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Closed-form solution for a cantilevered sectorial plate subjected to a twisting tip moment
Mechanics Research Communications 35 (2008) 491–496
Contents lists available at ScienceDirect
Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom
Closed-form solution for a cantilevered sectorial plate subjected to a twisting tip moment Benjamin T. Kennedy a, David C. Weggel a,*, David M. Boyajian a, R.E. Smelser b a The Department of Civil and Environmental Engineering, The University of North Carolina at Charlotte, 9201 University City Boulevard, Charlotte, NC 28223-0001, United States b The William States Lee College of Engineering, 310 Duke Centennial Hall, The University of North Carolina at Charlotte, 9201 University City Boulevard, Charlotte, NC 28223-0001, United States
a r t i c l e
i n f o
Article history: Received 28 January 2008 Available online 15 March 2008
a b s t r a c t A solution is presented for the deflection of a cantilevered sectorial plate with free radial edges subjected to a twisting moment at the tip. The resulting displacements for a specific plate are presented. The solution is verified using a finite element model. Ó 2008 Elsevier Ltd. All rights reserved.
Keywords: Sectorial plate Cantilever Twisting
1. Introduction Solutions to sectorial plates are readily available for those geometries with simply supported radial edges since the equations required for solution are similar to those for circular plates (Ugural, 1999). Apart from the usual classical applications, such a solution may also be useful for interface fracture investigations by extending the single contoured-cantilever beam (SCCB) specimen (Boyajian et al., 2005) to sectorial plate geometries. Sectorial plates having clamped radial edges have also been solved by superposition methods using full circular plate solutions (Conway and Huang, 1952). However, if the radial edges are free, as in a cantilevered sectorial plate, the problem becomes more complicated. Full circular plate solutions can no longer be easily manipulated to match the free boundary conditions. Nadai (1925) published solutions for an infinite sectorial plate with free radial edges subjected to two different tip moments. These solutions are valid only for regions of the plate sufficiently far from the restrained edge of the plate. Carrier and Shaw (1950) took one of Nadai’s solutions and applied eigenfunction expansions to enforce the fixed edge boundary conditions. Only a few terms of the eigenfunction expansion were required for an adequate solution, and the boundary conditions were satisfied in a relaxed way that did not adversely affect the solution. In the study of this reference, the authors discovered several errors in that solution. In this article, the authors correct these errors and verify the solution to Carrier and Shaw’s original problem using the finite element method. 2. Solution Fig. 1 shows the geometry, loading, and boundary conditions of the sectorial plate under study. A right-hand Cartesian coordinate system with the Z-axis pointing into the page and the X-axis bisecting the plate was chosen; in-plane angles measured by h, a 6 h 6 +a, are consistent with a circular-polar coordinate system. The applied twisting moment at the tip, M, is * Corresponding author. Tel.: +1 7046876189. E-mail address: [email protected] (D.C. Weggel). 0093-6413/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2008.03.001
492
B.T. Kennedy et al. / Mechanics Research Communications 35 (2008) 491–496
R
-α X
M
θ
r
+α
Y Fig. 1. Sectorial plate.
