# Influence of Non-Structural Localized Inertia on Free Vibration Response of Thin-Walled Structures by Variable Kinematic Beam Formulations

Authors: Alfonso Pagani; Francesco Zangallo; Erasmo Carrera
Journal: Shock and Vibration
 (2014)

---

## Abstract

Variable kinematic beam theories are used in this paper to carry out vibration analysis of isotropic thin-walled structures subjected to non-structural localized inertia. Arbitrarily enriched displacement fields for beams are hierarchically obtained by using the Carrera Unified Formulation (CUF). According to CUF, kinematic fields can be formulated either as truncated Taylor-like expansion series of the generalized unknowns or by using only pure translational variables by locally discretizing the beam cross-section through Lagrange polynomials. The resulting theories were, respectively, referred to as TE (Taylor Expansion) and LE (Lagrange Expansion) in recent works. If the finite element method is used, as in the case of the present work, stiffness and mass elemental matrices for both TE and LE beam models can be written in terms of the same fundamental nuclei. The fundamental nucleus of the mass matrix is opportunely modified in this paper in order to account for non-structural localized masses. Several beams are analysed and the results are compared to those from classical beam theories, 2D plate/shell, and 3D solid models from a commercial FEM code. The analyses demonstrate the ineffectiveness of classical theories in dealing with torsional, coupling, and local effects that may occur when
localized inertia is considered. Thus the adoption of higher-order beam models is mandatory. The results highlight the efficiency of the proposed models and, in particular, the enhanced capabilities of LE modelling approach, which is able to reproduce solid-like analysis with very low computational costs.


---

## Body

## 1. Introduction to Refined Beam Theories

In engineering practice, problems involving non-structural masses are of special interest [1]. An important example is that of aerospace engineering. In aerospace design, in fact, non-structural masses are commonly used in finite element (FE) models to incorporate the weight of the engines, fuel, and payload, see, for example, [2–5]. In this paper, the effects due to localized inertia on free vibration of thin-walled beams are investigated through one-dimensional (1D) higher-order models. A brief overview about the evolution of refined beam theories is given below.A number of refined beam theories have been proposed over the years to overcome the limitation of classical beam models such as those by Euler [6] (hereinafter referred to as EBBM) and Timoshenko [7, 8] (hereinafter referred to as TBM). If the rectangular cartesian coordinate system shown in Figure 1 is adopted and we consider bending on the x
y-plane, the kinematic field of EBBM can be written as follows:
(1)
u
=
u
0
,
v
=
v
0
-
x
∂
u
0
∂
y
,
where u and v are the displacement components of a point belonging to the beam domain along x and y coordinates, respectively. u
0 and v
0 are the displacements of the beam axis, whereas -
(
∂
u
0
/
∂
y
) is the rotation of the cross-section about the z-axis (i.e., ϕ
z) as shown in Figure 2(a). According to EBBM, the deformed cross-section remains plane and orthogonal to the beam axis.Figure 1
Coordinate frame of the beam model.Differences between Euler-Bernoulli (a) and Timoshenko (b) beam theories.
(a)
(b)EBBM neglects cross-sectional shear deformation phenomena. Generally, the shear stresses play an important role in several problems (e.g., short beams and composite structures) and neglecting these terms can lead to incorrect results. One may want to generalize (1) and overcome the EBBM assumption of the orthogonality of the cross-section. The improved displacement field results in the TBM are as follows:
(2)
u
=
u
0
,
v
=
v
0
+
x
ϕ
z
.
TBM constitutes an improvement over EBBM since the cross-section does not necessarily remain perpendicular to the beam axis after deformation and one degree of freedom (i.e., the unknown rotation ϕ
z) is added to the original displacement field (see Figure 2(b)).Classical beam models yield reasonably good results when slender, solid section, and homogeneous structures are subjected to bending. Conversely, the analysis of deep, thin-walled, and open section beams may require more sophisticated theories to achieve sufficiently accurate results, see [9]. One of the main problems of TBM is that the homogeneous conditions of the transverse stress components at the top/bottom surfaces of the beam are not fulfilled, as shown in Figure 3. One can impose, for instance, (2) in order to have null transverse strain component (γ
x
y
=
(
∂
u
/
∂
y
)
+
(
∂
v
/
∂
x
)) at x
=
±
(
b
/
2
). This leads to the third-order displacement field known as the Reddy-Valsov beam theory [10] as follows:
(3)
u
=
u
0
,
v
=
v
0
+
f
1
(
x
)
ϕ
z
+
g
1
(
x
)
∂
u
0
∂
y
,
where f
1
(
x
) and g
1
(
x
) are cubic functions of the x coordinate. It should be noted that although the model of (3) has the same number of degrees of freedom (DOFs) of TBM, it overcomes classical beam theory limitations by foreseeing a quadratic distribution of transverse stresses on the cross-section of the beam.Figure 3
Homogeneous condition of transverse stress components at the free boundaries.The above theories are not able to include any kinematics resulting from the application of torsional moments. The simplest way to include torsion consists of considering a rigid rotation of the cross-section around they-axis (i.e., ϕ
y), see Figure 4. The resulting displacement model is
(4)
u
=
z
ϕ
y
,
w
=
-
x
ϕ
y
,
where w is the displacement component along the z-axis. According to (4), a linear distribution of transverse displacement components is needed to detect the rigid rotation of the cross-section about the beam axis.Figure 4
Rigid torsion of the beam cross-section.Beam models that include all the capabilities discussed so far can be obtained by summing all these contributions. By considering the deformations also in they
z-plane, one has
(5)
u
=
u
0
+
z
ϕ
y
,
v
=
v
0
+
f
1
(
x
)
ϕ
z
+
f
2
(
z
)
ϕ
x
+
g
1
(
x
)
∂
u
∂
y
+
g
2
(
z
)
∂
w
∂
y
,
w
=
w
0
-
x
ϕ
y
,
where f
1
(
x
), g
1
(
x
), f
2
(
z
), and g
2
(
z
) are cubic functions. In the case of rectangular cross-section, the cubic functions from Vlasov’s theory are f
1
(
x
)
=
x
-
(
4
/
3
b
2
)
x
3, g
1
(
x
)
=
-
(
4
/
3
b
2
)
x
3,  f
2
(
z
)
=
z
-
(
4
/
3
h
2
)
z
3, and g
2
(
z
)
=
-
(
4
/
3
h
2
)
z
3, where b and h are the dimensions of the cross-section along the x- and z-axis, respectively. The beam models discussed so far are not able to account for manyhigher-order effects, such as the second-order in-plane deformations of the cross-section.Over the last century, many refined beam theories have been proposed to overcome the limitations of classical beam modelling. A commendable and comprehensive review on beam theories can be found in [11, 12]. Different approaches have been used to improve the beam models, which include the use of warping functions based on de Saint-Venant's solution [13–16], the variational asymptotic solution (VAM) [17–22], and the generalized beam theory (GBT) [23–28]. A displacement field which is able to take into account the cross-section deformation by means of warping functions is
(6)
u
=
u
0
,
v
=
v
0
+
f
(
x
)
ϵ
x
y
0
-
x
∂
u
0
∂
y
+
f
(
z
)
ϵ
y
z
0
-
z
∂
w
0
∂
y
,
w
=
w
0
,
where f
(
x
) and f
(
z
) are the warping functions, whereas ϵ
x
y
0 and ϵ
y
z
0 are the transverse shear strains measured on the beam axis. As a general guideline, one can state that the richer the kinematic field is, the more accurate the 1D model becomes [29]. The main disadvantages of a richer displacement field are the increase of equations to be solved and the choice of the terms to be added since this choice is generally problem dependent.CUF represents a tool to tackle the problem of the choice of the expansion terms. Letu
=
{
u
x
u
y
u
z
}
T be the transposed displacement vector. According to CUF, the generic displacement field can be expressed in a compact manner as an N-order expansion in terms of generic functions, F
τ, as follows:
(7)
u
(
x
,
y
,
z
)
=
F
τ
(
x
,
z
)
u
τ
(
y
)
,
τ
=
1,2
,
…
,
M
,
where F
τ are the functions of the coordinates x and z on the cross-section. u
τ is the vector of thegeneralized displacements, M stands for the number of terms used in the expansion. In line with (7), it is clear that (1) to (5) consist of MacLaurin expansions that uses 2D polynomials x
i
z
j as base functions, where i and j are positive integers. This class of models is referred to as TE (Taylor-Expansion). It should be noted that (1), (2), and (4) are particular cases of the linear (N
=
1) TE model, which can be expressed as
(8)
u
x
=
u
x
1
+
x
u
x
2
+
z
u
x
3
,
u
y
=
u
y
1
+
x
u
y
2
+
z
u
y
3
,
u
z
=
u
z
1
+
x
u
z
2
+
z
u
z
3
,
where the parameters on the right-hand sides (u
x
1, u
y
1, u
z
1, u
x
2, etc.) are the displacements and the rotations of the beam reference axis. Higher-order terms can be taken into account according to (7). For instance, the displacement fields of (3) and (5) can be seen as particular cases of the third-order (N
=
3) TE model as follows:
(9)
u
x
=
u
x
1
+
x
u
x
2
+
z
u
x
3
+
x
2
u
x
4
+
x
z
u
x
5
+
z
2
u
x
6
+
x
3
u
x
7
+
x
2
z
u
x
8
+
x
z
2
u
x
9
+
z
3
u
x
10
,
u
y
=
u
y
1
+
x
u
y
2
+
z
u
y
3
+
x
2
u
y
4
+
x
z
u
y
5
+
z
2
u
y
6
+
x
3
u
y
7
+
x
2
z
u
y
8
+
x
z
2
u
y
9
+
z
3
u
y
10
,
u
z
=
u
z
1
+
x
u
z
2
+
z
u
z
3
+
x
2
u
z
4
+
x
z
u
z
5
+
z
2
u
z
6
+
x
3
u
z
7
+
x
2
z
u
z
8
+
x
z
2
u
z
9
+
z
3
u
z
10
.
The possibility of dealing with any-order expansion makes the TE CUF able to handle arbitrary geometries, thin-walled structures, and local effects as it has been shown for static [30, 31], linearized stability analyses [32, 33], and free-vibrations of both metallic [34–36] and composite structures [37–39].Recently, a new class of CUF models has been developed by the first author and his coworkers [40, 41]. In this class of models, Lagrange-like polynomials are used to discretize the displacement field on the cross-section. These models are referred to as LE (Lagrange Expansion) and they have been used to develop a component-wise (CW) modelling approach in some recent works. Static analyses on isotropic [40] and composite structures [42, 43] have revealed the strength of LE models in dealing with open cross-sections, localized boundary conditions, and layer-wise descriptions of composite structures. Moreover, static and dynamic analyses of reinforced-shell wing structures [44, 45] as well as civil engineering framed constructions [46, 47] by 1D CW models have been carried out and the results have shown the enhanced capabilities of LE in obtaining 3D accuracy with very low computational costs. Furthermore, the LE models have shown their enhanced capabilities in dealing with load factors and localized inertia in static analysis of thin-walled beams in a companion paper [48].In the present paper, TE and LE models are extended to include localized inertia in the free vibration analysis of homogeneous isotropic thin-walled beams. In the next section, LE models are formulated by using CUF. 1D refined elements including localized inertia are then obtained by classical finite element method (FEM). Numerical results are subsequently discussed. Finally, the main conclusions are outlined.


