Dynamic Stiffness Matrices for Linear Members with Distributed Mass

Tamkang Journal of Science and Engineering, Vol. 4, No. 4, pp. 253-264 (2001)

253

Dynamic Stiffness Matrices for Linear Members with Distributed Mass Chih-Peng Yu1 and Jose M. Roesset2 1

Department of Construction Engineering Chaoyang University of Technology Wufeng, Taichung 413, Taiwan E-mail:[email protected] 2 Department of Civil Engineering Texas A&M University College Station, TX 77843, U.S.A. E-mail: [email protected]

Abstract The analytical determination of dynamic stiffness matrices in the frequency domain for linear structural members with distributed mass provides an efficient and accurate procedure for the dynamic analysis of frames. This formulation allows to account not only for the distributed masses but also for wave propagation effects within each member. It provides therefore a more economical solution and a more accurate one than the use of lumped or even consistent mass matrices, which would require dividing the member into various elements to reproduce these effects, particularly for high frequencies. This problem has been discussed recently in relation to the seismic analysis of structures (bridge piers in particular) subjected to vertical ground accelerations. It had always been a major consideration in the interpretation of dynamic non-destructive tests based on impact loads and wave propagation. In this paper the dynamic stiffness matrices for linear members with different formulations are presented together. Key Words: Dynamic Stiffness, Frequency Domain Solution, Continuous Formulation

1. Introduction The dynamic stiffness matrices in the frequency domain for various continuous linear elements with distributed mass, such as a rod, a shaft, and a beam or a beam-column, have been extensively investigated over the past decades. They are sometimes referred to as the ‘exact’ dynamic stiffness matrices. Many researchers have carried out independent studies applying these matrices to the study of different problems and have contributed to

specific topics. For example, the continuous formulation associated with flexural beam theory was used to investigate the accuracy of lumped masses and consistent mass matrices in predicting the natural frequencies of simple frames as early as 1969 by Latona [11]. These studies were later extended by Papaleontiou [15] who compared the responses of various frames to a combination of horizontal and vertical earthquake components using lumped, consistent and distributed masses. A textbook by Koloušek in 1973 [10] covered already

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Chih-Peng Yu and Jose M. Roesset

several aspects of the continuous solution of both prismatic and non-prismatic members. Banerjee et al. summarized a complete set of exact dynamic stiffness matrices for tapered members in 1985 and conducted over the past two decades a series of studies using formulations associated with more rigorous member theories including coupling effects due to torsion [1-3]. Doyle et al published a number of articles regarding the continuous dynamic stiffness formulations for various straight and curved linear members, as well as 2-D plate and shell elements [6-8,12]. Chen et al conducted a comprehensive study of the dynamic response of an axially loaded member on a viscoelastic foundation using a continuous dynamic stiffness formulation based on Timoshenko’s beam theory [4,5]. Yu et al used the continuous formulation to carry out a series of studies on various dynamic problems associated with seismic response as well as non-destructive dynamic testing of structural elements [17-19]. For prismatic members, the exact solution can usually be expressed in terms of exponential functions and thus the derivation of the elements of the exact dynamic stiffness matrix is straightforward. For non-prismatic members, the exact solution can only be obtained for very simple cases and it is normally impossible to obtain it for more sophisticated models. Instead of attempting to derive the exact dynamic stiffness matrices for non-prismatic members, approximations based on weighted residuals, such as the Rayleigh-Ritz method, can be used in the frequency domain. This approach provides often better approximations than the traditional discrete solutions. In this paper we will concentrate however on prismatic members, with various formulations, considering first a member under axial deformation, next a member under torsion, a beam and finally a pile under axial or lateral loads modeled as a member on an elastic, Winkler type, foundation. The basic advantage of using an exact dynamic stiffness formulation in the dynamic analysis of structures is that the number of elements used to model each member can be

minimized without sacrificing the accuracy of the solution. This is especially true for dynamic problems involving high frequencies or wave propagation phenomena as illustrated by the authors [19]. Owing to the fact that the distributed properties can be easily taken into account by the continuous formulation in the frequency domain, this type of formulation is particularly useful for dynamic analyses involving soil structure interaction and fluid structure interaction.

2. General Approach for the Exact Formulation The derivation of the dynamic stiffness matrix associated with the exact continuous solution is carried out using the same basic approach followed to obtain the static stiffness matrix. There are in general four steps as shown in figure 1. The governing equilibrium equation for a linear member is an ordinary linear differential equation of order 2m. The first step in constructing the dynamic stiffness matrix is to obtain the analytical solution for the displacement (solution of the homogeneous equation) in terms of 2m constants of integration. The general solution consists normally of exponential functions for members with uniform cross section and Bessel functions for specific types of tapered members. The second step is to obtain the expression of the nodal displacements in terms of the integration constants, which leads to a frequency-dependent spectral matrix [T 1 ] associated with the end displacement vector. In the third step, another spectral matrix [T 2 ] associated with the end force vector is established. Finally, eliminating the integration constants one can relate the end forces to the end displacements through the dynamic stiffness matrix. The computation of the dynamic stiffness matrices is straightforward from a numerical point of view. Explicit forms can be obtained for most simple cases. General expressions for problems with 2 and 4 degrees of freedom, corresponding to differential equations of order 2m = 2 and 2m = 4, respectively, are presented in the following.

