STRESS CONCENTRATION IN BENDING OF THIN PLATES BY MOMENT LOADS WITH CURVILINEAR WASHERS

M. G. PINSKII

Existing works on the question of stress concentration in the bending of plates consider the symmetric bending of an infinite plate with a circular opening [1], elliptical, triangular, and square openings [3], and also plates with a rigid washer of circular [2] or elliptical shape, free of external load.

In 1948 G. N. Savin and N. P. Fleishman [4] investigated the bending of a triangular isotropic plate with a reinforced circular opening, with a compound and a simple ring soldered into it, and bent by bending moments uniformly distributed along its edge.

We shall consider the bending of an infinite plate with a rigid washer, free of load and loaded by a moment, having both the indicated shapes and shapes approaching an equilateral triangle or a square with rounded vertices. Approximately such shapes are usually possessed in practice by various kinds of flanges of bushings for sleeve bearings, etc., fastened to thin walls.

We shall consider two cases of plate bending: under the action of moments applied and equilibrated at infinity (we shall call such bending symmetric), and bending under the action of a moment applied to the rigid washer (we shall call such bending nonsymmetric).

The deflection \(W\) of a thin isotropic plate, bent by external moment loads, satisfies, as is known [5], the biharmonic equation

\[ \Delta \Delta W = 0. \tag{1} \]

The solution of equation (1) can be represented by means of two analytic functions by the equality

\[ W(x,y)=2\,Re[\bar z\,\varphi(z)+\chi(z)]. \tag{2} \]

Here \(\bar z\) is the conjugate quantity of \(z=x+iy\).

If the general integral of the biharmonic equation is represented in the form (2), then the bending moments \(M_\rho, M_\theta\) and the twisting moment \(M_{\rho\theta}\), as well as the shearing forces, referred in the plane \(z\) to a curvilinear orthogonal coordinate system, which is mapped into the polar coordinate system \(\rho,\theta\) of the plane \(\zeta\) by means of the mapping function \(z=\omega(\zeta)\), can be represented by the following dependence:

\[ M_\theta - M_\rho + 2iM_{\rho\theta} = 4(1-\mu)De^{2i\alpha} \left[ \overline{\omega(\zeta)}\cdot \frac{\varphi'(\zeta)}{\omega'(\zeta)} + \Psi(\zeta) \right], \]

\[ M_\rho + M_\theta = -8(1+\mu)DRe\,\Phi(\zeta), \]

\[ Q_\rho - iQ_\theta = -8De^{i\alpha}\cdot \frac{\Phi'(\zeta)}{\omega'(\zeta)}. \tag{3} \]

Here \(\mu\) is Poisson’s ratio,

\[ D=\frac{Eh^3}{12(1-\mu^2)} \]

is the cylindrical rigidity,

\[ e^{2i\alpha}=-\frac{\zeta^2}{\rho^2}\cdot\frac{\omega'(\zeta)}{\overline{\omega'(\zeta)}}, \tag{3'} \]

\(\alpha\) is the angle between the direction of the normal to the curve and the \(X\)-axis. The functions

\[ \Phi(\zeta)=\varphi'[\omega(\zeta)], \]

\[ \Psi(\zeta)=\chi''[\omega(\zeta)] \]

are holomorphic and continuous in the region under consideration up to the boundary contour.

Consideration of the solution of the posed problem in Laurent series led us to very complicated and cumbersome calculations in computing the bending moments.

Solving the problem by means of integrals of Cauchy type leads to the goal more quickly and gives a solution convenient for applications.

