CALCULATION OF NATURAL BENDING VIBRATION FREQUENCIES OF BEAMS BEYOND THE ELASTIC LIMIT IN THE PRESENCE OF PLASTIC ZONES

N. K. Vitvitskii

(Presented by the scientific seminar of the Department of Higher Mathematics)

In the author’s work [9], where theoretical justifications of the method are given, it was indicated that in modern structures and facilities the occurrence of zones of plasticity is possible, caused by the appearance of supercritical loads.

Corresponding Member of the Academy of Sciences I. M. Rabinovich [4] pointed out that extending dynamic analysis beyond the elastic limit is necessary even when the stresses arising during the vibration of a structure have moderate values and lie below the elastic limit. An especially great need for such analyses arises in the design of protective structures intended to withstand the impact of a shell, an aerial bomb, or the air wave of an explosion. In such cases it is permissible to tolerate working stresses exceeding the elastic limit.

In determining the frequencies it is assumed that, when plastic zones are present in the beam, the plastically deformed material exerts its influence mainly on the change in the amplitude of vibration and only very weakly on the change in frequency. This also agrees with the circumstance that the dissipation of energy during vibrations, for a certain range of frequencies, may be regarded as independent of the vibration frequency and dependent on the amplitude of deformation and on temperature.

In order to popularize the method of calculating frequencies beyond the elastic limit among engineering and technical workers who, by the nature of their work, have to deal with such or analogous calculations, the author gives a scheme for the sequence of computations.

Let us consider the case in which an explosive wave or other similar factors cause (imitate) actions analogous to a uniformly distributed load of intensity \(p\) on a beam of length \(2l\), lying on two supports with hinged-fixed or freely supported ends. Since natural vibration frequencies are being considered, what is meant here is the indicated load as an impulse causing vibrations of the beam. Thus, we observe and calculate the frequencies directly—immediately after imitation of the explosion. From comparison of the bending moments of the external loads \(p\) and of the internal forces

elasticity of the beam material for the cross section of its elastoplastic zone [3], [5], we find the equation in the following form:

\[ \frac{2}{3}\,b\sigma_0(3h^2-z^2)=\frac{p}{2}(l^2-y^2), \tag{1} \]

where \(h\) and \(b\) are the dimensions of the beam profile (Fig. 1), and
\(\sigma_0\) is the tensile yield limit (the plastic constant).

Fig. 1. Axial section of the beam showing the applied uniformly distributed load.

Plastic deformation first arises in the beam in its outermost fibers \(z=\pm h\) of the critical section \(y=0\) at the critical load \(p_0\), which is obtained from formula (1) and is equal to

\[ p_0=\frac{8}{3}\,\frac{bh^2}{l^2}\,\sigma_0 . \tag{2} \]

The equation of the boundaries between the elastic and plastic zones in the section \(y0z\), that is, formula (1), is transformed to the canonical form

\[ \frac{z^2}{3h^2(1-\rho)}-\frac{u^2}{\dfrac{1-\rho}{\rho}}=1, \tag{3} \]

where

\[ u=\frac{y}{l}\,(-1\leq u\leq +1),\qquad \rho=\frac{p}{4b\sigma_0}\left(\frac{l}{h}\right)^2 . \]

Equation (3) represents a family of hyperbolas with semiaxes \(h\sqrt{3(1-\rho)}\) and \(\sqrt{\dfrac{1-\rho}{\rho}}\), where \(\rho\) is a parameter.

In Fig. 2 the plastic zones are hatched. Since

\[ \frac{p}{p_0}=\frac{3}{2}\left[\frac{p}{4b\sigma_0}\left(\frac{l}{h}\right)^2\right], \]

Fig. 2. Axial section of the beam showing the plastic zones.

and the expression in square brackets is \(\rho\), then

\[ \varkappa=\frac{3}{2}\rho, \tag{4} \]

where

\[ \varkappa=\frac{p}{p_0}. \]

For the elastic-plastic state of the cross section of the beam, the load intensity \(p\) must vary within a certain interval from \(p=p_{\min}\) to \(p=p_{\max}\), i.e. \(p_{\min}\leq p\leq p_{\max}\). It is not difficult to determine these limiting values \(p_{\min}\) and \(p_{\max}\). Indeed, for the critical section \(u=0\), at \(z=\pm h\) (onset of yielding), it follows from the zone equation (3) that \(\rho=\dfrac{2}{3}\). Then from (4) \(x=1\), consequently \(p_{\min}=p_0\) (Fig. 3).

Fig. 3. Axial section of the beam indicating the boundary between the elastic and elastic-plastic zones.