Dynamic Stiffness Matrices for Linear Members with Distributed Mass

255

Solution of the system of differential equations bj expressing the displacement in terms of integration constants or coefficients Cj u(x,ω) = ∑ Cjbj(x,ω)

Expression of the end displacement vector u in terms of a spectral matrix [T1] and the vector of constants C u = [T1] C

Expression of the end force vector F in terms of a spectral matrix [T2] and the vector of constants C F = [T2] C

Elimination of the vector of constants C to relate end displacements to end forces through the dynamic stiffness matrix F = [T2] [T1]-1 u Figure 1. Flowchart of symbolic computation of the exact dynamic stiffness matrix

For linear members with constant cross sectional properties, the solution of the homogeneous equation can be expressed in terms of 2m constants of integration and exponential functions. Substitution of these solutions into the second and fourth order linear differential equations leads normally to characteristic equations in the form of equations (1) and (2), respectively.

r + 2br + a = 0 2

2

(1) (2)

r + 2β r + α = 0 4

2

2

In the second order case, the end displacements can be expressed in terms of constants Cj and the characteristic roots rj as

) ) ) ⎧ u A = u ( 0) ⎫ ⎡ 1 {u} = ⎨ ) ) ⎬ = ⎢ r1 L ⎩u B = u ( L)⎭ ⎣e

1 ⎤ ⎧ C1 ⎫ ⎨ ⎬ = [T1 ]{C} e r2 L ⎥⎦ ⎩C2 ⎭ (3)

with r j = −b ± b 2 − a 2 , j = 1,2 The relation between the end forces and the constants Cj can be written in general as

) ) ) ⎧ FA = − F (0)⎫ ⎡ − K1 F =⎨ ) ) ⎬=⎢ r1L ⎩ FB = F ( L) ⎭ ⎣ K1e

{}

− K 2 ⎤ ⎧ C1 ⎫ ⎨ ⎬ = [T2 ]{C} K 2 e r2 L ⎥⎦ ⎩C 2 ⎭

(4) in which the K terms are stiffness coefficients. The two by two dynamic stiffness matrix [St] can then be obtained eliminating the constant Cj in equations (3) and (4) as ) ) F = [T2 ][T1]−1{u}

{}

⎡ K2er1L − K1er2 L K1 − K2 ⎤ ) ⎢ ⎥{u} er2 L − er1L ⎢⎣(K1 − K2 )e(r1 +r2 ) L K2er2 L − K1er1L ⎥⎦ ) = [St ]{u} (5)

=

1

It is worth noting that the dynamic stiffness matrix is symmetric when b is zero, that is r1 + r2 = 0. The displacement at an arbitrary point x can be computed using the exact shape function h(x) associated with the two nodal degrees of freedom. ) The exact shape functions for displacement u (x ) are

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Chih-Peng Yu and Jose M. Roesset

⎧C ⎫ e r2 x ⎨ 1 ⎬ = ⎩C 2 ⎭

{

}

) u ( x ) = e r1 x

(

e r2 ( L − x ) − e r1 ( L − x ) e r2 L − e r1 L

)(e

( r1 + r 2 ) x

)u)

A

+

(

e r2 x − e r1 x e r2 L − e r1 L

)u)

B

(6)

) ) = h A ( x )u A + h B ( x )u B

Similarly, for the fourth order case, the end displacements can be expressed in terms of constants Cj and the characteristic roots r j as

{v)} = [T1 ]{C1 with r j

2

C2

C3

C4 } = [T1 ]{C} T

(7)

= − β ± β 2 − α 2 , j = 1,2,3,4 ,

or, alternatively, setting r1 = r1 , r2 = − r1 , r3 = r2 ,

r4 = − r2 . The frequency-dependent matrix [T 1]

numerically multiplying the two four by four matrices [T2] and [T1]-1 as

[S ] = [T ][T ]

−1

f

[T1 ]4×4

⎡ 1 ⎢ D = ⎢ r11L ⎢ e ⎢ r1 L ⎣⎢ D1e

S f 13 = 4 (D1K2 − D2 K1 ) [D1 sinhr1L − D2 sinhr2 L] / ∆

− D1 e − r1 L − D1e − r1 L

D2 e r2 L D2e r2 L

⎤ ⎥ − D2 ⎥ e − r2 L ⎥ ⎥ − D2e − r2 L ⎦⎥ 1

(8) with Di defined as the magnitude ratio between the two types of displacements used in a fourth order equilibrium equation associated with a given member theory. Next, the end forces can be expressed in terms of the constants of integration Cj as 1

