Page 321 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

308 Introduction to the Finite Elennent Method Chap. 10

be expressed by the following matrix equation:
’
i’l ] ~\ 0 1 0 O 'P\'
Ol IP\'
1
le, 0 1 0 0 n P i
1

>-- — 1 ■ \ (10.2-4)
T 11 T ” T
1 ' ' -0 11 2 3-
\ie^\
With the matrix partitioned as shown, it is evident that p, and /?2 related to
and /0j by a unit matrix. After substituting pj = ¿ j and P2 1 9 easily
solve the last two rows of the matrix for and p^. The desired inverse of Eq.
(10.2-4) then becomes
' P\ ' 1 0 1 0
I
Pi 0 1 1 0 19,
-------. (10.2-5)
P?, ~ -Y ■2 I - T
2 1 I 1 J 19,
This equation enables the determination of the p- for each of the displace­
ments equated to unity with all the others equal to zero. That is, for ¿ ^(x) = 1
with all other displacements equal to zero, the first column of Eq. (10.2-5) gives

Pi = h Pi = 0, P3 = -3 , and ^4 2
Substituting these into Eq. (10.2-2) gives the shape function for the first configura­
tion of Fig. 10.2-2 of
<P,(.V) = 1 - 3^2 + 2 e
Similarly, the second column corresponding to = 1 gives
P\ ^ 0 ’ Pi"" P?>"" “ 2/, and p^ = /

and
^^(x) = i ^ - 2 i e + i e
The other two (p^{x) are obtained in a similar manner. In summary, we have the
following for the four beam shape functions:

^,{x) = i ^ - 2 i e + i e
( 10.2-6)
<p,{x) =

Generalized mass and generalized stiffness. By considering the dis­
placement in general to be the superposition of the four shape functions shown in