Separating the constant terms in the functions \(\Phi(\zeta)\) and \(\Psi(\zeta)\), we represent them in the form:

\[ \Phi(\zeta)=A+iB+\Phi_0(\zeta), \]

\[ \Psi(\zeta)=A'+iB'+\Psi_0(\zeta). \tag{4} \]

If the region of variation of the functions \(\Phi_0(\zeta)\) and \(\Psi_0(\zeta)\) is the region \(|\zeta|>1\), then we can represent them in the form:

\[ \Phi_0(\zeta)=\sum_{k=1}^{\infty} a_k\zeta^{-k}, \tag{5} \]

\[ \Psi_0(\zeta)=\sum_{k=1}^{\infty} a'_k\zeta^{-k}. \]

We shall determine the constants \(A, B, A'\), and \(B'\) from the conditions at infinity. Let a bending moment \(M\) act on the plate at infinity about the \(Y\)-axis, so that

\[ M_{x|\sim}=0,\qquad M_{y|\sim}=M\quad\text{and}\quad M_{xy|\sim}=0. \]

Then

\[ -2D[2(1+\mu)A+(1-\mu)A']=0, \]

\[ -2D[2(1+\mu)A-(1-\mu)A']=M, \]

\[ 2D(1-\mu)B'=0. \tag{6} \]

Whence we obtain:

\[ A=-\frac{M}{8D(1+\mu)},\qquad A'=\frac{M}{4D(1-\mu)},\qquad B'=0. \tag{7} \]

Since \(B\) is not determined by conditions (6), we may, without loss of generality, take \(B=0\). As is known [6], in the case where the deflection and the derivative of the deflection along the normal are specified on the boundary of the region (the second fundamental problem), the boundary condition can be written in the following form:

\[ \varphi'(z)+\overline{\varphi'(z)}-e^{2i\alpha}\left[\overline{z}\,\varphi''(z)+\chi''(z)\right]=K_1+iK_2, \tag{8} \]

where

\[ K_1+iK_2=\frac{1}{2}e^{i\alpha}\frac{d}{ds}\left[-\frac{\partial w}{\partial y}+i\frac{\partial w}{\partial x}\right]. \]

Taking (3′) into account, we give condition (7) for the transformed domain the form:

\[ \varphi'(z)+\overline{\varphi}'(\overline{z})-\frac{\zeta^2\omega(\zeta)}{\rho^2\omega'(\zeta)} \left[\overline{\omega}(\overline{\zeta})\varphi''(\zeta)+\chi''(\zeta)\right] =K_1+iK_2. \tag{9} \]

The complex-conjugate equality will be

\[ \overline{\varphi}'(\overline{z})+\varphi'(z)-\frac{\zeta^{-2}\overline{\omega}(\overline{\zeta})}{\rho^2\omega'(\zeta)} \left[\omega(\zeta)\overline{\varphi}''(\overline{\zeta})+\overline{\chi}''(\overline{\zeta})\right] =K_1-iK_2. \tag{9′} \]

Condition (9′) gives nothing new by itself, but it will be useful to us subsequently.

We had that \(\varphi[\omega(\zeta)]=\Phi(\zeta)\).

Consequently

\[ \varphi'[\omega(\zeta)]=\frac{\Phi'(\zeta)}{\omega'(\zeta)}. \tag{10} \]

Denoting the variable \(\zeta\) on the contour by \(\sigma=e^{i\theta}\) and taking (10) into account, we can represent equalities (9) and (9′) for contour points in the form:

\[ \Phi(\sigma)+\overline{\Phi}(\overline{\sigma})- \frac{\sigma^2}{\omega'(\sigma)} \left[\overline{\omega}(\overline{\sigma})\Phi'(\sigma)+\omega'(\sigma)\Psi(\sigma)\right] =(K_1+iK_2)_s, \tag{11} \]

\[ \overline{\Phi}(\overline{\sigma})+\Phi(\sigma)- \frac{\sigma^{-2}}{\omega'(\sigma)} \left[\omega(\sigma)\overline{\Phi}'(\overline{\sigma})+\overline{\omega}'(\overline{\sigma})\Psi(\overline{\sigma})\right] =(K_1-iK_2)_s. \tag{11′} \]

Let us write the following expressions for the transforming function and its derivatives:

\[ \begin{gathered} z=\omega(\zeta)=\zeta+m\zeta^{-n},\qquad \omega'(\zeta)=1-mn\zeta^{-(n+1)},\\ \overline{z}=\overline{\omega}(\overline{\zeta})=\zeta+m\overline{\zeta}^{-n},\qquad \overline{\omega}'(\overline{\zeta})=1-mn\overline{\zeta}^{-(n+1)}, \end{gathered} \tag{12} \]

where \(z\) is the complex variable in the plane under consideration,
\(\zeta\) is that in the mapped plane,
\(m\) is a real constant \(<1\),
\(n\) is a real number taking various integer values.

