Transactions on Modelling and Simulation vol 19, © 1998 WIT Press, www.witpress.com, ISSN 1743-355X 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
       The Dirac delta function and its use in plate
 
       bending problems under generalized loading
 
       A. El-Zafrany
 
       School of Mechanical Engineering, Cranfield University,
 
 
 
       Email: a. el-zafrany @ cranfield. ac. uk
 
 
 
 
       Abstract
 
 
       In this paper, after a brief introduction on the Dirac delta function and its
       interesting properties, some of its applications to the boundary element analysis
 
       of thick plates in bending are introduced. Modelling of concentrated shear forces
       and bending moments in boundary integral equations is firstly discussed. Then
 
       the treatment of domain loading terms for plate bending problems under line
       loading is explained, where line shear forces and bending moments are assumed
 
       acting on arbitrary curves defined on the plate midplane. Some numerical
       examples are demonstrated and the results were compared with corresponding
 
       analytical solutions. It is clear that, with the use of Dirac delta functions, the
       boundary element analysis of thick plates in bending, with general cases of
 
       loading, has been simplified, and very accurate results were obtained.
 
 
       1 Introduction
 
 
       The delta function was introduced by Dirac' as a quantity b(x)
 
       depending on a parameter jc, and satisfying the conditions:
 
 
 
 
 
 
 
 
 
 
 
 
 
 
                                                           
 
 
                                                                                  
 
                                                                      
 
                                                                                  
 
 
 
 
 
 
                                                              
                                                                                  
                                                                                  
                                                                                  
 
 
     
                                                                                 
                                                                         
                                                        
                                   
                   
                                   
 
 
                                                             Transactions on Modelling and Simulation vol 19, © 1998 WIT Press, www.witpress.com, ISSN 1743-355X 
 
 
       244 Boundary Element Research In Europe
        oo
       f&(x)dx = l, 6(a) =0 for x*Q (1)
 
       — oo
 
       with the following interesting property:
        oo
 
       ff(x)&(x-a)dx=f(a) (2)
       — oo
 
       Dirac considered that 6(x) is not a function of x according to the usual
 
       mathematical definition of a function, and he preferred for it to be
 
       classified as an improper function. This inspired mathematicians to use the
       term Generalized Functions for the 8 function and similar other functions,
 
       and Schwartz^ systemized the theory of such generalized functions. A
 
       simple presentation of their theory is also given by Hoskins.^
          The derivatives of delta functions are also treated as generalized
 
       functions. Although such derivatives are defined by means of an integral
 
       expression,^ it is acceptable to define the derivatives using first principles
       or limit rules. Jaswon & Symnf emphasized the fact that the use of delta
 
       functions provides a quick approach to results, specially boundary integral
 
       equation derivations, which could otherwise be obtained by limiting
       processes of classical analysis. El-Zafrany^ has simplified the definition
 
       of Dirac delta functions in multiple-dimensional space, and allocated
       physical or geometrical meanings to two-dimensional, three-dimensional,
 
       and axisymmetric delta functions.
 
          A Dirac delta function may be defined in a ^-dimensional space, in
       terms of one-dimensional delta functions, as follows:
 
 
 
 
 
 
       where x = ix^ x^ "' xA
       which represents a vector of Cartesian coordinates. If / is a finite
 
       function defined within a d-dimensional domain A, then:
 
                        = /(*,) (4)
 
       Q
       where *, is inside the domain £1 If the previous function has also finite
 
       partial derivatives up to the mth order, then it can be shown that:
 
 
 
 
 
 
 
 
 
 
 
                                                           
 
 
                                                                                  
 
                                                                      
 
                                                                                  
 
 
 
 
 
 
                                                             
                                                                                  
                                                                                  
                                                                                  
 
 
    
                                                                                 
                                                                        
                                                        
                                  
                   
                                  
 
 
                                                             Transactions on Modelling and Simulation vol 19, © 1998 WIT Press, www.witpress.com, ISSN 1743-355X 
 
                           Boundary Element Research In Europe 245
 
        /»    3tn &/*._*. \
                                      *"**> (5)
                I -\ tn-I                   m-l
 
 
 
       where l<,m.
           In this paper, the use of Dirac delta function in modelling generalized
 
       loading cases, and their applications to boundary element analysis of thick
 
       plates in bending are demonstrated. Modelling of concentrated shear
       forces and bending moments in boundary integral equations is firstly
 
       discussed. Then the treatment of domain loading terms for plate bending
 
       problems under line loading is explained, where shear forces and bending
       moments are assumed acting on arbitrary curves defined on the plate
 
       midplane.
 
 
       2 Plate bending example
 
 
       The boundary integral equations for thick plates in bending can be derived
 
       and expressed in the following fornf:
 
 
 
 
                                                                  (6)
 
 
 
 
                              +
                                r
                                                                  (7)
 
 
 
 
         i(6J, + (32(6,), + ^33^, + f (7i3 8. + T^Q, + 7-33^) dT
 
                                                                  (8)
 
 
 
 
       and the kernel functions in the domain loading integrals can be simplified
       as follows®:
 
 
                      * V|^2-, L, =11 - ^-*' V*|»' (9)
                  (l-v)A^ j a%«' ' [ (l-v)X
 
 
 
 
 
 
 
 
 
 
 
                                                           
 
 
                                                                                  
 
                                                                      
 
                                                                                  
 
 
 
 
 
 
                                                              
                                                                                  
                                                                                  
                                                                                  
 
 
     
                                                                                 
                                                                         
                                                        
                                   
                   
                                   
 
 
                                                             Transactions on Modelling and Simulation vol 19, © 1998 WIT Press, www.witpress.com, ISSN 1743-355X 
       246 Boundary Element Research In Europe
 
 
       where 4>* = — (logz-1), V*4>* = M^ (10)
                    on D 2 7t D
 
       and z = Ar, (^
 
 
 
       3 Case of concentrated forces and moments
 
 
       3.1 Representation in terms of domain loading intensity
 
 
       Concentrated forces and moments may be represented in terms of
       distributions over the domain, using Dirac delta functions, as explained
 
       next.
 
 
       (a) Case of a concentrated force
 
       Consider the case where a concentrated force F in the z direction is acting
       on a plate at midplane point (*,, y,) . Defining a small circle of radius e
 
       centred at (x^y^, as shown in Figure 1, then an equivalent loading
 
       intensity can be defined as follows:
                                               p
       q(x,y) = 0 for r > 0, q(x,y) = lim — - for r - 0
                                          €-0 7t€^
 
 
       where r = J(x -xrf + (y -yrf .
 
 
       Hence, it can be deduced from the definition of two-dimensional Dirac
       delta functions* that:
 
 
 
 
       (b) Case of a concentrated moment
 
       A concentrated moment Ty in the y direction, acting at (*,,?/), can be
 
       represented by a couple of two forces, as shown in Figure 2, where:
 
 
                         T
       FZ = -FI = lim -±
         *
       Using the previous analysis, the loading intensity equivalent to every force
 
       can be expressed as follows:
            x „ ., \ ,- r «(*-* y-y,)
        q(FJ = Fi6(*-jc,,y-yj) = - lim -^-
                                                AJC