Acta Mechanica 177, 191–197 (2005) DOI 10.1007/s00707-005-0223-5
Acta Mechanica Printed in Austria
Note
A refined beam theory based on the refined plate theory Y. Gao and M. Z. Wang, Beijing, P. R. China Received: May 26, 2004; revised: November 29, 2004 Published online: May 4, 2005 Ó Springer-Verlag 2005
Summary. Based on the refined plate theory, a refined theory of rectangular beams is derived by using the Papkovich-Neuber solution and Lur’e method without ad hoc assumptions. It is shown that the displacements and stresses of the beam can be represented by the angle of rotation and the deflection of the neutral surface. The solutions based on the new theory are the same as the exact solutions of elasticity theory. In three examples it is shown that the new theory provides as good or better results than Levinson’s beam theory when compared to those obtained from the linear theory of elasticity.
1 Introduction The beam theory has been studied for many years. In the early 18th century, Bernoulli and Euler presented the classical beam theory, Timoshenko [1] introduced the shear theory of beams, then Cowper [2] gave the shear coefficients. Since then, more and more works on the subject have been investigated by the researchers, e.g., Levinson [3], Fan and Widera [4], and Tullini and Savoia [5]. Cheng [6] gave a refined plate theory from the Boussinesq-Galerkin elasticity solution and the Lur’e method [7] without ad hoc assumptions. The final results obtained by his method can be justified by the satisfaction of all equations in the three-dimensional theory, and the only approximation in Cheng’s theory is the approximate specification of boundary conditions at the edges of plates; therefore, regarding Saint-Venant’s principle, Cheng’s theory is a very accurate one. A parallel development of Cheng’s theory by Barrett and Ellis [8] has been obtained for the plates under transverse surface loadings. It also presents a detailed discussion on the specification of boundary conditions in light of the work of Gregory and Wan [9], [10]. Wang and Shi [11] derived a new thick plate theory by using the Papkovich-Neuber solution and Lur’e method [7] without ad hoc assumptions, and derived a shear theory of plates from the refined plate theory. Moreover, from the nonuniqueness of the Papkovich-Neuber solution, a rigorous proof was given: the deformation of bending plates may be described by three generalized displacements on the neutral surface of the plate without loss in generality. Yin and Wang [12] extended it for the transversely isotropic plates by using the Elliott-Lodge solution. Recently, Gao and Wang [13], [14] extended [11] for the magnetoelastic beams and thermoelastic beams, and derived the refined theory of beams in the coupling fields.
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This paper presents the theory for an elastic beam of narrow rectangular cross-section using the plate method developed by Wang and Shi [11]. Accordingly, based on elasticity theory, the refined theory of rectangular beams is derived using the Papkovich-Neuber solution and Lur’e method without ad hoc assumptions. In the end, three examples are examined to illustrate the application of the new theory and to compare the results with the known exact and approximate solutions.
2 Lur’e method We consider a straight beam of narrow rectangular cross-section as a plane stress problem. In a fixed rectangular coordinate system, z is the coordinate normal to the neutral surface (x y plane) of the beam. We assume the beam length in x-direction is l, the beam width in y-direction is assumed as 1, the beam height in z-direction is h, and l h 1. In the absence of body forces, the equilibrium equations of the elastic plane stress problem expressed by the displacements ux and uz are expressed as r2 ux þ
1 þ m @h ¼ 0; 1 m @x
r2 uz þ
