free vibration of semi-rigid connected reddy–bickford

Sadhana Vol. 33, Part 6, December 2008, pp. 781–801. © Printed in India

Free vibration of semi-rigid connected Reddy–Bickfordpiles embedded in elastic soil

YUSUF YESILCE∗ and HIKMET H CATAL

Dokuz Eylul University, Civil Engineering Department, Engineering Faculty,35160, Buca, Izmir, Turkeye-mail: [email protected]

MS received 14 February 2008; revised 28 April 2008

Abstract. The literature on free vibration analysis of Bernoulli–Euler and Tim-oshenko piles embedded in elastic soil is plenty, but that of Reddy–Bickford pilespartially embedded in elastic soil with/without axial force effect is fewer. The soilthat the pile partially embedded in is idealized by Winkler model and is assumedto be two-layered. The pile part above the soil is called the first region and theparts embedded in the soil are called the second and the third region, respectively.It is assumed that the behaviour of the material is linear-elastic, that axial forcealong the pile length to be constant and the upper end of the pile that is semi-rigidsupported against rotation is modelled by an elastic spring. The governing dif-ferential equations of motion of the rectangular pile in free vibration are derivedusing Hamilton’s principle and Winkler hypothesis. The terms are found directlyfrom the solutions of the differential equations that describe the deformations ofthe cross-section according to the high-order theory. The models have six degreesof freedom at the two ends, one transverse displacement and two rotations, andthe end forces are a shear force and two end moments. Natural frequencies of thepile are calculated using transfer matrix and the secant method for non-trivial solu-tion of linear homogeneous system of equations obtained due to values of axialforces acting on the pile, total and embedded lengths of the pile, the linear-elasticrotational restraining stiffness at the upper end of the pile and to the boundary con-ditions of the pile. Two different boundary conditions are considered in the study.For the first boundary condition, the pile’s end at the first region is semi-rigid con-nected and not restricted for horizontal displacement and the end at the third regionis free and for the second boundary condition, the pile’s end at the first region issemi-rigid connected and restricted for horizontal displacement and the end at thethird region is fixed supported. The calculated natural frequencies of semi-rigidconnected Reddy–Bickford pile embedded in elastic soil are given in tables andcompared with results of Timoshenko pile model.

Keywords. Axial force effect; free vibration; Reddy–Bickford pile; semi-rigidconnected; transfer matrix.

∗For correspondence

781

782 Yusuf Yesilce and Hikmet H Catal

Figure 1. Cross-section displace-ments in different beam theories(Wang et al 2000) (a) Bernoulli–EulerBeam Theory (BET); (b) TimoshenkoBeam Theory (TBT); (c) Reddy–Bickford Beam Theory (RBT).

1. Introduction

The analysis of the piles embedded in elastic soil is similar to the analysis of beams on elasticfoundations.

The analysis of beams has been performed over the years mostly using Bernoulli–Eulerbeam theory (BET). The classical Bernoulli–Euler beam is well studied for slender beams,where the transverse shear deformation can be safely disregarded. This theory is based onthe assumption that plane sections of the cross-section remain plane and perpendicular tothe beam axis. The cross-sectional displacements are shown in (figure 1a), and expressedas

u(x, z, t) = −z · ∂w0(x, t)

∂x(1)

w(x, z, t) = w0(x, t), (2)

where w0(x, t) is the lateral displacement of the beam neutral axis, z is the distance from thebeam neutral axis.

For moderately thick beams Bernoulli–Euler beam theory can be modified in order to takeinto account the transverse shear effect in a simplified way. For example, the well-knownTimoshenko beam theory (TBT) predicts a uniform shear distribution, so necessitating theuse of a so-called shear factor (Cowper 1966, Gruttmann & Wagner 2001, Murthy 1970). Thecross-sectional displacements of Timoshenko beam theory are shown in (figure 1b) and the

Free vibration of semi-rigid connected Reddy–Bickford piles 783

equations for Timoshenko beam theory which relaxes the restriction on the angle of shearingdeformations are;

u(x, z, t) = z · φ(x, t) (3)

w(x, z, t) = w0(x, t), (4)

where φ(x, t) represents the rotation of a normal to the axis of the beam. Han et al (1999)presented a comprehensive study of Bernoulli–Euler, Rayleigh, Shear and Timoshenko beamtheories.

The real shear deformation distribution is not uniform along the depth of the beam, sothat Timoshenko beam theory is not recommended for composite beams, where the accuratedetermination of the shear stresses is required. Especially, it was found that the Timoshenkoshear deformation theory has some major numerical problems such as locking in the numericalanalysis for composite materials. The other problem was the need to supply an artificiallyderived shear correction factor. Although some remedies were devised, as a result, severalhigher-order theories have emerged. These theories, with small variations, are due to severalauthors relax the restriction on the warping of the cross-section and allow variation in thelongitudinal direction of the beam which is cubic (Bickford 1982, Heyliger & Reddy 1988,Levinson 1981, Wang et al 2000).

In this paper, Reddy–Bickford beam theory (RBT) is used, which seems a good compromisebetween accuracy and simplicity (Bickford 1982, Wang et al 2000). The cross-sectionaldisplacements of Reddy–Bickford beam theory are shown in (figure 1c) and according toReddy–Bickford beam theory, the displacements of the rectangular beam can be written as(Wang et al 2000, Reddy 2002, Reddy 2007)

u(x, z, t) = z · φ(x, t) − α · z3 ·[φ(x, t) + ∂w(x, t)

∂x

], (5)

w(x, z, t) = w0(x, t), (6)

where α = 43·h2 ; h is height of the beam.

