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It is proved that a function which is biharmonic in a half-strip and satisfies homogeneous Dirichlet conditions at the base of the strip oscillates.

SummaryA double series solution of the biharmonic equation is derived in the rectangular domain. The solution utilizes the finite double Fourier sine transform and the analysis shows how the transform may be evaluated for quite arbitrary boundary conditions, the solution then being found from the corresponding inversion formula.The method is applied to some classical boundary value problems which have been investigated by other means by earlier workers. It is felt that some of the series solutions given here are more amenable to computation than some of the earlier methods.In some of the problems considered, notably the elastostatics of the rectangular plate and the bending of clamped plates, the method reduces to solving an infinite set of linear equations. In such cases, a finite leading matrix of the coefficients has to be considered. In the examples considered, this technique produced results which converged to known solutions obtained by other methods. However, the mathematical criteria which permit the infinite system to be curtailed, have not been established.

A strongly implicit procedure is described which solves the system of 13 point finite difference equations associated with the biharmonic and similar fourth order elliptic equations. No factorization of the equation is required, and for the majority of problems, a universal set of iteration parameters provide rapid rates of convergence. In a comparison with another solution procedure for the biharmonic equation, the new method appears to reduce the computation required to about one-third.

Two methods for solving the biharmonic equation are compared. One method is direct, using eigenvalue-eigenvector decomposition. The other method is iterative, solving a Poisson equation directly at each iteration.

The biharmonic boundary value problem with Dirichlet boundary conditions is reduced to a coupled system of Poisson equations, which depends upon an arbitrary, positive coupling constant c. Since each of the Poisson equations is well-posed, the system may be solved by iteration. We show that the iterates may be represented as a linear combination of the eigenfunctions of the “Dirichlet eigenvalue problem” (a fourth order boundary value problem with the eigenvalue in the boundary condition). Convergence of the iterative scheme occurs when $0 < c < 2v_1$, where $v_1 $ is the smallest eigenvalue. By making use of an averaging scheme, convergence may be produced for any positive c. With the proper choice of c, the rate of convergence may be increased. This coupled equation approach includes the finite difference approach as a special case.

This paper presents a solution for the stresses in a perforated circular ring containing a ring of uniformly distributed circular holes. The stress function is constructed by using ring-periodic biharmonic functions. Even and odd solutions are formulated separately. The solution is illustrated by numerical examples of a ring containing six holes under two types of loading.

Two-dimensional viscous flow due to the motion of a semi-infinite flat plate moving perpendicular to its own plane in a region of fluid bounded by a rigid plane well is considered by means of the Stokes approximation. The stream function, being biharmonic, can be constructed by use of two harmonic functions. By mapping the flow region conformally onto the upper half of a complex plane, the problem can be reduced to determining the harmonic functions in this plane. From the first and second approximate solutions are obtained a lower and an upper bound, respectively, of the discharge of fluid through the gap between the plate and the well. The case in which the semi-infinite plate is held at rest and the flow is caused by the pressure difference through the gap is also considered.

A complete characterization is given for the fundamental solution of the complex-valued differential operator governing the linear behavior of a shallow, elastically isotropic hyperbolic paraboloidal shell. The only previously known fundamental solutions in shell theory have been those for the sphere and the shallow cylinder. Physically, the real and imaginary parts of these fundamental solutions represent the normal deflection and Airy’s stress function for an unbounded shell under a normal point load. The differential operator for the hyperbolic paraboloid consists of the biharmonic operator minus the imaginary unit i times the second order wave operator. Unlike the case with the sphere and the cylinder, this operator cannot be factored, and this accounts for the relative complexity of the analysis. The fundamental solution is expressed as a Fourier series in the polar angle, and the radially dependent Fourier coefficients are represented by a contour integral. From this, explicit series and asymptotic ...

A numerical procedure is proposed for the solution of the biharmonic equation for deflections of uniformly loaded plates with all edges clamped. It is shown that the quantity ∇2w at the plate boundary can be determined numerically by noting a linear relationship between ∇2w and w in finite-difference form at the boundary. According to the classical methods of solution, the quantity ∇2w at the clamped edge cannot be determined in advance. The new approach is based on the boundary vorticity method for the vorticity transport equation governing a steady two-dimensional incompressible fluid flow. By using the new procedure, it is now possible to replace the biharmonic equation for clamped plates by two second-order partial differential equations of elliptic type similar to those for simply supported plates with straight edges. The possibility of extending the procedure to various problems in plates and shells is pointed out. The study also serves to illustrate a close relationship between a class of fluid mechanics problems and a class of plate and shell problems.

The discrete biharmonic equation on a rectangular region and the discrete Poisson equation on an irregular region can be treated as modifications to matrix problems with very special structure. We show how to use the direct method of matrix decomposition to formulate an effective numerical algorithm for these problems. For typical applications the operation count is $O(N^3 )$ for an $N \times N$ grid. Numerical comparisons with other techniques are included.

Several forms of “Successive Overrelaxation (SOR)” are applied to a coupled pair of difference equations, equivalent to the biharmonic equation. The analysis and results yield a partial comparison amongst the methods. Computational results are also included.

