This paper presents the derivation of a new beam theory with the sixth-order differential equilibrium equations for the analysis of shear deformable beams. A sixth-order beam theory is desirable since the displacement constraints of some typical shear flexible beams clearly indicate that the boundary conditions corresponding to these constraints can be properly satisfied only by the boundary conditions associated with the sixth-order differential equilibrium equations as opposed to the fourth-order equilibrium equations in Timoshenko beam theory. The present beam theory is composed of three parts: the simple third-order kinematics of displacements reduced from the higher-order displacement field derived previously by the authors, a system of sixth-order differential equilibrium equations in terms of two generalized displacements $w$ and $ϕx$ of beam cross sections, and three boundary conditions at each end of shear deformable beams. A technique for the analytical solution of the new beam theory is also presented. To demonstrate the advantages and accuracy of the new sixth-order beam theory for the analysis of shear flexible beams, the proposed beam theory is applied to solve analytically three classical beam bending problems to which the fourth-order beam theory of Timoshenko has created some questions on the boundary conditions. The present solutions of these examples agree well with the elasticity solutions, and in particular they also show that the present sixth-order beam theory is capable of characterizing some boundary layer behavior near the beam ends or loading points.

1.
Zhang
,
Y. Y.
,
Wang
,
C. M.
, and
Tan
,
V. B. C.
, 2009, “
Assessment of Timoshenko Beam Models for Vibrational Behavior of Single-Walled Carbon Nanotubes Using Molecular Dynamics
,”
Adv. Appl. Math. Mech.
,
1
(
1
), pp.
89
106
.
2.
Gregory
,
R. D.
, and
Wan
,
F. Y. M.
, 1984, “
Decaying States of Plane Strain in a Semi-Infinite Strip and Boundary Conditions for Plate Theories
,”
J. Elast.
0374-3535,
14
, pp.
27
64
.
3.
Fan
,
H.
, and
Widera
,
G. E. O.
, 1992, “
On the Proper Boundary Conditions for a Beam
,”
ASME J. Appl. Mech.
0021-8936,
59
, pp.
915
922
.
4.
Gao
,
Y.
,
Xu
,
S. -P.
, and
Zhao
,
B. -S.
, 2007, “
Boundary Conditions for Elastic Beam Bending
,”
C. R. Mec.
1631-0721,
335
, pp.
1
6
.
5.
Dugundji
,
J.
, 2002, “
Cantilever Boundary Condition, Defections, and Stresses of Sandwich Beams
,”
AIAA J.
0001-1452,
40
, pp.
987
995
.
6.
Kim
,
J. -S.
,
Cho
,
M.
, and
Smith
,
E. C.
, 2008, “
An Asymptotic Analysis of Composite Beams With Kinematically Corrected End Effects
,”
Int. J. Solids Struct.
0020-7683,
45
, pp.
1954
1977
.
7.
Reissner
,
E.
, 1985, “
Reflection on the Theory of Elastic Plates
,”
Appl. Mech. Rev.
0003-6900,
38
, pp.
1453
1464
.
8.
Reissner
,
E.
, 1945, “
The Effect of Transverse Shear Deformation on the Bending of Elastic Plates
,”
ASME J. Appl. Mech.
0021-8936,
12
, pp.
66
77
.
9.
Karama
,
M.
,
Afaq
,
K. S.
, and
Mistou
,
S.
, 2003, “
Mechanical Behavior of Laminated Composite Beam by the New Multi-Layered Laminated Composite Structures Model With Transverse Shear Stress Continuity
,”
Int. J. Solids Struct.
0020-7683,
40
, pp.
1525
1546
.
10.
Levinson
,
M.
, 1980, “
An Accurate Simple Theory of Statics and Dynamics of Elastic Plates
,”
Mech. Res. Commun.
0093-6413,
7
, pp.
343
350
.
11.
Bickford
,
W. B.
, 1982, “
A Consistent Higher Order Beam Theory
,”
Dev. Theor. Appl. Mech.
0070-4598,
11
, pp.
137
150
.
12.
Shi
,
G.
, 2007, “
A New Simple Third-Order Shear Deformation Theory of Plates
,”
Int. J. Solids Struct.
0020-7683,
44
, pp.
4399
4417
.
13.
Franciosi
,
C.
, and
Tomasiello
,
S.
, 2007, “
Static Analysis of a Bickford Beam by Means of the DQEM
,”
Int. J. Mech. Sci.
0020-7403,
49
, pp.
122
128
.
14.
Gao
,
Y.
, and
Wang
,
M. Z.
, 2005, “
A Refined Beam Theory Based on the Refined Plate Theory
,”
Acta Mech.
0001-5970,
177
, pp.
191
197
.
15.
Gao
,
Y.
, and
Wang
,
M. Z.
, 2006, “
The Refined Theory of Deep Rectangular Beams Based on General Solution of Elasticity
,”
Sci. China, Ser. G
1672-1799,
49
(
3
), pp.
291
303
.
16.
Fan
,
H.
, and
Widera
,
G. E. O.
, 1994, “
On the Use of Variational Principles to Derive Beam Boundary Conditions
,”
ASME J. Appl. Mech.
0021-8936,
61
, pp.
470
471
.
17.
Yu
,
W.
, and
Hodges
,
D. H.
, 2004, “
Elasticity Solutions Versus Asymptotic Sectional Analysis of Homogeneous, Isotropic, Prismatic Beams
,”
ASME J. Appl. Mech.
0021-8936,
71
, pp.
15
23
.
18.
,
G. Z.
, and
Shi
,
G.
, 1991, “
A Refined Two-Dimensional Theory for Thick Cylindrical Shells
,”
Int. J. Solids Struct.
0020-7683,
27
, pp.
261
282
.
19.
Shi
,
G.
,
Lam
,
K. V.
, and
Tay
,
T. E.
, 1998, “
On Efficient Finite Element Modeling of Plates and Beams Based on Higher-Order Theory and a New Composite Beam Element
,”
Compos. Struct.
0263-8223,
41
, pp.
159
165
.
20.
Hu
,
H. -C.
, 1981,
Variational Principles in Elasticity and Their Applications
,
Science
,
Beijing
.
21.
Timoshenko
,
S. P.
, and
Goodier
,
J. N.
, 1970,
Theory of Elasticity
, 3rd ed.,
McGraw-Hill
,
New York
.
22.
Timoshenko
,
S. P.
, and
Gere
,
J.
, 1972,
Mechanics of Materials
,
Van Nostrand Reinhold
,
New York
.
23.
Murthy
,
V. V.
, 1981, “
An Improved Transverse Shear Deformation Theory for Laminate Anisotropic Plates
,”
NASA
Technical Report No. 1903.
24.
Reddy
,
J. N.
, 1984, “
A Simple Higher-Order Theory for Laminated Composite Plates
,”
ASME J. Appl. Mech.
0021-8936,
51
, pp.
745
752
.
25.
Shi
,
G.
, 2007, “
A Higher-Order Shear Beam Theory and Solution Technique for the Sixth-Order Differential Equilibrium Equations
,”
Proceeding of Chinese Conference of Theoretical and Applied Mechanics—2007
, Beijing, China, Paper No. 13-136.
You do not currently have access to this content.