3 Discrete and Nonlocal Beams 3.1 Hencky-Bar-Chain Model
3.3 Buckling and Vibrations Analyses of Discretized Beam
The governing equation can also be built from the continualization of Eq. (67), which again is approximated using a Padé approximant, as:
∂2M
∂x2 +P∂2w
∂x2 +μ
1−lc2 ∂2
∂x2 ∂2w
∂t2 =
1−lc2 d2 d x2
q (86)
The governing equation is no more local, and has been enriched with some nonlo- cal rotary inertia and an additional external load term. It is then possible to solve the paradox of apparent dependence of the length scale, with a nonlocal model coupled to modified nonlocal equilibrium equation. The nonlocal generalization of Newton’s second law has already been formulated in a different context by Milton and Willis [80] or Charlotte and Truskinovsky [81]. For the beam lattice model considered in this chapter (Hencky-Bar-Chain model), the nonlocal moment-curvature relationship and the governing equation are written by
M =E I ∂2w
∂x2
and ∂2M
∂x2
+P ∂2w
∂x2
+μ∂2w
∂t2 =q (87) where the angle bracket represents the nonlocal average operator associated with the pseudo-differential operator, given by Eq. (84) for the moment-curvature law, or Eq. (85) for its Padé approximant.
This correction Eq. (86) in the equilibrium equation may resolve the paradox of the dependence of length scale on the analysis type (buckling or vibrations analyses).
When correcting both the balance equation and the constitutive law by nonlocal terms, the nonlocal length scale calibrated from the beam lattice is no more dependent on the configuration.
wi+1+ β¯
n2 −2
wi+wi−1 =0 with β¯= P L2
E I (89)
The characteristic equation is obtained by replacing the displacement expressed as a power functionwi =Wλiin Eq. (89) which leads to
λ+1
λ =2− β¯
n2 (90)
which again is a palindromic equation. This last equation admits the following two solutions:
λ1,2=1− β¯
2n2 ± j 1−
1− β¯ 2n2
2
with j2= −1 and for
1− β¯ 2n2
<1 (91) It can be shown that the shape of solution wi changes for the other cases if 1−2nβ¯2≥1 (in term of hyperbolic functions). In this particular case, the solution is identically vanishing with the considered boundary conditions. The solution of the characteristic equation is therefore given, with the conditions Eq. (91), by
λ1,2=cosθ± jsinθ with θ =arccos
1− β¯ 2n2
(92) The general solution for the linear difference equation can be expressed in a trigonometric function:
wi =Acos(θi)+Bsin(θi) with θ =arccos
1− β¯ 2n2
(93) The two boundary conditions require a fundamental buckling mode wi = Bsin(θi)and then the formula for buckling load is given by:
sin(θn)=0 ⇒ θn =π ⇒ cosπ
n =1− β¯ 2n2
⇒ β¯=4n2sin2 π
2n
(94) Equation (94) is consistent with the buckling load reported by Wang [12, 13].
Challamel et al. [85] obtained the same result with an alternative method based on recursive formula and Chebyschev polynomials. More generally, the higher eigen- values are obtained fromθn=kπwith the exactk-th buckling load:
β¯k,n =4n2sin2 kπ
2n
(95) We are mainly interested in the calculation of the critical buckling load, i.e.k=1.
However, we could also compare the exact formulae associated with the higher modes, by approximating the Eringen’s nonlocal buckling solution via:
β¯k,n= (kπ)2 1+(knπ)22
lc
a
2 (96)
The comparison between Eqs. (95) and (96) shows that the best matching for the critical buckling load (k=1) is different from the highest buckling load (i.e., k=n):
k=1 ⇒ e20 = lc
a 2
= 1 12 and k=n ⇒ e20=
lc
a 2
=1 4 − 1
π2 (97)
Andrianov et al. [49] obtained both values ofe0reported in Eq. (97) via the match- ing of low frequency and high frequency of an axial discrete chain. As highlighted from the differential equations of the continualised nonlocal beam model, the value e0=1/
2√ 3
≈0.289 appears again for the buckling problem when n is large.