To determine \(p_{\max}\) in the critical section \((u=0)\), one must set \(z=0\), which will correspond to a plastic hinge; and from the zone equation (3) we find \(\rho=1\). Then from (4), \(x=\dfrac{3}{2}\); consequently, \(p_{\max}=\dfrac{3}{2}p_0\). Thus, when \(\dfrac{2}{3}<\rho<1\), then \(1<x<\dfrac{3}{2}\), i.e. \(p\) can take values only from \(p_0\) to \(\dfrac{3}{2}p_0\). For the case \(\rho=1\), or \(x=\dfrac{3}{2}\), which corresponds to \(p_{\max}=\dfrac{3}{2}p_0\), i.e. when a plastic hinge arises in the section \(u=0\). In this case the zone equation (3) is transformed into the equations of the asymptotes of the hyperbola \(z=\pm h\sqrt{3}\,u\). In Fig. 4 the zones for the plastic hinge are shown by hatching.

Fig. 4. Axial section of the beam indicating the zone in the case of a plastic hinge.

Equation (3) expresses two branches of a hyperbola, which are the boundaries between the plastic and elastic zones in the beam; inside them lies the plastic region, and outside them the elastic region (Fig. 2).

Since in what follows we shall use only the parameter \(x\), which will play the role of an independent variable when its values are varied over the interval \(0.05\), we shall establish the dependence between \(x\) and \(u_0\), where \(u_0\) is the boundary along the span of the outer part of the axial section of the beam between the elastic and elastic-plastic zones.

Substituting \(z=\pm h,\ u=u_0\) into equation (3), we then obtain

\[ |u_0|=\sqrt{1-\frac{1}{\chi}}. \tag{5} \]

The equation of the zones (3) can be given a more convenient form, explicit with respect to \(z\),

\[ z=h(2\chi)^{1/2}\left[\frac{3-2\chi}{2\chi}+u^2\right]^{1/2}. \tag{6} \]

As is known, for an arbitrary cross section of a beam, the bending stiffness and the mass per unit length of the elastic-plastic zone are expressed as follows:

\[ EI_x(u)=\frac{4}{3}Ebz^3, \]

\[ m(u)=4b\gamma z, \tag{7} \]

where \(E\) is Young’s modulus, and \(z\) is the applicate for the elastic region of the beam cross section, with \(-h\le z\le +h\).

Taking into account the equation of the zones (6), formulas (7) are transformed to the following form:

\[ EI_x(u)=(EI_x)_0(2\chi)^{3/2} \left[\frac{3-2\chi}{2\chi}+u^2\right]^{3/2}, \]

\[ m(u)=m_0(2\chi)^{1/2} \left[\frac{3-2\chi}{2\chi}+u^2\right]^{1/2}, \tag{8} \]

where

\[ (EI_x)_0=\frac{4}{3}Ebh^3,\quad m_0=\gamma S_0=4\gamma bh. \]

Using the approximating function\(^1\)

\[ Z(u)=u^4-6u^2+5, \]

which was computed earlier [9], and in the presence of (8), the formula for the natural frequencies of bending vibrations is transformed to the following form:

\[ \nu=\frac{48}{l^2}\sqrt{\frac{EI_x(\chi)}{m(\chi)}} \sqrt{ \frac{ (2\chi)^{3/2}\displaystyle\int_0^{u_0}(\lambda^2+u^2)^{3/2}(u^2-1)^2\,du +\displaystyle\int_{u_0}^{1}(u^2-1)^2\,du }{ (2\chi)^{1/2}\displaystyle\int_0^{u_0}(\lambda^2+u^2)P(u)\,du +\displaystyle\int_{u_0}^{1}P(u)\,du } }, \tag{9} \]

where

\[ P(u)=u^8-12u^6+46u^4-60u^2+25,\quad \lambda^2=\frac{3-2\chi}{2\chi}. \]

This formula (9) can be given a form more convenient for computing the frequencies,

\[ \nu=\frac{48}{l^2}\sqrt{\frac{EI_x(\chi)}{m(\chi)}} \sqrt{\frac{\Phi(\chi)+\varphi(u_0)}{F(\chi)+\psi(u_0)}}, \tag{10} \]

where

\[ \varphi(u_0)=\int_{u_0}^{1}(u^2-1)^2\,du =\frac{8}{15}-u_0+\frac{2}{3}u_0^3-\frac{1}{5}u_0^5, \]

\(^1\) The method of computing the approximating function and the derivation of the frequency formula are given in [7] and [8].

\[ \psi(u_0)=\int_{u_0}^{1} P(u)\,du=\frac{3968}{315}-25u_0+20u_0^3-\frac{46}{5}u_0^5+\frac{12}{7}u_0^7-\frac{1}{9}u_0^9, \]

\[ \Phi(x)=(2x)^{3/2}\int_{0}^{u_0}(\lambda^2+u^2)^{3/2}(u^2-1)^2\,du,\qquad F(x)=(2x)^{1/2}\int_{0}^{u_0}(\lambda^2+u^2)^{1/2}P(u)\,du. \]

