1 INTRODUCTION
Displacement-based finite element (DFE) method has extensively been used in computational solid mechanics. In this method, the displacement and slope are used as the nodal values in the modelling of beams. The main disadvantage of DFE is the discontinuity in the stress distribution. Furthermore, stress boundary conditions are not exactly satisfied which causes the inaccuracy of the approximated solution. To eliminate the mentioned problem, stress-based finite element (SFE) has been introduced (^{De Veubeke, 1965}; ^{De Veubeke, 1967}). In this method, stress distribution is approximated by assumed stress function and the transverse deflections and slopes are obtained by integration. Consequently, the considered method provides the continuities of not only transverse deflection but also stress at nodes. This technique was used for analyzing different problems, such as Kirchhoff plates (^{Morley, 1968}; ^{Punch and Atluri, 1986}), plane elastic problems (^{Watwood and Hartz, 1968}; ^{Wieckowski et al., 1999}) and elasto-plastic analysis (^{Wieckowski, 1995}; ^{Kuo et al., 2006}).
^{Kuo et al. (2006}) introduced CFE method for Euler- Bernoulli beam. In their work (^{Kuo et al., 2006}), a cantilever beam and a slewing beam were studied. After that, they used CFE (Kuo and Cleghorn, 2011) and SFE method (^{Kuo and Cleghorn, 2007}) to study a four-bar mechanism and a flexible slider crank mechanism with small strain but large rigid body motion, respectively.
Later, ^{Farid and Cleghorn (2012}) utilized CFE method for the first time to model the dynamics of a single-flexible-link spatial manipulator. They also obtained the dynamic equations of planar multi flexible-link manipulators and verified the results with the displacement finite element method (^{Farid and Cleghorn, 2014}). Furthermore, an improved curvature-based finite element method was developed in (^{Chen et al., 2015}) for the dynamic modelling of a high-speed planar parallel manipulator with flexible links. Also, the method was used for solving a sliding beam problem (^{Kuo, 2015}). The varying-length beam element was established for solving the considered problem.
To the best of our knowledge, the CFE method has been used for the analysis of the problems in which the beams are considered to be clamped-free. The main scope of the present research is to extend the CFE and to introduce CDFE method for vibration analysis of Euler-Bernoulli beams with different boundary conditions.
The paper is organized as follows: Section 2 introduces both stress-based finite element methods. In section 3, the shape functions of both CFE and CDFE methods are obtained for different boundary conditions in order to approximate the deflection in each element. In section 4, using Lagrange’s equation, equations of motion are obtained and the natural frequencies of beams are obtained. Finally, in section 5, numerical examples related to the static and dynamic responses of some beams are investigated.
2 STRESS-BASED FINITE ELEMENT METHODS
In Figure 1, the Euler-Bernoulli beam divided into Nelement is depicted. The transverse deflection, slope and the nodal variable at the left end of the eth element are designated with
In sequence, the shape functions in each of the curvature and the curvature derivative-based finite element methods are obtained.
2.1 Curvature-Based Finite Element Method (CFE)
The curvature distribution in the eth element, _{ me } (ξ), can be linearly approximated as
where, S _{1}(ξ) and S _{2}(ξ) are considered as
in which
The slope in the eth element, _{ ψe } an be obtained by integrating Eq. (1).
where,
Using the continuity of slope between the first and the second element, the constant,
In general, the constant
Integrating Eq. (4), the transverse deflection in the eth element can be obtained by the following equation.
In Eq. (8),
Using Eqs. (7-9), the deflection of the eth element is approximated as
In the above relation,
For e =1
For e = 3, 4, …, N
For e = 1, 2, … , N
For e = 2, 3, …, N
For e = 3, 4, …. , N
For e = 4, 5, …. , N
For i = e + 2, ..., N + 1
where, N is the total number of elements. Also,
2.2 Curvature Derivative-Based Finite Element Method (CDFE)
The curvature derivative distribution in the eth element, _{ ne } (ξ), can be linearly approximated as
where, S _{1}(ξ) and S _{2}(ξ) are defined in Eq. (2). The curvature distribution in the beam can be obtained by integrating Eq. (13).
The slope and transverse deflection of the eth element can be obtained by integrating Eq. (14) as
in which,
The deflection of the eth element in the CDFE method can be written as
in which, the shape functions
For e = 1
For e = 3, 4, …, N
For e = 1, 2, … , N
For e = 2, 3, …, N
For e = 3, 4, …. , N
For e = 4, 5, …. , N
For i = e + 2,..., N + 1
Furthermore,
In the appendix, the first five shape functions in the CFE and CDFE methods are given.
