CORE Report No. 2004-03
DYNAMICS OF FLUIDCONVEYING BEAMS
by
J. N. REDDY and C.M. WANG
Centre for Offshore Research and Engineering
National University of Singapore
DYNAMICS OF FLUID-CONVEYING BEAMS:
Governing Equations and Finite Element Models
J. N. REDDY
Department of Mechanical Engineering
Texas A&M University
College Station, TX 77843-3123
C. M. WANG
Centre for Offshore Research and Engineering
Department of Civil Engineering
National University of Singapore
10 Kent Ridge Crescent, Singapore 119260
August 2004
2
DYNAMICS OF FLUID-CONVEYING BEAMS:
Governing Equations and Finite Element Models
J. N. REDDY1
Computational Mechanics Laboratory, Department of Mechanical Engineering
Texas A&M University, College Station, TX 77843-3123, U.S.A.
C. M. WANG
Centre for Offshore Research and Engineering
Department of Civil Engineering, National University of Singapore
10 Kent Ridge Crescent, Singapore 119260
Abstract — Equations of motion governing the deformation of fluid-conveying
beams are derived using the kinematic assumptions of the (a) Euler—Bernoulli
and (b) Timoshenko beam theories. The formulation accounts for geometric
nonlinearity in the von Kármán sense and contributions of fluid velocity to the
kinetic energy as well as to the body forces. Finite element models of the resulting
nonlinear equations of motion are also presented.
Keywords — Analytical solution, Beams, Euler—Bernoulli beam theory, finite
element method, fluid-conveying beams, Timoshenko beam theory, transverse
motion, shear deformation.
1. INTRODUCTION
Fluid-conveying beams are found in many practical applications. They are
encountered, for example, in the form of exhaust pipes in engines, stacks of flue
gases, air-conditioning ducts, pipes carrying fluids (chemicals) in chemical and
power plants, risers in offshore platforms, and tubes in heat exchangers and power
plants. The fluid inside the pipe dynamically interacts with the pipe motion,
possibly causing the pipe to vibrate.
Studies of fluid-conveying pipes have been reported in a number of papers. A
survey of the subject by Paidoussis [1] indicates that more than 200 papers have
been written in the open literature. Here we shall not attempt to review the vast
literature on the dynamics of fluid-conveying pipes, but only cite few early papers
and some recent papers that have direct bearing on the present paper.
1
Author to whom correspondence should be sent. e-mail: [email protected]; Tel:
001-979-862-2417; Fax: 001-979-862-3989.
3
The early contributions to the literature are due to Ashley and Haviland [2],
Feodosyev [3], Housner [4], Benjamin [5], and Naguleswaran and Williams [6].
They all studied flexural vibration of a pipe conveying a fluid. Crandall et al.
[7] and Dimarogonas and Haddad [8] developed the equations of motion of fluid
conveying pipes using the kinematics of the Euler—Bernoulli beam theory.
Paidoussis and his coworkers [9-14] studied dynamics of pipes conveying fluid
using both the Euler—Bernoulli beam theory and the Timoshenko beam theory
(also see [15-20]). Semler, et al. [14] derived a complete set of geometrically
nonlinear equations of motion of fluid conveying pipes. They accounted for large
strains and assumed the kinematics of the Euler—Bernoulli beam theory. The
equations derived in [14] are more complete than seen in most other papers and
account for large strains and rotations (only in kinematic sense and not material
sense). However, they are valid only for the Euler—Bernoulli beams without rotary
inertia and are unduly complicated (and perhaps inconsistent because the stressstrain relations used do not account for the material density changes) for the
analysis of fluid-conveying pipes, especially when the pipe does not undergo large
deformation.
The present paper presents simple but complete derivation of the equations
of motion of fluid-conveying pipes with small strains but moderate rotations.
Derivations are presented for the Euler—Bernoulli beam theory and the Timoshenko
beam theory and they are based on energy considerations (i.e., using the dynamic
version of the principle of virtual displacements). The geometric nonlinearity in
the von Kármán sense is included and contributions of fluid velocity to the kinetic
energy as well as to the body forces are accounted for. The resulting nonlinear
equations agree with those of Semler, et al. [14] for the small strains case, and the
present equations include rotary inertia terms as well as transverse shear strains.
Finite element models of the equations of motion are also developed. Numerical
solutions using the finite element method will be presented in Part 2 of this paper
to bring out the effect of transverse shear deformation in the pipe and fluid velocity
on the transverse motion.
2. THE EULER—BERNOULLI BEAM THEORY
2.1 Displacements and Strains
The Euler—Bernoulli hypothesis requires that plane sections perpendicular to the
axis of the beam before deformation remain (a) plane, (b) rigid (not deform),
and (c) rotate such that they remain perpendicular to the (deformed) axis after
deformation (see Reddy [21—23]). The assumptions amount to neglecting the
Poisson effect and transverse strains. The bending of beams with moderately
large rotations but with small strains can be derived using the displacement field
∂w0
(2.1)
, w(x, z, t) = w0 (x, t)
∂x
where (u, w) are the total displacements along the coordinate directions (x, z), and
u(x, z, t) = u0 (x, t) − z
4
u0 and w0 denote the axial and transverse displacements of a point on the neutral
axis at time t.
Using the nonlinear strain-displacement relations and by omitting the large
strain terms but retaining only the square of ∂w0 /∂x (which represents the rotation
of a transverse normal line in the beam), we obtain
εxx
∂u0 1
+
=
∂x
2
Ã
∂w0
∂x
where
ε0xx
∂u0 1
+
=
∂x
2
!2
Ã
Ã
∂ 2 w0
+z −
∂x2
∂w0
∂x
!2
!
≡ ε0xx + zε1xx
, ε1xx = −
(2.2a)
∂ 2 w0
∂x2
(2.2b)
and all other strains are zero.
