NQ
MQ+dQ
dsdsM+dM
dsdsN+dN
dsdsu
kndsqnksqFigure 2: The free-body diagram of infinitesimal isolated segment
of pile.
differential, the equilibrium equations now take the following
forms:
푑푁
푑푠
=푘푠퐻퐷푢 + 푞휏,
푑푄
푑푠
=푘푛퐻푏V+푞푛,
푑푀
푑푠
=−푄,
(3)
where퐻={1,0,noncontactcontact,, the “one” indicates that Winkler
reaction force exists and the “zero” indicates that Winkler
reaction force does not exist.푆is the position coordinate;
퐷is perimeter of the cross-section.푏is the width of the
cross-section,푘푠is the Winkler modulus of vertical subgrade
reaction, and 푘푛is the Winkler modulus of horizontal
subgrade reaction.
For the sake of convenience of formula deducing, let
푋=[
[
푁
푄
푀
]
]
;푍=[
[
푢
V
휑
]
]
;푃=[
[
푞휏
푞푛
0
]
]
. (4)
The set of equations of equilibrium ( 3 )canberewrittenin
the following matrix form:푑푋
푑푠
=퐵⋅푋+퐿⋅푍+푃, (5)
where퐵=[000
000
0−10];퐿=[
푘푠퐻퐷 0 0
0푘푛퐻푏 0
000].
2.4. Geometric and Constitutive Equations.When the defor-
mation (푑푢,푑V, 푑휑) of the differential element (shown
inFigure 2)inducedbytheinternalforces(푁,푄,푀)is
considered given, the corresponding strains can be expressed
as푑푈
푑푠=(
푑푢
푑푠
,
푑V
푑푠
,
푑휑
푑푠
). (6)
According to the related theory of elastic beam, the internal
forces (푁,푄,푀)canberelatedtostrainsasinthefollowing
linear constitutive equation:푑푈
푑푠
=
[
[
[
[[
[
[
[
[
푑푢
푑푠
푑V
푑푠
푑휑
푑푠
]
]
]
]]
]
]
]
]
=
[
[
[[
[
[
[
[
1
퐸퐴
00
0
훼
퐺퐴
0
00
1
퐸퐼
]
]
]]
]
]
]
]
⋅[
[
푁
푄
푀
]
]
, (7)
where훼is the constant related to the shape of pile cross-
section (훼=6/5for rectangular cross-section;훼 = 10/9for
circular cross-section);퐴is the cross-sectional area;퐸퐼is the
flexural rigidity of the pile’s cross-section.
Considering the deformation,푑푈can be decomposed
into two parts. One part is the푑푍induced by the displace-
ment on its direction and another part is the projection of
other displacement onto this direction which takes the form
퐵푍푑푠,where퐵is the a undetermined third-order square
matrix. Then, the deformation푑푈canbeexpressedas푑푈=푑푍+퐵푍푑푠. (8)
Applying the principle of virtual work to the isolated dif-
ferential element of pile (shown inFigure 2),퐵can be
determined. We suppose that each point of the body is
givenaninfinitesimalvirtualdisplacement훿푍 satisfying
displacement boundary conditions where prescribed. The
virtual deformation associated with the infinitesimal virtual
displacement is훿푈. The virtual work of the external surface
forces is−∫푃푇
(훿푍)푑푠,where푃=푃+퐿푍.Thevirtual
work of the internal forces is∫푋푇푑(훿푈).Byequatingthe
external work to the internal work, we have−∫푃푇
(훿푍)푑푠 =
∫푋푇푑(훿푈).Substituting( 5 )and( 8 )intotheaboveequation
and simplifying yields∫푋푇(퐵푇−퐵)훿푍푑푠 = 0.Sincethis
equation is satisfied for arbitrary훿푍, the terms in the brackets
in the integral must vanish at every point which means that