THEORY OF ELASTICITY

V. M. TOLKACHEV

TRANSFER OF LOAD FROM A STRINGER OF FINITE LENGTH TO AN INFINITE AND A SEMI-INFINITE PLATE

(Presented by Academician Yu. N. Rabotnov on 18 VII 1963)

An infinite and a semi-infinite \((\operatorname{Im} z > 0,\ z = x + iy)\) elastic plate are considered, to which, on the segment \(0 \leq x \leq a\) of the \(x\)-axis, a stringer of length \(a\) is riveted, loaded at the end \(x = 0\) by a force \(P\) directed opposite to the \(x\)-axis. The distributed load \(q(x)\) transmitted from the stringer to the plate is sought.

Using the known solution of the problem of a concentrated force applied to an infinite plane and to the boundary of a half-plane \((^1)\), we find the deformation of the plate \(\varepsilon_x\) on the \(x\)-axis due to an arbitrary load \(q(x)\) directed along the \(x\)-axis.

Then, writing the equilibrium equation for a part of the stringer and using the compatibility condition for the deformations of the stringer and the plate, we obtain for both problems the following singular integral equation:

\[ \int_0^1 \frac{q(\tau)\,d\tau}{\tau-\tau_0} = \frac{\lambda \pi^2}{4} \int_{\tau_0}^1 q(\tau)\,d\tau, \tag{1} \]

whose solution satisfies the equilibrium condition for the entire stringer

\[ \frac{\lambda \pi}{2}\int_0^1 q(\tau)\,d\tau = 1. \tag{2} \]

Here \(\tau = x/a,\ \tau_0 = x_0/a,\ 0 \leq x,\ x_0 \leq a\).

Moreover, in the case of an infinite plate

\[ \lambda = \frac{16}{\pi(3-\nu)(1+\nu)} \frac{E\delta}{E_cF_c}\,a, \qquad q(\tau) = \frac{(3-\nu)(1+\nu)E_cF_c}{8E\delta} \frac{q(x)}{P}, \tag{3} \]

whereas in the case of a semi-infinite plate

\[ \lambda = \frac{2}{\pi} \frac{E\delta}{E_cF_c}, \qquad q(\tau) = \frac{E_cF_c}{E\delta} \frac{q(x)}{P}. \tag{4} \]

In formulas (3) and (4), \(E,\ \nu,\ \delta\) are, respectively, the modulus of elasticity, Poisson’s ratio, and thickness of the plate; \(E_c\) is the modulus of elasticity, and \(F_c\) the cross-sectional area of the stringer.

By the methods developed by L. I. Sedov \((^2)\), the original equation (1) is reduced to the equivalent Fredholm equation

\[ L[q] = A/\sin\theta + B\operatorname{ctg}\theta, \tag{5} \]

where the change of variable has been introduced

\[ \tau = \sin^2\frac{\sigma}{2}, \qquad \tau_0 = \sin^2\frac{\theta}{2}, \qquad q(\sigma) = q\!\left(\sin\frac{\sigma}{2}\right). \tag{6} \]

\(A\) is an arbitrary constant determined from condition (2),

\[ B=\frac{\lambda}{8}\int_0^\pi q(\sigma)\,\sigma\sin\sigma\,d\sigma, \tag{7} \]

\[ L[q]=q(\theta)-\frac{\lambda}{8}\int_0^\pi \sin\sigma\ln\left|\frac{\sin\dfrac{\sigma-\theta}{2}}{\sin\dfrac{\sigma+\theta}{2}}\right|q(\sigma)\,d\sigma . \tag{8} \]

The solution of equation (5) has, at the ends of the interval \(\theta=0;\ \pi\), a singularity of the type \(\operatorname{ctg}\theta\). With

\[ q(\theta)=q_1(\theta)+\frac{A\cos2\theta}{\sin\theta}+B\,\operatorname{ctg}\theta \tag{9} \]

we transform equation (5) to the form

\[ L(q_1)=f(\theta), \tag{10} \]

where

\[ f(\theta)=\frac{A}{4}(8+\lambda)\sin\theta+\left(A\sin2\theta+2B\sin\theta\right)\frac{\lambda}{8}\ln\operatorname{tg}\frac{\theta}{2}. \tag{11} \]