1 þ m @h ¼ 0; 1 m @z
ð1Þ
where r2 ¼ @ 2 =@x2 þ @ 2 =@z2 is the two-dimensional Laplacian operator, h ¼ @ux =@x þ @uz =@z, and m is Poisson’s ratio. Now let us consider the case that the beam is subject only to the transverse surface loadings, i.e., sxz ¼ 0;
rz ¼ q=2
ðz ¼ h=2Þ:
ð2Þ
The Papkovich-Neuber solution of the governing equations (1) can be obtained as ux ¼ P1
1þm @ ðP0 þ xP1 þ zP3 Þ; 4 @x
uz ¼ P3
1þm @ ðP0 þ xP1 þ zP3 Þ; 4 @z
where Pi ði ¼ 0; 1; 3Þ is a harmonic function, @ 2 Pi @ 2 : þ @ P ¼ 0 i ¼ 0; 1; 3; @ ¼ r2 Pi ¼ x x i @x @z2
ð3Þ
ð4Þ
The problem of beams may be decomposed into two fundamental problems: the extension of a beam by the symmetrical surface loadings and the bending of a beam by the anti-symmetrical surface loadings. In the case of the bending of a beam, the beam is subject only to antisymmetrical loadings and edge conditions, thus only odd functions of z are required for ux and even functions of z for uz . For the Lur’e method [7], satisfying these requirements and treating Eqs. (4) as an ordinary differential equation in z with constant coefficients, one obtains the following symbolic solution of Eqs. (4): P1 ðx; zÞ ¼
sinðz@x Þ g1 ðxÞ; @x
P3 ðx; zÞ ¼ cosðz@x Þg3 ðxÞ;
ð5Þ
where g1 and g3 are unknown functions of x yet to be determined, and sinðz@x Þ=@x and cosðz@x Þ have following symbolic expressions: sinðz@x Þ 1 2 2 1 4 4 1 1 ¼ z 1 z @x þ z @x ; cosðz@x Þ ¼ 1 z2 @x2 þ z4 @x4 : ð6Þ @x 3! 5! 2! 4!
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From Appendix A of [13], we know that the harmonic function P0 always can satisfy the following expression without loss in generality: P0 þ xP1 þ zP3 ¼ z cosðz@x Þf ð xÞ;
ð7Þ
where f ð xÞ ¼
Z
x
g1 ðtÞdt g3 ð xÞ:
0
Substituting Eqs. (5) and (7) into Eqs. (3), one obtains ux ¼
sinðz@x Þ 1þm z cosðz@x Þf 0 ; g1 þ @x 4
ð8Þ
1 þ m cosðz@x Þ z@x sinðz@x Þ : uz ¼ cosðz@x Þg3 þ 4
where the differential symbol ‘‘ 0 ’’ denotes differentiation with respect to x. The angle of rotation and the deflection of the neutral surface can be found to be @ux 1þm 0 1þm f f: ð9Þ ; w ¼ uz jz¼0 ¼ g3 þ ¼ g þ w¼ 1 4 4 @z z¼0 From Eqs. (8) and (9), the final expressions for the displacements are sinðz@x Þ 1þm sinðz@x Þ 0 ux ¼ z cosðz@x Þ f; wþ @x 4 @x
ð10Þ
1þm z@x sinðz@x Þf ; uz ¼ cosðz@x Þw 4 with f ð xÞ ¼
Z
x
wðtÞdt þ wð xÞ :
0
Equations (10) are the displacement expressions by w and w.
3 The refined theory of beams Using Hooke’s law, from Eqs. (10) the stress components rx , sxz and rz can be written as E 1 m sinðz@x Þ 4 sinðz@x Þ 0 00 rx ¼ z cosðz@x Þ f þ w ; 4 1 þ m @x 1 þ m @x 1þm sxz ¼ l cosðz@x Þðw w0 Þ þ z@x sinðz@x Þf 0 ; 2 E 1 m sinðz@x Þ 4 sinðz@x Þ 00 00 rz ¼ þ z cosðz@x Þ f þ w ; 4 1 þ m @x 1 þ m @x
ð11Þ
where E and l ¼ E=2ð1 þ mÞ are the Young’s modulus and the shear modulus of the beam, respectively. Substituting Eqs. (11) into the conditions (2), and neglecting higher-order terms by using Eqs. (6), we obtain
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2þm 2 2 m h @x w 1 þ h2 @x2 w0 ¼ 0; 1 8 8
2þm 2 2 0 m 2ð1 þ mÞ h @x w 1 þ h2 @x2 w00 ¼ q: 1 24 24 Eh
ð12Þ
Simplifying Eqs. (12), it can be seen that w w0
h2 ½ð2 þ mÞw00 þ mw000 ¼ 0; 8
2 0 2ð1 þ mÞ ðw w00 Þ ¼ q: 3 Eh
ð13Þ
If q is constant, Eqs. (13) turn out to be w w0
1 þ m 2 00 h w ¼ 0; 4
2 0 2ð1 þ mÞ ðw w00 Þ ¼ q: 3 Eh
ð14Þ
The bending moment M ðxÞ and the shear force Qð xÞ can be found to be M¼
Z
h=2
z rx dz;
Q¼
h=2
Z
h=2
ð15Þ
sxz dz: h=2
Assuming that q is a linear function, substituting Eqs. (11) into Eqs. (15) and neglecting higherorder terms by using Taylor series of the trigonometric functions (6), we get M¼
EI ð2 þ 5mÞ 8 þ 5m 0 w þ w00 ; 10ð1 þ vÞ 2 þ 5m
2 Q ¼ lhðw w0 Þ: 3
ð16Þ
Equations (13) and (16) make up the new refined beam theory. From the following three examples, it can be shown that the solutions derived by the new theory are the exact solutions of elasticity theory [15].