⎫ ⎧4(D2 K 1 + D1 K 2 ) ⎪ ⎪ = ⎨ ⎡(D1 + D 2 )(K 1 + K 2 ) cosh (r1 − r2 )L ⎤ ⎬ / ∆ − 2 ⎪ ⎢− (D − D )(K − K ) cosh (r + r )L ⎥ ⎪ 1 2 1 2 1 2 ⎦⎭ ⎩ ⎣

j =1, 2, 3, 4

1

2 4×4

(11)

⎡(D1 + D 2 ) sinh (r1 − r2 )L ⎤ S f 11 = 2 ⎢ ⎥ (D 2 K 1 − D1 K 2 ) / ∆ ⎣+ (D1 − D 2 ) sinh (r1 + r2 )L ⎦

S f 12

1

{F) }= [T ] {C

1

The explicit formulae associated with equation (11) can also be expanded as

can be written in a general form as

⎡ b j (0) ⎤ ⎢ ⎥ D j b j ( 0) ⎥ ⎢ = ⎢ b j ( L) ⎥ ⎢ ⎥ ⎣⎢ D j b j ( L) ⎦⎥

2

C2

C3 C4 } = [T2 ]4×4 {C}4×1 T

(9) The frequency-dependent matrix [T 2 ] can be arranged in a general form as

[T2 ]4×4

K2 − K1 − K2 ⎤ ⎡ K1 ⎢ ⎥ − R1 − R1 − R2 − R2 ⎥ ⎢ = ⎢− K1er1L K1e−r1L − K2er2 L − K2e−r2 L ⎥ ⎢ ⎥ r1L R1e−r1L R2er2 L R2e−r2 L ⎦⎥ ⎣⎢ R1e (10) with Ki and Ri the stiffness constants corresponding to the two sets of member forces. The computation of the dynamic stiffness matrix [Sf] can be carried out

S f 14 = 4 (D2 K1 − D1 K 2 )[cosh r1 L − cosh r2 L] / ∆ ⎡(D + D2 ) sinh (r1 − r2 )L ⎤ S f 22 = 2(R2 − R1 ) ⎢ 1 ⎥/∆ ⎣− (D1 − D2 ) sinh (r1 + r2 )L ⎦ S f 23 = 4 (R 2 − R1 ) D1 D 2 [cosh r1 L − cosh r2 L ] / ∆

S f 24 = 4 (R 2 − R1 ) [D1 sinh r2 L − D 2 sinh r1 L ] / ∆

S f 33 = S f 11 S f 34 = − S f 12

S f 44 = S f 22

[S ] is a symmetric matrix f

(12)

with

∆ = 8 D1 D2 + 2 ( D1 − D2 ) 2 cosh( r1 + r2 ) L − 2 ( D1 + D2 ) 2 cosh( r1 − r2 ) L As for the second order case, the displacements at an arbitrary position x can be computed using the exact shape functions associated with the corresponding degrees of freedom. The derivations of the exact shape functions in relation to the above fourth order dynamic stiffness matrix are summarized in appendix I.

Dynamic Stiffness Matrices for Linear Members with Distributed Mass

3.2 Love Theory

3. Dynamic Stiffness Matrices for an Axial Member 3.1 Elementary Theory Consider first an axial member of uniform cross sectional properties, modulus of elasticity E, cross-sectional area A, mass density ρ. The governing differential equation can be expressed as

∂F ∂ 2u ∂ 2u = EA 2 = ρA 2 ∂x ∂x ∂t

(13)

In the frequency domain, the governing differential equation at the specific frequency ω becomes

uˆ ′′ +

ω 2 ρA EA

uˆ = 0

(14)

with the prime representing the spatial derivative. The corresponding characteristic equation can then be expressed as

r +a =0 2

2

(15)

with the two roots

ρAω 2 , j = 1,2 r j = ±ia = ±i EA Based on the dynamic stiffness formulae illustrated in the previous section, the resulting dynamic stiffness matrix for this elementary axial member can be derived with Ki = ri EA as

⎡e iaL + e −iaL ⎢ −2 e iaL − e −iaL ⎢⎣ −1 ⎤ aEA ⎡cos aL = ⎢ −1 cos aL ⎥⎦ sin aL ⎣

[S a ] =

iaEA

257

⎤ −2 ⎥ e iaL + e −iaL ⎥⎦ (16)

From the point of view of wave propagation the elementary rod theory fails to depict the dispersion phenomenon of the longitudinally guided waves. A more realistic theory requires at least the consideration of the lateral contraction of the cross sections. The simplest modification is the so-called Love theory of rods in which the effect associated with Poisson's ratio is taken into account without increasing the number of degrees of freedom per node. The dispersion phenomenon predicted by the Love theory is only reasonable within a range of relatively low frequencies. For cases requiring more accurate dynamic behavior beyond the low frequency range, it may be necessary to employ the two-mode Mindlin-Herrmann theory in which an additional nodal degree of freedom is used to describe the lateral contraction.

