Nonlocal Theories For Bending, Buckling and Vibration of Beams - JN Reddy
Nonlocal Theories For Bending, Buckling and Vibration of Beams - JN Reddy
Nonlocal Theories For Bending, Buckling and Vibration of Beams - JN Reddy
http://www.elsevier.com/copyright
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Received 15 September 2006; received in revised form 23 October 2006; accepted 30 October 2006
Available online 18 June 2007
Abstract
Various available beam theories, including the Euler–Bernoulli, Timoshenko, Reddy, and Levinson beam theories, are
reformulated using the nonlocal differential constitutive relations of Eringen. The equations of motion of the nonlocal the-
ories are derived, and variational statements in terms of the generalized displacements are presented. Analytical solutions
of bending, vibration and buckling are presented using the nonlocal theories to bring out the effect of the nonlocal behavior
on deflections, buckling loads, and natural frequencies. The theoretical development as well as numerical solutions pre-
sented herein should serve as references for nonlocal theories of beams, plates, and shells.
2007 Elsevier Ltd. All rights reserved.
Keywords: Analytical solutions; Beam theories; Bending; Buckling; Nonlocal elasticity; Free vibration
1. Introduction
Most classical continuum theories are based on hyperelastic constitutive relations which assume that the
stress at a point are functions of strains at that point. On the other hand, the nonlocal continuum mechanics
assumes that the stress at a point is a function of strains at all points in the continuum. Such theories contain
information about the forces between atoms, and the internal length scale is introduced into the constitutive
equations as a material parameter. The nonlocal elasticity is initiated in the papers of Eringen [1–3] and Erin-
gen and Edelen [4].
The nonlocal theory of elasticity has been used to study lattice dispersion of elastic waves, wave propaga-
tion in composites, dislocation mechanics, fracture mechanics, surface tension fluids, etc. [7,8,12–19], and
Wang et al. [16] have applied the nonlocal elasticity constitutive equations to study vibration and buckling
of carbon nanotubes with the help of beam and shell theories. Wang et al. [16] neglected the nonlocal effect
in writing the shear stress–strain relation of the Timoshenko beam theory, and consequently the effect of
including nonlocal constitutive behavior amounted to using an equivalent shear correction factor.
*
Address: Engineering Science Programme, National University of Singapore, Singapore. Tel.: +1 979 862 2417.
E-mail address: jnreddy@tamu.edu
0020-7225/$ - see front matter 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijengsci.2007.04.004
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Following this brief introduction, a complete development of the classical and shear deformation beam the-
ories using the nonlocal constitutive differential equations is presented. The equations derived account for the
time dependent behavior as well as the axial compressive force. The equations are then solved for bending
deflections, buckling loads, and natural frequencies of simply supported beams.
2.1. Introduction
There are a number of beam theories that are used to represent the kinematics of deformation. To describe
the various beam theories, we introduce the following coordinate system. The x-coordinate is taken along the
length of the beam, z-coordinate along the thickness (the height) of the beam, and the y-coordinate is taken
along the width of the beam. In a beam theory, all applied loads and geometry are such that the displacements
(u1, u2, u3) along the coordinates (x, y, z) are only functions of the x and z coordinates and time t. Here it is
further assumed that the displacement u2 is identically zero.
In the following, equations of motion of various beam theories are presented using the dynamic version of
the principle of virtual displacements (see Reddy [10]). Since the principle of virtual work is independent of the
constitutive models, the equations of motion expressed in terms of the stress resultants are valid for local or
nonlocal theories. In Section 3, these equations are modified for the nonlocal effects by expressing the stress
resultants in terms of nonlocal parameter.
The following stress resultants are introduced for use in the coming sections:
Z Z
N¼ rxx dA; M ¼ zrxx dA
A A
Z Z Z ð2:1Þ
3 2
P¼ z rxx dA; Q ¼ rxz dA; R ¼ z rxz dA
A A A
The stress resultants P and R will appear only in the higher-order theories.