depicted by a double-headed arrow, the sense of which can be determined by application of the right-hand rule. The radius of the plate is R, and an arbitrary radial distance is denoted by r ranging from 0 6 r 6 R. The boundary conditions for the sectorial plate having free radial edges and a fixed arc at r = R, require that V h ðr; aÞ ¼ V h " ðr; aÞ ¼ M h ðr; aÞ ¼ M h ðr; aÞ ¼ 0 !# 1 o 2 o 1 o2 wðr; hÞ 1 owðr; hÞ r wðr; hÞ þ ð1 mÞ 2 Here V h ðr; hÞ ¼ D r oh or r oroh r oh 2 2 o wðr; hÞ 1 owðr; hÞ 1 o wðr; hÞ 2 þ 2 r wðr; hÞ ¼ þ or2 r or r oh2
ð1Þ
and " # 1 owðr; hÞ 1 o2 wðr; hÞ o2 wðr; hÞ þ 2 þm M h ðr; hÞ ¼ D r or r or2 oh2 Vh is the effective shear force per radial length, Mh is the bending moment per radial length, and D is the flexural rigidity of the plate. Along the edge r = R, the fixed condition is enforced wðR; hÞ ¼
owðR; hÞ ¼0 or
ð2Þ
w(R, h) is the deflection in the Z-direction. Nadai (1925) published a solution for an infinite plate that accounts for the free radial edge conditions w1 ¼
2y ln ar þ ð1 þ mÞxu M1 ð1 mÞN ð1 mÞ sin a þ ð3 þ mÞa
ð3Þ
w1 is the deflection in the Z-direction, M1 is the applied moment at the tip, m is Poisson’s ratio, N is the flexural rigidity of the plate, and a is an arbitrary, long radial length (less than infinity) that is used to normalize the solution. Nadai’s foregoing equation was written with notational differences to those used in this work. Table 1 lists these differences and the following equation gives an equivalent form of Nadai’s equation: r 1þm rh cos h ð4Þ w0 ¼ C r ln sin h þ R 2 2M Here C ¼ . D½ð1 mÞ2 sinð2aÞ þ 2að1 mÞð3 þ mÞ Since this solution is not valid for plate deflections near the fixed support, Carrier and Shaw (1950) employed eigenfunctions that met the free radial edge boundary conditions and would modify the boundary conditions at r = R. The eigenfunctions necessary to modify the solution to accommodate the fixed boundary conditions are Table 1 Notational differences Nadai (mixed coordinates)
Current work (polar coordinates)
Description
a u y x N = Eh3/[12(1 m2)] M1
2a h r sin(h) r cos(h) D = Et3/[12(1 m2)] M
Plate’s full angular dimension Angular coordinate Y-axis coordinate X-axis coordinate Flexural rigidity Applied tip moment
493
B.T. Kennedy et al. / Mechanics Research Communications 35 (2008) 491–496
wj ðr; hÞ ¼
r nj sinðnj hÞ þ bj sinððnj 2ÞhÞ R
ð5Þ
wj is part of the total deflection, and nj is the eigenvalue. The constant bj is obtained by imposing the boundary condition Vh(r, a) = 0 to yield bj ¼
nj ð1 mÞ sinðnj aÞ ½4 nj ð1 mÞ sinððnj 2ÞaÞ
ð6Þ
The radial distance r is normalized by the plate radius, R. In their original work, Carrier and Shaw chose to consider a plate of unit dimensions and normalized their solution. The boundary condition Mh(r, a) = 0 gives the characteristic equation for the values of nj nj ð1 vÞ cosðnj aÞ þ ð4 þ ð1 vÞðnj 2ÞÞbj cosððnj 2ÞaÞ ¼ 0
ð7Þ
Once the values of nj are known, the final deflection solution is wðr; hÞ ¼ w0 ðr; hÞ þ
1 X
aj wj ðr; hÞ
ð8Þ
j¼1
The boundary conditions along the fixed edge would be satisfied exactly if an infinite number of terms from the eigenexpansion were used. However, a different technique, one that satisfies the deflection and slope boundary conditions along the fixed edge in an ‘‘average” sense, is employed to reduce the number of terms required for an accurate deflection solution. The values of aj are then determined by imposing the two fixed boundary conditions in the following form: Z a wðR; hÞdh ¼ 0 ð9Þ Z0 a owðR; hÞ dh ¼ 0 ð10Þ or 0 This truncated eigenfunction expansion relaxes enforcement of the boundary conditions slightly; however, as will be shown through a finite element analysis of the plate, the method appears to be accurate enough for most applications. From these conditions, only two of the unknown values of aj may be computed; if more terms are desired, additional valid equations are required to supplement Eqs. (9) and (10). These equations may be obtained by imposing the deflection and/or slope boundary condition at specific points along the fixed support as demonstrated by the following example. 3. Example Consider a plate with the geometric and material properties as shown in Table 2. Substituting the values of a and m into Eq. (7), the first three positive, real roots nj are given in Table 3. Correspondingly, Eqs. (5) and (6) give the three eigenfunctions 4 r sin h 3þv R r 11:171 w2 ðr; hÞ ¼ ½sin ð11:171hÞ 1:756 sin ð9:171hÞ R r 13:409 ½sin ð13:409hÞ 1:062 sin ð11:409hÞ w3 ðr; hÞ ¼ R
w1 ðr; hÞ ¼
ð11Þ ð12Þ ð13Þ
Eq. (11) in Cartesian coordinates has the following linear form: w1 ðx; yÞ ¼
4 y Rð3 þ mÞ
This represents a rigid body rotation of the plate about the X-axis. It is noted that this eigenfunction for nj = 1 is always a solution of Eq. (7) regardless of the values of the constant terms.