Bernoulli–Euler beam theory does not consider the shear stress in the cross-section and theassociated strains. Thus, the shear angle is taken as zero through the height of the cross-section.Timoshenko beam theory assumes constant shear stress and shear strain in the cross-section.On the top and bottom edges of the beam the free surface condition is thus violated. The useof a shear correction factor, in various forms including the effect of Poisson’s ratio, does notcorrect this fault of the theory, but rather artificially adjusts the solutions to match the staticor dynamic behaviour of the beam. Reddy–Bickford beam theory and the other high-ordertheories remedy this physical mismatch at the free edges by assuming variable shear strainand shear stress along the height of the cross-section. Then there is no need for the shearcorrection factor. The high-order theory is more exact and represents much better physics ofthe problem. It results in a sixth-order theory compared to the fourth-order of the other less-accurate theories. This yields a six-degree-of-freedom element with six end forces, a shearforce, bending moment and a high-order moment, at the two ends of the beam element.

The determination of the natural frequencies is crucial in the dynamic analysis of piles thatare partially embedded in the soil. Previously, it was widely assumed that the upper ends ofpiles used to support offshore structures, marine and harbor structures or bridges are fullyrigidly connected. In reality, these connections are neither fully rigid nor flexible. They fallbetween fully rigid and flexible connections, depending on the cross-sectional and material


properties of the piles. A more reasonable way is to treat them as semi-rigid connections inthe structural analysis.

The analysis of a partially supported pile is similar to that of a beam that is elasticallysupported. Previously, numerous researchers studied the behaviour of beams supported byelastic foundations (Hetenyi 1955). Doyle & Pavlovic (1982) solved the partial differentialequation for free vibration of beams partially attached to elastic foundation using variableseparating method and neglecting axial force and shear effects. Boroomand & Kaynia (1991)studied dynamic analysis of pile–soil–pile interaction for vertical piles in a homogeneoussoil by using a Fourier expansion of variables. Aviles & Sanchez-Sesma (1983) studied theusefulness of a row of rigid piles as an isolating barrier for elastic waves. They formulatedthe problem as one of multiple scattering and diffraction. Aviles & Sanchez-Sesma (1988)presented a theoretical analysis to solve the problem of foundation isolation, using a row ofelastic piles as an isolating barrier for elastic waves. Liao & Sangrey (1978) employed anacoustic model for the use of rows of piles as passive isolation barriers to reduce groundvibrations. West & Mafi (1984) solved the partial differential equation for free vibration of anelastic beam on elastic foundation that is subjected to axial force by using initial value method.Yokoyama (1991) studied the free vibration motion of Timoshenko beam on two-parameterselastic foundation. Esmailzadeh & Ohadi (2000) investigated vibration and stability analysisof non-uniform Timoshenko beams under axial and distributed tangential loads. Catal (2002)calculated natural frequencies and relative stiffness of the pile for non-trivial solution of linearhomogeneous system of equations obtained due to the values of axial forces acting on thepile, the shape factors, and the boundary conditions of the pile with both ends free and bothends simply supported by using Timoshenko beam theory and transfer matrix. Further, heproceeded to determine the natural frequencies of Timoshenko piles partially embedded inthe soil, but semi-rigidly connected at the upper ends, using the method of initial values (Catal2006). Lin & Chang (2005) studied free vibration analysis of multi-span Timoshenko beamwith an arbitrary number of flexible constraints by transfer matrix method. Demirdag & Catal(2007) investigated spectral analysis of semi-rigid supported single storey frames modelledas Timoshenko column with attached mass. Demirdag (2008) studied elastically-supportedTimoshenko column with attached masses is under consideration to obtain its free vibrationnatural frequencies using two different algorithm; transfer matrix method and finite elementmethod. Yesilce & Catal (2008) calculated normalized natural frequencies of Timoshenkopile due to the different values of axial force using transfer matrix and considering rotatoryinertia.

2. The mathematical model and formulation

A rectangular pile partially embedded in the soil whose upper end is semi-rigid connectedagainst rotation is presented in (figure 2). The pile part above the soil is called the first regionand the parts embedded in the soil are called the second and the third region, respectively.

Using Hamilton’s principle and Eqs. (5) and (6); the equations of motion can be writtenfor each region as (Wang et al 2000, Eisenberger 2003):

− 68

105· EIx · ∂2φj (xj , t)

∂x2j

+ 16

105· EIx · ∂3wj(xj , t)

∂x3j

+ 8

15· AG

·[φj (xj , t) + ∂wj (xj , t)

∂xj

]= 0 (0 ≤ xj ≤ Lj) (j = 1, 2, 3) (7)


− m · ∂2w1(x1, t)

∂t2+ 8

15· AG ·

[∂φ1(x1, t)

∂x1+ ∂2w1(x1, t)

∂x21

]+ 16

105· EIx

· ∂3φ1(x1, t)

∂x31

− 1

21· EIx · ∂4w1(x1, t)

∂x41

−N · ∂2w1(x1, t)

∂x21

= 0 (0 ≤ x1 ≤ L1) (8)

− m · ∂2wk(xk, t)

∂t2+ 8

15· AG ·

[∂φk(xk, t)

∂xk

+ ∂2wk(xk, t)

∂x2k

]+ 16

105· EIx · ∂3φk(xk, t)

∂x3k

− 1

21·EIx · ∂

4wk(xk, t)

∂x4k

−CS(k−1) ·wk(xk, t)−N · ∂2wk(xk, t)

∂x2k

= 0(0 ≤ xk ≤ Lk) (k = 2, 3),

(9)

where wj(xj , t) is displacement function for j th region of the pile, φj (xj , t) represents therotation of a normal to the axis for j th region of the pile, m is mass per unit length of thepile, L1 is pile length above the soil, L2 is pile length embedded in the second region, L3 ispile length embedded in the third region, L is total length of the pile, N is the constant axialcompressive force, A is the cross-section area, Ix is moment of inertia, E, G are Young’smodulus and shear modulus of the pile, CS1 = CR1.b and CS2 = CR2.b in which CR1, CR2

Figure 2. Pile partially embedded in the elas-tic soil.


are the modulus of subgrade reaction for the second and the third regions, respectively andb is width of the pile, x1, x2 and x3 are pile positions for the first, the second and the thirdregions, t is time variable.