## 2. Higher-Order Models Based on Lagrange Polynomial Expansions

The degrees of freedom of the TE models (displacements andN-order derivatives of displacements) described above are defined along the axis of the beam. The unknown variables are only pure displacements if Lagrange polynomials are adopted as expansion functions (F
τ) in (7). This class of CUF models, which is referred to as LE, was recently introduced in [40], where triangular three-(L3) and six-node (L6) elements as well as quadrilateral four-(L4), nine-(L9), and 16-node (L16) elements were used to discretize the displacement unknowns on the cross-section of beam structures. In this paper, L9 elements are used. In the case of an L9 element, the cross-sectional interpolation functions are given by
(10)
F
τ
=
1
4
(
r
2
+
r
r
τ
)
(
s
2
+
s
s
τ
)
τ
=
1,3
,
5,7
,
F
τ
=
1
2
s
τ
2
(
s
2
-
s
s
τ
)
(
1
-
r
2
)
+
1
2
r
τ
2
(
r
2
-
r
r
τ
)
(
1
-
s
2
)
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
τ
=
2,4
,
6,8
,
F
τ
=
(
1
-
r
2
)
(
1
-
s
2
)
τ
=
9
,
where r and s vary from -
1 to +
1, whereas r
τ and s
τ are the coordinates of the nine points whose locations in the natural coordinate frame are shown in Figure 5. The displacement field given by an L9 element is therefore
(11)
u
x
=
F
1
u
x
1
+
F
2
u
x
2
+
⋯
+
F
9
u
x
9
u
y
=
F
1
u
y
1
+
F
2
u
y
2
+
⋯
+
F
9
u
y
9
u
z
=
F
1
u
z
1
+
F
2
u
z
2
+
⋯
+
F
9
u
z
9
,
where u
x
1
,
…
,
u
z
9 are the displacement variables of the problem and represent the translational displacement components of each of the nine points of the L9 element. The choice of using Lagrange-type polynomials to discretize the cross-section displacement field leads to the following advantages.(i)
Each of the problem unknowns (e.g.,u
x
1
,
…
,
u
z
9) has a precise physical meaning; that is, they are pure translational displacements. This is not true in the case of TE models.
(ii)
The beam model can be refined in a smart way; that is, unknown variables can be arbitrarily placed in subdomains over the cross-section area (e.g., close to loadings). This is realized by discretizing the beam cross-section with a number of L-elements as shown in Figure6.Figure 5
Cross-sectional L9 element in the natural plane.Figure 6
Two assembled L9 elements in actual geometry.


## 3. Finite Element Formulation

### 3.1. Preliminaries

Referring to the coordinate frame shown in Figure1, let the cross-section of the structure be denoted by Ω and let the beam boundaries over y be 0
≤
y
≤
L. The stress, σ, and strain, ϵ, components are grouped as follows:
(12)
σ
=
{
σ
y
y
σ
x
x
σ
z
z
σ
x
z
σ
y
z
σ
x
y
}
T
,
ϵ
=
{
ϵ
y
y
ϵ
x
x
ϵ
z
z
ϵ
x
z
ϵ
y
z
ϵ
x
y
}
T
.
In the case of small displacements with respect to a characteristic dimension of Ω, linear strain-displacement relations can be used as follows:
(13)
ϵ
=
D
u
,
where D is the following linear differential operator:
(14)
D
=
[
0
∂
∂
y
0
∂
∂
x
0
0
0
0
∂
∂
z
∂
∂
z
0
∂
∂
x
0
∂
∂
z
∂
∂
y
∂
∂
y
∂
∂
x
0
]
.
Constitutive laws were exploited to obtain stress components as follows:
(15)
σ
=
C
~
ϵ
.
In the case of isotropic material, the matrix C
~ is
(16)
C
~
=
[
λ
+
2
G
λ
λ
0
0
0
λ
λ
+
2
G
λ
0
0
0
λ
λ
λ
+
2
G
0
0
0
0
0
0
G
0
0
0
0
0
0
G
0
0
0
0
0
0
G
]
,
where G and λ are the Lamé’s parameters. If Poisson’s ratio ν and Young modulus E are used, one has G
=
E
/
(
2
(
1
+
ν
)
) and λ
=
ν
E
/
(
(
1
+
ν
)
(
1
-
2
ν
)
).


### 3.2. Fundamental Nuclei

The FE approach was adopted to discretize the structure along they-axis. This process is conducted via a classical finite element technique, where the displacement vector is given by
(17)
u
(
x
,
y
,
z
)
=
F
τ
(
x
,
z
)
N
i
(
y
)
q
τ
i
,
N
i stands for the shape functions and q
τ
i for the nodal displacement vector,
(18)
q
τ
i
=
{
q
u
x
τ
i
q
u
y
τ
i
q
u
z
τ
i
}
T
.
For the sake of brevity, the shape functions are not reported here. They can be found in many books on finite elements, for instance, in [49]. Elements with four nodes (B4) were adopted in this work; that is, a cubic approximation along the y axis was assumed. The choice of the cross-section discretization for the LE class (i.e., the choice of the type, the number, and the distribution of cross-section elements) or the theory order, N, for the TE class is completely independent of the choice of the beam finite element to be used along the axis of the beam.The stiffness and mass matrices of the elements were obtained via the principle of virtual displacements as follows:(19)
δ
L
int
⁡
=
∫
V
‍
δ
ϵ
T
σ
d
V
=
-
δ
L
ine
,
where L
int
⁡ stands for the strain energy, δ
L
ine is the work of the inertial loadings, and δ stands for the virtual variation. The virtual variation of the strain energy was rewritten using (13), (15), and (17) as follows:
(20)
δ
L
int
⁡
=
δ
q
τ
i
T
K
i
j
τ
s
q
s
j
,
where K
i
j
τ
s is the stiffness matrix in the form of the fundamental nucleus. Its components are provided below and they are referred to as K
r
c
i
j
τ
s, where r is the row number (r
=
1,2
,
3) and c is the column number (c
=
1,2
,
3). 								(21)
K
11
i
j
τ
s
=
(
λ
+
2
G
)
∫
Ω
F
τ
,
x
F
s
,
x
d
Ω
∫
l
‍
N
i
N
j
d
y
+
G
∫
Ω
‍
F
τ
,
z
F
s
,
z
d
Ω
∫
l
‍
N
i
N
j
d
y
+
G
∫
Ω
‍
F
τ
F
s
d
Ω
∫
l
‍
N
i
,
y
N
j
,
y
d
y
,
K
12
i
j
τ
s
=
λ
∫
Ω
‍
F
τ
,
x
F
s
d
Ω
∫
l
‍
N
i
N
j
,
y
d
y
+
G
∫
Ω
‍
F
τ
F
s
,
x
d
Ω
∫
l
‍
N
i
,
y
N
j
d
y
,
K
13
i
j
τ
s
=
λ
∫
Ω
‍
F
τ
,
x
F
s
,
z
d
Ω
∫
l
‍
N
i
N
j
d
y
+
G
∫
Ω
‍
F
τ
,
z
F
s
,
x
d
Ω
∫
l
‍
N
i
N
j
d
y
,
K
21
i
j
τ
s
=
λ
∫
Ω
‍
F
τ
F
s
,
x
d
Ω
∫
l
‍
N
i
,
y
N
j
d
y
+
G
∫
Ω
‍
F
τ
,
x
F
s
d
Ω
∫
l
‍
N
i
N
j
,
y
d
y
,
K
22
i
j
τ
s
=
G
∫
Ω
‍
F
τ
,
z
F
s
,
z
d
Ω
∫
l
‍
N
i
N
j
d
y
+
G
∫
Ω
‍
F
τ
,
x
F
s
,
x
d
Ω
∫
l
‍
N
i
N
j
d
y
+
(
λ
+
2
G
)
∫
Ω
‍
F
τ
F
s
d
Ω
∫
l
‍
N
i
,
y
N
j
,
y
d
y
,
K
23
i
j
τ
s
=
λ
∫
Ω
‍
F
τ
F
s
,
z
d
Ω
∫
l
‍
N
i
,
y
N
j
d
y
+
G
∫
Ω
‍
F
τ
,
z
F
s
d
Ω
∫
l
‍
N
i
N
j
,
y
d
y
,
K
31
i
j
τ
s
=
λ
∫
Ω
‍
F
τ
,
z
F
s
,
x
d
Ω
∫
l
‍
N
i
N
j
d
y
+
G
∫
Ω
‍
F
τ
,
x
F
s
,
z
d
Ω
∫
l
‍
N
i
N
j
d
y
,
K
32
i
j
τ
s
=
λ
∫
Ω
‍
F
τ
,
z
F
s
d
Ω
∫
l
‍
N
i
N
j
,
y
d
y
+
G
∫
Ω
‍
F
τ
F
s
,
z
d
Ω
×
∫
l
‍
N
i
,
y
N
j
d
y
,
K
33
i
j
τ
s
=
(
λ
+
2
G
)
∫
Ω
‍
F
τ
,
z
F
s
,
z
d
Ω
∫
l
‍
N
i
N
j
d
y
+
G
∫
Ω
‍
F
τ
,
x
F
s
,
x
d
Ω
∫
l
‍
N
i
N
j
d
y
+
G
∫
Ω
‍
F
τ
F
s
d
Ω
∫
l
‍
N
i
,
y
N
j
,
y
d
y
.
The fundamental nucleus has to be expanded according to the summation indexes τ and s in order to obtain the elemental stiffness matrix.The virtual variation of the work of the inertial loadings is(22)
δ
L
ine
=
∫
V
‍
ρ
δ
u
T
u
¨
d
V
,
where ρ stands for the density if the material, and u
¨ is the acceleration vector. Equation (22) is rewritten using (17) as follows:
(23)
δ
L
ine
=
δ
q
τ
i
T
∫
l
‍
N
i
N
j
d
y
∫
Ω
‍
ρ
F
τ
F
s
d
Ω
q
¨
s
j
=
δ
q
τ
i
T
M
i
j
τ
s
q
¨
s
j
,
where M
i
j
τ
s is the fundamental nucleus of the mass matrix. Its components are provided below and they are referred to as M
r
c
i
j
τ
s, where r is the row number (r
=
1,2
,
3) and c denotes column number (c
=
1,2
,
3). Consider the following:
(24)
M
11
i
j
τ
s
=
M
22
i
j
τ
s
=
M
33
i
j
τ
s
=
ρ
∫
l
‍
N
i
N
j
d
y
∫
Ω
‍
F
τ
F
s
d
Ω
M
12
i
j
τ
s
=
M
13
i
j
τ
s
=
M
21
i
j
τ
s
=
M
23
i
j
τ
s
=
M
31
i
j
τ
s
=
M
32
i
j
τ
s
=
0
.
It should be noted that no assumptions on the approximation order or on the choice of F
τ functions (TE or LE) have been made in formulating K
i
j
τ
s and M
i
j
τ
s. It is therefore possible to obtain refined beam models without changing the formal expression of the nuclei components. This is the key-points of CUF which allows, with only nine coding statements, the implementation of any-order of multiple class theories.In the present paper, the effect due to non-structural masses is also investigated. Localized inertia can in principle be arbitrarily placed into the 3D domain of the beam structure. In the framework of the CUF, this is easily realized by adding the following term to the fundamental nucleus of the mass matrix:(25)
m
i
j
τ
s
=
I
[
F
τ
(
x
m
,
z
m
)
F
s
(
x
m
,
z
m
)
N
i
(
y
m
)
N
j
(
y
m
)
]
m
~
,
where I is the 3
×
3 identity matrix and m
~ is the value of the non-structural mass, which is applied at point (
x
m
,
y
m
,
z
m
).The undamped dynamic problem can be derived by substituting (20) and (23) into (19). After global mass and stiffness FE matrices are assembled, one has
(26)
M
q
¨
+
K
q
=
0
.
Introducing harmonic solutions, it is possible to compute the natural frequencies ω
k by solving an eigenvalues problem as follows:
(27)
(
K
-
ω
k
2
M
)
q
k
=
0
,
where q
k is the kth eigenvector. The eigenvalue problem of (27) is solved by using ARPACK libraries [50], which are based on an algorithmic variant of the Arnoldi process [51] called Implicitly Restarted Arnoldi Method (IRAM) [52].