For \(n=1,2,3\ldots\), the function \(z=\omega(\zeta)\) maps the domain of the infinite plate with, respectively, an elliptic, a regular curvilinear triangular, quadrilateral, etc., internal contour with rounded vertices onto the exterior of the circle \(\gamma\) of unit radius in the \(\zeta\)-plane.

For \(n=0\) the mapped figures become a circle.

It is possible to choose such values of \(m\) that the mapped curvilinear triangles and squares will, with sufficient accuracy, represent rectilinear figures, differing from them by rounded vertices [9].

Increasing successively by one the exponent \(n\), we shall obtain mapped figures in the form of a curvilinear pentagon \((n=4)\), hexagon \((n=5)\), etc.

Taking (12) into account, we give equalities (11 and 11′) the form:

\[ \Phi_0(\sigma)+\overline{\Phi}_0(\overline{\sigma}) -\frac{\sigma^2}{1-mn\sigma^{n+1}} \left[(\sigma^{-1}+m\sigma^n)\Phi_0(\sigma) +\left(1-mn\sigma^{-(n+1)}\right)\Psi_0(\sigma)\right] = \]

\[ =(K_1+iK_2)_s-2A+(A'-iB')\,\frac{\sigma^2\left(1-mn\sigma^{-(n+1)}\right)}{1-mn\sigma^{n+1}}; \tag{13} \]

\[ \overline{\Phi}_0(\overline{\sigma})+\Phi_0(\sigma) -\frac{\sigma^{-2}}{1-mn\sigma^{-(n+1)}} \left[(\sigma+m\sigma^{-n})\overline{\Phi}'_0(\overline{\sigma}) +\left(1-mn\sigma^{n+1}\right)\overline{\Psi}_0(\overline{\sigma})\right] = \]
\[ =(K_1-iK_2)_s-2A+(A'+iB')\,\frac{\sigma^{-2}\left(1-mn\sigma^{n+1}\right)}{1-mn\sigma^{-(n+1)}}. \tag{14} \]

Multiplying equalities (13) and (14), respectively, by

\[ \frac{1}{2\pi i}\left(1-mn\sigma^{n+1}\right)\frac{d\sigma}{\sigma-\zeta} \quad\text{and}\quad \frac{1}{2\pi i}\left(1-mn\sigma^{-(n+1)}\right)\frac{d\sigma}{\sigma-\zeta} \]

and integrating the resulting relations along the circle \(\gamma\) in the direction opposite to the motion of the clock hand (7), we obtain:

\[ \frac{1}{2\pi i}\int_\gamma \Phi_0(\sigma)\frac{d\sigma}{\sigma-\zeta} -\frac{1}{2\pi i}\int_\gamma mn\sigma^{n+1}\Phi_0(\sigma)\frac{d\sigma}{\sigma-\zeta} +\frac{1}{2\pi i}\int_\gamma \overline{\Phi}_0(\overline{\sigma})\frac{d\sigma}{\sigma-\zeta} - \tag{15} \]
\[ -\frac{1}{2\pi i}\int_\gamma mn\sigma^{n+1}\overline{\Phi}_0(\overline{\sigma})\frac{d\sigma}{\sigma-\zeta} -\frac{1}{2\pi i}\int_\gamma mn\sigma\Phi'_0(\sigma)\frac{d\sigma}{\sigma-\zeta} - \]
\[ -\frac{1}{2\pi i}\int_\gamma m\sigma^{n+2}\Phi'_0(\sigma)\frac{d\sigma}{\sigma-\zeta} -\frac{1}{2\pi i}\int_\gamma \sigma^2\Psi_0(\sigma)\frac{d\sigma}{\sigma-\zeta} + \]
\[ +\frac{1}{2\pi i}\int_\gamma mn\sigma^{-(n-1)}\Psi_0(\sigma)\frac{d\sigma}{\sigma-\zeta} = \frac{1}{2\pi i}\int_\gamma \bigl((K_1+iK_2-2A)(1-mn\sigma^{n+1})\bigr)\frac{d\sigma}{\sigma-\zeta} + \]
\[ +\frac{1}{2\pi i}\int_\gamma (A'+iB')(\sigma^2-mn\sigma^{-(n-1)})\frac{d\sigma}{\sigma-\zeta}. \]

