymmetric beam with any parameters described using the membrane shell theory and the Vlasov theory assumptions. The beam model presented here is a generalisation of the model found in Wilde’s article [22]. The generalisation consisted in taken into account the curving of the axis formed by the shear centres of the beam cross sections. In most works on the vibrations of thin-walled beams known to the present authors, the equations describing this problem do not take into account the effects stemming from the curvilinear form of the beam axis and the axis defined by the shear centres. The Hamilton principle was used to derive equations describing the vibrations of the beam. In this article, the equations describing the vibrations of the beam were derived relative to an arbitrary rectilinear reference axis and the effects stemming from the curvilinear form of the axes were taken into account in the equations.

An approximation method using Legendre’s orthogonal polynomials (a special type of the Gegenbauer polynomials) for approximation was used to solve the obtained differential equations. This method, presented by Lewanowicz in, amongst others, work [9], so far has not been applied to this kind of problems. It is a generalisation of the method of approximating the solution of the Chebyshev series of the first kind [see Paszkowski Paszkowski, S. (1975). Numerical applications of Chebyshev polynomials. Warsaw: PWN; (in Polish).) and used by the present authors in earlier articles Ruta, P. & Szybiński, J. (2014). Nonlinear analysis of nonprismatic Timoshenko beam for different geometric nonlinearity models. International Journal of Mechanical Sciences. 101-102, 349-362. DOI: 10.1016/j. ijmecsci.2015.07.020. In the case of the free vibration problem, this method reduces the system of ordinary differential equations to an infinite system of algebraic equations.

A calculation example is provided to verify the derived equations and illustrate the capabilities of the presented method. In the example, the eigenproblem of a beam with a linearly variable web height is solved. Two static schemes were considered: (1) a cantilever beam and (2) a clamped-clamped beam. The effect of omitting certain parameters connected with the variability of the beam geometry was examined. The obtained eigenfrequencies were compared with the results yielded by FEM (Abaqus and SOFiSTiK).

The subject of the considerations is the problem of the spatial free vibrations of an arbitrarily supported, nonprismatic, thin-walled beam.

Figure 1

2.1

Description of model

The description of the model presented in this article is based on the membrane shell theory and the Vlasov assumptions used in the theory of thin-walled systems. The model is a generalisation of the model described by Wilde in [22]. The generalisation consists in taken into account the fact that the coordinates defining the position of the shear centres form a curvilinear axis and depend on the coordinate describing the position of the analyzed cross section. Also the beam axis is curvilinear. As a nomenclature different from that of Wilde’s work [22] is used in this article, the excerpts from the original work quoted in this section have been modified using the new nomenclature.

The displacement equations used here were derived relative to a rectilinear reference axis and to the system of coordinates x, y, z (Fig. 2). At this stage, no additional simplifying assumptions concerning the beam’s geometry (e.g. symmetry) were adopted.

Figure 2

The vector describing the location of a point on the central surface of the shell is expressed using the formula

(1)r=tex+y(t,s)ey+z(t,s)ez$$r=t{{e}_{x}}+y\left( t,s \right){{e}_{y}}+z\left( t,\,\,s \right){{e}_{z}}$$

where e _x, e _y, e _z, are the versors of the cartesian coordinate system basis.

The vectors of the covariant basis of the system of coordinates (t, s, n), determined on the central surface of the shell are defined as follows:

(2)at=ex+y,tey+z,tez,as=y,sey+z,sez,$${{a}_{t}}={{e}_{x}}+{{y}_{,t}}{{e}_{y}}+{{z}_{,t}}{{e}_{z}},{{a}_{s}}={{y}_{,s}}{{e}_{y}}+{{z}_{,s}}{{e}_{z}},$$

where f,s=∂f(t,s)/∂s,f,t=∂f(t,s)/∂t.$${{f}_{,s}}=\partial f\left( t,\,s \right)\text{/}\partial s\,\,,{{f}_{,t}}=\partial f\left( t,\,s \right)\text{/}\partial t\,.$$

Coordinate describes the arc length and so (y,s)2+(z,s)2=1..$${{\left( {{y}_{,s}} \right)}^{2}}+{{\left( {{z}_{,s}} \right)}^{2}}=1..$$

Therefore, one can introduce the following denotations y_,s=cosα, z_,s=sinα, where α is the angle included between the positive part of axis y and the tangent to the cross section when t=const. The third unit vector of the local basis defines the following relation:

(3)an=−at×as|at×as|=(1+β2)−12(βex+sinαey−cosαez)$${{a}_{n}}=\frac{-{{a}_{t}}\times {{a}_{s}}}{\left| {{a}_{t}}\times {{a}_{s}} \right|}={{\left( 1+{{\beta }^{2}} \right)}^{-\,\frac{1}{2}}}\left( \beta {{e}_{x}}+\sin \alpha \,{{e}_{y}}-\cos \alpha \,{{e}_{z}} \right)$$

where β=−y,tsinα+z,tcosα$$\beta =-{{y}_{,t}}\sin \,\alpha +{{z}_{,t}}\cos \alpha $$. Thus the matrix for the transformation of the Cartesian coordinate system basis to the covariant local basis of the system defined on the central surface is given by the formula

(4)Cl..k=[1y,tz,t0cosαsinαβ1+β2sinα1+β2−cosα1+β2]$$C_{l.}^{.k}=\left[ \begin{array}{*{35}{l}}1 & {{y}_{,t}} & {{z}_{,t}} \\0 & \cos \alpha & \sin \alpha \\\frac{\beta }{\sqrt{1+{{\beta }^{2}}}} & \frac{\sin \alpha }{\sqrt{1+{{\beta }^{2}}}} & \frac{-\cos \alpha }{\sqrt{1+{{\beta }^{2}}}} \\\end{array} \right]$$

For the local coordinate system tied to the central surface, the covariant surface metric tensor is expressed using the following formula:

(5)aαβ=[1+β2+γ2γγ1],|aαβ|=1+β2,$$\begin{align}& {{a}_{\alpha \beta }}=\left[ \begin{matrix}1+{{\beta }^{2}}+{{\gamma }^{2}} & \gamma \\\gamma & 1 \\\end{matrix} \right]\,\,, \\ & \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left| {{a}_{\alpha \beta }} \right|=1+{{\beta }^{2}}\,, \\ \end{align}$$

where γ=y,tcosα+z,tsinα.$$\gamma ={{y}_{,t}}\cos \alpha +{{z}_{,t}}\sin \alpha .$$

The covariant metric tensor assumes the following form:

(6)aαβ=[11+β2−γ1+β2−γ1+β21+β2+γ21+β2]$${{a}^{\alpha \beta }}=\left[ \begin{matrix}\frac{1}{1+{{\beta }^{2}}} & -\frac{\gamma }{1+{{\beta }^{2}}} \\-\frac{\gamma }{1+{{\beta }^{2}}} & \frac{1+{{\beta }^{2}}+{{\gamma }^{2}}}{1+{{\beta }^{2}}} \\\end{matrix} \right]$$

In general, the defined surface local coordinate system is not orthogonal. On the basis of the system defined earlier, one can define an orthogonal coordinate system as follows:

(7)aS=as,aN=anas⊥aT⊥an,|aT|=1$$\begin{array}{*{35}{l}}{{a}_{S}}={{a}_{s}}, {{a}_{N}}={{a}_{n}} \\{{a}_{s}}\bot {{a}_{T}}\bot {{a}_{n}},\,\,\left| {{a}_{T}} \right|=1 \\\end{array}$$

From the above relations, it follows that versor a_T lies in the plane tangent to the surface. Therefore, the vector

Figure 3

is defined as a_T = α₁a_t + α₂a_s. The values of coefficients α₁ and α₂ were calculated from the relation a_T ┴ a_S, |a_T| = 1. Ultimately, the matrix for the transformation of local coordinate system a_s, a_t to a system of orthogonal vectors a_S, a_T lying on the surface tangent to the central surface of the shell is defined using the formula

(8)CΩ..α=[11+β2−γ1+β201]$$C_{\Omega .}^{.\alpha }=\left[ \begin{matrix}\frac{1}{\sqrt{1+{{\beta }^{2}}}} & -\frac{\gamma }{\sqrt{1+{{\beta }^{2}}}} \\0 & 1 \\\end{matrix} \right]$$

and the inverse matrix has the form

(9)C¯α..Ω=[1+β2γ01].$$\bar{C}_{\alpha .}^{.\Omega }=\left[ \begin{matrix}\sqrt{1+{{\beta }^{2}}} & \gamma \\0 & 1 \\\end{matrix} \right]\,\,.$$

