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Elasticity-based beam vibrations for various support conditions
Accepted Manuscript Elasticity-Based Beam Vibrations for Various Support Conditions Alivia Plankis, Michael Lebsack, Paul R. Heyliger PII: DOI: Reference:
S0307-904X(15)00109-2 http://dx.doi.org/10.1016/j.apm.2015.02.023 APM 10444
To appear in:
Appl. Math. Modelling
Received Date: Revised Date: Accepted Date:
18 November 2013 20 December 2014 10 February 2015
Please cite this article as: A. Plankis, M. Lebsack, P.R. Heyliger, Elasticity-Based Beam Vibrations for Various Support Conditions, Appl. Math. Modelling (2015), doi: http://dx.doi.org/10.1016/j.apm.2015.02.023
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Elasticity-Based Beam Vibrations for Various Support Conditions Alivia Plankis, Michael Lebsack, and Paul R. Heyliger Department of Civil and Environmental Engineering, Colorado State University, Fort Collins, CO 80523, 970-491-6685, [email protected]
Abstract The accuracy of common beam theories in determining resonant frequencies and mode shapes for various support conditions is investigated in this paper. This study analyzed the influence of three assumptions employed in the Euler-Bernoulli, Rayleigh, and Timoshenko beam theories by comparing their results to a three-dimensional, elasticity-based approximation for three different support cases: fixed-free, fixed-fixed, and simply-supported. Each of the theories of interest were applied to both isotropic and orthotropic beams of varying length and compared with elasticity-based solutions to study the influence of slenderness ratio and anisotropic properties. The discrepancies caused by each of these effects are discussed and suggestions for the applicability of the common beam theories are provided.
1. Introduction The vibrating beam is a well studied mechanics problem with a multitude of practical applications in engineering [1-17]. These applications including the beam as standard structural elements [18] to more recent applications of beam configurations as sensors and actuators [19]. Problems involving a vibrating beam are typically approached using one of the common beam representations: the Euler-Bernoulli, Rayleigh, and Timoshenko theories. The use of these theories provides reasonable results for many engineering applications; however, all three rely upon restrictive kinematic assumptions. A main component that is missing from the collection of research on the vibrating beam is an investigation into when these assumptions begin to limit accuracy when compared with results obtained using elasticity theory with less restrictive constraints on the displacement fields. In order to confidently use these beam theories, their limitations should be well understood. Preprint submitted to Elsevier
March 4, 2015
In this paper a fourth method, which can be referred to as three-dimensional Ritz elasticity (3DRE) beam theory, to analyze vibrational beam mechanics is introduced that better approximates a fully three-dimensional elasticity theory. This approach will be employed to study the accuracy of the common beam theories and the point at which they fail to be acceptable analysis tools. When studying the effectiveness of common beam theories, three different boundary condition cases will be evaluated: fixed-free, fixed-fixed, and simply supported. Another common support condition in beam theory is the propped cantilever, but the modes for this beam are the anti-symmetric modes for the fixed-fixed beam [18] and hence not specifically considered in this study. It is extremely common in engineering practice to use beam frequencies that have been tabulated by various researchers for each of these boundary conditions using the simplified Euler-Bernoulli theory. These are the most widely available results and are a function of only beam length, density, elastic modulus, and beam cross-section. The intent of this study is to quantify levels of accuracy as the beam becomes stocky and the ratio between the elastic and shear moduli increases. The primary contribution of this study is to provide quantitative limits of typically used beam theories compared with results from an elasticity model that does not possess the strict kinematic limitations inherent in onedimensional representations of flexural beam deformation. Initial guidelines are also presented related to the influence of elasticity-based imposition of beam supports as opposed to the same supports modeled using beam theories. 2. Governing Equations This section will present the beam theories which were studied including the equations which govern each of these models. In accordance with common practice for beam theory discussion, dimensionless variables will be employed for the geometric and vibrational parameters. Also, a consistent coordinate system will be used for all of the calculations where the x axis is parallel to the width of the beam, the y axis provides the vertical dimension of the beam, and the z axis is used for the axial dimension of the beam. The variables u, v, and w will be used to represent the displacement along each of these axes, respectively. These are shown in Figure 1. 2.1. Euler-Bernoulli Beam Theory The Euler-Bernoulli beam theory, also known as the classical beam theory, Euler beam theory, Bernoulli beam theory, or Bernoulli-Euler beam 2
theory, is the most commonly used theory. It is very simple and provides acceptable results for many engineering problems. This theory includes both the strain energy from bending and the kinetic energy from transverse displacement, but does not consider the effects of rotary inertia or shear displacement. Ignoring these two contributions leads to overestimates of the natural frequencies of a vibrating beam, especially for higher modes of vibration and less slender beams when these two factors are not negligible [18]. The governing differential equation of motion for the Euler-Bernoulli model is given by [18] ρA
∂ 2 v(z, t) ∂ 4 v(z, t) + = f (z, t). ∂t2 ∂z 4
(1)
where ρ is the density of the material and A is the cross-sectional area of the beam. Only free vibration is considered in this paper, so the forcing function, f (z, t), is set to zero. To simplify the analysis the principle of separation of variables will be used to isolate an equation for transverse displacement that is dependent on the axial direction only and not time [18]. This new equation is referred to as the spatial solution and it is of the form vEB (z) = C1 sin az + C2 cos az + C3 sinh az + C4 cosh az.
(2)
In this equation, the parameter a is the dimensionless wave number [18] and the EB subscript has been added to the transverse displacement to distinguish this variable from those of other beam theories. For the EulerBernoulli theory the wave number is given by a = ρAω 2 .
(3)
where ω is the natural frequency of the beam. Using the spatial solution above along with appropriate boundary conditions for each of the support cases, a system of four equations results. The solution of this system of equations provides the frequency equation for each specific case. 2.2. Rayleigh Beam Theory The Rayleigh beam theory includes a correction that allows for rotary inertia in the beam [1]. The addition of this term provides a slight improvement over the Euler-Bernoulli model by slightly reducing the overestimation of the natural frequencies. The Rayleigh model still does not incorporate the contribution of shear deformation, and thus does not entirely eliminate errors for high frequency results [18]. 3
With the addition of the kinetic energy term from rotation, the governing differential equation of motion found using Hamilton’s principle for the Rayleigh beam model becomes [18] ρA
∂ 2 v(z, t) ∂ 4 v(z, t) ∂ 4 v(z, t) + − ρI = f (z, t). ∂t2 ∂z 4 ∂z 2 ∂t2
(4)
Here I is the second moment of the area of the cross-section about the centroidal axis. Similar to the Euler-Bernoulli model, the function f (z, t) is set to zero to represent free vibration. By separating the time and spatial variables in the governing equation of motion, the spatial solution for the Rayleigh beam theory is found to be [18] vR (z) = D1 sin a ¯z + D2 cos a ¯z + D3 sinh ¯bz + D4 cosh ¯bz. (5) In this model there are two dimensionless wave numbers, and they are defined by the following expressions [18]:
a ¯= ¯b =
v u u ρIω 2 t
2
v u u ρIω 2 t
−
2
+
s
ρIω 2 2
2
+ ρAω 2
(6)
+
s
ρIω 2 2
2
+ ρAω 2
(7)
The above equations can again be combined with the proper boundary conditions to solve for the frequency equations corresponding to each of the support conditions which were studied [18]. 2.3. Timoshenko Beam Theory The Timoshenko beam theory, sometimes known as the first order shear deformation theory (FSDT), expands upon the Euler-Bernoulli and Rayleigh theories by incorporating the effects of both rotary inertia and shear deformation. To include the influence of the latter, a new variable is used in this model. This variable, known as the shear factor (k 0 ) is necessary to correct for the variation in shear stress over the cross section. In his calculations, Timoshenko found that the change due to the addition of shear was four times greater than the change from the inclusion of rotary inertia [2]. Because of this, the Timoshenko beam theory is more popular than the Rayleigh model. This model results in far more accurate frequency values for non-slender beams and high frequency modes than either of the other two options. 4
By using shear factors determined for isotropic or orthotropic parallelepipeds [18, 15], the elastic strain energy from shear can be included in the Timoshenko beam model. Since there are two unknown variables for this theory, the transverse displacement w and the cross section rotation α, the use of Hamilton’s principle produces two differential equations of motion which are [18] ∂ 2 v(z, t) ∂α(z, t) ∂ 2 v(z, t) − k 0 GA − ρA 2 ∂t ∂z 2 ∂z
!
= f (z, t)
(8)
∂ 2 α(z, t) ∂ 2 α(z, t) ∂v(z, t) − − k 0 GA − α(z, t) = 0. (9) 2 2 ∂t ∂z ∂z The kinematic variables for this theory are shown in Figure 2. The function f (z, t) is set to zero again because only free vibration is considered for the analysis. For these equations, when the method of separation of variables is applied it is found that two different cases need to be considered for the spatial solutions [18]. These two cases are when the frequency is less than a critical value, or if it is above this value. In Han andpco-workers work, it was shown that this critical frequency value is equal to k 0 GA/ρI and is referred to as ωc [18]. For the case when ω < ωc , the form of the spatial solutions for the transverse displacement and total section rotation are given by
ρI
"
vT (z) Ψ(z)
#
+
=
"
E1 F1
"
E3 F3
#
#
sin a ˜z +
sinh ˜bz +
"
"
E2 F2
E4 F4
#
#
cos a ˜z
cosh ˜bz.
(10)
The expressions for the two dimensionless wave numbers for this case are given by the following expressions [18]: a ˜=
˜b =
v u u t
v u u t
1 I+ 0 kG
1 − I+ 0 kG
ρω 2 + 2
s
ρω 2 + 2
I−
s
1 k0 G
2 2 4 ρ ω
1 I− 0 kG
4
+ ρAω 2
2 2 4 ρ ω
4
+ ρAω 2
(11)
(12)
Although it seems that there are now eight unknown constants to solve for the Timoshenko beam model, the Ei and Fi values are related to each other 5
and thus only four unknowns need to be determined for the full solution [18]. For the case when ω > ωc the equations are identical except that ˜b is replaced with i˜b. When the boundary conditions corresponding to each of the support cases are input into the above equations, the frequency equations for the Timoshenko model can be determined similar to the previous two models [18]. 2.4. Elasticity-Based Solutions The proposed elasticity theory will remove a few of the assumptions which are included in the three main theories. The problem of the transversely vibrating beam typically cannot be solved exactly, so solutions using Ritz-based approximations are utilized in this advanced method. This allows for the inclusion of the Poisson effect and the ability to calculate accurate frequencies for non-slender beams and even anisotropic materials. There is no need to represent any rotational variables in this theory, since it replaces displacement functions of a single variable with displacements that vary with (x,y,z) at every location in the beam. The formulation for the mode shape and frequency solutions for this theory differs significantly from the three one-dimensional theories previously discussed. Rather than using governing differential equations and spatial solutions, Ritz-based approximations are used to solve Hamilton’s principle and thus determine frequencies. When used for displacement the general form for the three Ritz-based, direction component approximations can be written as w(x, y, z, t) = φw o (x, y, z) +
N X
cj (t)φw j (x, y, z)
(13)
j=1
The above represents the displacement component in the axial (z) direction, but the representations for the displacements in the x and y directions are of a similar form. Here the cj represent unknown constants that depend on time. For the problems considered in this study, harmonic motion is assumed so that this constant along with those of the other displacement components are written as cj (t) cos ω t, where ω is the natural frequency of vibration. The φo terms represent the simplest functions that satisfy the essential boundary conditions for that displacement direction. In the study discussed in this paper, the initial boundary conditions will be assumed to be zero, with no initial displacement or velocity. The φo terms are therefore all equal to zero for the considered case. In these approximation equations, 6
φj represents a selected approximate function for each respective direction. These functions must satisfy general requirements of independence as outlined by Reddy [20]. By using a large number of these approximation terms for each displacement component, very accurate solutions can be determined for the mode shapes and frequencies of vibration for a beam. Further, unlike the three beam theories, the only limit on the kinematics in the displacement field is on the number of terms used in the approximation. There is no fixed polynomial limitation on any of the variables other than this. These Ritz-based approximations are substituted into the equation for Hamilton’s principle which is shown below for this full elasticity theory. 0=−
Z tZ
1 {σ1 δ1 + σ2 δ2 + σ3 δ3 + σ4 δ4 + σ5 δ5 + σ6 δ6 }dV dt + δ 2 0 V
Z tZ
ρ(u˙ 2 + v˙ 2 + w˙ 2 )dV dt(14) V
0
where u˙ = ∂u/∂t and the conventional notation for stress (σ11 = σ1 , σ23 = σ4 , etc.) has been used [12]. When the substitution is performed and harmonic motion is assumed, the problem is reduced to a generalized eigenvalue problem expressed as follows.