\[ \frac{1}{2\pi i}\int_\gamma \overline{\Phi}_0(\overline{\sigma})\frac{d\sigma}{\sigma-\zeta} +\frac{1}{2\pi i}\int_\gamma \Phi_0(\sigma)\frac{d\sigma}{\sigma-\zeta} -\frac{1}{2\pi i}\int_\gamma mn\sigma^{-(n+1)}\overline{\Phi}_0(\overline{\sigma})\frac{d\sigma}{\sigma-\zeta} - \]
\[ -\frac{1}{2\pi i}\int_\gamma mn\sigma^{-(n+1)}\Phi'_0(\sigma)\frac{d\sigma}{\sigma-\zeta} -\frac{1}{2\pi i}\int_\gamma \sigma^{-1}\overline{\Phi}'_0(\overline{\sigma})\frac{d\sigma}{\sigma-\zeta} - \]
\[ -\frac{1}{2\pi i}\int_\gamma m\sigma^{-(n+2)}\overline{\Phi}'_0(\overline{\sigma})\frac{d\sigma}{\sigma-\zeta} -\frac{1}{2\pi i}\int_\gamma \sigma^{-2}\overline{\Psi}_0(\overline{\sigma})\frac{d\sigma}{\sigma-\zeta} + \]
\[ +\frac{1}{2\pi i}\int_\gamma mn\sigma^{n-1}\overline{\Psi}_0(\overline{\sigma})\frac{d\sigma}{\sigma-\zeta} = \frac{1}{2\pi i}\int_\gamma \bigl((K_1-iK_2-2A)(1-mn\sigma^{-(n+1)})\bigr)\frac{d\sigma}{\sigma-\zeta} + \]
\[ +\frac{1}{2\pi i}\int_\gamma (A'-iB')(\sigma^{-2}-mn)\frac{d\sigma}{\sigma-\zeta}. \tag{16} \]

Having evaluated each of the Cauchy-type integrals entering these equalities (15) and (16), we finally obtain:

\[ \frac{\zeta^{n+1}-mn}{\zeta^{n+1}}\Psi_0(\zeta) = \frac{1}{\zeta^2} -\frac{1}{2\pi i}\int_\gamma (K_1+iK_2)\frac{d\sigma}{\sigma-\zeta} +\frac{1-mn\zeta^{n+1}}{\zeta^2} \]

\[ -\frac{1+m\zeta^{n+1}}{\zeta}\Phi'_0(\zeta) +\sum_{k=1}^{n} m(n+k\zeta)a_k\zeta^{\,n-k-1} +\frac{a'_1}{\zeta}+\frac{a'_2}{\zeta^2} \tag{17} \]

\[ +mn a_{n+1}\zeta^{-2} +\frac{m(n+1)a_{n+1}}{\zeta} +\frac{m(n+2)a_{n+2}}{\zeta^2} -\sum_{1}^{n-1} mn a'_k\zeta^{-(n+k-1)}, \]

where

\[ K^\circ_1+iK^\circ_2=(K_1+iK_2-2A)(1-mn\zeta^{-(n+1)}) +(A'+iB')(\zeta^2-mn\zeta^{(n-1)}). \]