## 3.1. Preliminaries

Referring to the coordinate frame shown in Figure1, let the cross-section of the structure be denoted by Ω and let the beam boundaries over y be 0
≤
y
≤
L. The stress, σ, and strain, ϵ, components are grouped as follows:
(12)
σ
=
{
σ
y
y
σ
x
x
σ
z
z
σ
x
z
σ
y
z
σ
x
y
}
T
,
ϵ
=
{
ϵ
y
y
ϵ
x
x
ϵ
z
z
ϵ
x
z
ϵ
y
z
ϵ
x
y
}
T
.
In the case of small displacements with respect to a characteristic dimension of Ω, linear strain-displacement relations can be used as follows:
(13)
ϵ
=
D
u
,
where D is the following linear differential operator:
(14)
D
=
[
0
∂
∂
y
0
∂
∂
x
0
0
0
0
∂
∂
z
∂
∂
z
0
∂
∂
x
0
∂
∂
z
∂
∂
y
∂
∂
y
∂
∂
x
0
]
.
Constitutive laws were exploited to obtain stress components as follows:
(15)
σ
=
C
~
ϵ
.
In the case of isotropic material, the matrix C
~ is
(16)
C
~
=
[
λ
+
2
G
λ
λ
0
0
0
λ
λ
+
2
G
λ
0
0
0
λ
λ
λ
+
2
G
0
0
0
0
0
0
G
0
0
0
0
0
0
G
0
0
0
0
0
0
G
]
,
where G and λ are the Lamé’s parameters. If Poisson’s ratio ν and Young modulus E are used, one has G
=
E
/
(
2
(
1
+
ν
)
) and λ
=
ν
E
/
(
(
1
+
ν
)
(
1
-
2
ν
)
).


## 3.2. Fundamental Nuclei

The FE approach was adopted to discretize the structure along they-axis. This process is conducted via a classical finite element technique, where the displacement vector is given by
(17)
u
(
x
,
y
,
z
)
=
F
τ
(
x
,
z
)
N
i
(
y
)
q
τ
i
,
N
i stands for the shape functions and q
τ
i for the nodal displacement vector,
(18)
q
τ
i
=
{
q
u
x
τ
i
q
u
y
τ
i
q
u
z
τ
i
}
T
.
For the sake of brevity, the shape functions are not reported here. They can be found in many books on finite elements, for instance, in [49]. Elements with four nodes (B4) were adopted in this work; that is, a cubic approximation along the y axis was assumed. The choice of the cross-section discretization for the LE class (i.e., the choice of the type, the number, and the distribution of cross-section elements) or the theory order, N, for the TE class is completely independent of the choice of the beam finite element to be used along the axis of the beam.The stiffness and mass matrices of the elements were obtained via the principle of virtual displacements as follows:(19)
δ
L
int
⁡
=
∫
V
‍
δ
ϵ
T
σ
d
V
=
-
δ
L
ine
,
where L
int
⁡ stands for the strain energy, δ
L
ine is the work of the inertial loadings, and δ stands for the virtual variation. The virtual variation of the strain energy was rewritten using (13), (15), and (17) as follows:
(20)
δ
L
int
⁡
=
δ
q
τ
i
T
K
i
j
τ
s
q
s
j
,
where K
i
j
τ
s is the stiffness matrix in the form of the fundamental nucleus. Its components are provided below and they are referred to as K
r
c
i
j
τ
s, where r is the row number (r
=
1,2
,
3) and c is the column number (c
=
1,2
,
3). 								(21)
K
11
i
j
τ
s
=
(
λ
+
2
G
)
∫
Ω
F
τ
,
x
F
s
,
x
d
Ω
∫
l
‍
N
i
N
j
d
y
+
G
∫
Ω
‍
F
τ
,
z
F
s
,
z
d
Ω
∫
l
‍
N
i
N
j
d
y
+
G
∫
Ω
‍
F
τ
F
s
d
Ω
∫
l
‍
N
i
,
y
N
j
,
y
d
y
,
K
12
i
j
τ
s
=
λ
∫
Ω
‍
F
τ
,
x
F
s
d
Ω
∫
l
‍
N
i
N
j
,
y
d
y
+
G
∫
Ω
‍
F
τ
F
s
,
x
d
Ω
∫
l
‍
N
i
,
y
N
j
d
y
,
K
13
i
j
τ
s
=
λ
∫
Ω
‍
F
τ
,
x
F
s
,
z
d
Ω
∫
l
‍
N
i
N
j
d
y
+
G
∫
Ω
‍
F
τ
,
z
F
s
,
x
d
Ω
∫
l
‍
N
i
N
j
d
y
,
K
21
i
j
τ
s
=
λ
∫
Ω
‍
F
τ
F
s
,
x
d
Ω
∫
l
‍
N
i
,
y
N
j
d
y
+
G
∫
Ω
‍
F
τ
,
x
F
s
d
Ω
∫
l
‍
N
i
N
j
,
y
d
y
,
K
22
i
j
τ
s
=
G
∫
Ω
‍
F
τ
,
z
F
s
,
z
d
Ω
∫
l
‍
N
i
N
j
d
y
+
G
∫
Ω
‍
F
τ
,
x
F
s
,
x
d
Ω
∫
l
‍
N
i
N
j
d
y
+
(
λ
+
2
G
)
∫
Ω
‍
F
τ
F
s
d
Ω
∫
l
‍
N
i
,
y
N
j
,
y
d
y
,
K
23
i
j
τ
s
=
λ
∫
Ω
‍
F
τ
F
s
,
z
d
Ω
∫
l
‍
N
i
,
y
N
j
d
y
+
G
∫
Ω
‍
F
τ
,
z
F
s
d
Ω
∫
l
‍
N
i
N
j
,
y
d
y
,
K
31
i
j
τ
s
=
λ
∫
Ω
‍
F
τ
,
z
F
s
,
x
d
Ω
∫
l
‍
N
i
N
j
d
y
+
G
∫
Ω
‍
F
τ
,
x
F
s
,
z
d
Ω
∫
l
‍
N
i
N
j
d
y
,
K
32
i
j
τ
s
=
λ
∫
Ω
‍
F
τ
,
z
F
s
d
Ω
∫
l
‍
N
i
N
j
,
y
d
y
+
G
∫
Ω
‍
F
τ
F
s
,
z
d
Ω
×
∫
l
‍
N
i
,
y
N
j
d
y
,
K
33
i
j
τ
s
=
(
λ
+
2
G
)
∫
Ω
‍
F
τ
,
z
F
s
,
z
d
Ω
∫
l
‍
N
i
N
j
d
y
+
G
∫
Ω
‍
F
τ
,
x
F
s
,
x
d
Ω
∫
l
‍
N
i
N
j
d
y
+
G
∫
Ω
‍
F
τ
F
s
d
Ω
∫
l
‍
N
i
,
y
N
j
,
y
d
y
.
The fundamental nucleus has to be expanded according to the summation indexes τ and s in order to obtain the elemental stiffness matrix.The virtual variation of the work of the inertial loadings is(22)
δ
L
ine
=
∫
V
‍
ρ
δ
u
T
u
¨
d
V
,
where ρ stands for the density if the material, and u
¨ is the acceleration vector. Equation (22) is rewritten using (17) as follows:
(23)
δ
L
ine
=
δ
q
τ
i
T
∫
l
‍
N
i
N
j
d
y
∫
Ω
‍
ρ
F
τ
F
s
d
Ω
q
¨
s
j
=
δ
q
τ
i
T
M
i
j
τ
s
q
¨
s
j
,
where M
i
j
τ
s is the fundamental nucleus of the mass matrix. Its components are provided below and they are referred to as M
r
c
i
j
τ
s, where r is the row number (r
=
1,2
,
3) and c denotes column number (c
=
1,2
,
3). Consider the following:
(24)
M
11
i
j
τ
s
=
M
22
i
j
τ
s
=
M
33
i
j
τ
s
=
ρ
∫
l
‍
N
i
N
j
d
y
∫
Ω
‍
F
τ
F
s
d
Ω
M
12
i
j
τ
s
=
M
13
i
j
τ
s
=
M
21
i
j
τ
s
=
M
23
i
j
τ
s
=
M
31
i
j
τ
s
=
M
32
i
j
τ
s
=
0
.
It should be noted that no assumptions on the approximation order or on the choice of F
τ functions (TE or LE) have been made in formulating K
i
j
τ
s and M
i
j
τ
s. It is therefore possible to obtain refined beam models without changing the formal expression of the nuclei components. This is the key-points of CUF which allows, with only nine coding statements, the implementation of any-order of multiple class theories.In the present paper, the effect due to non-structural masses is also investigated. Localized inertia can in principle be arbitrarily placed into the 3D domain of the beam structure. In the framework of the CUF, this is easily realized by adding the following term to the fundamental nucleus of the mass matrix:(25)
m
i
j
τ
s
=
I
[
F
τ
(
x
m
,
z
m
)
F
s
(
x
m
,
z
m
)
N
i
(
y
m
)
N
j
(
y
m
)
]
m
~
,
where I is the 3
×
3 identity matrix and m
~ is the value of the non-structural mass, which is applied at point (
x
m
,
y
m
,
z
m
).The undamped dynamic problem can be derived by substituting (20) and (23) into (19). After global mass and stiffness FE matrices are assembled, one has
(26)
M
q
¨
+
K
q
=
0
.
Introducing harmonic solutions, it is possible to compute the natural frequencies ω
k by solving an eigenvalues problem as follows:
(27)
(
K
-
ω
k
2
M
)
q
k
=
0
,
where q
k is the kth eigenvector. The eigenvalue problem of (27) is solved by using ARPACK libraries [50], which are based on an algorithmic variant of the Arnoldi process [51] called Implicitly Restarted Arnoldi Method (IRAM) [52].