The last valuee0=
1/4 −1/π2 ≈0.386 that is valid for the highest buckling loads (same as axial dynamics) corresponds to that in Eringen [3] in which the non- local model is compared with the dispersive wave equations of the Born-Kármán model (Eringen [3]; see also Challamel et al. [37] for a two-length-scale nonlocal model).
For practical applications, the exact buckling load formulae restricted to the critical bucking case obtained fork=1 can be asymptotically expanded as follows,
β¯=4n2sin2 π
2n =π2
1− π2
12n2
+O 1
n4
(98) This formulae can be re-expressed in a more general equation. As mentioned by Seide [84], the dimensionless buckling load for the clamped-free, clamped-clamped or hinged-hinged boundary conditions can be formulated in a general form, which covers the previous structural case:
Pdiscrete
PE
=1− PEL2 12E I
1 n2 +O
1 n4
(99)
where PEis the buckling load of the Euler-Bernoulli column. A very similar length scale dependence would have been obtained from the buckling formulae of a hinge- hinge nonlocal Euler-Bernoulli beam model based on Eringen’s nonlocal theory [56]:
Pnonlocal
PE
= 1 1+π2lc
L
2 =1− π2 12n2 +O
1 n4
with lc
a = 1 2√
3 and PE = π2E I
L2 (100)
More generally, we find that for other classical boundary conditions such as clamped-free, clamped-clamped, hinged-hinged or clamped-hinged ends, the buck- ling load of the nonlocal column can be calculated from the following formulae (see Wang et al. [86], Reddy and Pang [87], Challamel et al. [78]):
Pnonlocal
PE
= 1
1+PE IElc2 =1− PEL2 12E I
1 n2 +O
1 n4
=1− PEa2 12E I +O
a4 L4
(101) The length scale dependence is of order 2 which is similar to the asymptotic formulae of the Hencky-Bar- Chain model. The lattice column presents a lower buckling load than that of the “local” Euler column. From Eq. (101), and in agreement with Salvadori [46] and Wang [13], the convergence rate of finite difference method shows that the error in the buckling load is of thea2-type (a =L/n) in the present homogeneous difference eigenvalue problem. The Finite Difference approximation (or its equivalent lattice formulation) for linear buckling problem provides a lower bound buckling load to its corresponding continuous local problem (the comparison between local and nonlocal continuum formulations yield the same tendency).
The eigenfrequencies of the Hencky-Chain-Bar can be exactly calculated in the same manner, as detailed for instance by Leckie and Lindberg [88] or more recently Santoro and Elishakoff [89] (for the equivalent finite difference formulation), from the following linear fourth-order difference Eq. (102):
wi+2−4wi+1+6wi−4wi−1+wi−2−2
n4wi =0 with 2=ω2μL4 E I (102) Similar to the buckling problem, the vibration mode can also assume a power functionwi =Wλi, and then by substituting the function into Eq. (102), one obtains the following characteristic equation:
1 λ+λ
2
−4 1
λ +λ
+4−2
n4 =0 (103)
This quartic equation generates the following four solutions [89]:
λ1,2=cosθ± jsinθ and λ3,4=2−cosθ±
(2−cosθ)2−1 with θ =arccos
1−
2n2
(104) For simply supported boundary conditions, the eigenmodes are obtained from the trigonometric shape functionwi =Bsin(θi). The fundamental eigenfrequency is then obtained by injecting the trigonometric solution in the fourth-order difference Eq. (102), as obtained by Leckie and Lindberg [88]:
=4n2sin2 π
2n =π2
1− π2
12n2
+O 1
n4
(105) This formulae is very similar in its form to the non-dimensional buckling formulae.
From Eq. (105), we then have the square of the fundamental frequency as ω2di scr ete
ω2E
=1− π2 6n2 +O
1 n4
with ωE2 = E I μ
π L
4
(106) As for the buckling problem, the lattice beam also presents lower natural frequen- cies than those of the continuous “local” beams.
ω2nonlocal
ω2E = 1 1+πlc
L
2 =1− π2 6n2 +O
1 n4
with lc
a = 1
√6 (107)
Eringen’s nonlocal model is an efficient continuous theory to get an approxima- tion of the fundamental frequency of the Hencky-Bar-Chain model while the Finite Difference approximation gives a lower bound of the asymptotic Euler-Bernoulli beam.