\[ (\zeta^{n+1}-mn)\Phi_0(\zeta) =-\zeta^{n+1}\frac{1}{2\pi i}\int_{\gamma} \frac{K^\circ_1-iK^\circ_2}{\sigma-\zeta}\,d\sigma +\overline{a'_1}\zeta^n+\overline{a'_2}\zeta^{n-1} \]

\[ +\sum_{k=1}^{n} m(n\zeta+K\zeta^{-2})\overline{a_k}\zeta^k +m(n+1)\overline{a_{n+1}} +m(n+2)\overline{a_{n+2}}\zeta^n, \tag{18} \]

where

\[ K^\circ_1-iK^\circ_2=(K_1-iK_2-2A)(1-mn\zeta^{-(n+1)}) +(A'-iB')(\zeta^{-2}-mn\zeta^{\,n-1}). \]

From the condition of single-valuedness of the displacements [5] it follows that:

\[ a'_1=\overline{a'_1}\quad \text{and}\quad a'_2=\overline{a'_2},\qquad a_1=\overline{a_1} =\frac{1}{2\pi i}\cdot \frac{\mathfrak{M}_x-i\mathfrak{M}_y}{8D}, \]

\[ a_2-ma_{n+1} =-\frac{1}{2\pi i}\int_{\gamma}(K^\circ_1+iK^\circ_2)\frac{d\sigma}{\sigma-\zeta}, \]

\[ a_k= -\frac{ \displaystyle \frac{1}{2\pi i}\int_{\gamma}(K^\circ_1-iK^\circ_2)\sigma^{k-1}\frac{d\sigma}{\sigma-\zeta} +m(k-1)\frac{1}{2\pi i}\int_{\gamma}(K^\circ_1-iK^\circ_2)\sigma^{n+k-1}\frac{d\zeta}{\zeta-\zeta} }{ 1-m^2(k-1)(n-k) }. \tag{19} \]

\[ k=2\ldots n_1-1\ldots (n_1\ge 3). \]

Here \(\mathfrak{M}_x\) and \(\mathfrak{M}_y\) are the components of the principal vector of the moment of the forces applied to the washer.

Fig. 1

Equations (17), (18), and (19) solve the problem of bending of plates in the presence of a washer free from external loading (symmetric bending, Fig. 1) and bending under the action of a moment applied to a rigid washer (unsymmetric bending, Fig. 2).

We compute the stresses by formula (8)

\[ (\sigma_\rho)_{\max}=\frac{6M_\rho}{h^2}. \]

Fig. 2

Bending of Plates with Curvilinear Washers Loaded by Balanced Moments

(Symmetric bending, Fig. 1)

a) Elliptical washer

In this case \(n=1\) and

\[ z=(\zeta+m\zeta^{-1}). \]

From (17), (18), and (19), omitting all intermediate calculations, by equations (3) we write the value of the bending moment along the contour of the washer:

\[ M_\rho(\theta)= \frac{M}{1+\mu} - \frac{2M}{1-2m\cos 2\theta+m^2} \left[ \left( \frac{m}{1+\mu} + \frac{\cos 2\alpha}{1-\mu} \right)(m-\cos\theta) - \frac{\sin 2\alpha}{1-\mu}\sin 2\theta \right]. \tag{2.1} \]

Here

\[ m=\frac{a-b}{a+b} \]

is the degree of compression of the elliptical washer; \(a\) and \(b\) are, respectively, the major and minor axes of the ellipse; \(\alpha\) is the angle between the \(X\)-axis and the major semi-axis of the elliptical washer. We investigated various degrees of compression of the elliptical washer \((m=0,\ 1/5,\ 1/3)\) for \(\alpha=0\) and \(\alpha=90^\circ\).

We represent the value of the bending moment on the contour of a steel plate \((\mu=0.3)\) in the form

\[ M_\rho(\theta)=K_{1\alpha}(\theta)M, \]

where \(M\) is the moment per unit length of the plate.