Table 2 Example constants R a t E m M
800 mm (31.5 in.) 0.2 rad (11.44°) 6.35 mm (0.25 in.) 70 103 MPa (10 106 psi) 0.33 110 N-mm (1 lb-in.)
494
B.T. Kennedy et al. / Mechanics Research Communications 35 (2008) 491–496
Table 3 Example njs and ajs j
Eigenvalue (nj)
Eigenfunction coefficient (aj)
1 2 3
1 11.171 13.409
1.826 103 7.722 105 1.176 105
Using the first three terms of the eigenexpansion, Eq. (8) gives the final solution as wðr; hÞ ¼ w0 ðr; hÞ þ a1 w1 ðr; hÞ þ a2 w2 ðr; hÞ þ a3 w3 ðr; hÞ
ð14Þ
The wjs are given by Eqs. (11)–(13). Imposing the fixed boundary conditions of Eqs. (9) and (10) together with the condition that the deflection at the point (R, a) is equal to zero gives the first three values of aj. These values are given in Table 3. Fig. 2 shows a finite element model of the sectorial plate created in ANSYS using triangular SHELL93 elements for the purpose of validating the analytical results. Through a convergence study, it was found that a model having 625 elements renders highly accurate results. Fig. 3 shows both the numerical and closed-form deflections along the lower radius where h = +a. The two solutions have been normalized by dividing by the maximum closed-form deflection, wmax. The two solutions correspond at the wall and diverge only slightly towards the tip of the plate. Fig. 4 shows the percent error at h = +a based on the following definition: %Error ¼
ðClosed-form solutionÞ ðNumerical solutionÞ 100% Maximum closed-form solution
ð15Þ
The error between the analytical and numerical results was defined in this manner to represent the error as a percentage of the maximum deflection in the problem. It also avoids the problem of division by zero at the fixed boundary. As expected, the greatest difference occurred near the tip of the plate with an error of 9.8%. This error rapidly decreases to within 5% over 98% of the plate’s radius.
Fig. 2. ANSYS model with only 16 elements shown for clarity.
0 -0.2
w/wmax w
-0.4 Numerical Closed Form
-0.6 -0.8 -1 -1.2 0
0.2
0.4
0.6 r/R
Fig. 3. Numerical vs. closed-form deflections at h = +a.
0.8
1
495
% Error E
B.T. Kennedy et al. / Mechanics Research Communications 35 (2008) 491–496
10 9 8 7 6 5 4 3 2 1 0 -1 0
0.1
0.2
0.3
0.4
0.5 r/R
0.6
0.7
0.8
0.9
1
w//wmax
Fig. 4. Percent error of plate deflections at h = +a.
1 0.8 0.6 0.4 0.2 0 -0.2 02 -0.4 -0.6 -0.8 -1
Numerical Closed Form
-1
0
θ/ α
1
Fig. 5. Numerical vs. closed-form deflections at r/R = 1/2.
