close

Вход

Log in using OpenID

embedDownload
Element for Beam Dynamic Analysis based on Analytical
Deflection Trial Function
Qiongqiong Cao1 , Min Ding1 , Xiugen Jiang1 , Jinsan Ju1, Hongzhi Wang1 , Peng Zhang1,2
1. College of Water Resources & Civil Engineering, China Agricultural University, Beijing
100083, China; 2 China Aerospace Construction Group Co., Ltd, Beijing 100071, China
Correspondence should be addressed to Xiugen Jiang; jiangxj@cau.edu.cn
Abstract: For beam dynamic finite element analys is, according to differential equation of motion
of beam with distributed mass, general analytical solution of displacement equation for the beam
vibration is obtained. By applying displacement element construction principle, the general
solution of displacement equation is conversed to the mode expressed by beam end displacements.
And taking the mode as displacement trial function, element stiffness matrix and element mass
matrix for beam flexural vibration and axial vibration are established respectively by applying
principle of minimum potential energy. After accurate integral, explicit form of element matrix is
obtained. The comparison results show that the series of relative error between the solution of
analytical trial function element and theoretical solution is about 1×10
-9
and the accuracy and
efficiency are superior to that of interpolation trial function element. The reason is that
interpolation trial function can’t accurately simulate the displacement mode of vibrating beam.
The accuracy of dynamic stiffness matrix method is almost identical with that of analytical trial
function. But the application of dynamic stiffness matrix method in engineering is limited. The
beam dynamic element obtained in this paper is analytical and accurate and can be applied in
practice.
Keywords : beam, dynamic, finite element, analytical trial function, element construction
1 Introduction
Dynamic structural analysis is essential in structure engineering design. It is especially important
for large-scale structure in earthquake area, such as high-rise buildings, dam, hydropower station,
oil pipelines, gas pipelines, etc. Dynamic problem of beam structure is common in engineering.
Vibration happens at beam under earthquake, gas-liquid flows, and impact. Resonance occurs
when frequency of external load is close to natural frequency of structure. It is a great threat for
structure safety. Therefore, perfecting accuracy and efficiency of dynamic structural analysis is to
ensure structure safety and reliability.
A large number of theories and methods for dynamic structural analysis have been suggested.
The analysis methods include direct solving method, energy method, numerical method, etc.
Clough[1] has obtained analytical solution of displacement equation for flexural vibrating beam,
and presented the first three order vibration mode and the corresponding frequency of cantilever
beam and simply supported beam. Jin
[2]
provided analytical solution of Timoshenko beam
1
clamped two ends and subjected to uniformly distributed load according to different jump
condition. Guo
[3]
established structure element property matrix by energy principle considering
flexural and torsional deformation of T-beam, and the effect of bridge local member (such as
diaphragm plate). Fang[4] attained beam natural vibration frequency by analyzing dynamic
property of external prestressing beam using energy method, and the solution has a better match
with numerical solution. Lou[5] derived the approximation analysis technique for dynamic
characteristics of the prestressed beam by applying the mode perturbation method. Carrer et al.
[6]
analyzed the dynamic behavior of Timoshenko beam by using boundary element method. Wu et
al. [7] analyzed the dynamic behavior of two-dimension frame with stiffening bar random
distributed by using elastic-rigid composite beam element. De Rosa et al.
[8]
studied the dynamic
behavior of slender beam with concentrated mass at beam end, and numerical solution of
frequency equation was obtained. Among them, numerical analysis represented by finite element
is the main and efficient method for dynamic analys is (such as natural vibration analysis, and
forced vibration analysis). Finite element method was firstly proposed by Clough[9] in an article
about plane elastic problem, and it is perfect in theory as a numerical method.
At present, the methods of constructing dynamic element for beam include dynamic stiffness
matrix method, Galerkin method, Ritz method, energy variation method, etc. Dynamic stiffness
matrix can accurately solve differential equation of motion according to initial displacement field
without any assumption, and then accurate results can be obtained irrespective of element number.
This method was proposed by Kolousek[10]. To gain more accurate results, stiffness matrix of
tensile torsion bar and Euler beam about frequency, that is dynamic stiffness matrix, was firstly
derived from analytical solution when he studied vibration characteristics of plane truss. A lot of
work on the research and development of dynamic stiffness matrix method was also done by
Long[11], Hashemi[12], Chen[13], Jayatunga[14], Banerjee[15] et al. Shavezipur[16] put forward an
accurate finite element method. In this method, closed form solution of differential equation of
beam not coupling flexural vibration torsional vibration was obtained by merging Galerkin
weighted residual method and dynamic stiffness matrix (DSM). Result of dynamic stiffness matrix
