# The Analysis of Curved Beam Using B-Spline Wavelet on Interval Finite Element Method

Authors: Zhibo Yang; Xuefeng Chen; Yumin He; Zhengjia He; Jie Zhang
Journal: Shock and Vibration
 (2014)

---

## Abstract

A B-spline wavelet on interval (BSWI) finite element is developed for curved beams, and the static and free vibration behaviors of curved beam (arch) are investigated in this paper. Instead of the traditional polynomial interpolation, scaling functions at a certain scale have been adopted to form the shape functions and construct wavelet-based elements. Different from the process of the direct wavelet addition in the other wavelet numerical methods, the element displacement field represented by the coefficients of wavelets expansions is transformed from wavelet space to physical space by aid of the corresponding transformation matrix. Furthermore, compared with the commonly used Daubechies wavelet, BSWI has explicit expressions and excellent approximation properties, which guarantee satisfactory results. Numerical examples are performed to demonstrate the accuracy and efficiency with respect to previously published formulations for curved beams.


---

## Body

## 1. Introduction

Curved beams, which are also called arches in some fields, are well established due to their capacity of transferring loads through the combined action of bending and stretching. They are more efficient than straight beams. However, compared with the analysis of straight beams, the analysis of curved beams is more complex due to the presence of bending stretching coupling. Besides, the effects of shear deformation and rotatory inertia also increase the complexity. Static analysis neglecting these effects can lead to inaccuracies especially when the ratio of length to thickness is small. It is the same with the free vibration analysis, which results in erroneous frequencies and mode shapes for higher modes [1]. Thus, it is difficult to solve this problem exactly due to the aforementioned reasons. Although the energy methods such as Castigliano’s theorem can be used to obtain the deflections, they are only useful in solving some simple problems [2]. The Rayleigh-Ritz is also an alternative method for the analysis of arches. However, its accuracy depends on the selection of displacement function and the chosen displacement function must satisfy the boundary condition, which turns out to restrict the application of the Rayleigh-Ritz method seriously. As an improvement of the Rayleigh-Ritz method, the finite element method (FEM) is widely used now in solving complex boundary problems.In the finite element analysis of curved structures, the use of curved beam element is an efficient alternative to the use of large number of straight beam element to approximate the geometric shape of arches. However, many kinds of successful shape functions used in straight beam result in slow convergence or poor performance when they are applied to arches [3]. In the early studies [4, 5], there was a main view that it is the ignorance of the explicit rigid body motion that results in the slow convergence and poor performance of elements. Meck [6] showed that the unsatisfactory performance of curved elements is not due to the neglect of rigid body motions but due to the neglect of coupling required between normal and tangential displacements to satisfy the condition of inextensibility. Then he suggested the use of an independent interpolation for normal and tangential displacements to get a good result. Yamamoto and Ohtsubo [7] verified that cubic and quartic are the suitable orders of shape functions for curved beam element.Based on their conclusions, a great deal of work was devoted to the development and improvement of curved beam element. Friedman and Kosmatka presented a curved beam element with two nodes and three degrees of freedom per node in [8]; their study took the shear deformable into account. Krishnan and Suresh [9] used a simple cubic linear element to analyze the static and free vibration of arches. By aid of additional internal nodeless degrees of freedom, a higher order hybrid-mixed curved beam element was developed by Kim et al. [10, 11], which enhances the numerical performance of element. In order to improve the element performance, Raveendranath et al. [3, 12] introduced coupled polynomial displacement field into arches analysis. By selecting the trigonometric function as shape function, Leung and Zhu [13] formulated Fourier’s element for the analysis of arch vibration. Using the first-order shear deformation theory, Sabir et al. [14] studied the effect of shear deformation on the vibration of circular arches by FEM. And a structural analysis of a curved beam element defined in global coordinates was established by Gimena et al. [15].The selection of the discrete interpolation function is a key issue in the development of FEM. Wavelet analysis is a new method developed in recent years. In the field of structural analysis, several kinds of wavelet transform functions have been used as interpolation functions to develop wavelet-based finite element formulations for their extraordinary characteristics of integrating the advantages in functional analysis, Fourier’s transform, spline analysis, harmonic analysis, and numerical analysis [16–18]. However, most of the wavelet functions used now lack an explicit expression, which would cause trouble and numerical error when finite element solving equation is formulated [19–21]. Compared with the interpolation wavelet function basis used now, B-spline wavelet on the interval (BSWI) basis has the desirable characteristics of compact support, smoothness, and symmetry in addition to the multiresolution analysis [22, 23]. Moreover, it has an explicit expression, which will not lead to any trouble for differentiation and integration. Furthermore, as a type of generalized spline FEM, BSWI element inherits the superiority of spline for structural analysis [24].Considering the conclusions given by [6, 7], the objective of this study is to present a BSWI (4-order 3-scale) Timoshenko curved beam element for arch analysis. The scaling functions of BSWI wavelets are selected because the scaling functions have excellent analyzing and computing capabilities. The BSWI curved beam element formulation is derived by using the Hamilton principle with the first-order shear deformation theory. The coupling between normal and tangential displacements is considered, and the normal displacement, tangential displacements, and rotation are interpolated by BSWI scaling function, respectively, according to the first-order shear deformation theory. At last, various numerical examples are performed to demonstrate the accuracy and efficiency of the present method with respect to previously published formulations for curved beam.


## 2. Theoretical Development for Curved Beam

### 2.1. The Geometry of Curved Beam

The development of a shear deformable curved beam theory is done along the same line as the development for straight beam (Timoshenko beam). The main differences are that the development of the curved beam is performed in a natural coordinate system and there exists a coupling between normal and tangential displacements. These differences make curved beam require three functions (normal, tangential, and rotational) that are coupled in the differential equations rather than two functions (tangential and rotational) for Timoshenko’s straight beam. Figure1 depicts the geometry of a curved beam having a general cross-section of area A and moment of inertia I about the area’s centroid and curvature of R. In Figure 1, h indicates the thickness of beam, L is the span of beam, and u, w, and θ present the normal, tangential, and rotational displacements, respectively. We define a natural coordinate system as one in which the x-coordinate is coincident with the centroidal curved axis and y-coordinate is coincident with the principal axes of the cross-section. Similar to Timoshenko’s beam, the centroidal axis is the line about which the cross-sections are rotating during bending and therefore represents a zero stress point at each cross-section and is known as the neutral axis. Two parameters are used to depict arches in this study according to the classifications given by [25] the following.(i)
Slenderness ratio (R
/
h):
(1)
thick arch (R
/
h
<
40);
(2)
moderately thick arch (R
/
h
=
40);
(3)
thin arch (R
/
h
>
40).
(ii)
Subtended angle (α (deg.)):
(1)
shallow arch (α
<
40);
(2)
moderately deep arch (α
=
40);
(3)
deep arch (40
<
α
<
180);
(4)
very deep arches (α
>
180).Figure 1
The geometry of curved beam.