The simplest beam theory is the Euler–Bernoulli beam theory (EBT), which is based on the displacement
field
owE
u1 ¼ uðx; tÞ z ; u2 ¼ 0; u3 ¼ wE ðx; tÞ ð2:2Þ
ox
where (u, wE) are the axial and transverse displacements of the point (x, 0) on the mid-plane (i.e., z = 0) of the
beam and the superscript ‘E’ denotes the quantities in the Euler–Bernoulli beam theory. The only nonzero
strain of the Euler–Bernoulli beam theory is
ou o2 w E ou o2 w E
eExx ¼ z 2 e0xx þ zjE ; e0xx ¼ ; jE ¼ ð2:3Þ
ox ox ox ox2
where f(x, t) and q(x, t) are the axial and transverse distributed forces (measured per unit length) and N E is the
applied axial compressive force. We obtain the following Euler–Lagrange equations in 0 < x < L
oN o2 u
þ f ¼ m0 2 ð2:5Þ
ox ot
2 E
oM o E ow
E
o2 wE o4 wE
þ q N ¼ m0 m2 ð2:6Þ
ox2 ox ox ot2 ox2 ot2
The boundary conditions involve specifying one element of each of the following three pairs at x = 0 and
x = L:
u or N
oM E owE o3 wE
wE or NE þ m2 VE ð2:7Þ
ox ox ox ot2
owE
or ME
ox
The Timoshenko beam theory (TBT) (see Reddy [10,11]), which is based on the displacement field
where /T denotes the rotation of the cross-section and the superscript ‘T’ denotes the quantities in the Tim-
oshenko beam theory. The nonzero strains of the Timoshenko beam theory are
ou o/T owT
eTxx ¼ þz e0xx þ zjT ; 2eTxz ¼ þ / T cT
ox ox ox
ð2:9Þ
ou o/T owT
e0xx ¼ ; j ¼T
; T
c ¼ þ /T
ox ox ox
Here jT denotes the bending strain and cT is the transverse shear strain. The Timoshenko beam theory re-
quires shear correction factors to compensate for the error due to this constant shear stress assumption.
The shear correction factors depend not only on the material and geometric parameters but also on the load
and boundary conditions.
The principle of virtual displacements for the Timoshenko beam is given by
Z Z
T L
ou odu owT odwT o/T od/T
0¼ m0 þ þ m2 N de0xx M T djT QT dcT
0 0 ot ot ot ot ot ot
T T
T T ow odw
þ f du þ qdw þ N dx dt ð2:10Þ
ox ox
The boundary conditions involve specifying one element of each of the following two pairs:
u or N
owT
wT or QT N T VT ð2:13Þ
ox
/T or MT
Levinson [6] and Reddy [9] (also see Heyliger and Reddy [5]) employed the following displacement field to
develop a refined beam theory:
owR
u1 ¼ uðx; tÞ þ z/R ðx; tÞ c1 z3 /R þ ; u2 ¼ 0; u3 ¼ wR ðx; tÞ ð2:14Þ
ox
where c1 = 4/(3h2) and h being the height of the beam. Although the displacement field in (2.14) was derived
using the local shear stress–strain constitutive relation, it is assumed to be a valid displacement field. The non-
zero strains of the refined beam theory are
ou o/R o2 w R
eR
xx ¼ þ zð1 c1 z2 Þ c1 z3 2
e0xx þ zjR þ z3 qR
ox R ox ox
ow R
2eR
xz ¼ ð1 c 2 z 2
Þ þ / cR þ z2 bR ð2:15Þ
ox
R
o/R o/ o2 wR
jR ¼ ; qR ¼ c1 þ
ox ox ox2
where c2 = 4/h2. The displacement field of the refined beam theory accommodates a quadratic variation of the
transverse shear strain that vanishes on the top and bottom faces, z = ±h/2, of a beam. Thus, there is no need to
use shear correction factors in the Reddy beam theory. The virtual strain energy of the Reddy beam theory is
Z L
dU R ¼ N de0xx þ M R djR þ P R dqR þ QR dcR þ RR dbR dx ð2:16Þ
0
The principle of virtual displacements for this higher-order theory has the form
Z T Z L
ou odu owR odwR o/R od/R
0¼ m0 þ þ m2
0 0 ot ot ot ot ot ot
R R 2 R
o/ od/ o dw R
od/ o/R o2 wR
þ c1 m4 þ m4 þ
ot ot ox ot ot ot ox ot
R 2 R R 2 R
2 R
o/ ow od/ o dw b R djR þ c1 P R o dw Q
b R dcR
þ c1 m6 þ þ N de0xx M
ot ox ot ot ox ot ox2
R R
R R ow odw
þ f du þ qdw þ N dx dt ð2:17Þ
ox ox
where
b R ¼ M R c1 P R ;
M b R ¼ Q R c2 R R
Q ð2:18Þ
The Euler–Lagrange equations of motion (in addition to Eq. (2.5)) are
3 R
o2 w R o3 /R 2 o/ o4 w R
m0 2 c1 m4 þ c1 m 6 þ
ot ox ot2 ox ot2 ox2 ot2
b R o owR
o2 P R o Q
þ c1 þ NR þq¼0 ð2:19Þ
ox2 ox ox ox
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2 R
o2 /R o/ o3 wR bR
oM
b 2 2 þ c1 m
m b4 þ þ bR ¼ 0
Q ð2:20Þ
ot ot2 ox ot2 ox
The boundary conditions for bending variables involve specifying one element of each of the following pairs:
u or N
R
R
b R N R ow þ c1 oP m o2 /R o3 wR
wR or Q b 4 2 þ c1 m6 VR
ox ox ot ox ot2
ð2:21Þ
/T or bR
M
owR
or c1 P R
ox
The displacement field of the Levinson beam theory is the same as in the Reddy beam theory
L 3 L owL
u1 ¼ uðx; tÞ þ z/ ðx; tÞ c1 z / þ ; u2 ¼ 0; u3 ¼ wL ðx; tÞ ð2:22Þ
ox
However, Levinson [6] used a vector approach to derive the equations of equilibrium, and therefore they are
the same as those of the Timoshenko beam theory. On the other hand, Reddy [9] derived variationally con-
sistent equations of motion associated with the displacement field in Eq. (2.14) using the principle of virtual
displacements. Thus the Reddy and Levinson beam theories have the same displacement and strain fields, i.e.,
jL ¼ jR ; qL ¼ qR ; cL ¼ cR ; bL ¼ bR ð2:23Þ
but, the two theories have different equations of equilibrium or motion. The equations of Levinson’s beam
theory cannot be derived from the principle of total potential energy.