3 BEAMS WITH DIFFERENT BOUNDARY CONDITIONS
In this section, the unknown constants in Eqs, (10) and (18) are obtained by considering the boundary conditions. In CFE method, two of the boundary conditions are used to determine the constants ψ _{0} and w _{0}, the other boundary conditions are incorporated as constraints. In CDFE method, the constant m _{0}, ψ _{0} and w _{0} are obtained by using three boundary conditions and the other one is imposed as constraint.
Therefore, the deflection of the elements in the CFE and CDFE methods can be written in terms of nodal variables as
In what follows, the shape functions,
3.1 Clamped Free (CFE)
For the clamped free beam, the deflection and slope of the first node are zero and the boundary conditions are written as
Thus, ψ
_{0} and w
_{0} are zero and the shape function
3.2 Clamped Free (CDFE)
For the clamped free beam, the constants w _{0}, ψ _{0} and m _{0} in Eq. (18), are obtained using the following conditions
Constants ψ _{0} and w _{0} are zero and the following relation for m _{0} is derived
Therefore, the shape functions can be presented in the form of Eq. (21), where
3.3 Pinned-Pinned (CFE)
In this case, the boundary conditions are given as
Considering the first boundary condition, constant w _{0} is zero. Incorporating, the second boundary condition, constant ψ _{0} is obtained as
By substituting Eq. (27), to Eq. (10), the deflection of the nodes is obtained in which the shape function,
3.4 Pinned-Pinned (CDFE)
Since the deflection and the curvature at the left side of the beam are zero, constants w _{0} and m _{0} are zero. Constant ψ _{0} can be obtained by considering zero deflection at the left side of the beam as
In this case, the deflection of the beam can be written in the form of Eq. (21), where
3.5 Pinned-Guided (CFE)
For the pinned-guided case, the boundary condition are written as
Considering the boundary conditions, the unknown parameter, w _{0} is zero and the parameter ψ _{0} is derived as
Using Eqs. (31), and (10), the nodes’ displacement of the pinned-guided beam is derived where,
3.6 Pinned-Guided (CDFE)
Considering the following conditions
Constants w _{0} and ψ _{0} are zero and m _{0}is derived obtained as
In this case, the shape functions can be derived as given in Eq. (32).
3.7 Clamped-Pinned (CFE)
Considering zero deflection and slope for the first node, the shape functions are obtained similar to the clamped free beam in the CFE method. The zero displacement at the right end is considered as a constraint where can be obtained by multiplying the matrix Γ by the vector of curvature. The matrix Γ is given as
3.8 Clamped-Pinned (CDFE)
Using the following conditions
Constants w _{0} and ψ _{0} are zeros and m _{0} is found as
By substituting Eq. (37), to Eq. (18), the deflection of the nodes is obtained in the form of Eq. (28).
3.9 Clamped-Guided (CFE)
In this case, the shape functions are similar to the clamped-free beam in CFE method. Also, the zeros slope at the right end of the beam is considered as a constraint. In this case, the matrix Γ is defined as
3.10 Clamped-Guided (CDFE)
In this case, the constants w _{0}, ψ _{0} and m _{0} are obtained using the following conditions
Constants w _{0} and ψ _{0} are zero and m _{0} is derived as
The shape functions are similar to Eq. (32).
3.11 Clamped-Clamped (CFE)
For beams with this boundary condition, the shape functions are similar to those of the clamped-free beam in CFE method. Furthermore, the constraints are zero displacement and zero slope at the right end of the beam which can be obtained by multiplication the matrix Γ to the curvature vector. In this case, matrix Γ can be presented as
3.12 Clamped-Clamped (CDFE)
In this case, the conditions are
Constants w _{0} and ψ _{0} are zero and m _{0} is obtained as
The zero slope at the right side of the beam is considered as a constraint which, can be obtained by multiplying matrix Γ by curvature derivative vector
The shape functions can be seen in Eq. (28).
4 FREQUENCY EQUATION
In this section, using Lagrange’s equation and the assumed deflection of the eth element in terms of nodal curvatures and curvature derivatives in CFE and CDFE methods, respectively, the mass matrix and stiffness matrix can be obtained.