4 Examples and comparison Fan and Widera [16] employed the asymptotic expansion approach and derived the proper new boundary conditions of a beam for the outer expansion without a consideration of the inner solution by adopting Gregory and Wan’s technique [9], [10]. It is interesting to note that the new boundary conditions in the stress data case are consistent with the conventional boundary conditions [15]. To illustrate the applications of the new theory developed in the previous sections, we present the following three typical examples by using the boundary conditions given by Fan and Widera [16]. Results for three examples are given for both the new theory and Levinson’s beam theory, and are compared with the known exact solutions and the solutions by Levinson [3].
4.1 The end-loaded cantilever beam Consider a cantilever beam of uniform cross-section loaded by a transverse shear force of magnitude Q0 at x ¼ 0 and clamped at x ¼ l. For the present theory, the boundary conditions are 8 þ 5m 0 w ð0Þ þ w00 ð0Þ ¼ 0; 2 þ 5m
2 lh½wð0Þ w0 ð0Þ ¼ Q0 ; 3
wðlÞ ¼ wðlÞ ¼ 0:
ð17Þ
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A refined beam theory
From Eqs. (14) and (17) one obtains as the solution for the deflection curve of the neutral surface Q0 l 3 x3 x Q0 l3 h2
x : ð18Þ 3 þ 3 2 ð1 þ mÞ 1 w¼ l l 6EI l 4EI l2 The solution based on the new theory is the exact solution of elasticity theory [15] and is the same as the solution by Levinson [3].
4.2 The uniformly loaded and simply supported beam The second example is that of a beam of uniform cross-section which is simply supported at x ¼ l and which carries a uniformly distributed load of intensity q ¼ q0 . For the new theory, the boundary conditions are 8 þ 5m 0 w ðlÞ þ w00 ðlÞ ¼ 0; 2 þ 5m
wðlÞ ¼ 0:
ð19Þ
From Eqs. (14) and (19), the solution for the deflection curve of the neutral surface is q0 l4 x4 x2 5 q 0 l 4 h2 x2 m 6 þ 5 þ 1 þ 1 : wð xÞ ¼ 8 24EI l4 l2 10EI l2 l2 The central deflection of Eq. (20) is 2 5q0 l4 12 5 h 1þ m 2 ; 1þ wð0Þ ¼ 25 8 24EI l
ð20Þ
ð21Þ
whereas the theory of Levinson solution is 5q0 l4 12 h2 wð0Þ ¼ 1 þ ð1 þ mÞ 2 : 25 24EI l
ð22Þ
The solution based on the new theory is the exact solution of elasticity theory [15]. From Eqs. (21) and (22), for this problem both of the beam theories equally overestimate the ‘‘shear correction’’ term at the center of the beam less than 14% if 0 m 0:5.