[M 11 ] 0 0 [K 11 ] [K 12 ] [K 13 ] {c} {c} 2 21 22 23 [M 22 ] 0 {d} (15) {d} =ω 0 [K ] [K ] [K ] 0 0 [M 33 ] {e} [K 31 ] [K 23 ] [K 33 ] {e}
The elements of the stiffness ([K]) and mass ([M ]) matrices are related to the Ritz approximations used and some basic material properties of the beam that is considered. The stiffness matrix is symmetric, so there are only six different equations for those elements. The mass matrix is diagonalized so there are only three different forms. All of these relationships are listed below.
11 Kij
=
∂φu ∂φuj ∂φu ∂φuj ∂φu ∂φuj C11 i + C55 i + C66 i ∂x ∂x ∂z ∂z ∂y ∂y
Z
V
12 Kij
=
13 Kij =
22 Kij
=
Z
V
Z
V
∂φu ∂φvj ∂φu ∂φvj + C66 i C12 i ∂x ∂y ∂y ∂x
!
Z
V
∂φu ∂φw ∂φu ∂φw j j C13 i + C55 i ∂x ∂z ∂z ∂x
!
!
21 dV = Kji
!
(16)
(17)
31 dV = Kji
∂φv ∂φvj ∂φv ∂φvj ∂φv ∂φvj + C44 i + C66 i C22 i ∂y ∂y ∂z ∂z ∂x ∂x 7
dV
(18) dV
(19)
23 Kij
33 Kij =
Z
V
=
Z
V
∂φv ∂φw ∂φv ∂φw j j + C44 i C23 i ∂y ∂z ∂z ∂y
!
32 dV = Kji
∂φw ∂φw ∂φw ∂φw ∂φw ∂φw j j j C33 i + C44 i + C55 i ∂z ∂z ∂y ∂y ∂x ∂x
!
dV
(20)
(21)
Z
ρφui φuj dV
(22)
Mij22 =
Z
ρφvi φvj dV
(23)
Mij33 =
Z
w ρφw i φj dV
(24)
Mij11
=
V
V
V
The general form for the Ritz approximation functions will change for each of the different support conditions that will be considered. The primary base for all of the functions employed are polynomial-based, because these functions provide simplicity when evaluating integrals over a parallelepiped. Other functions are used as supplements when appropriate. As is the case for the class of method used in this study, in which global approximation functions are used to describe the displacements, and related methods such as the finite element method, it is critical to determine the number of terms used in the solution series that provide accurate results. In the related method of resonant ultrasound, Migliori and Sarrao [21] discuss the number of terms necessary to provide good agreement with experiment for solids that have three dimensions that are relatively close in magnitude. In the case of the beam, in which one dimension is much larger than the other two, this usually has to be determined by convergence studies on the actual geometry. This procedure is described below. There is a key difference between the support conditions as usually interpreted from beam theories as they are applied to elasticity-based theories including the 3DRE theory used in this study. Beam theory support conditions are usually applied to constraints of either displacement or force/moment resultants along a neutral surface. Hence, the displacement variables or their derivatives are specified at one specific location of the axial coordinate z. In the case of 3DRE, it is an entire surface that needs to be constrained in terms of either displacements or tractions. For example, a free end under beam theory would be associated with zero vertical force and zero resultant
8
out-of-plane moment. In 3DRE, these are replaced by specifying a tractionfree surface at the free end. This difference in how support conditions are satisfied have other implications as will be seen in the results that follow. When any combination of values for j, k, and l are considered, these power functions can represent any type of vibrational mode. If all possible combinations are evaluated, the results will include flexural vibration, shear vibration, and torsional vibration modes about all three axes. For the purposes of the investigation performed for this paper, only the transverse flexural modes are of interest. All other displacement patterns can be removed from consideration. To simplify and reduce the calculations required, group theory is used. This process splits all of the possible Ritz functions into eight different symmetry groups [22, 23]. Parallelepipeds contain three symmetry planes which intersect each other at right angles. Using these symmetry planes, the possible displacement patterns can be separated by considering specific combinations of odd or even behavior in (x,y,z) for each direction about these planes. For example, half of the groups follow the pattern that the u displacement is odd in x, even in y, and even in z. Therefore only odd integers will be considered for j and only even integers for k and l. The v and w deformation patterns can be split similarly, and through this certain modes of vibration can be isolated. Therefore, for the calculations performed for this paper, only two groups are considered, each of which represent the transverse flexural vibration modes [22]. This isolation of the flexural modes was possible for all three support conditions that were studied. For the cantilever beam case, identical approximations can be used for u, v, and w since the boundary constraints (u=v=w=0 at the fixed support) are identical for all three direction components. The approximations used are given by φi = xj y k z l l > 0. (25) For this support case, the number of Ritz terms used to express each of the displacement components is adjusted by summing the coefficients of the powers in x, y, and z such that j + k + l ≤ 16, which provides very accurate frequency results. This number is obtained by successively increasing this term (beginning at 8 terms) until there is no change in the first five significant figures of the results presented for the majority of modes studied for any beam with a clamped boundary condition. For the fixed-fixed beam case, the same approximation form can again be used for all three direction components, u, v, and w for the fixed-fixed beam case since the boundary conditions are the same for all three directions 9
at both of the fixed ends. The approximations in this case were of the form φi = xj y k z l (z − L).
(26)
The Ritz-based approximations for a simply-supported beam are significantly different than for the other two support cases considered, because the constraints are not the same for all three direction components for a hinged end. Therefore, the same general form of approximation terms cannot be employed for all three directions. Another considerable difference for the simply-supported case comes from the inability to satisfy the necessary boundary conditions through a power series. In order to satisfy these conditions and still have the ability to simply evaluate the functions over a parallelepiped, trigonometric functions are utilized for the axial dimension. Sine functions are used for the transverse and out-of-plane displacements, and cosine terms will be employed for the axial displacement. The type of trigonometric function that is used for each displacement component is determined by studying the displacement patterns from a finite element analysis of simply-supported beam. With all of these considerations, the form of the approximations are as follows. lπz L lπz φvi = xj y k sin L lπz j k φw = x y cos i L φui = xj y k sin
(27) (28) (29)
This class of approximations has a fairly long history in plate and beam mechanics, and in fact was used by Pagano in his development of exact solutions for simply supported composite plates [24]. The constraint on the number of terms was set such that j + k + l ≤ 14. A square and solid cross-section is used in the results that follow. This is certainly not the only configuration that could be considered using the elasticity beam model, and in fact modern beams are more likely to be thinwalled or non-rectangular. The solid square is used primarily to eliminate additional variables based on the beam cross-section and also because the element matrices used in 3DRE beam theory are easily computed in closed form for this section. The conclusions drawn at the end of this study should be viewed with this restriction in mind.
10
3. Results and Discussion The resonant frequencies for the lowest five modes were the primary results that were investigated in this vibrational mechanics study. These values were found for all combinations of the various lengths, materials, and support conditions considered. To demonstrate this behavior, we consider a beam with a square cross-section of 0.01 x 0.01 meters. The elastic constants for the isotropic case are taken to be those of steel, with C11 = 269.231 GPa and C44 = 76.923 GPa along with a density of 7830 kg/m3 . The elastic constants for the orthotropic elastic constants of graphite-magnesium give as C11 = C22 = 28.18 (all GPa), C33 = 174.3, C12 = 10.67, C13 = C23 = 12.20, C66 = 8.76 with a density of 1738 kg/m3 [25]. All of the calculated frequencies are presented in tabular form. The results for a cantilever beam are listed in Table 1. For a fixed-fixed support case, the results are provided in Table 2. Lastly, the frequencies for a simply-supported beam case are given in Table 3. There are two columns presented for the 3DRE model: one for the full set of elastic constants, and the second where all values of Poisson ratio are set equal to zero. For a few of the cases involving a fixed support, the higher mode frequencies from the Timoshenko beam model were not able to be calculated. Even without some of these values trends were still able to be determined and conclusions drawn. From the values found the influence of slenderness, anisotropy, and the Poisson effect could be investigated. The effect of each of these factors for the different support conditions are discussed in the remainder of this section. 3.1. Effects of Slenderness The slenderness ratio of a beam is dimensionless and is defined using the symbol S = √L . Its value has a significant influence on the accuracy I/A
of frequency calculations for two of the common beam theories. As the slenderness ratio decreases to a point where the solid is closer to a cube than a typical beam, the error between the elasticity solution and both the Euler-Bernoulli and Rayleigh results dramatically increases. For the Timoshenko beam model, the length of a beam has a much smaller effect on the percent error from the full elasticity theory. This is especially true for the fundamental mode of an isotropic beam. The support case for a beam affects the level of accuracy for each beam theory as the slenderness ratio changes, but these general trends are consistent throughout. For the cantilever beam and fixed-fixed beam conditions, all three beam theories approach a similar error as the slenderness increases. Here we use the word “error” to indicate the difference in frequencies obtained using each 11
Table 1: Frequencies of a Cantilevered Beam L
2 (S=6.928)
5 (S=17.32)
10 (S=34.64)
20 (S=69.28)
40 (S=138.56)
100 (S=346.41)
Mode
E-B
Rayleigh
1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5
1282 8037 22500 44100 72900 205.2 1286 3601 7056 11660 51.30 321.5 900.2 1764 2916 12.82 80.37 225.0 441.0 729.0 3.206 20.09 56.26 110.3 182.2 0.5130 3.215 9.002 17.64 29.16
1224 6205 14100 22570 31180 203.6 1221 3211 5812 8812 51.20 317.2 872.5 1667 2673 12.82 80.10 223.2 434.6 712.2 3.206 20.08 56.15 109.8 181.2 0.5130 3.214 8.998 17.63 29.13
Isotropic Timoshenko 1093 4147 8676 11250 14600 199.1 1074 2560 4255 6056 50.90 305.3 803.5 1459 2222 12.80 79.29 218.1 416.8 668.7 3.205 20.02 55.81 108.6 178.0 0.5129 3.213 8.990 17.60 29.05
3DRE (ν = 0)
3DRE
E-B
Rayleigh
1100 4285 8899 11310 13160 199.3 1089 2635 4447 6277 50.92 307.1 815.6 1501 2383 12.80 79.57 220.1 425.5 712.1 3.205 20.08 56.24 110.6 189.2 0.5131 3.222 9.056 17.90 30.85
1149 4314 8919 11320 13400 209.1 1121 2682 4501 6313 53.45 318.1 836.1 1526 2408 13.44 82.62 226.4 433.7 719.1 3.365 20.86 57.91 112.8 191.0 0.5386 3.348 9.328 18.27 31.14
2484 15570 43600 85440 141200 397.5 2491 6976 13670 22600 99.38 622.8 1744 3418 5649 24.84 155.7 436.0 854.4 1412 6.211 38.93 109.0 213.6 353.1 0.9938 6.228 17.44 34.18 56.49
2372 12020 27330 43730 60420 394.5 2366 6222 11260 17070 99.19 614.6 1690 3231 5179 24.83 155.2 432.5 841.9 1380 6.211 38.89 108.8 212.8 351.0 0.9938 6.228 17.43 34.15 56.44
Orthotropic Timoshenko 3DRE (ν = 0) 1704 5434 365.8 1659 3604 5571 7557 97.20 542.8 1325 2241 3220 24.71 148.9 400.1 737.0 1137 6.203 38.55 106.5 204.9 330.9 0.9936 6.219 17.37 33.94 55.87
1729 5548 9726 10820 11670 366.8 1708 3708 5677 7700 97.28 549.3 1360 2336 3337 24.72 150.7 405.5 757.3 1220 6.204 38.67 107.5 209.0 352.5 0.9939 6.236 17.51 34.54 59.36
3DRE 1739 5557 9802 10870 11720 369.5 1713 3715 5689 7714 98.04 551.9 1364 2342 3343 24.91 151.5 407.1 759.5 1222 6.254 38.90 107.9 209.6 353.1 1.002 6.272 17.58 34.63 59.44
Table 2: Frequencies of a Fixed-Fixed Beam L
2 (S=6.928)
5 (S=17.32)
10 (S=34.64)
20 (S=69.28)
40 (S=138.56)
100 (S=346.41)
Mode
E-B
Rayleigh
Isotropic Timoshenko
3DRE (ν = 0)
3DRE
E-B
Rayleigh
1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5
8160 22490 44100 7289 108900 1306 3599 7056 11660 17420 326.4 899.7 1764 2916 4356 81.60 224.9 441.0 729.0 1089 20.40 56.23 110.3 182.2 272.2 3.264 8.997 17.64 29.16 43.56
7277 16030 25230 34160 42890 1280 3351 6118 9302 12710 324.8 883.0 1695 2727 3943 81.50 223.9 436.5 716.3 1060 20.39 56.17 110.0 181.4 270.4 3.264 8.996 17.63 29.14 43.51
3947 7711 1054 2416 4029 5760 7567 306.3 785.8 1422 2160 2968 80.25 216.6 413.4 661.9 954.2 20.31 55.69 108.4 177.5 262.3 3.262 8.984 17.59 29.03 43.29
4068 7952 12670 17870 23040 1070 2471 4157 5962 7873 310.1 798.1 1449 2213 3068 81.19 219.5 419.2 674.3 975.4 20.57 56.45 109.8 180.7 267.4 3.323 9.150 17.96 29.78 44.75