The governing equations of this higher-order theory are the same as in the Timoshenko beam theory
oQL o L ow
L
o2 w L
þq N ¼ m0 2 ð2:24Þ
ox ox ox ot
oM L o2 / L
QL ¼ m2 2 ð2:25Þ
ox ot
3. Nonlocal theories
According to Eringen [1–3], Eringen (1983) the stress field at a point x in an elastic continuum not only
depends on the strain field at the point (hyperelastic case) but also on strains at all other points of the body.
Eringen attributed this fact to the atomic theory of lattice dynamics and experimental observations on phonon
dispersion. Thus, the nonlocal stress tensor r at point x is expressed as
Z
r¼ Kðjx0 xj; sÞtðx0 Þdx0 ð3:1Þ
V
where t(x) is the classical, macroscopic stress tensor at point x and the kernel function K(|x 0 x|, s) represents
the nonlocal modulus, |x 0 x| being the distance (in Euclidean norm) and s is a material constant that depends
on internal and external characteristic lengths (such as the lattice spacing and wavelength, respectively). The
macroscopic stress t at a point x in a Hookean solid is related to the strain e at the point by the generalized
Hooke’s law
tðxÞ ¼ CðxÞ : eðxÞ ð3:2Þ
where C is the fourth-order elasticity tensor and: denotes the ‘double-dot product’.
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The constitutive Eqs. (3.1) and (3.2) together define the nonlocal constitutive behavior of a Hookean solid.
Eq. (3.1) represents the weighted average of the contributions of the strain field of all points in the body to the
stress field at a point. However, the integral constitutive relation in (3.1) makes the elasticity problems difficult
to solve. However, it is possible (see Eringen, 1983) to represent the integral constitutive relations in an equiv-
alent differential form as
e0 a
ð1 s2 ‘2 r2 Þr ¼ t; s ¼ ð3:3Þ
‘
where e0 is a material constant, and a and ‘are the internal and external characteristic lengths, respectively.
Using Eqs. (3.2) and (3.3), we can express stress resultants in terms of the strains in different beam theories.
As opposed to the linear algebraic equations between the stress resultants and strains in a local theory, the
nonlocal theory results in differential relations involving the stress resultants and the strains. In the following,
we present these relations for homogeneous isotropic beams under the assumption that the nonlocal behavior
is negligible in the thickness direction. Then the nonlocal constitutive relation in Eq. (3.3), with Eq. (3.2) for
the macroscopic stress, takes the following special relations for beams:
o2 rxx o2 rxz
rxx l 2
¼ Ee xx ; rxz l 2
¼ 2Gexz ðl ¼ e20 a2 Þ ð3:4Þ
ox ox
where E and G are Young’s modulus and shear modulus, respectively. When the nonlocal parameter l is zero,
we obtain the constitutive relations of the local theories.
In all theories, the axial force–strain relation is the same and it is given by
o2 N
N l 2 ¼ EAe0xx ð3:5Þ
ox
where we have used the relations
Z Z
A¼ dA; z dA ¼ 0 ð3:6Þ
A A
Thus, the x-axis is taken along the geometric centroid of the beam.
Euler–Bernoulli Beam Theory. In this theory we only have N and ME. The constitutive relations are given by
o2 M E
ME l ¼ EIjE ð3:7Þ
ox2
where I denotes the second moment of area about the y-axis.
Timoshenko Beam Theory. In the Timoshenko beam theory we have MT and QT, in addition to N. Then
constitutive relations are given by
o2 M T T o2 QT
MT l ¼ EIjT
; Q l ¼ GAK s cT ð3:8Þ
ox2 ox2
Here Ks denotes the shear correction factor.
