Engineering Transactions,

**8**, 4, pp. 805-843, 1960### Ogólna Teoria Stateczności Prętów Cienkościennych z Uwzględnieniem Odkształcalności Przekroju Poprzecznego Cześć II. Pręty o Złożonym Przekroju Poprzecznym

The object of the present paper is the theory of stability of thin-walled bars with composite cross-section (the subdivision into simple and composite being done in the Ref. [1]). All the assumptions of the elementary theory of stability are preserved except that of indeformability of the cross-section. It is assumed in addition that the compression is axial, the cross-section – polygonal, the state of material elastic, the wall thickness constant, the support

simple and the end cross-sections - free to warp and prevented from deformation by means of rigid diaphragms. The solution method is similar to that of the Ref. [1].

The cross-section is divided into elements whose features are those of simple cross-sections. Starting from (2.11) (Fig. 4), the equations of equilibrium of the deformed wall (2.13) are derived. In this equation ξ is the displacement of the wall y in its plane, ξ, n, ϕ - including the given wall, b, Δt(r) the displacement components of the element the shear force due to the loading of the element by the shear forces facting along the edges of contact with the neighbouring elements and p, the load acting on the wall in its plane. The displacements ξ, can always be expressed in terms of ξ, n, ϕ. The second group of equations are the equations of equilibrium at the edges (2.16), the third those of compatibility of axial deformations e of each element along the edges of contact (2.21). The moments Mne and the loads p, are determined in terms of ξ, n, ϕ and the rotation angles of the edges y due to the deformation of the cross- section are found from the Eqs. (2.17)-(2.19) derived in [1] (Figs. 5 and 6).

The boundary conditions (2.22) are always satisfied by assuming the sin nnx/l type of variability for the geometric unknowns ξ and cos nnx/l for the unknowns t. The Sec. 3 contains solution examples for symmetric and anti-symmetric buckling of a profile, Fig. 8. The states of the unit unknowns are illustrated in the symmetric case by Figs. 10 and 11 and in the antisymmetric case by Figs. 15 and 16. Thus a system of homogeneous equations is obtained (Tables 1 and 2); setting the determinant of this system equal to zero we obtain a transcendental equation from which the critical value of the coefficient k= σo/E can be calculated (0, representing the axial compressive stress and E-YOUNG's modulus). In Sec. 4 a simplified method is described in which the deformability of some walls (external walls in general) is disregarded, the walls being treated as bars (Fig. 18). By means of the equations of equilibrium (4.1) the Eqs (4.2) and (4.3) are derived for two important practical cases (Figs. 19 and 20).

In Sec. 5 the example of Sec. 3 is solved by means of the simplified methods (Tables 3 and 4). The numerical results for symmetric buckling are collected in Table 6; the reduction of the thickness 8, of the upper flange causes a rapid growth of the influence of deformability of the cross-section on the EULERIAN critical stress. Table 10 shows the results of calculations for cross-section with limited deformability (Fig. 23). It is seen that the flexural deformability of the walls 4 and 5 is of essential importance, its prevention (cases c and d) causing the profile to work as an indeformable one (case e). Sec. 6 discusses the possibilities of further simplifications consisting in assuming indeformable elements. We start from the equations of equilibrium of the elements (6.3), from which the loads (6.4) can be found, Fig. 25. As an ex- ample, the cases of torsional buckling of the profiles of Fig. 27 are solved, the channels being assumed to be indeformable, the web thus remaining the only deformable element. The reduction of the critical force is by 44% and 69%, respectively. The influence of deformability of the cross-section on the critical load is much greater in general in composite cross-sections than in simple ones.

simple and the end cross-sections - free to warp and prevented from deformation by means of rigid diaphragms. The solution method is similar to that of the Ref. [1].

The cross-section is divided into elements whose features are those of simple cross-sections. Starting from (2.11) (Fig. 4), the equations of equilibrium of the deformed wall (2.13) are derived. In this equation ξ is the displacement of the wall y in its plane, ξ, n, ϕ - including the given wall, b, Δt(r) the displacement components of the element the shear force due to the loading of the element by the shear forces facting along the edges of contact with the neighbouring elements and p, the load acting on the wall in its plane. The displacements ξ, can always be expressed in terms of ξ, n, ϕ. The second group of equations are the equations of equilibrium at the edges (2.16), the third those of compatibility of axial deformations e of each element along the edges of contact (2.21). The moments Mne and the loads p, are determined in terms of ξ, n, ϕ and the rotation angles of the edges y due to the deformation of the cross- section are found from the Eqs. (2.17)-(2.19) derived in [1] (Figs. 5 and 6).

The boundary conditions (2.22) are always satisfied by assuming the sin nnx/l type of variability for the geometric unknowns ξ and cos nnx/l for the unknowns t. The Sec. 3 contains solution examples for symmetric and anti-symmetric buckling of a profile, Fig. 8. The states of the unit unknowns are illustrated in the symmetric case by Figs. 10 and 11 and in the antisymmetric case by Figs. 15 and 16. Thus a system of homogeneous equations is obtained (Tables 1 and 2); setting the determinant of this system equal to zero we obtain a transcendental equation from which the critical value of the coefficient k= σo/E can be calculated (0, representing the axial compressive stress and E-YOUNG's modulus). In Sec. 4 a simplified method is described in which the deformability of some walls (external walls in general) is disregarded, the walls being treated as bars (Fig. 18). By means of the equations of equilibrium (4.1) the Eqs (4.2) and (4.3) are derived for two important practical cases (Figs. 19 and 20).

In Sec. 5 the example of Sec. 3 is solved by means of the simplified methods (Tables 3 and 4). The numerical results for symmetric buckling are collected in Table 6; the reduction of the thickness 8, of the upper flange causes a rapid growth of the influence of deformability of the cross-section on the EULERIAN critical stress. Table 10 shows the results of calculations for cross-section with limited deformability (Fig. 23). It is seen that the flexural deformability of the walls 4 and 5 is of essential importance, its prevention (cases c and d) causing the profile to work as an indeformable one (case e). Sec. 6 discusses the possibilities of further simplifications consisting in assuming indeformable elements. We start from the equations of equilibrium of the elements (6.3), from which the loads (6.4) can be found, Fig. 25. As an ex- ample, the cases of torsional buckling of the profiles of Fig. 27 are solved, the channels being assumed to be indeformable, the web thus remaining the only deformable element. The reduction of the critical force is by 44% and 69%, respectively. The influence of deformability of the cross-section on the critical load is much greater in general in composite cross-sections than in simple ones.

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Copyright © Polish Academy of Sciences & Institute of Fundamental Technological Research (IPPT PAN).

#### References

A. CHUDZIKIEWICZ, Ogólna teoria stateczności prętów cienkościennych z uwzględnieniem odkształcalności przekroju poprzedzanego. Część I. Pręty o prostym przekroju poprzecznym, Rozpr. inzyn. 3, 8 (1960).

[in Russian]

A. CHUDZIKIEWICZ, Wpływ odkształcalności przekroju poprzedzanego na sile krytyczną wyboczenia skrętnego pręta dwuteowego, Rozpr. inzyn. 2, 8 (1960).