It can be shown that the frequency domain differential equation associated with the Love theory results in a form similar to that of the elementary case with an additional inertial term from lateral contraction motion as [9]

ρI ⎞ ) ⎛ Fˆ ′ = E ⎜1 − ω 2ν 2 P ⎟ Au ′′ EA ⎠ ⎝ ) ) = E eff (ω ) Au ′′ = −ω 2 ρAu (17) where ν is Poisson's ratio and IP is the polar moment of inertia of the cross section. The final result for the Love theory is then clearly identical to those derived for the elementary one except for the replacement of Young's modulus E with a newly defined effective Young's modulus Eeff for each frequency. 3.3 Mindlin-Herrmann (two-mode) Theory In this two-mode theory, an additional degree of freedom to represent the lateral contraction is included in the formulation. Following the derivation proposed by Mindlin and Herrmann [13] with u and ψ standing for the uniform axial displacement and the contraction strain at the outer fiber of the cross section, the frequency domain governing equations and boundary conditions associated with an axial element with uniform circular cross sections can be written as

) ) ) ) k 2µI Pψ ′′ − 4k12 (λ + µ) Aψ − 2k12λAu′ = −ω 2 ρI Pψ )

) ) 2λAψ ′ + ( λ + 2 µ ) Au ′′ = −ω 2 ρAu

(18a) (18b)

Fˆ = 2λAψˆ + ( λ + 2 µ ) Auˆ ′ or specified uˆ Qˆ = k 2 µI Pψˆ ′ or specified ψˆ ′

(18c) (18d)

with λ and µ Lame's constants of elasticity, Fˆ and Qˆ the corresponding member forces, axial force and contraction moment, respectively. k and k1 are adjustment coefficients defined to match certain characteristic frequencies of the theory with the 3-D exact theory. In reference [13], the values of the two adjustment parameters were selected based on matching the limiting value of the first mode wave and setting a common tangent point for dispersion curves of different Poisson’s ratios. The proposed formulae for k and k1 were

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Chih-Peng Yu and Jose M. Roesset

c R2 ⎛ 0.862 + 1.14ν ⎞ k = 2 ≅⎜ ⎟ cS ⎝ 1 +ν ⎠ 2 2 k1 = 0.422( 2 − k )

2

2

(19b)

(19a) The governing differential equations can be rearranged combining equations (18a) and (18b) in terms of the axial displacement uˆ as ) IV ⎤ ⎡ 2 ⎥ ⎢k µI P u ⎢ ⎛ 2 Wx ⎞ ⎥ ) ) ( λ + 2 µ ) ⎢ ⎜ k µI P ( λ + 2 µ ) A ⎟ ) ⎥ u ′′ − k12 [EAu ′′ + W x u ] = 0 + 4( λ + µ ) ⎢ ⎜ ⎥ ⎟ ⎠ ⎥ ⎢ ⎝ + Wc ) Wx ⎥ ⎢ + Wc ( λ + 2 µ ) A u ⎦⎥ ⎣⎢ (20) with two inertial terms associated with the axial and contracting directions being

Wx = ω 2 ρA and Wc = ω 2 ρI P The characteristic equation can be written as

r 4 + 2 βr 2 + α 2 = 0

(21)

∂ 2θ ∂ 2θ ∂ 2θ ρ = GJ = I (23) P ∂x 2 ∂x 2 ∂t 2 with T the twisting moment and J the torsional stiffness factor dependent on the form and dimensions of the cross section. Owing to the similarity in the governing equations between the axial and torsional vibrations, the dynamic stiffness matrix and shape functions for the torsional vibration of a uniform member can be obtained with substitution of K i = ri GJ into equations (5) and (6). The resulting dynamic stiffness matrix is thus T′ = C

[ST ] =

−1 ⎤ aGJ ⎡cos aL ⎢ cos aL⎥⎦ sin aL ⎣ − 1

with a 2 =

(24)

ω 2 GJ ρI P

5. Dynamic Stiffness Matrices for a Flexural Member

with

2β = and

⎛ Wx 1 (λ + 2 µ ) ⎜ ⎝ A

α2 =

−

4 k12 ( 3λ + 2 µ ) IP k2

(

Wx Wc k 2 µI P ( λ + 2 µ ) A

A ⎞⎟ + ⎠

Wc k 2 µI P

− 4k12 (λ + µ )

)

The final result would be in the form shown in equations (12) with the magnitude ratio of the ) lateral contraction strain ψ to the axial ) displacement u being defined as

Di =

−1 2 λA

((λ + 2µ ) Ar + ) i

Wx ri

(22a)

and the two spectral constants defined for the matrix [T2 ] are

K i = W x ri

(22b)

Ri = k 2 µI P Di ri

(22c)

4. Dynamic Stiffness Matrices for a Member in Torsion 4.1 Elementary Theory The governing differential equation of a member of uniform cross sectional properties with shear modulus G, polar moment of inertia IP, mass density ρ, and torsional rigidity C, subjected to torsion can be written as