### 2.2. Energy Functional of Curved Beam

The strain vectorε
=
[
ε
0
,
κ
,
γ
0
]
T of the curved beam is determined using the definitions for strain in a natural coordinate system, and the nonzero strain components are obtained from the generalized shell theory [10, 11]:
(1)
ε
0
=
∂
u
∂
x
-
v
R
,
κ
=
∂
θ
∂
x
,
γ
0
=
u
R
+
∂
v
∂
x
-
θ
.
For the sake of simplicity, (1) is depicted in the polar coordinate system as
(2)
ε
0
=
1
R
∂
u
∂
α
-
v
R
,
κ
=
1
R
∂
θ
∂
α
,
γ
0
=
u
R
+
1
R
∂
v
∂
α
-
θ
,
where the physical means of α can be found in Figure 1. Using a vector form to rewrite (2), the relations between displacement and strain are
(3)
ε
=
B
ε
d
,
where
(4)
B
ε
=
[
1
R
∂
∂
α
-
1
R
0
0
0
1
R
∂
∂
α
1
R
1
R
∂
∂
α
-
1
]
,
d
=
[
u
,
v
,
θ
]
T
.The strain energy is given by(5)
U
=
1
2
∫
V
ε
T
D
ε
d
V
.
By making use of (3), the strain energy of curved beam is obtained:
(6)
U
=
R
2
∫
0
α
E
(
B
ε
d
)
T
D
B
ε
d
d
α
,
where D
=
[
E
A
E
I
k
G
A
], E and G are Young’s and shear moduli, k is the shear modifying factor, and α
E is used to depict ending angle of arch.Similar to the process of obtaining the strain energy, the kinetic energy of curved beam is obtained by(7)
T
=
1
2
∫
V
ρ
[
(
∂
u
∂
t
)
2
+
(
∂
v
∂
t
)
2
+
(
∂
θ
∂
t
)
2
]
d
V
=
R
2
∫
0
α
E
∂
d
T
∂
t
[
ρ
A
ρ
A
ρ
I
]
∂
d
∂
t
d
α
,
where t depicts time and ρ is the mass density of beam material. Also, the work of the external forces is given by
(8)
W
=
R
∫
0
α
E
d
T
W
d
α
,
W
=
[
f
q
m
]
T
,
where f, q, and m are the distributed axial force, radial force, and moment along the length of beam, respectively. By aid of (6)–(8), the total energy of curved beam is formulated as follows:
(9)
Π
=
U
-
T
-
W
.
Usually, the kinetic energy part is neglected in static analysis, and part of the work from the external forces is neglected when free vibration analysis is needed. Based on (9) and Hamilton’s principle, the corresponding terms of BSWI curved beam element static and free vibration analysis are obtained in Section 4.


## 2.1. The Geometry of Curved Beam

The development of a shear deformable curved beam theory is done along the same line as the development for straight beam (Timoshenko beam). The main differences are that the development of the curved beam is performed in a natural coordinate system and there exists a coupling between normal and tangential displacements. These differences make curved beam require three functions (normal, tangential, and rotational) that are coupled in the differential equations rather than two functions (tangential and rotational) for Timoshenko’s straight beam. Figure1 depicts the geometry of a curved beam having a general cross-section of area A and moment of inertia I about the area’s centroid and curvature of R. In Figure 1, h indicates the thickness of beam, L is the span of beam, and u, w, and θ present the normal, tangential, and rotational displacements, respectively. We define a natural coordinate system as one in which the x-coordinate is coincident with the centroidal curved axis and y-coordinate is coincident with the principal axes of the cross-section. Similar to Timoshenko’s beam, the centroidal axis is the line about which the cross-sections are rotating during bending and therefore represents a zero stress point at each cross-section and is known as the neutral axis. Two parameters are used to depict arches in this study according to the classifications given by [25] the following.(i)
Slenderness ratio (R
/
h):
(1)
thick arch (R
/
h
<
40);
(2)
moderately thick arch (R
/
h
=
40);
(3)
thin arch (R
/
h
>
40).
(ii)
Subtended angle (α (deg.)):
(1)
shallow arch (α
<
40);
(2)
moderately deep arch (α
=
40);
(3)
deep arch (40
<
α
<
180);
(4)
very deep arches (α
>
180).Figure 1
The geometry of curved beam.


## 2.2. Energy Functional of Curved Beam

The strain vectorε
=
[
ε
0
,
κ
,
γ
0
]
T of the curved beam is determined using the definitions for strain in a natural coordinate system, and the nonzero strain components are obtained from the generalized shell theory [10, 11]:
(1)
ε
0
=
∂
u
∂
x
-
v
R
,
κ
=
∂
θ
∂
x
,
γ
0
=
u
R
+
∂
v
∂
x
-
θ
.
For the sake of simplicity, (1) is depicted in the polar coordinate system as
(2)
ε
0
=
1
R
∂
u
∂
α
-
v
R
,
κ
=
1
R
∂
θ
∂
α
,
γ
0
=
u
R
+
1
R
∂
v
∂
α
-
θ
,
where the physical means of α can be found in Figure 1. Using a vector form to rewrite (2), the relations between displacement and strain are
(3)
ε
=
B
ε
d
,
where
(4)
B
ε
=
[
1
R
∂
∂
α
-
1
R
0
0
0
1
R
∂
∂
α
1
R
1
R
∂
∂
α
-
1
]
,
d
=
[
u
,
v
,
θ
]
T
.The strain energy is given by(5)
U
=
1
2
∫
V
ε
T
D
ε
d
V
.
By making use of (3), the strain energy of curved beam is obtained:
(6)
U
=
R
2
∫
0
α
E
(
B
ε
d
)
T
D
B
ε
d
d
α
,
where D
=
[
E
A
E
I
k
G
A
], E and G are Young’s and shear moduli, k is the shear modifying factor, and α
E is used to depict ending angle of arch.Similar to the process of obtaining the strain energy, the kinetic energy of curved beam is obtained by(7)
T
=
1
2
∫
V
ρ
[
(
∂
u
∂
t
)
2
+
(
∂
v
∂
t
)
2
+
(
∂
θ
∂
t
)
2
]
d
V
=
R
2
∫
0
α
E
∂
d
T
∂
t
[
ρ
A
ρ
A
ρ
I
]
∂
d
∂
t
d
α
,
where t depicts time and ρ is the mass density of beam material. Also, the work of the external forces is given by
(8)
W
=
R
∫
0
α
E
d
T
W
d
α
,
W
=
[
f
q
m
]
T
,
where f, q, and m are the distributed axial force, radial force, and moment along the length of beam, respectively. By aid of (6)–(8), the total energy of curved beam is formulated as follows:
(9)
Π
=
U
-
T
-
W
.
Usually, the kinetic energy part is neglected in static analysis, and part of the work from the external forces is neglected when free vibration analysis is needed. Based on (9) and Hamilton’s principle, the corresponding terms of BSWI curved beam element static and free vibration analysis are obtained in Section 4.