2.2 Virtual Work
We assume that the beam is hollow and subjected to a transverse distributed
load of q(x, t) along the length of the beam, and suppose that a fluid with velocity
v(t) is being conveyed by the beam. The distributed load may include the weight
of the beam as well as the fluid, and also the hydrostatic pressure if the beam is
submerged in water. The forces on the beam due to the centripetal and tangential
accelerations of the fluid can be accounted for in the potential energy of the loads
(see Fig. 2.1). The fluid velocity v(t) also contributes to the kinetic energy of the
system. Since we are primarily interested in deriving the equations of motion and
the nature of the boundary conditions of the beam that experiences a displacement
field of the form in Eq. (2.1), we will not consider specific geometric or force
boundary conditions here.
Tangential Force:
m f v&
z, w
Centripetal Force:
∂ 2 w0
v2
mf
≈ m f v2
R
∂x 2
∂w
θ =− 0
u0
∂x
Deformed
v cos θ
?
Radius of curvature, R
v w0
Undeformed
θ =−
?
v
v sin θ
v
∂w0
∂x
x, u
Fig. 2.1 An element of fluid-conveying beam in the Euler—Bernoulli beam theory.
5
The dynamic version of the principle of virtual displacements (i.e. Hamilton’s
principle for deformable bodies) is given by
Z T
0
[(δU − δV ) − δK] dt = 0
(2.3)
where δU is the virtual work due to internal forces, δV the virtual work due to
external forces (including those of flowing fluid), and δK is the virtual kinetic
energy in the beam as well as the fluid inside the beam. We have
δU =
δV =
Z LZ
0
Z L∙
0
Ap
³
´
(1)
σxx δε(0)
xx + zδεxx dAdx
qδw0 − mf v 2
∂ 2 w0
(sin θ δu0 + cos θ δw0 )
∂x2
¸
− mf v̇ (cos θ δu0 − sin θ δw0 ) dx
Z LZ
"Ã
(2.4a)
!
(2.4b)
#
∂δ ẇ0
+ ẇ0 δ ẇ0 dAdx
∂x
0
Ap
"
Ã
!Ã
!#
Z LZ
∂δ ẇ0
2 ∂ ẇ0
+
ρf v · δv + z
dAdx
∂x
∂x
0
Af
δK =
ρp
u̇0 − z
∂ ẇ0
∂x
!Ã
δ u̇0 − z
(2.4c)
where ρp is the mass density of the beam material; ρf is the mass density of the
fluid inside the beam; Ap and Af are the cross-sectional areas of the beam material
and beam fluid passage, respectively; mf = ρf Af and mp = ρp Ap are the mass
densities of the fluid and beam, respectively, per unit lenght; and v is the fluid
velocity vector (see Fig. 2.1)
v = (v cos θ + u̇0 ) î + (−v sin θ + ẇ0 ) ĵ, θ = −
∂w0
∂x
(2.5)
2.3 Euler—Lagrange Equations
Substituting for δΠ and δK from Eqs. (2.4a) and (2.4b) into (2.3), we obtain
0=
Z T Z L (³
0
´
(1)
ˆ
ˆ ∂ ẇ0 ∂δ ẇ0
Nxx δε(0)
xx + Mxx δεxx − mp (u̇0 δ u̇0 + ẇ0 δ ẇ0 ) − (Ip + If )
∂x ∂x
0
)
− mf [(v cos θ + u̇0 ) δ (v cos θ + u̇0 ) + (v sin θ − ẇ0 ) δ (v sin θ − ẇ0 )] dxdt
−
Z L∙
0
qδw0 − mf v
2w
0
2
∂x
2∂
¸
(sin θ δu0 + cos θ δw0 ) − mf v̇ (cos θ δu0 − sin θ δw0 ) dx
(2.6a)
6
=
Z T Z L∙
0
0
Ã
´ ∂ 2u
∂ 2 w0
∂Nxx ³
∂ 2 w0
0
+ mp + mf
+
v
−
+
m
v
sin
θ
f
∂x
∂t2
∂x∂t
∂x2
¸
!
+ mf v̇ cos θ δu0 dxdt
Z TZ L
(
Ã
!
³
´ ∂2w
∂ 4 w0
∂w0
0
ˆ
ˆ
Nxx + mp + mf
− (Ip + If ) 2 2
+
∂x
∂t2
∂x ∂t
0
0
2
∂ w0
− q + mf v2 cos θ
− mf v̇ sin θ
∂x2
Ã
!
Ã
!¸)
∙
∂ 2 u0 ∂w0 ∂w0
∂ 2 w0 ∂u0 ∂ 2 w0
−
+
+ sin θ
δw0 dxdt
+ mf v cos θ 2
∂x∂t
∂t ∂x2
∂x∂t
∂t ∂x2
"
Z T(
∂δw0
∂Mxx ∂w0
∂ 3 w0
+
+
+
Nxx − (Iˆp + Iˆf )
Nxx δu0 − Mxx
∂x
∂x
∂x
∂x∂t2
0
∂ 2 Mxx
∂
−
−
2
∂x
∂x
#
− mf v (sin θu̇0 + cos θẇ0 ) δw0
)L
dt
(2.6b)
0
where all the terms involving [ · ]T0 vanish on account of the assumption that all
variations and their derivatives are zero at t = 0 and t = T , and the variables
introduced in arriving at the last expression are defined as follows:
mp =
mf =
Z
Z
½
Ap
Af
Nxx
Mxx
¾
=
Z
Ap
½ ¾
1
σxx dA
z
ρp dA = ρp Ap , Iˆp =
ρf dA = ρf Af , Iˆf =
Z
Ap
Z
(2.7a)
ρp z 2 dA = ρp Ip
(2.7b)
ρf z 2 dA = ρf If
(2.7c)
Af
Thus, the Euler—Lagrange equations of motion are
Ã
!
´ ∂ 2u
∂ 2 w0
∂Nxx ³
∂ 2 w0
0
−
+ mp + mf
+
v
+
m
v
sin
θ
+ mf v̇ cos θ = 0 (2.8)
f
∂x
∂t2
∂x∂t
∂x2
Ã
!
4
³
´ ∂ 2w
∂ 2 Mxx
∂ ∂w0
0
ˆp + Iˆf ) ∂ w0
−
N
−
+
m
+
m
−
(
I
xx
p
f
∂x2
∂x ∂x
∂t2
∂x2 ∂t2
Ã
!
Ã
!¸
∙
∂w0 ∂ 2 w0 ∂ 2 u0
∂ 2 w0 ∂u0 ∂ 2 w0
−
+mf v cos θ 2
+ sin θ
+
∂x∂t
∂t ∂x2
∂t ∂x2
∂x∂t
∂ 2 w0
+mf v2 cos θ
− mf v̇ sin θ = q (2.9)
∂x2
Equations (2.8) and (2.9) represent coupled nonlinear equations among (u0 , w0 ).
To the authors’ knowledge, this is the first time that the more complete version of
equations of motion for fluid conveying beams are derived systematically for the
case of small strains and moderate rotations.
7
2.4 Simplified Cases
The well-known equations governing Euler—Bernoulli beams without the fluid
are obtained by setting v = 0 and mf = 0 in Eqs. (2.8) and (2.9):
∂ 2 Mxx
−
∂x2
∂ 2 u0
∂Nxx
+ mp 2 = 0
∂x
∂t
Ã
!
2
4
∂ w0
∂ w0
∂ ∂w0
−
Nxx + mp 2 − Iˆp 2 2 = q
∂x ∂x
∂t
∂x ∂t
(2.10)
−
(2.11)
where the stress resultants Nxx and Mxx are related to the displacements (u0 , w0 )
by
⎡
Ã
!2 ⎤
1
∂u
∂w
∂ 2 w0
0
0
⎦ , Mxx = −Ep Ip
Nxx = Ep Ap ⎣
+
(2.12)
∂x
2 ∂x
∂x2
If we assume that θ = −∂w0 /∂x is small compared to unity, then cos θ ≈ 1 and
sin θ ≈ θ, and Eqs. (2.8) and (2.9) become (for constant material and geometric
parameters)
³
´
³
00
00 ´
0
0
³
00 ´
0
mp + mf ü0 + mf v̇ − Ep Ap u0 + w0 w0 − mf vw0 ẇ0 + vw0 = 0
³
´
(2.13)
00
0
0
00
mp + mf ẅ0 − (Iˆp + Iˆf )ẅ0 + mf v̇w0 + 2mf v ẇ0 + mf v 2 w0
∙
0000
00
0
00
0
0
00
+Ep Ip w0 − Ep Ap w0 u0 + 1.5w0 (w0 )2 + w0 u0
h
00
0
³
00
0
−mf v u̇0 w0 + w0 ẇ0 w0 + u̇0
¸
´i
=q
(2.14)
where the prime ’ denotes partial differentiation with respect to x and the
superposed dot denotes partial differentiation with respect to time t. Most of
the terms in the above equations agree with those derived by Semler, et al.
[14]. The large strain terms and terms due to externally applied tension (T0 )
and pressurization (P ) are extra in [14]. On the other hand, Semler, et al. [14] do
not account for the rotary inertia term and terms resulting from the approximation
sin θ ≈ θ. It is not clear why the latter is neglected in [14].
By omitting all of the nonlinear terms, we obtain
³
´
³
³
´
00
mp + mf ü0 + mf v̇ − Ep Ap u0 = 0 (2.15)
´
00
0
0
00
0000
mp + mf ẅ0 − (Iˆp + Iˆf )ẅ0 + mf v̇w0 + 2vẇ0 + v2 w0 + Ep Ip w0 = q (2.16)
The linearized equation of motion (2.16), without the rotary inertia term, can be
found in the books by Crandall et al. [7] and Dimarogonas and Haddad [8]. They
derived Eq. (2.16) using the Eulerian formulation for the fluid flow and neglecting
the axial component of displacement [hence, Eq. (2.15) is omitted]. The first
term in (2.16) denotes bulk acceleration, the second term denotes the bulk rotary
8
inertia term, the third term is due to change in flow velocity, the fourth term
is the Coriolis acceleration, and the fifth term is the contribution of centripetal
acceleration. The linearized equations indicate that the axial motion is no longer
damped while the transverse motion is damped by the velocity of the fluid.
3. THE TIMOSHENKO BEAM THEORY
3.1 Displacements and Strains
In the Timoshenko beam theory, the normality condition of the Euler—Bernoulli
hypothesis is relaxed (i.e., the rotation is no longer equal to −∂w0 /∂x; see Fig.
3.1). The rotation of a transverse normal is treated as an independent variable.
Consequently, the transverse shear strain γxz is no longer zero but a constant
through the depth of the beam. The displacement field of the Timoshenko beam
theory is (see Reddy [21])
u(x, z, t) = u0 (x, t) + zφ(x, t),
w(x, z, t) = w0 (x, t)
(3.1)
where φ denotes the rotation of a transverse normal about the y axis.
The nonlinear strain-displacement relations are
εxx
2εxz
Ã
∂u0 1 ∂w0
+
=
∂x
2 ∂x
∂w0
+ φ ≡ γxz
=
∂x
where
ε0xx
∂u0 1
+
=
∂x
2
!2
Ã
Ã
∂φ
+z
∂x
!
≡ ε0xx + zε1xx
(3.2a)
(3.2b)
∂w0
∂x
!2
, ε1xx =
∂φ
∂x
(3.2c)
and all other strains are zero.
γxz
x
φx
−
dw0
dx
w0
−
dw0
dx
z
u0
Fig. 3.1 Kinematics of the Timoshenko beam theory.