2.0 1.5
% Error
1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 -1
0
1
θ/ α Fig. 6. Percent error of plate deflections at r/R = 1/2.
Fig. 5 shows both the numerical and closed-form deflections along the arc r/R = 1/2. The two solutions are once again normalized by dividing the deflection by the maximum closed-form deflection. The results are nearly identical. Fig. 6 shows the percent error as defined by Eq. (15) along this same arc. A maximum error of 1.6% occurs at the edges and decreases to zero toward the center of the plate. 4. Conclusion The solution for the deflections of a cantilevered sectorial plate twisted by a tip moment is presented. This solution corrects errors in the Carrier and Shaw (1950) paper. This corrected solution provides an analytical solution that can be used to
496
B.T. Kennedy et al. / Mechanics Research Communications 35 (2008) 491–496
validate numerical models for plate elements. Despite the truncated eigenfunction expansion, the closed-form solution shows good agreement with the current finite element model. The highest error of 9.8% occurs near the tip. The error then drops to less than 5% over approximately 98% of the plate. References Boyajian, D.M., Davalos, J.F., Ray, I., 2005. Appraisal of the novel single contoured-cantilever beam. Materials and Structures/Matériaux et Constructions, RILEM 38 (275), 11–16. Carrier, G.F., Shaw, F.S., 1950. Some problems in the bending of thin plates. In: Elasticity – Proceedings of the Third Symposium in Applied Mathematics. The American Mathematical Society, pp. 125–128. Conway, H.D., Huang, M.K., 1952. The bending of uniformly loaded sectorial plates with clamped edges. Journal of Applied Mechanics, Transactions of the ASME 74, 5–8. Nadai, A., 1925. Die elastischen Platten. Julius Springer, Berlin. pp. 202–204. Ugural, A.C., 1999. Stresses in Plates and Shells, second ed. McGraw Hill Companies, Inc., New York. pp. 200–202.
Contents lists available at ScienceDirect
Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom
Closed-form solution for a cantilevered sectorial plate subjected to a twisting tip moment Benjamin T. Kennedy a, David C. Weggel a,*, David M. Boyajian a, R.E. Smelser b a The Department of Civil and Environmental Engineering, The University of North Carolina at Charlotte, 9201 University City Boulevard, Charlotte, NC 28223-0001, United States b The William States Lee College of Engineering, 310 Duke Centennial Hall, The University of North Carolina at Charlotte, 9201 University City Boulevard, Charlotte, NC 28223-0001, United States
a r t i c l e
i n f o
Article history: Received 28 January 2008 Available online 15 March 2008
a b s t r a c t A solution is presented for the deflection of a cantilevered sectorial plate with free radial edges subjected to a twisting moment at the tip. The resulting displacements for a specific plate are presented. The solution is verified using a finite element model. Ó 2008 Elsevier Ltd. All rights reserved.