4157 8016 12740 18070 23230 1106 2527 4225 6039 7937 321.5 823.1 1483 2254 3117 84.22 227.3 430.8 690.6 995.9 21.33 58.52 113.0 185.5 273.6 3.435 9.475 18.44 30.58 45.75
15810 43580 85440 141200 211000 2530 6973 13670 22600 33760 632.4 1743 3418 5649 8439 158.1 435.8 854.4 1412 2110 39.53 109.0 213.6 353.1 527.5 6.324 17.43 34.18 56.49 84.39
14100 31060 48890 66190 83110 2479 6492 11850 18020 24620 629.2 1711 3285 5284 7640 157.9 433.7 845.7 1388 2054 39.51 108.8 213.0 351.5 523.9 6.324 17.43 34.16 56.45 84.30
12
Orthotropic Timoshenko 3DRE (ν = 0) 4500 8885 1508 3136 5015 6938 8885 524.6 1231 2081 3002 3963 149.8 389.4 713.3 1096 1520 38.98 105.7 202.8 326.8 474.3 6.310 17.35 33.88 55.75 82.83
4650 9193 10570 17040 21820 1537 3222 5142 7136 9139 531.3 1253 2135 3089 4120 151.6 394.4 725.2 1119 1565 39.43 107.0 205.4 332.5 484.0 6.394 17.59 34.36 56.80 84.53
3DRE 4661 9211 10680 16720 21770 1544 3230 5155 7151 9156 534.2 1258 2143 3099 4129 152.4 396.7 728.1 1123 1570 39.66 107.6 206.3 333.8 485.7 6.431 17.69 34.51 57.02 84.83
Table 3: Frequencies of a Simply-Supported Beam L
2 (S=6.928)
5 (S=17.32)
10 (S=34.64)
20 (S=69.28)
40 (S=138.56)
100 (S=346.41)
Mode
E-B
Rayleigh
1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5
3600 14400 32400 57600 90000 576.0 2304 5184 9216 14400 144.0 576.0 1296 2304 3600 36.00 144.0 324.0 576.0 900.0 9.000 36.00 81.00 144.0 225.0 1.440 5.760 12.96 23.04 36.00
3279 10670 19190 27810 36320 566.7 2166 4553 7459 10670 143.4 566.7 1250 2166 3279 35.96 143.4 321.0 566.7 877.7 8.997 35.96 80.81 143.4 223.6 1.440 5.759 12.95 23.02 35.96
Isotropic Timoshenko 2713 7394 12230 17010 21730 541.7 1882 3601 5471 7394 141.7 541.7 1142 1882 2713 35.85 141.7 312.6 541.7 820.9 8.990 35.85 80.25 141.7 219.4 1.440 5.76 12.94 22.98 35.85
3DRE (ν = 0)
3DRE
E-B
Rayleigh
2705 7367 12200 17000 21750 541.2 1878 3589 5451 7367 141.6 541.2 1140 1878 2705 35.85 141.6 312.4 541.2 819.9 8.990 35.85 80.24 141.6 219.3 1.440 5.756 12.94 23.00 35.85
2689 7287 12060 16830 21580 540.4 1869 3562 5398 7287 141.6 540.4 1137 1869 2689 35.84 141.6 312.2 540.4 818.1 8.990 35.84 80.22 141.6 219.2 1.440 5.756 12.94 22.98 35.84
6975 27900 62770 111600 174400 1116 4464 10040 17860 27900 279.0 1116 2511 4464 6975 69.75 279.0 627.7 1116 1744 17.44 69.75 156.9 279.0 435.9 2.790 11.16 25.11 44.64 69.75
6352 20670 37180 53880 70360 1098 4196 8822 14450 20670 277.8 1098 2423 4196 6352 69.67 277.8 622.0 1098 1700 17.43 69.67 156.6 277.8 433.1 2.790 11.16 25.10 44.61 69.67
Orthotropic Timoshenko 3DRE (ν = 0) 3854 8857 13710 18490 21010 950.7 2841 4869 6879 8857 266.5 950.7 1853 2841 3854 68.92 266.5 569.7 950.7 1385 17.38 68.92 152.8 266.5 406.6 2.788 11.14 25.00 44.30 68.92
of the beam theories in comparison with elasticity theory. The existence of a single fixed support for a beam leads to often small but still noticeable differences in frequency for even very slender beams. This is caused in part by neglecting the Poisson effect and will be discussed in Section 3.3. Although the frequency results from the beam theories do not approach the full elasticity solution, they still demonstrate the slenderness trends discussed above. As the slenderness value decreases, the results for both the Euler-Bernoulli and Rayleigh models begin to deviate from the asymptotic error percentage at an exponential rate as can be seen in Figures 3 and 5. The error values from the Rayleigh beam model are less than those from Euler-Bernoulli model calculations, as was expected, but the slenderness ratio of a beam still has a meaningful influence on the accuracy of the Rayleigh model. This effect is significantly greater for higher modes of vibration for both beam theories. While the errors for the fundamental mode of a 1x1x2 cantilever beam are 11.6% and 6.5% for Euler-Bernoulli and Rayleigh, respectively, these errors grow to 444% and 133% for the fifth mode of vibration. The errors for a fixed-fixed beam are even larger for higher modes of vibration. The error values for the fifth mode are shown in Figures 4 and 6. The small
13
3824 8865 10450 16620 20930 946.4 2818 4836 6856 8865 266.1 946.4 1840 2818 3824 68.89 266.1 567.9 946.4 1377 17.38 68.89 152.7 266.1 405.7 2.788 11.14 25.00 44.28 68.89
3DRE 3813 8843 10570 16100 21040 945.1 2810 4822 6837 8844 266.0 945.1 1836 2810 3813 68.88 266.0 567.4 945.1 1374 17.38 68.88 152.7 266.0 405.4 2.788 11.14 24.99 44.28 68.88
effect that slenderness has upon the Timoshenko frequency results can also be seen in these figures. While the other two beam theories diverge exponentially as the beam length falls below ten, the Timoshenko model gives results that agree extremely well with elasticity theory throughout the entire range of slenderness ratios considered. The error increases slightly more for very stout beams and higher modes of vibration, but this increase is small in comparison to the other two theories. The error values for a simply-supported beam as the beam length varies are shown in Figures 7 and 8. The main difference for this support case is that the frequency values from the three beam theories approach the full elasticity solutions rather than a common error value. The error increases significantly for the Euler-Bernoulli and Rayleigh beam theories as a beam becomes less slender, while the influence is minimal for the Timoshenko model. Slenderness is also still a greater factor for higher modes of vibration than for the fundamental mode. All of these results follow expected trends. As the slenderness of a beam decreases, the displacement pattern is known to deviate from the kinematic model that restricts the axial displacement to be a linear function in thickness assumed by each of the three beam theories. For example, in the extreme case where the beam length approaches the width or becomes less than the width, the deformation pattern becomes more similar to that of a plate than a beam. It is therefore clear that the one-dimensional displacement patterns of the common beam theories cannot be applied to very short and thick beams. Three-dimensional displacement patterns were determined using the elasticity model. These modal patterns demonstrate obvious warping of the beam surfaces through the width and height for stocky beams. When a beam is slender, the displacement is dominant in only one dimension as the three common beam theories assume, but this deteriorates rapidly as the slenderness decreases. The first five of the three-dimensional elasticity mode shapes were plotted for both a slender beam (1x1x40) and a stout beam (1x1x2) for each of the support cases. Figures 9, 10, 11, 12, 13, and 14 show these mode shapes for the cantilever, clamped-clamped, and simply-supported cases, respectively. The deformation pattern deviates from beam theory expectations in Figure 10 where the cross-sectional plane is no longer plane throughout displacement; this is especially noticeable on the free face of the fifth mode shape. In the fourth and fifth mode shape for both the fixed-fixed and simply-supported cases, the Poisson effect begins to have a significant influence for short beams. In these plots, obvious warping through the width of the beam occurs due to the shrinking and expansion 14
Figure 1: The coordinate axes (x,y,z) for the general beam and the displacement components (u,v,w) in the respective directions.
15
influences of the Poisson ratio. The displacement patterns in these cases can no longer be accurately modeled through a one-dimensional plot of the centerline since the deformation of the outer surfaces is significantly different than what occurs at the centerline of the beam. For most beams, the lower modes of vibration are nearly always flexural, but it was found that for very non-slender beams this was not the case. For the very short beam case, the lowest three modes followed flexural displacements for all three support conditions, but the fourth and fifth lowest frequencies corresponded to very different deformation types. For the fixed-fixed case, the fifth flexural mode did not appear until the 14th lowest mode of vibration. This finding provides another strong reason for more accurate frequency and mode shape calculations when dealing with nonslender beams since non-flexural modes of vibration may become dominant far earlier than anticipated. 3.2. Effects of Anisotropy The accuracy of frequency values for the orthotropic material, graphitemagnesium, decreases even more rapidly than for steel as the slenderness decreases. Figures 15, 16, and 17 display this trend. For example, the error is more than four times greater for the orthotropic fixed-fixed beam than the isotropic case. The error increase for an anisotropic beam becomes insignificant for very slender beams, displaying that the slenderness ratio provides the dominant influence on frequency results. Anisotropic material properties can exacerbate the loss of accuracy at low slenderness ratios, especially for beam theories without shear deformation, because the shear modulus in the transverse-axial plane can be an order of magnitude or more smaller than the longitudinal modulus. The Timoshenko model was exceptionally adaptable to the orthotropic material. The frequency errors were similar in magnitude for both the isotropic and orthotropic material as can be noted from Figures 15, 16, and 17. The shear coefficient for both the isotropic and orthotropic materials were calculated using the methodology of Puchegger et al. [15]. 3.3. Effects of Poisson Ratio There is a significant difference between the Ritz-based elasticity method and the three beam theories for even very slender beams when at least one support in a beam is fixed. One possible reason for this difference is because of the restraint of the Poisson effect at the fixed support. When considering a three-dimensional model, a perfectly fixed support completely restricts the Poisson effect. For example, in a cantilevered beam, the largest forces 16
would exist at the fixed support and thus the influence of the Poisson effect would cause the beam to shrink or expand through the width to counteract the large axial compression and tension forces. A perfectly fixed end does not allow this. The displacements and forces cannot act as is expected by typical beam theories. With a hinged or free support, there is no restriction of the Poisson effect and the beam is free to displace freely since there are no restraining forces transverse to the long direction of the beam. Therefore, ignoring the Poisson effect in a beam with only free or hinged supports would have minimal effect on the accuracy of the frequency results. To investigate this explanation, the more accurate Ritz-based elasticity program was used to determine frequency results with both the normal material properties as well as elastic stiffness values found using ν = 0. In the latter case, the components of the elastic stiffness tensor were calculated directly as the the inverse of the diagonal compliances with C12 = C13 = C23 = 0 and a shear modulus that was half of the elastic modulus for the steel. For the orthotropic case, a similar procedure but the shear moduli were treated as independent constants n the latter case, the components of the elastic stiffness tensor were calculated directly as the the inverse of the diagonal compliances with C12 = C13 = C23 = 0 and a shear modulus that was half of the elastic modulus for the steel. For the orthotropic boron-aluminum, a similar procedure was used (for example, C33 = 1/E3 , where E3 is the elastic modulus in the direction of the boron fibers) but the shear moduli were treated as independent constants and hence were unaffected. This allowed for the influence of the Poisson effect to be isolated from the other factors that cause errors in the typical beam theories. The results of this investigation are shown in Figure 18, and indicate that a consistent error exists between the models with and without the Poisson effect that is independent of beam slenderness. The error in the fundamental mode is near 4.7% for a cantilevered beam, 3.4% for a fixedfixed beam, but well under one percent for a simply-supported beam. The percent difference does vary somewhat with the length of the beam, but the change is minimal. The result for the simply-supported beam case followed expectations and showed that neglecting the Poisson effect has a minimal influence for this specific support condition. Although the impact of neglecting the Poisson effect seems to be independent of slenderness, the error is related to the mode of vibration. The error decreased for higher modes of vibration where there is at least one fixed support. For the simply-supported case, the opposite pattern occurs. The influence of the Poisson effect is still minor for all modes of vibration, but the error does tend to increase for higher modes of vibration. Since 17
