eliminated in ( 9 ) through the governing differential equilib-
rium equations of ( 3 )and( 4 ). Therefore, the system virtual
forceequationcanbewrittenas
−훿P푇U+∫
퐿훿푀(푥)
푀(푥)
IE
푑푥 +∫
퐿훿푉푠(푥)
푉푠(푥)
GA
푑푥
+∫
퐿푑훿푉푠(푥)
푑푥
(
1
푘 1
)(−
푑^2 푀(푥)
푑푥^2
+푝푦(푥)+
푑푉푠(푥)
푑푥
)푑푥
+∫
퐿푑^2 훿푀(푥)
푑푥^2
(
1
푘 1
)(
푑^2 푀(푥)
푑푥^2
−푝푦(푥)−
푑푉푠(푥)
푑푥
)푑푥
+∫
퐿푑^2 훿푀(푥)
푑푥^2
(
1
푘 2
)(
푑^2 푀(푥)
푑푥^2
−푝푦(푥))푑푥=0.
(10)
In order to move all differential operators to the bending
moment푀(푥)and the shear-layer section shear force푉푠(푥),
integration by parts is applied once to the forth term and
twice to the fifth and sixth terms of ( 10 ), respectively, hence
resulting in the following expression:
∫
퐿훿푀(푥)(
푀(푥)
IE
+(
1
푘 1
+
1
푘 2
)(
푑^4 푀(푥)
푑푥^4
−
푑^2 푝푦(푥)
푑푥^2
)
−
1
푘 1
푑^3 푉푠(푥)
푑푥^3
)푑푥
+∫
퐿훿푉푠(푥)(
푉푠(푥)
GA
+
1
푘 1
(
푑^3 푀(푥)
푑푥^3
−
푑푝푦(푥)
푑푥
−
푑^2 푉푠(푥)
푑푥^2
))푑푥
+[((
1
푘 1
+
1
푘 2
)(
푑^2 푀(푥)
푑푥^2
−푝푦(푥))−
1
푘 1
푑푉푠(푥)
푑푥
)
×
푑훿푀(푥)
푑푥
]
퐿0+[((
1
푘 1
+
1
푘 2
)(
푑푝푦(푥)
푑푥
−
푑^3 푀(푥)
푑푥^3
)+
1
푘 1
푑^2 푉푠(푥)
푑푥^2
)
×훿푀(푥)]
퐿0+[
1
푘 1
(푝푦(푥)+
푑푉푠(푥)
푑푥
−
푑^2 푀(푥)
푑푥^2
)훿푉푠(푥)]
퐿0−훿P푇U=0.
(11)Considering the shear-force definition of ( 2 )andfollow-
ing the Cartesian sign convention, ( 11 )canbewrittenas∫
퐿훿푀(푥)(
푀(푥)
IE
+(
1
푘 1
+
1
푘 2
)(
푑^4 푀(푥)
푑푥^4
−
푑^2 푝푦(푥)
푑푥^2
)
−
1
푘 1
푑^3 푉푠(푥)
푑푥^3
)푑푥
+∫
퐿훿푉푠(푥)(
푉푠(푥)
GA
+
1
푘 1
(
푑^3 푀(푥)
푑푥^3
−
푑푝푦(푥)
푑푥
−
푑^2 푉푠(푥)
푑푥^2
))푑푥
−훿푃 1 (푈 1 −((
1
푘 1
+
1
푘 2
)(푝푦(푥)−
푑^2 푀(푥)
푑푥^2
)
+
1
푘 1
푑푉푠(푥)
푑푥
)
푥=0)
−훿푃 2 (푈 2 −(
1
푘 1
푑^2 푉푠(푥)
푑푥^2
+(
1
푘 1
+
1
푘 2
)
푑푝푦(푥)
푑푥
−(
1
푘 1
+
1
푘 2
)
푑^3 푀(푥)
푑푥^3
)
푥=0)
−훿푃 3 (푈 3 −((
1
푘 1
+
1
푘 2
)(푝푦(푥)−
푑^2 푀(푥)
푑푥^2
)
+
1
푘 1
푑푉푠(푥)
푑푥
)
푥=퐿)
−훿푃 4 (푈 4 −(
1
푘 1
푑^2 푉푠(푥)
푑푥^2
+(
1
푘 1
+
1
푘 2
)
푑푝푦(푥)
푑푥
−(
1
푘 1
+
1
푘 2
)
푑^3 푀(푥)
푑푥^3
)
푥=퐿)
−훿푃 5 (푈 5 −
1
푘 1
(푝푦(푥)+
푑푉푠(푥)
푑푥
−
푑^2 푀(푥)
푑푥^2
)
푥=0)
−훿푃 6 (푈 6 −
1
푘 1
(푝푦(푥)+
푑푉푠(푥)
푑푥
−
푑^2 푀(푥)
푑푥^2
)
푥=퐿)
=0.
(12)
Due to the arbitrariness of훿푀(푥)and훿푉푠(푥),thegov-
erning differential compatibility equations of the beam and
shear-layer components are obtained; thus푀(푥)
IE
+(
1
푘 1
+
1
푘 2
)(
푑^4 푀(푥)
푑푥^4
−
푑^2 푝푦(푥)
푑푥^2
)
−
1
푘 1
푑^3 푉푠(푥)
푑푥^3
=0: for푥∈(0,퐿),