9
3.2 Virtual Work
The expressions for the total virtual work done and the virtual kinetic energy
in the Timoshenko beam theory are given by
δΠ =
Z LZ
0
+
+
δK =
Ap
Z L
0
Z L
0
Z LZ
0
+
h
³
´
i
(1)
σxx δε(0)
xx + zδεxx + σxz γxz dAdx −
mf v
2w
0
2
∂x
2∂
Ap
Z LZ
0
Af
h³
´³
u̇0 + z φ̇
³
0
qδw0 dx
(sin θ δu0 + cos θ δw0 ) dx
mf v̇ (cos θ δu0 − sin θ δw0 ) dx
ρp
Z L
(3.3a)
´
i
δ u̇0 + zδ φ̇ + ẇ0 δ ẇ0 dAdx
´
ρf v · δv + z 2 φ̇δ φ̇ dAdx
(3.3b)
where v is the fluid velocity vector given in Eq. (2.5).
3.3 Euler—Lagrange Equations
Substituting for δΠ and δK from Eqs. (3.3a) and (3.3b) [with v given by Eq.
(2.5)] into (2.3), we obtain
0=
Z T Z L(
0
(1)
ˆ
ˆ
Nxx δε(0)
xx + Mxx δεxx + Qx δγxz − mp (u̇0 δ u̇0 + ẇ0 δ ẇ0 ) − (Ip + If )φ̇δ φ̇
0
− qδw0 + mf v2
∂ 2 w0
(sin θ δu0 + cos θ δw0 ) + mv̇ (cos θ δu0 − sin θ δw0 )
∂x2
)
− mf [(v cos θ + u̇0 ) δ (v cos θ + u̇0 ) + (v sin θ − ẇ0 ) δ (v sin θ − ẇ0 )] dxdt
(3.4a)
Ã
=
Z T Z L∙
+
Z T Z L(
0
0
!
´ ∂ 2u
∂ 2 w0
∂ 2 w0
∂Nxx ³
0
+ mp + mf
+
v
−
+
m
v
sin
θ
f
∂x
∂t2
∂x∂t
∂x2
0
"
#
¸
Z TZ L
2φ
∂Mxx
∂
+ Qx + (Iˆp + Iˆf ) 2 δφ dxdt
+ mf v̇ cos θ δu0 dxdt +
−
∂x
∂t
0
0
0
∂
∂Qx
−
−
∂x
∂x
∙
Ã
Ã
!
³
´ ∂ 2w
∂w0
∂ 2 w0
0
2
Nxx + mp + mf
−
q
+
m
v
cos
θ
f
∂x
∂t2
∂x2
∂ 2 w0 ∂u0 ∂ 2 w0
−
+ mf v cos θ 2
∂x∂t
∂t ∂x2
)
− mf v̇ sin θ δw0 dxdt
!
Ã
∂ 2 u0 ∂w0 ∂w0
+
+ sin θ
∂x∂t
∂t ∂x2
!¸
10
+
Z T"
0
∂w0
∂ 3 w0
ˆ
ˆ
Nxx − (Ip + If )
Nxx δu0 + Mxx δφ + Qx δw0 +
∂x
∂x∂t2
− mf v (sin θu̇0 + cos θẇ0 ) δw0
where
Qx = Ks
Z
Ap
#L
dt
(3.4b)
σxz dA
(3.4c)
0
Ks being the shear correction coefficient. For circular pipes with outer radius a
and inner radius b, the shear correction coefficient is given by (see Cowper [24]
and Wang, et al. [25])
Ks =
6(1 + ν)(1 + s)2
b
,
s
=
(7 + 6ν)(1 + s)2 + (20 + 12ν)s2
a
(3.5)
The Euler—Lagrange equations of motion of the Timoshenko beam theory are
Ã
´ ∂ 2u
∂ 2 w0
∂ 2 w0
∂Nxx ³
0
+ mp + mf
+
v
+
m
v
sin
θ
−
f
∂x
∂t2
∂x∂t
∂x2
Ã
!
+ mf v̇ cos θ = 0 (3.6)
!
³
´ ∂ 2w
∂ ∂w0
∂Qx
0
−
Nxx + mp + mf
−
2
∂x
∂x ∂x
∂t
Ã
!
Ã
!¸
∙
2
2
2
2
∂w0 ∂ w0 ∂ u0
∂ w0 ∂u0 ∂ w0
−
+mf v cos θ 2
+ sin θ
+
∂x∂t
∂t ∂x2
∂t ∂x2
∂x∂t
∂ 2 w0
+mf v 2 cos θ
− mf v̇ sin θ = q (3.7)
∂x2
−
∂ 2φ
∂Mxx
+ Qx + (Iˆp + Iˆf ) 2 = 0 (3.8)
∂x
∂t
Equations (3.6)—(3.8) represent coupled nonlinear equations among (u0 , w0 , φ), and
they cannot be be found in the literature.
3.4 Simplified Cases
The equations governing Timoshenko beams without the fluid are obtained by
setting v = 0 and mf = 0 in Eqs. (3.6)—(3.8):
∂ 2 u0
∂Nxx
+ mp 2 = 0
∂x!
∂t
Ã
∂Qx
∂ ∂w0
∂ 2 w0
−
−
Nxx + mp 2 = q
∂x
∂x ∂x
∂t
∂Mxx
∂ 2φ
−
+ Qx + Iˆp 2 = 0
∂x
∂t
−
(3.9)
(3.10)
(3.11)
11
where the stress resultants (Nxx , Mxx , Qx ) are known in terms of the displacements
(u0 , w0 , φ) by the relations
⎡
∂u0 1
+
= Ep Ap ⎣
∂x
2
Ã
!2 ⎤
⎦,
Ã
!