Assuming that the motion is harmonic we substitute for wj(zj , t) and φj (rj , t) the follow-ing:

wj(zj , t) = wj(zj ) · sin(ω · t) (10)

φj (zj , t) = φj (zj ) · sin(ω · t) (j = 1, 2, 3) (11)

and obtain a system of two coupled ordinary equation for each region as:

− 68

105· EIx

L2· d2φj (zj )

dz2j

+ 16

105· EIx

L3· d3wj(zj )

dz3j

+ 8

15· AG

·[φj (zj ) + 1

L· dwj (zj )

dzj

]= 0 (j = 1, 2, 3) (12)

m · ω2 · w1 (z1) + 8

15· AG

L·[dφ1(z1)

dz1+ 1

L· d2w1(z1)

dz21

]+ 16

105· EIx

L3

· d3φ1(z1)

dz31

− 1

21· EIx

L4· d4w1(z1)

dz41

− Nr · π2 · EIx

L4· d2w1(z1)

dz21

= 0 (13)

m · ω2 · wk(zk) + 8

15· AG

L·[dφk(zk)

dzk

+ 1

L· d2wk(zk)

dz2k

]+ 16

105

· EIx

L3· d3φk(zk)

dz3k

− 1

21· EIx

L4· d4wk(zk)

dz4k

− CS(k−1) · wk(zk) − Nr

· π2 · EIx

L4· d2wk(zk)

dz2k

= 0 (k = 2, 3), (14)

where z = xL

, dimensionless position parameter; Nr = N ·L2

π2·EIx, non-dimensionalized multi-

plication factor for the axial compressive force and ω is natural frequency of the pile.It is assumed that the solution is

wj(zj ) = Cj · ei·sj ·zj (15)

φj (zj ) = Pj · ei·sj ·zj (j = 1, 2, 3) (16)

and substituting Eqs. (15) and (16) into Eqs. (12), (13) and (14) results in(

8

15· AG + 68

105· EIx

L2· s2

j

)· Pj

+(

8

15· AG

L· sj · i − 16

105· EIx

L3· s3

j · i

)· Cj = 0 (j = 1, 2, 3) (17)


(8

15· AG

L· s1 · i − 16

105· EIx

L3· s3

1 · i

)· P1

+(

m · ω2 − 8

15· AG

L2· s2

1 − 1

21· EIx

L4· s4

1 + Nr · π2 · EIx

L4· s2

1

)· C1 = 0

(18)

(8

15· AG

L· sk · i − 16

105· EIx

L3· s3

k · i

)· Pk

+(

m · ω2 − 8

15· AG

L2· s2

k − 1

21· EIx

L4· s4

k − CS(k−1) + Nr · π2 · EIx

L4· s2

k

)

· Ck = 0 (k = 2, 3), (19)

where i = √−1.Eqs. (17), (18) and (19) can be written in matrix form for the two unknowns Pj and Cj and

the non-trivial solution will be obtained when the determinant of the coefficient matrix willbe zero for each region, i.e.

[− 4

525· (EIx)

2

L6

]· s6

1 +[

68

105· Nr · π2 · (EIx)

2

L6− 8

15· AG · EIx

L4

]· s4

1

+(

68

105· EIx

L2· m · ω2 + 8

15· Nr · π2 · EIx · AG

L4

)· s2

1

+ 8

15· AG · m · ω2 = 0 (20)

[− 4

525· (EIx)

2

L6

]· s6

k +[

68

105· Nr · π2 · (EIx)

2

L6− 8

15· AG · EIx

L4

]· s4

k

+(

68

105· EIx

L2· (m · ω2 − CS(k−1)

)+ 8

15· Nr · π2 · EIx · AG

L4

)· s2

k

+ 8

15· AG · (m · ω2 − CS(k−1)) = 0. (21)

Thus, we have a sixth-order equation with the unknowns for each region, resulting in sixvalues and the general solution for each region can be written as:

wj(zj , t) = [Cj1 · ei·sj1·zj + Cj2 · ei·sj2·zj + Cj3 · ei·sj3·zj + Cj4 · ei·sj4·zj

+Cj5 · ei·sj5·zj + Cj6 · ei·sj6·zj ] · sin(ω · t) (22)

φj (zj , t) = [Pj1 · ei·sj1·zj + Pj2 · ei·sj2·zj + Pj3 · ei·sj3·zj + Pj4 · ei·sj4·zj

+Pj5 · ei·sj5·zj + Pj6 · ei·sj6·zj ] · sin(ω · t) (j = 1, 2, 3). (23)


The thirty-six constants, Cj1, . . . , Cj6 and Pj1, . . . , Pj6, will be found from Eqs. (17), (18),(19) and boundary conditions.

For each region, the expression for bending rotation w′j (zj , t) is given by

w′j (zj , t) = 1

L· dwj (zj )

dzj

· sin(ω · t) (j = 1, 2, 3). (24)

For each region, the shear force function Qj(zj , t) can be obtained by using Eqs. (22) and(23) as:

Qj(zj , t) = −8 · AG

15·(

φj (zj ) + 1

L· dwj (zj )

dzj

)· sin(ω · t)

+ Nr · π2 · EIx

L3· dwj (zj )

dzj

· sin(ω · t)

+ EIx

21 · L3· d3wj(zj )

dz3j

· sin(ω · t)

− 16 · EIx

105 · L2· d2φj (zj )

dz2j

· sin(ω · t) (j = 1, 2, 3). (25)

Similarly, the bending moment function Mj(zj , t) can be obtained by using Eqs. (22) and(23) as:

Mj(zj , t) =(

− EIx

21 · L2· d2wj(zj )

dz2j

− Nr · π2 · EIx

L2· wj(zj )

+16 · EIx

105 · L· dφj (zj )

dzj

)· sin(ω · t). (26)

For each region, the higher-order moment function Mhj(zj , t) can be obtained as:

Mhj(zj , t) =(

16 · EIx

105 · L2· d2wj(zj )

dz2j

− 68 · EIx

105 · L· dφj (zj )

dzj

)· sin(ω · t) (j = 1, 2, 3).

