- Email: [email protected]
An improved third order beam theory
Journal of Sound and Vibration (1990) 142(3), 527-528
LETTERS TO THE EDITOR AN IMPROVED THIRD ORDER BEAM THEORY 1.
INTRODUCTION
third order beam theories were introduced to eliminate the shear correction coefficient in the Timoshenko theory. When the axial displacement is expanded in polynomials across the thickness, the order of the polynomial defines the order of the beam theory. Wang and Dickson [l] found that the polynomial expansion of the transverse displacement across the thickness is not essential. In the first order theories, the Timoshenko theory has advantages over the Euler theory because the assumption that a plane normal to the beam axis remains normal during deformation is removed without introducing further complexity of the governing equations. Among the third order theories [2-41, the Reddy theory [4] is variationally consistent. Reddy used two functions, w(x, 1) and 9(x, t), to describe the beam displacement and assumed the shear strain to be ($ + w’)( 1 - 4z2/ Zr’), where x is along the beam axis and z across the thickness, and the prime denotes a derivative with respect to x. The resulting governing equations are inertially coupled, involving derivatives of displacements, the time derivatives of which also appear in the natural boundary conditions. It is pointed out in what follows that if the assumption of the shear distribution is relaxed then the inertia coupling in the derivatives of displacements disappears and the natural boundary conditions do not involve time derivatives. The numerical performance is also improved at no extra cost. The
2. Let the displacement
DISPLACEMENT
ASSUMPTIONS
field for a beam of height h be u* = 0,
24,= ZJ,+ 238,
u3 = w,
(1)
where x and z are the axial and thickness co-ordinates of the beam, w(x, t) is the transverse displacement, and I(I(x, 1) and 0(x, t) are the generalized displacements representing the linear and cubic variations of u,(x, z, t) across the thickness. The non-vanishing strains are E, = z$‘+ z38’
and
yxv = lj + w’+ 3z2e.
(2)
The relationship 8 = -4($+
w’)/3Z?
(3)
is implied in the Reddy theory. Hamilton’s principle [5] gives the following appropriate natural and essential boundary conditions respectively: M = E(Zz$‘+ I‘@‘), Q=G(AJI+3Z,e+Aw’);
P = E(z&‘+
Z#), (4)
MS+ + P60 + QSw = 0.
Also the equilibrium equations M’ - Q = p(
Z,ij + Zz,di),
P’-
R = p( Z,j; + Z&,
Q’=pA+-q,
(5)
are obtained, where R = G(3Z21(,+9Z,8+3Z2w’), q(x, t) is the applied transverse excitation, and Z, = j z” dA, n = 2,4,6, in which A is the cross-sectional area and dots denote time derivatives. 527 0022460X/90/210527+02
%03.00/O
@ 1990
Academic
Press
Limited
528
LETTERS TO THE EDITOR 3. EXAMPLES
For a simply supported beam of length L, (Tj}-l(
(6)
~~~~~~~}sin~r,
and one has, from equation (5), EZ&+GA, EZqa2,+3GZ2,
EZ,cu2,+3GZ,, EZ,&+9GZ,,
(7)
GAan,
where (Y,= n7r/ L. For rectangular cross-sections, Z,= Ah2/12,
Z., = 3Zh2/20
and
Z6= 3Zh4/ 112.
(8)
When the thickness shear mode is of interest, n = 0, 35-45A+3hZ=0,
A = p~‘r’/G,
r2 = h2/12,
(9)
the smaller root of which is A =0*8229. When compared with the exact value [6], h = O-8225, the present theory is seen to perform better than the Reddy theory, which gives A = 0.8235. 4.
DISCUSSION
Due to the absence of the second order term in u(x, z, t), the present theory is suitable for beam sections having one axis of symmetry. The moments of area Z,,take good account of non-rectangular beam sections. Although the distribution of the shear stress across the thickness is still parabolic, the optimal distribution will be found through the variational principle; in some cases, the shear stress may not be zero along the upper and lower edges, due to the truncation of the power expansion. However, the theory ensures that the modes obtained are optimal under the third order theories. Department of Civil & Structural Engineering, University of Hong Kong, Hong Kong
A. Y. T. LEUNG
(Received 6 December 1989)
REFERENCES 1. J. T. S. WANG and J. N. DICKSON 1979 American Institute of Aeronautics and Astronautics Journal 17, 535-537. Elastic beams of various orders. 2. M. LEVINSON 1981 Journal of Sound and Vibration 74, 81-87. A new rectangular beam theory. 3. W. B. BICKFORD 1982 Developments in Theoretical and Applied Mechanics 11, 137-150. A consistent higher order beam theory. 4. P. R. HEYLIGER and J. N. REDDY 1988 Journal of Sound and Vibration 126,309-326. A higher order beam finite element for bending and vibration problems. 5. R. W. CLOUGH and J. PENZIEN 1975 Dynamics of Structures. Tokyo: McGraw-Hill. 6. H. LAMB 1945 Hydrodynamics. New York: Dover.
