푉푠(푥)
GA
+
1
푘 1
(
푑^3 푀(푥)
푑푥^3
−
푑푝푦(푥)
푑푥
−
푑^2 푉푠(푥)
푑푥^2
)
=0: for푥∈(0,퐿).(14)
Equations ( 13 )and( 14 ) form a set of governing differ-
ential compatibility equations of the system. It is noted that
the compatibility equations of the lower and upper spring
deformations are not involved in the virtual force equation
since their conjugate-work forces(퐷 1 (푥)and퐷 2 (푥)) are
eliminated in ( 10 ). However, they can be obtained simply by
considering the geometrical deformations of the lower and
the upper springs in Figure1(a)as
Δ 1 (푥)−V푠(푥)=0,Δ 2 (푥)−(V퐵(푥)−V푠(푥))=0,(15)
whereV푠(푥)is the shear-layer vertical displacement. Consid-
ering the deformation-force relations of ( 5 )andenforcing
the equilibrium equations of ( 3 )and( 4 )andcompatibility
equations of ( 15 ), ( 13 ), and ( 14 ) are reduced to
휅(푥)−
푑^2 V퐵(푥)
푑푥^2
=0, (16)
훾(푥)−
푑V푠(푥)
푑푥
=0. (17)
It now becomes clear that ( 13 )and( 14 ) simply state the
definitions of the beam section curvature and shear-layer
section shear strain, respectively.
To m a k e u s e o f ( 13 )and( 14 ), there is a need to establish
therelationbetween푉푠(푥)and푀(푥). This could be accom-
plished by recalling the governing differential equation of the
foundation surface subjected to a continuously distributed
load as given by Kerr [ 11 ]:
(1 +
푘 1
푘 2
)퐷 2 (푥)−
GA
푘 2
푑^2 퐷 2 (푥)
푑푥^2
=푘 1 V퐵(푥)−GA
푑^2 V퐵(푥)
푑푥^2
.
(18)
Enforcing the equilibrium equations of ( 3 )and( 4 )as
well as compatibility equations of ( 15 ), recalling the curvature
definition of ( 16 ), and considering the deformation-force
relations of ( 5 ), ( 18 ) relates the first derivative of the shear-
layer section shear force to the bending moment and its
fourth-order derivative as
푑푉푠(푥)
푑푥=
GA
IE
푀(푥)+
GA
푘 2
푑^4 푀(푥)
푑푥^4
−
GA
푘 2
푑^2 푝푦(푥)
푑푥^2
. (19)
Differentiating ( 19 )twiceandsubstitutinginto( 13 )yield
the following sixth-order differential equation:
푑^6 푀(푥)
푑푥^6+휆 1
푑^4 푀(푥)
푑푥^4
+휆 2
푑^2 푀(푥)
푑푥^2
+휆 3 푀(푥)
=
푑^4 푝푦(푥)
푑푥^4
+휆 1
푑^2 푝푦(푥)
푑푥^2
: for푥∈(0,퐿),(20)
where휆 1 = −((푘 1 +푘 2 )/GA);휆 2 =푘 2 /IE and휆 3 =
−푘 1 푘 2 /IEGA.
It is noted that when the upper-spring modulus푘 2
approaches infinite, ( 20 ) is reduced to a fourth-order gov-
erning differential compatibility equation of the beam on
Winkler-Pasternak foundation as given by Limkatanyu et al.
[ 30 ] and when the shear-layer section modulus GA is equal
to zero, ( 20 ) becomes a fourth-order governing differential
compatibility equation of the beam on Winkler foundation
as given by Limkatanyu et al. [ 29 ]. Furthermore, when
compared to the governing differential equation derived by
Avramidis and Morfidis [ 19 ]usingtheprincipleofstationary
potential energy (equivalent to the virtual displacement
principle), it becomes clear that ( 20 ) and the one derived
by Avramidis and Morfidis [ 19 ]aredual.Thisillustrates
the dualism of the virtual displacement and virtual force
principles.
The end-boundary compatibility conditions are obtained
by accounting for the arbitrariness of훿Pin ( 12 )as푈 1 =
1
푘 1
(
푑푉푠(푥)
푑푥
)
푥=0−(
1
푘 1
+
1
푘 2
)(
푑^2 푀(푥)
푑푥^2
)
푥=0+(
1
푘 1
+
1
푘 2
)(푝푦(푥))푥=0,
푈 2 =
1
푘 1
(
푑^2 푉푠(푥)
푑푥^2
)
푥=0−(
1
푘 1
+
1
푘 2
)(
푑^3 푀(푥)
푑푥^3
)
푥=0+(
1
푘 1
+
1
푘 2
)(
푑푝푦(푥)
푑푥
)
푥=0,
푈 3 =
1
푘 1
(
푑푉푠(푥)
푑푥
)
푥=퐿−(
1
푘 1
+
1
푘 2
)(
푑^2 푀(푥)
푑푥^2
)
푥=퐿+(
1
푘 1
+
1
푘 2
)(푝푦(푥))푥=퐿,
푈 4 =
1
푘 1
(
푑^2 푉푠(푥)
푑푥^2
)
푥=퐿−(
1
푘 1
+
1
푘 2
)(
푑^3 푀(푥)
푑푥^3
)
푥=퐿+(
1
푘 1
+
1
푘 2
)(
푑푝푦(푥)
푑푥
)
푥=퐿,
푈 5 =
1
푘 1
(
푑푉푠(푥)
푑푥
−
푑^2 푀(푥)
푑푥^2
)
푥=0+
1
푘 1
(푝푦(푥))푥=0,
푈 6 =
1
푘 1
(
푑푉푠(푥)
푑푥
−
푑^2 푀(푥)
푑푥^2
)
푥=퐿+
1
푘 1
(푝푦(푥))푥=퐿.
(21)
It is observed that the homogeneous and particular con-
tributions to the end displacements are clearly separated in
( 21 ). This observation is unique to the proposed formulation
and very useful in determining the equivalent fixed-end force
vectorduetotheelementload푝푦(푥).