4.3 The linearly loaded cantilever beam As a third example, we consider a cantilever beam of uniform cross-section which carries a linearly distributed load qðxÞ ¼ q0 x and clamped at x ¼ l, where q0 is a constant. For the present theory, the boundary conditions are 8 þ 5m 0 w ð0Þ þ w00 ð0Þ ¼ 0; 2 þ 5m
2 lh½wð0Þ w0 ð0Þ ¼ 0; 3
wðlÞ ¼ wðlÞ ¼ 0:
Using Eqs. (13) and (23) we derive
q0 l5 x5 x q 0 l 5 h2 x3 x ; wð xÞ ¼ 5 þ4 þ ð2 þ mÞ 1 3 þ 3m 1 l l 120EI l5 48EI l2 l whereas the solution from the theory of Levinson is
ð23Þ
ð24Þ
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wð xÞ ¼
Y. Gao and M. Z. Wang
q0 l5 x5 x q0 l5 h2 x3 x þ 4 þ ð 1 þ m Þ þ 7 : 5 4 3 l l 120EI l5 120EI l2 l3
ð25Þ
The solution based on the new theory is the exact solution of elasticity theory [15] and is the same as the solution by Levinson [3] if m ¼ 2=3. Eqs. (18), (20) and (24) that we give from the new theory are the exact solutions for the neutral surface given for the case of plane stress by the linear theory of elasticity [15]. The new theory provides as good or better results than Levinson’s beam theory when compared to those obtained from the linear theory of elasticity.
5 Conclusion Based on the plate method developed by Wang and Shi [11], the refined theory for beams has been deduced directly from the elasticity theory by using the Papkovich-Neuber solution and Lur’e method without any ad hoc assumptions. The solutions based on the new theory are the same as the exact solutions of elasticity theory. By using the boundary conditions given by Fan and Widera [16], the three examples studied indicate that the refined beam theory for the loaded beams can be justified by comparing its form with that of other well-known beam theories.
Acknowledgements Support from the National Natural Science Foundation of China (Project nos. 10172003 and 10372003) is acknowledged.
References [1] Timoshenko, S. P.: On the correction for shear of the differential equation for transverse vibration of prismatic bars. Phil. Mag. 41, 744–746 (1921). [2] Cowper, G. R.: The shear coefficients in Timoshenko’s beam theory. ASME J. Appl. Mech. 33, 335–340 (1966). [3] Levinson, M.: A new rectangular beam theory. J. Sound Vibr. 74, 81–87 (1981). [4] Fan, H., Widera, G. E. O.: Refined engineering beam theory based on the asymptotic expansion approach. AIAA J. 29, 444–449 (1991). [5] Tullini, N., Savoia, M.: Elasticity interior solution for orthotropic strips and the accuracy of beam theories. ASME J. Appl. Mech. 66, 368–373 (1999). [6] Cheng, S.: Elasticity theory of plates and a refined theory. ASME J. Appl. Mech. 46, 644–650 (1979). [7] Lur’e, A. I.: Three-dimensional problems of the theory of elasticity. New York: Interscience 1964. [8] Barrett, K. E., Ellis, S.: An exact theory of elastic plates. Int. J. Solids Struct. 24, 859–880 (1988). [9] Gregory, R. D., Wan, F. Y. M.: Decaying states of plane strain in a semi-infinite strip and boundary conditions for plate theory. J. Elasticity 14, 27–64 (1984). [10] Gregory, R. D., Wan, F. Y. M.: On plate theories and Saint-Venant’s principle. Int. J. Solids Struct. 21, 1005–1024 (1985). [11] Wang, W., Shi, M. X.: Thick plate theory based on general solutions of elasticity. Acta Mech. 123, 27–36 (1997). [12] Yin, H. M., Wang, W.: A refined theory of transversely isotropic plates. Acta Sci. Natl. Univ. Pek. 37, 23–33 (2001).
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[13] Gao, Y., Wang, M. Z.: The refined theory of magnetoelastic rectangular beams. Acta Mech. 173, 147–161 (2004). [14] Gao, Y., Wang, M. Z.: A refined theory of thermoelastic beams under steady temperature. Eng. Mech. (in press). [15] Timoshenko, S. P., Goodier, J. C.: Theory of elasticity. New York: McGraw-Hill 1970. [16] Fan, H., Widera, G. E. O.: On the proper boundary conditions for a beam. ASME J. Appl. Mech. 59, 915–922 (1992). Authors’ addresses: Y. Gao, College of Science, China Agricultural University, Beijing 100083, P. R. China (E-mail:
[email protected]); M. Z. Wang, Department of Mechanics and Engineering Science, Peking University, Beijing 100871, P. R. China