If

\[ q_1(\theta)=f(\theta)+\sum_{n=1}^{\infty}C_n\sin n\theta, \tag{12} \]

then the solution of equation (10) reduces to the solution of the infinite linear system of equations

\[ C_k' - \lambda\sum_{n=1}^{\infty} C_n'\alpha_{kn}=b_k \quad (k=1,2,3,\ldots). \tag{13} \]

Here

\[ \alpha_{kn}=n\left\{[(k-n)^2-1][(k+n)^2-1]\right\}^{-1} \tag{14} \]

for \(k+n\) even, and \(\alpha_{kn}=0\) for \(k+n\) odd,

\[ C_n=-\frac{A\lambda^2 C_n'}{4}\quad (n=1,3,5,\ldots);\qquad C_n=-\frac{B\lambda^2 C_n'}{4}\quad (n=2,4,6,\ldots), \tag{15} \]

\[ b_1=\frac{8+\lambda}{3\lambda}-\frac{1}{12};\qquad b_3=-\frac{8+\lambda}{45\lambda}-\frac{104}{108\cdot24}; \]

\[ b_k=-\frac{8+\lambda}{\lambda k^2(k^2-4)} +\frac{2\left(1+\frac{1}{3}+\cdots+\frac{1}{k-4}\right)}{(k^2-1)(k^2-9)} -\frac{3k^2(k+1)-2(k-1)}{2(k^2-4)(k^2-1)k^2(k+3)} \quad (k=5,7,\ldots); \tag{16} \]

\[ b_2=-\frac{1}{12};\qquad b_k=\frac{2\left(1+\frac{1}{3}+\cdots+\frac{1}{k+3}\right)}{k^2(k^2-4)} -\frac{3k+2}{2k^2(k^2-1)(k+2)} \quad (k=4,6,\ldots). \tag{17} \]

By virtue of (14), system (13) splits into two independent systems for the even and odd coefficients \(C_k'\) and is completely regular for any finite values of the parameters \(\lambda\).

For large \(\lambda\), \(|C_k| \leqslant 1.756/\lambda\) for odd \(k\), and \(|C'_k| \leqslant 1/\lambda\) for even \(k\).

If system (13) is solved, then the solution of the problem is found from formula (9), taking (12) and (15) into account. For the constants \(A\) and \(B\) we have the formulas

\[ A=\frac{32}{\pi^2\lambda\chi}, \qquad B=AB_1; \tag{18} \]

\[ B_1=\frac{\pi^2\lambda}{128}\, \frac{\chi}{ 1+\frac{\lambda}{4}-\frac{\lambda^2\omega}{32} -\frac{\lambda^3}{8}\sum_{2,4,\ldots}^{\infty} \frac{C'_n n}{(n^2-1)^2} }; \tag{19} \]

\[ \chi=8+\lambda-\lambda^2 C'_1-\frac{\lambda}{3}, \qquad \omega=0.603599\ldots \tag{20} \]

For the stresses in the stringer \(P(\theta)\), referred to the stresses \(P(0)\) at the point \(\theta=0\), we have the formula

\[ \frac{P(\theta)}{P(0)} =1-\frac{\theta}{\pi} -\frac{1}{\pi\chi}\left\{ B_1(8+\lambda)\sin\theta +\frac{1}{2}(8-\chi)\sin 2\theta +\frac{\lambda}{2}\left(\sin\theta-\frac{1}{3}\sin 3\theta-B_1\sin 2\theta\right)\ln \operatorname{tg}\frac{\theta}{2} \right. \]

\[ \left. +B_1\lambda\omega_1(\theta) -\lambda^2\sum_{2,3,4,\ldots}^{\infty} C'_n\delta_n \left[ \frac{\sin(n-1)\theta}{n-1} -\frac{\sin(n+1)\theta}{n+1} \right] \right\}; \tag{21} \]

\[ \delta_n=1 \ \text{for } n=3,5,\ldots; \qquad \delta_n=B_1 \ \text{for } n=2,4,6,\ldots; \]

\[ \omega_1(\theta)=\int_0^\theta \ln \operatorname{tg}\frac{\theta}{2}\,d\theta =-2\sum_{1,3,5}^{\infty}\frac{\sin n\theta}{n^2}. \tag{22} \]

Received
9 VII 1963

REFERENCES

¹ N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, 4th ed., Publishing House of the Academy of Sciences of the USSR, 1954. ² L. I. Sedov, Plane Problems of Hydrodynamics and Aerodynamics, 1950.