Stress analysis for a circular cylindrical shell having two circular cutouts of equal size is worked out. The following four cases are treated. A circular cylindrical shell having two circular cutouts of equal size (a) in a longitudinal direction under uniform tension, (b) in a circumferential direction under uniform tension, (c) in a longitudinal direction under internal pressure and (d) in a circumferential direction under internal pressure. The method of Green's general biharmonic analysis is extended to this problem and the boundary conditions along the edge of cutouts are satisfied by the point-matching method. Numerical solutions are obtained for three kinds of diameter of circular cutout and four kinds of center-distance of the cutouts. Especially, in the cases of (a) and (b), the results are compared with those of the flat plate.

Die Kelvin-Almansi-Darstellung einer Bipotentialfunktion vermittelst zweier harmonischer Funktionen wird zur Losung elastischer Warmespannungsprobleme fur stationare Temperaturfelder angewendet.

An analogue technique is described giving good approximate solutions of the biharmonic equation. The method uses two plane sheets of conducting paper linked by an array of resistors. The technique is applied to problems involving the slow flow of a viscous fluid past uniformly spaced parallel fibres. Calculated values of the Kozeny constant from the results show much better agreement with the practical and theoretical values than those derived from previous analogue solutions.

A solution is presented for the computation of the transient thermoelastic stresses in a hollow cylinder with temperature boundary conditions given as a circumferential variation of surface heat transfer coefficient. The temperature distribution is solved explicitly. The problem is set up using the Airy stress function which leads to the biharmonic equation. This approach requires the satisfaction of three Michell integrals at the inner boundary in order to ensure single-valued displacements and rotation. An iterative method is described in which these integrals are all simultaneously satisfied and thus provide the necessary non-zero boundary conditions for the solution of the biharmonic equation which is rapidly solved by Gaussian elimination. Results are presented for the general case where the temperature is a function of r and θ. The computer program is checked by assuming a constant value of the surface heat transfer coefficients. In this case a closed form solution is obtained.

The Kirchhoff assumption in thin elastic plates results in a biharmonic equation for the lateral deflection and a C 1 deflection field is therefore required in the finite element method for their approximate solution. By considering the thin plate as a three dimensional solid and by discarding the Kirchhoff assumption, the continuity requirement for the displacements is reduced to C 0 . The stiffness matrix produced in this way becomes, however, violently ill-conditioned as the thickness t of the structure is reduced. It is shown here that the factor 1/t 2 causing this ill-conditioning can be removed from the stiffness matrix and consequently from its condition number by relating the thickness t to the diameter of the element h, without losing the rate of convergence provided by the degree of the shape functions inside the element. This is used here to construct a well-conditioned 9-degrees-of-freedom plate bending element which is only C 0 but which converges quadratically to the C 1 solution (Kirchhoff solution) of thin plates. Addition of shear to C 1 elements is also considered.

This paper deals with the formulation of a Monte Carlo method for the numerical solution of a class of boundary value problems involving the homogeneous biharmonic equation. Isotropic random walk and a mean value relation have been used. In der vorliegenden Arbeit wird ein Monte-Carlo-Verfahren fur die numerische Losung zur homogenen biharmonischen Gleichung gehoriger Randwertaufgaben formuliert. Dabei werden das Modell einer isotropen Irrfahrt und eine Mittelwertrelation benutzt.

We deduce properties concerning singular curves of harmonic functions generated by the Bergman-Whittaker operator and obtain an analogue of the Eisenstein theorem for a certain class of such harmonic functions.

The “two-group” partial differential equations for reactors of axial symmetry, and with anisotropic neutron diffusion properties, are converted into finite difference equations. These are then represented by the current flow in resistance networks, the fast- and slow-neutron fluxes being represented by the network voltages. The resulting resistance-network analogue is similar to the “cascaded” network for the solution of the biharmonic equation, but does not involve some of the approximations made in this earlier technique. A convenient and fast measuring technique is described which is used in obtaining the correct solution by an iterative process of readjusting currents fed into the analogue. Experimental work has confirmed that this method leads to solutions in which the neutron fluxes can be determined to better than 1%, and the critical neutron multiplication constant k to better than 0·1% This work also sheds some light on the truncation error of the solution, and on the convergence rate of the iteration process. Design data for an (r,z) analogue with unequal meshes are given, and the technique for critical size determination is outlined.

The problem of finding thermal stresses and displacements in long cylinders is reduced to obtaining a particular integral for the Poisson equation ▿2V=kT, where T is the temperature distribution and k is a constant, and to solving the biharmonic equation ▿4U=0 when the first derivatives of the function U are known on the boundary of the cross-section of the body. These two differential equations are related by the conditions that on the boundary the first derivatives of U are given in terms of the first derivatives of V. The biharmonic equation is solved by use of two analytic functions [Muschelisvili, ‘ Bulletin de l'Academie des Sciences de Russie, IV Serie, Tome 13, 2nd partie, 1919; ‘ Mathematische Annalen,’ vol. cvii. 1933]. The region within which the solution is-desired is mapped conformally on the unit circle by a rational function, and from the boundary values on the unit circle the required analytic functions are determined directly by the use of the Cauchy integral formula. The methods...