Reddy Beam Theory. In the Reddy beam theory we have MR, PR, QR, and RR. The constitutive relations of
the Reddy beam theory are
o2 M R
MR l ¼ EIjR þ EJ qR
ox2
o2 P R
P R l 2 ¼ EJ jR þ EKqR
ox ð3:9Þ
o2 QR
R
Q l 2
¼ GAcR þ GIbR
ox
o 2 RR
R l 2 ¼ GIcR þ GJ bR
R
ox
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where
Z
ðA; I; J ; KÞ ¼ ð1; z2 ; z4 ; z6 ÞdA ð3:10Þ
A
are the second, fourth, and sixth order moments of area about the y-axis.
Levinson Beam Theory. The stress resultants of the Levinson beam theory are exactly like in the Timo-
shenko beam theory and they are given by
o2 M L
ML l ¼ EIjL þ EJ qL
ox2 ð3:11Þ
o2 QL
QL l 2 ¼ GAcL þ GIbL
ox
The equations of motion of each beam theory now can be expressed in terms of the displacements (u, w, /).
This requires the use of force- and moment-deflection relationships in (3.5), (3.7), (3.8), (3.9), and (3.11) to
replace the stress resultants appearing in the equations of motion of each theory.
First, the equation of motion governing the axial displacement is derived for the nonlocal theory, as it is
common to all beam theories. Substituting for the first derivative of the axial force N from Eq. (2.5) into
Eq. (3.5), we obtain
ou o3 u of
N ¼ EA þ l m0 ð4:1Þ
ox ox ot2 ox
Substituting N from Eq. (4.1) into the equation of motion (2.5), we obtain
2
o ou o2 f ou o4 u
EA þ f l 2 ¼ m0 l 2 2 ð4:2Þ
ox ox ox ot2 ox ot
Eqs. (4.1) and (4.2) are valid for all nonlocal beam theories. The axial equation of motion of the conventional
(i.e., local) beam theory can be obtained from Eq. (4.2) by setting l = 0.
Substituting for the second derivative of ME from Eq. (2.6) into Eq. (3.7), we obtain
E o2 wE o E ow
E
o2 wE o4 w E
M ¼ EI þl N q þ m0 2 m2 2 2 ð4:3Þ
ox2 ox ox ot ox ot
Substituting ME from Eq. (4.3) into Eq. (2.6), we obtain
o2 o2 w E o2 o E ow
E
o2 w E o4 wE o E ow
E
EI þ l N q þ m 0 m 2 þ q N
ox2 ox2 ox2 ox ox ot2 ox2 ot2 ox ox
o2 w E o4 wE
¼ m0 m2 ð4:4Þ
ot2 ox2 ot2
The equation of motion of the conventional Euler–Bernoulli beam theory is obtained from Eq. (4.4) by setting
l = 0.
Substituting for the second derivative of MT from Eq. (4.5) into the first equation in (3.8), we obtain
T o/T o T ow
T
o2 w T o3 /T
M ¼ EI þ l q þ N þ m0 2 þ m2 ð4:6Þ
ox ox ox ot ox ot2
Next, substituting for the second derivative of QT from Eq. (2.11) into the second equation in (3.8), we obtain
T T owT o o T ow
T
o2 wT
Q ¼ GAK s / þ þl q þ N þ m0 2 ð4:7Þ
ox ox ox ox ot
Now substituting for MT and QT from Eqs. (4.6) and (4.7), respectively, into Eqs. (2.11) and (2.12), we obtain
o owT o owT
GAK s /T þ þq NT
ox ox ox ox
2 2
o o owT o wT o4 wT
l 2 q NT ¼ m0 l ð4:8Þ
ox ox ox ot2 ox2 ot2
o o/T owT o2 / T o4 /T
EI GAK s /T þ ¼ m2 2 lm2 2 2 ð4:9Þ
ox ox ox ot ox ot
From Eq. (3.9) the nonlocal constitutive equations for the stress resultants of the Reddy beam theory are
2 bR
b R l o M ¼ EbI jR þ E b
M J qR
ox2
2 bR
b R l o Q ¼ GAcR þ GIbR
Q ð4:10Þ
ox2
2 R
oP
P R l 2 ¼ EJ jR þ EKqR
ox
where (additional variables introduced here are used in the subsequent equations)
bI ¼ I c1 J ; b
J ¼ J c1 K; ^ 2 ¼ m2 c1 m4 ;
m b 4 ¼ m4 c1 m6
m
ð4:11Þ
A ¼ A c2 I; I ¼ I c2 J ; e ¼ A c2 I
A
b R from Eqs. (2.19) and (2.20), we obtain
Eliminating Q
3 R
bR
o2 M o2 P R o R ow
R
o2 wR o3 / R o/ o4 wR
¼ c 1 q þ N þ m0 þ m2 c m
1 4 þ ð4:12Þ
ox2 ox2 ox ox ot2 ox ot2 ox ot2 ox2 ot2
Substituting the above result in the first of Eq. (4.10), we arrive at
R R 2 R
b ¼ EbI o/ o/ o w
M R
c1 E b
J þ
ox ox ox2
3 R
o2 P R o R ow
R
o2 w R o3 /R o/ o4 w R
þ l c1 q þ N þ m 0 þ m2 c m
1 4 þ ð4:13Þ
ox2 ox ox ot2 ox ot2 ox ot2 ox2 ot2