5.1 Elementary (Bernoulli beam) Theory Consider first a uniform beam with Young’s modulus E, cross-sectional area A, moment of inertia I, mass density ρ. The governing equilibrium equations and boundary conditions for the transverse displacement can be written in the frequency domain as

∂Mˆ + Vˆ = 0 ∂x ∂Vˆ = −ω 2 ρAvˆ ∂x with

Mˆ = EIvˆ ′′ Vˆ = − Mˆ ′ = − EIvˆ ′′′ The characteristic equation is obtained combining the two governing equations as

EIr 4 − ω 2 ρA = 0

(25)

The final result for the dynamic stiffness matrix can be obtained from equation (12) with

Di = ri K i = ω 2 ρA ri Ri = EI where the characteristic equation gives

ri2 = ± ω 2 ρA EI , i = 1,2

Dynamic Stiffness Matrices for Linear Members with Distributed Mass

259

The dynamic stiffness matrix of this elementary beam can also be expressed in a simpler form as

ri2 = − β ± β 2 − α 2 , i = 1,2

S b11 = EI (sin rL cosh rL + cos rL sinh rL ) r 3 / ∆

with

S b12 = EI (sin rL sinh rL ) r / ∆ 2

S b13 = − EI (sin rL + sinh rL ) r 3 / ∆

S b14 = EI (cosh rL − cos rL ) r 2 / ∆

S b 22 = EI (sin rL cosh rL − cos rL sinh rL ) r / ∆

S b23 = − S b14

S b 24 = − EI (sinh rL − sin rL ) r / ∆

S b33 = S b11 S b34 = − S b12 S b44 = S b22

[Sb ] is a symmetric matrix with

∆ = 1 − cos rL cosh rL r 2 = ω 2 ρA EI The Bernoulli beam theory is inappropriate when dealing with higher modes of flexural vibration as well as the analysis of beams with deep cross sections. In such cases, effects due to rotational inertia and shear deformation of the cross section need to be taken into account. The associated beam theories are sometimes referred to as Rayleigh beam and Timoshenko beam theories, respectively. 5.2 Rayleigh Beam Theory When considering the rotational inertia of the cross section, the frequency domain differential equations can be expressed as

∂Mˆ + Vˆ = −ω 2 ρIvˆ ′ ∂x ∂Vˆ = −ω 2 ρAvˆ ∂x with

Mˆ = EIvˆ ′′ Vˆ = − Mˆ ′ − ω 2 ρIvˆ′ The corresponding characteristic equation is thus

2β =

ω 2 ρI EI

and α 2 = −

ω 2 ρA EI

As in the Bernoulli beam case, the dynamic stiffness matrix associated with Rayleigh theory can also be expressed in a simpler form than the general form shown in equation (12) as has been done in reference [11]. 5.3 Timoshenko Beam (two-mode) Theory When the effect due to shear deformation is also included in the formulation with shear modulus G and shear coefficient κ, an additional degree of freedom, bending rotation of the cross section ϕ, is introduced. The frequency domain equilibrium equations and the boundary conditions can then be written as

∂Mˆ + Vˆ = −ω 2 ρIϕˆ ∂x ∂Vˆ = −ω 2 ρAvˆ ∂x with

Mˆ = EIϕˆ ′ Vˆ = κGA( vˆ ′ − ϕˆ ) The characteristic equation obtained using the transverse displacement vˆ is thus

(κ

)r

ω 2 ρI + ⎛⎜ κGA − 1⎞⎟ω 2 ρA = 0 ⎠ ⎝ The dynamic stiffness matrix can be obtained from equation (12) using

EIr 4 +

EI ω 2 ρA + ω 2 ρI GA

Di = ri +

2

ω 2 ρA 1 κGA ri

K i = ω 2 ρA ri Ri = EIDi ri where the characteristic equation leads to

ri2 = − β ± β 2 − α 2 , i = 1,2 with

2β =

ω 2 ρA κGA

+

ω 2 ρI

ω 2 ρA ω 2 ρI

and α 2 = EI ( κGA − 1) EI

EIr 4 + ω 2 ρIr 2 − ω 2 ρA = 0

5.4 Effect of Axial Force

When employing equation (12) to derive the final result of the dynamic stiffness matrix, the Rayleigh theory leads to an identical matrix to that of Bernoulli theory with the definition for the characteristic root ri changed to

Finally, an axially loaded linear member subjected to lateral loads is considered. The governing differential equations in the frequency domain can be derived as shown in reference [18] as

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Chih-Peng Yu and Jose M. Roesset

) ∂M ) ) ) ) + V = EIϕˆ ′′ + κGA(v ′ − ϕ ) = −ω 2 ρIϕ ∂x (25a)

∂Yˆ ) ) ) = κGA(vˆ ′′ − ϕ ′) + Nv ′′ = −ω 2 ρAv ∂x (25b) where M, Y, V represent the bending moment, transverse and shear forces of the cross section, respectively. A positive N represents a tension force. Combining the above two equations and neglecting higher order terms,

