RESEARCH NOTES

Theoretical basis of boundary solutions for the linear theory of structures

Artur Portela

LNEC, 101 Av do Brasil, I799 Lisbon Codex, Portugal [Received March 1980; revised September 1980)

Weigh ted residuals Theorem of virtual work

Let A be a domain and S its boundary subdivided into two portions S1 and &. Consider a differential equation:

LU’pg=O inA (1)

with the boundary conditions:

Cu’-h=O on S1 (2)

Du’-l =0 on SZ (3)

where U’ is-the exact solution, L, C and D are differential operators and g, h and 1 are prescribed functions.

As the set of differential equations (l)-(3) has to be zero at each point of the domain A and its boundary, it follows that:

Let A be a domain and S its boundary. Consider a symmetric stress field, ab, which is assumed

equilibrated by a system of surface and body forces with densities 0 and f respectively on S and in A. For the sake of simplicity, the stress field u$ is assumed to admit first derivatives everywhere in A. Therefore, ub belongs to the class of equilibrated elastic fields and satisfies the equations:

I (Lu’-g) WA dA +

s (Cu’- h) Ws, dS

A S,

+ (Du’-Z)WSIdS=O s

(4)

82

a:i,itfi=O inA (9)

0; = +Q = iji on S (10)

where Izi is the ith component of the normal unit vector n outwardly directed, of the boundary S.

Let Oij be an approximating function, Oii z Ob, SO that the residuals are defined as:

where WA, Ws and Ws are arbitrary weighting functions, such that the integrals ?n (4) are capable of being evaluated.

Statement (4) is the starting point for the weighted residual method.

It is much easier to determine an approximate solution u, u = u’, by means of numerical methods, than to find the exact solution u’.

Substituting U’ by u in (l), (2) and (3), one obtains:

EA=LU-g#O inA (5)

Es, = Cu-h#O on S1 (6)

Es2=Du-I#0 on S2 (7)

where EA, Es and Es, are the residuals, i.e. the approxi- mating error f unctions.

The weighted residual method minimizes such approxi- mating error functions, distributing them over the domain A and its boundary S, according to some weighting func- tions WA, W,, and WQ hence leading to the well known different approximatmg methods.

Therefore, equation (4) becomes:

EAi=Gii,iffi#O in A (11)

Esj = Oj - Oj # 0 on S (12)

The weighted residual statement is:

/EAiWAj tkSjWSi dS= O (13)

A S

or

I (Uij,i +fi) WAj dA t

s (Uj - Oj) Wsj dS = 0 (14)

A S

where WAN and Wsi are arbitrary weighting functions. Integrating equation (14) by parts leads to:

sf$WAj dA-So0WAj.i dA +SW,i0h?Zi dS

A A S

t s

(ui-Oi) Wsi dS=O

s

f E,WadA+

s Es,W,y, dS+

s ESzW~2dS=0 (8)

A Sl S,

which is the general weighted residual equation.

Now, consider in A a strain field E$ generated by a displacement field II* on S. For simplicity, the displacement field ~7 is assumed to admit first derivatives everywhere in A. Therefore, ~7 belongs to the class of compatible elastic fields and the following equations hold:

E$ = f(UTj t Uif: i) in A (16)

ui* = tii* on S on S (17)

(15)

0307-904X/81/010057-03/$02.00 0 1981 IPC Business Press Appt. Math. Modelling, 1981, Vol 5, February 57

Research Note

Note that equation (16) implies the assumption of geo- metrical linearity.

Now, defining the arbitrary weighting functions as:

Waj = lli* in A (18)

wsj = -ui* on S (19)