The values of \(K_{1\alpha}\) are given in Fig. 3.

b) Triangular washer \((n=2)\)

Analogously to the preceding case, we obtain:

\[ M_\rho(\theta)= \frac{4M}{1-4m\cos 3\theta+4m^2} \left[ \frac{1-4m^2}{2(1+\mu)} - \frac{2m}{1-\mu}\cos(\theta+2\alpha) + \frac{1}{1-\mu}\cos 2(\theta-\alpha) \right]. \tag{2.2} \]

For \(m=\frac{1}{4}\), a regular triangle with rounded vertices is mapped onto the circle. The curvature at the vertices of the triangle is equal to \(8/R\), and the radius of curvature at the vertices is \(R/8\).

For \(\alpha=0\). The value of the bending moment on the contour of a triangular washer of a steel plate will be represented as

\[ M_\rho(\theta)=K_2(\theta)\cdot M. \]

The values of \(K_2(\theta)\) are given in Table No. 1 and in Fig. 4.

Table 1

c) Square washer \((n=3)\)

The value of the bending moment on the contour of a square washer will be represented as

\[ M_\rho(\theta)= \frac{2M}{1-6m\cos4\theta+9m^2} \left[ \frac{1-9m^2}{2(1+\mu)} + \frac{(1-3m)\cos2\alpha}{(1-\mu)(1-m)}\cos2\theta + \frac{(1+3m)\sin2\alpha}{(1-\mu)(1+m)}\sin2\theta \right]. \tag{2.3} \]

For \(m=\frac{1}{9}\), the square washer has the form of a rectilinear square with rounded vertices. The radius of curvature at the vertices for \(m=\frac{1}{9}\) is equal to \(\frac{2}{9}R\). \(\alpha\) is the angle between the diagonal of the square and the \(X\)-axis.

For \(\alpha=0\) (Fig. 5)

\[ M_\rho(\theta)= \frac{9M}{2} \left[ \frac{4}{9(1+\mu)} + \frac{3}{4(1-\mu)}\cos2\theta \right]. \tag{2.4} \]

For \(\alpha=45^\circ\) (Fig. 6).

\[ M_\rho(\theta)= \frac{9M}{2} \left[ \frac{4}{9(1+\mu)} + \frac{6}{5(1-\mu)}\sin\theta \right]. \tag{2.5} \]

In both cases the bending moment will be represented in the form

\[ M_\rho(\theta)=K_3(\theta)\cdot M, \]

where \(K_3^{(\theta)}\) is a coefficient whose values for a steel plate \((\mu = 0.3)\) with a square washer \(\left(m = \dfrac{1}{9}\right)\) are given in Table 2 and in Figs. 5 and 6.

Fig. 3. Concentration coefficient \(K_1\) on the contour of an elliptical washer with various degrees of compression of the ellipse \((m)\).

Fig. 4. Distribution of bending moments along the contour of a triangular washer at \(\alpha = 0^\circ\), \((\mu = 0.3)\), \(m = \dfrac{1}{4}\).

Table 2

It should be noted that if $m=0$ is substituted into formulas (2.1), (2.2), (2.3), then we obtain the solution for a plate reinforced with a circular washer.

Fig. 5. Distribution of bending moments along the contour of a square washer for $\alpha = 0^\circ$ $(\mu = 0.3)$

The agreement of these results was to be expected, since the mapping function (12) for $m=0$ does not depend on $n$, but maps the circle onto itself. In this case the bending moment on the contour of the circular washer has the form

\[ M_t(\theta)=\frac{2M}{1+\mu}\cos\theta . \]

Fig. 6. Distribution of bending moments along the contour of a square washer for $\alpha = 45^\circ$ $(\mu = 0.3)$

This result coincides with the result obtained by S. G. Lekhnitskii [1] for a plate with a circular washer.

Bending of plates with curvilinear washers loaded by a moment

a) Elliptical washer

In this case a moment \(\mathfrak{M}\) acts on the washer, while the moments on the outer boundary (infinity) tend to zero.