## 3. Scaling Functions on the Interval[
0
,
1
]

B-spline on interval[
0
,
1
] is given by Quak et al. [26, 27]. Since there should be at least one inner wavelet on the interval [
0
,
1
], the following condition must be satisfied:
(10)
2
j
≥
2
m
-
1
,
where m and j are the order and scale of BSWI, respectively. According to the 0 scale mth order B-spline functions and the corresponding wavelets given by Goswami et al. [27], the j scale mth order BSWI, simply denoted as BSWI m
j, scaling functions ϕ
m
,
k
j
(
ξ
) can be evaluated by the following formulas:
(11)
ϕ
m
,
k
j
(
ξ
)
=
{
ϕ
m
,
k
l
(
2
j
-
l
ξ
)
,
k
=
-
m
+
1
,
…
,
-
1
(
0
boundary
scaling
functions
)
ϕ
m
,
2
j
-
m
-
k
l
(
1
-
2
j
-
l
ξ
)
,
k

=

2
j

-

m
+
1
,
…
,
2
j
-
1
(
1
boundary
scaling
functions
)
ϕ
m
,
0
l
(
2
j
-
l
ξ
-
2
-
l
k
)
,
k
=
0
,
…
,
2
j
-
m
(
inner
scaling
functions
)
.
Therefore, the scaling functions on the interval [
0
,
1
] can be written in the vector form as follows:
(12)
Φ
=
[
ϕ
m
,
-
m
+
1
j
(
ξ
)
ϕ
m
,
-
m
+
2
j
(
ξ
)
⋯
ϕ
m
,
2
j
-
1
j
(
ξ
)
]
,
where ξ belongs to the interval [
0
,
1
]. Figure 2 presents the BSWI43, which is used in this study as the shape function of curved beam element.Figure 2
BSWI43 scaling functions on the interval [
0,1
].


## 4. Finite Element Formulation Using B-Spline on Interval

Using Hamilton’s principle, the following equations of motions can be derived:(13a)
δ
Π
=
∫
t
1
t
2
(
δ
U
-
δ
T
-
δ
W
)
d
t
=
0
.
For static analysis,
(13b)
δ
Π
=
∫
t
1
t
2
(
δ
U
-
δ
W
)
d
t
=
0
.
For free vibration analysis,
(13c)
δ
Π
=
∫
t
1
t
2
(
δ
U
-
δ
T
)
d
t
=
0
.For the finite element formulation, the normal displacement, tangential displacements, and rotation should be interpolated by BSWI43 scaling functions, respectively, to follow the first-order shear deformation theory. A displacement field assumption as shown in (14) is made firstly:
(14)
u
=
Φ
T
u
,
v
=
Φ
T
v
,
θ
=
Φ
T
θ
,
where T
=
[
Φ
T
(
ξ
1
)
Φ
T
(
ξ
2
)
⋯
Φ
T
(
ξ
n
+
1
)
]
-
T is the BSWI element transform matrix. The element displacement field represented by the coefficients of wavelets expansions is transformed from wavelet space to physical space by aid of this transform matrix. In order to give a clear expression, we use some necessary notations here:
(15)
χ
0,0
=
α
E
∫
0
1
T
T
Φ
T
Φ
T
d
ξ
,
χ
0,1
=
∫
0
1
T
T
Φ
T
d
Φ
d
ξ
T
d
ξ
,
χ
1,0
=
∫
0
1
T
T
d
Φ
T
d
ξ
Φ
T
d
ξ
,
χ
1,1
=
1
α
E
∫
0
1
T
T
d
Φ
T
d
ξ
d
Φ
d
ξ
T
d
ξ
,
χ
1,2
=
1
α
E
2
∫
0
1
T
T
d
Φ
T
d
ξ
d
2
Φ
d
ξ
2
T
d
ξ
,
χ
2,1
=
1
α
E
2
∫
0
1
T
T
d
2
Φ
T
d
ξ
2
d
Φ
d
ξ
T
d
ξ
,
χ
2,2
=
1
α
E
3
∫
0
1
T
T
d
2
Φ
T
d
ξ
2
d
2
Φ
d
ξ
2
T
d
ξ
.
Substituting (14) with (13b), a finite element formulation of curved beam static analysis can be obtained:
(16)
K
Δ
=
P
,
K
=
1
R
∫
0
α
E
B
ε
T
D
B
ε
d
α
,
P
=
R
∫
0
α
E
(
Φ
T
)
T
[
f
q
m
]
d
α
,
where Δ
=
[
u
v
θ
]
T is the vector formulation of element displacement, K is the stiffness matrix, and P is the force vector. The details of the stiffness matrix are
(17)
K
=
1
R
×
[
E
A
χ
1,1

+

k
G
A
χ
0,0
-
E
A
χ
1,0

+

k
G
A
χ
0,1
-
R
k
G
A
χ
0,0
-
E
A
χ
0,1

+

k
G
A
χ
1,0
E
A
χ
0,0

+

k
G
A
χ
1,1
-
R
k
G
A
χ
1,0
-
R
k
G
A
χ
0,0
-
R
k
G
A
χ
0,1
E
I
χ
1,1

+

R
2
k
G
A
χ
0,0
]
.
Substituting (14) with (13c), a finite element formulation of arch free vibration analysis is obtained as a generalized eigenproblem:
(18)
(
K
-
ω
2
M
)
X
=
0
,
where ω is the natural frequency and X is the mode shape of arches. The mass matrix Mis defined as
(19)
M
=
R
∫
0
α
E
(
Φ
T
)
T
[
ρ
A
ρ
A
ρ
I
]
Φ
T
d
α
.


## 5. Numerical Examples

The validity and efficiency of the element formulated in the previous section are verified through numerical examples in this part, and static and dynamic numerical examples are given, respectively.


### 5.1. Static Analysis

#### 5.1.1. A Quarter Circular Cantilever Arch

The model consists of a quarter circular cantilever arch, where the origin of coordinate is at the fixed end, as shown in Figure3. The quarter ring is subjected to a radial tip load. This moderately deep arch is idealized with one BSWI element and analyzed for a wide range of thick to thin beams by changing the slenderness ratio of R/h. The exact solution for this problem is derived using Castigliano’s energy theorem as [28](20a)
u
C
=
P
R
3
2
E
I
-
P
R
2
k
G
A
-
P
R
2
E
A
,
(20b)
v
C
=
π
P
R
3
4
E
I
+
π
P
R
4
k
G
A
+
π
P
R
4
E
A
,
(20c)
θ
C
=
P
R
2
E
I
.Figure 3
Geometry and loading of a circular cantilever arch.In Table1, MFE method [3] and the solution given by Lee and Sin [28] are used to contrast with the present method. In order to compare them with the exact solutions given by (20a)–(20c), we transform the data into the form of a ratio between numerical result and exact solution. It is noteworthy that the presence of BSWI is analogous to MFE, and the accuracy of v and θ is better than MFE in a wide range of thick to thin beams, even in the thick condition. Compared with the solution given by [28], the present method also shows its superiority.Table 1
Comparisons of the present solutions of the one quarter ring with Castigliano’s energy solution and other methods.
Slenderness ratio (R/h)
MFE [3]
Lee and Sin [28]
Present
u
/
u
C
v
/
v
C
θ
/
θ
C
u
/
u
C
v
/
v
C
θ
/
θ
C
u
/
u
C
v
/
v
C
θ
/
θ
C
5
1.0124
0.9996
0.9997
0.99916
1.01494
1.00000
1.0129
1.0000
1.0000
10
1.0036
1.0000
0.9999
0.99968
1.00354
1.00000
1.0052
1.0000
1.0000
20
1.0010
1.0000
1.0000
0.99981
1.00072
1.00000
1.0013
1.0000
1.0000
100
1.0001
1.0000
0.9997
0.99986
0.99982
1.00000
1.0000
1.0000
1.0000
200
1.0001
1.0000
0.9999
0.99986
0.99979
1.00000
1.0000
0.9999
1.0000
1000
1.0001
1.0000
0.9999
0.99986
0.99978
1.00000
0.9999
0.9998
0.9995