Let us assume that the centre lines of the beam cross sections undergo deformation only in the direction consistent with the direction of the longitudinal axis x (the so-called rigid cross section hypothesis). This leads to the following equation describing the displacement of an arbitrary point of the thin-walled beam:

(10)u=Uxex+Uzez+Uzez==ux(t,s)ex+[uy(t)−(z(t,s)−Cz(t))θ(t)]ey+[uz(t)+(y(t,s)−Cy(t))θ(t)]ez$$\begin{align}& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,u={{U}_{x}}{{e}_{x}}+{{U}_{z}}{{e}_{z}}+{{U}_{z}}{{e}_{z}}= \\ & ={{u}_{x}}\left( t,\,s \right){{e}_{x}}+\left[ {{u}_{y}}\left( t \right)-\left( z\left( t,\,s \right)-{{C}_{z}}\left( t \right) \right)\theta \left( t \right) \right]{{e}_{y}} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,+\left[ {{u}_{z}}\left( t \right)+\left( y\left( t,\,s \right)-{{C}_{y}}\left( t \right) \right)\theta \left( t \right) \right]{{e}_{z}} \\ \end{align}$$

where θ(t) is the angle of rotation relative to the axis marked out by the points whose coordinates (C_y(t), C_z(t)) define the centre of shear, u_y(t), u_z(t) are the components of the displacements of the cross section in the shear centre measured along the axis of the global coordinate system and u_x(t,s) is the measure of the displacements of the cross section along axis x, taking into account wrappings relative to plane x = const.

The linear components of the deformation tensor are expressed using the formula

(11)γαβ=12(aα⋅u,β+aβ⋅u,α).$${{\gamma }_{\alpha \beta }}=\frac{1}{2}\left( {{a}_{\alpha }}\cdot {{u}_{,\beta }}+{{a}_{\beta }}\cdot {{u}_{,\alpha }} \right)\,.$$

Hence,

(12)γtt=ux,t+y,t(uy,t−(z−Cz)θ,t+Cz,tθ)+z,t(uz,t+(y−Cy)θ,t−Cy,tθ)γts=γst=12(ux,s+y,s(uy,t−(z−Cz)θ,t+Cz,tθ)+z,s(uz,t+(y−Cy)θ,t−Cy,tθ))γss=0$$\begin{align}& \begin{matrix}{{\gamma }_{tt}}={{u}_{x,t}}+{{y}_{,t}}\left( {{u}_{y,t}}-\left( z-{{C}_{z}} \right){{\theta }_{,t}}+{{C}_{z,t}}\theta \right)+{{z}_{,t}}\left( {{u}_{z,t}}+\left( y-{{C}_{y}} \right){{\theta }_{,t}}-{{C}_{y,t}}\theta \right) \\{{\gamma }_{ts}}={{\gamma }_{st}}=\frac{1}{2}\left( {{u}_{x,s}}+{{y}_{,s}}\left( {{u}_{y,t}}-\left( z-{{C}_{z}} \right){{\theta }_{,t}}+{{C}_{z,t}}\theta \right)+{{z}_{,s}}\left( {{u}_{z,t}}+\left( y-{{C}_{y}} \right){{\theta }_{,t}}-{{C}_{y,t}}\theta \right) \right) \\\end{matrix} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{\gamma }_{ss}}=0 \\ \end{align}$$

The physical deformations are expressed using the formula

(13)γΩΦ=CΩ..αCΦ..βγαβ$${{\gamma }_{\Omega \Phi }}=C_{\Omega .}^{.\alpha }C_{\Phi .}^{.\beta }\,{{\gamma }_{\alpha \beta }}$$

Using formula (9), one gets

(14)γTT=(1+β2)−1γtt−2γ(1+β2)−1γtsγTS=γST=(1+β2)−12γts,γSS=0.$$\begin{matrix}{{\gamma }_{TT}}={{\left( 1+{{\beta }^{2}} \right)}^{-1}}{{\gamma }_{tt}}-2\gamma {{\left( 1+{{\beta }^{2}} \right)}^{-1}}{{\gamma }_{ts}} \\{{\gamma }_{TS}}={{\gamma }_{ST}}={{\left( 1+{{\beta }^{2}} \right)}^{-\,\frac{1}{2}}}{{\gamma }_{ts}},{{\gamma }_{SS}}=0\,. \\\end{matrix}$$

According to the Vlasov assumption, strain γ_TS=0. Thus it follows from formula (14) that also strain γ_ts=0. This assumption ultimately leads to the following relation:

(15)ux,s=−(y,s(uy,t−(z−Cz)θ,t+Cz,tθ)+z,s(uz,t+(y−Cy)θ,t−Cy,tθ))$${{u}_{x,s}}=-\left( {{y}_{,s}}\left( {{u}_{y,t}}-\left( z-{{C}_{z}} \right){{\theta }_{,t}}+{{C}_{z,t}}\theta \right)+{{z}_{,s}}\left( {{u}_{z,t}}+\left( y-{{C}_{y}} \right){{\theta }_{,t}}-{{C}_{y,t}}\theta \right) \right)$$

Integrating both sides of equation (15) over variable s and taking into account the fact that on the right side of this equality, only functions z and y depend on s, one gets

(16)ux=u0x−yuy,t−zuy,t−zuz,t−(yCz,t−zCy,t)θ−ωθ,t$$\begin{align}& {{u}_{x}}={{u}_{0x}}-y{{u}_{y,t}}-z{{u}_{y,t}}-z{{u}_{z,t}}-\left( y{{C}_{z,t}}-z{{C}_{y,t}} \right)\theta -\omega {{\theta }_{,t}} \\ & \\ \end{align}$$

where u_0x=u_0x(t) is an integration constant independent of variable s and ω = ω(t,s) is a sectorial coordinate described by the following relation:

(17)ω(t,s)=∫[−(z−Cz)y,s+(y−Cy)z,s]ds$$\omega \left( t,\,\,s \right)=\int{\left[ -\left( z-{{C}_{z}} \right){{y}_{,s}}+\left( y-{{C}_{y}} \right){{z}_{,s}} \right]ds}$$

Then performing the differentiation of function u_x (16) over variable t, one gets

(18)ux,y=u0x,t−y,tuy,t−yuy,tt−z,tuz,t−zuz,tt−(y,tCz,t+yCz,tt−z,tCy,t−zCy,tt)θ−(yCz,t−zCy,t)θ,t−ω,tθ,t−ωθ,tt$$\begin{align}& {{u}_{x,y}}={{u}_{0x,t}}-{{y}_{,t}}{{u}_{y,t}}-y{{u}_{y,tt}}-{{z}_{,t}}{{u}_{z,t}}-z{{u}_{z,tt}}-\left( {{y}_{,t}}{{C}_{z,t}}+y{{C}_{z,tt}}-{{z}_{,t}}{{C}_{y,t}}-z{{C}_{y,tt}} \right)\theta - \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( y{{C}_{z,t}}-z{{C}_{y,t}} \right){{\theta }_{,t}}-{{\omega }_{,t}}{{\theta }_{,t}}-\omega {{\theta }_{,tt}} \\ \end{align}$$

Substituting u_x,t into formula (12)₁ defining γ_tt, and taking into account the fact that t = x, ultimately, one gets

(19)γxx=u0x,x−yuy,xx−zuz,xx−ωθ,xx−ψθ,x−δθ,$${{\gamma }_{xx}}={{u}_{0x,x}}-y{{u}_{y,xx}}-z{{u}_{z,xx}}-\omega {{\theta }_{,xx}}-\psi {{\theta }_{,x}}-\delta \theta \,\,,$$

where

(20)ψ=ω,x+y,x(z−Cz)−z,x(y−Cy)+(yCz,x−zCy,x),δ=yCz,xx−zCy,xx.$$\begin{align}& \psi ={{\omega }_{,x}}+{{y}_{,x}}\left( z-{{C}_{z}} \right)-{{z}_{,x}}\left( y-{{C}_{y}} \right)+\left( y{{C}_{z,x}}-z{{C}_{y,x}} \right)\,, \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\delta =y{{C}_{z,xx}}-z{{C}_{y,xx}}. \\ \end{align}$$

The dependences between strains and stresses are defined using the formula

(21)nαβ=DHαβϱλγϱλ$${{n}^{\alpha \beta }}=D{{H}^{\alpha \beta \varrho \lambda }}{{\gamma }_{\varrho \lambda }}$$

where Hαβϱλ=12[aαβaβϱ+aαϱaβλ+u(εαϱεβλ+εαλεβϱ)],$${{H}^{\alpha \beta \varrho \lambda }}=\frac{1}{2}\left[ {{a}^{\alpha \beta }}{{a}^{\beta \varrho }}+{{a}^{\alpha \varrho }}{{a}^{\beta \lambda }}+u\left( {{\varepsilon }^{\alpha \varrho }}{{\varepsilon }^{\beta \lambda }}+{{\varepsilon }^{\alpha \lambda }}{{\varepsilon }^{\beta \varrho }} \right) \right]\,,$$, D=E′g,E′=E1−v2,$$D={E}'g,{E}'=\frac{E}{1-{{v}^{2}}}\,\,,$$ν is the Poisson ratio, E is Young’s modulus, g is the wall thickness and ε_αβ is the Ricci tensor. In the considered case, relation (21), after taking into account the fact that γ_xs = γ_ss = 0 is, comes down to