many of these errors are relatively small, some of these differences may be attributable in part to the decrease in relative accuracy of the Ritz method for higher modes. These results are shown for a beam with a slenderness ratio of 69 in Figure 19. It is possible that the change along beam length of the sign of the curvature balances out the restraint at the fixed end since the higher modes have more sign changes (and hence a Poisson displacement that also changes sign along the beam length). The frequency error associated with non-zero Poisson ratio also appears to be connected with the material properties of a beam when a fixed support is present, and in fact this may give a hint as to the source of this behavior. The percent difference between the zero Poisson ratio and full elasticity results is far less for the orthotropic material studied than for the isotropic material. The error is approximately six times greater for a steel beam than for beam of graphite-magnesium if there is a fixed support. The probable reason for this behavior lies in the magnitude of the values of the Poisson ratio. In the isotropic case, the values for the steel for the two axial-transverse Poisson ratio are νzx = νzy = 0.3. However, for the graphite-magnesium, these values are νzx = νzy = 0.046. Such small values of Poisson ratio are somewhat typical for orthotropic solids whose longitudinal modulus is so much larger than its transverse modulus, whereas for isotropic materials these moduli are identical. Hence, there is a much stronger Poisson effect being restrained at the fixed supports for the isotropic beam, leading to larger differences in frequency. The results also show that this error difference does not exist for the simply-supported beam. For this support case, the error values with or without the Poisson effect are virtually identical. 4. Conclusions The results in this work are somewhat limited in that only solid and hollow square cross-sections were used to determine initial differences in frequency between the various theories. More extensive study of the influence of the beam cross-section are likely to provide additional useful results. Based on the geometries, constitutive laws, and kinematic assumptions made during the course of this study, the following conclusions can be suggested: • The Euler-Bernoulli and Rayleigh theories give frequencies that are significantly different from the 3DRE values for beams with clamped support. The differences between the Timoshenko and 3DRE frequencies for this type of support are noticeable but much smaller. One possible reason for the former is the lack of representation of the Poisson 18
restraint at these supports. The frequency differences with elasticity models are 4% or greater for the entire range of slenderness values studied. • When the influence of the Poisson ratio was isolated, it was found that neglecting this effect leads to frequency differences greater than 3% for beams with s > 300 when a fixed support exists. • When the Euler-Bernoulli or Rayleigh theories are applied to an orthotropic material, the frequency differences for s < 50 more than double in comparison to the differences for an isotropic material. This trend was found for all three support cases. • For higher modes of vibration and a slenderness ratio of 138, the frequency differences are above 2% for Euler-Bernoulli and Rayleigh, mildly contradicting the assumption that these theories are adequate for beams with s > 100. This loss of accuracy for higher modes was consistent for all three support conditions studied. • When the slenderness ratio falls below 50 the differences in frequency results when compared with elasticity theory increases rapidly for both the Euler-Bernoulli and Rayleigh beam models, with differences up to 90%. • For all three support cases studied, the frequency differences for the Timoshenko model are small when compared with the 3DRE results for zero Poisson ratio and have little relationship to slenderness or anisotropic properties. When compared with the full set of elastic constants with a fixed support, the Timoshenko results are generally lower than the elasticity solution - indicating that there may be errors in both kinetic and elastic energy terms that combine to yield frequencies under the elasticity values. • For the simply-supported case, the Poisson effect has a minimal influence. The Timoshenko model gives excellent results for all slenderness ratios considered. • For slenderness ratios over 40, the differences in frequency between the Euler-Bernoulli and Rayleigh theories are below 5 percent for all materials and support conditions. This value could be considered a reasonable upper bound for the applicability of these two beam theories. 19
Based on these results, applications of beam theory frequency results for clamped support conditions may be problematic regardless of slenderness ratio or material type. For a simply-supported beam, the Timoshenko beam model performs extremely well. The accuracy of frequency results only becomes poor for very low slenderness values combined with high modes of vibration. The Euler-Bernoulli and Rayleigh beam theories deteriorate in accuracy at much higher slenderness ratios. A common standard is that Euler-Bernoulli is acceptable for slenderness ratios greater than 100, but this standard is rarely quantified. It was found here that for a slenderness ratio of 138, the differences in frequency with 3DRE results for higher modes and orthotropic materials is near 2% for Euler-Bernoulli and above 1% for Rayleigh. The Timoshenko beam model is the clear choice for a simplysupported beam unless the slenderness ratio is extremely high. For fixedfree or fixed-fixed beams, it may be the case that more complex analysis is required to calculate accurate resonant frequencies and mode shapes. 5. Acknowledgements The funding for this research was provided by a grant from Mountain Plains Consortium. References [1] Timoshenko SP. History of Strength of Materials. New York: McGrawHill Book Company, Inc.; 1953. [2] Timoshenko SP. On the correction for shear of the differential equation for transverse vibrations of bars of uniform cross-section. Philos Mag 1921;41:744-746. [3] Timoshenko SP. On the transverse vibrations of bars of uniform crosssection. Philos Mag 1922;125:125-131. [4] Traill-Nash RW, Collar AR. The effects of shear flexibility and rotatory inertia on the bending vibrations of beams. Q J Mech Appl Math 1953;6:186-213. [5] Levinson M. A new rectangular beam theory. J Sound Vibrat 1981;74:81-87. [6] Anderson RA. Flexural vibration of uniform beams according to the Timoshenko theory. J Appl Mech 1953;75:504-510. 20
[7] Dolph CL. On the Timoshenko theory of transverse beam vibrations. Q Appl Math 1954;12:175-187. [8] Stephen NG. On the variation of Timoshenko’s shear coefficient with frequency. J Appl Mech 1978;45:695-697. [9] Huang TC. The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions. J Appl Mech 1961;28:579-584. [10] Thomas J, Abbas BAH. Finite element model for dynamic analysis of Timoshenko beam. J Sound Vibrat 1975;41:291-299. [11] Schramm U, Kitis L, Kan W, Pilkey WD. On the shear deformation coefficient in beam theory. Fin Elem Anal Des 1994;16:141-162. [12] Heyliger PR, Jilani A. The free vibrations of inhomogeneous elastic cylinders and spheres. Int. J. Solids Structures 1992;29(22):2689-2708. [13] King JL. The free transverse vibrations of anisotropic beams. J Sound Vibrat 1985;98:575-585. [14] Levinson M. Further results of a new beam theory. J Sound Vibrat 1981;77:440-444. [15] Puchegger S, Loidl D, Kromp K, Peterlik H. Hutchinson’s shear coefficient for anisotropic beams. J Sound Vibrat 2003;266:207-216. [16] Cowper, GR. The shear coefficient in Timoshenko’s beam theory. J Appl Mech 1966; 33:335-340. [17] Bank, LC. Shear coefficients for thin-walled composite beams. Composite Structures 1987; 8: 47-61. [18] Han SM, Benaroya H, Wei T. Dynamics of transversely vibrating beams using four engineering beam theories. J Sound Vibrat 1999;225(5):935988. [19] Plankis, A and Heyliger, P. Off-grid MEMS sensors configurations for transportation applications. MPC Report MPC-13-257, Mountain Plains Consortium, 2013. [20] Reddy, JN. Energy Principles and Variational Methods in Mechanics, Wiley, New York, 2002. 21
[21] Migliori, A and Sarrao, JL. Resonant ultrasound spectroscopy, Wiley, New York, 1997. [22] Ohno I. Free vibration of a rectangular parallelepiped crystal and its application to determination of elastic constants of orthorhombic crystals. J Phys Earth 1976;24:355-379. [23] Mochizuki, E. Application of group theory to free oscillations of an anisotropic rectangular parallelepiped. J. Phys. Earth 1987; 35: 159170. [24] Pagano, N. J. Exact solutions for rectangular bidirectional composite and sandwich plates. J. Comp. Mater. 1970; 4: 20-34. [25] Ledbetter HM, Datta SK, Kyono T. Elastic constants of a graphitemagnesium composite. J Appl Phys 1989;65(9):3411-3416.
22
Figure 2: An undeformed beam (top) and the deformed beam (bottom) as viewed from the side are shown to highlight the rotational kinematic variable α for the Timoshenko theory and the difference with the Euler-Bernoulli theory.
23
Percent Error from Full Elasticity Solution, %
12 Euler−Bernoulli Rayleigh Timoshenko
10 8 6 4 2 0 −2 −4 −6
0
10
20
30
40 50 60 Beam Length
70
80
90
100
Figure 3: Frequency Error for the First Mode of a Cantilevered Beam
Percent Error from Full Elasticity Solution, %
450 Euler−Bernoulli Rayleigh Timoshenko
400 350 300 250 200 150 100 50 0 −50
0
10
20
30
40 50 60 Beam Length
70
80
90
100
Figure 4: Frequency Error for the Fifth Mode of a Cantilevered Beam
24
Percent Error from Full Elasticity Solution, %
100 Euler−Bernoulli Rayleigh Timoshenko
80
60
40
20
0
−20
0
10
20
30
40 50 60 Beam Length
70
80
90
100
Figure 5: Frequency Error for the First Mode of a Fixed-Fixed Beam
Percent Error from Full Elasticity Soltuion, %
400 Euler−Bernoulli Rayleigh Timoshenko
350 300 250 200 150 100 50 0 −50 0
10
20
30
40 50 60 Beam Length
70
80
90
100
Figure 6: Frequency Error for the Fifth Mode of a Fixed-Fixed Beam
25
Percent Error from Full Elasticity Solution, %
35 Euler−Bernoulli Rayleigh Timoshenko
30
25
20
15
10
5
0
0
10
20
30
40 50 60 Beam Length
70
80
90
100
Figure 7: Frequency Error for the First Mode of a Simply-Supported Beam
Percent Error from Full Elasticity Soltuion, %
350 Euler−Bernoulli Rayleigh Timoshenko
300
250
200
150
100
50
0
0
10
20
30
40 50 60 Beam Length
70
80
90
100
Figure 8: Frequency Error for the Fifth Mode of a Simply-Supported Beam
26
Figure 9: First Five Mode Shapes for a 1x1x40 Isotropic Fixed-Free Beam
27
Figure 10: First Five Mode Shapes for a 1x1x2 Isotropic Fixed-Free Beam
Figure 11: First Five Mode Shapes for a 1x1x40 Isotropic Fixed-Fixed Beam
28
Figure 12: First Five Mode Shapes for a 1x1x2 Isotropic Fixed-Fixed Beam
Figure 13: First Five Mode Shapes for a 1x1x40 Isotropic Simply-Supported Beam
29
Figure 14: First Five Mode Shapes for a 1x1x2 Isotropic Simply-Supported Beam
Percent Error from Full Elasticity Solution, %
45 E−B Iso E−B Ortho Rayleigh Iso Rayleigh Ortho Timo Iso Timo Ortho
40 35 30 25 20 15 10 5 0 −5
0
10
20
30
40 50 60 Beam Length
70
80
90
100
Figure 15: Error Comparison Between Isotropic and Orthotropic Materials for a Cantilevered Beam
Percent Error from Full Elasticity Solution, %
250 E−B Iso E−B Ortho Rayleigh Iso Rayleigh Ortho Timo Iso Timo Ortho
200
150
100
50
0
−50
0
10
20
30
40 50 60 Beam Length
70
80
90
100
Figure 16: Error Comparison Between Isotropic and Orthotropic Materials for a FixedFixed Beam
30
Percent Error from Full Elasticity Solution, %
90 E−B Iso E−B Ortho Rayleigh Iso Rayleigh Ortho Timo Iso Timo Ortho
80 70 60 50 40 30 20 10 0
0
10
20
30
40 50 60 Beam Length
70
80
90
100
Figure 17: Error Comparison Between Isotropic and Orthotropic Materials for a SimplySupported Beam
Percent Error for Zero Poisson Ratio, %
5
4
3
Fixed−Free Fixed−Fixed Simply−Supported
2
1
0
−1
0
10
20
30
40 50 60 Beam Length
70
80
90
100
Figure 18: Influence of Poisson Effect for Varying Slenderness Ratios
31
Percent Error for Zero Poisson Ratio, %
5 Fixed−Free Fixed−Fixed Simply−Supported
4
3
2
1
0
−1
1
1.5
2
2.5 3 3.5 Mode of Vibration
4
4.5
5
Figure 19: Influence of Poisson Effect for Varying Modes of Vibration: The Isotropic Beam with Slenderness Ratio of 69
32
S0307-904X(15)00109-2 http://dx.doi.org/10.1016/j.apm.2015.02.023 APM 10444
To appear in:
Appl. Math. Modelling
Received Date: Revised Date: Accepted Date:
18 November 2013 20 December 2014 10 February 2015
Please cite this article as: A. Plankis, M. Lebsack, P.R. Heyliger, Elasticity-Based Beam Vibrations for Various Support Conditions, Appl. Math. Modelling (2015), doi: http://dx.doi.org/10.1016/j.apm.2015.02.023
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Elasticity-Based Beam Vibrations for Various Support Conditions Alivia Plankis, Michael Lebsack, and Paul R. Heyliger Department of Civil and Environmental Engineering, Colorado State University, Fort Collins, CO 80523, 970-491-6685, [email protected]
Abstract The accuracy of common beam theories in determining resonant frequencies and mode shapes for various support conditions is investigated in this paper. This study analyzed the influence of three assumptions employed in the Euler-Bernoulli, Rayleigh, and Timoshenko beam theories by comparing their results to a three-dimensional, elasticity-based approximation for three different support cases: fixed-free, fixed-fixed, and simply-supported. Each of the theories of interest were applied to both isotropic and orthotropic beams of varying length and compared with elasticity-based solutions to study the influence of slenderness ratio and anisotropic properties. The discrepancies caused by each of these effects are discussed and suggestions for the applicability of the common beam theories are provided.