#### 5.1.2. A Pinched Ring

Babu and Prathap [29] considered that the pinched ring is the best example to demonstrate the behavior of the elements in a deep arch configuration. Figure 4 depicts a pinched ring, on which the same radial loads are applied at the top and bottom of the ring. The physical model is idealized with one BSWI element and analyzed for a wide range of thick to thin beams by changing the slenderness ratio of R
/
h. The exact solution of points A and B is derived by means of Castigliano’s energy theorem as [28]
(21)
v
AC
⁡
=
P
R
3
E
I
π
2
-
8
4
π
+
π
P
R
4
k
G
A
+
π
P
R
4
E
A
,
(22)
v
BC
=
-
(
P
R
3
E
I
4
-
π
2
π
+
P
R
2
k
G
A
-
P
R
2
E
A
)
.Figure 4
Geometry and loading of a pinched ring.Numerical results calculated by BSWI are shown in Table2, and, as a reference, the solution given by Lee and Sin [28] is also shown there. It can be seen that the present method has a satisfying accuracy in a wide range of slenderness ratio. Although there are some slight accuracy losses with tangential displacement of point B when slenderness ratio is big, it is still in an acceptable confine. It should be noticed that the present method shows a good presence for the calculation of tangential displacement of point A.Table 2
Comparisons of the present solutions of the pinched ring with Castigliano’s energy solution and the solution given by [28].
Slenderness ratio (R/h)
Lee and Sin [28]
Present
v
A
/
v
AC
⁡
v
B
/
v
BC
v
A
/
v
AC
⁡
v
B
/
v
BC
2.5
0.98730
0.98533
1.00000
1.08543
5
0.99574
0.99556
1.00000
1.03484
10
0.99834
0.99828
1.00000
1.00869
20
0.99904
0.99897
1.00000
1.00209
100
0.99923
0.99918
0.99998
1.00005
200
0.99927
0.99924
0.99991
0.99988
1000
0.99925
0.99922
0.99777
0.99662


#### 5.1.3. A Nearly Straight Cantilever Curved Beam

In order to validate the reducibility of the present curved beam element to straight beam configuration, a numerical example as shown in Figure5 is given. This arch has a very large radius and short span subjected to a concentrated load. The properties are R
=
2000 m, L
=
100 m, h
=
10 m, P
=
1 N, E
=
1
×
10
6 Pa, Poisson’s ratio μ
=
0.3, and width of beam b
=
30 m. It should be mentioned that the properties presented above are defined in SI (System International), but all closed-form unit systems are suitable in this example; they will not influence the solved values. This structure is modeled using one BSWI element. For this type of thin beam (R
/
h
>
40), the exact solutions of tip displacement v for the three kinds of load conditions shown in Figure 5 are given by [3]
(23)
v
a
=
P
L
3
3
E
I
(
1
+
6
I
(
1
+
μ
)
k
L
2
A
)
,
(24)
v
b
=
q
L
4
8
E
I
(
1
+
8
I
(
1
+
μ
)
k
L
2
A
)
,
(25)
v
c
=
q
L
4
30
E
I
(
1
+
10
I
(
1
+
μ
)
k
L
2
A
)
.
Equations (23)–(25) correspond to Figures 5(a)–5(c), respectively. Table 3 presents the comparison of numerical results and the exact solution in which the relative errors are all less than 1%. Thus, the present method has a good reducibility to straight beam configuration.Table 3
Numerical results for the tip displacement (m) of a nearly straight cantilever curved beam.
Load
Present(×
10
-
5)
Exact solution(×
10
-
5)
Relative error
Concentrated
1.338
1.344
0.44%
Uniform distributed
50.255
50.520
0.53%
Linear distributed
13.419
13.507
0.65%Geometry and loading of a nearly straight cantilever curved beam: (a) concentrated load, (b) uniform distributed load, and (c) linear distributed load.
(a)
(b)
(c)


### 5.2. Free Vibration Analysis

#### 5.2.1. A Hinged Arch

Figure6 gives the geometry of a hinged arch which is widely analyzed in previously published works. The following properties of arch are employed for the computation: R
=
12 in, h
=
0.25 in, A
=
0.1563 in2, I
=
8.138
×
10
-
4 in4, k
=
0.8497, E
=
3.04
×
10
7 psi, μ
=
0.3, and ρ
=
0.02763 slugs ft/in4.Figure 6
Geometry of a hinged arch.In order to test the accuracy of the present method for the change of subtended angle, the whole results for shallow arch, moderately deep arch, deep arch, and very deep arches, as classified in [25], are given in Table 4. With a various subtended angle α ranging from 10 to 350 degrees, Table 4 shows the comparisons of fundamental natural frequency (rad/s) of the hinged arch. The present solution is computed by four BSWI elements. Compared with the solutions given by CHM2 [11], THICK-2 [13], MFE [30], and E1.1b [31], the present element shows good consistency with them. In order to represent this clearly, Figure 7 is presented. In Figure 7, numerical results are normalized by those obtained by THICK-2, which could be seen as a relative accurate solution. It is clear that the present solution is bounded by E1.1b and CHM2. Specifically, the current solution has a good agreement with the solution given by THICK-2. Thus the accuracy of the present method is validated.Table 4
Fundamental frequency (rad/s) of a hinged arch with various subtended angles.
Angles (deg.)
CHM2 [11]
THICK-2 [13]
MFE [30]
E1.1b [31]
Present
10
5845.78
5841.74
5852.32
5874.30
5881.64
20
2836.20
2827.63
2829.66
2823.10
2829.13
30
2370.01
2339.82
2373.23
2345.20
2348.11
60
564.05
560.25
567.71
561.20
560.62
90
230.31
229.66
232.94
230.40
229.69
120
115.82
115.64
117.50
116.30
115.64
150
64.52
64.43
76.24
64.93
64.42
180
37.91
37.86
38.71
38.24
37.85
210
22.80
22.77
23.42
23.05
22.76
240
13.69
13.66
14.19
13.87
13.66
270
7.94
7.92
8.39
8.06
7.92
300
4.20
4.18
4.65
4.27
4.19
330
1.70
1.69
2.28
1.73
1.69
350
0.50
0.49
1.38
0.50
0.50Figure 7
Normalized fundamental frequency for a hinged arch.


#### 5.2.2. A Thin Circular Ring

The geometry of a circular ring has been given in Figure4. Different from the problem considered for a pinched ring mentioned above, there exists no external force for the free vibration of circular ring. Timoshenko [32] gave the exact solution of the ith order natural vibration of a thin ring:
(26)
ω
i
=
E
I
i
2
(
1
-
i
2
)
2
ρ
A
R
4
(
1
+
i
2
)
.
Considering the classifications given by [25], we use the slenderness ratio R
/
h
>
40 to depict thin rings. Thus the slenderness ratio ranging from 50 to 1000 is studied in Table 5. The other properties for the model are R
=
0.3048 m, E
=
1.31
×
10
11 Pa, μ
=
0.3, and ρ
=
1741 kg/m3. Two BSWI elements are used to idealize the model. Table 5 presents the first four order mode frequencies for ring, including the rigid body mode. The numerical results obtained using BSWI agree well with the exact solutions.Table 5
Frequencies (rad/s) of a thin circular ring with various slenderness ratios.
Slenderness ratio (R/h)
Method
Mode
Rigid body
1
2
3
50
Present
0.021
70.158
198.397
380.314
Timoshenko [32]
0
70.169
198.469
380.546
100
Present
0.021
35.085
99.237
190.301
Timoshenko [32]
0
35.085
99.234
190.273
200
Present
0.021
17.545
49.640
95.240
Timoshenko [32]
0
17.542
49.617
95.137
500
Present
0.020
7.023
19.899
38.229
Timoshenko [32]
0
7.017
19.847
38.055
800
Present
0.020
4.394
12.474
23.968
Timoshenko [32]
0
4.386
12.404
23.784
1000
Present
0.019
3.518
10.002
19.212
Timoshenko [32]
0
3.508
9.923
19.027