∂φ
∂w0
Nxx
+φ
Mxx = Ep Ip , Qx = Gp Ap Ks
∂x
∂x
(3.12)
Under the assumption that θ = −∂w0 /∂x is small compared to unity (cos θ ≈ 1
and sin θ ≈ θ), Eqs. (2.22)—(2.24) are reduced to (for constant material and
geometric parameters)
³
´
∂w0
∂x
³
00
00 ´
0
0
³
00 ´
0
mp + mf ü0 + mf v̇ − Ep Ap u0 + w0 w0 − mf vw0 ẇ0 + vw0 = 0 (3.13)
³
³
´
0
0
00
mp + mf ẅ0 + mf v̇w0 + 2mf vẇ0 + mf v2 w0
00 ´
0
h
00
0
0
00
0
00 i
−Gp Ap Ks φ + w0 − Ep Ap w0 u0 + 1.5(w0 )2 w0 + w0 u0
∙
00
0
³
00
0
−mf v u̇0 w0 + w0 ẇ0 w0 + u̇0
00
³
´¸
0
= q (3.14)
´
(Iˆp + Iˆf )φ̈ − Ep Ip φ + Gp Ap Ks φ + w0 = 0 (3.15)
The linear equations associated with (3.13)—(3.15) are
³
´
0
0
³
´
00
mp + mf ü0 + mf v̇ − Ep Ap u0 = 0 (3.16)
00
³
0
00 ´
mp + mf ẅ0 + mf v̇w0 + 2mf v ẇ0 + mf v 2 w0 − Gp Ap Ks φ + w0 = q (3.17)
³
´
00
0
(Iˆp + Iˆf )φ̈ − Ep Ip φ + Gp Ap Ks φ + w0 = 0 (3.18)
Most papers on fluid-conveying Timoshenko beams [17—19] do not give the
governing equations but somehow account for various terms in (3.16)—(3.18).
4. FINITE ELEMENT MODELS
4.1 The Euler—Bernoulli Beam Model
The finite element model of the equations of motion (2.8) and (2.9) can be
constructed using the virtual work statement (2.6a). The virtual work statement
over a typical element (xa , xb ) can be written as
0=
⎡
Z T Z xb (
0
xa
∂u0 1
+
Ep Ap ⎣
∂x
2
³
´
Ã
!2 ⎤ Ã
∂w0 ⎦ ∂δu0
∂x
∂w0 ∂δw0
+
∂x
∂x ∂x
!
∂ 2 w0 ∂ 2 δw0
+ Ep Ip
∂x2 ∂x2
∂ ẇ0 ∂δ ẇ0
∂x ∂x
− mf v [cos θ δ u̇0 − (u̇0 sin θ + ẇ0 cos θ) δθ − sin θδ ẇ0 ] − qδw0
)
2
2 ∂ w0
+ mf v
(sin θ δu0 + cos θ δw0 ) + mf v̇ (cos θ δu0 − sin θ δw0 ) dxdt
∂x2
(4.1)
− mp + mf (u̇0 δ u̇0 + ẇ0 δ ẇ0 ) − (Iˆp + Iˆf )
12
which is equivalent to the following two statements:
0=
⎡
Z T Z xb (
0
xa
Ã
∂u0 1
+
Ep Ap ⎣
∂x
2
Ã
∂w0
∂x
!2 ⎤
´
∂δu0 ³
⎦
+ mp + mf ü0 δu0
∂x
!
)
∂ 2 w0
∂ 2 w0
+v
+ mf v sin θ
δu0 + mf v̇ cos θ δu0 dxdt
∂x∂t
∂x2
0=
+
⎡
Z T Z xb (
0
xa
Z T Z xb (³
0
Ã
∂u0 1
+
Ep Ap ⎣
∂x
2
∂w0
∂x
´
!2 ⎤
)
∂w0 ∂δw0
∂ 2 w0 ∂ 2 δw0
⎦
+ Ep Ip
dxdt
2
2
∂x ∂x
∂x
∂x
∂ ẅ0 ∂δw0
mp + mf ẅ0 δw0 + (Iˆp + Iˆf )
∂x ∂x
∙
¸
∂ 2 w0
∂δw0
+ cos θ
δw0
+ mf v − (u̇0 sin θ + ẇ0 cos θ)
∂x
∂x∂t
)
∂ 2 w0
+ mf v2
cos θδw0 − mf v̇ sin θ δw0 − qδw0 dxdt
∂x2
xa
(4.2)
(4.3)
We assume finite element approximations of the form (see Reddy [21—23])
u0 (x, t) =
2
X
uj (t)ψj (x) ,
w0 (x, t) =
j=1
4
X
¯ j (t)ϕj (x)
∆
(4.4)
j=1
¯ 4 (t) ≡ −θ(xb , t)
∆
(4.5)
and ψj (x) are the linear Lagrange interpolation functions, and ϕj (x) are the
Hermite cubic interpolation functions (see Fig. 4.1).
Substituting Eq. (4.5) for u0 (x, t) and w0 (x, t), and δu0 (x) = ψi (x) and
δw0 (x) = ϕi (x) (to obtain the ith algebraic equation of the model) into the weak
forms (4.3) and (4.4), we obtain
∙ 11
¸( 1) ∙
¸ ( 1 ) ∙ 11
¸½ 1¾
½ 1¾
¨
˙
M
0
0
C12
K
K12
∆
F
∆
∆
+
+
=
22
21
22
21
22
2
2
2
¨
˙
0
M
C
C
K
K
∆
F2
∆
∆
(4.6a)
where
¯I
∆1i = ui , ∆2I = ∆
(4.6b)
¯ 1 (t) ≡ w0 (xa , t),
∆
¯ 2 (t) ≡ −θ(xa , t),
∆
¯ 3 (t) ≡ w0 (xb , t),
∆
∆5
Q2
∆2
∆3
∆1
1
he
(a)
2
∆6
∆4
Q5
Q6
Q3
Q1
1
2
Q4
he
(b)
Fig. 4.1 A typical beam finite element with displacement and force degrees
of freedom.