Substituting the second derivative of Qb R from Eq. (2.19) into the second equation in Eq. (4.10), we arrive at
the result
b R ¼ GAe /R þ ow
R
o o2 P R o R ow
R
Q þl c1 þ N q
ox ox ox2 ox ox
o o2 w R o3 /R 2 o3 / R o4 w R
þl m 0 2 þ c1 m 4 c1 m6 þ ð4:14Þ
ox ot ox ot2 ox ot2 ox2 ot2
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b R and Q
Now we use M b R from Eqs. (4.13) and (4.14) and the identity
3 R
o2 R o2 P R o3 /R o/ o4 wR
c1 2 P l 2 ¼ c1 EJ c1 EK þ ð4:15Þ
ox ox ox3 ox3 ox4
to rewrite the equations of motion (2.19) and (2.20) in terms of the generalized displacements. We obtain
R
e o/ o2 w R o R ow
R
o2 o R ow
R
GA þ N þqþl 2 N q
ox ox2 ox ox ox ox ox
3 R
o3 /R o/ o4 wR
þ c1 EJ c1 EK þ
ox3 ox3 ox4
o2 w R o3 /R 2 o2 o/R o2 wR
¼ m0 2 þ c1 m4 c1 m 6 þ
ot ox ot2 ox ot ot ox ot
4 R 5 R 5 R
ow o/ 2 o/ o6 w R
l m0 2 2 þ c1 m4 3 2 c1 m6 þ ð4:16Þ
ox ot ox ot ox3 ot2 ox4 ot2
2 R
b o2 /R b o/ o3 wR e R owR
EI c1 E J þ GA / þ
ox2 ox2 ox3 ox
2 R 2 R 3 R
o/ o/ ow
¼m b 2 2 c1 m b4 2
þ
ot ot ox ot2
o4 / R o4 /R o5 w R
l m b 2 2 2 c1 m b4 þ ð4:17Þ
ox ot ox2 ot2 ox3 ot2
b i (i = 2, 4) are defined in Eq. (4.11).
where m
5. Variational statements
In view of the fact that the stress resultants are known in terms of the generalized displacements, the
statement of Hamilton’s principle can be expressed in terms of the displacements for the nonlocal theo-
ries of beams. This facilitates the direct derivation of the equations of motion in terms of the generalized
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displacements of a beam theory using Hamilton’s principle. We note that it is not possible to construct the
underlying quadratic functionals for nonlocal beam theories. These variational statements can be used to
develop displacement finite element models.
To write the variational statements, we substitute the expressions for the stress resultants in terms of the
generalized displacements into the principle of virtual displacements of each beam theory (see Eqs. (2.4),
(2.10), (2.17), and (2.10), respectively, for the four theories). The details are presented next.
The Hamilton principle for the Timoshenko beam theory has the form
Z T Z L T
ou oduT owT odwT o/T od/T ouT oduT o/T od/T
0¼ m0 þ þ m2 EA EI
0 0 ot ot ot ot ot ot ox ox ox ox
T
T
T T
ow odw ow odw
GAK s /T þ d/T þ þ f duE þ qdwT þ N T
ox ox ox ox
3 T 2 T
ou of odu T
o T ow
T
ow o3 /T od/T
l m0 þl q N m0 2 m2
ox ot2 ox ox ox ox ot ox ot2 ox
2
o o owT o wT odwT
þl q NT m0 2 d/T þ dx dt
ox ox ox ot ox
Z T
L
þ ½N T duT þ V T dwT þ M T d/T 0 dt ð5:5Þ
0
The Euler–Lagrange equations resulting from the above statement are given by Eqs. (4.2), (4.8), and (4.9). The
natural boundary conditions are
T ouT o3 uT of
N EA þ l m0 ¼0 ð5:6Þ
ox ox ot2 ox
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owT owT o o2 wT o2 wT
V T GAK s /T þ NT l NT q þ m 0 ¼0 ð5:7Þ
ox ox ox ox2 ot2
o/T o2 wT o2 wT o3 / T
M T EI þ l NT q þ m 0 þ m 2 ¼0 ð5:8Þ
ox ox2 ot2 ox ot2
Virtual work principle for the Levinson beam theory is the same as that for the Timoshenko beam theory,
except for the constitutive relations
Z T Z L L
ou oduL owL odwL o/L od/L ouL oduL
0¼ m0 þ þ m2 EA
0 0 ot ot ot ot ot ot ox ox
L L L
o/ od/ L ow L
L odw L
o/ o2 wL o/L
EI GA / þ d/ þ þ c1 EJ þ
ox ox ox ox ox ox2 ox
2 3 L
ow L
odw L
o ow L
o wL
o / o/L
þ f duL þ qdwL þ N L l q þ NL þ m0 2 þ m2
ox ox ox ox ot ox ot2 ox
o o owL o2 wL odwL
mu q þ NL þ m0 2 d/L þ dx dt
ox ox ox ot ox
Z T
L
þ ½N L duL þ V L dwL þ M L d/L 0 dt ð5:12Þ
0
The Euler–Lagrange equations resulting from the above statement are given by Eqs. (4.2), (4.21), and (4.22).