## 4. Numerical Results

This section investigates the efficiency of the present approach applied to modal analysis of beam structures with localized inertia. The attention is focused on the capability of the approach to deal with refined solutions with low computational costs. The results were compared with classical beam theories and FE models from the commercial code MSCNastran
©. Common C- and I-shaped beams as well as hollow-rectangular boxes were considered. The adopted material was an aluminium alloy and it had the following characteristics: Young modulus E equal to 75 GPa, Poisson’s ratio ν equal to 0.33, and density ρ
=
2700 Kg/m3. For all the problems addressed, ten four-node (B4) cubic 1D Lagrangian finite elements were used along the beam axis for both TE and LE models.


### 4.1. C-Section Beam

The analysis of a cantilever C-shaped beam was carried out as the first assessment. The cross-section of the structure is shown in Figure7. The geometrical data were as follows: b
=
0.5 m, a
=
h
=
2
b, and t
=
0.1 m. The length of the beam, L, was equal to 20 m.Figure 7
Cross-section of the C-shaped beam.Table1 shows the first ten natural frequencies for each implemented model. Columns 2 and 3 refer to the classical models EBBM and TBM. The natural frequencies by fourth-order (N
=
4), sixth-order (N
=
6), and eighth-order (N
=
8) refined TE beam model are given in columns 4 to 6. Two different distributions of L9 elements above the cross-section were considered (6L9 and 10L9) in the case of LE modelling approach, as shown in Figure 8, and the results are given in the 7th and 8th columns of Table 1. In columns 9 and 10, MSC Nastran
© shell and solid models are given for comparison purposes, and they are,respectively, referred to as FEM-2D and FEM-3D in the table. FEM-2D model was obtained by using 4-node QUAD elements, whereas FEM-3D model was obtained by using 8-node CHEXA elements. For each model, the number of DOFs is given in the last row of Table 1. The following comments can be made:(1)
As expected, classical models are not able to provide torsional and coupled modes.
(2)
A higher than fourth-order TE model is necessary to correctly detect torsional modes.
(3)
As indicated in Table1, where superscripts are used to denote the kind of each vibrational mode, bending, torsional, and coupled bending/torsional modes are detected by LE models in accordance with MSC Nastran
© 2D and 3D models. An appropriate distribution of the L9 elements above the cross-section is effective in improving the accuracy of the solution.
(4)
The 10 L9 LE model provides a solid-like solution with a high reduction of the number of the DOFs.Table 1
First10 natural frequencies (Hz) for the cantilever C-shaped beam.
Mode
Classical and refined models based on TE
LE models
MSCNastran
©
EBBM
TMB
N
=
4
N
=
6
N
=
8
6 L9
10 L9
FEM-2D
FEM-3D
1
1.75b
1.75b
1.76b
1.76b
1.76b
1.76b
1.76b
1.74b
1.75b
2
2.99b
2.98b
2.93b
2.76b
2.65b
2.56b
2.56b
2.49b
2.54b
3
10.97b
10.90b
10.90b
7.66t
6.73t
6.29t
6.27t
6.09t
6.23t
4
18.67b
18.35b
12.25t
10.89c
10.89c
10.87c
10.87b
10.71b
10.83b
5
30.63b
30.20b
17.46t
13.83t
12.37t
11.46t
11.41t
10.99t
11.34t
6
51.81b
49.83b
29.94b
24.87t
22.54t
21.18t
21.10t
20.29c
20.97t
7
59.75b
58.25b
34.66t
29.93b
29.87t
29.25t
29.19t
28.82t
29.06t
8
65.87a
65.87a
45.76t
32.49b
30.21b
29.89b
29.86b
29.55b
29.75b
9
98.20b
93.73b
56.94c
41.26c
37.12c
34.93c
34.76c
33.44c
34.53c
10
100.22b
94.43b
59.22c
57.05c
53.02c
50.23c
49.88c
48.06c
49.51c
DOFs
93
155
1395
2604
4185
3627
5859
38250
177000
aAxial; bbending; ccoupled; ttorsionalDistribution of L9 elements above the cross-section of the C-shaped beam.
(a)
6 L9
(b)
10 L9A second beam configuration was considered and a non-structural mass was added on the C-shaped beam at the coordinates (0,L, h), as Figure 9 shows. The weight of the non-structural mass was equal to 4000 Kg. For this load case, Table 2 reports the main vibrational frequencies by different models for clamped-free boundary conditions. The number of the DOFs for each model implemented is also given in the last row of the table. Figures 10 and 11 show the 8th mode shape by both LE and FEM-3D models, and the effects due to the localized non-structural mass are highlighted. Table 3 shows the natural frequencies of the C-shaped beam undergoing clamped-clamped boundary conditions. For this case, the same localized inertia as above was placed at y
=
(
3
/
4
)
L as shown in Figure 9. The following comments arise.(1)
As expected, the application of the non-structural mass is responsible for bending/torsional mode couplings.
(2)
Classical models and lower-order TE models are not effective for the problem under consideration.
(3)
The proposed LE model is able to detect the MSCNastran
© shell and solid results.
(4)
The number of the DOFs is very low in the case of CUF models if compared both to the shell and solid models.Table 2
First10 natural frequencies (Hz) for the cantilever C-shaped beam with localized inertia at the tip cross-section (Figure 9).
Mode
Classical and refined models based on TE
LE models
MSCNastran
©
EBBM
TMB
N
=
4
N
=
6
N
=
8
6 L9
10 L9
FEM-2D
FEM-3D
1
1.15b
1.15b
1.15b
1.13c
1.12c
1.11c
1.11c
1.10c
1.11c
2
1.97b
1.96b
1.89b
1.73c
1.65c
1.58c
1.58c
1.53c
1.57c
3
8.67b
8.63b
7.62t
5.70t
5.16t
4.90t
4.88t
4.88t
4.56t
4
14.69b
14.48b
11.37t
9.28t
8.37t
7.77t
7.73t
7.39t
7.67t
5
25.76b
25.44b
12.60c
10.09c
9.82c
9.73c
9.72c
9.52c
9.86c
6
41.50b
40.48b
25.28c
20.64c
18.23c
16.92c
16.81c
16.08c
16.63c
7
49.00b
48.07b
30.46c
23.37c
22.83c
22.71c
22.64c
22.59c
22.23c
8
54.78b
53.78b
35.77c
29.79c
28.69c
27.42c
27.16c
26.09c
26.71c
9
85.83b
81.86b
42.93c
33.48c
30.51c
29.70c
29.64c
29.27c
29.51c
10
89.33b
85.37b
51.67c
40.65c
38.24c
39.12c
38.16c
37.90c
34.35c
DOFs
93
155
1395
2604
4185
3627
5859
38250
177000
bBending; ccoupled; ttorsional.Table 3
First10 natural frequencies (Hz) for the clamped-clamped C-shaped beam with localized inertia at y
=
(
3
/
4
)
L (Figure 9).
Mode
Classical and refined models based on TE
LE models
MSCNastran
©
EBBM
TMB
N
=
4
N
=
6
N
=
8
6 L9
10 L9
FEM-2D
FEM-3D
1
10.09b
9.99b
9.50b
7.13c
6.34c
5.90c
5.86c
5.55c
5.57c
2
17.18b
16.70b
11.62b
10.32c
10.21c
10.14c
10.13c
9.86c
9.91c
3
24.52b
24.01b
19.39t
14.82t
13.33t
12.49t
12.41t
11.99t
12.45t
4
41.73b
39.38b
26.06t
23.00t
22.54t
22.34t
22.30t
22.02t
21.76t
5
51.56b
49.97b
31.58c
28.12c
27.59c
26.33c
26.13c
25.15c
26.26c
6
85.41b
79.17b
45.77c
31.47c
28.50c
27.99c
27.96c
27.63c
27.81c
7
91.67b
87.63b
49.97c
47.70c
44.95c
42.40c
41.92c
39.70c
41.22c
8
108.39b
105.88b
64.95c
50.60c
48.43c
48.14c
47.96c
46.89c
46.82c
9
142.65b
135.53b
76.50c
61.84c
56.29c
52.94c
52.18c
48.26c
48.43c
10
154.97b
139.71b
83.10c
63.53c
62.81c
62.63c
62.41c
61.33c
56.38c
DOFs
93
155
1395
2604
4185
3627
5859
37995
176115
bBending; ccoupled; ttorsional.Figure 9
Position of the non-structural mass at the tip cross-section of the C-shaped beam.Eighth modal shape of the C-shaped beam.
(a)
10 L9 LE model (29.861 Hz)
(b)
FEM-3D model (29.756 Hz)Eighth modal shape of the C-shaped beam with a non-structural mass at the tip cross-section (Figure9).
(a)
10 L9 LE model (27.169 Hz)
(b)
FEM-3D model (26.716 Hz)