13
for i = 1, 2 and I = 1, 2, 3, 4. The element coefficients are
Mij11
22
MIJ
12
CiJ
21
CIj
22
CIj
11
Kij
12
KiJ
21
KIj
22
KIJ
=
Z xb ³
´
mp + mf ψi ψj dx
xa
Z xb "³
#
´
dϕI dϕJ
=
mp + mf ϕI ϕJ + (Iˆp + Iˆf )
dx
dx dx
xa
Z xb
Z xb
dϕJ
∂w0 dϕJ
=
mf v sin θψi
mf v
dx ≈ −
ψi
dx
dx
∂x
dx
xa
xa
Z xb
Z xb
dϕI
∂w0 dϕI
ψj dx ≈
ψj dx
=−
mf v sin θ
mf v
dx
∂x
dx
xa
xa
Ã
!
Ã
!
Z xb
Z xb
dϕJ
dϕJ
dϕI
dϕI
ϕI −
ϕJ dx ≈
ϕI −
ϕJ dx
=
mf v cos θ
mf v
dx
dx
dx
dx
xa
xa
Z xb
dψi dψj
=
Ep Ap
dx
dx dx
xa
"Ã
!
#
∂w0 dψi dϕJ
d2 ϕJ
1 Z xb
2
+ mf v sin θψi
=
Ep Ap
dx
2 xa
∂x dx dx
dx2
"Ã
!
#
1 Z xb
∂w0 dψi dϕJ
d2 ϕJ
2 ∂w0
≈
− mf v
ψi
Ep Ap
dx
2 xa
∂x dx dx
∂x
dx2
Z xb
∂w0 dϕI dψj
21
12
dx , KIj
=
Ep Ap
= 2KjI
∂x dx dx
xa
⎡
⎤
=
Z xb
xa
+
≈
Fi1 = −
≈
xa
xa
−
FI2 =
Z xb
Z xb
x
a
Z x
b
xa
xa
Z xb
xa
!2
mf v 2 cos θϕI
d2 ϕJ
dx
dx2
⎡
Ã
!2 ⎤
⎦
1 Z xb ⎣
d2 ϕI d2 ϕJ
∂w0
Ep Ip 2
dx
+
Ep Ap
2
dx dx
2 xa
∂x
Z xb
Z xb
Ã
d2 ϕI d2 ϕJ
∂w0
1 Z xb ⎣
Ep Ip 2
dx
+
E
A
p
p
dx dx2
2 xa
∂x
mf v 2
⎦
dϕI dϕJ
dx
dx dx
dϕI dϕJ
dx
dx dx
Z xb
dϕI dϕJ
dϕJ
dx +
ϕI dx
mf v̇
dx dx
dx
xa
mf v̇ cos θψi dx + Qi ≈ −
q ϕI dx −
q ϕI dx +
Z xb
xa
Z xb
xa
Z xb
xa
mf v̇ψi dx + Qi
mf v̇ sin θϕI dx + Q̂I
mf v̇
∂w0
ϕI dx + Q̂I
∂x
(4.7)
for (i, j = 1, 2) and (I, J = 1, 2, 3, 4); Qi and Q̂I denote element end contributions
from the extensional and bending terms, respectively. Note that the coefficient
matrices [K 12 ], [K 21 ] and [K 22 ] are functions of the unknown w0 (x, t). Stiffness
coefficients are also given for the case in which cos θ and sin θ are approximated
as cos θ ≈ 1 and sin θ ≈ θ.
14
4.2 The Timoshenko Beam Model
The finite element model of the Timoshenko beam equations can be constructed
using the virtual work statement in Eq. (3.4a), where the axial force Nxx , the
shear force Qx , and bending moment Mxx are known in terms of the generalized
displacements (u0 , w0 , φ) by Eq. (3.12).
The virtual work statement (3.4a) is equivalent to the following three
statements:
0=
+
0=
⎧
Z T "Z xb ⎨
0
xa
⎡
∂δu0 ⎣ ∂u0 1
+
E
A
p
p
⎩
∂x
∂x
2
Z T Z xb "
0
!2 ⎤⎫
#
⎬
e
e
⎦ dx − Q δu0 (xa , t) − Q δu0 (xb , t) dt
1
4
⎭
Ã
!
#
∂w0
∂x
(mp + mf )ü0 + mf v sin θ v
xa
Z T "Z xb
0
Ã
xa
⎧
∂ 2 w0 ∂ 2 w0
+
+ mf v̇ cos θ δu0 dxdt
∂x2
∂x∂t
(4.8)
⎤⎫
⎡
Ã
!
Ã
!2
∂δw0 ⎨
∂w0
∂w0 ⎣ ∂u0 1 ∂w0 ⎦⎬
+ φ + Ep Ap
+
Gp Ap K
dx
⎭
∂x ⎩
∂x
∂x ∂x
2 ∂x
−
Z xb
Z T Z xb ("
xa
δw0 q
#
dx − Qe2 δw0 (xa , t) − Qe5 δw0 (xb , t)
dt
#
∂ 2 w0
∂ 2 w0
+
− mf v̇ sin θ δw0
(mp + mf )ẅ0 + mf v 2 cos θ
+
m
v
cos
θ
f
∂x2
∂x∂t
xa
0
)
∂δw0
− mf v (sin θ u̇0 + cos θ ẇ0 )
dxdt
(4.9)
∂x
Ã
!