is more accurate, but analytical solution of differential equation can’t be derived when structure
load or displacement boundary condition is too complex. Then dynamic stiffness matrix method is
not suitable any more. According to principle of virtual displacement, a large number of research
achievements on dynamic analysis of thin-walled open section beam, elastic foundation beam and
composite beam have been conducted by Chopra
[21]
Pagani
, et al. Nabi and Ganesan
[22]
[17]
, Hu
[18]
, Wang
[19]
[20]
, Mohammad Hashemi
,
put forward a finite element method based on free vibration
analys is theory of composite beam with the first order shear deformation. Zhao[23] developed the
dynamic analysis of a unified stochastic variational principle and the corresponding stochastic
finite element method via the instantaneous minimum potential energy principle and the small
parameter perturbation technique. On the basis of energy variation principle, Wang[24] derived
governing differential equation and natural boundary condition of dynamic response for I-shaped
2
beam, and obtained closed solution of the corresponding generalized dynamic displacement.
Accuracy and efficiency of beam element depend on beam displacement trial function by
applying potential energy variation principle to constructing beam displacement element. For
current beam element, cubic polynomial displacement mode is used to obtain a series of static and
dynamic element widely applied to the software, such as ANSYS, NASTRAN, etc. For dynamic
analys is, vibrating beam displacement mode has a big difference from polynomial mode. Precision
requirements cannot be met by taking polynomial function as vibrating beam displacement trail
function. The basic analytical solution is used as the element trial function in analytical trial
function method. Discrete finite element method takes advantage of analytical solution. It
embodies the superiority of trial function using basic analytical solution.
This paper focuses on constructing element for beam dynamic analysis using analytical
deflection trail function based on variational method of principle of minimum potential energy and
displacement element construction theory. The fruits are useful to beam dynamic analysis.
2 Displacement Trial Function for Beam Element
Selecting displacement trial function is one of the main content of constructing displacement
element. Appropriate displacement trial function should be in accordanc e with element
deformation behavior and be easy for integral of element energy functional. Element accuracy is
determined by the accuracy of displacement trial function. The corresponding functional integral
has a direct effect on element calculation efficiency and accuracy.
With regard to beam element, displacement mode based on interpolation function is used for
element displacement trial function of all kinds of problems. Linear polynomial Lagrange
interpolation function meeting the continuity condition at C0 is used for axial displacement. Cubic
polynomial Hermit interpolation function meeting the continuity condition at C 1 is used for
flexural deformation.
Take flexural deformation for example, for the static problems of uniform cross-section beam
d 4 wx 
 q is stiffness equilibrium equation about
subjected to uniform distributed load, EI
dx 4
deflection w(x). Here, EI is beam section flexural stiffness, and q is the uniform distributed load.
The accurate solution of this equation is a cubic polynomial. Cubic polynomial Hermit
interpolation function meeting the continuity condition of beam end displacement is actual
displacement of element. And the corresponding potential energy functional has analytic,
derivable, integrable solution. For this reason, static beam element derived from Hermit
interpolation shape function is accurate element.
However, for dynamic and non-linear straight beam and every non-linear beam, deflection
equation is not cubic polynomial due to the change of equilibrium differential equation mode.
Therefore, element constructed by Hermit interpolation trial function is an approximate element.
The key of constructing accurate element for all kinds of problems of beam is to search analytical
3
trial function having functional integrability.
3 Analytical Trial Function of Displacement for Vibrating Beam
Element
The method to construct analytical trial function of displacement for vibrating beam element is as
follows:
(1) To deduce the general solution of displacement equation for beam vibration containing
undetermined parameters according to differential equation of equilibrium for beam vibration;
(2) To determine the undetermined parameters in displacement equation according to
displacement condition of vibrating beam end;
(3) To write out displacement equation for beam vibration expressed by beam end displacement,
and then to obtain displacement trial function for vibrating beam.
To construct beam dynamic element, local coordinate as shown in Fig.1 is established. It is in
accordance with general beam element. The positive direction of linear displacement of beam end
and force is in accordance with coordinate direction, and the positive direction of rotation angle
and moment is in accordance with clockwise direction.
y
V1
1
w(x)
2
V2
x N2
N1 O
M1
M2
Fig.1 Coordinate system and parameter positive direction
3.1 Governing equation for Beam Free Vibration and Displace ment General
Solution
3.1.1 Beam Flexural Vibration Equation
Dynamic equilibrium governing equation for beam flexural vibration is expressed as
EI
 4 v  x, t 
 2 v  x, t 