N κGA

and

N EA

, versus 1 give

then ⎡ ω 2 ρI ⎤ ) ) ⎡ EI 2 ⎤) ω ρA + ω 2 ρI − N ⎥ v ′′ + ω 2 ρA⎢ EIv IV + ⎢ − 1⎥ v = 0 ⎣ κGA ⎦ ⎥⎦ ⎣⎢ κGA

(26)

6.1 Second Order Cases In this paper, the three second order cases correspond to the elementary axial and torsional theories and the Love theory. Using the axial member as an example, the consideration of a distributed resisting force fS(x) = -ka(x)u and a distributed viscous damping force fD(x) = -ca(x) u& due to soil springs and dashpots leads to the governing differential equation in the frequency domain as

(

) ) ) Fˆ ′ = EAu ′′ = − ρAω 2u − KV u

)

(29)

with KV (ω ) = k a (ω ) + iωca (ω ) As a result, the corresponding dynamic stiffness matrix is actually of an identical form to that for a frame member except that the expression for the characteristic root rj is changed to

leading to the characteristic equation

r 4 + 2 βr 2 + α 2 = 0

(27)

with

2β =

ω 2 ρA κGA

+

ω 2 ρI EI

−

N EI

ρI α 2 = ωEIρA ( ωκGA − 1) 2

and

r j = ±i

ρAω 2 − K V EA

(30a)

or, when Love theory is considered, in a more general form as

2

(28)

As a result, the dynamic stiffness matrix is the same as that for the previously illustrated Timoshenko beam case changing only the definition of the constant β.

6. Dynamic Stiffness Matrices for Members on an Elastic Foundation When dealing with dynamic analysis of structures involving soil structure interaction, a simple model for piles can be derived using a member on an elastic, Winkler type, foundation as suggested by Novak [14]. More rigorous studies accounting properly for the coupling of forces and displacements in the soil along the pile have shown that the Winkler approximation is indeed a reasonable one for most typical piles [16]. In such a model, the effect of the soil pile interaction can be considered easily by accounting for the soil resistance as a distributed traction on the member surface. The soil resistance is normally computed using dynamic stiffness formulae for the soil derived from various approaches [14,16]. Under the assumptions of this simple model, the dynamic stiffness formulation for a linear member considering also an elastic foundation can be obtained in a straightforward manner using the formulation for a normal frame member constructed in the previous sections.

r j = ±i

ρAω 2 − K V E eff A

(30b)

Similarly, for the torsional element, the consideration of torsional soil springs kT and dashpots cT results in the same dynamic stiffness matrix as that previously derived with only the replacement of the definition for r as

GJω 2 − K T r j = ±i ρI P

(31)

with K T (ω ) = kT (ω ) + iωcT (ω ) 6.2 Fourth Order Cases The fourth order cases correspond to the axial member associated with the two-mode theory and the flexural members with the various considerations. In these fourth order member theories, the consideration of the distributed resisting forces due to soil springs and dashpots simply gives the following modifications to the formulations.

Dynamic Stiffness Matrices for Linear Members with Distributed Mass

ω 2 ρA ⇒ (ω 2 ρA − K V ) for the axial member (32a) ω ρI P ⇒ (ω ρI P − Kψ ) for the axial member 2

2

(32b) ω ρA ⇒ (ω ρA − K f ) for the flexural member 2

2

(32c) ω ρI ⇒ (ω ρI − K r ) for the flexural member (32d) with 2

2

K V (ω ) = k a (ω ) + iωc a (ω )

Kψ (ω ) = kψ (ω ) + iωcψ (ω ) K f (ω ) = k f (ω ) + iωc f (ω ) K r (ω ) = k r (ω ) + iωcr (ω ) where KV, Kψ, Kf and Kr are the dynamic stiffness functions standing for the restraining effects in the axial, lateral contracting, transverse, and bending rotational directions due to the soil pile interaction, respectively.

7. Conclusions Over the years, the continuous formulation has been proven to be an efficient approach used to obtain accurate results for the dynamic response of linear structures. The major advantage of using an exact continuous formulation is that it takes into account exactly the distributed properties and thus the size of the element can be as large as the structural geometry permits. Owing to the fact that the Fourier transform converts one-dimensional partial differential equations in the time domain to simpler ordinary differential equations in the frequency domain, the frequency domain formulation gives relatively simpler (spectral) relationships between displacements and forces than those derived in the traditional time domain formulations. As a result, the computation of the dynamic stiffness and the corresponding shape functions can be achieved in quite a straightforward manner by implementing these spectral relationships and spectral constitutive equations into computer programs. The dynamic stiffness matrices presented in this work are expressed in a sense that formulae are composed of specific spectral constants. Therefore, these formulae can be applied to cases associated with other member theories with the same number of nodal degrees of freedom, that is 1 or 2 degrees of freedom per node in the second and fourth

261

formulations, respectively. In fact, the computation of continuous dynamic stiffness matrices for member models with any nodal number of degree of freedom can be performed numerically directly from the matrix multiplication of the two spectral constant matrices. Explicit forms of the dynamic stiffness matrices can be derived by expanding terms, but they are too complicated to be derived for models with a large number of nodal degrees of freedom. The consideration of an elastic foundation results in the same dynamic stiffness matrices as those derived without an elastic foundation except that the definition of certain inertial terms has to be modified to account for the restraining effect associated with soil structure interaction. Once a computer program has been implemented based on the formulations of a frame member, it can also be applied to study the dynamic behavior of piles without too much additional work.