(27)

3. Obtaining the transfer matrices of the pile

The position for each region is written due to the values of the transverse displacementwj(zj , t), bending rotation w′

j (zj , t), rotation of normal φj (zj , t), shear force Qj(zj , t),bending moment Mh(zj , t) and higher-order moment function Mhj(zj , t) at the locations ofzj and t for Reddy–Bickford pile, as (j = 1, 2, 3):

〈Sj (zj , t)〉T = 〈wj(zj ) w′j (zj ) φj (zj ) Mj(zj ) Mhj (zj ) Qj (zj )〉T · sin(ω.t), (28)

where {Sj (zj , t)} shows the position vector for each region.


All terms of the position vector in Eq. (28), is written reducing the sin(ωt) terms as:

{S1(z1)} =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

w1(z1)

w′1(z1)

φ1(z1)

M1(z1)

Mh1(z1)

Q1(z1)

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭

=

⎡⎢⎢⎢⎢⎢⎢⎢⎣

A11(z1) A12(z1) A13(z1) A14(z1) A15(z1) A16(z1)

A21(z1) A22(z1) A23(z1) A24(z1) A25(z1) A26(z1)

A31(z1) A32(z1) A33(z1) A34(z1) A35(z1) A36(z1)

A41(z1) A42(z1) A43(z1) A44(z1) A45(z1) A46(z1)

A51(z1) A52(z1) A53(z1) A54(z1) A55(z1) A56(z1)

A61(z1) A62(z1) A63(z1) A64(z1) A65(z1) A66(z1)

⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

C11

C12

C13

C14

C15

C16

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭

(29)

{S2(z2)} =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

w2(z2)

w′2(z2)

φ2(z2)

M2(z2)

Mh2(z2)

Q2(z2)

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭

=

⎡⎢⎢⎢⎢⎢⎢⎢⎣

B11(z2) B12(z2) B13(z2) B14(z2) B15(z2) B16(z2)

B21(z2) B22(z2) B23(z2) B24(z2) B25(z2) B26(z2)

B31(z2) B32(z2) B33(z2) B34(z2) B35(z2) B36(z2)

B41(z2) B42(z2) B43(z2) B44(z2) B45(z2) B46(z2)

B51(z2) B52(z2) B53(z2) B54(z2) B55(z2) B56(z2)

B61(z2) B62(z2) B63(z2) B64(z2) B65(z2) B66(z2)

⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

C21

C22

C23

C24

C25

C26

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭

(30)

{S3(z3)} =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

w3(z3)

w′3(z3)

φ3(z3)

M3(z3)

Mh3(z3)

Q3(z3)

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

C11(z3) C12(z3) C13(z3) C14(z3) C15(z3) C16(z3)

C21(z3) C22(z3) C23(z3) C24(z3) C25(z3) C26(z3)

C31(z3) C32(z3) C33(z3) C34(z3) C35(z3) C36(z3)

C41(z3) C42(z3) C43(z3) C44(z3) C45(z3) C46(z3)

C51(z3) C52(z3) C53(z3) C54(z3) C55(z3) C56(z3)

C61(z3) C62(z3) C63(z3) C64(z3) C65(z3) C66(z3)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

C31

C32

C33

C34

C35

C36

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭

(31)

Eqs. (29), (30) and (31) can be written in closed form, as [(j = 1, . . . , 6), (m =1, . . . , 6), (n = 1, . . . , 6)]:

{S1(z1)} = [Ajm(z1)] · {C1n} (32)

{S2(z2)} = [Bjm(z2)] · {C2n} (33)

{S3(z3)} = [Cjm(z3)] · {C3n}. (34)

The coefficient vectors {C1n}, {C2n}, {C3n} are obtained from Eqs. (32), (33) and (34) forz1 = 0, z2 = 0 and z3 = 0 with the condition that the matrices [Ajm(z1)], [Bjm(z2)] and[Cjm(z3)] are not singular. Substituting these coefficient vectors into the Eqs. (32), (33) and(34) respectively, gives,

{S1(z1)} = [F1(z1)] · {S1(0)} (35)

{S2(z2)} = [F2(z2)] · {S2(0)} (36)

{S3(z3)} = [F3(z3)] · {S3(0)} (37)


where;

[F1(z1)] = [Ajm(z1)] · [Ajm(0)]−1 (38)

[F2(z2)] = [Bjm(z2)] · [Bjm(0)]−1 (39)

[F3(z3)] = [Cjm(z3)] · [Cjm(0)]−1 (40)

are the transfer matrices for the first, the second and the third regions of the pile, respectively.The transfer matrices for the three regions can be combined to yield one global transfer

matrix using the characteristics of transfer matrix. Thus, the position vector of the pile end atthe third region can be related to the position vector of the pile end at the first region as

{S1(z1 = 1)} =[F1

(z1 = L1

L

)]·[F2

(z2 = L2

L

)]·[F3

(z3 = L3

L

)]· {S3(z3 = 0)}

(41)

{S1(z1 = 1)} = [F(z = 1)] · {S3(z3 = 0)}, (42)

where [F(z = 1)] shows the global transfer matrix that transfers the values of the positionvector of the pile end at the third region to the values of the position vector of the pile end atthe first region and can be written as:

[F(z = 1)] =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

F11 F12 F13 F14 F15 F16

F21 F22 F23 F24 F25 F26

F31 F32 F33 F34 F35 F36

F41 F42 F43 F44 F45 F46

F51 F52 F53 F54 F55 F56

F61 F62 F63 F64 F65 F66

⎤⎥⎥⎥⎥⎥⎥⎥⎦

, (43)

where the term Fjm shows the terms of the global transfer matrix [F(z = 1)].