#### 5.2.3. A 90° Arch with Different Boundaries

In this example, free vibrations of a 90° arch with different boundaries are analyzed to test the adaptability of BSWI element. The dimensionless frequency of the arch is defined asC
i
=
ω
i
α
2
R
2
ρ
A
/
E
I. The main property of arch is R
/
r, where r is the radius of gyration I
/
A. Arches (R
/
r
=
100) with different boundaries are studied, and the numerical results are presented in Table 6. The results show that the present method has a good agreement with the references.Table 6
First and second lowest dimensionless frequencies of 90° arch with different boundaries.
Boundary condition
Method
Mode
1
2
Hinged-hinged
Raveendranath et al. [3]
33.83
78.72
Krishnan and Suresh [9]
33.93
79.42
Sabir et al. [14]
33.79
79.02
Present
33.87
78.95
Clamped-clamped
Raveendranath et al. [3]
55.34
102.28
Krishnan and Suresh [9]
55.82
104.28
Sabir et al. [14]
55.45
103.59
Present
55.54
103.00
Clamped-hinged
Raveendranath et al. [3]
43.81
90.60
Krishnan and Suresh [9]
44.05
91.82
Sabir et al. [14]
43.81
91.29
Present
43.91
91.00


#### 5.2.4. A Three-Span Clamped Arch

This example is a curved beam with three subspans of equal span angle and different radiuses of curvature are depicted in Figure8. The total span angle is 180° and each subspan angle is 60°. In this example, three BSWI elements are used to idealize this model. In contrast, the dimensionless frequency of the arch (R
/
r
=
100) is selected as C
i
=
ω
i
R
2
ρ
A
/
E
I, which is slightly different from the C
i used in Section 5.2.3. The problem presented here has also been solved by applying the finite element method (FEM) [9, 33] and the wave analysis method [34]. In Table 7, the dimensionless frequencies obtained from the present approach are compared with those of [9, 33, 34]. It is noticed that the natural frequencies obtained from the BSWI agree well with these solutions, especially with [34], which can be seen as a relative accurate reference solution for this problem [34].Table 7
Dimensionless frequencies of the three-span clamped arch.
Mode
Krishnan and Suresh [9]
Maurizi et al. [33]
Kang et al. [34]
Present
1
2.701
2.680
2.6833155
2.682
2
4.828
4.824
4.8337572
4.828
3
9.543
9.536
9.5647244
9.549
4
14.535
14.527
14.5850042
14.551
5
21.751
21.749
21.8646005
21.795Figure 8
Geometry of a three-span clamped arch.


## 5.1. Static Analysis

### 5.1.1. A Quarter Circular Cantilever Arch

The model consists of a quarter circular cantilever arch, where the origin of coordinate is at the fixed end, as shown in Figure3. The quarter ring is subjected to a radial tip load. This moderately deep arch is idealized with one BSWI element and analyzed for a wide range of thick to thin beams by changing the slenderness ratio of R/h. The exact solution for this problem is derived using Castigliano’s energy theorem as [28](20a)
u
C
=
P
R
3
2
E
I
-
P
R
2
k
G
A
-
P
R
2
E
A
,
(20b)
v
C
=
π
P
R
3
4
E
I
+
π
P
R
4
k
G
A
+
π
P
R
4
E
A
,
(20c)
θ
C
=
P
R
2
E
I
.Figure 3
Geometry and loading of a circular cantilever arch.In Table1, MFE method [3] and the solution given by Lee and Sin [28] are used to contrast with the present method. In order to compare them with the exact solutions given by (20a)–(20c), we transform the data into the form of a ratio between numerical result and exact solution. It is noteworthy that the presence of BSWI is analogous to MFE, and the accuracy of v and θ is better than MFE in a wide range of thick to thin beams, even in the thick condition. Compared with the solution given by [28], the present method also shows its superiority.Table 1
Comparisons of the present solutions of the one quarter ring with Castigliano’s energy solution and other methods.
Slenderness ratio (R/h)
MFE [3]
Lee and Sin [28]
Present
u
/
u
C
v
/
v
C
θ
/
θ
C
u
/
u
C
v
/
v
C
θ
/
θ
C
u
/
u
C
v
/
v
C
θ
/
θ
C
5
1.0124
0.9996
0.9997
0.99916
1.01494
1.00000
1.0129
1.0000
1.0000
10
1.0036
1.0000
0.9999
0.99968
1.00354
1.00000
1.0052
1.0000
1.0000
20
1.0010
1.0000
1.0000
0.99981
1.00072
1.00000
1.0013
1.0000
1.0000
100
1.0001
1.0000
0.9997
0.99986
0.99982
1.00000
1.0000
1.0000
1.0000
200
1.0001
1.0000
0.9999
0.99986
0.99979
1.00000
1.0000
0.9999
1.0000
1000
1.0001
1.0000
0.9999
0.99986
0.99978
1.00000
0.9999
0.9998
0.9995


### 5.1.2. A Pinched Ring

Babu and Prathap [29] considered that the pinched ring is the best example to demonstrate the behavior of the elements in a deep arch configuration. Figure 4 depicts a pinched ring, on which the same radial loads are applied at the top and bottom of the ring. The physical model is idealized with one BSWI element and analyzed for a wide range of thick to thin beams by changing the slenderness ratio of R
/
h. The exact solution of points A and B is derived by means of Castigliano’s energy theorem as [28]
(21)
v
AC
⁡
=
P
R
3
E
I
π
2
-
8
4
π
+
π
P
R
4
k
G
A
+
π
P
R
4
E
A
,
(22)
v
BC
=
-
(
P
R
3
E
I
4
-
π
2
π
+
P
R
2
k
G
A
-
P
R
2
E
A
)
.Figure 4
Geometry and loading of a pinched ring.Numerical results calculated by BSWI are shown in Table2, and, as a reference, the solution given by Lee and Sin [28] is also shown there. It can be seen that the present method has a satisfying accuracy in a wide range of slenderness ratio. Although there are some slight accuracy losses with tangential displacement of point B when slenderness ratio is big, it is still in an acceptable confine. It should be noticed that the present method shows a good presence for the calculation of tangential displacement of point A.Table 2
Comparisons of the present solutions of the pinched ring with Castigliano’s energy solution and the solution given by [28].
Slenderness ratio (R/h)
Lee and Sin [28]
Present
v
A
/
v
AC
⁡
v
B
/
v
BC
v
A
/
v
AC
⁡
v
B
/
v
BC
2.5
0.98730
0.98533
1.00000
1.08543
5
0.99574
0.99556
1.00000
1.03484
10
0.99834
0.99828
1.00000
1.00869
20
0.99904
0.99897
1.00000
1.00209
100
0.99923
0.99918
0.99998
1.00005
200
0.99927
0.99924
0.99991
0.99988
1000
0.99925
0.99922
0.99777
0.99662


