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.
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.
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.
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
The vector describing the location of a point on the central surface of the shell is expressed using the formula
where
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:
where
Coordinate describes the arc length and so
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
where
For the local coordinate system tied to the central surface, the covariant surface metric tensor is expressed using the following formula:
where
The covariant metric tensor assumes the following form:
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:
From the above relations, it follows that versor
is defined as
and the inverse matrix has the form
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
where
The linear components of the deformation tensor are expressed using the formula
Hence,
The physical deformations are expressed using the formula
Using formula (9), one gets
According to the Vlasov assumption, strain
Integrating both sides of equation (15) over variable
where
Then performing the differentiation of function
Substituting
where
The dependences between strains and stresses are defined using the formula
where
Let us consider cross-sectional forces acting in cross section
where
In order to find component
where
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
where
In order to simplify the notation in formula (28), denotation
In order to derive displacement equations describing the vibration problem for a thin-walled beam, we shall use the Hamilton principle
where
where
the potential energy of the system can be presented in the form
where
The kinetic energy is expressed using the formula
where
where
Parameter
Using Hamilton’s principle (29), one gets the following system of displacement equations describing the thin-walled beam vibration problem:
and associated with the system of differential equations (38)–(41), equations determining the boundary conditions,
where
where
where
where
where
Solutions
where
or using the following recurrence formula:
Legendre polynomials
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.
and
and functions
where
and
Expansion coefficients
where
When calculating coefficients with negative subscripts, the following relations are also used if 2
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
where the elements of the matrices are defined using formulas
In the case of the fourth order (
where
The system, satisfied at k ≥ 4, should be completed with 14 equations describing boundary conditions. In order to formulate the conditions, the following relations defining the values of the Legendre’ polynomials and their derivatives in points
Exemplary equations describing boundary condition
In order to illustrate the proposed method as well as to verify it and show its effectiveness, a numerical example is provided. In the example, the eigenproblem was solved using Legendre polynomials for approximation. The sought functions were approximated with 15 series terms. The Wolfram
Nonprismatic monosymmetric I-beams with a linearly variable web height are analysed. Two static systems, that is, a clamped-clamped beam and a cantilever beam, are considered. The diagrams of the beams and the dimensions of their cross sections are shown in Fig. 4. The material parameters of the beams are
Formula (28) derived in this article was used to calculate the geometrical characteristics of the cross section. Various size of approximation base was used for checking the convergence of the presented method. The sought functions (47) were approximated with 10, 15 and 20 series terms. The obtained results are presented in Tables 1 and 2.
Eigenfrequencies in case of clamped-clamped support (C-C) [Hz].
Mode number | Paper dim = 10 | Paper dim = 15 | Paper dim = 20 | FEM SOFiSTiK beam element 7DOF d = 20 | FEM SOFiSTiK beam element 7DOF d = 25 | FEM SOFiSTiK beam element 7DOF d = 50 | FEM Abaqus shell elements (quad) |
---|---|---|---|---|---|---|---|
1 | 10.17 | 10.15 | 10.15 | 10.17 | 10.16 | 10.15 | 10.05 |
2 | 18.43 | 18.39 | 18.39 | 17.93 | 17.92 | 17.89 | 18.18 |
3 | 26.70 | 26.68 | 26.68 | 26.83 | 26.75 | 26.64 | 26.11 |
4 | 48.69 | 48.70 | 48.69 | 47.62 | 47.45 | 47.20 | 46.79 (lf) |
5 | 52.71 | 50.72 | 50.72 | 49.49 | 49.41 | 49.31 | 48.18 |
6 | 53.11 | 52.92 | 52.92 | 51.86 | 51.57 | 51.17 | 49.32 |
7 | 90.71 | 82.42 | 82.34 | 83.58 | 82.66 | 81.43 | 67.38 (lf) |
8 | 96.80 | 93.59 | 93.59 | 94.05 | 93.45 | 92.64 | 85.73 (lf) |
Eigenfrequencies in case of clamped-free support (C-F) [Hz].