1. Introduction The vibrating beam is a well studied mechanics problem with a multitude of practical applications in engineering [1-17]. These applications including the beam as standard structural elements [18] to more recent applications of beam configurations as sensors and actuators [19]. Problems involving a vibrating beam are typically approached using one of the common beam representations: the Euler-Bernoulli, Rayleigh, and Timoshenko theories. The use of these theories provides reasonable results for many engineering applications; however, all three rely upon restrictive kinematic assumptions. A main component that is missing from the collection of research on the vibrating beam is an investigation into when these assumptions begin to limit accuracy when compared with results obtained using elasticity theory with less restrictive constraints on the displacement fields. In order to confidently use these beam theories, their limitations should be well understood. Preprint submitted to Elsevier
March 4, 2015
In this paper a fourth method, which can be referred to as three-dimensional Ritz elasticity (3DRE) beam theory, to analyze vibrational beam mechanics is introduced that better approximates a fully three-dimensional elasticity theory. This approach will be employed to study the accuracy of the common beam theories and the point at which they fail to be acceptable analysis tools. When studying the effectiveness of common beam theories, three different boundary condition cases will be evaluated: fixed-free, fixed-fixed, and simply supported. Another common support condition in beam theory is the propped cantilever, but the modes for this beam are the anti-symmetric modes for the fixed-fixed beam [18] and hence not specifically considered in this study. It is extremely common in engineering practice to use beam frequencies that have been tabulated by various researchers for each of these boundary conditions using the simplified Euler-Bernoulli theory. These are the most widely available results and are a function of only beam length, density, elastic modulus, and beam cross-section. The intent of this study is to quantify levels of accuracy as the beam becomes stocky and the ratio between the elastic and shear moduli increases. The primary contribution of this study is to provide quantitative limits of typically used beam theories compared with results from an elasticity model that does not possess the strict kinematic limitations inherent in onedimensional representations of flexural beam deformation. Initial guidelines are also presented related to the influence of elasticity-based imposition of beam supports as opposed to the same supports modeled using beam theories. 2. Governing Equations This section will present the beam theories which were studied including the equations which govern each of these models. In accordance with common practice for beam theory discussion, dimensionless variables will be employed for the geometric and vibrational parameters. Also, a consistent coordinate system will be used for all of the calculations where the x axis is parallel to the width of the beam, the y axis provides the vertical dimension of the beam, and the z axis is used for the axial dimension of the beam. The variables u, v, and w will be used to represent the displacement along each of these axes, respectively. These are shown in Figure 1. 2.1. Euler-Bernoulli Beam Theory The Euler-Bernoulli beam theory, also known as the classical beam theory, Euler beam theory, Bernoulli beam theory, or Bernoulli-Euler beam 2
theory, is the most commonly used theory. It is very simple and provides acceptable results for many engineering problems. This theory includes both the strain energy from bending and the kinetic energy from transverse displacement, but does not consider the effects of rotary inertia or shear displacement. Ignoring these two contributions leads to overestimates of the natural frequencies of a vibrating beam, especially for higher modes of vibration and less slender beams when these two factors are not negligible [18]. The governing differential equation of motion for the Euler-Bernoulli model is given by [18] ρA
∂ 2 v(z, t) ∂ 4 v(z, t) + = f (z, t). ∂t2 ∂z 4
(1)
where ρ is the density of the material and A is the cross-sectional area of the beam. Only free vibration is considered in this paper, so the forcing function, f (z, t), is set to zero. To simplify the analysis the principle of separation of variables will be used to isolate an equation for transverse displacement that is dependent on the axial direction only and not time [18]. This new equation is referred to as the spatial solution and it is of the form vEB (z) = C1 sin az + C2 cos az + C3 sinh az + C4 cosh az.
(2)
In this equation, the parameter a is the dimensionless wave number [18] and the EB subscript has been added to the transverse displacement to distinguish this variable from those of other beam theories. For the EulerBernoulli theory the wave number is given by a = ρAω 2 .
(3)
where ω is the natural frequency of the beam. Using the spatial solution above along with appropriate boundary conditions for each of the support cases, a system of four equations results. The solution of this system of equations provides the frequency equation for each specific case. 2.2. Rayleigh Beam Theory The Rayleigh beam theory includes a correction that allows for rotary inertia in the beam [1]. The addition of this term provides a slight improvement over the Euler-Bernoulli model by slightly reducing the overestimation of the natural frequencies. The Rayleigh model still does not incorporate the contribution of shear deformation, and thus does not entirely eliminate errors for high frequency results [18]. 3
With the addition of the kinetic energy term from rotation, the governing differential equation of motion found using Hamilton’s principle for the Rayleigh beam model becomes [18] ρA
∂ 2 v(z, t) ∂ 4 v(z, t) ∂ 4 v(z, t) + − ρI = f (z, t). ∂t2 ∂z 4 ∂z 2 ∂t2
(4)
Here I is the second moment of the area of the cross-section about the centroidal axis. Similar to the Euler-Bernoulli model, the function f (z, t) is set to zero to represent free vibration. By separating the time and spatial variables in the governing equation of motion, the spatial solution for the Rayleigh beam theory is found to be [18] vR (z) = D1 sin a ¯z + D2 cos a ¯z + D3 sinh ¯bz + D4 cosh ¯bz. (5) In this model there are two dimensionless wave numbers, and they are defined by the following expressions [18]:
a ¯= ¯b =
v u u ρIω 2 t
2
v u u ρIω 2 t
−
2
+
s
ρIω 2 2
2
+ ρAω 2
(6)
+
s
ρIω 2 2
2
+ ρAω 2
(7)
The above equations can again be combined with the proper boundary conditions to solve for the frequency equations corresponding to each of the support conditions which were studied [18]. 2.3. Timoshenko Beam Theory The Timoshenko beam theory, sometimes known as the first order shear deformation theory (FSDT), expands upon the Euler-Bernoulli and Rayleigh theories by incorporating the effects of both rotary inertia and shear deformation. To include the influence of the latter, a new variable is used in this model. This variable, known as the shear factor (k 0 ) is necessary to correct for the variation in shear stress over the cross section. In his calculations, Timoshenko found that the change due to the addition of shear was four times greater than the change from the inclusion of rotary inertia [2]. Because of this, the Timoshenko beam theory is more popular than the Rayleigh model. This model results in far more accurate frequency values for non-slender beams and high frequency modes than either of the other two options. 4
By using shear factors determined for isotropic or orthotropic parallelepipeds [18, 15], the elastic strain energy from shear can be included in the Timoshenko beam model. Since there are two unknown variables for this theory, the transverse displacement w and the cross section rotation α, the use of Hamilton’s principle produces two differential equations of motion which are [18] ∂ 2 v(z, t) ∂α(z, t) ∂ 2 v(z, t) − k 0 GA − ρA 2 ∂t ∂z 2 ∂z
!
= f (z, t)
(8)
∂ 2 α(z, t) ∂ 2 α(z, t) ∂v(z, t) − − k 0 GA − α(z, t) = 0. (9) 2 2 ∂t ∂z ∂z The kinematic variables for this theory are shown in Figure 2. The function f (z, t) is set to zero again because only free vibration is considered for the analysis. For these equations, when the method of separation of variables is applied it is found that two different cases need to be considered for the spatial solutions [18]. These two cases are when the frequency is less than a critical value, or if it is above this value. In Han andpco-workers work, it was shown that this critical frequency value is equal to k 0 GA/ρI and is referred to as ωc [18]. For the case when ω < ωc , the form of the spatial solutions for the transverse displacement and total section rotation are given by
ρI
"
vT (z) Ψ(z)
#
+
=
"
E1 F1
"
E3 F3
#
#
sin a ˜z +
sinh ˜bz +
"
"
E2 F2
E4 F4
#
#
cos a ˜z
cosh ˜bz.
(10)
The expressions for the two dimensionless wave numbers for this case are given by the following expressions [18]: a ˜=
˜b =
v u u t
v u u t
1 I+ 0 kG
1 − I+ 0 kG
ρω 2 + 2
s
ρω 2 + 2
I−
s
1 k0 G
2 2 4 ρ ω
1 I− 0 kG
4
+ ρAω 2
2 2 4 ρ ω
4
+ ρAω 2
(11)
(12)
Although it seems that there are now eight unknown constants to solve for the Timoshenko beam model, the Ei and Fi values are related to each other 5
and thus only four unknowns need to be determined for the full solution [18]. For the case when ω > ωc the equations are identical except that ˜b is replaced with i˜b. When the boundary conditions corresponding to each of the support cases are input into the above equations, the frequency equations for the Timoshenko model can be determined similar to the previous two models [18]. 2.4. Elasticity-Based Solutions The proposed elasticity theory will remove a few of the assumptions which are included in the three main theories. The problem of the transversely vibrating beam typically cannot be solved exactly, so solutions using Ritz-based approximations are utilized in this advanced method. This allows for the inclusion of the Poisson effect and the ability to calculate accurate frequencies for non-slender beams and even anisotropic materials. There is no need to represent any rotational variables in this theory, since it replaces displacement functions of a single variable with displacements that vary with (x,y,z) at every location in the beam. The formulation for the mode shape and frequency solutions for this theory differs significantly from the three one-dimensional theories previously discussed. Rather than using governing differential equations and spatial solutions, Ritz-based approximations are used to solve Hamilton’s principle and thus determine frequencies. When used for displacement the general form for the three Ritz-based, direction component approximations can be written as w(x, y, z, t) = φw o (x, y, z) +
N X
cj (t)φw j (x, y, z)
(13)
j=1
The above represents the displacement component in the axial (z) direction, but the representations for the displacements in the x and y directions are of a similar form. Here the cj represent unknown constants that depend on time. For the problems considered in this study, harmonic motion is assumed so that this constant along with those of the other displacement components are written as cj (t) cos ω t, where ω is the natural frequency of vibration. The φo terms represent the simplest functions that satisfy the essential boundary conditions for that displacement direction. In the study discussed in this paper, the initial boundary conditions will be assumed to be zero, with no initial displacement or velocity. The φo terms are therefore all equal to zero for the considered case. In these approximation equations, 6
φj represents a selected approximate function for each respective direction. These functions must satisfy general requirements of independence as outlined by Reddy [20]. By using a large number of these approximation terms for each displacement component, very accurate solutions can be determined for the mode shapes and frequencies of vibration for a beam. Further, unlike the three beam theories, the only limit on the kinematics in the displacement field is on the number of terms used in the approximation. There is no fixed polynomial limitation on any of the variables other than this. These Ritz-based approximations are substituted into the equation for Hamilton’s principle which is shown below for this full elasticity theory. 0=−
Z tZ
1 {σ1 δ1 + σ2 δ2 + σ3 δ3 + σ4 δ4 + σ5 δ5 + σ6 δ6 }dV dt + δ 2 0 V
Z tZ
ρ(u˙ 2 + v˙ 2 + w˙ 2 )dV dt(14) V
0
where u˙ = ∂u/∂t and the conventional notation for stress (σ11 = σ1 , σ23 = σ4 , etc.) has been used [12]. When the substitution is performed and harmonic motion is assumed, the problem is reduced to a generalized eigenvalue problem expressed as follows.