### 5.1.3. A Nearly Straight Cantilever Curved Beam

In order to validate the reducibility of the present curved beam element to straight beam configuration, a numerical example as shown in Figure5 is given. This arch has a very large radius and short span subjected to a concentrated load. The properties are R
=
2000 m, L
=
100 m, h
=
10 m, P
=
1 N, E
=
1
×
10
6 Pa, Poisson’s ratio μ
=
0.3, and width of beam b
=
30 m. It should be mentioned that the properties presented above are defined in SI (System International), but all closed-form unit systems are suitable in this example; they will not influence the solved values. This structure is modeled using one BSWI element. For this type of thin beam (R
/
h
>
40), the exact solutions of tip displacement v for the three kinds of load conditions shown in Figure 5 are given by [3]
(23)
v
a
=
P
L
3
3
E
I
(
1
+
6
I
(
1
+
μ
)
k
L
2
A
)
,
(24)
v
b
=
q
L
4
8
E
I
(
1
+
8
I
(
1
+
μ
)
k
L
2
A
)
,
(25)
v
c
=
q
L
4
30
E
I
(
1
+
10
I
(
1
+
μ
)
k
L
2
A
)
.
Equations (23)–(25) correspond to Figures 5(a)–5(c), respectively. Table 3 presents the comparison of numerical results and the exact solution in which the relative errors are all less than 1%. Thus, the present method has a good reducibility to straight beam configuration.Table 3
Numerical results for the tip displacement (m) of a nearly straight cantilever curved beam.
Load
Present(×
10
-
5)
Exact solution(×
10
-
5)
Relative error
Concentrated
1.338
1.344
0.44%
Uniform distributed
50.255
50.520
0.53%
Linear distributed
13.419
13.507
0.65%Geometry and loading of a nearly straight cantilever curved beam: (a) concentrated load, (b) uniform distributed load, and (c) linear distributed load.
(a)
(b)
(c)


## 5.1.1. A Quarter Circular Cantilever Arch

The model consists of a quarter circular cantilever arch, where the origin of coordinate is at the fixed end, as shown in Figure3. The quarter ring is subjected to a radial tip load. This moderately deep arch is idealized with one BSWI element and analyzed for a wide range of thick to thin beams by changing the slenderness ratio of R/h. The exact solution for this problem is derived using Castigliano’s energy theorem as [28](20a)
u
C
=
P
R
3
2
E
I
-
P
R
2
k
G
A
-
P
R
2
E
A
,
(20b)
v
C
=
π
P
R
3
4
E
I
+
π
P
R
4
k
G
A
+
π
P
R
4
E
A
,
(20c)
θ
C
=
P
R
2
E
I
.Figure 3
Geometry and loading of a circular cantilever arch.In Table1, MFE method [3] and the solution given by Lee and Sin [28] are used to contrast with the present method. In order to compare them with the exact solutions given by (20a)–(20c), we transform the data into the form of a ratio between numerical result and exact solution. It is noteworthy that the presence of BSWI is analogous to MFE, and the accuracy of v and θ is better than MFE in a wide range of thick to thin beams, even in the thick condition. Compared with the solution given by [28], the present method also shows its superiority.Table 1
Comparisons of the present solutions of the one quarter ring with Castigliano’s energy solution and other methods.
Slenderness ratio (R/h)
MFE [3]
Lee and Sin [28]
Present
u
/
u
C
v
/
v
C
θ
/
θ
C
u
/
u
C
v
/
v
C
θ
/
θ
C
u
/
u
C
v
/
v
C
θ
/
θ
C
5
1.0124
0.9996
0.9997
0.99916
1.01494
1.00000
1.0129
1.0000
1.0000
10
1.0036
1.0000
0.9999
0.99968
1.00354
1.00000
1.0052
1.0000
1.0000
20
1.0010
1.0000
1.0000
0.99981
1.00072
1.00000
1.0013
1.0000
1.0000
100
1.0001
1.0000
0.9997
0.99986
0.99982
1.00000
1.0000
1.0000
1.0000
200
1.0001
1.0000
0.9999
0.99986
0.99979
1.00000
1.0000
0.9999
1.0000
1000
1.0001
1.0000
0.9999
0.99986
0.99978
1.00000
0.9999
0.9998
0.9995


## 5.1.2. A Pinched Ring

Babu and Prathap [29] considered that the pinched ring is the best example to demonstrate the behavior of the elements in a deep arch configuration. Figure 4 depicts a pinched ring, on which the same radial loads are applied at the top and bottom of the ring. The physical model is idealized with one BSWI element and analyzed for a wide range of thick to thin beams by changing the slenderness ratio of R
/
h. The exact solution of points A and B is derived by means of Castigliano’s energy theorem as [28]
(21)
v
AC
⁡
=
P
R
3
E
I
π
2
-
8
4
π
+
π
P
R
4
k
G
A
+
π
P
R
4
E
A
,
(22)
v
BC
=
-
(
P
R
3
E
I
4
-
π
2
π
+
P
R
2
k
G
A
-
P
R
2
E
A
)
.Figure 4
Geometry and loading of a pinched ring.Numerical results calculated by BSWI are shown in Table2, and, as a reference, the solution given by Lee and Sin [28] is also shown there. It can be seen that the present method has a satisfying accuracy in a wide range of slenderness ratio. Although there are some slight accuracy losses with tangential displacement of point B when slenderness ratio is big, it is still in an acceptable confine. It should be noticed that the present method shows a good presence for the calculation of tangential displacement of point A.Table 2
Comparisons of the present solutions of the pinched ring with Castigliano’s energy solution and the solution given by [28].
Slenderness ratio (R/h)
Lee and Sin [28]
Present
v
A
/
v
AC
⁡
v
B
/
v
BC
v
A
/
v
AC
⁡
v
B
/
v
BC
2.5
0.98730
0.98533
1.00000
1.08543
5
0.99574
0.99556
1.00000
1.03484
10
0.99834
0.99828
1.00000
1.00869
20
0.99904
0.99897
1.00000
1.00209
100
0.99923
0.99918
0.99998
1.00005
200
0.99927
0.99924
0.99991
0.99988
1000
0.99925
0.99922
0.99777
0.99662


## 5.1.3. A Nearly Straight Cantilever Curved Beam

In order to validate the reducibility of the present curved beam element to straight beam configuration, a numerical example as shown in Figure5 is given. This arch has a very large radius and short span subjected to a concentrated load. The properties are R
=
2000 m, L
=
100 m, h
=
10 m, P
=
1 N, E
=
1
×
10
6 Pa, Poisson’s ratio μ
=
0.3, and width of beam b
=
30 m. It should be mentioned that the properties presented above are defined in SI (System International), but all closed-form unit systems are suitable in this example; they will not influence the solved values. This structure is modeled using one BSWI element. For this type of thin beam (R
/
h
>
40), the exact solutions of tip displacement v for the three kinds of load conditions shown in Figure 5 are given by [3]
(23)
v
a
=
P
L
3
3
E
I
(
1
+
6
I
(
1
+
μ
)
k
L
2
A
)
,
(24)
v
b
=
q
L
4
8
E
I
(
1
+
8
I
(
1
+
μ
)
k
L
2
A
)
,
(25)
v
c
=
q
L
4
30
E
I
(
1
+
10
I
(
1
+
μ
)
k
L
2
A
)
.
Equations (23)–(25) correspond to Figures 5(a)–5(c), respectively. Table 3 presents the comparison of numerical results and the exact solution in which the relative errors are all less than 1%. Thus, the present method has a good reducibility to straight beam configuration.Table 3
Numerical results for the tip displacement (m) of a nearly straight cantilever curved beam.
Load
Present(×
10
-
5)
Exact solution(×
10
-
5)
Relative error
Concentrated
1.338
1.344
0.44%
Uniform distributed
50.255
50.520
0.53%
Linear distributed
13.419
13.507
0.65%Geometry and loading of a nearly straight cantilever curved beam: (a) concentrated load, (b) uniform distributed load, and (c) linear distributed load.
(a)
(b)
(c)