and introducing (18) and (19) into (15) we obtain:

~~~; dA~~~~j~~idA+J’~~oidS=O (20)

A A s

i and j are repeated subscripts. Therefore: * * *

Oijllj,i = OjiUi,j = OijUi,j (21)

Hence :

UijU,~i = ~(OijtUji)U,~i = fUijU,~j t ~ OjiUiTi = Uij~i::

(22)

and, thus, (20) becomes:

Equation (23) states the relation between a compatible elastic field and an equilibrated elastic one, both defined in A with boundary S. This relation is nothing other than the theorem of virtual work.

Some observations can be made on the virtual work theorem:

(i) The stress field Uij and the strain field E$ do not depend on each other. This is a natural consequence of the way they have been introduced in the weighted residual equation. (ii) The equilibrated field Oij is statically undetermined. Consequently, aij is any stress field that satisfies equations (9) and (10). (iii) The work principle replaces either the equilibrium equations or the strain-displacement equations. This means that the equilibrium equations can be deduced from the virtual work equations and the strain-displacement equa- tions, and the strain-displacement equations from the equilibrium equations and the virtual work equations. This can be found elsewhere.’

Generalization of the virtual work theorem to exact fields

When establishing the theorem of virtual work we con- sidered two independent fields, one compatible and the other one equilibrated. Now we shah relate two independent exact structural fields, i.e. each one of the fields is simul- taneously compatible and equilibrated.

Let A be a domain and S its boundary subdivided in two portions Sr and S2.

Consider an exact field which is assumed equilibrated by a system of surface and body forces with densities 6 and f respectively on Sz and in A, and compatible with the displacements U prescribed on Sr. Therefore, the structural field belongs to the class of exact fields and satisfies the equations:

Ut,i t fi= 0 inA (24)

ejj = +(Uf,j + Uj,i) in A (25)

0ij = CijklEk[ in A (26)

u; = uj on Sr (27)

u; = u;j$ = aj on Sz (28)

where Cij,, denotes the elastic constants of the material, which is anisotropic in general.

Let uij and uj be approximating functions, Uij x Uk and uj = u,!, so that the residuals are defined as:

EAj=Uij,itfj#O inA (29)

Es,j=Uj -Uj#O on Sr (30)

Es,j = Oj ~ Gi # 0 on Sz (31)

The weighted residual equation is:

I (Uij,i tfi) WAj dA +

s (Uj-Uj) Ws,j dS

A s, .

t (Uj~Oj)dS=O I

(32)

s*

where WA, Ws, and Ws, are arbitrary weighting functions. Integrating by parts equation (32) we obtain:

s fjWAj dA ~

c aijWAj,i dA t

I WAjuijni dS

A a s

t s (Uj ~ aj) WS, dS +

s (Uj -Uj) WS, dS = 0

8, S, (33)

Now, consider another exact field which satisfies the equations:

U$,i +fi=O in A (34)

Ei:l= i(Uzj t UiTi) in A (35)

U,$ = Cijk-Ek*l inA (36) vi* = iii* on Sr (37)

*_ * -* aj - Uijni = Oj on Sz (38)

Note that equations (25) and (35) imply the assumption of geometrical linearity.

Defining two of the arbitrary weighting functions as:

WAj =UT in A (39)

Ws, j = pUi* on Sz (40)

and introducing (39) and (40) into (33) we obtain:

AGUE dA- S~~~U~i dA t~~~~j dS

A

i’ I

S,

+ UT6jdSt (Uj-Uj) WsljdS=O (41)

S, 8,

Therefore, by virtue of the statements (2 1) and (22) we can write:

* * OijUj, i = Uijfij (42)

Assuming physical linearity, i.e. assuming that strain energy density is a quadratic function of the strains, we can write :

* * UijEij = UijFij (43)

Such is the internal form of Betti’s reciprocity theorem.

58 Appl. Math. Modelling, 1981, Vol 5, February

Research Note

In A and on S the exact field (24)-(28) is defined. The fundamental structural field is defined in A,, i.e. a

structural field satisfying the equations:

I& = 61(x -xi) in A, (50)

e; = S(U& + L&) in A, (51)

el$ = Cijkf Elt in A, (52)

where sl(x -Xi) is the Dirac delta generalized function defined at a point Xi in the I direction. Functions a$, E$ and ulF are the values of the field observed at a point x, due to a unit load acting at xi in the I direction. These functions are the Kelvin solution (empty-space) or the Mindlin solu- tion (half-space).

Now, equation (48) can be rewritten as:

s, s,

By virtue of the selective properties of the Dirac delta function, we can write for any point xi in the domain A:

Introducing (42) and (43) into (41) we obtain:

If;U~dA-jO;FjjdAtSiii*“idS+~U~~~dS

A A S, S,

+ s

(UjpUj) It!sljdS=O (44)

1

By virtue of (21) and of geometrical linearity:

$ 11 11 ‘! U*E.. dA = U$i(Ui,i + Uj,i) dA = s

U$Uj,i dA (45)

A A A

Therefore, introducing (45) into (44) and integrating by parts once more, we obtain:

bU7 dA +SUt,iUj dA -bjU,? dS

A A S

+ (Ui-iii) WS,i dS t s

UTUj dS

S, S,

+ t&dS=O s

(46)

By virtue of the arbitrariness of the weighting functions, we can define:

Ws,j=Ui* on S, (47)

Introducing (47) into (46) we find that:

U$,iUi dA + SJJ j-1 J s f.U* dA t U*O. dS t UTCj dS

;\ A

=~~~j dS t~~~~j dS (48)

S, S2

Now, considering (34) equation (48) becomes:

our dA t~~Uj dS+ju;oi dS

A

which expresses the relation between two independent exact structural fields defined in A with boundary S=Sr t SZ. Such relation is in fact the external form of Betti’s reci- procity theorem, and is itself a boundary integral equation which can be used as the starting point for boundary solutions.

Extended generalization of the work theorem: fundamental structural field and boundary integral equation

When stating equation (48) we considered two distinct exact structural fields in the same domain A with boundary S. Now we relate the two exact fields in the case where one of the fields is defined in an infinite domain and the other field is defined in a finite domain.

Hence, let A_ be an infmite domain which contains the domain A and its boundary S subdivided in two portions S, and &.

-U~*,),+JU~u,dS+JU~UjdS

S, S,

=~Up~j dS t~U~~j dSt~U;lf, dA (54)

8, S, A

or, in a compact way:

-Ucxj)l +kzUj dS =/U;ej dS +/U$fi dA (55)

S S A

then applying boundary conditions iii on Sr and Oi on SZ. The boundary values UJ and u$ are the tractions and the

displacements in the j direction due to a unit load acting at xi in the I direction.

When the point xi is on the boundary, equation (55) takes the form:

-CC~~)U(,~)/ + UjlUj dS = UjlUj dS + J-* s* I

Uzfi dA

S S A

(56)

and is a boundary integral equation,2 which, from a struc- tural point of view, involves two independent exact fields and itself generalizes the work theorem.

Discretization of equation (55) will not be discussed here and can be found elsewhere.2, 3

References

Oliveira, E. R. A. ‘Advanced theory of structures’, CEB Inter- national Course on Structural Concrete, LNEC, Lisbon, Portugal, 1973 Brebbia, C. A. ‘The boundary element method for engineers’, Pentech Press, London, 1978 Portela, A. and Romaozinho, T. ‘Topics on the boundary element method’, Portuguese edition of LNEC, Lisbon, Portugal, 1979

Appl. Math. Modelling, 1981, Vol 5, February 59