The natural boundary conditions of the Levinson beam theory are
L ouL o3 uL of
N EA þ l m0 ¼0 ð5:13Þ
ox ox ot2 ox
owL owL o o2 w L o2 w L
V L þ NL GA /L þ þl NL q þ m 0 ¼0 ð5:14Þ
ox ox ox ox2 ot2
L
L o/L 2 L
Lo w o2 w L o3 /L o/ o2 wL
M EI þl N q þ m0 2 þ m2 c1 EJ þ ¼0 ð5:15Þ
ox ox2 ot ox ot2 ox ox2
Here we consider exact solutions of bending, natural vibration, and buckling of simply supported beams.
The boundary conditions of simply supported beams are
w ¼ 0 and M ¼ 0 at x ¼ 0; L ð6:1Þ
The following expansions of the generalized displacements w and / satisfy the boundary conditions in
Eq. (6.1):
X1
npx ixn t X1
npx ixn t
wðx; tÞ ¼ W n sin e ; /ðx; tÞ ¼ Un cos e ð6:2Þ
n¼1
L n¼1
L
For bending, we set N and all time derivatives to zero and take the distributed load be of the form
X1 Z
npx 2 L npx
qðxÞ ¼ Qn sin ; Qn ¼ qðxÞ sin dx ð6:3Þ
n¼1
L L 0 L
In particular, the coefficients Qn associated with various types of loads are given below (see Reddy [11]):
4q
qðxÞ ¼ q0 ; Qn ¼ 0 ; n ¼ 1; 3; 5; . . .
np
q0 x 2q nþ1
qðxÞ ¼ ; Qn ¼ 0 ð1Þ ; n ¼ 1; 2; 3; . . .
L np ð6:4Þ
2Q0 px0
qðxÞ ¼ Q0 dðx x0 Þ; Qn ¼ sin ; n ¼ 1; 2; 3; . . .
a L
px
qðxÞ ¼ q0 sin ; Q1 ¼ q0 ; Qn ¼ 0; n ¼ 2; 3; . . .
L
For buckling, we set q and all time derivatives to zero, and for free vibration, we set N and q to zero.
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Substitution of the expansions for w and q from Eqs. (6.2) and (6.3) into the equation of motion (4.18), we
obtain
np2
np4
E np
2
2
kn N þ xn m0 þ m2 EI W n þ kn Qn ¼ 0 ð6:5Þ
L L L
for any n, where
np2
kn ¼ 1 þ l ð6:6Þ
L
Bending. The static deflection is given by setting N E and x2n to zero
X1
kn Qn L4 npx
wE ðxÞ ¼ 4 4
sin ð6:7Þ
n¼1
n p EI L
Substitution of the expansions for w, / and q from Eqs. (6.2) and (6.3) into the equations of motion (4.8)
and (4.9), we obtain
np
np
np2
GAK s Un þ W n þ kn Qn þ kn N T W n þ kn m0 x2n W n ¼ 0 ð6:11Þ
L L L
np2
np
EI Un GAK s Un þ W n þ kn m2 x2n Un ¼ 0 ð6:12Þ
L L
Bending. For static bending, we obtain
X
1
Qn L4 npx
T
w ðxÞ ¼ kn K n sin ð6:13Þ
n¼1
n4 p4 EI L
X
1
Qn L3 npx
/T ðxÞ ¼ kn 3 3
cos ð6:14Þ
n¼1
n p EI L
where
EI
Kn ¼ ð1 þ n2 p2 XÞ; X¼ ð6:15Þ
GAK s L2
Author's personal copy
Note that both inclusion of transverse shear strains and nonlocal constitutive equations increase the deflec-
tion. For the simply supported beam considered here, inclusion of transverse shear has no effect on the rota-
tion /T.
Buckling. The critical buckling load is given by
1 p2 EI
NT ¼ ð6:16Þ
k1 K1 L2
Clearly, the nonlocal parameter k as well as the transverse shear deformation has the effect of reducing the
critical buckling load.