Appendix I. Derivation of Exact Shape Functions for the Fourth Order Case Following the derivation shown in the section 2 of this paper, the constants of integration Cj can be expressed in terms of the nodal displacements as, ⎡ d11 ⎢ {C j } = [T1 ]−1{v)} = 1 ⎢⎢d21 ∆ d31 ⎢ ⎢⎣d 41

) d12 e−r1L d 21 − e−r1L d22 ⎤ ⎧v1A ⎫ ⎥⎪) ⎪ d22 er1L d11 − er1L d12 ⎥ ⎪v2 A ⎪ ⎨) ⎬ d32 e−r2L d41 − e−r2L d42 ⎥ ⎪v1B ⎪ ⎥ ) d42 er2L d31 − er2L d32 ⎥⎦ ⎪⎩v2 B ⎪⎭ (A1)

in which

[

]

[

]

[

]

[

]

d11 = D2 2D1 − (D1 + D2 )e−(r1 −r2 ) L − (D1 − D2 )e−(r1+r2 ) L d21 = D2 2D1 − (D1 + D2 )e(r1 −r2 ) L − (D1 − D2 )e(r1 +r2 ) L d31 = D1 2D2 − (D1 + D2 )e(r1 −r2 ) L + (D1 − D2 )e−(r1 +r2 ) L

d41 = D1 2D2 − (D1 + D2 )e−(r1 −r2 ) L − (D1 − D2 )e(r1 +r2 ) L

[

d12 = 2D2 − (D1 + D2 )e −(r1 −r2 ) L + (D1 − D2 )e −(r1 +r2 ) L

[

d 22 = − 2D2 − (D1 + D2 )e( r1 −r2 ) L + (D1 − D2 )e(r1 +r2 ) L

] ]

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Chih-Peng Yu and Jose M. Roesset

[

d 32 = 2D1 − ( D1 + D2 )e ( r1 −r2 ) L − ( D1 − D2 )e −( r1 +r2 ) L

[

d 42 = − 2D1 − (D1 + D2 )e−(r1 −r2 ) L − (D1 − D2 )e(r1 +r2 ) L

]

]

∆ = 8D1 D 2 + 2( D1 − D 2 ) 2 cosh(r1 + r2 ) L − 2( D1 + D 2 ) 2 cosh(r1 − r2 ) L Thus, the first displacement function can be obtained as

) v1 ( x ) = C1e r1 x + C2e − r1 x + C3e r2 x + C4e − r2 x

( (

)

) ⎧ d 11 e r1 x + d 21 e − r1 x + d 31 e r2 x + d 41 e − r2 x v1 A ⎪ ) + d 12 e r1 x + d 22 e − r1 x + d 32 e r2 x + d 42 e − r2 x v 2 A 1 ⎪⎪ = ×⎨ ∆ ⎪ + d 11 e r1 ( L − x ) + d 21 e r1 ( L − x ) + d 31 e r2 ( L − x ) + d 41 e − r2 ( L − x ) ⎪ ⎪⎩− d 12 e r1 ( L − x ) + d 22 e − r1 ( L − z ) + d 32 e r2 ( L − x ) + d 42 e − r2 ( L − x )

(

(

)

⎫ ⎪ ⎪⎪ ) ⎬ v1B ⎪ ) ⎪ v 2 B ⎪⎭

) )

Defining fi(x) as the shape functions, the displacement function can be rearranged as

v1 ( x) = f1 ( x)v1A + f 2 ( x)v2 A + f3 ( x)v1B + f 4 ( x)v2 B

[

(A3)

]

f1( x) = 4 D1D2C1( x) + D12SS2 (L − x) + D22SS1(L − x) ∆ f 2 ( x) = 4 [S1 ( x) + D1SC2 ( L − x) + D2 SC1 ( L − x)] ∆

[

]

f3 ( x) = 4 D1D2C1( L − x) + D12SS2 ( x) + D22SS1( x) ∆ f 4 ( x ) = 4 [S 1 ( L − x ) + D 1 SC 2 ( x ) + D 2 SC 1 ( x ) ] ∆

with

C1 ( x) = cosh r1 x + cosh r2 x − cosh r1 ( L − x) cosh r2 L − cosh r2 ( L − x) cosh r1 L SS1 ( x ) = sinh r1 x sinh r2 L SS 2 ( x ) = sinh r2 x sinh r1 L S1 ( x ) = K1 sinh r2 x + K 2 sinh r1 x SC1 ( x ) = sinh r1 x cosh r2 L − cosh r2 x sinh r1 L SC 2 ( x ) = sinh r2 x cosh r1 L − cosh r1 x sinh r2 L Similarly, the second displacement function v2(x) can be derived as

v2 ( x) = g1 ( x)v1A + g 2 ( x)v2 A + g3 ( x)v1B + g 4 ( x)v2 B

(A4)