## 5.2. Free Vibration Analysis

### 5.2.1. A Hinged Arch

Figure6 gives the geometry of a hinged arch which is widely analyzed in previously published works. The following properties of arch are employed for the computation: R
=
12 in, h
=
0.25 in, A
=
0.1563 in2, I
=
8.138
×
10
-
4 in4, k
=
0.8497, E
=
3.04
×
10
7 psi, μ
=
0.3, and ρ
=
0.02763 slugs ft/in4.Figure 6
Geometry of a hinged arch.In order to test the accuracy of the present method for the change of subtended angle, the whole results for shallow arch, moderately deep arch, deep arch, and very deep arches, as classified in [25], are given in Table 4. With a various subtended angle α ranging from 10 to 350 degrees, Table 4 shows the comparisons of fundamental natural frequency (rad/s) of the hinged arch. The present solution is computed by four BSWI elements. Compared with the solutions given by CHM2 [11], THICK-2 [13], MFE [30], and E1.1b [31], the present element shows good consistency with them. In order to represent this clearly, Figure 7 is presented. In Figure 7, numerical results are normalized by those obtained by THICK-2, which could be seen as a relative accurate solution. It is clear that the present solution is bounded by E1.1b and CHM2. Specifically, the current solution has a good agreement with the solution given by THICK-2. Thus the accuracy of the present method is validated.Table 4
Fundamental frequency (rad/s) of a hinged arch with various subtended angles.
Angles (deg.)
CHM2 [11]
THICK-2 [13]
MFE [30]
E1.1b [31]
Present
10
5845.78
5841.74
5852.32
5874.30
5881.64
20
2836.20
2827.63
2829.66
2823.10
2829.13
30
2370.01
2339.82
2373.23
2345.20
2348.11
60
564.05
560.25
567.71
561.20
560.62
90
230.31
229.66
232.94
230.40
229.69
120
115.82
115.64
117.50
116.30
115.64
150
64.52
64.43
76.24
64.93
64.42
180
37.91
37.86
38.71
38.24
37.85
210
22.80
22.77
23.42
23.05
22.76
240
13.69
13.66
14.19
13.87
13.66
270
7.94
7.92
8.39
8.06
7.92
300
4.20
4.18
4.65
4.27
4.19
330
1.70
1.69
2.28
1.73
1.69
350
0.50
0.49
1.38
0.50
0.50Figure 7
Normalized fundamental frequency for a hinged arch.


### 5.2.2. A Thin Circular Ring

The geometry of a circular ring has been given in Figure4. Different from the problem considered for a pinched ring mentioned above, there exists no external force for the free vibration of circular ring. Timoshenko [32] gave the exact solution of the ith order natural vibration of a thin ring:
(26)
ω
i
=
E
I
i
2
(
1
-
i
2
)
2
ρ
A
R
4
(
1
+
i
2
)
.
Considering the classifications given by [25], we use the slenderness ratio R
/
h
>
40 to depict thin rings. Thus the slenderness ratio ranging from 50 to 1000 is studied in Table 5. The other properties for the model are R
=
0.3048 m, E
=
1.31
×
10
11 Pa, μ
=
0.3, and ρ
=
1741 kg/m3. Two BSWI elements are used to idealize the model. Table 5 presents the first four order mode frequencies for ring, including the rigid body mode. The numerical results obtained using BSWI agree well with the exact solutions.Table 5
Frequencies (rad/s) of a thin circular ring with various slenderness ratios.
Slenderness ratio (R/h)
Method
Mode
Rigid body
1
2
3
50
Present
0.021
70.158
198.397
380.314
Timoshenko [32]
0
70.169
198.469
380.546
100
Present
0.021
35.085
99.237
190.301
Timoshenko [32]
0
35.085
99.234
190.273
200
Present
0.021
17.545
49.640
95.240
Timoshenko [32]
0
17.542
49.617
95.137
500
Present
0.020
7.023
19.899
38.229
Timoshenko [32]
0
7.017
19.847
38.055
800
Present
0.020
4.394
12.474
23.968
Timoshenko [32]
0
4.386
12.404
23.784
1000
Present
0.019
3.518
10.002
19.212
Timoshenko [32]
0
3.508
9.923
19.027


### 5.2.3. A 90° Arch with Different Boundaries

In this example, free vibrations of a 90° arch with different boundaries are analyzed to test the adaptability of BSWI element. The dimensionless frequency of the arch is defined asC
i
=
ω
i
α
2
R
2
ρ
A
/
E
I. The main property of arch is R
/
r, where r is the radius of gyration I
/
A. Arches (R
/
r
=
100) with different boundaries are studied, and the numerical results are presented in Table 6. The results show that the present method has a good agreement with the references.Table 6
First and second lowest dimensionless frequencies of 90° arch with different boundaries.
Boundary condition
Method
Mode
1
2
Hinged-hinged
Raveendranath et al. [3]
33.83
78.72
Krishnan and Suresh [9]
33.93
79.42
Sabir et al. [14]
33.79
79.02
Present
33.87
78.95
Clamped-clamped
Raveendranath et al. [3]
55.34
102.28
Krishnan and Suresh [9]
55.82
104.28
Sabir et al. [14]
55.45
103.59
Present
55.54
103.00
Clamped-hinged
Raveendranath et al. [3]
43.81
90.60
Krishnan and Suresh [9]
44.05
91.82
Sabir et al. [14]
43.81
91.29
Present
43.91
91.00


### 5.2.4. A Three-Span Clamped Arch

This example is a curved beam with three subspans of equal span angle and different radiuses of curvature are depicted in Figure8. The total span angle is 180° and each subspan angle is 60°. In this example, three BSWI elements are used to idealize this model. In contrast, the dimensionless frequency of the arch (R
/
r
=
100) is selected as C
i
=
ω
i
R
2
ρ
A
/
E
I, which is slightly different from the C
i used in Section 5.2.3. The problem presented here has also been solved by applying the finite element method (FEM) [9, 33] and the wave analysis method [34]. In Table 7, the dimensionless frequencies obtained from the present approach are compared with those of [9, 33, 34]. It is noticed that the natural frequencies obtained from the BSWI agree well with these solutions, especially with [34], which can be seen as a relative accurate reference solution for this problem [34].Table 7
Dimensionless frequencies of the three-span clamped arch.
Mode
Krishnan and Suresh [9]
Maurizi et al. [33]
Kang et al. [34]
Present
1
2.701
2.680
2.6833155
2.682
2
4.828
4.824
4.8337572
4.828
3
9.543
9.536
9.5647244
9.549
4
14.535
14.527
14.5850042
14.551
5
21.751
21.749
21.8646005
21.795Figure 8
Geometry of a three-span clamped arch.


