3. ‘‘Exact’’ Element Stiffness Matrix:
Natural Approach
In this paper, the “exact” element stiffness matrix is derived
simply by inverting the “exact” element flexibility equation.
This is feasible since the system does not experience any
rigid-body motion (neither rigid-body translation nor rigid-
body rotation) due to the presence of supporting foundation.
Thus, the exact element flexibility matrix is at the core of
the element formulation and requires the “exact”moment
interpolation functions. The analytical solution to the sixth-
order governing differential compatibility equation of ( 20 )is
central to obtain the exact moment interpolation functions.
For the sake of simplicity, the applied distributed load푝푦(푥)
is assumed to be uniform along the whole length of the
beam. Thus, only homogeneous solution is required to derive
the exact moment interpolation functions. This merit comes
from the fact that the force terms on the right-hand side
of ( 20 ) disappear as long as푝푦(푥)varies uniformly along
the whole length of the beam, thus rendering the proposed
flexibility-based model attractive and unique.
Thanks to the comprehensive investigation performed by
Morfidis [ 34 ] and Avramidis and Morfidis [ 19 ]onallpossible
solutions to the similar sixth-order differential equation, the
general solution of ( 20 )canbewrittenas
푀(푥)=휑 1 (푥)푐 1 +휑 2 (푥)푐 2 +휑 3 (푥)푐 3 +휑 4 (푥)푐 4
+휑 5 (푥)푐 5 +휑 6 (푥)푐 6 ,
(22)
where휑 1 (푥),휑 2 (푥),휑 3 (푥),휑 4 (푥),휑 5 (푥),and휑 6 (푥)are real
functions and their forms depend on the values of the system
parameters (휆 1 ,휆 2 ,and휆 3 ) as given in AppendixA;and
푐 1 ,푐 2 ,푐 3 ,푐 4 ,푐 5 ,and푐 6 are constants of integration to be
determined by imposing force boundary conditions. These
six force boundary conditions are
−[
푑푀
푑푥
]
푥=0=푃 1 ,−푀( 0 )=푃 2 ,
[
푑푀
푑푥
]
푥=퐿=푃 3 ,푀(퐿)=푃 4 ,
−[푉푠]푥=0=푃 5 ,[푉푠]푥=퐿=푃 6.
(23)
The first four boundary conditions on the beam ends
can be imposed directly while the last two on the shear-
layer ends cannot be enforced at the first stage since the
beam-section bending moment푀(푥)istheonlyvariablefield
in the governing differential compatibility equation of ( 20 ).
This difficulty can be overcome by establishing the relation
between푉푠(푥)and푀(푥).
Recalling the shear-layer compatibility condition of ( 14 )
and the푉푠(푥) − 푀(푥)relation of ( 19 ), the shear-layer shear
force푉푠(푥)can be expressed in terms of the beam-section
bending moment푀(푥)and its derivatives as푉푠(푥)=휗 1 (
푑^5 푀(푥)
푑푥^5
−
푑^3 푝푦(푥)
푑푥^3
)
+휗 2 (
푑^3 푀(푥)
푑푥^3
−
푑푝푦(푥)
푑푥
)+휗 3
푑푀(푥)
푑푥
,
(24)
where휗 1 =GA^2 /푘 1 푘 2 ,휗 2 =−GA/푘 1 ,and휗 3 =GA^2 /IE푘 1.
By imposing force boundary conditions of ( 23 ), the
moment interpolation relation can be expressed as푀(푥)=N퐵퐵(푥)P, (25)whereN퐵퐵(푥) = ⌊푁퐵퐵1(푥) 푁퐵퐵2(푥) 푁퐵퐵3(푥) 푁퐵퐵4(푥)
푁퐵퐵5(푥) 푁퐵퐵6(푥)⌋ is an array containing the moment
interpolation functions. Imposing the relation of ( 24 )and
differential equilibrium equations of ( 3 )and( 4 ), the shear-
layer shear force푉푠(푥),thelower-springforce퐷 1 (푥),andthe
upper-spring force퐷 2 (푥)can be expressed in terms ofPas푉푠(푥)=N푉푠퐵(푥)P,퐷 1 (푥)=N퐷 1 퐵(푥)P,퐷 2 (푥)=N퐷 2 퐵(푥)P,(26)
whereN푉푠퐵(푥) = ⌊푁푉푠퐵1(푥) 푁푉푠퐵2(푥) 푁푉푠퐵3(푥) 푁푉푠퐵4(푥)
푁푉푠퐵5(푥) 푁푉푠퐵6(푥)⌋is an array containing the shear-layer
shear-force interpolation functions;N퐷 1 퐵(푥) = ⌊푁퐷 1 퐵1(푥)
푁퐷 1 퐵2(푥) 푁퐷 1 퐵3(푥) 푁퐷 1 퐵4(푥) 푁퐷 1 퐵5(푥) 푁퐷 1 퐵6(푥)⌋ is an
array containing the lower-spring force interpolation func-
tions; andN퐷 2 퐵(푥) = ⌊푁퐷 2 퐵1(푥) 푁퐷 2 퐵2(푥) 푁퐷 2 퐵3(푥)
푁퐷 2 퐵4(푥) 푁퐷 2 퐵5(푥) 푁퐷 2 퐵6(푥)⌋is an array containing the
upper-spring force interpolation functions.
Applying the virtual force expression of ( 9 ), substituting
( 25 )-( 26 ), and accounting for the arbitrariness of훿Pyield the
following element flexibility equation:FP=U+U푝푦, (27)whereFis the element flexibility matrix, defined asF=F퐵퐵+F푉푠푉푠+F퐷 1 퐷 1 +F퐷 2 퐷 2 , (28)whereF퐵퐵,F푉푠푉푠,F퐷 1 퐷 1 ,andF퐷 2 퐷 2 are the beam, the shear-
layer, the lower-spring, and the upper-spring contributions
to the element flexibility matrix, respectively:F퐵퐵=∫
퐿N퐵퐵푇(
1
IE
)N퐵퐵푑푥,
F푉푠푉푠=∫
퐿N푉푠퐵푇(
1
GA
)N푉푠퐵푑푥,
F퐷 1 퐷 1 =∫
퐿N퐷 1 퐵푇(
1
푘 1
)N퐷 1 퐵푑푥,
F퐷 2 퐷 2 =∫
퐿