Direct approach based on Betty's reciprocal theorem is employed to obtain a general formulation of Kirchhoff plate bending problems in terms of the boundary integral equation (BIE) method. For spatial discretization a collocation method with linear boundary elements (BEs) is adopted. Analytical formulas for regular and divergent integrals calculation are presented. Numerical calculations that illustrate effectiveness of the proposed approach have been done.

The BIE method and its numerical realization BEM are powerful tools for analysis of the wide range problems in mechanics and engineering [

During the last decade, the BEM has been established as a robust numerical method for solution of the elastic thin plate problems [

Another problem which gives many troubles in application of the BEM to the Kirchhoff’s plates bending is the divergent integrals with singularities of the following type: weakly singular

In this paper, we present a general direct BEM approach for thin Kirchhoff’s plate bending problems. For evaluation of the regular integrals we use analytical formulas which give accurate results and need less time for calculation. For divergent integrals regularization we use special method developed in [

Let us consider thin elastic plate with thickness

Bending of a thin plate.

We consider element that cuts out from the plate by two pairs of planes which are parallel to the planes

Elements of the bending plate.

From (

The plate deflection is related to other parameters of the plate by relations

Stress-strain state of the plate is defined by stress and strain tensors. Due to Kirchhoff’s hypothesis they are related to the deflection by the equations

Moments and shear force are expressed through stress tensor in the form

Generalized shear force may be introduced by the relation

Correct statement of the plate bending problem supposes that (

For arbitrary part of the boundary we have (see Figure

Arbitrary part of the boundary.

In the case if boundary contains corner points, corner forces

Action of the generalized shear force on the corner pint.

Under the Kirchhoff assumptions boundary conditions can be defined as follows:

Each of the boundary conditions (

The boundary-value problems (

Let us transform (

In the same way can be transformed integral

With taking into account representations (

From Betty’s reciprocal relation (

Taking into account that Dirac’s delta function

The slope of the element perpendicular to the middle surface of the plate

Further in order to construct BIE limit transition to the boundary

For convenience and compactness of the BIE consideration we introduce the following contour and area potentials:

Now with taking into account (

Fundamental solution

Solution of (

Fundamental solutions

The fundamental solutions

Fundamental solutions

Finally the fundamental solutions

Analysis of (

For

In order to get integral representations for the deflection of the plate

For any boundary-value problem (

One has

This system of the BIE is the system of Fredholm integral equations of the first kind with smooth kernels. Such integral equations correspond the so-called ill-posed problems [

One has

This system of the BIE consists of the Fredholm integral equation of the first kind with smooth kernels and singular integral equation. Specific features of the Fredholm integral equation solution have been considered above. Singular integral equations can be solved using technique developed in [

One has

Integral operations in this system of integral equations contain kernels with different singularities. Integrals with logarithmic singularity

In each specific case corresponding system of BIE can be easily constructed based on (

The BEM can be treated as the approximate method for the BIE solution, which includes approximation of the functions and the domain where they are defined by discrete finite dimensional model. Let us construct discrete model the plate boundary

We shall divide the boundary

On each FE we introduce a local coordinate system

Having constructed BE model of the area

In general case BEs can be of different shape and size with different shape functions defined on them. The simplest are linear BE with piecewise constant shape functions as it is shown in Figure

Linear BE.

For linear BE and piecewise constant shape functions corresponding integrals can be calculated analytically. It can significantly simplify calculations, reduce time, and increase accuracy and stability of the calculation process. Advantages of application curvilinear BE and high-order shape functions consist in more accurate approximation of the boundary and functions, but it leads to complication of calculations. Some time that circumstance can devalue the above-mentioned advantages. For more information regarding advantages and disadvantages of different BEs and function approximation refer to [

Let us divide boundary

In order to calculate integrals over domain in (

Discretization of the domain

Therefore, finite dimensional system of the BEM equations has the form

System of (

For any specific external load and boundary conditions from system (

There are two approaches to calculation of integrals (

Numerically integrals over boundary elements are usually calculated using the Gaussian quadrature formulas [

The Gaussian quadrature formulas can be effectively applied for calculation of the integrals without singularities, in the case of integrals (

Let us introduce the system of coordinates that related to the BE with number

In the above mentioned system of coordinates, the local coordinates of the points

Therefore, fundamental solutions on the singular boundary element have the form [

Integrals in (

In (

In the system of coordinates presented in Figure

In the BIE can appear also domain integrals (

In order to calculate integrals over domain

We have considered here some benchmark examples that correspond to bending of the thin plates of different shape. In all examples the ratio