#
Z T Z xb "
∂δφ ∂φ
∂w0
+ Gp Ap Kδφ
+ φ + (Iˆp + Iˆf )φ̈ δφ dxdt
Ep Ip
0=
∂x ∂x
∂x
xa
0
−
Z T
0
[Qe3 δφ(xa , t) + Qe6 δφ(xb , t)] dt
(4.10)
where δu0 , δw0 , and δφ are the virtual displacements. The Qei have the same
physical meaning as in the Euler—Bernoulli beam element, and their relationship
to the horizontal displacement u0 , transverse deflection w0 , and rotation φ is
Qe1 (t) = −Nxx (xa , t),
"
∂w0
Qe2 (t) = − Qx + Nxx
∂x
Qe3 (t)
= −Mxx (xa , t),
Qe4 = Nxx (xb , t)
#
x=xa
Qe6 (t)
,
"
∂w0
Qe5 = Qx + Nxx
∂x
= Mxx (xb , t)
#
x=xb
(4.11)
An examination of the virtual work statements (4.10a)—(4.10c) suggests that
u0 (x, t), w0 (x, t), and φ(x, t) are the primary variables and therefore must be
carried as nodal degrees of freedom. In general, u0 , w0 , and φ need not be
approximated by polynomials of the same degree. However, the approximations
15
should be such that possible deformation modes (i.e. kinematics) are represented
correctly.
Suppose that the displacements are approximated as
u0 (x, t) =
m
X
(1)
ue (t)ψ (x),
j
j
n
X
w0 (x, t) =
j=1
(2)
wje (t)ψj (x), φ(x) =
j=1
p
X
(3)
se (t)ψ (x)
j
j
j=1
(4.12)
where
(α = 1, 2, 3) are Lagrange interpolation functions of degree (m − 1),
(n − 1), and (p − 1), respectively. At the moment, the values of m, n, and p are
arbitrary, that is, arbitrary degree of polynomial approximations of u0 , w0 , and φ
(1)
(2)
may be used. Substitution of (4.12) for u0 , w0 , and φ, and δu0 = ψi , δw0 = ψi ,
(3)
and δφx = ψi into Eqs. (4.10a)—(4.10c) yields the finite element model
(α)
ψj (x)
⎡
M11
⎢
⎣ 0
0
⎤⎧
⎫
⎡
21
Cij
=
=
12
Kij
Z xb
x
Z xab
xa
Z xb
xa
=−
22
=
Cij
11
Kij
⎤⎧
K12
K22
K32
(2)
(1) dψj
mf v sin θψi
dx
xa
Z xb
xa
≈
=
Z xb
xa
xa
Z xb
xa
Mij22
dx,
(4.13)
=
(3) (3)
(Iˆp + Iˆf )ψi ψj dx
Z xb
Z xb
=
(1) (1)
(mp + mf )ψi ψj
mf v sin θ
⎛
(2)
dψi
dx
dx ≈ −
(1)
ψj
(2)
(2) dψj
⎝
mf v cos θ ψi
⎛
dx
dx ≈
−
(2)
Z xb
xa
Z xb
Z xb
(1)
E A ∂w0 dψi
⎣ p p
2
∂x
dx
(2)
dψj
dx
dx
(2)
(2)
∂w0 dψi
(1)
ψj dx
mf v
∂x dx
xa
⎞
(2)
dψi
(2) ⎠
ψj
dx
dx
⎞
(1)
⎡
xa
(2) (2)
(mp + mf )ψi ψj
∂w0 (1) dψj
ψ
dx
mf v
∂x i dx
(2)
dψ
(2) dψj
(2)
− i ψj ⎠ dx
mf v ⎝ψi
dx
dx
(1)
dψi dψj
Ep Ap
dx,
dx dx
21
Kij
⎫
0 ⎪
⎬
⎨u⎪
⎥
23
K ⎦⎪w⎪
K33 ⎩ s ⎭
= ⎪ F2 ⎪
⎩ F3 ⎭
Mij33 =
12
Cij
⎡
0 ⎪
K11
⎬
⎨ u̇ ⎪
⎥
⎢ 21
0 ⎦ ⎪ ẇ ⎪ + ⎣ K
0 ⎩ ṡ ⎭
0
⎧
⎫
1
⎪
⎨F ⎪
⎬
where
Mij11
⎫
⎤⎧
C12
C22
0
⎪
0
0
⎬
⎨ ü ⎪
⎥
⎢ 21
0 ⎦ ⎪ ẅ ⎪ + ⎣ C
M33 ⎩ s̈ ⎭
0
0
M22
0
=
+ mf v
Z xb
xa
2
(1)
(2)
∂w0 dψi dψj
Ep Ap
dx
∂x dx dx
⎤
2 (2)
(1) d ψj ⎦
sin θψi
dx
2
dx
16
22
Kij
≈
Z xb
=
Z xb
xa
xa
+
≈
23
Kij
=
33
Kij
=
Fi1
xa
xa
≈−
Fi2 =
≈
Fi3 =
mf v
2
2 (2)
(2) d ψj
cos θψi
2
dx
!2
(2)
dψi dψj
dx
dx dx
Ã
!2
(2)
dψi dψj
dx
dx dx
(2)
mf v
⎛
(2)
x
x
a
Z xb ³
(2)
(2)
Z xb
dψj
(2) dψj
dx +
dx
mf v̇ψi
dx dx
dx
xa
(2)
2 dψi
(2)
⎝Ep Ip
Z xab
(2)
dx
dψ
(3)
32
Gp Ap Ks i ψj dx = Kji
dx
Z xb
x
Z xab
⎤
Ã
(2)
dψ dψj
dw0
1 Z xb
dx +
Gp Ap Ks i
Ep Ap
dx dx
2 xa
dx
xa
Z xb
=−
(2)
Z xb
Z xb
(2)
(2)
dψi dψj
dw0
1 Z xb
dx +
Gp Ap Ks
Ep Ap
dx dx
2 xa
dx
xa
xa
(2)
(1)
E A ∂w0 dψi dψj
∂w0 (1) d2 ψj ⎦
⎣ p p
dx
− mf v 2
ψ
2 ∂x dx dx
∂x i dx2
Z xb
Z xb
−
⎡
⎞
(3)
(3)
dψi
dψj
(3) (3)
+ Gp Ap Ks ψi ψj ⎠ dx
dx dx
(1)
mf v̇ cos θψi
(1)
mf v̇ψi
(1)
(1)
dx + Qe1 ψi (xa ) + Qe4 ψi (xb )
(1)
(1)
dx + Qe1 ψi (xa ) + Qe4 ψi (xb )
´
(2)
q + mf v̇ sin θ ψi
(2)
(2)
dx + Qe2 ψi (xa ) + Qe5 ψi (xb )
(2)
(2)
(2)
ψi q dx + Qe2 ψi (xa ) + Qe5 ψi (xb )
xa
(3)
(3)
Qe3 ψi (xa ) + Qe6 ψi (xb )
(4.14)
(α)
The choice of the approximation functions ψi dictates different finite element
models. When all field variables are interpolated with linear functions, the
element is known to experience shear and membrane locking. To avoid shear and
membrane locking (see Reddy [23]), one must use reduced integration to evaluate
the transverse shear coefficients as well as the nonlinear terms.