From (6) it follows that \(A=B=A'=B'=0\). The coefficients \(a_1\) and \(a'_1\) are determined by the equality

\[ a_1=a'_1=-\frac{\mathfrak{M}}{16\pi D}. \]

From (17) and (18) we write the values of the functions

\[ \Phi_0(\zeta)=\frac{a'_1 \zeta}{\zeta^2-m}, \tag{3.1} \]

\[ \frac{\zeta^2-m}{\zeta}\,\Psi_0(\zeta) = \frac{1-m\zeta^2}{\zeta}\,\Phi_0'(\zeta) - (1+m\zeta)\Phi_0''(\zeta). \]

Fig. 7. Variation of \(K_1\) for elliptical washers with various degrees of compression \((m)\)

Fig. 8. Coefficient \(K_1\) on the contour of elliptical washers with various degrees of compression of the ellipse \((m)\)

From (3) and (4), taking (3.1) into account, we obtain the following expression for the bending moment on the washer contour:

\[ M_e(\theta) = -\frac{\mathfrak{M}}{2\pi} \cdot \frac{(1-m)\cos\theta}{1-2m\cos 2\theta+m^2} \tag{3.2} \]

or represent it in the form

\[ M_e(\theta)=K_{1H}(\theta)\cdot \frac{\mathfrak{M}}{2\pi}. \tag{3.3} \]

Here \(K_{1H}\) is a coefficient whose values for various \(m\) (the degree of compression of the ellipse) and \(\theta\) are given in Figs. 7 and 8.

For \(m>0\), the major axis of the ellipse coincides with the \(X\)-axis \((\alpha=0^\circ)\). (Figs. 9, 10, 11).

For \(m<0\), the major axis of the ellipse is perpendicular to the \(X\)-axis \((\alpha=90^\circ)\). (Figs. 12, 13, 14).

b) Triangular washer \((n = 2)\).

Repeating the same calculations as in case (a), we write

\[ (\zeta^3 - 2m)\Phi_0(\zeta) = a_1' \zeta + m\overline{a_1'}\zeta, \]

Fig. 9. Stress concentration \(K_1\) in the formula \(M_p = K_1 \dfrac{M}{2\pi}\) on the contour of an elliptical washer, \(m = \dfrac{1}{5}\), under nonsymmetric bending by a moment

\[ \frac{\zeta^3 - 2m}{\zeta^2}\Psi_0(\zeta) = \frac{1 - 2m\zeta^2}{\zeta^2}\Phi_0(\zeta) - \frac{1 + m\zeta^2}{\zeta}\overline{\Phi_0'(\zeta)} + ma_1 + \frac{a_1'}{\zeta}. \tag{3.4} \]

Fig. 10. Stress concentration \(K_{1H}\) in the formula \(M_p = K_{1H}\dfrac{M}{2\pi}\) on the contour of an elliptical washer \(m = \dfrac{1}{3}\) under nonsymmetric bending by a moment

The bending moment on the contour (3.4) of the washer will be

\[ M_p(\theta) = \frac{\mathfrak{M}}{2\pi}\cdot \frac{(1 - 2m^2)\cos\theta - m\cos 2\theta} {1 - 4m\cos 3\theta + 4m^2}. \tag{3.5} \]

Fig. 11. Stress concentration \(K_1\) in the formula \(M_p = K_1\dfrac{M}{2\pi}\) on the contour of an elliptical washer \(\left(m = \dfrac{2}{5}\right)\) under bending

; or

\[ M_t(\theta)=K_{2н}(\theta)\cdot \frac{\mathfrak{M}}{2\pi}, \tag{3.6} \]

where \(K_{2н}(\theta)\) for \(m=\dfrac{1}{4}\) are given in Table 3 and in Fig. 15.