Plane-stress solutions are reported for perforated plates with triangular hole patterns under uniform edge loading, hollow hexagons with internal pressure, and a hollow square in uniform shear. Pertinent features are described of a specially developed computer program based on the approximate solution of the biharmonic equation in polar coordinates by the point-matching method. The program is tailored to the solution of two-dimensional elasticity problems that involve solid or perforated structures with boundary conditions in terms of stresses and displacements.

The parameter differentiation method of Joseph is applied to the title problem. General forms of the operator derivatives suitable to the biharmonic eigenvalue problem are found. Applications to the simply-supported parallelogram plate and to the clamped elliptical plate are given, and comparisons are made with other exact and approximate calculations.

A stress function theory for linearly viscoelastic solids is developed in this paper. If body forces and inertial forces are neglected, the stresses, strains, and displacements are biharmonic functions of the space variables. The displacements are expressed in terms of harmonic functions by an extension of the Neuber‐Papkovitch theory. Axially symmetric stresses and strains are given in terms of a single biharmonic function. As the analysis does not involve boundary conditions, it is valid for problems involving moving leads and moving boundaries. The theory may be used to formulate stress analysis problems in terms of Volterra integral equations, using creep or relaxation functions as the kernel. The integral equations governing the stresses in a viscoelastic propellant grain subjected to the combined effects of internal pressure and erosion are derived.

Summary A rectangular slab spans between simple supports and is stiffened by beams of uniform cross-section (Figure 1). Complete composite action is assumed to exist between the slab and the beams. Each panel of the slab can carry point loads, acting at any point of the panel, and a uniformly distributed transverse load, and the beams carry loads which are either uniformly distributed along their lengths or are concentrated so as to act as point loads. The magnitudes of the loads carried by each panel and by each beam will generally not be the same. It is assumed that the transverse deflexion of the structure and the stress function that describes forces in the plane of the slab are governed by independent biharmonic equations. To these equations series solutions are taken and the coefficients which they contain are determined by satisfying the conditions of compatibility and continuity at the slab-beam junctions. The results of experiments with models are described and brief details are given of computer programmes which may be used to carry out the computations involved.

Setting the Reynolds number equal to zero, in a method for solving the Navier-Stokes equations numerically, results in a fast numerical method for biharmonic problems. The equation is treated as a system of two second order equations and a simple smoothing process is essential for convergence. An application is made to a crack-type problem.

Saint Venant’s principle is formulated as a biharmonic eigenvalue problem for the symmetrical truncated wedge x ≥ 0, |y| ≤ yb (yb = 1 + x tan ω) which is stress-free along the lateral edges, and is loaded by self-equilibrating shear and normal tractions τko(y), σko(y) along edge x = 0. It is found that for the wedge angle 2ω = 0 the law of decay is of the type e−αkτ; for 2ω ≠ 0 the law of decay is of type r−μk; μk is minimum for 2ω = π. The indexes k attached to τo, σo indicate that we deal with systems of characteristic tractions which produce characteristic decay rates. Practical implications of the results, as they apply to structural design, are discussed in a separate paper, “Some Aspects of Saint Venant’s Principle.”

An integral equation method is developed for the analysis of thin elastic plates. The integral equations are based on a Green's formula and are adapted for the solution of the biharmonic problem of plate analysis. By means of this method, a numerical solution is obtained for a simply-supported, uniformly loaded corner plate. Deflections and bending moments along the diagonal and shear forces along the edges are found. Values of these quantities previously calculated by means of a finite difference technique for a right-angled plate are confirmed by this new method and the analysis is extended to corner plates with different angles at the corner. Furthermore, for a right-angled corner plate, detailed contour plots of bending and twisting moments inside the plate are presented. With a digital computer these results are obtained without undue difficulty, and it appears that the integral equation method is well suited to this type of problem.

Maximum and minimum principles for plate bending problems where the boundary is part clamped and part simply supported are derived in a unified manner from the canonical theory of complementary variational principles. Two new error bounds for approximate variational solutions are also presented.

The stress developed in an elastic cylinder of finite length undergoing thermal expansion with one end clamped is expressed in terms of a series expansion of a biharmonic function, appropriate derivatives of which give the displazements and stresses within the cylinder. The coefficients in this series are determined by a least-squares fit to the boundary conditions at the ends of the cylinder and values of the stress on various surfaces are found as functions of the height-to-radius ratio. All components of the stress tensor become infinite at the circumference on the clamped end. A tabulation is included of quantities of interest in any cylindrical problem in which the curved surface is a free surface.

This note concerns the maximum principle which applies to solutions of partial differential equations of elliptic type. This principle asserts that the maximum of a solution occurs on the boundary of a region. Consideration of the ratio of solutions of an elliptic equation shows that the ratio satisfies the same maximum principle. This result is then used to obtain a maximum principle relating to biharmonic functions. These maximum principles give inequalities which biharmonic functions must satisfy. The relations and concepts developed in this note have application in elasticity and in hydrodynamics.