(22)nxx=E′g(1+β2)−2γxx.$${{n}^{xx}}={E}'g{{\left( 1+{{\beta }^{2}} \right)}^{-2}}{{\gamma }_{xx}}\,.$$

Let us consider cross-sectional forces acting in cross section x = const. The force vector defining the membrane forces was determined from the relation

(23)t=nαβnαaβ$$t={{n}^{\alpha \beta }}{{n}_{\alpha }}{{a}_{\beta }}$$

where n_a is a covariant element of the unit vector normal to the plane and perpendicular to the line tangent to the cross section. Thus n is equal to a_T, and so the following equalities hold: nx=(1+β2)12,ns=0$${{n}_{x}}={{\left( 1+{{\beta }^{2}} \right)}^{\frac{1}{2}}},{{n}_{s}}=0$$. Hence,

(24)t=nxx(1+β2)12at+nxs(1+β2)12as$$t={{n}^{xx}}{{\left( 1+{{\beta }^{2}} \right)}^{\frac{1}{2}}}{{a}_{t}}+{{n}^{xs}}{{\left( 1+{{\beta }^{2}} \right)}^{\frac{1}{2}}}{{a}_{s}}$$

In order to find component t acting along axis x, one projects vector t onto axis x by scalar multiplying it by vector e_x. Ultimately, one gets

(25)tx=t⋅ex=E′g*[u0x,x−yuy,xx−zuz,xx−ωθ,xx−ψθ,x−δθ]$${{t}_{x}}=t\cdot {{e}_{x}}={E}'g*\left[ {{u}_{0x,\text{x}}}-y{{u}_{y,\text{xx}}}-z{{u}_{z,\text{xx}}}-\omega {{\theta }_{,\text{xx}}}-\psi {{\theta }_{,x}}-\delta \theta \right]$$

where g*=(1+β2)−32g$$g*={{\left( 1+{{\beta }^{2}} \right)}^{-\,\frac{3}{2}}}g$$expresses the so-called reduced thickness. The internal forces are defined using the formulas

(26)N=∫stxds,My=∫sztxds,Mz=∫sytxds,B=∫sωtxds$$N=\int_{s}{{{t}_{x}}\,ds,}\,\,\,\,{{M}_{y}}=\int_{s}{z\,{{t}_{x}}\,ds,}\,\,\,{{M}_{z}}=\int_{s}{y\,\,{{t}_{x}}\,ds,}\,\,\,\,B=\int_{s}{\omega {{t}_{x}}\,ds}$$

where integration is performed over the contour of the cross section. Substituting relation (25) into formula (26) and performing the integrations, one gets the following formulas for the cross-sectional forces (under the assumption that the considered beam is monosymmetric and that axis z is the symmetry axis of the cross sections):

(27)N=E′[A*u0x,x−Sy*uz,xx]My=E′[Sy*u0x,x−Iy*uz,xx]Mz=E′[−Iz*uy,xx−Iψz*θ,x−Iδz*θ]B=E′[−Iω*θ,xx−Iψω*θ,x−Iδω*θ]$$\begin{matrix}N={E}'\left[ A*{{u}_{0x,x}}-S_{y}^{*}{{u}_{z,xx}} \right] \\{{M}_{y}}={E}'\left[ S_{y}^{*}{{u}_{0x,x}}-I_{y}^{*}{{u}_{z,xx}} \right] \\{{M}_{z}}={E}'\left[ -I_{z}^{*}{{u}_{y,xx}}-I_{\psi z}^{*}{{\theta }_{,x}}-I_{\delta z}^{*}\theta \right] \\B={E}'\left[ -I_{\omega }^{*}{{\theta }_{,xx}}-I_{\psi \omega }^{*}{{\theta }_{,x}}-I_{\delta \omega }^{*}\theta \right] \\\end{matrix}$$