### 4.2. I-Shaped Beam

A cantilever I-shaped beam was considered as the second example. The cross-section geometry is shown in Figure12. The dimensions a and b were, respectively, equal to 0.2 m and 0.3 m, whereas the thickness t was 0.05 m. The length of the structure, L, was 3 m.Figure 12
Cross-section of the I-shaped beam.The first ten natural frequencies, calculated according to different models, are given in Table4. Classical and refined TE as well as LE models are considered. Two different LE models were implemented and they differed in the cross-section L9 elements discretization, as shown in Figure 13. In columns 8 to 10, MSC Nastran
© beam, shell, and solid models are given for comparison purposes. MSC Nastran
© 1D model (hereinafter referred to as FEM-1D) was obtained with 21 two-node CBAR elements, whereas shell and solid elements were constructed using the same elements as in the previous analysis case, and they are referred to as FEM-2D and FEM-3D, respectively.Table 4
First10 natural frequencies (Hz) for the I-shaped beam.
Mode
Classical and refined models based on TE
LE models
MSCNastran
©
EBBM
TMB
N
=
6
N
=
8
7 L9
12 L9
FEM-1D
FEM-2D
FEM-3D
1
15.67b
15.65b
15.73b
15.73b
15.75b
15.72b
15.63b
15.11b
15.68b
2
35.28b
35.01b
34.93b
34.90b
34.87b
34.86b
34.73b
34.73b
34.80b
3
97.91b
96.08b
96.16t
88.74t
77.68t
76.58t
96.26b
68.45t
75.61t
4
217.27b
206.05b
97.06b
96.99b
97.13b
96.88b
196.99b
93.04b
96.63b
5
272.62b
265.09b
198.27b
197.28b
196.28b
196.10b
262.81b
191.98b
195.71b
6
439.19a
439.01a
265.24b
264.79b
246.78t
243.49t
439.09a
220.31t
240.63t
7
529.90b
507.02b
298.13t
277.27t
265.12t
264.20b
486.79b
253.52b
263.49b
8
592.06b
533.06b
440.66a
440.64a
440.64a
440.62a
497.53b
413.86c
440.23a
9
866.98b
811.08b
491.81b
487.36b
453.27t
448.06t
788.63b
439.82a
443.55t
10
1117.81b
950.08b
502.74t
495.78t
482.74b
481.97b
828.65b
464.45b
480.95b
DOFs
93
155
2604
4185
4185
6975
126
12200
127800
aAxial; bbending; ccoupled; ttorsional.Distribution of L9 elements above the cross-section of the I-shaped beam.
(a)
7 L9
(b)
12 L9The consistent correspondence between the 12 L9 LE model and the solid model was further investigated by means of the Modal Assurance Criterion (MAC), whose graphic representation is shown in Figure15(a). The MAC is defined as a scalar representing the degree of consistency between two distinct modal vectors (see [53]) as follows:
(28)
MA
C
i
j
=
|
{
ϕ
A
i
}
T
{
ϕ
B
j
}
|
2
{
ϕ
A
i
}
T
{
ϕ
A
i
}
{
ϕ
B
j
}
{
ϕ
B
j
}
T
,
where {
ϕ
A
i
} is the ith eigenvector of model A, whereas {
ϕ
B
j
} is the jth eigenvector of model B. The modal assurance criterion takes on values from zero (representing no consistent correspondence) to one (representing a consistent correspondence). The following remarks can be made:(1)
Unlike classical beam theories, higher-order TE and LE models are able to foresee torsional and coupled modes.
(2)
Unlikely CUF 1D models, FEM-1D model is not able to produce the solid-like solution.
(3)
The cross-section discretization refinement for LE models is an effective method that leads to the 3D solution.
(4)
Bending, torsional, and coupled modes are detected by LE models in accordance with FEM-2D and FEM-3D models with a high reduction of the number of the DOFs.
(5)
The MAC analysis underlines the perfect correspondence between the LE model and the solid one for the considered load case.In the second load case, a non-structural mass was added at the tip cross-section of the I-shaped beam, as shown in Figure14. The weight of the localized inertia was equal to 1000 Kg and it was applied at (a, L, b). Table 5 gives the first 10 vibrational frequencies for classical, TE, LE, and MSC Nastran
© models. The application of the non-structural mass produces an increase in the number of the coupled modes, as expected. The MAC matrix between the 12 L9 model and the FEM-3D model for the load case under consideration is also calculated and it is represented in Figure 15(b). It should be underlined that(1)
Classical models are ineffective in detecting the effects due to non-structural masses that are not placed in correspondence of the shear center;
(2)
The increase in the order for TE models provides greater accuracy on the evaluation of the coupled frequencies;
(3)
Even though local inertia is considered, LE models can produce solid-like solution, as shown by the MAC matrix. In fact, a very good agreement was found for the FEM-3D and LE models. LE analyses are therefore clearly able to detect 3D-elasticity solutions. However, some mixings in modes 3 and 4 are evident.Table 5
First10 natural frequencies (Hz) for the I-shaped beam with localized inertia at the tip cross-section (Figure 14).
Mode
Classical and refined models based on TE
LE models
MSCNastran
©
EBBM
TMB
N
=
4
N
=
6
N
=
8
7 L9
12 L9
FEM-2D
FEM-3D
1
3.70b
3.69b
3.67b
3.66b
3.65b
3.66b
3.63b
2.60b
3.61b
2
8.33b
8.29b
8.07c
8.01c
7.95c
8.17c
7.84c
5.74b
7.78c
3
61.86b
61.46b
53.74c
50.74t
47.77t
52.63t
45.30t
41.95t
38.50c
4
103.26b
102.06b
72.37t
64.48t
58.75t
80.77t
55.33t
53.62t
48.73t
5
171.03b
163.96b
101.98c
95.58c
91.37c
104.66c
88.99c
85.71c
84.95c
6
237.21b
231.18b
157.05t
151.08t
145.40t
150.93t
139.02t
129.29t
132.30t
7
458.27b
441.52b
224.05c
213.21c
204.64c
199.42c
191.89c
186.50c
184.73c
8
486.61b
444.07b
281.72c
266.67c
257.43c
254.03c
247.97c
230.50c
243.84c
9
679.49b
649.65b
392.70t
368.10t
343.10t
371.59t
316.84t
292.44t
292.90t
10
841.12b
777.59b
435.90c
420.36c
408.36c
398.96c
389.35c
366.42c
370.94c
DOFs
93
155
1395
2604
4185
4185
6975
12200
127800
bBending; ccoupled; ttorsional.Figure 14
Position of the non-structural mass at the tip cross-section of the I-shaped beam.MAC values between the12 L9 and FEM-3D models of the I-shaped beam.
(a)
Without localized inertia
(b)
With localized inertia


### 4.3. Hollow-Rectangular Beam

The cantilever hollow-square cross-section beam shown in Figure16 was investigated as the last example. The geometrical data were as follows: a
=
0.8 m, h
=
0.2 m, and t
=
0.01 m. The length of the beam, L, was equal to 3.2 m. Table 6 reports the first 10 natural frequencies, together with the number of DOFs for each model implemented. Columns 2 and 3 show the results from the classical beam theories. The TE models from the fourth-order to the eighth-order Taylor-like expansion are considered in columns 4 to 6. The LE models are reported in columns 7 and 8. The LE models were obtained by adopting two different discretizations above the cross-section of the beam, as Figure 17 shows. In Figure 17, a different notation is adopted with respect to Figures 8 and 13. The nodes of the L-elements are, in fact, not depicted in Figure 17, for the sake of clearness. In columns 9 and 10 of Table 6, MSC Nastran
© shell and solid models are given for comparison purposes and they are referred to as FEM-2D and FEM-3D, respectively. For this analysis case, the mesh discretizations for FEM-2D and FEM-3D models are shown in Figure 18. Figure 19 shows a comparison between some selected modal shapes by the 16L9 LE and FEM-3D models.Table 6
First10 natural frequencies (Hz) for the hollow-rectangular box beam.
Mode
Classical and refined models based on TE
LE models
MSCNastran
©
EBBM
TMB
N
=
4
N
=
6
N
=
8
10 L9
16 L9
FEM-2D
FEM-3D
1
25.48b
25.36b
25.15b
24.73b
24.52b
24.53b
24.07b
24.42b
23.89b
2
75.70b
72.79b
72.58b
72.42b
72.35b
62.08d
57.02d
55.34d
54.55d
3
158.03b
153.20b
137.79b
127.58b
90.87d
72.62b
72.67b
70.43s
69.59s
4
411.74a
357.95b
146.19t
134.29d
111.68s
100.61s
74.19s
73.98s
72.51b
5
435.12b
406.88b
330.86b
265.87s
127.00s
103.42s
77.60s
74.31b
73.12s
6
435.23b
411.74a
346.16b
274.47d
131.29s
109.38s
80.88s
75.35s
75.03s
7
833.29b
745.94b
413.68a
344.10b
139.88s
118.52d
84.76s
81.38s
80.48s
8
1089.50b
808.29b
428.76t
361.71s
141.44d
119.24s
96.36s
89.73s
89.56s
9
1235.23a
1146.76b
549.40b
388.13d
152.86s
122.01b
97.79s
93.33s
92.30s
10
1338.69b
1284.52b
690.19t
413.49a
166.72s
133.77s
109.87s
101.87s
101.58s
DOFs
93
155
1395
2604
4185
5580
8928
15000
38400
aAxial; bbending; ccoupled; ddifferential bending; ttorsional; sshell-like.Figure 16
Cross-section of the hollow-rectangular box beam.Distribution of L9 elements above the cross-section of the hollow-rectangular box beam.
(a)
10 L9
(b)
16 L9Mesh discretization for the FEM-2D and FEM-3D MSCNastran
© models.
(a)
FEM-2D
(b)
FEM-3DModal shapes of the hollow-rectangular box beam.
(a)
Mode 2,16 L9 LE model (57.023 Hz)
(b)
Mode 2, FEM-3D model (54.551 Hz)
(c)
Mode 10,16 L9 LE model (109.87 Hz)
(d)
Mode 10, FEM-3D model (101.58 Hz)The vibrations of the box beam were further analyzed and a non-structural mass with a weight of500 Kg was added at the coordinates (a, L, −h/2), as shown in Figure 20(a) (Case A). A comparison between LE and MSC Nastran
© models is given in Table 7. The correspondence between LE 16L9 and FEM-3D models is underlined through MAC analyses, which are shown in Figure 21. MAC matrices for both the cases without and with localized inertia (Case A) are provided in Figure 21. Some modal shapes by LE and FEM-3D models for the load case considered are shown in Figure 22.Table 7
First10 natural frequencies (Hz) for the hollow-rectangular box beam with localized inertia at the tip cross-section (Figure 20(a), Case A).
Mode
LE models
MSCNastran
©
10 L9
16 L9
FEM-2D
FEM-3D
1
6.20d
6.04d
6.11d
5.98d
2
18.54d
18.23d
18.94d
17.97d
3
43.77d
39.75d
38.66d
38.02d
4
44.07d
43.20d
69.36d
52.90d
5
95.47d
74.18s
70.43s
69.60s
6
100.62s
77.59s
73.98s
73.12s
7
103.41s
79.15s
78.77s
74.72s
8
109.34s
84.75s
81.38s
80.47s
9
118.77s
93.54s
89.90s
89.07s
10
119.60s
96.37s
93.33s
90.49s
DOFs
5580
8928
15000
38400
bBending; ccoupled; ddifferential bending; ttorsional; sshell-like.Position of the non-structural mass at the tip cross-section of the hollow-rectangular box beam.
(a)
Case A
(b)
Case BMAC values between the16L9 and FEM-3D models of the hollow-rectangular box beam.
(a)
Without localized inertia
(b)
With localized inertia (Case A)Modal shapes of the hollow-rectangular box beam with localized inertia at the tip cross-section (Figure20(a), Case A).
(a)
Mode 1,16 L9 LE model (6.04 Hz)
(b)
Mode 1, FEM-3D model (5.98 Hz)
(c)
Mode 4,16 L9 LE model (43.20 Hz)
(d)
Mode 4, FEM-3D model (52.90 Hz)In the last load case, two non-structural masses were added at (a, L, −h/2) and (0, L, h/2), as shown in Figure 20(b) (Case B). For this case, the weight of each non-structural mass was equal to 250 Kg. The results from LE and MSC Nastran
© analyses are given in Table 8 for Case B. The following statements hold:(1)
As it is also clear from previous analyses, CUF 1D models are able to foresee cross-sectional deformations, and thus local shell-like modes of the box beam are detected.
(2)
At least an eight-order TE model (N
=
8) is necessary to correctly detect shell-like natural frequencies.
(3)
The LE models accurately detect the solid solution. Both global and local modes are correctly found with a significant reduction of computational costs.
(4)
For the case with no localized inertia, MAC matrix shows a perfect correspondence between LE and FEM-3D models. However, modes 3 and 4 as well as modes 8 and 9 are exchanged between the two models.
(5)
Good agreement is evident from the MAC matrix even though non-structural masses are applied. In this case, no mode exchanges appear but some mixings are evident since coupled phenomena occured.Table 8
First10 natural frequencies (Hz) for the hollow-rectangular box beam with two localized masses at the tip cross-section (Figure 20(b), Case B).
Mode
LE models
MSCNastran
©
10 L9
16 L9
FEM-2D
FEM-3D
1
6.86b
6.86b
7.02b
6.84b
2
13.68d
12.08d
11.79d
11.66d
3
19.96b
19.89b
20.54b
19.75b
4
55.69s
54.27d
66.47s
54.22d
5
62.60d
60.69s
66.78d
64.12s
6
65.02s
62.80s
70.43s
69.21s
7
94.04b
73.64s
73.98s
71.08s
8
97.95s
74.55s
81.38s
73.56s
9
101.2s
77.86s
83.15s
80.21s
10
100.4s
84.96s
85.66s
80.67s
DOFs
5580
8928
15000
38400
bBending; ccoupled; ddifferential bending; ttorsional; sshell-like.


## 4.1. C-Section Beam

The analysis of a cantilever C-shaped beam was carried out as the first assessment. The cross-section of the structure is shown in Figure7. The geometrical data were as follows: b
=
0.5 m, a
=
h
=
2
b, and t
=
0.1 m. The length of the beam, L, was equal to 20 m.Figure 7
Cross-section of the C-shaped beam.Table1 shows the first ten natural frequencies for each implemented model. Columns 2 and 3 refer to the classical models EBBM and TBM. The natural frequencies by fourth-order (N
=
4), sixth-order (N
=
6), and eighth-order (N
=
8) refined TE beam model are given in columns 4 to 6. Two different distributions of L9 elements above the cross-section were considered (6L9 and 10L9) in the case of LE modelling approach, as shown in Figure 8, and the results are given in the 7th and 8th columns of Table 1. In columns 9 and 10, MSC Nastran
© shell and solid models are given for comparison purposes, and they are,respectively, referred to as FEM-2D and FEM-3D in the table. FEM-2D model was obtained by using 4-node QUAD elements, whereas FEM-3D model was obtained by using 8-node CHEXA elements. For each model, the number of DOFs is given in the last row of Table 1. The following comments can be made:(1)
As expected, classical models are not able to provide torsional and coupled modes.
(2)
A higher than fourth-order TE model is necessary to correctly detect torsional modes.
(3)
As indicated in Table1, where superscripts are used to denote the kind of each vibrational mode, bending, torsional, and coupled bending/torsional modes are detected by LE models in accordance with MSC Nastran
© 2D and 3D models. An appropriate distribution of the L9 elements above the cross-section is effective in improving the accuracy of the solution.
(4)
The 10 L9 LE model provides a solid-like solution with a high reduction of the number of the DOFs.Table 1
First10 natural frequencies (Hz) for the cantilever C-shaped beam.
Mode
Classical and refined models based on TE
LE models
MSCNastran
©
EBBM
TMB
N
=
4
N
=
6
N
=
8
6 L9
10 L9
FEM-2D
FEM-3D
1
1.75b
1.75b
1.76b
1.76b
1.76b
1.76b
1.76b
1.74b
1.75b
2
2.99b
2.98b
2.93b
2.76b
2.65b
2.56b
2.56b
2.49b
2.54b
3
10.97b
10.90b
10.90b
7.66t
6.73t
6.29t
6.27t
6.09t
6.23t
4
18.67b
18.35b
12.25t
10.89c
10.89c
10.87c
10.87b
10.71b
10.83b
5
30.63b
30.20b
17.46t
13.83t
12.37t
11.46t
11.41t
10.99t
11.34t
6
51.81b
49.83b
29.94b
24.87t
22.54t
21.18t
21.10t
20.29c
20.97t
7
59.75b
58.25b
34.66t
29.93b
29.87t
29.25t
29.19t
28.82t
29.06t
8
65.87a
65.87a
45.76t
32.49b
30.21b
29.89b
29.86b
29.55b
29.75b
9
98.20b
93.73b
56.94c
41.26c
37.12c
34.93c
34.76c
33.44c
34.53c
10
100.22b
94.43b
59.22c
57.05c
53.02c
50.23c
49.88c
48.06c
49.51c
DOFs
93
155
1395
2604
4185
3627
5859
38250
177000
aAxial; bbending; ccoupled; ttorsionalDistribution of L9 elements above the cross-section of the C-shaped beam.
(a)
6 L9
(b)
10 L9A second beam configuration was considered and a non-structural mass was added on the C-shaped beam at the coordinates (0,L, h), as Figure 9 shows. The weight of the non-structural mass was equal to 4000 Kg. For this load case, Table 2 reports the main vibrational frequencies by different models for clamped-free boundary conditions. The number of the DOFs for each model implemented is also given in the last row of the table. Figures 10 and 11 show the 8th mode shape by both LE and FEM-3D models, and the effects due to the localized non-structural mass are highlighted. Table 3 shows the natural frequencies of the C-shaped beam undergoing clamped-clamped boundary conditions. For this case, the same localized inertia as above was placed at y
=
(
3
/
4
)
L as shown in Figure 9. The following comments arise.(1)
As expected, the application of the non-structural mass is responsible for bending/torsional mode couplings.
(2)
Classical models and lower-order TE models are not effective for the problem under consideration.
(3)
The proposed LE model is able to detect the MSCNastran
© shell and solid results.
(4)
The number of the DOFs is very low in the case of CUF models if compared both to the shell and solid models.Table 2
First10 natural frequencies (Hz) for the cantilever C-shaped beam with localized inertia at the tip cross-section (Figure 9).
Mode
Classical and refined models based on TE
LE models
MSCNastran
©
EBBM
TMB
N
=
4
N
=
6
N
=
8
6 L9
10 L9
FEM-2D
FEM-3D
1
1.15b
1.15b
1.15b
1.13c
1.12c
1.11c
1.11c
1.10c
1.11c
2
1.97b
1.96b
1.89b
1.73c
1.65c
1.58c
1.58c
1.53c
1.57c
3
8.67b
8.63b
7.62t
5.70t
5.16t
4.90t
4.88t
4.88t
4.56t
4
14.69b
14.48b
11.37t
9.28t
8.37t
7.77t
7.73t
7.39t
7.67t
5
25.76b
25.44b
12.60c
10.09c
9.82c
9.73c
9.72c
9.52c
9.86c
6
41.50b
40.48b
25.28c
20.64c
18.23c
16.92c
16.81c
16.08c
16.63c
7
49.00b
48.07b
30.46c
23.37c
22.83c
22.71c
22.64c
22.59c
22.23c
8
54.78b
53.78b
35.77c
29.79c
28.69c
27.42c
27.16c
26.09c
26.71c
9
85.83b
81.86b
42.93c
33.48c
30.51c
29.70c
29.64c
29.27c
29.51c
10
89.33b
85.37b
51.67c
40.65c
38.24c
39.12c
38.16c
37.90c
34.35c
DOFs
93
155
1395
2604
4185
3627
5859
38250
177000
bBending; ccoupled; ttorsional.Table 3
First10 natural frequencies (Hz) for the clamped-clamped C-shaped beam with localized inertia at y
=
(
3
/
4
)
L (Figure 9).
Mode
Classical and refined models based on TE
LE models
MSCNastran
©
EBBM
TMB
N
=
4
N
=
6
N
=
8
6 L9
10 L9
FEM-2D
FEM-3D
1
10.09b
9.99b
9.50b
7.13c
6.34c
5.90c
5.86c
5.55c
5.57c
2
17.18b
16.70b
11.62b
10.32c
10.21c
10.14c
10.13c
9.86c
9.91c
3
24.52b
24.01b
19.39t
14.82t
13.33t
12.49t
12.41t
11.99t
12.45t
4
41.73b
39.38b
26.06t
23.00t
22.54t
22.34t
22.30t
22.02t
21.76t
5
51.56b
49.97b
31.58c
28.12c
27.59c
26.33c
26.13c
25.15c
26.26c
6
85.41b
79.17b
45.77c
31.47c
28.50c
27.99c
27.96c
27.63c
27.81c
7
91.67b
87.63b
49.97c
47.70c
44.95c
42.40c
41.92c
39.70c
41.22c
8
108.39b
105.88b
64.95c
50.60c
48.43c
48.14c
47.96c
46.89c
46.82c
9
142.65b
135.53b
76.50c
61.84c
56.29c
52.94c
52.18c
48.26c
48.43c
10
154.97b
139.71b
83.10c
63.53c
62.81c
62.63c
62.41c
61.33c
56.38c
DOFs
93
155
1395
2604
4185
3627
5859
37995
176115
bBending; ccoupled; ttorsional.Figure 9
Position of the non-structural mass at the tip cross-section of the C-shaped beam.Eighth modal shape of the C-shaped beam.
(a)
10 L9 LE model (29.861 Hz)
(b)
FEM-3D model (29.756 Hz)Eighth modal shape of the C-shaped beam with a non-structural mass at the tip cross-section (Figure9).
(a)
10 L9 LE model (27.169 Hz)
(b)
FEM-3D model (26.716 Hz)


## 4.2. I-Shaped Beam

A cantilever I-shaped beam was considered as the second example. The cross-section geometry is shown in Figure12. The dimensions a and b were, respectively, equal to 0.2 m and 0.3 m, whereas the thickness t was 0.05 m. The length of the structure, L, was 3 m.Figure 12
Cross-section of the I-shaped beam.The first ten natural frequencies, calculated according to different models, are given in Table4. Classical and refined TE as well as LE models are considered. Two different LE models were implemented and they differed in the cross-section L9 elements discretization, as shown in Figure 13. In columns 8 to 10, MSC Nastran
© beam, shell, and solid models are given for comparison purposes. MSC Nastran
© 1D model (hereinafter referred to as FEM-1D) was obtained with 21 two-node CBAR elements, whereas shell and solid elements were constructed using the same elements as in the previous analysis case, and they are referred to as FEM-2D and FEM-3D, respectively.Table 4
First10 natural frequencies (Hz) for the I-shaped beam.
Mode
Classical and refined models based on TE
LE models
MSCNastran
©
EBBM
TMB
N
=
6
N
=
8
7 L9
12 L9
FEM-1D
FEM-2D
FEM-3D
1
15.67b
15.65b
15.73b
15.73b
15.75b
15.72b
15.63b
15.11b
15.68b
2
35.28b
35.01b
34.93b
34.90b
34.87b
34.86b
34.73b
34.73b
34.80b
3
97.91b
96.08b
96.16t
88.74t
77.68t
76.58t
96.26b
68.45t
75.61t
4
217.27b
206.05b
97.06b
96.99b
97.13b
96.88b
196.99b
93.04b
96.63b
5
272.62b
265.09b
198.27b
197.28b
196.28b
196.10b
262.81b
191.98b
195.71b
6
439.19a
439.01a
265.24b
264.79b
246.78t
243.49t
439.09a
220.31t
240.63t
7
529.90b
507.02b
298.13t
277.27t
265.12t
264.20b
486.79b
253.52b
263.49b
8
592.06b
533.06b
440.66a
440.64a
440.64a
440.62a
497.53b
413.86c
440.23a
9
866.98b
811.08b
491.81b
487.36b
453.27t
448.06t
788.63b
439.82a
443.55t
10
1117.81b
950.08b
502.74t
495.78t
482.74b
481.97b
828.65b
464.45b
480.95b
DOFs
93
155
2604
4185
4185
6975
126
12200
127800
aAxial; bbending; ccoupled; ttorsional.Distribution of L9 elements above the cross-section of the I-shaped beam.
(a)
7 L9
(b)
12 L9The consistent correspondence between the 12 L9 LE model and the solid model was further investigated by means of the Modal Assurance Criterion (MAC), whose graphic representation is shown in Figure15(a). The MAC is defined as a scalar representing the degree of consistency between two distinct modal vectors (see [53]) as follows:
(28)
MA
C
i
j
=
|
{
ϕ
A
i
}
T
{
ϕ
B
j
}
|
2
{
ϕ
A
i
}
T
{
ϕ
A
i
}
{
ϕ
B
j
}
{
ϕ
B
j
}
T
,
where {
ϕ
A
i
} is the ith eigenvector of model A, whereas {
ϕ
B
j
} is the jth eigenvector of model B. The modal assurance criterion takes on values from zero (representing no consistent correspondence) to one (representing a consistent correspondence). The following remarks can be made:(1)
Unlike classical beam theories, higher-order TE and LE models are able to foresee torsional and coupled modes.
(2)
Unlikely CUF 1D models, FEM-1D model is not able to produce the solid-like solution.
(3)
The cross-section discretization refinement for LE models is an effective method that leads to the 3D solution.
(4)
Bending, torsional, and coupled modes are detected by LE models in accordance with FEM-2D and FEM-3D models with a high reduction of the number of the DOFs.
(5)
The MAC analysis underlines the perfect correspondence between the LE model and the solid one for the considered load case.In the second load case, a non-structural mass was added at the tip cross-section of the I-shaped beam, as shown in Figure14. The weight of the localized inertia was equal to 1000 Kg and it was applied at (a, L, b). Table 5 gives the first 10 vibrational frequencies for classical, TE, LE, and MSC Nastran
© models. The application of the non-structural mass produces an increase in the number of the coupled modes, as expected. The MAC matrix between the 12 L9 model and the FEM-3D model for the load case under consideration is also calculated and it is represented in Figure 15(b). It should be underlined that(1)
Classical models are ineffective in detecting the effects due to non-structural masses that are not placed in correspondence of the shear center;
(2)
The increase in the order for TE models provides greater accuracy on the evaluation of the coupled frequencies;
(3)
Even though local inertia is considered, LE models can produce solid-like solution, as shown by the MAC matrix. In fact, a very good agreement was found for the FEM-3D and LE models. LE analyses are therefore clearly able to detect 3D-elasticity solutions. However, some mixings in modes 3 and 4 are evident.Table 5
First10 natural frequencies (Hz) for the I-shaped beam with localized inertia at the tip cross-section (Figure 14).
Mode
Classical and refined models based on TE
LE models
MSCNastran
©
EBBM
TMB
N
=
4
N
=
6
N
=
8
7 L9
12 L9
FEM-2D
FEM-3D
1
3.70b
3.69b
3.67b
3.66b
3.65b
3.66b
3.63b
2.60b
3.61b
2
8.33b
8.29b
8.07c
8.01c
7.95c
8.17c
7.84c
5.74b
7.78c
3
61.86b
61.46b
53.74c
50.74t
47.77t
52.63t
45.30t
41.95t
38.50c
4
103.26b
102.06b
72.37t
64.48t
58.75t
80.77t
55.33t
53.62t
48.73t
5
171.03b
163.96b
101.98c
95.58c
91.37c
104.66c
88.99c
85.71c
84.95c
6
237.21b
231.18b
157.05t
151.08t
145.40t
150.93t
139.02t
129.29t
132.30t
7
458.27b
441.52b
224.05c
213.21c
204.64c
199.42c
191.89c
186.50c
184.73c
8
486.61b
444.07b
281.72c
266.67c
257.43c
254.03c
247.97c
230.50c
243.84c
9
679.49b
649.65b
392.70t
368.10t
343.10t
371.59t
316.84t
292.44t
292.90t
10
841.12b
777.59b
435.90c
420.36c
408.36c
398.96c
389.35c
366.42c
370.94c
DOFs
93
155
1395
2604
4185
4185
6975
12200
127800
bBending; ccoupled; ttorsional.Figure 14
Position of the non-structural mass at the tip cross-section of the I-shaped beam.MAC values between the12 L9 and FEM-3D models of the I-shaped beam.
(a)
Without localized inertia
(b)
With localized inertia