Keywords: Sectorial plate Cantilever Twisting
1. Introduction Solutions to sectorial plates are readily available for those geometries with simply supported radial edges since the equations required for solution are similar to those for circular plates (Ugural, 1999). Apart from the usual classical applications, such a solution may also be useful for interface fracture investigations by extending the single contoured-cantilever beam (SCCB) specimen (Boyajian et al., 2005) to sectorial plate geometries. Sectorial plates having clamped radial edges have also been solved by superposition methods using full circular plate solutions (Conway and Huang, 1952). However, if the radial edges are free, as in a cantilevered sectorial plate, the problem becomes more complicated. Full circular plate solutions can no longer be easily manipulated to match the free boundary conditions. Nadai (1925) published solutions for an infinite sectorial plate with free radial edges subjected to two different tip moments. These solutions are valid only for regions of the plate sufficiently far from the restrained edge of the plate. Carrier and Shaw (1950) took one of Nadai’s solutions and applied eigenfunction expansions to enforce the fixed edge boundary conditions. Only a few terms of the eigenfunction expansion were required for an adequate solution, and the boundary conditions were satisfied in a relaxed way that did not adversely affect the solution. In the study of this reference, the authors discovered several errors in that solution. In this article, the authors correct these errors and verify the solution to Carrier and Shaw’s original problem using the finite element method. 2. Solution Fig. 1 shows the geometry, loading, and boundary conditions of the sectorial plate under study. A right-hand Cartesian coordinate system with the Z-axis pointing into the page and the X-axis bisecting the plate was chosen; in-plane angles measured by h, a 6 h 6 +a, are consistent with a circular-polar coordinate system. The applied twisting moment at the tip, M, is * Corresponding author. Tel.: +1 7046876189. E-mail address: [email protected] (D.C. Weggel). 0093-6413/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2008.03.001
492
B.T. Kennedy et al. / Mechanics Research Communications 35 (2008) 491–496
R
-α X
M
θ
r
+α
Y Fig. 1. Sectorial plate.
depicted by a double-headed arrow, the sense of which can be determined by application of the right-hand rule. The radius of the plate is R, and an arbitrary radial distance is denoted by r ranging from 0 6 r 6 R. The boundary conditions for the sectorial plate having free radial edges and a fixed arc at r = R, require that V h ðr; aÞ ¼ V h " ðr; aÞ ¼ M h ðr; aÞ ¼ M h ðr; aÞ ¼ 0 !# 1 o 2 o 1 o2 wðr; hÞ 1 owðr; hÞ r wðr; hÞ þ ð1 mÞ 2 Here V h ðr; hÞ ¼ D r oh or r oroh r oh 2 2 o wðr; hÞ 1 owðr; hÞ 1 o wðr; hÞ 2 þ 2 r wðr; hÞ ¼ þ or2 r or r oh2
ð1Þ
and " # 1 owðr; hÞ 1 o2 wðr; hÞ o2 wðr; hÞ þ 2 þm M h ðr; hÞ ¼ D r or r or2 oh2 Vh is the effective shear force per radial length, Mh is the bending moment per radial length, and D is the flexural rigidity of the plate. Along the edge r = R, the fixed condition is enforced wðR; hÞ ¼
owðR; hÞ ¼0 or
ð2Þ
w(R, h) is the deflection in the Z-direction. Nadai (1925) published a solution for an infinite plate that accounts for the free radial edge conditions w1 ¼
2y ln ar þ ð1 þ mÞxu M1 ð1 mÞN ð1 mÞ sin a þ ð3 þ mÞa
ð3Þ