where

(28)A*=∫SdA*,Sy*=∫SzdA*,Iy*=∫Sz2dA*,Iz*=∫Sy2dA*,Iω*=∫Sω2dA*,Iψz*=∫SyψdA*,Iψω*=∫SωψdA*,Iδz*=∫SyψdA*,Iδω*=∫SωδdA*.$$\begin{matrix}A*=\int_{S}{dA*,\,S_{y}^{*}=}\,\int_{S}{zdA*}, \\I_{y}^{*}=\int_{S}{{{z}^{2}}dA*},\,I_{z}^{*}=\int_{S}{{{y}^{2}}dA*}, \\I_{\omega }^{*}=\int_{S}{{{\omega }^{2}}dA*}, \\I_{\psi z}^{*}=\int_{S}{y\psi dA*,\,\,I_{\psi \omega }^{*}=\int_{S}{\omega \psi dA*},} \\I_{\delta z}^{*}=\int_{S}{y\psi dA*,\,\,I_{\delta \omega }^{*}=\int_{S}{\omega \delta dA*}.} \\\end{matrix}$$

In order to simplify the notation in formula (28), denotation dA*=g*ds$$dA*=g*ds$$, in which g^* is reduced thickness g*=(1+β2)−32g$$g*={{\left( 1+{{\beta }^{2}} \right)}^{-\,\frac{3}{2}}}g$$, was used.

Solutions u0x(x),uy(x),uz(x),u(x)$${{u}_{0x}}\left( x \right),{{u}_{y}}\left( x \right),{{u}_{z}}\left( x \right),u\left( x \right)$$of the system of differential equations (38)–(41) will be sought as expansions of the functions into series relative to the Legendre polynomials [8]

where pk(x)$${{p}_{k}}\left( x \right)$$are the Legendre polynomials defined using the formula

or using the following recurrence formula:

Legendre polynomials Pk(x)$${{P}_{k}}\left( x \right)$$are a special kind of Gegenbauer polynomials Ck(λ)(x)$$C_{k}^{\left( \lambda \right)}\left( x \right)$$and dependence Pk(x)=Ck(12)(x)$${{P}_{k}}\left( x \right)=C_{k}^{\left( \frac{1}{2} \right)}\left( x \right)$$holds between them.

Let us use the following theorem, derived and presented by Lewanowicz in, amongst others, [9], to determine the coefficients of series (47). This theorem describes the method of determining the solutions of differential equations (in the form of the Gegenbauer series) with variable coefficients.

Theorem: If the function f satisfies a linear differential equation of order n ˃ 0.

and

and functions (Q0(x)f)(n),(Q0(x)f)(n−1),...,Q0(x)f,P$${{\left( {{\mathbf{Q}}_{0}}\left( x \right)f \right)}^{\left( n \right)}},{{\left( {{\mathbf{Q}}_{0}}\left( x \right)f \right)}^{\left( n-1 \right)}},...,{{\mathbf{Q}}_{0}}\left( x \right)f,P$$have specific Gegenbauer series coefficients, then for each integral k, the following identity holds true:

where ϱ_ijm (k) are functions of integral variable k.

and a_k [h] is the kth coefficient of the expansion of the function h(x) into a series relative to Gegenbauer polynomials Ckλ<!-- λ -->xλ<!-- λ -->≠<!-- ≠ -->0$C_{k}^{\left( \lambda \right)}\left( x \right)\ \ \left( \lambda \ne 0 \right)$(a proof of this theorem can be found in [9]). As already mentioned, in a special case, when λ = 1 ⁄ 2, the Gegenbauer polynomials become Legendre polynomials Pk(x)=Ck(12)(x).$${{P}_{k}}\left( x \right)=C_{k}^{\left( \frac{1}{2} \right)}\left( x \right).$$.

Expansion coefficients c_k[Q_i(x) f(x)] of the function product Q_i(x) f(x) are calculated from the formula given below ([9]), assuming that matrices Q_i(x) contain exclusively polynomials. If this assumption is not satisfied, one should beforehand apply an appropriate polynomial approximation to matrices Q_i(x). For λ = 1 ⁄ 2,

where

When calculating coefficients with negative subscripts, the following relations are also used if 2λ = m is a non-negative integer.

The problem of free vibrations described by equations (38)–(41) can be described by the following matrix differential equation of the fourth order:

In the considered problem, matrices P_i occurring in formula (59) assume the following form:

where the elements of the matrices are defined using formulas