Mode number | Paper dim = 10 | Paper dim = 15 | Paper dim = 20 | FEM SOFiSTiK beam element 7DOF d = 20 | FEM SOFiSTiK beam element 7DOF d = 25 | FEM SOFiSTiK beam element 7DOF d = 50 | FEM Abaqus shell elements (quad) |
---|---|---|---|---|---|---|---|
1 | 2.20 | 2.19 | 2.20 | 2.15 | 2.15 | 2.15 | 2.19 |
2 | 8.33 | 7.68 | 7.60 | 7.68 | 7.69 | 7.69 | 7.55 |
3 | 13.44 | 12.96 | 12.91 | 12.26 | 12.25 | 12.24 | 12.77 |
4 | 14.42 | 13.88 | 13.85 | 13.66 | 13.67 | 13.68 | 13.77 |
5 | 24.31 | 23.18 | 23.13 | 23.97 | 23.95 | 23.93 | 22.75 |
6 | 35.95 | 32.68 | 32.54 | 31.07 | 31.00 | 30.90 | 30.89 |
7 | 44.18 | 50.31 | 50.21 | 52.13 | 51.91 | 51.60 | 48.68 lf) |
8 | 61.28 | 59.55 | 59.30 | 57.68 | 57.46 | 57.17 | 52.96 (lf) |
In order to verify the model and the effectiveness of the presented method of solving the derived equations, the considered problem was solved using FEM. The Abaqus program, in which 0.02 m × 0.02 m rectangular shell elements were adopted for the calculations, and the SOFiSTiK program, in which the beams were solved using thin-walled beam elements with 7 degrees of freedom (DOFs), were used for the calculations. In the SOFiSTiK calculations, various numbers of elements were used. The beams were divided into 20, 25 and 50 elements. The obtained results for the above two cases and the results yielded by FEM are presented in Tables 1 and 2.
In the case of the solutions obtained using the shell elements (Abaqus), ‘additional’ local effects, consisting in local deformations of the medium, appear in the eigenforms corresponding to higher frequencies. In the tables, the forms are denoted as
To examine the impact of considering the so-called reduced thickness
sections,
Eigenfrequencies for different models in case of clamped-clamped support (C-C) [Hz].
Mode number | Paper | Paper | Paper |
---|---|---|---|
1 | 10.15 | 10.16 | 9.99 |
2 | 18.39 | 18.41 | 18.63 |
3 | 26.68 | 26.71 | 26.55 |
4 | 48.70 | 48.76 | 48.21 |
5 | 50.72 | 50.78 | 51.45 |
6 | 52.92 | 52.97 | 52.92 |
7 | 82.42 | 82.52 | 81.84 |
8 | 93.59 | 93.70 | 94.47 |
Eigenfrequencies for different models in case of clamped-free support (C-F) [Hz].
Mode number | Paper | Paper | Paper |
---|---|---|---|
1 | 2.19 | 2.20 | 2.21 |
2 | 7.68 | 7.68 | 7.60 |
3 | 12.96 | 12.97 | 13.36 |
4 | 13.88 | 13.89 | 13.88 |
5 | 23.18 | 23.21 | 22.44 |
6 | 32.68 | 32.71 | 33.91 |
7 | 50.31 | 50.37 | 48.72 |
8 | 59.55 | 59.61 | 61.33 |
The following conclusions can be drawn from the results presented in the examples:
using the proposed method, one can solve problems relating to the dynamics of nonprismatic thin-walled beams with any geometry parameters. Numerical examples show that the method is highly accurate and efficient. Only approximation with 15 series terms for each calculated function has to be used to obtain sufficient results (60 series terms for all functions). In case of FEM, the total number of finite elements required for analysis is circa 25. It equals 175 degrees of freedom. Moreover, the presented semi-analytical method is useful for further calculation which is difficult in case of FEM.
in the considered free vibration problem, the adoption of the simplified way of determining the geometrical characteristics of the cross section has no significant effect on the values of the eigenfrequencies of the considered systems;
the differences between the eigenfrequency values determined using, respectively, the Wilde model and the generalised model are not significantly large (for the first eight eigenfrequencies, they amount to 0–1.6% for the clamped-clamped beam and to 0–3.8% for the clamped-free beam). A more detailed analysis shows that the omission of the derivatives of functions