5. PRELIMINARY NUMERICAL RESULTS
5.1 Static Nonlinear Analysis
First, we present results for the case of geometrically nonlinear analysis to
illustrate the effect of transverse shear deformation on the nonlinear deflections.
We use linear approximation of the axial displacement u0 and Hermite cubic
approximation of w0 in the Euler—Bernoulli beam theory. In the Timoshenko
beam theory, linear interpolation of both u0 and w0 is used. Reduced integration
is used to alleviate the shear and membrane locking (see Reddy [22, 23]).
17
In the static nonlinear analysis, solid beams of rectangular section are used
(b = 1, L = 100, and L/h = 10 or L/h = 100). The load parameter (P = q0 /∆q)
versus nondimensional deflection (w̄ = wEh3 /∆qL3 ) plots for a clamped-clamped
(i.e., both ends are clamped) beam are shown in Fig. 5.1 (∆q is the load
increment). The beam is loaded with uniformly distributed load of intensity q0 .
A mesh of two elements (EBT or TBT elements) in the half beam are used. The
effect of shear deformation is to increase the deflection (i.e., the kinematics of
the Timoshenko beam theory make the beam more flexible). The load parameter
(P = F0 L2 /EI) versus nondimensional deflection (w/L) plots for a are shown in
Fig. 5.2 for a cantilever beam (two Timoshenko beam elements in the full beam
are used) under a point load F0 at the free end.
5.2 Linear Transient Analysis
Linear transient analysis is presented using a rectangular channel cross-section
beams with the following data (see Fig. 5.3)
B = 1.5 × 10−3 m, H = 10.0 × 10−3 m, b = 7 × 10−3 m, h = 1.5 × 10−3 m
E = 2.5 × 108 GPa, G = 108 GPa, ρb = 850 kg/m3 , ρf = 103 kg/m3
Figure 5.3 contains plots of the center deflection versus time for a simply
supported beam under the weight of the beam and fluid but for v = 0, while
Fig. 5.4 contains plots for v = 1 and v = 2. Clearly, the fluid has the effect
of increasing the period of vibration (or reducing the frequency of oscillation).
Additional investigation into the parametric effects of the material density and
fluid density as well as the magnitude of the velocity is warranted. These results
will appear elsewhere.
6. CONCLUDING REMARKS
In this paper the complete set of equations of motion governing fluidconveying beams are derived using the dynamic version of the principle of virtual
displacements. Equations for both the Euler—Bernoulli and Timoshenko beam
theories are developed, and they account for the von Kármán nonlinear strains,
rotary inertia, forces due to the flowing fluid in the beam, and kinetic energy of the
flowing fluid. The resulting equations of motion contain all of the terms derived by
others in the literature for the small strain case, but they also contain additional
terms that were neglected. Finite element models of the governing equations of
both theories are also presented. Preliminary numerical results are presented but
more complete set of results will appear in a separate report.
Acknowledgement. The first author gratefully acknowledges the support of
Department of Civil Engineering at the National University of Singapore for his
stay as the Visiting Professor.
18
5.0
w =w
4.5
L/h = 10 (TBT)
4.0
Nondim. deflection w
Eh 3
, ∆q = Load increment
∆qL3
L/h = 100 (EBT)
3.5
3.0
2.5
L/h = 100 (TBT)
2.0
1.5
L/h = 10 (EBT)
1.0
0.5
0.0
0
10
20
30
Nondim. load
40
50
60
q0 / ∆q
Fig. 5.1 Load versus deflection curves for a clamped beam under uniform
transverse load.
F0
1.50
a/h = 10
Nondim. deflection w
1.25
w=
1.00
2
FL
w
, P = 0
L
EI
0.75
0.50
a/h = 100
0.25
0.00
0 1 2 3 4 5 6 7 8 9 10 11 12
Nondim. load P
Fig. 5.2 Load versus deflection curves for a cantilever beam with an end point
load.
19
h
0.05
TBT (with RI)
EBT (with RI)
Transient response
without fluid
b
H
EBT (without RI)
0.04
Deflection, w
B
0.03
0.02
0.01
0.00
0.0
0.2
0.4
0.6
0.8
1.0
Time, t
Fig. 5.3 Center deflection versus time for a simply supported beam under
uniform transverse load (v = 0).
0.05
EBT (without RI), v = 0
EBT (with RI), v = 1
EBT (with RI), v = 2
Deflection, w
0.04
0.03
0.02
0.01
0.00
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Time, t
Fig. 5.4 Center deflection versus time for a simply supported beam under
uniform transverse load (v = 1 m/s and v = 2 m/s).
20
REFERENCES
1. Paidoussis, M. P., “Pipes Conveying Fluid: A Model Dynamical Problem,”
Proceedings of the Canadian Congress of Applied Mechanics, Winnipeg,
Manitoba, Canada, pp. 1—33, 1991; published as: Paidoussis, M. P. and
Li, G. X., “Pipes Conveying Fluid: A Model Dynamical Problem,” Journal
of Fluids and Structures, 7, 137—204, 1993.
2. Ashley, H. and Haviland, G., “Bending Vibrations of a Pipe Line Containing
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