[M 11 ] 0 0 [K 11 ] [K 12 ] [K 13 ] {c} {c} 2 21 22 23 [M 22 ] 0 {d} (15) {d} =ω 0 [K ] [K ] [K ] 0 0 [M 33 ] {e} [K 31 ] [K 23 ] [K 33 ] {e}
The elements of the stiffness ([K]) and mass ([M ]) matrices are related to the Ritz approximations used and some basic material properties of the beam that is considered. The stiffness matrix is symmetric, so there are only six different equations for those elements. The mass matrix is diagonalized so there are only three different forms. All of these relationships are listed below.
11 Kij
=
∂φu ∂φuj ∂φu ∂φuj ∂φu ∂φuj C11 i + C55 i + C66 i ∂x ∂x ∂z ∂z ∂y ∂y
Z
V
12 Kij
=
13 Kij =
22 Kij
=
Z
V
Z
V
∂φu ∂φvj ∂φu ∂φvj + C66 i C12 i ∂x ∂y ∂y ∂x
!
Z
V
∂φu ∂φw ∂φu ∂φw j j C13 i + C55 i ∂x ∂z ∂z ∂x
!
!
21 dV = Kji
!
(16)
(17)
31 dV = Kji
∂φv ∂φvj ∂φv ∂φvj ∂φv ∂φvj + C44 i + C66 i C22 i ∂y ∂y ∂z ∂z ∂x ∂x 7
dV
(18) dV
(19)
23 Kij
33 Kij =
Z
V
=
Z
V
∂φv ∂φw ∂φv ∂φw j j + C44 i C23 i ∂y ∂z ∂z ∂y
!
32 dV = Kji
∂φw ∂φw ∂φw ∂φw ∂φw ∂φw j j j C33 i + C44 i + C55 i ∂z ∂z ∂y ∂y ∂x ∂x
!
dV
(20)
(21)
Z
ρφui φuj dV
(22)
Mij22 =
Z
ρφvi φvj dV
(23)
Mij33 =
Z
w ρφw i φj dV
(24)
Mij11
=
V
V
V
The general form for the Ritz approximation functions will change for each of the different support conditions that will be considered. The primary base for all of the functions employed are polynomial-based, because these functions provide simplicity when evaluating integrals over a parallelepiped. Other functions are used as supplements when appropriate. As is the case for the class of method used in this study, in which global approximation functions are used to describe the displacements, and related methods such as the finite element method, it is critical to determine the number of terms used in the solution series that provide accurate results. In the related method of resonant ultrasound, Migliori and Sarrao [21] discuss the number of terms necessary to provide good agreement with experiment for solids that have three dimensions that are relatively close in magnitude. In the case of the beam, in which one dimension is much larger than the other two, this usually has to be determined by convergence studies on the actual geometry. This procedure is described below. There is a key difference between the support conditions as usually interpreted from beam theories as they are applied to elasticity-based theories including the 3DRE theory used in this study. Beam theory support conditions are usually applied to constraints of either displacement or force/moment resultants along a neutral surface. Hence, the displacement variables or their derivatives are specified at one specific location of the axial coordinate z. In the case of 3DRE, it is an entire surface that needs to be constrained in terms of either displacements or tractions. For example, a free end under beam theory would be associated with zero vertical force and zero resultant
8
out-of-plane moment. In 3DRE, these are replaced by specifying a tractionfree surface at the free end. This difference in how support conditions are satisfied have other implications as will be seen in the results that follow. When any combination of values for j, k, and l are considered, these power functions can represent any type of vibrational mode. If all possible combinations are evaluated, the results will include flexural vibration, shear vibration, and torsional vibration modes about all three axes. For the purposes of the investigation performed for this paper, only the transverse flexural modes are of interest. All other displacement patterns can be removed from consideration. To simplify and reduce the calculations required, group theory is used. This process splits all of the possible Ritz functions into eight different symmetry groups [22, 23]. Parallelepipeds contain three symmetry planes which intersect each other at right angles. Using these symmetry planes, the possible displacement patterns can be separated by considering specific combinations of odd or even behavior in (x,y,z) for each direction about these planes. For example, half of the groups follow the pattern that the u displacement is odd in x, even in y, and even in z. Therefore only odd integers will be considered for j and only even integers for k and l. The v and w deformation patterns can be split similarly, and through this certain modes of vibration can be isolated. Therefore, for the calculations performed for this paper, only two groups are considered, each of which represent the transverse flexural vibration modes [22]. This isolation of the flexural modes was possible for all three support conditions that were studied. For the cantilever beam case, identical approximations can be used for u, v, and w since the boundary constraints (u=v=w=0 at the fixed support) are identical for all three direction components. The approximations used are given by φi = xj y k z l l > 0. (25) For this support case, the number of Ritz terms used to express each of the displacement components is adjusted by summing the coefficients of the powers in x, y, and z such that j + k + l ≤ 16, which provides very accurate frequency results. This number is obtained by successively increasing this term (beginning at 8 terms) until there is no change in the first five significant figures of the results presented for the majority of modes studied for any beam with a clamped boundary condition. For the fixed-fixed beam case, the same approximation form can again be used for all three direction components, u, v, and w for the fixed-fixed beam case since the boundary conditions are the same for all three directions 9
at both of the fixed ends. The approximations in this case were of the form φi = xj y k z l (z − L).
(26)
The Ritz-based approximations for a simply-supported beam are significantly different than for the other two support cases considered, because the constraints are not the same for all three direction components for a hinged end. Therefore, the same general form of approximation terms cannot be employed for all three directions. Another considerable difference for the simply-supported case comes from the inability to satisfy the necessary boundary conditions through a power series. In order to satisfy these conditions and still have the ability to simply evaluate the functions over a parallelepiped, trigonometric functions are utilized for the axial dimension. Sine functions are used for the transverse and out-of-plane displacements, and cosine terms will be employed for the axial displacement. The type of trigonometric function that is used for each displacement component is determined by studying the displacement patterns from a finite element analysis of simply-supported beam. With all of these considerations, the form of the approximations are as follows. lπz L lπz φvi = xj y k sin L lπz j k φw = x y cos i L φui = xj y k sin
(27) (28) (29)
This class of approximations has a fairly long history in plate and beam mechanics, and in fact was used by Pagano in his development of exact solutions for simply supported composite plates [24]. The constraint on the number of terms was set such that j + k + l ≤ 14. A square and solid cross-section is used in the results that follow. This is certainly not the only configuration that could be considered using the elasticity beam model, and in fact modern beams are more likely to be thinwalled or non-rectangular. The solid square is used primarily to eliminate additional variables based on the beam cross-section and also because the element matrices used in 3DRE beam theory are easily computed in closed form for this section. The conclusions drawn at the end of this study should be viewed with this restriction in mind.
10
3. Results and Discussion The resonant frequencies for the lowest five modes were the primary results that were investigated in this vibrational mechanics study. These values were found for all combinations of the various lengths, materials, and support conditions considered. To demonstrate this behavior, we consider a beam with a square cross-section of 0.01 x 0.01 meters. The elastic constants for the isotropic case are taken to be those of steel, with C11 = 269.231 GPa and C44 = 76.923 GPa along with a density of 7830 kg/m3 . The elastic constants for the orthotropic elastic constants of graphite-magnesium give as C11 = C22 = 28.18 (all GPa), C33 = 174.3, C12 = 10.67, C13 = C23 = 12.20, C66 = 8.76 with a density of 1738 kg/m3 [25]. All of the calculated frequencies are presented in tabular form. The results for a cantilever beam are listed in Table 1. For a fixed-fixed support case, the results are provided in Table 2. Lastly, the frequencies for a simply-supported beam case are given in Table 3. There are two columns presented for the 3DRE model: one for the full set of elastic constants, and the second where all values of Poisson ratio are set equal to zero. For a few of the cases involving a fixed support, the higher mode frequencies from the Timoshenko beam model were not able to be calculated. Even without some of these values trends were still able to be determined and conclusions drawn. From the values found the influence of slenderness, anisotropy, and the Poisson effect could be investigated. The effect of each of these factors for the different support conditions are discussed in the remainder of this section. 3.1. Effects of Slenderness The slenderness ratio of a beam is dimensionless and is defined using the symbol S = √L . Its value has a significant influence on the accuracy I/A
of frequency calculations for two of the common beam theories. As the slenderness ratio decreases to a point where the solid is closer to a cube than a typical beam, the error between the elasticity solution and both the Euler-Bernoulli and Rayleigh results dramatically increases. For the Timoshenko beam model, the length of a beam has a much smaller effect on the percent error from the full elasticity theory. This is especially true for the fundamental mode of an isotropic beam. The support case for a beam affects the level of accuracy for each beam theory as the slenderness ratio changes, but these general trends are consistent throughout. For the cantilever beam and fixed-fixed beam conditions, all three beam theories approach a similar error as the slenderness increases. Here we use the word “error” to indicate the difference in frequencies obtained using each 11
Table 1: Frequencies of a Cantilevered Beam L
2 (S=6.928)
5 (S=17.32)
10 (S=34.64)
20 (S=69.28)
40 (S=138.56)
100 (S=346.41)
Mode
E-B
Rayleigh
1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5
1282 8037 22500 44100 72900 205.2 1286 3601 7056 11660 51.30 321.5 900.2 1764 2916 12.82 80.37 225.0 441.0 729.0 3.206 20.09 56.26 110.3 182.2 0.5130 3.215 9.002 17.64 29.16
1224 6205 14100 22570 31180 203.6 1221 3211 5812 8812 51.20 317.2 872.5 1667 2673 12.82 80.10 223.2 434.6 712.2 3.206 20.08 56.15 109.8 181.2 0.5130 3.214 8.998 17.63 29.13
Isotropic Timoshenko 1093 4147 8676 11250 14600 199.1 1074 2560 4255 6056 50.90 305.3 803.5 1459 2222 12.80 79.29 218.1 416.8 668.7 3.205 20.02 55.81 108.6 178.0 0.5129 3.213 8.990 17.60 29.05
3DRE (ν = 0)
3DRE
E-B
Rayleigh
1100 4285 8899 11310 13160 199.3 1089 2635 4447 6277 50.92 307.1 815.6 1501 2383 12.80 79.57 220.1 425.5 712.1 3.205 20.08 56.24 110.6 189.2 0.5131 3.222 9.056 17.90 30.85
1149 4314 8919 11320 13400 209.1 1121 2682 4501 6313 53.45 318.1 836.1 1526 2408 13.44 82.62 226.4 433.7 719.1 3.365 20.86 57.91 112.8 191.0 0.5386 3.348 9.328 18.27 31.14
2484 15570 43600 85440 141200 397.5 2491 6976 13670 22600 99.38 622.8 1744 3418 5649 24.84 155.7 436.0 854.4 1412 6.211 38.93 109.0 213.6 353.1 0.9938 6.228 17.44 34.18 56.49
2372 12020 27330 43730 60420 394.5 2366 6222 11260 17070 99.19 614.6 1690 3231 5179 24.83 155.2 432.5 841.9 1380 6.211 38.89 108.8 212.8 351.0 0.9938 6.228 17.43 34.15 56.44
Orthotropic Timoshenko 3DRE (ν = 0) 1704 5434 365.8 1659 3604 5571 7557 97.20 542.8 1325 2241 3220 24.71 148.9 400.1 737.0 1137 6.203 38.55 106.5 204.9 330.9 0.9936 6.219 17.37 33.94 55.87
1729 5548 9726 10820 11670 366.8 1708 3708 5677 7700 97.28 549.3 1360 2336 3337 24.72 150.7 405.5 757.3 1220 6.204 38.67 107.5 209.0 352.5 0.9939 6.236 17.51 34.54 59.36
3DRE 1739 5557 9802 10870 11720 369.5 1713 3715 5689 7714 98.04 551.9 1364 2342 3343 24.91 151.5 407.1 759.5 1222 6.254 38.90 107.9 209.6 353.1 1.002 6.272 17.58 34.63 59.44
Table 2: Frequencies of a Fixed-Fixed Beam L
2 (S=6.928)
5 (S=17.32)
10 (S=34.64)
20 (S=69.28)
40 (S=138.56)
100 (S=346.41)
Mode
E-B
Rayleigh
Isotropic Timoshenko
3DRE (ν = 0)
3DRE
E-B
Rayleigh
1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5
8160 22490 44100 7289 108900 1306 3599 7056 11660 17420 326.4 899.7 1764 2916 4356 81.60 224.9 441.0 729.0 1089 20.40 56.23 110.3 182.2 272.2 3.264 8.997 17.64 29.16 43.56