LETTERS TO THE EDITOR AN IMPROVED THIRD ORDER BEAM THEORY 1.
INTRODUCTION
third order beam theories were introduced to eliminate the shear correction coefficient in the Timoshenko theory. When the axial displacement is expanded in polynomials across the thickness, the order of the polynomial defines the order of the beam theory. Wang and Dickson [l] found that the polynomial expansion of the transverse displacement across the thickness is not essential. In the first order theories, the Timoshenko theory has advantages over the Euler theory because the assumption that a plane normal to the beam axis remains normal during deformation is removed without introducing further complexity of the governing equations. Among the third order theories [2-41, the Reddy theory [4] is variationally consistent. Reddy used two functions, w(x, 1) and 9(x, t), to describe the beam displacement and assumed the shear strain to be ($ + w’)( 1 - 4z2/ Zr’), where x is along the beam axis and z across the thickness, and the prime denotes a derivative with respect to x. The resulting governing equations are inertially coupled, involving derivatives of displacements, the time derivatives of which also appear in the natural boundary conditions. It is pointed out in what follows that if the assumption of the shear distribution is relaxed then the inertia coupling in the derivatives of displacements disappears and the natural boundary conditions do not involve time derivatives. The numerical performance is also improved at no extra cost. The
2. Let the displacement
DISPLACEMENT
ASSUMPTIONS
field for a beam of height h be u* = 0,
24,= ZJ,+ 238,
u3 = w,
(1)
where x and z are the axial and thickness co-ordinates of the beam, w(x, t) is the transverse displacement, and I(I(x, 1) and 0(x, t) are the generalized displacements representing the linear and cubic variations of u,(x, z, t) across the thickness. The non-vanishing strains are E, = z$‘+ z38’
and
yxv = lj + w’+ 3z2e.
(2)
The relationship 8 = -4($+
w’)/3Z?
(3)
is implied in the Reddy theory. Hamilton’s principle [5] gives the following appropriate natural and essential boundary conditions respectively: M = E(Zz$‘+ I‘@‘), Q=G(AJI+3Z,e+Aw’);
P = E(z&‘+
Z#), (4)
MS+ + P60 + QSw = 0.
Also the equilibrium equations M’ - Q = p(
Z,ij + Zz,di),
P’-
R = p( Z,j; + Z&,
Q’=pA+-q,
(5)
are obtained, where R = G(3Z21(,+9Z,8+3Z2w’), q(x, t) is the applied transverse excitation, and Z, = j z” dA, n = 2,4,6, in which A is the cross-sectional area and dots denote time derivatives. 527 0022460X/90/210527+02
%03.00/O
@ 1990
Academic
Press
Limited
528
LETTERS TO THE EDITOR 3. EXAMPLES
For a simply supported beam of length L, (Tj}-l(
(6)
~~~~~~~}sin~r,
and one has, from equation (5), EZ&+GA, EZqa2,+3GZ2,
EZ,cu2,+3GZ,, EZ,&+9GZ,,
(7)
GAan,
where (Y,= n7r/ L. For rectangular cross-sections, Z,= Ah2/12,
Z., = 3Zh2/20
and
Z6= 3Zh4/ 112.
(8)
When the thickness shear mode is of interest, n = 0, 35-45A+3hZ=0,
A = p~‘r’/G,
r2 = h2/12,
(9)
the smaller root of which is A =0*8229. When compared with the exact value [6], h = O-8225, the present theory is seen to perform better than the Reddy theory, which gives A = 0.8235. 4.
DISCUSSION
Due to the absence of the second order term in u(x, z, t), the present theory is suitable for beam sections having one axis of symmetry. The moments of area Z,,take good account of non-rectangular beam sections. Although the distribution of the shear stress across the thickness is still parabolic, the optimal distribution will be found through the variational principle; in some cases, the shear stress may not be zero along the upper and lower edges, due to the truncation of the power expansion. However, the theory ensures that the modes obtained are optimal under the third order theories. Department of Civil & Structural Engineering, University of Hong Kong, Hong Kong
A. Y. T. LEUNG
(Received 6 December 1989)
REFERENCES 1. J. T. S. WANG and J. N. DICKSON 1979 American Institute of Aeronautics and Astronautics Journal 17, 535-537. Elastic beams of various orders. 2. M. LEVINSON 1981 Journal of Sound and Vibration 74, 81-87. A new rectangular beam theory. 3. W. B. BICKFORD 1982 Developments in Theoretical and Applied Mechanics 11, 137-150. A consistent higher order beam theory. 4. P. R. HEYLIGER and J. N. REDDY 1988 Journal of Sound and Vibration 126,309-326. A higher order beam finite element for bending and vibration problems. 5. R. W. CLOUGH and J. PENZIEN 1975 Dynamics of Structures. Tokyo: McGraw-Hill. 6. H. LAMB 1945 Hydrodynamics. New York: Dover.