4. Obtaining natural frequencies for semi-rigid connected pile

The behaviour of the pile end that is semi-rigid supported against rotation at the first regionis modelled by an elastic spring. The rotational spring rigidities are related with fixity factorthat is defined as below (Monforton & Wu 1963):

f = 1

1 + 3·EIx

Cθ ·L, (44)

where Cθ is the rotational restraining stiffness at the upper end of the pile in the first region.Bending moment function at semi-rigid connected end is written as a linear function of

rotational restraining stiffness and bending rotation as (Wang et al 2000):

M1

(z1 = L1

L

)= −Cθ · φ1

(z1 = L1

L

), (45)


Figure 3. (a) Pile whose end at the first region is semi-rigid connected and not restricted for horizontaldisplacement and the end at the third region is free (The first model). (b) Pile whose end at the firstregion is semi-rigid connected and restricted for horizontal displacement and the end at the third regionis fixed supported (The second model).

where M1(z1 = L1

L

)and φ1

(z1 = L1

L

)are the bending moment and the rotation of normal for

the first region.Similarly, high-order moment function at semi-rigid connected end is written as a lin-

ear function of rotational restraining stiffness and rotation of normal as (Wang et al2000):

Mh1

(z1 = L1

L

)= 0, (46)

where Mh1(z1 = L1

L

)is the high-order moment for the first region.

Using Eq. (42), the position vectors at free and the fixed supported ends are transferredto the position vectors at the semi-rigid connected end by Eqs. (47) and (48), respec-tively. For the first model, the pile’s end at the first region is semi-rigid connected andnot restricted for horizontal displacement and the end at the third region is free as in fig-ure 3a. For the second model, the pile’s end at the first region is semi-rigid connected andrestricted for horizontal displacement and the end at the third region is fixed supported as infigure 3b.

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

w1(z1 = L1

L

)w′

1

(z1 = L1

L

)φ1(z1 = L1

L

)Cθ · w′

1

(z1 = L1

L

)Cθ · φ1

(z1 = L1

L

)0

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

=

⎡⎢⎢⎢⎢⎢⎢⎢⎣

F11 F12 F13 F14 F15 F16

F21 F22 F23 F24 F25 F26

F31 F32 F33 F34 F35 F36

F41 F42 F43 F44 F45 F46

F51 F52 F53 F54 F55 F56

F61 F62 F63 F64 F65 F66

⎤⎥⎥⎥⎥⎥⎥⎥⎦

·

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

w3(z3 = 0)

w′3(z3 = 0)

φ3(z3 = 0)

000

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭

(47)


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

0

w′1

(z1 = L1

L

)φ1(z1 = L1

L

)Cθ · w′

1

(z1 = L1

L

)Cθ · φ1

(z1 = L1

L

)Q1

(z1 = L1

L

)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

=

⎡⎢⎢⎢⎢⎢⎢⎢⎣

F11 F12 F13 F14 F15 F16

F21 F22 F23 F24 F25 F26

F31 F32 F33 F34 F35 F36

F41 F42 F43 F44 F45 F46

F51 F52 F53 F54 F55 F56

F61 F62 F63 F64 F65 F66

⎤⎥⎥⎥⎥⎥⎥⎥⎦

·

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

000

M3(z3 = 0)

Mh3(z3 = 0)

Q3(z3 = 0)

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭

(48)

Eqs. (47) and (48) can be written in matrix form as:

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

F11 F12 F13 −1 0 0F21 F22 F23 0 −1 0F31 F32 F33 0 0 −1F41 F42 F43 0 −Cθ 0F51 F52 F53 0 0 −Cθ

F61 F62 F63 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

·

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

w3(z3 = 0)

w′3(z3 = 0)

φ3(z3 = 0)

w1(z1 = L1

L

)w′

1

(z1 = L1

L

)φ1(z1 = L1

L

)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭

=

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

000000

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭

(49)

⎡⎢⎢⎢⎢⎢⎢⎢⎣

F14 F15 F16 0 0 0F24 F25 F26 −1 0 0F34 F35 F36 0 −1 0F44 F45 F46 −Cθ 0 0F54 F55 F56 0 −Cθ 0F64 F65 F66 0 0 −1

⎤⎥⎥⎥⎥⎥⎥⎥⎦

·

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

M3(z3 = 0)

Mh3(z3 = 0)

Q3(z3 = 0)

w′1

(z1 = L1

L

)φ1(z1 = L1

L

)Q1

(z1 = L1

L

)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

000000

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭

. (50)

For non-trivial solutions of this problem, following relations are written by using Eqs. (49)and (50), as:

∣∣∣∣∣∣∣∣∣∣∣∣∣

F11 F12 F13 −1 0 0F21 F22 F23 0 −1 0F31 F32 F33 0 0 −1F41 F42 F43 0 −Cθ 0F51 F52 F53 0 0 −Cθ

F61 F62 F63 0 0 0

∣∣∣∣∣∣∣∣∣∣∣∣∣

= 0 (51)

∣∣∣∣∣∣∣∣∣∣∣∣∣

F14 F15 F16 0 0 0F24 F25 F26 −1 0 0F34 F35 F36 0 −1 0F44 F45 F46 −Cθ 0 0F54 F55 F56 0 −Cθ 0F64 F65 F66 0 0 −1

∣∣∣∣∣∣∣∣∣∣∣∣∣

= 0. (52)


5. Numerical analysis and discussions

In this paper, for numerical analysis, two models that are partially embedded in Winklersoil and whose ends above the soil are semi-rigid connected by an elastic spring having therotational spring rigidities of Cθ , as one being not restricted for horizontal displacement andthe other being restricted for horizontal displacement, respectively, are considered. For twoexamples, natural frequencies of the pile, ωi(i = 1, 2, 3) are calculated by using computerprograms prepared by authors. Natural frequencies are found by determining values for whichthe determinant of the coefficient matrix is equal to zero. There are various methods forcalculating the roots of the frequency equation. One commonly used and simple technique isthe secant method in which a linear interpolation is employed. The eigenvalues, the naturalfrequencies, are determined by a trial and error method based on interpolation and the bisectionapproach. One such procedure consists of evaluating the determinant for a range of frequencyvalues, ωi . When there is a change of sign between successive evaluations, there must be aroot lying in this interval. The iterative computations are determined when the value of thedeterminant changed sign due to a change of 10−4 in the value of ωi .