Fig. 12. Stress concentration \(K_1\) on the contour of an elliptical washer under asymmetric bending \((m=1/5)\)

Fig. 13. Stress concentration \(K_1\) on the contour of an elliptical washer \((m=1/3)\) under asymmetric bending

Table 3

Fig. 14. Stress concentration \(K_1\) on the contour of an elliptical washer \(\left(m=\dfrac{2}{5}\right)\) under asymmetric bending

Fig. 15. Stress concentration \(K_2\) in the formula \(M_\rho = K_2 \dfrac{M}{2\pi}\) on the contour of a triangular washer \(\left(m=\dfrac{1}{4}\right)\) under nonsymmetric bending by a moment

c) Square washer \((n=3)\)

The values of the functions will be:

\[ \frac{\zeta^4-3m}{\zeta^4}\,\Phi_0'(\zeta) = a_1'\zeta^2+2ma_1\zeta, \tag{3.7} \]

\[ \frac{\zeta^4-3m}{\zeta^4}\,\Psi_0(\zeta) = \frac{1-3m\zeta^4}{\zeta^2}\,\Phi_0(\zeta) - \frac{1+m\zeta^4}{\zeta}\,\Phi_0'(\zeta) + 2ma_1\zeta + \frac{a_1'}{\zeta}. \]

We shall write the value of the bending moment on the contour of the washer in the form:

\[ M_\rho(\theta) = \frac{\mathfrak{M}}{2\pi}\, \frac{(1-6m^2)\cos\theta-m\cos3\theta} {1-6m\cos4\theta+9m^2} \tag{3.8} \]

or

\[ M_\rho(\theta) = k_{3\text{н}}(\theta)\cdot\frac{\mathfrak{M}}{2\pi}, \tag{3.9} \]

The values of the coefficient \(K_{3\text{н}}(\theta)\) for \(m=\dfrac{1}{9}\) are given in Table 4 and in Figs. 16 and 17.

Fig. 16. Stress concentration \(K_3\) in the formula \(M_\rho = K_3 \dfrac{M}{2r}\) on the contour of a square washer \(\left(m=\dfrac{1}{9}\right)\) under bending \((\alpha=0^\circ)\)

Table 4

\[ \alpha = 45^\circ \]

Fig. 17

It should be noted that for \(m=0\) expressions (3.2), (3.5), and (3.8) coincide, and we obtain the solution for the case of a circular washer (Fig. 18),

Fig. 18. Stress concentration \(K_1\) in the formula \(M_\rho = K_1 \dfrac{M}{2\pi}\) on the contour of a circular washer \((M=0)\) under nonsymmetric bending by a moment.

In conclusion, the author takes the opportunity to express gratitude to his teacher, Prof. Dr. V. I. Blokh, for the valuable comments he made and for his attention to the work.

References

1. S. G. Lekhnitskii. On certain cases of bending of an isotropic plate weakened by a circular hole. Vestnik inzhenerov i tekhnikov, No. 12, pp. 725–727, 1936.

2. S. G. Lekhnitskii. On certain cases of bending of an anisotropic plate weakened by a circular hole. Vestnik inzhenerov i tekhnikov, No. 4, pp. 249–252, 1937.

3. M. M. Fridman. Bending of a thin isotropic plate with a curvilinear hole. PMM, vol. IX, issue 4, pp. 334–338, 1945.

4. G. N. Savin and N. P. Fleishman. Bending of thin plates with a circular opening whose edge is reinforced by an elastic ring. Academy of Sciences of the Ukrainian SSR, No. 6, 1948.

5. S. G. Lekhnitskii. On certain questions connected with the theory of bending of thin plates. PMM, vol. II, issue 2, pp. 181–210, 1938.

6. S. G. Lekhnitskii. PMM, vol. II, issue 2, pp. 181–210, 1938. Anisotropic Plates, 1947.

7. N. I. Muskhelishvili. Some Basic Problems of the Mathematical Theory of Elasticity. Publishing House of the Academy of Sciences of the USSR, pp. 251–266, 1949.

8. S. P. Timoshenko. Plates and Shells, pp. 252–256, 1948.

9. M. I. Naiman. Proceedings of TsAGI, issue 313, 1937.