## 5.2.1. A Hinged Arch

Figure6 gives the geometry of a hinged arch which is widely analyzed in previously published works. The following properties of arch are employed for the computation: R
=
12 in, h
=
0.25 in, A
=
0.1563 in2, I
=
8.138
×
10
-
4 in4, k
=
0.8497, E
=
3.04
×
10
7 psi, μ
=
0.3, and ρ
=
0.02763 slugs ft/in4.Figure 6
Geometry of a hinged arch.In order to test the accuracy of the present method for the change of subtended angle, the whole results for shallow arch, moderately deep arch, deep arch, and very deep arches, as classified in [25], are given in Table 4. With a various subtended angle α ranging from 10 to 350 degrees, Table 4 shows the comparisons of fundamental natural frequency (rad/s) of the hinged arch. The present solution is computed by four BSWI elements. Compared with the solutions given by CHM2 [11], THICK-2 [13], MFE [30], and E1.1b [31], the present element shows good consistency with them. In order to represent this clearly, Figure 7 is presented. In Figure 7, numerical results are normalized by those obtained by THICK-2, which could be seen as a relative accurate solution. It is clear that the present solution is bounded by E1.1b and CHM2. Specifically, the current solution has a good agreement with the solution given by THICK-2. Thus the accuracy of the present method is validated.Table 4
Fundamental frequency (rad/s) of a hinged arch with various subtended angles.
Angles (deg.)
CHM2 [11]
THICK-2 [13]
MFE [30]
E1.1b [31]
Present
10
5845.78
5841.74
5852.32
5874.30
5881.64
20
2836.20
2827.63
2829.66
2823.10
2829.13
30
2370.01
2339.82
2373.23
2345.20
2348.11
60
564.05
560.25
567.71
561.20
560.62
90
230.31
229.66
232.94
230.40
229.69
120
115.82
115.64
117.50
116.30
115.64
150
64.52
64.43
76.24
64.93
64.42
180
37.91
37.86
38.71
38.24
37.85
210
22.80
22.77
23.42
23.05
22.76
240
13.69
13.66
14.19
13.87
13.66
270
7.94
7.92
8.39
8.06
7.92
300
4.20
4.18
4.65
4.27
4.19
330
1.70
1.69
2.28
1.73
1.69
350
0.50
0.49
1.38
0.50
0.50Figure 7
Normalized fundamental frequency for a hinged arch.


## 5.2.2. A Thin Circular Ring

The geometry of a circular ring has been given in Figure4. Different from the problem considered for a pinched ring mentioned above, there exists no external force for the free vibration of circular ring. Timoshenko [32] gave the exact solution of the ith order natural vibration of a thin ring:
(26)
ω
i
=
E
I
i
2
(
1
-
i
2
)
2
ρ
A
R
4
(
1
+
i
2
)
.
Considering the classifications given by [25], we use the slenderness ratio R
/
h
>
40 to depict thin rings. Thus the slenderness ratio ranging from 50 to 1000 is studied in Table 5. The other properties for the model are R
=
0.3048 m, E
=
1.31
×
10
11 Pa, μ
=
0.3, and ρ
=
1741 kg/m3. Two BSWI elements are used to idealize the model. Table 5 presents the first four order mode frequencies for ring, including the rigid body mode. The numerical results obtained using BSWI agree well with the exact solutions.Table 5
Frequencies (rad/s) of a thin circular ring with various slenderness ratios.
Slenderness ratio (R/h)
Method
Mode
Rigid body
1
2
3
50
Present
0.021
70.158
198.397
380.314
Timoshenko [32]
0
70.169
198.469
380.546
100
Present
0.021
35.085
99.237
190.301
Timoshenko [32]
0
35.085
99.234
190.273
200
Present
0.021
17.545
49.640
95.240
Timoshenko [32]
0
17.542
49.617
95.137
500
Present
0.020
7.023
19.899
38.229
Timoshenko [32]
0
7.017
19.847
38.055
800
Present
0.020
4.394
12.474
23.968
Timoshenko [32]
0
4.386
12.404
23.784
1000
Present
0.019
3.518
10.002
19.212
Timoshenko [32]
0
3.508
9.923
19.027


## 5.2.3. A 90° Arch with Different Boundaries

In this example, free vibrations of a 90° arch with different boundaries are analyzed to test the adaptability of BSWI element. The dimensionless frequency of the arch is defined asC
i
=
ω
i
α
2
R
2
ρ
A
/
E
I. The main property of arch is R
/
r, where r is the radius of gyration I
/
A. Arches (R
/
r
=
100) with different boundaries are studied, and the numerical results are presented in Table 6. The results show that the present method has a good agreement with the references.Table 6
First and second lowest dimensionless frequencies of 90° arch with different boundaries.
Boundary condition
Method
Mode
1
2
Hinged-hinged
Raveendranath et al. [3]
33.83
78.72
Krishnan and Suresh [9]
33.93
79.42
Sabir et al. [14]
33.79
79.02
Present
33.87
78.95
Clamped-clamped
Raveendranath et al. [3]
55.34
102.28
Krishnan and Suresh [9]
55.82
104.28
Sabir et al. [14]
55.45
103.59
Present
55.54
103.00
Clamped-hinged
Raveendranath et al. [3]
43.81
90.60
Krishnan and Suresh [9]
44.05
91.82
Sabir et al. [14]
43.81
91.29
Present
43.91
91.00


## 5.2.4. A Three-Span Clamped Arch

This example is a curved beam with three subspans of equal span angle and different radiuses of curvature are depicted in Figure8. The total span angle is 180° and each subspan angle is 60°. In this example, three BSWI elements are used to idealize this model. In contrast, the dimensionless frequency of the arch (R
/
r
=
100) is selected as C
i
=
ω
i
R
2
ρ
A
/
E
I, which is slightly different from the C
i used in Section 5.2.3. The problem presented here has also been solved by applying the finite element method (FEM) [9, 33] and the wave analysis method [34]. In Table 7, the dimensionless frequencies obtained from the present approach are compared with those of [9, 33, 34]. It is noticed that the natural frequencies obtained from the BSWI agree well with these solutions, especially with [34], which can be seen as a relative accurate reference solution for this problem [34].Table 7
Dimensionless frequencies of the three-span clamped arch.
Mode
Krishnan and Suresh [9]
Maurizi et al. [33]
Kang et al. [34]
Present
1
2.701
2.680
2.6833155
2.682
2
4.828
4.824
4.8337572
4.828
3
9.543
9.536
9.5647244
9.549
4
14.535
14.527
14.5850042
14.551
5
21.751
21.749
21.8646005
21.795Figure 8
Geometry of a three-span clamped arch.


## 6. Conclusions

A B-spline wavelet on interval curved beam element is constructed in this paper. This method gives satisfactory results for static and free vibration behaviors of arches with varied curvatures, thicknesses, and boundaries. The reason for getting acceptable results can be attributed to the fact that the present element is developed in generalized shell theory, which is adapted to obtain the couple of normal, tangential, and rotational displacement. Another reason for getting acceptable results can be attributed to the numerical properties of B-spline wavelet on interval. By means of the numerical examples, the accuracy and efficiency of the present element are validated. It can be seen that the proposed method can obtain good results for static and free vibration analysis. The methodology and results presented here can help in understanding the more complicated behavior of the curved shell element.



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*Source: 738162-2014-02-25.xml*