Vibration. The natural frequencies x2n can be computed from
np2
np4
m0 m2 2 4
kn xn m0 Kn þ m2 kn x2n þ EI ¼0 ð6:17Þ
GAK s L L
Table 1
Comparison of non-dimensional maximum center deflection [ w ¼ w 102 ðEI=Q0 L3 Þ] in simply supported beams subjected to point load
6
Q0 at the center (Q0 = 1, L = 10, E = 30 · 10 , m = 0.3, 100 term series)
L/h l EBT TBTa RBT LBT
100 0.0 1.9444 1.9449 1.9449 1.9450
0.5 2.0278 2.0282 2.0282 2.0283
0.1 2.1111 2.1115 2.1115 2.1116
1.5 2.1944 2.1949 2.1949 2.1949
2.0 2.2778 2.2782 2.2782 2.2782
2.5 2.3611 2.3615 2.3615 2.3615
3.0 2.4444 2.4448 2.4448 2.4448
3.5 2.5277 2.5282 2.5282 2.5281
4.0 2.6111 2.6115 2.6115 2.6114
4.5 2.6944 2.6948 2.6948 2.6947
5.0 2.7777 2.7782 2.7782 2.7780
Substitution of the expansions for w, / and q from Eqs. (6.2) and (6.3) into the equations of motion (4.16)
and (4.17), we obtain
np
np
np2
np3
An Un þ W n þ kn Qn þ kn N R W n þ c1 EJ Un
L h L
L i L
np np np
þ kn m0 W n c1 m4 Un þ c21 m6 Un þ W n x2n ¼ 0 ð6:19Þ
L L L
Table 2
Comparison of non-dimensional maximum center deflection [ w ¼ w 102 ðEI=q0 L4 Þ] in simply supported beams subjected to uniform
6
load q0 (q0 = 1, L = 10, E = 30 · 10 , m = 0.3, 100 term series)
L/h l EBT TBTa RBT LBT
100 0.0 1.3130 1.3134 1.3134 1.3135
0.5 1.3809 1.3813 1.3813 1.3814
0.1 1.4487 1.4492 1.4492 1.4493
1.5 1.5165 1.5170 1.5170 1.5172
2.0 1.5844 1.5849 1.5849 1.5851
2.5 1.6522 1.6528 1.6528 1.6530
3.0 1.7201 1.7207 1.7207 1.7209
3.5 1.7879 1.7886 1.7886 1.7888
4.0 1.8558 1.8565 1.8565 1.8567
4.5 1.9236 1.9244 1.9244 1.9246
5.0 1.9914 1.9923 1.9923 1.9925
np2
np2
b np np
E I U n Bn U n þ W n þ k n mc2 Un c1 m
b4 Un þ W n x2n ¼ 0 ð6:20Þ
L L L L
where
np2
np2
An ¼ GA c2 GI þ c21 EK; Bn ¼ GA c2 GI c1 Eb
J ð6:21Þ
L L
Table 3
Comparison of non-dimensional critical buckling loads [N 0cr ¼ N ðL2 =EIÞ] of simply supported beams (L = 10, E = 30 · 106, m = 0.3)
L/h l EBT TBTa RBT LBT
100 0.0 9.8696 9.8671 9.8671 9.8675
0.5 9.4055 9.4031 9.4031 9.4035
0.1 8.9830 8.9807 8.9807 8.9811
1.5 8.5969 8.5947 8.5947 8.5950
2.0 8.2426 8.2405 8.2405 8.2408
2.5 7.9163 7.9143 7.9143 7.9146
3.0 7.6149 7.6130 7.6130 7.6133
3.5 7.3356 7.3337 7.3337 7.3340
4.0 7.0761 7.0743 7.0743 7.0746
4.5 6.8343 6.8325 6.8325 6.8328
5.0 6.6085 6.6068 6.6068 6.6070
" #
2 b A þc B J
1 p E I 1 1 1 ^
NR ¼
I
ð6:24Þ
k1 L2 B1 þ EbI p 2
L
np3
np2
np np 2 np 2 2 2
A n þ c1 EJ kn xn c1 m ^ 4 Un þ An þ kn xn m0 þ c1 m6 Wn ¼0 ð6:25Þ
L L L L L
np2 h
np
npi
EbI 2
Bn þ kn x n m
~ 2 Un þ Bn kn x2n c1 m
b4 Wn ¼0 ð6:26Þ
L L L
Table 4
pffiffiffiffi
Comparison of non-dimensional fundamental natural frequencies x ¼ x1 L2 mEI0 of simply supported beams (L = 10, E = 30 · 106,
m = 0.3, q = 1)
L/h l EBT TBTa RBT LBT
100 0.0 9.8696 9.8683 9.8683 9.8685
0.5 9.6347 9.6335 9.6335 9.6337
0.1 9.4159 9.4147 9.4147 9.4149
1.5 9.2113 9.2101 9.2101 9.2103
2.0 9.0195 9.0183 9.0183 9.0185
2.5 8.8392 8.8380 8.8380 8.8382
3.0 8.6693 8.6682 8.6682 8.6683
3.5 8.5088 8.5077 8.5077 8.5079
4.0 8.3569 8.3558 8.3558 8.3560
4.5 8.2129 8.2118 8.2118 8.2120
5.0 8.0761 8.0750 8.0750 8.0752
~2 ¼ m
where m b 4 . By setting the determinant of the coefficient matrix of the above equations, we obtain a
^ 2 c1 m
quadratic polynomial for x2n . If we neglect the higher-order inertias (i.e., m2 = m4 = m6 = 0), we obtain
np3
np
np np 2 np 2
An þ c 1 EJ Un þ An þ kn xn m0 W n ¼ 0 EbI
2
þ Bn Un Bn Wn ¼0
L L L L L
from which we obtain
" #
EbI
np4 c1 EJBn þ An EbI
x2n ¼ 2 ð6:27Þ
m0 kn L EbI np
L
þ Bn
Substitution of the expansions for w, / and q from Eqs. (6.2) and (6.3) into the equations of motion (4.21)
and (4.22), we obtain
np
np
2
L np
GA Un þ W n þ kn Qn þ kn N W n þ kn m0 x2n W n ¼ 0 ð6:28Þ
L L L
np2
np2
np
EIUn GA c1 EJ Un þ W n þ kn m2 x2n Un ¼ 0 ð6:29Þ
L L L
Bending. For static bending, we obtain
!