## 4.3. Hollow-Rectangular Beam

The cantilever hollow-square cross-section beam shown in Figure16 was investigated as the last example. The geometrical data were as follows: a
=
0.8 m, h
=
0.2 m, and t
=
0.01 m. The length of the beam, L, was equal to 3.2 m. Table 6 reports the first 10 natural frequencies, together with the number of DOFs for each model implemented. Columns 2 and 3 show the results from the classical beam theories. The TE models from the fourth-order to the eighth-order Taylor-like expansion are considered in columns 4 to 6. The LE models are reported in columns 7 and 8. The LE models were obtained by adopting two different discretizations above the cross-section of the beam, as Figure 17 shows. In Figure 17, a different notation is adopted with respect to Figures 8 and 13. The nodes of the L-elements are, in fact, not depicted in Figure 17, for the sake of clearness. In columns 9 and 10 of Table 6, MSC Nastran
© shell and solid models are given for comparison purposes and they are referred to as FEM-2D and FEM-3D, respectively. For this analysis case, the mesh discretizations for FEM-2D and FEM-3D models are shown in Figure 18. Figure 19 shows a comparison between some selected modal shapes by the 16L9 LE and FEM-3D models.Table 6
First10 natural frequencies (Hz) for the hollow-rectangular box beam.
Mode
Classical and refined models based on TE
LE models
MSCNastran
©
EBBM
TMB
N
=
4
N
=
6
N
=
8
10 L9
16 L9
FEM-2D
FEM-3D
1
25.48b
25.36b
25.15b
24.73b
24.52b
24.53b
24.07b
24.42b
23.89b
2
75.70b
72.79b
72.58b
72.42b
72.35b
62.08d
57.02d
55.34d
54.55d
3
158.03b
153.20b
137.79b
127.58b
90.87d
72.62b
72.67b
70.43s
69.59s
4
411.74a
357.95b
146.19t
134.29d
111.68s
100.61s
74.19s
73.98s
72.51b
5
435.12b
406.88b
330.86b
265.87s
127.00s
103.42s
77.60s
74.31b
73.12s
6
435.23b
411.74a
346.16b
274.47d
131.29s
109.38s
80.88s
75.35s
75.03s
7
833.29b
745.94b
413.68a
344.10b
139.88s
118.52d
84.76s
81.38s
80.48s
8
1089.50b
808.29b
428.76t
361.71s
141.44d
119.24s
96.36s
89.73s
89.56s
9
1235.23a
1146.76b
549.40b
388.13d
152.86s
122.01b
97.79s
93.33s
92.30s
10
1338.69b
1284.52b
690.19t
413.49a
166.72s
133.77s
109.87s
101.87s
101.58s
DOFs
93
155
1395
2604
4185
5580
8928
15000
38400
aAxial; bbending; ccoupled; ddifferential bending; ttorsional; sshell-like.Figure 16
Cross-section of the hollow-rectangular box beam.Distribution of L9 elements above the cross-section of the hollow-rectangular box beam.
(a)
10 L9
(b)
16 L9Mesh discretization for the FEM-2D and FEM-3D MSCNastran
© models.
(a)
FEM-2D
(b)
FEM-3DModal shapes of the hollow-rectangular box beam.
(a)
Mode 2,16 L9 LE model (57.023 Hz)
(b)
Mode 2, FEM-3D model (54.551 Hz)
(c)
Mode 10,16 L9 LE model (109.87 Hz)
(d)
Mode 10, FEM-3D model (101.58 Hz)The vibrations of the box beam were further analyzed and a non-structural mass with a weight of500 Kg was added at the coordinates (a, L, −h/2), as shown in Figure 20(a) (Case A). A comparison between LE and MSC Nastran
© models is given in Table 7. The correspondence between LE 16L9 and FEM-3D models is underlined through MAC analyses, which are shown in Figure 21. MAC matrices for both the cases without and with localized inertia (Case A) are provided in Figure 21. Some modal shapes by LE and FEM-3D models for the load case considered are shown in Figure 22.Table 7
First10 natural frequencies (Hz) for the hollow-rectangular box beam with localized inertia at the tip cross-section (Figure 20(a), Case A).
Mode
LE models
MSCNastran
©
10 L9
16 L9
FEM-2D
FEM-3D
1
6.20d
6.04d
6.11d
5.98d
2
18.54d
18.23d
18.94d
17.97d
3
43.77d
39.75d
38.66d
38.02d
4
44.07d
43.20d
69.36d
52.90d
5
95.47d
74.18s
70.43s
69.60s
6
100.62s
77.59s
73.98s
73.12s
7
103.41s
79.15s
78.77s
74.72s
8
109.34s
84.75s
81.38s
80.47s
9
118.77s
93.54s
89.90s
89.07s
10
119.60s
96.37s
93.33s
90.49s
DOFs
5580
8928
15000
38400
bBending; ccoupled; ddifferential bending; ttorsional; sshell-like.Position of the non-structural mass at the tip cross-section of the hollow-rectangular box beam.
(a)
Case A
(b)
Case BMAC values between the16L9 and FEM-3D models of the hollow-rectangular box beam.
(a)
Without localized inertia
(b)
With localized inertia (Case A)Modal shapes of the hollow-rectangular box beam with localized inertia at the tip cross-section (Figure20(a), Case A).
(a)
Mode 1,16 L9 LE model (6.04 Hz)
(b)
Mode 1, FEM-3D model (5.98 Hz)
(c)
Mode 4,16 L9 LE model (43.20 Hz)
(d)
Mode 4, FEM-3D model (52.90 Hz)In the last load case, two non-structural masses were added at (a, L, −h/2) and (0, L, h/2), as shown in Figure 20(b) (Case B). For this case, the weight of each non-structural mass was equal to 250 Kg. The results from LE and MSC Nastran
© analyses are given in Table 8 for Case B. The following statements hold:(1)
As it is also clear from previous analyses, CUF 1D models are able to foresee cross-sectional deformations, and thus local shell-like modes of the box beam are detected.
(2)
At least an eight-order TE model (N
=
8) is necessary to correctly detect shell-like natural frequencies.
(3)
The LE models accurately detect the solid solution. Both global and local modes are correctly found with a significant reduction of computational costs.
(4)
For the case with no localized inertia, MAC matrix shows a perfect correspondence between LE and FEM-3D models. However, modes 3 and 4 as well as modes 8 and 9 are exchanged between the two models.
(5)
Good agreement is evident from the MAC matrix even though non-structural masses are applied. In this case, no mode exchanges appear but some mixings are evident since coupled phenomena occured.Table 8
First10 natural frequencies (Hz) for the hollow-rectangular box beam with two localized masses at the tip cross-section (Figure 20(b), Case B).
Mode
LE models
MSCNastran
©
10 L9
16 L9
FEM-2D
FEM-3D
1
6.86b
6.86b
7.02b
6.84b
2
13.68d
12.08d
11.79d
11.66d
3
19.96b
19.89b
20.54b
19.75b
4
55.69s
54.27d
66.47s
54.22d
5
62.60d
60.69s
66.78d
64.12s
6
65.02s
62.80s
70.43s
69.21s
7
94.04b
73.64s
73.98s
71.08s
8
97.95s
74.55s
81.38s
73.56s
9
101.2s
77.86s
83.15s
80.21s
10
100.4s
84.96s
85.66s
80.67s
DOFs
5580
8928
15000
38400
bBending; ccoupled; ddifferential bending; ttorsional; sshell-like.


## 5. Conclusions

The effects due to non-structural localized inertia on the vibration of isotropic thin-walled beams have been investigated in this work. The ineffectiveness of classical beam theories and related FEs in dealing with non-structural masses which are not placed in the shear center of the cross-section has been demonstrated. In fact, localized inertia has been shown to be responsible for torsion, bending/torsion couplings, and local phenomena such as cross-sectional deformations. For these reasons, the use of higher-order beam theories is mandatory.In this paper, higher-order beam theories have been developed by using the Carrera Unified Formulation (CUF), which is a hierarchical approach allowing for the automatic formulation of FE arrays deriving from arbitrarily enriched kinematic fields. By using an appropriate indexing notation, stiffness and mass matrices are in fact written in terms of fundamental nuclei which depend neither on the approximation order nor the class of the beam theory implemented. Accordingly, in the present work, non-structural localized masses have been introduced and arbitrarily placed over the 3D beam domain by exploiting the higher-order capabilities of CUF. Two CUF beam classes have been considered for the proposed analyses, TE and LE. In TE class, beam theories are formulated by using Taylor-like polynomials to discretize the cross-sectional generalized displacements. On the other hand, LE models have only pure displacement unknowns and Lagrange polynomials are used to locally refine the cross-section displacement field.A number of homogeneous thin-walled beams subjected to localized inertia have been addressed in the proposed study and the results compared to those from 2D plate/shell and 3D solid FE models by the commercial code MSCNastran
©. The analyses highlight the enhanced capabilities of CUF theories with respect to classical ones when dealing with localized inertia effects. In particular, it is demonstrated that LE models yield 3D results with very low computational efforts.



---

*Source: 141982-2014-04-27.xml*