(A2)

Dynamic Stiffness Matrices for Linear Members with Distributed Mass

263

with the exact shape functions

g 1(x) = 4D1 D 2[S2 (x) + D1SC1 (L − x) + D 2 SC2(L − x)] ∆

[

]

g 2 ( x ) = 4 D1 D 2 C1 ( x ) + D1 2 SS 1 ( L − x ) + D 2 2 SS 2 ( L − x ) ∆

g 3 ( x ) = 4 D1 D 2 [S 2 ( L − x ) + D1 SC1 ( x ) + D 2 SC 2 ( x ) ] ∆

[

]

g 4 ( x ) = 4 D1 D 2 C1 ( L − x ) + D1 2 SS 1 ( x ) + D 2 2 SS 2 ( x ) ∆ where

S 2 ( x ) = K1 sinh r1 x + K 2 sinh r2 x

References [1] Banerjee, J. R. and William, F. W.," Exact Bernoulli-Euler Dynamic Stiffness Matrix for a Range of Tapered Beams," International Journal for Numerical Methods in Engineering, Vol. 21, pp. 2289-2302 (1985). [2] Banerjee, J. R. and William, F. W., " An exact dynamic stiffness matrix for coupled extentional-torsional vibration of structural members ", Computers and Structures, Vol. 50, pp. 161-166 (1994). [3] Banerjee, J. R. and William, F. W.," Coupled bending-torsional dynamic stiffness matrix for an axially loaded timoshenko beam elements", International Journal of Solids and Structures, Vol. 31, pp. 749-762 (1994). [4] Chen, Y.-H. and Sheu, J.-T., " Axial loaded damped timoshenko beam on viscoelastic foundation", Int. J. for Numerical Methods in Engineering, Vol. 36, pp. 1013-1027 (1993). [5] Chen, Y.-H. and Sheu, J.-T., " Beam length and dynamic stiffness", International Computer Methods in Applied Mechanics and Engineering, Vol. 129, pp. 311-318 (1996). [6] Doyle, J. F., Wave Propagation in Structures - An FFT-Based Spectral Analysis Methodology, Springer-Verlay (1989). [7] Doyle, J. F., Wave Propagation in Structures - Spectral Analysis Using Fast Discrete Fourier Transform, Springer-Verlay, 2nd. Ed. (1997). [8] Gopalakrishnan, S. and Doyle, J. F., " Wave propagation in connected waveguides of

[9] [10] [11]

[12]

[13]

[14]

[15]

[16]

[17]

varying cross-section ", Journal of Sound and Vibration, Vol. 175, pp. 374-363 (1994). Graff, K. F., Wave Motion in Elastic Solids, Dover, U. S. A. (1975). Koloušek, V., Dynamics in Engineering Structures, Butterworths, London (1973). Latona, R. W. "Analysis of Building Frames for Natural Frequencies and Natural Mode Shapes," Inter American Program, Civil Engineering Department, MIT (1969). Martin, M., Gopalakrishnan, S., and Doyle, J. F., " Wave Propagation in Multiply Connected Deep Waveguides ", Journal of Sound and Vibration, Vol. 174, pp. 521-538 (1993). Mindlin, R. D. and Herrmann, G., " A One-Dimensional Theory of Compressional Waves in an Elastic Rod," Proceedings of the First U.S. National Congress of Applied Mechanics, pp. 187-191 (1951). Novak, M., "Dynamic Stiffness and Damping of Piles", Canadian Geotechnical Journal, Vol. 2, pp. 574-598 (1974). Papaleontiou, C. G., "Dynamic Analysis of Building Structures under Combined Horizontal and Vertical Vibrations," Doctoral Dissertation, University of Texas at Austin (1992). Sanchez-Salinero, I., "Static and Dynamic Stiffnesses of Single Piles," Department of Civil Engineering, The University of Texas at Austin, Geotechnical Engineering Report GR82-31 (1982). Yu, C. P., "Determination of Pile Lengths Using Flexural Waves," Civil Engineering Department, The University of Texas at Austin, Report GR95-3, Geotechnical Engineering Center (1995).

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[18] Yu, C. P., "Effect of Vertical Earthquake Components on Bridge Responses," Doctoral Dissertation, University of Texas at Austin, TX, U. S. A. (1996). [19] Yu, C. P. and Roësset, J. M., "Dynamic Analysis of Structures Using Continuous Solutions in Frequency Domain," Proceedings of EASEC-6 Conference on Structural Engineering & Construction, Taipei, Taiwan, Vol.2, pp 1369-1375 (1998).

Manuscript Received: Apr. 1, 2001 And Accepted: Jun. 13, 2001