X1
GAL2 þ EbI n2 p2 Qn L4 npx
L
w ðxÞ ¼ kn 2 2 2 4 p4 EI
sin ð6:30Þ
n¼1 GAL c1 EJn p n L
X 1
c1 n2 p2 EJ Qn L3 npx
/L ðxÞ ¼ kn 1 2 3 3
cos ð6:31Þ
n¼1 GAL n p EI L
5.0
EBT for all L/h ratios
4.0
Non-local parameter,
3.0
2.0
0.0
1.0
Transverse deflection,
5.0
Non-local parameter,
3.0 Vibration
2.0 Buckling
1.0
All shear deformation
theories for L/h= 10
0.0
6.0 7.0 8.0 9.0 10.0
Buckling load/Fundamental frequency
Fig. 2. Comparison of non-dimensional critical buckling load and fundamental frequency versus nonlocal parameter.
where K is that defined in Eq. (6.15) except for X replaced with X ¼ EbI =GAL2 and Ks = 1. If we neglect the first
term, we obtain
1
np4
np2
x2n ¼ EI; Hn ¼ m0 þ m2 þ Xm0 ð6:34Þ
kn H n L L
Note that the rotary inertia m2, nonlocal parameter kn, and transverse shear strain parameter Kn have the ef-
fect of decreasing frequencies of vibration.
In this section the analytical solutions developed in the previous sections are presented. The following
parameters are used in computing the numerical values:
E ¼ 30 106 ; m ¼ 0:3; L ¼ 10; h ¼ varied; q¼1 ð6:35Þ
The following non-dimensional quantities are used:
EI EI
w ¼ w 102 ; w ¼ w 102
Q0 L3 q0 L 4
rffiffiffiffiffiffi ð6:36Þ
L2 m0
N 0cr ¼N ; ¼ x1 L
x 2
EI EI
The numerical results for bending under point load at the center and uniformly distributed load are tabulated
in Tables 1 and 2, respectively. The analytical solutions are obtained using 100 terms in the series (6.2). Tables
3 and 4 contain the non-dimensional critical buckling loads and fundamental frequencies versus the nonlocal
parameter l ¼ e20 a2 are presented in Tables 3 and 4, respectively.
In general, the effect of transverse shear strains and the nonlocal parameter l ¼ e20 a2 is to increase the
deflections and reduce the buckling loads as well as natural frequencies, as can be seen from the results pre-
sented in Tables 1–4 and Figs. 1 and 2. The effect of the nonlocal parameter on buckling load is more than on
frequency (see Fig. 2). The effect is negligible for L/h ratios less than 20. The Timoshenko, Reddy, and Lev-
inson beam theories yield solutions that are almost the same for all values of l.
7. Conclusions
Equations of motion of various beam theories are derived based on Eringen’s differential constitutive equa-
tions of nonlocal elasticity. Hamilton’s variational statements of the beam theories are also presented to facil-
itate direct development of displacement finite element models of the nonlocal beam theories. The equations of
Author's personal copy
motion are then analytically solved for bending deflections, buckling, and natural vibration of simply sup-
ported beams to bring out the effect of nonlocal parameter. The inclusion of the nonlocal effect increases
the magnitudes of deflections and decreases buckling loads and natural frequencies. As shown in this study,
the nonlocal effect is considerably different and more pronounced than the using a shear correction coefficient.
Acknowledgement
The author gratefully acknowledges the support of this work by the Oscar S. Wyatt Endowed Chair.
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