For each example, CS1 = 15000 kN/m3 and CS2 = 60000 kN/m3. The length of the pile istaken as 15 m and 30 m.

The all numerical results of this paper are obtained based on uniform, rectangular Reddy–Bickford and Timoshenko piles with the following data:

h = 0·50 m; b = 0·30 m; EIx = 6·5625 × 103 kNm2;AG = 121500 kN; m = 0·50 kN.sec2/m;

for the axial force effect Nr = 0·25 and 1·00, fixity factors are taken as f = 0·25 andf = 0·75.

The all numerical results are given for the following three models: Timoshenko model withtwo values for shear correction factor k and Reddy–Bickford model. Many values for theshear correction factor k were suggested, but in this paper, the original values suggested byTimoshenko k = 5

6 and k = 1417 are used (Gruttmann & Wagner 2001).

Natural frequencies of the pile are obtained from the solution of Eqs. (51) and (52) accordingto the boundary conditions, by using the computer program for Nr = 0·25 and Nr = 1·0 withthe values of L, L1/L, L2/L and L3/L taken from table 1.

For L = 15 m, the frequency values obtained for the first three modes of Reddy–Bickfordand Timoshenko piles whose end at the first region is semi-rigid connected and not restrictedfor horizontal displacement and the end at the third region is free are presented in table 2;for L = 30 m, the frequency values obtained for the first three modes of the same piles arepresented in table 3 being compared with the frequency values obtained for Nr = 0·25 and1·00, f = 0·25 and 0·75.

Table 1. Values of L1, L2 and L3 with respect to values of L, L1/L, L2/L and L3/L.

L1/L = 0·50 L2/L = 0·30 L3/L = 0·20 L1/L = 0·50 L2/L = 0·20 L3/L = 0·30

L(m) L1(m) L2(m) L3(m) L(m) L1(m) L2(m) L3(m)

15 7·5 4·5 3·0 15 7·5 3·0 4·530 15·0 9·0 6·0 30 15·0 6·0 9·0


Tabl

e2.

The

first

thre

ena

tura

lfre

quen

cies

ofR

eddy

–Bic

kfor

dan

dT

imos

henk

opi

les

who

seen

dat

the

first

regi

onis

sem

i-ri

gid

conn

ecte

dan

dno

tre

stri

cted

for

hori

zont

aldi

spla

cem

enta

ndth

een

dat

the

thir

dre

gion

isfr

ee,L

=15

m.

L=

15m

f=

0·25

L1/L

=0·5

0L

2/L

=0·3

0L

3/L

=0·2

0L

1/L

=0·5

0L

2/L

=0·2

0L

3/L

=0·3

0ωi

(rad/sec)N

r=

0·25

Nr=

1·00

Nr=

0·25

Nr=

1·00

TB

TT

BT

TB

TT

BT

TB

TT

BT

TB

TT

BT

RB

T(k

=5/

6)(k

=14

/17

)R

BT

(k=

5/6)

(k=

14/17

)R

BT

(k=

5/6)

(k=

14/17

)R

BT

(k=

5/6)

(k=

14/17

)

ω1

6·127

46·0

826

6·082

36·2

982

6·198

56·1

981

6·134

06·0

891

6·088

86·3

051

6·205

46·2

050

ω2

34·00

9733

·8781

33·87

3332

·8500

32·69

5432

·6906

34·03

7733

·9054

33·90

0632

·8792

32·72

4032

·7191

ω3

88·89

1488

·9223

88·90

2887

·5489

87·62

4187

·6048

88·94

3688

·9725

88·95

2987

·6051

87·67

8187

·6587

f=

0·75

L1/L

=0·5

0L

2/L

=0·3

0L

3/L

=0·2

0L

1/L

=0·5

0L

2/L

=0·2

0L

3/L

=0·3

0

Nr=

0·25

Nr=

1·00

Nr=

0·25

Nr=

1·00

ωi

(rad/sec)

TB

TT

BT

TB

TT

BT

TB

TT

BT

TB

TT

BT

RB

T(k

=5/

6)(k

=14

/17

)R

BT

(k=

5/6)

(k=

14/17

)R

BT

(k=

5/6)

(k=

14/17

)R

BT

(k=

5/6)

(k=

14/17

)

ω1

7·657

17·3

756

7·375

17·0

486

6·656

56·6

560

7·665

27·3

834

7·382

97·0

574

6·665

06·6

645

ω2

40·41

8239

·1142

39·10

8039

·2134

37·77

7737

·7716

40·45

0239

·1444

39·13

8139

·2468

37·80

9337

·8031

ω3

96·99

0795

·2717

95·24

9295

·6900

93·90

8093

·8857

97·04

1095

·3205

95·29

7995

·7448

93·96

1093

·9386


Tabl

e3.

The

first

thre

ena

tura

lfre

quen

cies

ofR

eddy

–Bic

kfor

dan

dT

imos

henk

opi

les

who

seen

dat

the

first

regi

onis

sem

i-ri

gid

conn

ecte

dan

dno

tre

stri

cted

for

hori

zont

aldi

spla

cem

enta

ndth

een

dat

the

thir

dre

gion

isfr

ee,L

=30

m.