7277 16030 25230 34160 42890 1280 3351 6118 9302 12710 324.8 883.0 1695 2727 3943 81.50 223.9 436.5 716.3 1060 20.39 56.17 110.0 181.4 270.4 3.264 8.996 17.63 29.14 43.51
3947 7711 1054 2416 4029 5760 7567 306.3 785.8 1422 2160 2968 80.25 216.6 413.4 661.9 954.2 20.31 55.69 108.4 177.5 262.3 3.262 8.984 17.59 29.03 43.29
4068 7952 12670 17870 23040 1070 2471 4157 5962 7873 310.1 798.1 1449 2213 3068 81.19 219.5 419.2 674.3 975.4 20.57 56.45 109.8 180.7 267.4 3.323 9.150 17.96 29.78 44.75
4157 8016 12740 18070 23230 1106 2527 4225 6039 7937 321.5 823.1 1483 2254 3117 84.22 227.3 430.8 690.6 995.9 21.33 58.52 113.0 185.5 273.6 3.435 9.475 18.44 30.58 45.75
15810 43580 85440 141200 211000 2530 6973 13670 22600 33760 632.4 1743 3418 5649 8439 158.1 435.8 854.4 1412 2110 39.53 109.0 213.6 353.1 527.5 6.324 17.43 34.18 56.49 84.39
14100 31060 48890 66190 83110 2479 6492 11850 18020 24620 629.2 1711 3285 5284 7640 157.9 433.7 845.7 1388 2054 39.51 108.8 213.0 351.5 523.9 6.324 17.43 34.16 56.45 84.30
12
Orthotropic Timoshenko 3DRE (ν = 0) 4500 8885 1508 3136 5015 6938 8885 524.6 1231 2081 3002 3963 149.8 389.4 713.3 1096 1520 38.98 105.7 202.8 326.8 474.3 6.310 17.35 33.88 55.75 82.83
4650 9193 10570 17040 21820 1537 3222 5142 7136 9139 531.3 1253 2135 3089 4120 151.6 394.4 725.2 1119 1565 39.43 107.0 205.4 332.5 484.0 6.394 17.59 34.36 56.80 84.53
3DRE 4661 9211 10680 16720 21770 1544 3230 5155 7151 9156 534.2 1258 2143 3099 4129 152.4 396.7 728.1 1123 1570 39.66 107.6 206.3 333.8 485.7 6.431 17.69 34.51 57.02 84.83
Table 3: Frequencies of a Simply-Supported Beam L
2 (S=6.928)
5 (S=17.32)
10 (S=34.64)
20 (S=69.28)
40 (S=138.56)
100 (S=346.41)
Mode
E-B
Rayleigh
1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5
3600 14400 32400 57600 90000 576.0 2304 5184 9216 14400 144.0 576.0 1296 2304 3600 36.00 144.0 324.0 576.0 900.0 9.000 36.00 81.00 144.0 225.0 1.440 5.760 12.96 23.04 36.00
3279 10670 19190 27810 36320 566.7 2166 4553 7459 10670 143.4 566.7 1250 2166 3279 35.96 143.4 321.0 566.7 877.7 8.997 35.96 80.81 143.4 223.6 1.440 5.759 12.95 23.02 35.96
Isotropic Timoshenko 2713 7394 12230 17010 21730 541.7 1882 3601 5471 7394 141.7 541.7 1142 1882 2713 35.85 141.7 312.6 541.7 820.9 8.990 35.85 80.25 141.7 219.4 1.440 5.76 12.94 22.98 35.85
3DRE (ν = 0)
3DRE
E-B
Rayleigh
2705 7367 12200 17000 21750 541.2 1878 3589 5451 7367 141.6 541.2 1140 1878 2705 35.85 141.6 312.4 541.2 819.9 8.990 35.85 80.24 141.6 219.3 1.440 5.756 12.94 23.00 35.85
2689 7287 12060 16830 21580 540.4 1869 3562 5398 7287 141.6 540.4 1137 1869 2689 35.84 141.6 312.2 540.4 818.1 8.990 35.84 80.22 141.6 219.2 1.440 5.756 12.94 22.98 35.84
6975 27900 62770 111600 174400 1116 4464 10040 17860 27900 279.0 1116 2511 4464 6975 69.75 279.0 627.7 1116 1744 17.44 69.75 156.9 279.0 435.9 2.790 11.16 25.11 44.64 69.75
6352 20670 37180 53880 70360 1098 4196 8822 14450 20670 277.8 1098 2423 4196 6352 69.67 277.8 622.0 1098 1700 17.43 69.67 156.6 277.8 433.1 2.790 11.16 25.10 44.61 69.67
Orthotropic Timoshenko 3DRE (ν = 0) 3854 8857 13710 18490 21010 950.7 2841 4869 6879 8857 266.5 950.7 1853 2841 3854 68.92 266.5 569.7 950.7 1385 17.38 68.92 152.8 266.5 406.6 2.788 11.14 25.00 44.30 68.92
of the beam theories in comparison with elasticity theory. The existence of a single fixed support for a beam leads to often small but still noticeable differences in frequency for even very slender beams. This is caused in part by neglecting the Poisson effect and will be discussed in Section 3.3. Although the frequency results from the beam theories do not approach the full elasticity solution, they still demonstrate the slenderness trends discussed above. As the slenderness value decreases, the results for both the Euler-Bernoulli and Rayleigh models begin to deviate from the asymptotic error percentage at an exponential rate as can be seen in Figures 3 and 5. The error values from the Rayleigh beam model are less than those from Euler-Bernoulli model calculations, as was expected, but the slenderness ratio of a beam still has a meaningful influence on the accuracy of the Rayleigh model. This effect is significantly greater for higher modes of vibration for both beam theories. While the errors for the fundamental mode of a 1x1x2 cantilever beam are 11.6% and 6.5% for Euler-Bernoulli and Rayleigh, respectively, these errors grow to 444% and 133% for the fifth mode of vibration. The errors for a fixed-fixed beam are even larger for higher modes of vibration. The error values for the fifth mode are shown in Figures 4 and 6. The small
13
3824 8865 10450 16620 20930 946.4 2818 4836 6856 8865 266.1 946.4 1840 2818 3824 68.89 266.1 567.9 946.4 1377 17.38 68.89 152.7 266.1 405.7 2.788 11.14 25.00 44.28 68.89
3DRE 3813 8843 10570 16100 21040 945.1 2810 4822 6837 8844 266.0 945.1 1836 2810 3813 68.88 266.0 567.4 945.1 1374 17.38 68.88 152.7 266.0 405.4 2.788 11.14 24.99 44.28 68.88
effect that slenderness has upon the Timoshenko frequency results can also be seen in these figures. While the other two beam theories diverge exponentially as the beam length falls below ten, the Timoshenko model gives results that agree extremely well with elasticity theory throughout the entire range of slenderness ratios considered. The error increases slightly more for very stout beams and higher modes of vibration, but this increase is small in comparison to the other two theories. The error values for a simply-supported beam as the beam length varies are shown in Figures 7 and 8. The main difference for this support case is that the frequency values from the three beam theories approach the full elasticity solutions rather than a common error value. The error increases significantly for the Euler-Bernoulli and Rayleigh beam theories as a beam becomes less slender, while the influence is minimal for the Timoshenko model. Slenderness is also still a greater factor for higher modes of vibration than for the fundamental mode. All of these results follow expected trends. As the slenderness of a beam decreases, the displacement pattern is known to deviate from the kinematic model that restricts the axial displacement to be a linear function in thickness assumed by each of the three beam theories. For example, in the extreme case where the beam length approaches the width or becomes less than the width, the deformation pattern becomes more similar to that of a plate than a beam. It is therefore clear that the one-dimensional displacement patterns of the common beam theories cannot be applied to very short and thick beams. Three-dimensional displacement patterns were determined using the elasticity model. These modal patterns demonstrate obvious warping of the beam surfaces through the width and height for stocky beams. When a beam is slender, the displacement is dominant in only one dimension as the three common beam theories assume, but this deteriorates rapidly as the slenderness decreases. The first five of the three-dimensional elasticity mode shapes were plotted for both a slender beam (1x1x40) and a stout beam (1x1x2) for each of the support cases. Figures 9, 10, 11, 12, 13, and 14 show these mode shapes for the cantilever, clamped-clamped, and simply-supported cases, respectively. The deformation pattern deviates from beam theory expectations in Figure 10 where the cross-sectional plane is no longer plane throughout displacement; this is especially noticeable on the free face of the fifth mode shape. In the fourth and fifth mode shape for both the fixed-fixed and simply-supported cases, the Poisson effect begins to have a significant influence for short beams. In these plots, obvious warping through the width of the beam occurs due to the shrinking and expansion 14
Figure 1: The coordinate axes (x,y,z) for the general beam and the displacement components (u,v,w) in the respective directions.
15
influences of the Poisson ratio. The displacement patterns in these cases can no longer be accurately modeled through a one-dimensional plot of the centerline since the deformation of the outer surfaces is significantly different than what occurs at the centerline of the beam. For most beams, the lower modes of vibration are nearly always flexural, but it was found that for very non-slender beams this was not the case. For the very short beam case, the lowest three modes followed flexural displacements for all three support conditions, but the fourth and fifth lowest frequencies corresponded to very different deformation types. For the fixed-fixed case, the fifth flexural mode did not appear until the 14th lowest mode of vibration. This finding provides another strong reason for more accurate frequency and mode shape calculations when dealing with nonslender beams since non-flexural modes of vibration may become dominant far earlier than anticipated. 3.2. Effects of Anisotropy The accuracy of frequency values for the orthotropic material, graphitemagnesium, decreases even more rapidly than for steel as the slenderness decreases. Figures 15, 16, and 17 display this trend. For example, the error is more than four times greater for the orthotropic fixed-fixed beam than the isotropic case. The error increase for an anisotropic beam becomes insignificant for very slender beams, displaying that the slenderness ratio provides the dominant influence on frequency results. Anisotropic material properties can exacerbate the loss of accuracy at low slenderness ratios, especially for beam theories without shear deformation, because the shear modulus in the transverse-axial plane can be an order of magnitude or more smaller than the longitudinal modulus. The Timoshenko model was exceptionally adaptable to the orthotropic material. The frequency errors were similar in magnitude for both the isotropic and orthotropic material as can be noted from Figures 15, 16, and 17. The shear coefficient for both the isotropic and orthotropic materials were calculated using the methodology of Puchegger et al. [15]. 3.3. Effects of Poisson Ratio There is a significant difference between the Ritz-based elasticity method and the three beam theories for even very slender beams when at least one support in a beam is fixed. One possible reason for this difference is because of the restraint of the Poisson effect at the fixed support. When considering a three-dimensional model, a perfectly fixed support completely restricts the Poisson effect. For example, in a cantilevered beam, the largest forces 16
would exist at the fixed support and thus the influence of the Poisson effect would cause the beam to shrink or expand through the width to counteract the large axial compression and tension forces. A perfectly fixed end does not allow this. The displacements and forces cannot act as is expected by typical beam theories. With a hinged or free support, there is no restriction of the Poisson effect and the beam is free to displace freely since there are no restraining forces transverse to the long direction of the beam. Therefore, ignoring the Poisson effect in a beam with only free or hinged supports would have minimal effect on the accuracy of the frequency results. To investigate this explanation, the more accurate Ritz-based elasticity program was used to determine frequency results with both the normal material properties as well as elastic stiffness values found using ν = 0. In the latter case, the components of the elastic stiffness tensor were calculated directly as the the inverse of the diagonal compliances with C12 = C13 = C23 = 0 and a shear modulus that was half of the elastic modulus for the steel. For the orthotropic case, a similar procedure but the shear moduli were treated as independent constants n the latter case, the components of the elastic stiffness tensor were calculated directly as the the inverse of the diagonal compliances with C12 = C13 = C23 = 0 and a shear modulus that was half of the elastic modulus for the steel. For the orthotropic boron-aluminum, a similar procedure was used (for example, C33 = 1/E3 , where E3 is the elastic modulus in the direction of the boron fibers) but the shear moduli were treated as independent constants and hence were unaffected. This allowed for the influence of the Poisson effect to be isolated from the other factors that cause errors in the typical beam theories. The results of this investigation are shown in Figure 18, and indicate that a consistent error exists between the models with and without the Poisson effect that is independent of beam slenderness. The error in the fundamental mode is near 4.7% for a cantilevered beam, 3.4% for a fixedfixed beam, but well under one percent for a simply-supported beam. The percent difference does vary somewhat with the length of the beam, but the change is minimal. The result for the simply-supported beam case followed expectations and showed that neglecting the Poisson effect has a minimal influence for this specific support condition. Although the impact of neglecting the Poisson effect seems to be independent of slenderness, the error is related to the mode of vibration. The error decreased for higher modes of vibration where there is at least one fixed support. For the simply-supported case, the opposite pattern occurs. The influence of the Poisson effect is still minor for all modes of vibration, but the error does tend to increase for higher modes of vibration. Since 17
