671017.pdf

problems. This could be accomplished by first following the
evolution of beam and foundation mechanical parameters
andthenupdatingtheforceinterpolationfunctionsaccord-
ingly. Another interesting topic worth investigating in future
works is the derivation of consistent mass and geometric
stiffness matrices based on the force interpolation functions.

Appendices

A. Homogenous Solution to the

Sixth-Order Governing Differential

Compatibility Equation( 20 )

The homogeneous form of ( 20 )canbewrittenas

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)

=0: for푥∈(0,퐿).

(A.1)

For simplicity, the following auxiliary variables are intro-
duced instead of terms of system parameters휆 1 ,휆 2 ,and휆 3 :

훼=

(−(휆^21 /3) + 휆 2 )

3

,

훽=

((2휆^31 /27) − (휆 1 휆 2 /3) + 휆 3 )

2

,

Δ=훼^3 +훽^2 ,

(A.2)

Φ 1 =−

(√^3 −훽 +√Δ+√^3 −훽 −√Δ+(2휆 1 /3))

2

,

Φ 2 =

√3(√^3 −훽 +√Δ−√^3 −훽 −√Δ)

2

.

(A.3)

There are many solution types to (A.1) depending on the
sign of the auxiliary parameterΔ. Thanks to the thorough
investigations of all possible solution cases, Avramidis and
Morfidis [ 19 ]andMorfidis[ 20 ]suggestthemostcommon
solution case corresponding to the positive sign of the
auxiliary parameterΔas follows.

Solution Case (whenΔ=훼^3 +훽^2 >0).The homogeneous
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 ,

(A.4)

where

휑 1 (푥)=푒Γ^1 퐿,휑 2 (푥)=푒−Γ^1 퐿,

휑 3 (푥)=푒Γ^3 퐿cosΓ 2 퐿, 휑 4 (푥)=푒Γ^3 퐿sinΓ 2 퐿,

휑 5 (푥)=푒−Γ^3 퐿cosΓ 2 퐿, 휑 6 (푥)=푒−Γ^3 퐿sinΓ 2 퐿,

(A.5)

Γ 1 =√√^3 −훽 +√Δ+√^3 −훽 −√Δ−

휆 1

3

,

Γ 2 =

√(√Φ

2

1 +Φ

2

2 −Φ^1 )

2

,

Γ 3 =

√(√Φ

2

1 +Φ

2

2 +Φ 1 )

2

.

(A.6)

B. Nodal Displacements due to푝푦(푥)

The nodal displacements due to the uniformly distributed

load푝푦(푥) = 푝 0 may be written as

푈1푝푦=푈3푝푦=(

1

푘 1

+

1

푘 2

)푝 0 ,

푈2푝푦=푈4푝푦=0,

푈5푝푦=푈6푝푦=

푝 0

푘 1

.

(B.1)

Acknowledgments

This study was partially supported by the Thai Ministry of

University Affairs (MUA), by the Thailand Research Fund

(TRF) under Grant MRG4680109 and Grant RSA5480001,

by STREAM Research Group under Grant ENG-51-2-7-11-

022-S, Faculty of Engineering, Prince of Songkla University,

and by the National Research Foundation of Korea (NRF)

Grant funded by the Korean government (MEST) (NRF-2011-

0028531). Any opinions expressed in this paper are those of

theauthorsanddonotreflecttheviewsofthesponsoring

agencies. Special thanks are due to the senior lecturer Mr.

Wiwat Sutiwipakorn for reviewing and correcting the English

of this paper and Mr. Paitoon Ponbunyanon and Mr. Attpon

Sangkeaw for preparing the numerical example. In addition,

theauthorswouldalsoliketothankthreeanonymous

reviewers for their valuable and constructive comments.

References

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[2] E. Ghavanloo, F. Daneshmand, and M. Rafiei, “Vibration and

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