128 ELASTOSTATICS

5.5 MATRIX FORMULATION

We can now work with matrices instead of using the indicial notation.

In order to do so we can define u as the displacement, p as the traction

and b as the body force vectors, such that

(5.34)

At a point i these displacements are called u

1

. In addition we can

define the following two matrices:

u =

«1

"2

«3

·. P =

Pi

Pi

.Ps.

, b =

<

*i ;

^2

Λ j

u =

«Tl

«21

«?1

«12

«?2

"32

«ί

3

"

«23

«Ϊ3

, P* =

"pTi

P*I

_P!I

ph

P*2

P*2

phi

P*3

P33J

(5.35)

where the u

lk

and p

lk

are the displacements and forces in the k

directions due to a unit force at the point

1

under consideration, acting

in the / direction.

Let us

first

write formula (5.27) before any boundary conditions are

applied, i.e.

cM+ pr

k

u

k

dr= ur

kPk

dr+ uf

k

b

k

aQ (5.36)

Jr Jr Jß

This equation can now be expressed in matrix form as follows,

c

f

Ui+

p*udr= u*pdr+ u*bdß (5.37)

This formulation is valid for a point i. Note that p*, u* and b are

known and c

t

will soon be determined. The unknowns are the values

of u and p over the boundary.

BOUNDARY ELEMENTS

We will now assume that the boundary is divided into elements. These

boundary elements can be constant, linear, quadratic or higher order

and they can be triangles or quadrilaterals. Figure

5.3(a)

shows the

ELASTOSTATICS 129

body divided into a series of constant elements. For these elements the

unknowns are assumed to be at the centre of the element and to have a

constant value over the element. The elements of Figure

5.3(b)

are

such that the functions vary linearly over them. Finally in Figure

5.3(c)

the variation of displacements and stresses over each element is

quadratic. In addition the geometry of the element is described by a

quadratic function.

Triangles

Nodes

Boundary

elements

Triangles

Nodes

b)

Elements

Triangles

Nodes

( c )

Elements

Figure

5.3

Three-dimensional

bodies divided into

elements:

(a)

constant;

(b)

linear;

and

(c) quadratic

130 ELASTOSTATICS

Over each element the functions for u and p can be approximated

using the following interpolation functions:

U = 0

T

U

n

=

Φ

Ί

p=

Ψ

τ

ρ

η

=

Φ

1

Φ

Ί

u

n

(5.38)

P

n

(5.39)

Note that the functions for u and p are generally different. For

consistence it may be better to take the functions for p of one order

less than those for u. The interpolation functions are the ones

discussed in Chapter 3. u

n

, p

n

are the nodal displacements and

tractions.

We can substitute the above functions into (5.37) and obtain the

following equation for the point i:

c,u,+ Σ (| P*4>

T

drju

n

= Σ (f

u*y

T

drV

M Λ

+

Σ U*

j=*

JO™

+

bdß (5.40)

where the summation from j = 1 to N indicates summation over the

N elements on the surface and Γ, is the surface of the) element. The

summation from; = 1 to M is carried out over the internal cells and

Q

m

is the volume of each of them.

The integrals in equation (5.40) are usually solved numerically and

the functions Ψ and φ are expressed in some of the homogeneous

system of coordinates described in Chapter

3.

The integrals need then

to be written in terms of the homogeneous system, to be called ξ, η.

Hence

dr = (absolute value of |G|) άξάη

(5.41)

where G has been defined previously.

Due to the difficulties in integrating equation (5.40) it is usual to

Start Free Trial

No credit card required