many of these errors are relatively small, some of these differences may be attributable in part to the decrease in relative accuracy of the Ritz method for higher modes. These results are shown for a beam with a slenderness ratio of 69 in Figure 19. It is possible that the change along beam length of the sign of the curvature balances out the restraint at the fixed end since the higher modes have more sign changes (and hence a Poisson displacement that also changes sign along the beam length). The frequency error associated with non-zero Poisson ratio also appears to be connected with the material properties of a beam when a fixed support is present, and in fact this may give a hint as to the source of this behavior. The percent difference between the zero Poisson ratio and full elasticity results is far less for the orthotropic material studied than for the isotropic material. The error is approximately six times greater for a steel beam than for beam of graphite-magnesium if there is a fixed support. The probable reason for this behavior lies in the magnitude of the values of the Poisson ratio. In the isotropic case, the values for the steel for the two axial-transverse Poisson ratio are νzx = νzy = 0.3. However, for the graphite-magnesium, these values are νzx = νzy = 0.046. Such small values of Poisson ratio are somewhat typical for orthotropic solids whose longitudinal modulus is so much larger than its transverse modulus, whereas for isotropic materials these moduli are identical. Hence, there is a much stronger Poisson effect being restrained at the fixed supports for the isotropic beam, leading to larger differences in frequency. The results also show that this error difference does not exist for the simply-supported beam. For this support case, the error values with or without the Poisson effect are virtually identical. 4. Conclusions The results in this work are somewhat limited in that only solid and hollow square cross-sections were used to determine initial differences in frequency between the various theories. More extensive study of the influence of the beam cross-section are likely to provide additional useful results. Based on the geometries, constitutive laws, and kinematic assumptions made during the course of this study, the following conclusions can be suggested: • The Euler-Bernoulli and Rayleigh theories give frequencies that are significantly different from the 3DRE values for beams with clamped support. The differences between the Timoshenko and 3DRE frequencies for this type of support are noticeable but much smaller. One possible reason for the former is the lack of representation of the Poisson 18
restraint at these supports. The frequency differences with elasticity models are 4% or greater for the entire range of slenderness values studied. • When the influence of the Poisson ratio was isolated, it was found that neglecting this effect leads to frequency differences greater than 3% for beams with s > 300 when a fixed support exists. • When the Euler-Bernoulli or Rayleigh theories are applied to an orthotropic material, the frequency differences for s < 50 more than double in comparison to the differences for an isotropic material. This trend was found for all three support cases. • For higher modes of vibration and a slenderness ratio of 138, the frequency differences are above 2% for Euler-Bernoulli and Rayleigh, mildly contradicting the assumption that these theories are adequate for beams with s > 100. This loss of accuracy for higher modes was consistent for all three support conditions studied. • When the slenderness ratio falls below 50 the differences in frequency results when compared with elasticity theory increases rapidly for both the Euler-Bernoulli and Rayleigh beam models, with differences up to 90%. • For all three support cases studied, the frequency differences for the Timoshenko model are small when compared with the 3DRE results for zero Poisson ratio and have little relationship to slenderness or anisotropic properties. When compared with the full set of elastic constants with a fixed support, the Timoshenko results are generally lower than the elasticity solution - indicating that there may be errors in both kinetic and elastic energy terms that combine to yield frequencies under the elasticity values. • For the simply-supported case, the Poisson effect has a minimal influence. The Timoshenko model gives excellent results for all slenderness ratios considered. • For slenderness ratios over 40, the differences in frequency between the Euler-Bernoulli and Rayleigh theories are below 5 percent for all materials and support conditions. This value could be considered a reasonable upper bound for the applicability of these two beam theories. 19
Based on these results, applications of beam theory frequency results for clamped support conditions may be problematic regardless of slenderness ratio or material type. For a simply-supported beam, the Timoshenko beam model performs extremely well. The accuracy of frequency results only becomes poor for very low slenderness values combined with high modes of vibration. The Euler-Bernoulli and Rayleigh beam theories deteriorate in accuracy at much higher slenderness ratios. A common standard is that Euler-Bernoulli is acceptable for slenderness ratios greater than 100, but this standard is rarely quantified. It was found here that for a slenderness ratio of 138, the differences in frequency with 3DRE results for higher modes and orthotropic materials is near 2% for Euler-Bernoulli and above 1% for Rayleigh. The Timoshenko beam model is the clear choice for a simplysupported beam unless the slenderness ratio is extremely high. For fixedfree or fixed-fixed beams, it may be the case that more complex analysis is required to calculate accurate resonant frequencies and mode shapes. 5. Acknowledgements The funding for this research was provided by a grant from Mountain Plains Consortium. References [1] Timoshenko SP. History of Strength of Materials. New York: McGrawHill Book Company, Inc.; 1953. [2] Timoshenko SP. On the correction for shear of the differential equation for transverse vibrations of bars of uniform cross-section. Philos Mag 1921;41:744-746. [3] Timoshenko SP. On the transverse vibrations of bars of uniform crosssection. Philos Mag 1922;125:125-131. [4] Traill-Nash RW, Collar AR. The effects of shear flexibility and rotatory inertia on the bending vibrations of beams. Q J Mech Appl Math 1953;6:186-213. [5] Levinson M. A new rectangular beam theory. J Sound Vibrat 1981;74:81-87. [6] Anderson RA. Flexural vibration of uniform beams according to the Timoshenko theory. J Appl Mech 1953;75:504-510. 20
[7] Dolph CL. On the Timoshenko theory of transverse beam vibrations. Q Appl Math 1954;12:175-187. [8] Stephen NG. On the variation of Timoshenko’s shear coefficient with frequency. J Appl Mech 1978;45:695-697. [9] Huang TC. The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions. J Appl Mech 1961;28:579-584. [10] Thomas J, Abbas BAH. Finite element model for dynamic analysis of Timoshenko beam. J Sound Vibrat 1975;41:291-299. [11] Schramm U, Kitis L, Kan W, Pilkey WD. On the shear deformation coefficient in beam theory. Fin Elem Anal Des 1994;16:141-162. [12] Heyliger PR, Jilani A. The free vibrations of inhomogeneous elastic cylinders and spheres. Int. J. Solids Structures 1992;29(22):2689-2708. [13] King JL. The free transverse vibrations of anisotropic beams. J Sound Vibrat 1985;98:575-585. [14] Levinson M. Further results of a new beam theory. J Sound Vibrat 1981;77:440-444. [15] Puchegger S, Loidl D, Kromp K, Peterlik H. Hutchinson’s shear coefficient for anisotropic beams. J Sound Vibrat 2003;266:207-216. [16] Cowper, GR. The shear coefficient in Timoshenko’s beam theory. J Appl Mech 1966; 33:335-340. [17] Bank, LC. Shear coefficients for thin-walled composite beams. Composite Structures 1987; 8: 47-61. [18] Han SM, Benaroya H, Wei T. Dynamics of transversely vibrating beams using four engineering beam theories. J Sound Vibrat 1999;225(5):935988. [19] Plankis, A and Heyliger, P. Off-grid MEMS sensors configurations for transportation applications. MPC Report MPC-13-257, Mountain Plains Consortium, 2013. [20] Reddy, JN. Energy Principles and Variational Methods in Mechanics, Wiley, New York, 2002. 21
[21] Migliori, A and Sarrao, JL. Resonant ultrasound spectroscopy, Wiley, New York, 1997. [22] Ohno I. Free vibration of a rectangular parallelepiped crystal and its application to determination of elastic constants of orthorhombic crystals. J Phys Earth 1976;24:355-379. [23] Mochizuki, E. Application of group theory to free oscillations of an anisotropic rectangular parallelepiped. J. Phys. Earth 1987; 35: 159170. [24] Pagano, N. J. Exact solutions for rectangular bidirectional composite and sandwich plates. J. Comp. Mater. 1970; 4: 20-34. [25] Ledbetter HM, Datta SK, Kyono T. Elastic constants of a graphitemagnesium composite. J Appl Phys 1989;65(9):3411-3416.
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Figure 2: An undeformed beam (top) and the deformed beam (bottom) as viewed from the side are shown to highlight the rotational kinematic variable α for the Timoshenko theory and the difference with the Euler-Bernoulli theory.
23
Percent Error from Full Elasticity Solution, %
12 Euler−Bernoulli Rayleigh Timoshenko
10 8 6 4 2 0 −2 −4 −6
0
10
20
30
40 50 60 Beam Length
70
80
90
100
Figure 3: Frequency Error for the First Mode of a Cantilevered Beam
Percent Error from Full Elasticity Solution, %
450 Euler−Bernoulli Rayleigh Timoshenko
400 350 300 250 200 150 100 50 0 −50
0
10
20
30
40 50 60 Beam Length
70
80
90
100
Figure 4: Frequency Error for the Fifth Mode of a Cantilevered Beam
24
Percent Error from Full Elasticity Solution, %
100 Euler−Bernoulli Rayleigh Timoshenko
80
60
40
20
0
−20
0
10
20
30
40 50 60 Beam Length
70
80
90
100
Figure 5: Frequency Error for the First Mode of a Fixed-Fixed Beam
Percent Error from Full Elasticity Soltuion, %
400 Euler−Bernoulli Rayleigh Timoshenko
350 300 250 200 150 100 50 0 −50 0
10
20
30
40 50 60 Beam Length
70
80
90
100
Figure 6: Frequency Error for the Fifth Mode of a Fixed-Fixed Beam
25
Percent Error from Full Elasticity Solution, %
35 Euler−Bernoulli Rayleigh Timoshenko
30
25
20
15
10
5
0
0
10
20
30
40 50 60 Beam Length
70
80
90
100
Figure 7: Frequency Error for the First Mode of a Simply-Supported Beam
Percent Error from Full Elasticity Soltuion, %
350 Euler−Bernoulli Rayleigh Timoshenko
300
250
200
150
100
50
0
0
10
20
30
40 50 60 Beam Length
70
80
90
100
Figure 8: Frequency Error for the Fifth Mode of a Simply-Supported Beam
26
Figure 9: First Five Mode Shapes for a 1x1x40 Isotropic Fixed-Free Beam
27
Figure 10: First Five Mode Shapes for a 1x1x2 Isotropic Fixed-Free Beam
Figure 11: First Five Mode Shapes for a 1x1x40 Isotropic Fixed-Fixed Beam
28
Figure 12: First Five Mode Shapes for a 1x1x2 Isotropic Fixed-Fixed Beam
Figure 13: First Five Mode Shapes for a 1x1x40 Isotropic Simply-Supported Beam
29
Figure 14: First Five Mode Shapes for a 1x1x2 Isotropic Simply-Supported Beam
Percent Error from Full Elasticity Solution, %
45 E−B Iso E−B Ortho Rayleigh Iso Rayleigh Ortho Timo Iso Timo Ortho
40 35 30 25 20 15 10 5 0 −5
0
10
20
30
40 50 60 Beam Length
70
80
90
100
Figure 15: Error Comparison Between Isotropic and Orthotropic Materials for a Cantilevered Beam
Percent Error from Full Elasticity Solution, %
250 E−B Iso E−B Ortho Rayleigh Iso Rayleigh Ortho Timo Iso Timo Ortho
200
150
100
50
0
−50
0
10
20
30
40 50 60 Beam Length
70
80
90
100
Figure 16: Error Comparison Between Isotropic and Orthotropic Materials for a FixedFixed Beam
30
Percent Error from Full Elasticity Solution, %
90 E−B Iso E−B Ortho Rayleigh Iso Rayleigh Ortho Timo Iso Timo Ortho
80 70 60 50 40 30 20 10 0
0
10
20
30
40 50 60 Beam Length
70
80
90
100
Figure 17: Error Comparison Between Isotropic and Orthotropic Materials for a SimplySupported Beam
Percent Error for Zero Poisson Ratio, %
5
4
3
Fixed−Free Fixed−Fixed Simply−Supported
2
1
0
−1
0
10
20
30
40 50 60 Beam Length
70
80
90
100
Figure 18: Influence of Poisson Effect for Varying Slenderness Ratios
31
Percent Error for Zero Poisson Ratio, %
5 Fixed−Free Fixed−Fixed Simply−Supported
4
3
2
1
0
−1
1
1.5
2
2.5 3 3.5 Mode of Vibration
4
4.5
5
Figure 19: Influence of Poisson Effect for Varying Modes of Vibration: The Isotropic Beam with Slenderness Ratio of 69
32