m
0
x 4
t 2
[1]
(1)
In Eq.(1), v x, t  is the displacement response for beam flexural vibration.
By applying the method of separation of variables, analytical solution of deflection equation for
beam flexural vibration is obtained[1] :
w x   C1 sin x  C2 cosx  C3shx  C4 chx
(2)
In Eq.(2), flexural vibration parameter  is connected with circular frequency for beam flexural
4
vibration  w ,  w  
2
EI
, EI is beam section flexural stiffness, m is beam mass per unit
m
length.
3.1.2 Beam Axial Vibration Equation
Dynamic equilibrium governing equation for beam axial vibration is expressed as [1]
d 2u
 2u
EA 2  m 2  0
dx
t
(3)
In Eq.(3), u  x, t  is the displacement response for beam axial vibration.
By applying the method of separation of variables, analytical solution of deflection equation for
beam axial vibration is obtained[1]:
u x   D1 cos x  D2 sin x
(4)
In Eq.(4), axial vibration parameter  is connected with circular frequency for beam axial
vibration  u ,  u  
EA
, EA is beam section tensile (compressive) rigidity.
m
3.2 Analytical Trial Function of Displacement for Dynamic Beam Element
3.2.1 Analytical Trial Function for Beam Flexural Vibration
(1) Parameter determination of displacement function
Eq.(2) can be further expressed as
4
 w x    i  x Ci   C
(5)
i 1
where
C  C1
C2 C3 C4 
T
  1 x  2 x  3 x  4 x 
 sin x cosx shx chx
(6)
Beam end displacement is defined as  w   1 1 2  2 
e
T
According to Eq.(5), beam end displacement becomes
 w e
 2 0 3 0  4 0  C1 
 1 0
  0    0    0    0 C 
2
3
4
2

 1
   AC
 1 l 
 2 l 
3 l 
 4 l   C3 



  1l    2 l   3 l    4 l  
C4 
If   l , with Eq.(6), [A] can be expressed as
5
(7)
0
1

 
0
A  
 sin 
cos 

  cos   sin 
0

0 
ch 

 sh 
1

sh
 ch
With Eq.(7), undetermined parameter of displacement function is expressed as
C  A1 w e
(8)
To perform matrix inversion, then
A1
 c11 c12
c
c
  21 22
c31 c32

c41 c42
c13
c23
c33
c43
c14 
c24 
c34 

c44 
where
c11  c31 
cos  sinh   cosh sin 
cos  cosh  sinh  sin   1
, c12  
S
S
c13  c33  
c21 
cosh sin   cos  sinh 
1  sinh  sin   cos  cosh
, c22  c42 
,
S
S
c23  c43 
c32 
sin   sinh 
cosh  cos
, c14  c34  
,
S
S
cosh  cos 
sinh   sin 
, c24  c44 
,
S
S
1  sinh  sin   cos  cosh
cos  cosh  sin  sinh   1
, c41 
,
S
S
S  2cos  cosh   1
(2) Element displacement shape function
By substituting Eq.(8) into Eq.(5), displacement function for element flexural vibration expressed
by beam end displacement can be obtained:
w x  A1 w e
(9)
If
Nw   A1
(10)
w x  N w  w e
(11)
Eq.(9) becomes
N w  is flexural vibration displacement trial function for dynamic beam element. With Eq.(10),
it can be expressed as
Nw   A1  N1
6
N2
N3
N4 
(12)
where N 1 , N 2 , N 3 , N 4 are deflection shape functions. After matrix operation, they can be
expressed as

1 sinh  sin x     cosh cosx     cosx  
 N1  
S  cos coshx     sin  sinh x     coshx 


1  cosh sin x     sinh  cosx     sin x  
N 2 
S  sinh x   sin  coshx     cos sinh x   


 N  1  sin x sinh   cosx     cosx cosh 
 3 S  coshx     sin  sinh x   coshx cos 




1  sin x cosh  sin x     cosx sinh  
N 4 
S  sinh x     sinh x cos  coshx sin  

(13)
3.2.2 Analytical Trial Function for Beam Axial Vibration
(1) Parameter determination of displacement function
Eq.(4) can be further expressed as
4
u  x    i  x Di   D
(14)
i 1
where
D  D1
    1 x 
 cosx
D2 
T
 2 x 
sin x
Beam end axial displacement is defined as  u   u1
e
(15)
T
u2  .
According to Eq.(14), beam end axial displacement is obtained
 1 0  2 0  D1 
    BD
 1 l   2 l  D2 
 u e  
If
(16)
  l , with Eq.(15), [B] can be expressed as
B  
1
cos
0 
sin  
With Eq.(16), undetermined parameter of axial displacement function is expressed as
D  B1u e
(17)
 1
  cos
 sin 

(18)
To perform matrix inversion, then
B
1
(2) Element displacement shape function
7
0 
1 
sin  
By substituting Eq.(17) into Eq.(14), the following can be obtained
u x   B1 u e
(19)
Nu    B1
(20)
u x  Nu  u e
(21)
If
Eq.(19) becomes
N u 
is axial vibration displacement trial function for dynamic beam element. With Eq.(19), it
can be expressed as
Nu    B1  N5
N6 
(22)
where N 5 , N 6 are axial vibration displacement shape functions. After matrix operation, they can
be expressed as
cos x  sin x cos

N

5

sin 


 N  sin x
 6
sin 
(23)
4 Element Potential Energy Functional and Variation
Finite element formulation of dynamic beam element is constructed by principle of minimum
potential energy and analytical trial function.
4.1 Potential Energy Functional
In the light of potential energy, functional of total potential energy is given by
p  U  V

1 l
1 l
1 l
1 l
2
2
2 2









EA

x
dx

EI

x
dx

m


x
dx

m  w2 w2  x dx
u
w
u
u




0
0
0
0
2
2
2
2
(24)
where U is element strain energy, V is potential energy of inertia force.
With element displacement, the following can be obtained
1 l
1 l
eT
T
e
eT
T
e
EA u  N u  N u  u  dx   EI w  N w  N w  w  dx

2 0
2 0
1 l
1 l
eT
T
e
eT
T
e
  m u2  u  N u  N u  u  dx   m  w2  w  N w  N w  w  dx
2 0
2 0
p 
4.2 Functional Variation and Transformation
8
(25)
In the light of principle of minimum potential energy, that is
 p
 
e
 0 , element equilibrium
equation is obtained:
K e  e   2 M e  e  0
Where
K e and M e are
(26)
dynamic stiffness matrix and mass matrix of beam element,
respectively.
K e  0EAN u T N u dx  0EI N w T N w dx
l
l
M w e  0m N u T N u dx  0m N w T N w dx
l
l
5 Element Matrix
Flexural vibration and axial vibration are independent of one another irrespective of large
deformation. Element matrixes at flexural vibration and axial vibration are obtained by applying
variation on flexural vibration displacement and axial vibration displacement, respectively.
5.1 Ele ment Matrix for Beam Flexural Vibration
5.1.1 Stiffness Matrix
After matrix operation, element dynamic stiffness matrix for beam flexural vibration is expressed
as
K w e
 k w11 k w12
k
k
 H  w21 w22
k w31 k w32

k w41 k w42
k w13
k w23
k w33
k w43
k w14 
k w24 
k w34 

k w44 
where
k wij 
1
H

l
0
 N wj
 dx , H 
N wi
EI 3e 2
2
2
, G  16 1  cos ch   2 cos ch 
G
After integral, each stiffness matrix coefficient can be obtained, such as
9
 cosh sin   cos sin   sinh  cosh  2 
 2e  e 4   1
k w11  
2


2
cosh

sin

cos




3
 4 e  e cosh sinh  sin   cosh cos 




 21  e cos sin  cosh sinh   4e cos    sinh  cosh 
 6e  e  1cos sinh   1  e  8e  cos cosh
4
2
2
4
4
2
2
 cos sinh  cosh  cos2  sinh   sin  

 4 e   e 3 
2
2

  cosh  sin   cos  cosh

2
4
2
4
 4e  e  1 cosh   1  e sin  sinh 






5.1.2 Mass Matrix
Mass matrix of element for beam flexural vibration is given by
 mw11
m
M w   P  w21
mw31

mw41
mw12
mw13
mw22
mw32
mw23
mw33
mw42
mw43
mw14 
mw24 
mw34 

mw44 
where
mwij 
m e 2 
1 l
P

,
, G  16 1  cos 2 ch 2   2 cos ch 
N
N
dx
wi
wj
G
P 0
After integral, each mass matrix coefficient can be obtained, such as





4 e   e 3 cos2  sinh   cos sinh  cosh  sin  cosh2   sin 

2
4
2
 4e  e  1 cosh 
 1  e 4  sin  sinh   2 cos sinh  cosh 

 e 4   10e 2   1 cos sinh   8e 2   e 4   1 cos cosh
mw11   -1 
3

2
 4 e  e cosh sin  sinh   cos   4 cos cosh

2

 2e  e  1 sin  cos  sinh  cosh 
 3e 2  cos2  4 sinh  cosh   

2
4
2
4
2
 2 6e  1  e sin  cos cosh   e  14e  1 sin  cosh
















5.2 Ele ment Matrix for Beam Axial Vibration
After matrix operation, element dynamic stiffness matrix for beam axial vibration is expressed as
K u e
 cos  sin   
 2 cos 2   1
 EAc 
  cos   sin 
 2 cos 2   1




 cos   sin  
2cos 2   1 

cos  sin    

2cos 2   1 
Mass matrix of element for beam axial vibration is given by
10












 sin  cos   
2

1
M u   m  2ccoscos sin


2
 2c cos   1




 cos   sin  
2ccos 2   1 

sin  cos    
2ccos 2   1 
5.3 Dynamic Beam Element Matrix
Because of the independence of axial vibration and flexural vibration, 6 by 6 matrix is obtained by
adding two rows and two columns to the above 4 by 4 matrix considering axial vibration without
coupling of axial vibration and flexural vibration. It is element stiffness matrix.
Element stiffness matrix is given by
0
0
k11

k 22
k 23


k33
K e  


symmetry


k14
0
0
k 25
0
k35
k 44
0
k55
0
k 26 
k36 

0
k56 

k66 
where
k11  
EAc cos  sin    
EAc  cos   sin  
, k14 
, k 22  Hk w11 ,
2
2 cos   1
2 cos 2   1




k 23  Hk w12 ,k 25  Hk w13 ,k 26  Hk w14 ,k33  Hk w 22 , k35  Hk w23 , k36  Hk w24 ,
k 44  
EAc cos  sin    
, k55  Hk w33 , k56  Hk w34 , k 66  Hk w 44
2 cos 2   1


Element mass matrix is given by
M e
0
0
m11

m22
m23


m33



symmetry


m14
0
0
m25
0
m35
m44
0
m55
0 
m26 
m36 

0 
m56 

m66 
where
m11 
m sin  cos    
m  cos   sin  
, m14 
, m22  Pm w11 , m23  Pm w12
2
2c cos   1
2c cos 2   1




m25  Pmw13 , m26  Pm w14 , m33  Pm w 22 , m35  Pmw 23 , m36  Pmw 24 ,
11
m44 
m sin  cos    
, m55  Pmw33 , m56  Pmw34 , m66  Pm w 44
2c cos 2   1