w1 is the deflection in the Z-direction, M1 is the applied moment at the tip, m is Poisson’s ratio, N is the flexural rigidity of the plate, and a is an arbitrary, long radial length (less than infinity) that is used to normalize the solution. Nadai’s foregoing equation was written with notational differences to those used in this work. Table 1 lists these differences and the following equation gives an equivalent form of Nadai’s equation: r 1þm rh cos h ð4Þ w0 ¼ C r ln sin h þ R 2 2M Here C ¼ . D½ð1 mÞ2 sinð2aÞ þ 2að1 mÞð3 þ mÞ Since this solution is not valid for plate deflections near the fixed support, Carrier and Shaw (1950) employed eigenfunctions that met the free radial edge boundary conditions and would modify the boundary conditions at r = R. The eigenfunctions necessary to modify the solution to accommodate the fixed boundary conditions are Table 1 Notational differences Nadai (mixed coordinates)
Current work (polar coordinates)
Description
a u y x N = Eh3/[12(1 m2)] M1
2a h r sin(h) r cos(h) D = Et3/[12(1 m2)] M
Plate’s full angular dimension Angular coordinate Y-axis coordinate X-axis coordinate Flexural rigidity Applied tip moment
493
B.T. Kennedy et al. / Mechanics Research Communications 35 (2008) 491–496
wj ðr; hÞ ¼
r nj sinðnj hÞ þ bj sinððnj 2ÞhÞ R
ð5Þ
wj is part of the total deflection, and nj is the eigenvalue. The constant bj is obtained by imposing the boundary condition Vh(r, a) = 0 to yield bj ¼
nj ð1 mÞ sinðnj aÞ ½4 nj ð1 mÞ sinððnj 2ÞaÞ
ð6Þ
The radial distance r is normalized by the plate radius, R. In their original work, Carrier and Shaw chose to consider a plate of unit dimensions and normalized their solution. The boundary condition Mh(r, a) = 0 gives the characteristic equation for the values of nj nj ð1 vÞ cosðnj aÞ þ ð4 þ ð1 vÞðnj 2ÞÞbj cosððnj 2ÞaÞ ¼ 0
ð7Þ
Once the values of nj are known, the final deflection solution is wðr; hÞ ¼ w0 ðr; hÞ þ
1 X
aj wj ðr; hÞ
ð8Þ
j¼1
The boundary conditions along the fixed edge would be satisfied exactly if an infinite number of terms from the eigenexpansion were used. However, a different technique, one that satisfies the deflection and slope boundary conditions along the fixed edge in an ‘‘average” sense, is employed to reduce the number of terms required for an accurate deflection solution. The values of aj are then determined by imposing the two fixed boundary conditions in the following form: Z a wðR; hÞdh ¼ 0 ð9Þ Z0 a owðR; hÞ dh ¼ 0 ð10Þ or 0 This truncated eigenfunction expansion relaxes enforcement of the boundary conditions slightly; however, as will be shown through a finite element analysis of the plate, the method appears to be accurate enough for most applications. From these conditions, only two of the unknown values of aj may be computed; if more terms are desired, additional valid equations are required to supplement Eqs. (9) and (10). These equations may be obtained by imposing the deflection and/or slope boundary condition at specific points along the fixed support as demonstrated by the following example. 3. Example Consider a plate with the geometric and material properties as shown in Table 2. Substituting the values of a and m into Eq. (7), the first three positive, real roots nj are given in Table 3. Correspondingly, Eqs. (5) and (6) give the three eigenfunctions 4 r sin h 3þv R r 11:171 w2 ðr; hÞ ¼ ½sin ð11:171hÞ 1:756 sin ð9:171hÞ R r 13:409 ½sin ð13:409hÞ 1:062 sin ð11:409hÞ w3 ðr; hÞ ¼ R
w1 ðr; hÞ ¼
ð11Þ ð12Þ ð13Þ
Eq. (11) in Cartesian coordinates has the following linear form: w1 ðx; yÞ ¼
4 y Rð3 þ mÞ
This represents a rigid body rotation of the plate about the X-axis. It is noted that this eigenfunction for nj = 1 is always a solution of Eq. (7) regardless of the values of the constant terms.
Table 2 Example constants R a t E m M
800 mm (31.5 in.) 0.2 rad (11.44°) 6.35 mm (0.25 in.) 70 103 MPa (10 106 psi) 0.33 110 N-mm (1 lb-in.)