The capability of the nonlocal beam theory to predict the eigenfrequencies of the Hencky-Bar-Chain model is also valid for higher frequencies. The frequency spectrum can be obtained fromθn=kπwherekrepresents thek-th eigenfrequency.
The exact formulae valid for all eigenfrequencies of the Hencky- Bar-Chain model includes the one of the fundamental frequency:
k,n =4n2sin2 kπ
2n
⇒ 2k,n=16n4sin4 kπ
2n
(108) The nonlocal continuous approximation obtained from Eringen’s beam theory:
2k,n = (kπ)4 1+(knπ)22
lc
a
2 (109)
The comparison between Eqs. (108) and (109) also shows that the best match for the fundamental frequency (k=1) and the highest natural frequency (k=n) have a discrepancy:
k=1 ⇒ e20= lc
a 2
=1 6 and k=n ⇒ e02=
lc
a 2
= π2 16− 1
π2 (110)
The fundamental eigenfrequency (k=1) can be calibrated frome0=1/√ 6
≈ 0.408, wherease0=
π2/16 −1/π2 ≈0.718 is valid for the highest frequencies (particularlyk=n).
It is possible to consider the coupling between buckling and vibrations. In this case, the difference eigenvalue problem is governed by the following fourth-order linear difference equation:
wi+2−4wi+1+6wi−4wi−1+wi−2−2 n4wi
+β¯
n2(wi+1−2wi+wi−1)=0 (111) Following the procedure detailed for the free vibration of the Hencky-Bar-Chain, the vibration mode can be assumed aswi =Wλi, and then by substituting the shape function into Eq. (111), one obtains the characteristic palindromic equation:
1 λ+λ
2
−
4− β¯ n2
1 λ+λ
+4−2β¯ n2 −2
n4 =0 (112)
Again, this quartic equation admits four solutions:
λ1,2 =cosθ± jsinθ and λ3,4 =2− β¯
2n2 −cosθ± 2− β¯
2n2 −cosθ 2
−1
with θ=arccos
⎛
⎝1− β¯ 4n2 − 1
2n2 β¯2
4 +2
⎞
⎠ (113)
These solutions include the free vibration case treated by Leckie and Lindberg [88] and Santoro and Elishakoff [89].
For simply supported boundary conditions, the eigenmodes could be assumed as the trigonometric shape functionwi=Bsin(θi)and then we have
sin(θn)=0 ⇒ θ =π n
⇒ β¯ 2 + β¯2
4 +2 =4n2sin2π
2n (114)
The frequency-normal force equation can be equivalently reformulated in a Dunkerley type interaction formulae valid for the Hencky-Bar-Chain system:
4n2sin22nπ
2
+ β¯
4n2sin22nπ =1 (115)
Dunkerley’s line, or Melan’s formula is known to relate linearly the square fre- quency and the load parameter (see Tarnai [90]). Even for the continuous beam problem, it is also known that the linearity between the square frequency and the load parameter valid for the simply supported boundary conditions does not exactly apply to other boundary conditions, even though it provides a closed approximation (see for instance Massonnet [91] and Galef [92]).
This exact relationship between load and frequency for the Hencky-Bar-Chain system can be generally approximated using the asymptotic expansion as follows
π2
2
=1− π2 6n2 − β¯
π2
1− π2 12n2
+O
1 n4
(116) This last formulae valid for the Hencky-Bar-Chain system can now be matched to that obtained from Eringen’s method based on the single length scalelc=e0a= e0L/n(see recently Wang et al. [79]):
π2
2
= 1 1+e20πn22
− β¯
π2 =1−e20π2 n2 − β¯
π2 +O 1
n4
(117) The equivalent length scale coefficient of Eringen’s model can be calculated from the discrete beam model, by comparing Eq. (116) with Eq. (117), thus leading to a load-dependent nonlocal length scale:
e20= 1 6− β¯
12π2 (118)
In this chapter, the nonlocal length scalee0is calibrated from the exact relationship between normal force and frequency in the Hencky-Bar-Chain model, a method which differs from that in Wang et al. [79] who derived the same relationship by continualizing the difference equations of the lattice beam.