6 Example and Comparison
To verify the dynamic beam element constructed by analytical trial function method, the
comparisons between the calculation results of this element, general beam element, and theoretical
solution are conducted.
6.1 Calculation Model
(1) Four kinds of typical beams
Free vibration of four kinds of typical beams are analyzed, including cantilever beam, simply
supported beam, one end clamped and another simply supported beam, and clamped-clamped
beam.
(2) Structure and material parameters
The parameters of calculation model are as follows: beam dimension is 6m×0.2m×0.3m,
material elastic modulus E=210GPa, Poisson’s ratio   0.3 , material density   7800 kg/m
3
.
(3) Theoretical solution
According to theoretical equation of free vibration for distributed mass beam, base frequencies
of free vibration for all kinds of beam are obtained by using analytic method[1].
(4) Dynamic stiffness matrix method
According to governing equation of free vibration for distributed mass beam, stiffness
coefficient [11] of beam vibration is provided, and then base frequencies of free vibration for all
kinds of beam are obtained.
(5) Finite element model of general beam element
By using the software ANSYS, finite element model of beam vibration is established and modal
analys is is conducted. Element BEAM3 is used to simulate beam. BEAM3 is general
two-dimension elastic beam element and uniaxial element bearing tension, pressure and bend.
Every node has three degrees of freedom. That is X-axis linear displacement, Y-axis linear
displacement and Z-axis angular displacement. Cubic polynomial interpolating function is used as
its displacement trial function. Beam is divided into one beam element. Block Lanczos method is
used for modal extraction.
(6) Model in this paper
Equilibrium equation of beam flexural free vibration is given by
 t   0
K  t   M  
(27)
where K   structure stiffness matrix with boundary displacement restraint, M   is structure
mass matrix with boundary displacement restraint. According to natural vibration governing
12
equation of beam, beam natural vibration is harmonic vibration. The following assumption can be
obtained
t   Y sin t
(28)
Substituting Eq.(28) into Eq.(27) yields
M 1  K  Y   2Y
(29)
Assuming    2 , then E    M 1  K  
The following equation can be obtained
E  Y  Y
(30)
Beam natural vibration analys is is converted to eigenvalue and eigenvector analysis of
matrix E    M 1  K   .
Iteration method is used to calculate base frequency of beam vibration. Through circular
calculations, the iteration stops when frequency relative error reaches  y 
 i 1   i
 10 8 .
i