494
B.T. Kennedy et al. / Mechanics Research Communications 35 (2008) 491–496
Table 3 Example njs and ajs j
Eigenvalue (nj)
Eigenfunction coefficient (aj)
1 2 3
1 11.171 13.409
1.826 103 7.722 105 1.176 105
Using the first three terms of the eigenexpansion, Eq. (8) gives the final solution as wðr; hÞ ¼ w0 ðr; hÞ þ a1 w1 ðr; hÞ þ a2 w2 ðr; hÞ þ a3 w3 ðr; hÞ
ð14Þ
The wjs are given by Eqs. (11)–(13). Imposing the fixed boundary conditions of Eqs. (9) and (10) together with the condition that the deflection at the point (R, a) is equal to zero gives the first three values of aj. These values are given in Table 3. Fig. 2 shows a finite element model of the sectorial plate created in ANSYS using triangular SHELL93 elements for the purpose of validating the analytical results. Through a convergence study, it was found that a model having 625 elements renders highly accurate results. Fig. 3 shows both the numerical and closed-form deflections along the lower radius where h = +a. The two solutions have been normalized by dividing by the maximum closed-form deflection, wmax. The two solutions correspond at the wall and diverge only slightly towards the tip of the plate. Fig. 4 shows the percent error at h = +a based on the following definition: %Error ¼
ðClosed-form solutionÞ ðNumerical solutionÞ 100% Maximum closed-form solution
ð15Þ
The error between the analytical and numerical results was defined in this manner to represent the error as a percentage of the maximum deflection in the problem. It also avoids the problem of division by zero at the fixed boundary. As expected, the greatest difference occurred near the tip of the plate with an error of 9.8%. This error rapidly decreases to within 5% over 98% of the plate’s radius.
Fig. 2. ANSYS model with only 16 elements shown for clarity.
0 -0.2
w/wmax w
-0.4 Numerical Closed Form
-0.6 -0.8 -1 -1.2 0
0.2
0.4
0.6 r/R
Fig. 3. Numerical vs. closed-form deflections at h = +a.
0.8
1
495
% Error E
B.T. Kennedy et al. / Mechanics Research Communications 35 (2008) 491–496
10 9 8 7 6 5 4 3 2 1 0 -1 0
0.1
0.2
0.3
0.4
0.5 r/R
0.6
0.7
0.8
0.9
1
w//wmax
Fig. 4. Percent error of plate deflections at h = +a.
1 0.8 0.6 0.4 0.2 0 -0.2 02 -0.4 -0.6 -0.8 -1
Numerical Closed Form
-1
0
θ/ α
1
Fig. 5. Numerical vs. closed-form deflections at r/R = 1/2.
2.0 1.5
% Error
1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 -1
0
1
θ/ α Fig. 6. Percent error of plate deflections at r/R = 1/2.
Fig. 5 shows both the numerical and closed-form deflections along the arc r/R = 1/2. The two solutions are once again normalized by dividing the deflection by the maximum closed-form deflection. The results are nearly identical. Fig. 6 shows the percent error as defined by Eq. (15) along this same arc. A maximum error of 1.6% occurs at the edges and decreases to zero toward the center of the plate. 4. Conclusion The solution for the deflections of a cantilevered sectorial plate twisted by a tip moment is presented. This solution corrects errors in the Carrier and Shaw (1950) paper. This corrected solution provides an analytical solution that can be used to
496
B.T. Kennedy et al. / Mechanics Research Communications 35 (2008) 491–496
validate numerical models for plate elements. Despite the truncated eigenfunction expansion, the closed-form solution shows good agreement with the current finite element model. The highest error of 9.8% occurs near the tip. The error then drops to less than 5% over approximately 98% of the plate. References Boyajian, D.M., Davalos, J.F., Ray, I., 2005. Appraisal of the novel single contoured-cantilever beam. Materials and Structures/Matériaux et Constructions, RILEM 38 (275), 11–16. Carrier, G.F., Shaw, F.S., 1950. Some problems in the bending of thin plates. In: Elasticity – Proceedings of the Third Symposium in Applied Mathematics. The American Mathematical Society, pp. 125–128. Conway, H.D., Huang, M.K., 1952. The bending of uniformly loaded sectorial plates with clamped edges. Journal of Applied Mechanics, Transactions of the ASME 74, 5–8. Nadai, A., 1925. Die elastischen Platten. Julius Springer, Berlin. pp. 202–204. Ugural, A.C., 1999. Stresses in Plates and Shells, second ed. McGraw Hill Companies, Inc., New York. pp. 200–202.