A differential segment푑푥taken from the beam on
Kerr-type foundation is shown in Figure1(b).Thevertical
equilibrium of the infinitesimal beam segment푑푥is written
as
푑푉퐵(푥)
푑푥+푝푦(푥)−퐷 2 (푥)=0, (1)
where푉퐵(푥)is the beam-section shear force;푝푦(푥)is the
transverse distributed load; and퐷 2 (푥)is the interactive force
in the upper spring and acts at the bottom face of the beam.
Considering the moment equilibrium, this yields
푑푀(푥)
푑푥+푉퐵(푥)=0, (2)
where푀(푥)is the beam-section bending moment. Following
the Euler-Bernoulli beam theory, only flexural contributions
are considered in the paper. Enforcing the beam shear
equilibrium of ( 2 ), ( 1 )and( 2 ) can be combined into a single
relation; thus,
푑^2 푀(푥)
푑푥^2−푝푦(푥)+퐷 2 (푥)=0. (3)
A differential segment푑푥taken from the shear layer
resting on the lower foundation springs is shown in Fig-
ure1(c). The vertical equilibrium of the infinitesimal shear-
layer segment푑푥canbewrittenas
푑푉푠(푥)
푑푥+퐷 2 (푥)−퐷 1 (푥)=0, (4)
where푉푠(푥)is the shear-layer section shear force and퐷 1 (푥)
istheinteractiveforceinalowerspring.Equations( 3 )and
( 4 ) form a set of governing differential equilibrium equations
of the system and are coupled through the upper-spring
interactive force퐷 2 (푥).
It is noteworthy to remark that this system is internally
statically indeterminate and the internal forces cannot be
determined simply by equilibrium conditions since there are
4internalforceunknownfields,푀(푥),푉푠(푥),퐷 1 (푥),and
퐷 2 (푥), at any system section while only two equilibrium
equations are available.
2.2. Deformation-Force Relations.The system sectional
deformations can be related to their conjugate-work forces
as follows:
휅(푥)=푀(푥)
IE
,훾푠(푥)=
푉푠(푥)
GA
,
Δ 1 (푥)=
퐷 1 (푥)
푘 1
,Δ 2 (푥)=
퐷 2 (푥)
푘 2
,
(5)
where 휅(푥)is the beam-section curvature;훾푠(푥) is the
shear-layer section shear strain;Δ 1 (푥)is the lower-spring
deformation;Δ 2 (푥)is the upper-spring deformation; IE is
the flexural rigidity; GA is the shear-layer section modulus;
푘 1 is the lower-spring modulus; and푘 2 is the upper-spring
modulus. Following the comprehensive work by Worku [ 16 ],
the three foundation parameters (푘 1 ,푘 2 ,andGA)canbe
related to the elastic modulus, Poisson ratio, and depth of the
soil continuum underneath the beam.
2.3. Differential Compatibility Equations and End Compatibil-
ity Conditions: The Virtual Force Principle.The virtual force
equation is an integral expression of the system compatibility
equations and can be expressed in the general form as훿푊∗=훿푊int∗+훿푊ext∗ =0, (6)where훿푊∗is the system total complementary virtual work;
훿푊int∗is the system internal complementary virtual work; and
훿푊ext∗ is the system external complementary virtual work.
In the case of the beam-Kerr foundation system,훿푊int∗
and훿푊ext∗ canbeexpressedas훿푊int∗ =∫
퐿훿푀(푥)휅(푥)푑푥 +∫
퐿훿푉푠(푥)훾푠(푥)푑푥
+∫
퐿훿퐷 1 (푥)Δ 1 (푥)푑푥 +∫
퐿훿퐷 2 (푥)Δ 2 (푥)푑푥
훿푊ext∗ =−∫
퐿훿푝푦(푥)V퐵(푥)푑푥 − 훿P푇U,
(7)
whereV퐵(푥)is the beam vertical displacement; the vectorP=
{푃 1 푃 2 푃 3 푃 4 푃 5 푃 6 }푇contains shear forces and moments
acting at beam ends and shear forces acting at the shear-
layer ends; and the vectorU ={푈 1 푈 2 푈 3 푈 4 푈 5 푈 6 }푇contains their conjugate-work displacements and rotations
at the beam ends and displacements at the shear-layer ends.
At the moment, external force quantity,훿푝푦(푥), is arbitrarily
chosentobezero.Thus,( 6 )becomes훿푊∗=∫
퐿훿푀(푥)휅(푥)푑푥 +∫
퐿훿푉푠(푥)훾푠(푥)푑푥
+∫
퐿훿퐷 1 (푥)Δ 1 (푥)푑푥 +∫
퐿훿퐷 2 (푥)Δ 2 (푥)푑푥
−훿P푇U=0.
(8)
Eliminating the internal deformation fields through the
deformation-force relations, ( 8 )canbewrittenas훿푊∗=∫
퐿훿푀(푥)
푀(푥)
IE
푑푥 +∫
퐿훿푉푠(푥)
푉푠(푥)
GA
푑푥
+∫
퐿훿퐷 1 (푥)
퐷 1 (푥)
푘 1
푑푥 +∫
퐿훿퐷 2 (푥)
퐷 2 (푥)
푘 2
푑푥
−훿P푇U=0.
(9)
The lower and upper spring forces(퐷 1 (푥)and퐷 2 (푥))
and their virtual counterparts(훿퐷 1 (푥)and훿퐷 2 (푥))can be