6.2 Result Comparison and Analysis
6.2.1 Element Accuracy
Vibration analysis is conducted by dividing beam into one element. Table 1 shows base frequency
of beam free vibration, and the relative error between calculated solution and theoretical solution.
Table 1 Base frequency of beam free vibration and its comparison (rad/s)
Dynamic stiffness matrix
Interpolation trial
Analytical trial function
method
function element
element
Theoretical
Beam type
Cantilever beam
solution
Absolute
Relative
value
error (%)
value
error (%)
2.89E-07
44.0740
0.423
43.8875389 43.8875390
Absolute Relative
Simply supported beam 123.1941891 123.1941888 -8.59E-08 136.5902
Absolute
Relative
value
error (%)
43.8875391 3.12E-07
10.874 123.1941889 -6.37E-08
One end clamped and
another simply supported 192.4528355 192.4528348 -1.50E-07 255.4366
32.727 192.4528349 -1.53E-07
beam
Clamped-clamped beam 279.2673996 279.2673991 -1.32E-07 283.4156
1.485
279.2673992 -1.27E-07
Note: Calculation of eigenvalue and eigenvector can’t be conducted due to all DOFs of two ends of clamped-cl amped beam restrained. If
the beam is divided into one element, semi-structure is used for vibration simulation of clamped-clamped beam.
As shown in Table 1, the following results can be obtained:
(1) By applying analytical trial function element to the analysis, a more accurate solution can be
attained when the relative error is less than 10 -8. The series of relative error between this solution
-9
and theoretical solution is about 1×10 . It can be supposed that this error be calculation error of
iteration efficiency.
(2) By applying interpolation trial function element to the analysis, the maximum relative error
13
between the solution and theoretical solution reaches 30%. It indicates that dynamic beam element
constructed by taking polynomial function as displacement trial function has bigger error.
(3) The solution of dynamic stiffness matrix method[11] is close to theoretical solution. The
relative error is almost identical with that calculated by analytical trial function method. The
reason is that their displacement curves for vibrating beam are both derived from dynamic
equilibrium governing equation. Element load value and node balance condition need to be
analyzed in dynamic stiffness matrix method. Dynamic stiffness matrix of beam element can be
directly obtained in analytical trial function method. The latter is more simple and intuitive and
widely applied.
6.2.2 Element efficiency
The beam is divided into 1 element, 2 elements, 10 elements and 20 elements respectively. Beam
vibration numerical simulation is conducted by applying interpolation trial function method. The
relative error of calculated base frequency of free vibration for every typical beam and theoretical
solution is obtained. Table 2 presents the results.
Table 2 Comparison of different element results (relative error, %)
Element number for
Beam type
Element number for interpolation trial function
analytical trial function
1
1
2
10
20
Cantilever beam
3.12E-07
0.4249
-0.0003
-0.0489
-0.0489
Simply supported beam
-6.37E-08
10.8739
0.2909
-0.1018
-0.1018
-1.53E-07
32.7269
0.8037
-0.1169
-0.1202
-1.27E-07
1.4854
1.4854
-0.1255
-0.1278
One end clamped and another
simply supported beam
Clamped-clamped beam
As shown in Table 2, the result of interpolation trial function is closer to theoretical solution
with the increasing of element number. But it is obvious that base frequency becomes smaller and
its difference with theoretical solution is larger. The reason is that the more elements are, the lower
structure calculation stiffness is. The difference with structure actual stiffness is also larger. Then
the calculated base frequency further deviates theoretical solution. It indicates that actual
deformation curve for vibrating beam is not polynomial, and polynomial function can’t be taken as
displacement trial function for vibrating beam.
7 Conclusions
Based on the results of this investigation, the following conclusions can be drawn:
(1) The result of interpolation trial function element for simulating beam vibration is not in
accordance with theoretical solution, and the relative error is larger. With element number
increasing, base frequency becomes smaller and deviates theoretical solution. It indicates that
vibrating beam displacement mode is different from polynomial mode, and the precision
14
requirement cannot be met by taking polynomial function as displacement trial function for
vibrating beam.
(2) The solution of dynamic stiffness matrix method for simulating beam vibration is close to
theoretical solution. But in this method, element load value and node balance condition need to be
specially analyzed, and it is not easy to derive non-structural question. Its application in
engineering is limited.
(3) Dynamic stiffness matrix of beam element is obtained by applying analytical trial function
put forward in this paper. Base frequencies of four typical beams are attained by analyzing
eigenvalue and eigenvector, and are compared with theoretical solution. The results show that the
-9
series of relative error is about 1×10 , and it is actually calculation error of iteration efficiency.
The dynamic beam element in the light of analytical trial function put forward in this paper is
high-precision element.
Acknowledgments
Support for this research by National Natural Science Funds No. 51279206, Beijing Natural
Science Foundation No.3144029, Chinese Universities Scientific Fund No. 2011JS126, and the
Specialized Research Fund for the Doctoral Program of Higher Education of China No.
20110008120017.
References
[1]
A.K. Clough and J. Penzien. Dynamic of Structures[M ]. New York: McGraw-Hill, 1995, 293-297.
[2]
Jin Qhanlin. An analytical solution of dynamic response for the rigid perfectly plastic Timoshenko beam[J].
Chinese Journal of Theoretical and Applied Mechanics, 1984, 16(5): 504-511.
[3]
Guo Xiangrong, Chen Huai, Zeng Qingyuan. Dynamic characteristics analysis model of prestressed concrete
T-type beam[J]. Chinese Journal of Computational Mechanics, 2000, 17(2): 176-183.
[4]
Fang Deping, Wang Quanfeng. Dynamic behavior analysis of an externally prestressed beam with energy
method[J]. Journal of Vibration and Shock, 2012, 31(1): 177-181.
[5]
Lou M englin, Hong Tingting. Analytical approach for dynamic characteristics of prestressed beam with
external tendons[J]. Journal of Tongji University(Natural Science), 2006, 34(10): 1284-1288.
[6]
J.A.M . Carrer, S.A. Fleischfresser, L.F.T. Garcia, W.J. M ansur. Dynamic analysis of Timoshenko beams by
the boundary element method[J]. Engineering Analysis with Boundary Elements, 2013, 37(12): 1602-1616.
[7]
Wu Jiajang. Use of the elastic-and-rigid-combined beam element for dynamic analysis of a two-dimensional
frame with arbitrarily distributed rigid beam segments[J]. Applied Mathematical Modelling, 2011, 35(3):
1240-1251.
[8]
M .A. De Rosa, C. Franciosi, M .J. M aurizi. On the dynamic behaviour of slender beams with elastic ends
carrying a concentrated mass[J]. Computers & Structures, 1996, 58(6), 17: 1145-1159.
[9]
R.W. Clough. The finite element method in plane stress analysis[J]. Proceedings of 2nd ASCE Conference on
Electronic Computation, Pittsburgh, Pa., 1960.
[10] Kolousek V. Anwendung des Gesetzes der virtuellen Ver schiebungen and des in der Reziprozitatssatzes[J].
Stab weks Dynamik Lngenieur Archiv, 1941,12: 363-370.
15
[11] Long Yuqiu, Bao Shihua. Structural M echanics Ⅱ(the second edition), Beijing: Higher Education Press,
1996, 240-241.
[12] S. M . Hashemi, M . J. Richard. Free Vibrational analysis of axially loaded bending-torsion coupled beams: a
dynamic finite element[J]. Computers and Structures, 2000, 77(6): 711-724.
[13] Shilin Chen, M . Geradin, E. Lamine. An improved dynamic stiffness method and modal analysis for beam
like structures[J]. Computers and Structures, 1996, 60( 5): 725- 731.
[14] J.R. Banerjee, H. Su, C. Jayatunga. A dynamic stiffness element for free vibration analysis of composite
beams and its application to aircraft wings[J]. Computers & Structures, 2008, 86(6): 573-579.
[15] J.R. Banerjee, S. Guo, W.P. Howson. Exact dynamic stiffness matrix of a bending-torsion coupled beam
including warping[J]. Computers & Structures, 1996, 59(4): 613-621.
[16] M . Shavezipur, S.M . Hashemi. Free vibration of triply coupled centrifugally stiffened nonuniform beams,
using a refined dynamic finite element method[J]. Aerospace Science and Technology, 2009, 13(1): 59-70.
[17] A.K. Chopra. Dynamics of Structures[M ]. Englewood cliffs, NJ:Prentice Hall, 2000, 505-507.
[18] Hu Xuanli, Dai Zongmiao. On the dynamic stress concentrations in orthotropic plates with an arbitrary
cutout[J]. Chinese Journal of Applied Mechanics, 1998, 15(1): 12-17+142.
[19] Wang Xiangqiu, Yang Linde, Gao Wenhua, Dynamic fem analysis for the integration ballast structure based
on variation principle[J]. Journal of Vibration and Shock, 2005, 24(4): 99-102+143.
[20] S.M . Hashemi, M .J. Richard. A dynamic finite element (DFE) method for free vibrations of
bending-torsion coupled beams[J]. Aerospace Science and Technology, 2000, 4(1): 41-45
[21] A. Pagani, E. Carrera, M . Boscolo, J.R. Banerjee.Refined dynamic stiffness elements applied to free
vibration analysis of generally laminated composite beams with arbitrary boundary conditions[J]. Composite
Structures, 2014, 110: 305-316.
[22] S. M ohamed Nabi, N. Ganesan. A generalized element for the free vibration analysis of composite beams[J].
Computers & Structures, 1994, 51(5): 607-610.
[23] Zhao Lei, Chen Qiu. Dynamic analysis of the stochastic variational principle and stochastic finite element
method for structures with random parameters[J]. Chinese Journal of Computational Mechanics,1998, 15(3):
263-274.
[24] Wang Genhui, Gan Yanan ,Wang Zhenbo. Energy variational method for the dynamic response of
thin-walled I-beams with wide flange[J]. Engineering Mechanics, 2010, 27(8): 15-20.
16
1/--pages
Пожаловаться на содержимое документа