4.1 Mass Matrix
The kinetic energy of the eth beam element can be written as
where, the density and the cross area of the beam are designated with constants ρ and A, respectively. Using Eqs. (21) and (45), the kinetic energy can be rewritten as
Thus, the components of the eth element mass matrix are
Also, the kinetic energy of a beam carrying a concentrated mass, m _{0} attached at the eth global node is given as
Therefore, the corresponding components of the eth element mass matrix can be obtained as
4.2 Stiffness Matrix
The potential energy of the eth element of the Euler beam can be written as
in which, EI ^{e} is the flexural stiffness of the eth element. Considering the transverse deflection of the eth element, the component of the eth element stiffness matrix can be obtained as
where,
If linear and torsional springs with stiffness _{ kl } and _{ kt } are attached to the eth global node, the corresponding component of the stiffness matrix can be obtained as
Remark: The size of the total mass and stiffness matrices of the spring-mass-beam system is (e + 1) × (e +1). The ij mponent of the assembled mass and stiffness matrix is obtained by summation of all the ij component of elemental mass and stiffness matrices.
4.3 Load Vector
The virtual work of a discrete load, _{ Fk } acting at the eth node can be written as
While the virtual displacement of each node is as
Using Eqs. (53) and (54), the generalized force can be written as
where, the vector Λ is defined as
The generalized force vector associated to a concentrated moment, _{ Mk } at the eth node can be written as
where, the vector Λ for the moment is obtained as
Furthermore, it can be shown that the generalized force vector due to a continuous force, f(ξ) and a continuous moment, M(ξ) in the eth element can be obtained from Eqs. (59) and (60), respectively.
The ith column of the assembled load vectors is obtained by summation the ith column of the elements.
4.4 Natural Frequency
Using the obtained assembled mass and stiffness matrices, the dynamic equation of a beam without constraint can be written as
The natural frequencies of these beams can be obtained from the following eigenvalue relation
For the beams with constraints, by incorporating the constraints, the resulting differential algebraic equations can be written as
in which, the vector of reaction force is presented by p.
The natural frequencies for these beams can be obtained by solving the following equation
5 NUMERICAL EXAMPLES
In this section, some numerical examples are presented and the results are verified using DFE method. For this purpose, the beams in the presented examples are assumed to be made of steel bar of 0.1m × 0.1m rectangular cross section for which ρ = 7800/m ^{3} and E = 200GPA. Also, the length of the beam is considered to be 𝓁 = 1m.
The first five natural frequencies of the beams with different boundary conditions are obtained with DFE, CFE and CDFE methods and are shown in Table 1. The number of elements in each case is determined.
type | ω _{ 1 } | ω _{ 2 } | ω _{ 3 } | ω _{ 4 } | ω _{ 5 } | |
---|---|---|---|---|---|---|
Clamped-free | exact | 51.39 | 322.09 | 901.86 | 1767.29 | 2921.47 |
DFE(10) | 51.39 | 322.10 | 902.09 | 1768.98 | 2928.83 | |
DFE(20) | 51.39 | 322.09 | 901.88 | 1767.42 | 2921.97 | |
CFE(20) | 51.39 | 322.09 | 901.88 | 1767.41 | 2922.04 | |
CDFE(10) | 51.39 | 322.09 | 901.87 | 1767.36 | 2922.38 | |
CDFE (20) | 51.39 | 322.09 | 901.86 | 1767.30 | 2922.48 | |
Pinned-pinned | exact | 144.27 | 577.08 | 1298.43 | 2308.32 | 3606.75 |
DFE(10) | 144.27 | 577.14 | 1299.12 | 2312.14 | 3620.99 | |
DFE(20) | 144.27 | 577.08 | 1298.47 | 2308.57 | 3607.69 | |
CFE(20) | 144.27 | 577.08 | 1298.47 | 2308.59 | 3608.74 | |
CDFE (10) | 144.27 | 577.08 | 1298.45 | 2308.61 | 3609.16 | |
CDFE (20) | 144.27 | 577.08 | 1298.43 | 2308.32 | 3606.74 | |
Pinned-guided | exact | 36.06 | 324.60 | 901.68 | 1767.31 | 2921.47 |
DFE(10) | 36.06 | 324.61 | 901.92 | 1769.04 | 2929.13 | |
DFE(20) | 36.06 | 324.61 | 901.96 | 1767.43 | 2922.03 | |
