Engineering Transactions,

**8**, 4, pp. 671-677, 1960### Skręcanie Prętów o Przekroju w Kształcie Wielokąta Foremnego Wzory Obliczeniowe

The accurate solution of the problem of torsion of a regular polygonal bar is expressed by means of infinite series, inconvenient for practical purposes. In this connection formulae for torsional rigidity and maximum shear stresses have been worked out such as to allow the direct use in any practical computation. These approximate formulae express the torsional rigidity in terms of the radius of the circumscribed circle [Eq. (1.1)] or the polar moment of inertia

[Eq-(1.4)]. The numerical coefficients appearing in the formulae are given in Tabs. 2 and 5, respectively. The relative errors due to the approximate character of these values are collated in Tabs. 3 and 6, respectively.

From the numerical values of the coefficients appearing in the Eq. (1.4) it follows that for n >0 the torsional rigidity may be expressed in the form of the product of the polar moment of inertia and the shear modulus, the error thus committed being less than 0,906%. For the maximum shear stress the Eq. (2.4) is proposed, where the approximate values of the numerical coefficient are given for various n in Table 11. It is seen that for n-> the numerical coefficient of the Eq. (2.4) tends to 1 and passes through a certain maximum which is reached for one lying between 10 and 15. Table 13 gives the upper bound of the error that is committed by using the Eq. (2.6) instead of (2.4). The relation between Tmax and the torque is expressed by Eq. (2.8).

The relevant approximate values of the coefficients are collated in Table 14, the approximation error being given for various n in Table 15. The computation of all the approximate values of the coefficients and the appraisal of the errors are based on the relations derived in Ref. [1].

[Eq-(1.4)]. The numerical coefficients appearing in the formulae are given in Tabs. 2 and 5, respectively. The relative errors due to the approximate character of these values are collated in Tabs. 3 and 6, respectively.

From the numerical values of the coefficients appearing in the Eq. (1.4) it follows that for n >0 the torsional rigidity may be expressed in the form of the product of the polar moment of inertia and the shear modulus, the error thus committed being less than 0,906%. For the maximum shear stress the Eq. (2.4) is proposed, where the approximate values of the numerical coefficient are given for various n in Table 11. It is seen that for n-> the numerical coefficient of the Eq. (2.4) tends to 1 and passes through a certain maximum which is reached for one lying between 10 and 15. Table 13 gives the upper bound of the error that is committed by using the Eq. (2.6) instead of (2.4). The relation between Tmax and the torque is expressed by Eq. (2.8).

The relevant approximate values of the coefficients are collated in Table 14, the approximation error being given for various n in Table 15. The computation of all the approximate values of the coefficients and the appraisal of the errors are based on the relations derived in Ref. [1].

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

#### References

W. ZAPALOWICZ, Torsion of Prismatic Bars of Regular Polygonal Cross-section, Arch. Mech. stos., 5, 11 (1959).

M. T. HUBER, Teoria sprężystości, t. 1, PAU, Kraków 1948.

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