The last two quadratures are taken by the substitution of the variable \(u=\lambda \tan t\). After taking the quadratures and carrying out certain other intermediate calculations, Tables 1 and 2 were compiled for

\[ \sigma_1(x)=\Phi(x)+\varphi(u_0), \tag{11} \]

\[ \sigma_2(x)=F(x)+\psi(u_0). \tag{12} \]

Table 1

Table 2

The frequency formula (10), in the presence of (11) and (12), is simplified and takes the following form:

\[ \nu=\frac{\nu_0}{l^2}\sqrt{\frac{EI_x(x)}{m(x)}}. \tag{13} \]

where

\[ \nu_0 = 48 \sqrt{\frac{\sigma_1(\chi)}{\sigma_2(\chi)}} . \tag{14} \]

For the dimensionless quantity \(\nu_0\), according to formula (14), a graph has been constructed (Fig. 5), and its values are entered in Table 3. Since in formula (13) \(EI_x(x)=\dfrac{4}{3}Ebz^3\) and \(m(x)=4\gamma bz\) correspond to the middle cross section of the beam, we shall have

\[ \nu = \frac{\nu_0}{l^2}\, z \sqrt{\frac{E}{3\gamma}}, \tag{15} \]

where \(z\) also corresponds to the middle cross section of the beam. But then \(z\) from formula (6) at \(u=0\), i.e., for the dangerous section in the middle of the beam, will be expressed in the following form:

\[ z = h\sqrt{3-2\chi}. \tag{16} \]

If the value (16) found for \(z\) is substituted into (15), we shall obtain the following:

\[ \nu = \frac{h}{l^2}\sqrt{\frac{E}{3\gamma}}\,\nu_\chi, \tag{17} \]

where

\[ \nu_\chi = \nu_0\sqrt{3-2\chi}. \tag{18} \]

In formula (17), for the given beam the factor

\[ \frac{h}{l^2}\sqrt{\frac{E}{3\gamma}} = \mathrm{const}, \]

and \(\nu_\chi\) is a dimensionless majorizing function that yields the frequencies beyond the elastic limit.

According to formula (18), Table 3 and the graph (Fig. 6) have been compiled.

Fig. 5. Graph of the function
\[ \nu_0 = 48\sqrt{\frac{\sigma_1(\chi)}{\sigma_2(\chi)}} . \]

Here the error of the first approximation is \(0.07\%\), and of the second, \(0.01\%\). The problem of calculating the natural frequencies when the beam is loaded by a uniformly distributed load only over a certain part of the span is solved analogously.

In conclusion it should be noted that, in practice, the calculation of frequencies beyond the elastic limit (according to the hypothesis) reduces to finding the frequencies of a beam with cutouts (Fig. 7).

Fig. 6. Graph of the majorizing function
\[ \nu_\chi=\nu_0\sqrt{3-2\chi}. \]

Fig. 7. Axial section of a beam with cutouts corresponding to removal of the plastic mass.

Table 3

REFERENCES

1. I. V. Anan’ev. Handbook for Calculating Natural Vibrations of Elastic Systems. OGIZ Gostekhizdat, Moscow, 1946.

2. I. M. Babakov. Theory of Vibrations. State Publishing House of Technical-Theoretical Literature, Moscow, 1958.

3. A. Nadai. Plasticity (Mechanics of the Plastic State of Matter). Translated from English. Foreign Literature Publishing House, Moscow, 1936.

4. I. M. Rabinovich. On the dynamic calculation of structures beyond the elastic limit. Studies on the Dynamics of Structures, collection of articles. Stroiizdat, Leningrad, 1947.

5. V. V. Sokolovskii. Theory of Plasticity. Publishing House of the Academy of Sciences of the USSR, Moscow, 1946.

6. A. P. Filippov. Vibrations of Elastic Systems. Publishing House of the Academy of Sciences of the Ukrainian SSR, Kiev, 1956.

7. N. K. Vitvitskii. An approximate method for determining the frequencies of natural bending, torsional, and coupled vibrations of beams of variable cross-section. Proceedings of TEMIIT, vol. XXV, TSU Publishing House, Tomsk, 1958.

8. N. K. Vitvitskii. Determination of the natural frequencies of torsional vibrations of beams of variable cross-section. Proceedings of TEMIIT, vol. XXVIII, TSU Publishing House, Tomsk, 1959.

9. N. K. Vitvitskii. Determination of the frequencies of natural bending vibrations of beams beyond the elastic limit in the presence of plastic zones. Proceedings of TEMIIT, vol. XXXIV, TSU Publishing House, Tomsk, 1962.