CFE(20) | 36.06 | 324.61 | 901.92 | 1767.21 | 2921.97 | |
CDFE (10) | 36.06 | 324.60 | 901.69 | 1767.39 | 2922.37 | |
CDFE (20) | 36.06 | 324.60 | 901.68 | 1767.31 | 2921.48 | |
Clamped-pinned | exact | 225.37 | 730.36 | 1523.85 | 2605.88 | 3976.44 |
DFE(10) | 225.38 | 730.49 | 1524.97 | 2616.37 | 3995.44 | |
DFE(20) | 225.37 | 730.37 | 1523.93 | 2606.27 | 3977.92 | |
CFE(20) | 225.37 | 730.37 | 1523.92 | 2606.23 | 3977.70 | |
CDFE (10) | 225.37 | 730.36 | 1523.89 | 2606.33 | 3979.89 | |
CDFE (20) | 225.37 | 730.36 | 1523.85 | 2605.88 | 3976.47 | |
Clamped-guided | exact | 81.76 | 441.83 | 1091.04 | 2028.80 | 3255.09 |
DFE(10) | 81.76 | 441.85 | 1091.45 | 2031.41 | 3265.66 | |
DFE(20) | 81.76 | 441.83 | 1091.07 | 2028.98 | 3255.88 | |
CFE(20) | 81.76 | 441.83 | 1091.07 | 2028.96 | 3255.78 | |
CDFE (10) | 81.76 | 441.83 | 1091.05 | 2028.95 | 3256.43 | |
CDFE (20) | 81.76 | 441.83 | 1091.04 | 2028.80 | 3255.10 | |
Clamped-clamped | exact | 327.04 | 901.52 | 1767.32 | 2921.47 | 4364.17 |
DFE(10) | 327.05 | 901.74 | 1769.06 | 2929.18 | 4389.14 | |
DFE(20) | 327.04 | 901.52 | 1767.44 | 2922.03 | 4366.14 | |
CFE(20) | 327.04 | 901.52 | 1767.43 | 2921.97 | 4365.80 | |
CDFE (10) | 327.04 | 901.51 | 1767.43 | 2922.21 | 4369.31 | |
CDFE (20) | 327.04 | 901.52 | 1767.32 | 2921.47 | 4364.15 |
Now, two examples for the static analysis of beams are presented. In the first example, deflection, slope and curvature distribution of a simply support beam caring a uniformly distributed load w = 10KN/m is obtained using DFE, CFE and CDFE methods with different number of elements. The results are shown in Figures 2 to 4.
For a clamped-clamped beam with uniformly distributed load w = 10KN/m, deflection, slope and its curvature distributions are plotted in Figures 5 to 7.
It can be seen from Figures 2 to 7 that the deflection and slope distribution in the DFE, CFE and CDFE methods with two elements have the same accuracy. The curvature distribution in CDFE with two elements is close to the results of DFE method with ten elements which confirm the effectiveness of the CDFE method in comparison with DFE method.
Now, the dynamic response of an Euler-Bernoulli beam with CFE and CDFE methods are investigated. In the first example, midpoint deflection of a clamped free beam under a suddenly applied concentrated load w = 10KN at point x = 3 𝓁/4 is shown in Figure 8.
The second example is related to the dynamic response of a clamped free beam with a spring at its right end (k = 6000KN/m). The deflection of the midpoint of the beam in the presence of a suddenly distributed uniform load w = 10KN/m is depicted in Figure 9.
As can be seen, CFE and CDFE methods have the same accuracies in comparison with DFE method. Since the number of nodal variables in CFE and CDFE methods is less than that of DFE method, the computational cost is reduced. Thus, the proposed methods are more efficient for dynamic analysis of beams and can be used for the dynamic analysis of different problems in solid mechanics.
6 CONCLUSION
This study focused on the dynamic analysis of Euler-Bernoulli beams using curvature and curvature derivative-based finite element methods. In curvature based finite element method (CFE) instead of interpolating displacement of Euler Bernoulli beam in usual displacement based finite element method (DFE), second derivative of displacement is interpolated. CFE method previously was used by a few researchers for dynamic analysis of clamped beams. In this research, CFE method was modified for static and dynamic analysis of beams with various boundary conditions.
In addition, a new method called CDFE (curvature derivative-based finite element) which is somehow a modification of CFE, was proposed. CDFE method, which interpolates the derivative of curvature instead of curvature, was used for beams with different boundary conditions.
The results were compared with those obtained by DFE method and the effectiveness of the CFE and CDFE methods was shown. In comparison with DFE method, the proposed methods have the following advantages: