Differential quadrature analysis of deflection, buckling, and free vibration of beams and rectangular plates

Computers & Scrucrures Vol. 48, No. 3, pp. 473479, Printed in Gleat Britain.

1993 0

0045.7949/93 56.00 + 0.00 1993 Pergamon Prrrs Ltd

DIFFERENTIAL QUADRATURE ANALYSIS OF DEFLECTION, BUCKLING, AND FREE VIBRATION OF BEAMS AND RECTANGULAR PLATES X. WANG, C. W. BERT and A. G. STRIZ School of Aerospace and Mechanical Engineering, The University of Oklahoma, 865 Asp Avenue, Room 212, Norman, OK 73019-0601, U.S.A. (Received 30 Mmch 1992) Abstract-A new approach to apply the differential quadrature method to the deflection, buckling, and free vibration analysis of beams and plates with various boundary conditions is presented. A different method for application of the beam boundary conditions, proposed earlier by Wang and Bert, is extended to clamped-fixed, simply supported-fixed, and fixed-fixed beams and excellent results are obtained. It is found that the differential quadrature method gives Iess accurate results for the buckling load when the plate aspect ratio (u/b) exceeds 2.45 for the cases considered and possible reasons for this are discus&.

In the present study, the new approach is extended to C-F, SS-F, and F-F beams with good results. C-C beams are not yet treated by this approach since the ‘normal’ approach is most sumful for this latter case if 6 is small enough. Unfortunately, however, it is not easy to extend the present method to plates with one or more free edges. In these cases, a combination of the new approach and the general approach may be used. In the following, deflection, buckling, and free vibration analyses by the DQ method are performed in great depth for beams and plates with various boundary conditions and aspect ratios. Comparisons with analytical solutions are made, in most cases, to show the accuracy and limitation of the DQ method.

INTRODUCZION

The differential quadrature (DQ) method, introduced by Bellman and Casti [l] and elaborated upon further by Civan and Sliepcevich[2], has proven to be a rather efficient approximate technique for the direct solution of partial differential equations. Various problems in structural mechanics have been solved successfully by this method D-91. It has been found that the DQ method is ~mpu~tionally efficient and gives excellent results for the problems under investigation. To apply the DQ method to the analysis of structural components, two points, separated by a very small distance 6, are placed at each boundary point; boundary conditions are then applied to both points. This method is called the ‘normal’ approach here, since it can be used for any boundary condition. Although the method is ~n~ptually simple, the value of S and the relative spacing of the grid points affect the accuracy of the solutions. In some cases, for example, anisotropic plates with simply supported edges, the solution is very sensitive to the relative spacing of the grid points [S]. The authors have proposed a new approach for applying the DQ method to static and free vibrational analyses of beams and plates elsewhere [9]. The method is called DQMU, because only one point at each end is used to achieve uniform grid point spacing, and is very simple and eompu~tionally efficient. Exprience shows that the relative spacing of grid points has little effect on the tlnal result for the problems considered. Unfortunately, however, the new approach is only applicable to SS-SS and C-SS (or SS-C) beams (where SS is simply supported, C is clamped, and F is fixed) and their combinations for orthotropic plates or cylindrical shell panels.

DIFFERENTIALQUADRATURE MJCIHOD The DQ method approbate the partial derivative of a function at a given discrete point as a weighted linear sum of function values at all given discrete points. In one dimension (with position coordinate x), for example, the first derivative of a defle&ion function W(X) at a given discrete point i, w;(x), is approximated as

wi=Avwj,

i=1,2 ,...,

N,

(1)

where A, are known as the weighting coefficients of the first derivative, and repeated index j means summation over all values of that index (i.e., 1 to N). Coefficients A, can be determined by requiring that eqn (1) be exact for all polynomials of degree less than or equal to (N - 1). Details on determination of A, and their values for equally spaced grid points can be found in [2,3]. If the function is smooth enough,

473

474

x.

WANG

the second-order derivative can be formulated in the same way by simply replacing wj and wi in eqn (1) with wi and w; [9]. Thus, one has w; = Avw; = A,A,w,=

B,wj,

(2)

et

a/.

To apply w; = wI= 0, similar * procedures are used in formulating C$ and Dif. However, only one of the last two can be applied in a similar way, while the other should be used to express w, in terms of wi at the inner grid points. For example, if w; = 0 is used in formulating the Cz, one obtains

where Bii are the weighting coefficients of the second derivative. Weighting coefficients of the third- and fourth-order derivatives (Cii and Dij) can be obtained in the same way. Once the A, are determined, B,, C,, and Dii can be obtained easily by matrix multiplication [8,9]. APPLYING

BOUNDARY

CONDITIONS

A different method of applying boundary conditions was proposed earlier for SS-SS and CSS beams and extended to plates [9]. The essence of the new approach is that the boundary conditions are applied during formulation of the weighting coefficients for the inner grid points. It was found that the method is very simple and convenient. Excellent results for both beams and plates were obtained for appropriate boundary conditions. For C-F, SS-F, and F-F beams, slight modifications are necessary. To elaborate on the method further, consider a C-F beam, which has the boundary conditions w,=w;=O

and

[A+]{w”} = {w”‘}.

Substituting

(7)

eqn (5) into eqn (7) yields

(~“‘1= [A~IP*l{w}= [C*l{w}.

(8)

Finally (~‘7 = [A]{w”‘} = [A][C*]{w} = [D*](w).

(9)

Using eqn (8) and taking w; = 0, one has

w;=wF=O. Thus

Using eqn (1) in matrix form and taking w, = 0, one obtains an equivalent equation for the first derivative [9] Substituting eqn (10) into eqns (5) and (9), one obtains equations valid at inner grid points, namely 60

= Pl{w)

(11)

and {w’9 = [A*]{w} = {w’}.

(4)

Using eqn (2) and taking w; = 0, one has {w”} = [A*]{w’} = [A*][A*]{w}

= [B*](w).

(5)

mw1.

(12)

Similar procedures can be followed for SS-F and F-F beams. Experience shows that there is not much effect on the final results whether wi = 0 or wg = 0 is used to express w, in terms of wi (i = 2,3, . . . , N - 1) for a cantilever beam. For the results listed in Table 1,

Table 1. Summary of numerical results for a cantilever beam Deflection at free end (G) N=5

DQMU

DQMN Exact

N=6 N=7

Wit

N=7 N=8 N=9

Buckling load (N=)

Fundamental frequency (5)

0.125 (0.0%) 0.125 (0.0%) 0.125 (0.0%)

2.3478 (-4.8%) 2.4827 (0.62%) 2.4696 (0.09%)

3.3868 (-3.7%) 3.5562 (1.15%) 3.5137 (-0.05%)

0.12498 :!O.Ol6%) NA 0.125

2.4494 (-0.73%) 2.4589 (-0.34%) 2.4651 (-0.09%) 2.4674

3.5003 (-0.44%) 3.5341 (0.53%) 3.5240 (0.24%) 3.5156

t Results depend on the value of 6.

415

DQ analysis of de&&on, buckling, and free vibration of beams and plates

Table 2. Fundamental frequency (~9) of flexural vibration of a SS-F and a F-F beam

DQMU Boundary conditions

SS-F F-F

N=7

IV=8

N=9

15.510 (0.6%)

15.436 (0.1%)

15.416 (-0.01%)

15.418

22.1017 (1.5%)

22.6080 (1.O%)

22.3625 (-0.05%)

22.373

w” = 0 is used for a C-F beam; while for the results listed in Table 2, w”’ = 0 is used for a SS-F and a F-F beam. As can be seen, both give good results. As pointed out earlier, the method is not amenable to a C-C beam. In such a case, two boundary nodes, separated by a very small distance 6, are placed at each end and boundary conditions are then applied at both nodes [3, lo]. To distinguish it from the present method, this ‘normal’ approach is called DQMN. If w; = wb_ , = 0 are used to express w, and wN_, in terms of wi at inner grid points, equations similar to eqns (11) and (12) can be obtained for a c-c beam.

APPLICATIONS The governing differential equation for a slender beam undergoing small deflections can be written as follows: d2 a where EI, q, N, and rn are, respectively, the flexural rigidity, distributed lateral loading, axial compressive load, and mass per unit length. For simplicity, EI, q, and m are assumed constant, and X = x/l. In terms of differential quadrature at inner points, one obtains the following. 1. Bernoulli-Euler beam de$ection l&w, = - (l”/EI)q.

(14)

Exact[ll]

where D is the flexurai rigidity, h is the plate thickness, w is normal deflection, and p is density. In writing eqn (17) , the plate is assumed to be very thin, homogeneous, isotropic, linearly elastic, and undergoing small deflections. 1. Plate hflection Using nondimensional coordinates, X and Y, the governing equation in terms of differential quadrature is given by & wW+ 2/E&BJ, w, $ fl”& w, = (a’/D)qv

(18)

where /I = a/b is the plate aspect ratio and superscripts x and y indicate that the ~~~n~ng weighting coefficients are formulated in the x and y directions, respectively. In other words, different boundary conditions may be encountered in the x and y directions. 2. 3uckikg of a plate un&r ainiaxia/ compression Figure 1 shows a rectangular plate under uniaxial compression. The corresponding governing equation in terms of differential quadrature is

3. Transverse ~~rution of thin, ~otrop~~, rectangular plates Applying differential quadrature to the governing equation, one has

2. Slender column ~ckling b,w,=

-(12N/EI)Btiwj=

-NBvwj.

=

(15)

(@la2WZ/D)Wij=

i%‘Wg.

(20)

3. Slender beam free vibration V

P), wj = (m1402/EI)wi = cS2w,.

06)

Rectangular plates The governing equation for a rectangular plate, subjected to a lateral ~s~but~ loading q(x, y), in-plane loads N,, N,,, and NXy, and a normal inertia load, can be expressed as

---cl

Y

0 =q(x,y)+N,w,+N,w,,+2N,,w,,.

(17)

a

X

Fig. 1. Rectangular plate under uniaxial compression. Coordinate axes and numerical scheme for the sides are shown.

476

et al.

X. WANG

Table 3. Numerical results for SS-SASS-SS

rectangular plates subjected to uniform loading (v = 0.3)

%mt

clcr,Lt

cJf&axt

bla

DQMU

Analytical [ 121

DQMU

Analytical [ 121

DQMU

Analytical [12]

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 3.0 4.0 5.0 cc

0.04438 0.05319 0.06172 0.06983 0.07739 0.08437 0.09075 0.09653 0.10175 0.10643 0.11061 0.13349 0.13986 0.14156 0.14219

0.0443 0.0530 0.0616 0.0697 0.0770 0.0843 0.0906 0.0964 0.1017 0.1064 0.1106 0.1336 0.1400 0.1416 0.1422

0.04789 0.05549 0.06269 0.06940 0.07556 0.08117 0.08624 0.09079 0.09485 0.09847 0.10168 0.11878 0.12336 0.12456 0.12500

0.0479 0.0554 0.0627 0.0694 0.0755 0.0812 0.0862 0.0908 0.0948 0.0985 0.1017 0.1189 0.1235 0.1246 0.1250

0.04789 0.0493 1 0.05007 0.05032 0.05019 0.04981 0.04924 0.04857 0.04783 0.04706 0.04629 0.04060 0.03842 0.03774 0.03750

0.0479 0.0493 0.0501 0.0503 0.0502 0.0498 0.0492 0.0486 0.0479 0.0471 0.0464 0.0406 0.0384 0.0375 0.0375

t wmax= s,

qa4/Eh’; IU, = i61,qa2; My = flyqa2.

It should he emphasized that all equations are formulated only at all inner grid points and various boundary conditions (free edges for plates are not included) have been applied when Bti and Dij are calculated. Therefore, results can be obtained by solving the corresponding equations directly. It can be seen that the formulation in this paper is very convenient and efficient in computer programming. Plates with various boundary conditions can be solved for using the same procedure by simply entering different weighting coefficients. RESULTS AND DISCUSSION

In order to simplify the presentation, the edge conditions for plates are denoted by letters SS (simply supported) and C (clamped) according to the assigned edge numbers, as shown in Fig. 1. For example, SS-C-C-SS denotes that the plate is simply supported at x = 0 and y = b and clamped at y = 0 and x = a. DQMU and DQMN [9] represent the differential quadrature method by using, respectively, the present (uniform grid spacing without 6) and ‘normal’ (with 6) approaches in applying boundary Table 4. Numerical results for C-C-CC

DQMN 0.01381 0.01647 0.01883 0.02087 0.02258 0.02399 0.025 11 0.02600 0.02670 0.02723 0.02764

rectangular plates subjected to uniform loading (v = 0.3 and 6 = 0.00005)

flxt at the plate center

caxt bla 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

conditions; and DQMC is simply the combination of the two. It should be mentioned that equal spacing is used for inner grid points for all cases. Table 1 summarizes nondimensionalized results for a cantilever beam obtained by DQMU for N = 5, 6, and 7, i.e., the maximum deflection W, the buckling load NC,, and the fundamental natural frequency 6. It is seen that an exact solution is obtained by the present method for a beam subjected to a uniformly distributed load when N 2 5. The results for buckling load and fundamental frequency by DQMU with N = 7 compare very well with the exact solution. In this case, as well as in the cases of the SS-SS beam and the CSS (or SS-C) beam [9], results obtained by DQMU are all slightly better than those obtained by DQMN [lo]. The fundamental frequencies (6) of flexural vibration of a SSF and a F-F beam are shown in Table 2, as determined by the present method. All results again compare well with the analytical solution [l 11, and show good convergence with increasing N for N < 9. Unfortunately, however, the present method is not amenable to problems involving plates when one or more free edges are present, since the

iC3,t at the plate center

Analytical [ 121

DQMN

Analytical [ 12)

DQMN

Analytical [ 121

0.0138 0.0164 0.0188 0.0209 0.0266 0.0240 0.025 1 0.0260 0.0267 0.0272 0.0277

0.02290 0.02669 0.02997 0.03272 0.03497 0.03677 0.03817 0.03926 0.04008 0.04069 0.04114

0.023 1 0.0264 0.0299 0.0327 0.0349 0.0368 0.038 1 0.0392 0.0401 0.0407 0.0412

0.02290 0.023 14 0.02284 0.02216 0.02127 0.02026 0.01925 0.01827 0.01736 0.01654 0.01582

0.023 1 0.023 1 0.0228 0.0222 0.0212 0.0203 0.0193 0.0182 0.0174 0.0165 0.0158

t w,,x = %ax qa4/Eh’; M, = mxqa2; IU, = myqa2.

DQ analysis of de&&ion, buckling, and free vibration of beams and plates

477

Table 5. Numerical results for SS-C-SS-C rectangular plates subjected to uniform loading (v = 0.3 and S = 0.00005) JZXt at the plate center

%a,t

il;i,t at the plate center

hia

DQMC

Analytical f 121

DQMC

Analytical [12]

DQMC

Analytical [12]

1.0

0.02094

0.0209

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

0.02760 0.03488 0.04253 0.05035 0.058 13 0.06571 0.07296 0.0798 1 0.08618 0.09207 0.1272 0.1379 0.1409 0.1422

0.0274 0.0348 0.0424 0.0502 0.0582 0.0658 0.0730 0.0799 0.0863 0.0922 0.1276 0.1422

0.02437 0.03085 0.03769 0.04468 0.05166 0.05845 0.06495 0.07107 0.07676 0.08199 0.08676 0.1141 0.1220 0.1241 0.1250

0.0244 0.0307 0.0376 0.0444 0.05 14 0.0585 0.0650 0.0712 0.0768 0.082 1 0.0869 0.1144 0.1250

0.03325 0.03695 0.04008 0.04260 0.04454 0.04594 0.04687 0.04741 0.04763 0.04759 0.04736 0.04218 0.03903 0.03790 0.03750

0.0332 0.0371 0.0400 0.0426 0.0448 0.0460 0.0469 0.0475 0.0477 0.0476 0.0474 0.0419 0.0375

:‘x 3:o 4.0 5.0 00

boundary conditions are not, in general, satisfied by the present method. Therefore, DQMC or DQMN may be used for such cases, although the boundary conditions cannot be applied as conveniently in those cases. Numerical results for SS-SS-SS-SS plates subjected to uniformly distributed loading are listed in Table 3. A 7 x 7 grid (5 x 5 inner points) is used in conjunction with DQMU for all aspect ratios. It is interesting to see that all results (not only the maximum deflections but also the bending moments at the plate center) compare very well with the analytical solutions [12], although the same number of grid points is used in both directions for all aspect ratios. Similar data for C-C-C-C plates subjected to a uniformly distributed loading are presented in Table 4. A 9 x 9 grid (5 x 5 inner grid points) is used in conjunction with DQMU for all aspect ratios. The method is very successful in this case. However, results depend on the 6 value used with DQMN. Conceptually, the smaller the value of d, the better the solution. If 6 is too small, say lo-‘, however, the round-off error becomes predominant to make the solution worse, because the values of the weighting coefficients may differ by several orders of magnitude from each other in such a case. In the present paper, S is chosen as 0.00005, determined by trial and error. When this value is used in analyzing a C-C beam,

excellent results are obtained for maximum deflection G (0.002603 l), buckling load Ecr (39.352), and fundamental frequency (22.3732), as compared with the corresponding exact solutions (0.0026042, 39.4784, and 22.3733, respectively). Numerical data for SS-C-SSC plates subjected to a uniformly distributed loading are shown in Table 5 for various aspect ratios. The results are obtained by DQMC with 5 x 5 inner grid points. It should be emphasized that B@and eji were also obtained for a C-C beam before they were used in analyzing the plate problems. In that case, the method could be easily combined with DQMU and to make the computations more efficient. It can be seen that very good results are obtained by DQMC, not only for the deflections but also for the bending moments. The fundamental frequencies of flexural vibration of rectangular plates were surnrna~z~ in Table 5 for various boundary conditions (without free edges) and aspect ratios. When S = 0.00005, results for C-C-C-C obtained by DQMN (5 x 5 inner grid points) agree very well with the analytical solution by Leissa [l 1f. Equally good results are obtained by DQMC for cases of SS-C-SS-C and C-C-C-SS plates. Results by DQMU for SS-SS-SASS, SS-C-SS-SS, and C-C-SS-SS plates are reported elsewhere [9$ It is seen that the DQMC method is rather convenient and successful. It is somewhat less

Table 6. Fundamental frequency (cj) of flexural vibration of plates with various boundary conditions (r52 = pha4w2/D and 6 = 0.00005) c-c-c-c 0 ;j: 1.00 1.50 2.50

SS-GSSC

C-CC-SS

DQMN

Analytical [1 I]

DQMC

Analytical [I l]

DQMC

Analytical [ 111

27.02 23.66 36.01 60.80 147.9

23.648 27.010 35.992 60.772 147.80

17.38 12.15 28.96 56.35 145.5

17.373 12.135 28.95 56.35 145.48

25.73 23.32 31.68 47.96 106.6

23.440 25.861 31.829 48.167 107.07

418

x. WANG et al. Table 7. Buckling load coefficient K of plates under uniaxial compression with various boundary conditions (6 = 0.00005, K = (b/n)Z(N,),/D) ss-ss-ss-ss

SS-C-!B-C

a/b

DQMU

Analytical [131

DQMC

Analytical [13]

0.2 0.4 0.8 1.0 1.414 2.0

21.03 8.410 4.202 4.08 4.50 4.00

27.04 8.41 4.20 4.0 4.49 4.0

27.56 9.47 7.31 7.856 6.991 6.981

27.86 9.49 1.44 7.69 7.04 6.99

convenient, however, when one or more free edges are involved. Buckling loads are determined by DQMC for various aspect ratios. Table 7 lists the dimensionless buckling load coefficient K for plates under tmiaxial compression for SS-SS-SSSS and SS-C-SS-C boundary conditions. Here, results obtained by DQMU and DQMC, respectively, compare well with the analytical solutions by Iyengar [13] for aspect ratios up to 2.0. However, the results obtained by DQMU and DQMC become poor when a/b is greater than 2.45. In such cases, the plates buckle into three or more half waves in the x direction (the loading direction). The DQ method cannot describe this periodic behavior accurately because it utilizes a polynomial fitting for the whole plate. In deflection calculations (see results listed in Tables 3-5), no periodic phenomena exist and, thus, good results are obtained even for a/b = GO.It would be expected that poor results would be obtained by DQMC for higher mode frequencies. For completeness, Table 8 lists the nondimensionalized buckling load (RX), of plates under uniaxial compression with various boundary conditions. No comparisons are made due to lack of analytical solutions. It can be seen that boundary conditions in the y direction (perpendicular to the loading direction), too, affect the buckling load. When the load direction (x direction) is clamped, as expected, the buckling load is the largest for plates clamped as well in the perpendicular direction (y direction). However, for other boundary conditions, the tendency of the buckling load seems to also depend upon the aspect ratio. Table 8. Nondimensional buckling load [(Rx),] of rectangular plates under uniaxial compression with various boundary conditions [(iv,), = (a*/D)(N,),] 0 BCs C-C-C-C c-.%-C-ss C-CC-ss cxxSss c-C-s&s-C c-SS-SSSS SS-C-SS-ss

0.2 40.34 40.15 40.02 20.83 21.14 20.95 10.57

0.4

0.8

1.0

44.09 42.70 42.82 23.72 25.00 23.52 13.61

69.90 54.98 60.21 42.01 28.41 36.13 33.63

99.10 66.36 78.27 61.68 60.28 48.03 55.92

CONCLUSIONS

In conclusion, the new approach of applying the differential quadrature method offers an even more compact and convenient procedure for static and free vibrational analyses of structural components and provides excellent results. In cases of clamped edges, 6 can be chosen to be a very small value (0.00005 in the present study), determined by trial and error, to give excellent results for deflection, buckling load, and fundamental frequency. For plate buckling analysis, it should be pointed out that the DQ method can give excellent results only for a certain range of aspect ratios. skilful typing of the manuscript by Susan Wierimaa is gratefully acknowledged.

Acknowledgement-The

REFERENcEs

1. R. E. Belhnan and J. Casti, Differential quadrature and long-term integration. J. Math. Anal. Applic. 34, 235-238 (1971). 2. F. Civan and C. M. Sliepcevich, Differential quadrature for multidimensional problems. J. Math. Anal. Appfic. 101, 423443 (1984). 3. C. W. Bert, S. K. Jang and A. G. Striz, Two new

approximate methods for analyzing free vibration of structural components. AIAA Jnl 26, 612-618 (1988). 4. S. K. Jang, C. W. Bert and A. G. Striz, Application of differential quadrature to static analysis of structural comtmnents. Int. J. Numer. Meth. Enrma -_ 28, 561-577 (1989). 5. A. G. Striz, S. K. Jang and C. W. Bert, Nonlinear bending analysis of thin circular nlates bv differential quadrature. ?hin- Walled Struct. 8, 51-62(1988). 6. C. W. Bert, A. G. Striz and S. K. Jang, Nonlinear deflection of rectangular plates by differential quadrature. Znt. Con/. Comput. Engng Sci., Atlanta, GA, Vol. I, Chap. 23, pp. iii.l-iii.4 (1988). 7. C. W. Bert, S. K. Jang and A. G. Striz, Nonlinear bending analysis of orthotropic rectangular plates by the method of differential quadrature. Comput. Mech. 5, 217-226 (1989). 8. A. N. Sherboume and M. D. Pandey, Differential quadrature method in the buckling analysis of beams and composite plates. Comput. Struct. 40, 903-913 (1991). 9. X. Wang and C. W. Bert, A new approach in applying differential quadrature to static and free vibrational analyses of beams and plates. J. Sound Vibr. 162, 566-572 (1993).

DQ analysis of de&&on, buckling, and free vibration of beams and plates 10. S. K. Jang, Application of differential quadrature to the analysis of structural components. Ph.D. d&sertation, The U~~~ty of Okiahoma, Norman, OK (1987). II. A. W. Leissa, The free vibration of rectangular plates. J. Sound Vibr. 31, 257-293 (1973).

479

12. S. Timoshenko and S. WoinowskyXrieger, Theory of P+‘u~es and &&s, 2nd Edn. McGraw-Hill, New York (1959). 13. N. G. R. Iyengar, StructuralStability of Cobms and Plates. Ellis Horwood, Chichester (1988).