L=

30m

f=

0·25

L1/L

=0·5

0L

2/L

=0·3

0L

3/L

=0·2

0L

1/L

=0·5

0L

2/L

=0·2

0L

3/L

=0·3

0ωi

(rad/sec)N

r=

0·25

Nr=

1·00

Nr=

0·25

Nr=

1·00

TB

TT

BT

TB

TT

BT

TB

TT

BT

TB

TT

BT

RB

T(k

=5/

6)(k

=14

/17

)R

BT

(k=

5/6)

(k=

14/17

)R

BT

(k=

5/6)

(k=

14/17

)R

BT

(k=

5/6)

(k=

14/17

)

ω1

1·748

91·7

438

1·743

81·7

885

1·776

21·7

761

1·749

01·7

439

1·743

91·7

886

1·776

31·7

762

ω2

9·946

49·9

296

9·929

09·6

504

9·629

59·6

290

9·946

59·9

297

9·929

19·6

505

9·629

69·6

290

ω3

26·99

0326

·9513

26·94

8426

·6369

26·60

0226

·5973

26·99

0426

·9514

26·94

8526

·6370

26·60

0326

·5974

f=

0·75

L1/L

=0·5

0L

2/L

=0·3

0L

3/L

=0·2

0L

1/L

=0·5

0L

2/L

=0·2

0L

3/L

=0·3

0

Nr=

0·25

Nr=

1·00

Nr=

0·25

Nr=

1·00

ωi

(rad/sec)

TB

TT

BT

TB

TT

BT

TB

TT

BT

TB

TT

BT

RB

T(k

=5/

6)(k

=14

/17

)R

BT

(k=

5/6)

(k=

14/17

)R

BT

(k=

5/6)

(k=

14/17

)R

BT

(k=

5/6)

(k=

14/17

)

ω1

2·167

12·1

284

2·128

42·0

088

1·955

11·9

551

2·167

22·1

285

2·128

52·0

089

1·955

21·9

552

ω2

11·65

0911

·4670

11·46

6311

·3302

11·12

7311

·1265

11·65

1011

·4671

11·46

6411

·3303

11·12

7411

·1266

ω3

29·24

9828

·9423

28·93

9028

·8924

28·57

2328

·5690

29·24

9928

·9424

28·93

9128

·8925

28·57

2428

·5691


For L = 15 m, the frequency values obtained for the first three modes of Reddy–Bickfordand Timoshenko piles whose end at the first region is semi-rigid connected and restricted forhorizontal displacement and the end at the third region is fixed supported are presented intable 4; for L = 30 m, the frequency values obtained for the first three modes of the same pilesare presented in table 5 being compared with the frequency values obtained for Nr = 0·25and 1·00, f = 0·25 and 0·75.

The first two natural frequency values of Reddy–Bickford pile are higher than the first twonatural frequency values of Timoshenko pile for both models and for all values of Nr , L andf . For both models, the third natural frequency values of Reddy–Bickford pile are lower forf = 0·25 and L = 15 m; are higher for f = 0·75 and L = 30 m, than the third naturalfrequency values of Timoshenko pile. The third natural frequency values of Reddy–Bickfordpile are higher than the third natural frequency values of Timoshenko pile for both models,L = 30 m and all values of f . The differences between Reddy–Bickford beam theory andTimoshenko beam theory are more prominent for the higher frequencies.

For all boundary conditions, the differences between natural frequency values of Timo-shenko pile models with k = 5

6 and k = 1417 are small.

For f = 0·25 and for the condition of the other variables (L, L1/L, L2/L and L3/L ratios)are constant, as the axial compressive force acting to Reddy–Bickford pile and Timoshenkopile whose end at the first region is semi-rigid connected and not restricted for horizontaldisplacement and the end at the third region is free, is increased, the first natural frequencyvalues of Reddy–Bickford pile and Timoshenko pile are increased but the second and the thirdnatural frequency values are decreased. The natural frequency values of Reddy–Bickford pileand Timoshenko pile are decreased as the axial force is increased for f = 0·25 in the secondmodel and for f = 0·75 at all boundary conditions. These results indicate that, the boundaryconditions (especially the end at the third region) of Reddy–Bickford pile and Timoshenkopile are important for the effect of axial force.

As the total length of the pile is increased, a decrease is observed in natural frequencyvalues for both beam theories; for all boundary conditions and the condition of the othervariables (f, Nr , L1/L, L2/L and L3/L ratios) are being constant. This result indicates thatthe increase in the length of the pile leads to a reduction in natural frequency values for bothbeam theories.

A decrease is observed in natural frequency values of the first three modes of the pile forthe condition of total pile length, Nr and L1/L ratio being constant and of L2/L ratio beinggreater than L3/L ratio, for both support conditions and both beam theories. This decrease ismore prominent for the short piles.

Natural frequencies values are different for two combinations of boundary conditions. Forthe condition of all variables are constant, the first three natural frequency values of the pilewhose end at the first region is semi-rigid connected and restricted for horizontal displacementand the end at the third region is fixed supported are higher than the first three natural frequencyvalues of the pile whose end at the first region is semi-rigid connected and not restricted forhorizontal displacement and the end at the third region is free. This result indicates that thetypes of supporting affect the natural frequency values of the pile.

6. Summary and conclusion

In this study, starting from the governing differential equations of motion in free vibration,transfer matrices are developed by using Reddy–Bickford beam theory and the iterative-


Tabl

e4.

The

first

thre

ena

tura

lfre

quen

cies

ofR

eddy

–Bic

kfor

dan

dT

imos

henk

opi

les

who

seen

dat

the

first

regi

onis

sem

i-ri

gid

conn

ecte

dan

dre

stri

cted

for

hori

zont

aldi

spla

cem

enta

ndth

een

dat

the

thir

dre

gion

isfix

edsu

ppor

ted,

L=

15m

.

L=

15m

f=

0·25

L1/L

=0·5

0L

2/L

=0·3

0L

3/L

=0·2

0L

1/L

=0·5

0L

2/L

=0·2

0L

3/L

=0·3

0ωi

(rad/sec)

Nr=

0·25

Nr=

1·00

Nr=

0·25

Nr=

1·00

TB

TT

BT

TB

TT

BT

TB

TT

BT

TB

TT

BT

RB

T(k

=5/

6)(k

=14

/17

)R

BT

(k=

5/6)

(k=

14/17

)R

BT

(k=

5/6)

(k=

14/17

)R

BT

(k=

5/6)

(k=

14/17

)

ω1

23·50

2723

·4191

23·41

5922

·0924

22·03

0622

·0275

23·52

3923

·4399

23·43

6622

·1163

22·05

2422

·0492

ω2

72·22

6172

·1754

72·15

8570

·8519

70·66

2570

·6458

72·27

5372

·2225

72·20

5570

·9039

70·71

2570

·6957

ω3

139·2

844

139·6

384

139·5

974

137·8

130

138·2

810

138·2

401

139·3

032

139·6

627

139·6

217

137·8

316

138·3

044

138·2

635

f=

0·75

L1/L

=0·5

0L

2/L

=0·3

0L

3/L

=0·2

0L

1/L

=0·5

0L

2/L

=0·2

0L

3/L

=0·3

0

Nr=

0·25

Nr=

1·00

Nr=

0·25

Nr=

1·00

ωi

(rad/sec)

TB

TT

BT

TB

TT

BT

TB

TT

BT

TB

TT

BT

RB

T(k

=5/

6)(k

=14

/17

)R

BT

(k=

5/6)

(k=

14/17

)R

BT

(k=

5/6)

(k=

14/17

)R

BT

(k=

5/6)

(k=

14/17

)

ω1

28·37

5227

·1491

27·14

4627

·2392

25·99

2125

·9877

28·39

9327

·1718

27·16

7327

·2642

26·01

5826

·0114

ω2

78·14

4176

·3510

76·33

1076

·6952

74·93

4574

·9148

78·19

3976

·3984

76·37

8376

·7481

74·98

4874

·9650

ω3

144·8

557

143·3

242

143·2

794

143·4

875

142·0

368

141·9

921

144·8

906

143·3

586

143·3

137

143·5

151

142·0

662

142·0

215


Tabl

e5.

The

first

thre

ena

tura

lfr

eque

ncie

sof

Red

dy–B

ickf

ord

and

Tim

oshe

nko

pile

sw

hose

end

atth

efir

stre

gion

isse

mi-

rigi

dco

nnec

ted

and

rest

rict

edfo

rho

rizo

ntal

disp

lace

men

tand

the

end

atth

eth

ird

regi

onis

fixed

supp

orte

d,L

=30

m.

L=

30m

f=

0·25

L1/L

=0·5

0L

2/L

=0·3

0L

3/L

=0·2

0L

1/L

=0·5

0L

2/L

=0·2

0L

3/L

=0·3

0ωi

(rad/sec)N

r=

0·25

Nr=

1·00

Nr=

0·25

Nr=

1·00

TB

TT

BT

TB

TT

BT

TB

TT

BT

TB

TT

BT

RB

T(k

=5/

6)(k

=14

/17

)R

BT

(k=

5/6)

(k=

14/17

)R

BT

(k=

5/6)

(k=

14/17

)R

BT

(k=

5/6)

(k=

14/17

)

ω1

6·848

26·8

402

6·839

86·4

948

6·488

66·4

882

6·848

36·8

403

6·839

96·4

949

6·488

66·4

883

ω2

21·72

8321

·6986

21·69

6321

·3219

21·29

7521

·2952

21·72

8421

·6987

21·69

6421

·3220

21·29

7621

·2953

ω3

44·66

9444

·6216

44·61

4044

·2449

44·20

7544

·2001

44·66

9544

·6217

44·61

4144

·2450

44·20

7644

·2002

f=

0·75

L1/L

=0·5

0L

2/L

=0·3

0L

3/L

=0·2

0L

1/L

=0·5

0L

2/L

=0·2

0L

3/L

=0·3

0

Nr=

0·25

Nr=

1·00

Nr=

0·25

Nr=

1·00

ωi

(rad/sec)

TB

TT

BT

TB

TT

BT

TB

TT

BT

TB

TT

BT

RB

T(k

=5/

6)(k

=14

/17

)R

BT

(k=

5/6)

(k=

14/17

)R

BT

(k=

5/6)

(k=

14/17

)R

BT

(k=

5/6)

(k=

14/17

)

ω1

8·065

77·9

103

7·909

87·7

722

7·613

77·6

132

8·065

87·9

104

7·909

97·7

723

7·613

87·6

133

ω2

23·23

0422

·9722

22·96

9622

·8494

22·59

3522

·5909

23·23

0522

·9723

22·96

9722

·8495

22·59

3622

·5910

ω3

46·32

9145

·9953

45·98

7145

·9199

45·59

4545

·5864

46·32

9245

·9954

45·98

7245

·9200

45·59

4645

·5865


based computer programs are developed for solution of linear-homogeneous frequency equa-tion set relating to free vibration of different supported two piles partially embedded inelastic soil. Variation in free vibration natural frequencies for the first three modes of thepile is investigated for Nr = 0·25 and Nr = 1·0, f = 0·25 and f = 0·75, due to sup-porting conditions of pile ends and different lengths of the pile. Natural frequency val-ues obtained from Reddy–Bickford beam theory are compared with the results of Tim-oshenko beam theory. As shown, the differences between Reddy–Bickford beam theoryand Timoshenko beam theory become more prominent for free vibration of piles embed-ded in elastic soil. So that, to be on safer side it is recommended to use the higher-ordertheory.

Notation

The following symbols are used in this paper.

A cross-section area of the pileb width of the pileCR1 modulus of subgrade reaction for the second regionCR2 modulus of subgrade reaction for the third regionCθ rotational restraining stiffness at the upper end of the pile in the first regionE Young’s modulusf fixity factor[F(z = 1)] global transfer matrixFjm terms of the global transfer matrixG shear modulush height of the pileIx moment of inertiak shape factor due to cross-section geometryL total length of the pileL1 pile length above the soilL2 pile length embedded in the second regionL3 pile length embedded in the third regionm mass per unit length of the pileMh(zj , t) bending moment function for j th regionMhj(zj , t) higher-order moment function for j th regionN axial compressive forceNr non-dimensionalized multiplication factor for the axial compressive forceQj(zj , t) shear force function for j th region{Sj (zj , t)} position vector for j th regionz dimensionless position parametert time variablewj(zj , t) transverse displacement function for j th regionw0(x, t) lateral displacement of the beam neutral axisw′

j (zj , t) bending rotation function for j th regionxj position for j th regionφj (zj , t) rotation of a normal to the axis of the pile for j th regionω natural frequency


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free vibration of semi-rigid connected reddy–bickford

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