First let us consider circular simply supported over-the-contour plate that subjected to action the uniformly distributed over the domain

Expressions for

Dependence of the boundary data

Analysis of the data presented in Figure

In order to compare traditional and proposed here analytical approach, evaluation of integrals (

We consider here some benchmark examples for rectangular plate with different boundary conditions. In the first example plate was loaded by concentrated force applied to the point in its center. In all other examples the plate was loaded by uniformly distributed over the domain

Loading pattern | Number of BE | Points | Boundary conditions | |||||

20 | 1 | — | 0.947 | — | 0.813 | 3.291 | 0.845 | |

36 | 1 | — | 0.987 | — | 0.977 | 4.224 | 0.938 | |

44 | 1 | — | 0.994 | — | 0.991 | 4.657 | 0.955 |

Loading pattern | Number of BE | Points | Boundary conditions | |||||

20 | 1 | — | 0.940 | — | 0.849 | 3.831 | 0.950 | |

36 | 1 | — | 0.985 | — | 0.980 | 3.952 | 0.983 | |

44 | 1 | — | 0.993 | — | 0.993 | 4.154 | 0.988 |

Loading pattern | Number of BE | Points | Boundary conditions | |||||

20 | 1 | — | — | 0.956 | 0.997 | 4.251 | 0.995 | |

36 | 1 | — | — | 0.990 | 0.997 | 5.462 | 0.996 | |

44 | 1 | — | — | 0.995 | 0.999 | 5.564 | 0.998 |

Loading pattern | Number of BE | Points | Boundary conditions | |||||

20 | 1 | — | 0.935 | — | 0.997 | 2.631 | 0.969 | |

2 | — | — | 0.945 | 0.968 | ||||

36 | 1 | — | 0.985 | — | 0.990 | 4.529 | 0.989 | |

2 | — | — | 0.987 | 0.997 | ||||

44 | 1 | — | 0.993 | — | 0.992 | 4.761 | 0.992 | |

2 | — | — | 0.994 | 1.000 |

Loading pattern | Number of BE | Points | Boundary conditions | |||||

20 | 1 | — | — | 0.945 | 0.967 | 3.671 | 0.970 | |

2 | — | 0.937 | — | 0.926 | ||||

36 | 1 | — | — | 0.987 | 0.998 | 4.239 | 0.999 | |

2 | — | 0.985 | — | 0.995 | ||||

44 | 1 | — | — | 0.994 | 0.999 | 4.281 | 0.996 | |

2 | — | 0.993 | — | 0.993 |

Loading pattern | Number of BE | Points | Boundary conditions | |||||

20 | 1 | 0.947 | 0.995 | — | — | 4.171 | 0.936 | |

2 | — | 0.937 | — | 0.817 | ||||

36 | 1 | 0.980 | 0.998 | — | — | 4.219 | 0.976 | |

2 | — | 0.984 | — | 0.973 | ||||

44 | 1 | 0.993 | 0.999 | — | — | 4.461 | 0.983 | |

2 | — | 0.992 | — | 0.995 |

Columns 4–7 of Tables

Integrals in (

In order to visualize results from Tables

Dependence of the boundary data and time of calculations on number of the BE.

Dependence of the boundary data and time of calculations on number of the BE.

Dependence of the boundary data and time of calculations on number of the BE.

Dependence of the boundary data and time of calculations on number of the BE.

Dependence of the boundary data and time of calculations on number of the BE.

Dependence of the boundary data and time of calculations on number of the BE.

The diagram of the T-shaped plate together with boundary conditions is presented in Figure

T-shaped plate.

In Figure

From this data follows that results obtained by the BEM and by the FEM are in a good agreement but time of calculation by the BEM is significantly less.

Direct BIEM based on Betty’s theorem is applied here for solution of thin elastic plate bending problems for different boundary conditions and load. Analytical integration of the regular and divergent integrals over the BE is applied, and effective formulas for calculation of coefficients of the system of linear algebraic equations for the BEM have been developed. The main advantage of the proposed approach consists in significant reduction of the calculation time comparison with traditional approach based on Gaussian’s quadratures. Numerical examples demonstrate effectiveness of the proposed here approach. In all presented examples it was demonstrated high accuracy, in good agreement with existing analytical solutions and significant reduction of the time of calculations in comparison with traditional approaches.

The author is very grateful to his former Ph.D. student Dr. Alexander Lukin from Kharkov State University for help in this paper preparation.