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Kirsch problem and the lower natural frequencies of a clamped square-plate
Ocean Engng, Vol. 24, No. 10, pp. 985-988, 1997 © 1997 ElsevierScience Ltd. All fights reserved Printed in Great Britain 00294018/97 $17.00 + 0.00
Pergamon PII: S0029-8018(97)00051-0
TECHNICAL NOTE KIRSCH PROBLEM AND THE LOWER NATURAL FREQUENCIES OF A CLAMPED SQUARE-PLATE P. A. A. Laura* and V. Sonzognit *Department of Engineering, UniversidadNacional del Sur and Institute of Applied Mechanics (CONICET), 8000 Bahia Blanca, Argentina tlNTEC (CONICET-UNL)and Facultad Regional Santa Fe (UTN), 3000 Santa Fe, Argentina (Received 10 October 1996; accepted in final form 6 November 1996)
Abstraet--A clamped square-plateis subjectedto a uniform state of stress in the x-direction. The present study deals with the basic question of determining the variation of the lower natural frequencies of transverse vibration of a plate as a small diameter, central hole is made in it, introducing a stress concentrationfield. © 1997 Elsevier Science Ltd.
1. INTRODUCTION Consider a thin plate of infinite width and length that has a circular hole at its center. The plate is subjected to a uniform tensile stress Sx' in the x-direction, see Fig. 1. It was shown in a classical paper by G. Kirsch (Timoshenko and Goodier, 1951) that the components of the plane-stress tensor are given by Sx'(
O'r~- ~ -
1-
Oo = ~ -
1+
~ t2)
/
"l-~-
Sx'(
4Ri2 T ) 1-~-+--cos20
-~-
1+3
cos 20
Ux
~t
Fig. 1. Kirsch problem. 985
(1)
.~s~
986
P. A. A. Laura and V. Sonzogni ~'r0 =
2
~ - 3
sin 20.
At the edge of the hole, r --- Ri, one has tr0 = Sx'(1 - 2 cos 20);O'r = ~'r0= 0
(2)
and, consequently, at 0 = ¢r/2 or 37r/2 the tangential stress tr0 is three times the uniform stress applied at the ends of the plate. At 0 = 0 or ¢r one has o-0 = - Sx', a compressive stress equal in magnitude to the applied tensile stress (Durelli et al., 1958). In the case of a plate of finite size with a very small, central, circular hole the basic conclusions are essentially the same. Consider now the situation in which a square plate is clamped at its four edges and subjected to a uniform stress, applied in the x-direction, at the ends of the plate. Determination of the natural frequencies of transverse vibration is a straightforward task. If, owing to service conditions (passage of ducts, cables, etc.) one makes a hole at the plate center, one has a non uniform stress field. From a designer's viewpoint it is a matter of basic structural dynamics to know, especially when the plate or slab is subjected to dynamic loads, if the natural frequencies change with respect to those encountered when the structure is simply connected. Following a previous study (Sonzogni and Laura, 1996) use will be made of a very efficient finite-element code (SAMCEF, 1994). 2. FINITE ELEMENT DETERMINATIONS The numerical results were obtained using a SAMCEF finite-element code using hybrid elements of triangular and rectangular shape (dements type 55 and 56 of the SAMCEF library). All calculations were performed with a finite-element net containing 1694 elements and 1468 nodes (Fig. 2). It is important to note that some verifications were performed using 2718 elements and 2344 nodes. The eigenvalues determined using this net practically agreed with the results obtained using the previously mentioned arrangement. Poisson's ratio was taken as equal to 0.30 in all calculations. 3. NUMERICAL RESULTS AND CONCLUSIONS Table 1 depicts values of the natural frequency coefficients of a clamped square-plate of side a determined by means of the Rayleigh-Ritz method and solving a 36 × 36 determinantal equation (Leissa, 1973). These eigenvalues are quite accurate from an engineering viewpoint (less than 1% higher than the exact values). If a small, central hole is made in the plate, the fundamental frequency decreases its value by a small amount, but starts increasing as the hole diameter becomes larger (Leissa, 1969). This is the so-called 'dynamic stiffening' effect (Laura et al., 1995). Table 2 shows values of ~'~i in the case of a solid, clamped square-plate subjected to a uniform, tensile stress resultant Sx = Sx'.h, while Table 3 deals with two cases of plates with central circular holes: (a) Ri/a = 0.05 and (b) RJa = 0.10. From the analysis of Table 2 and Table 3 one concludes that for a particular mode shape and a particular value of the applied stress parameter Sx a21D, the corresponding frequency values decrease as a hole is made in the plate (Ri/a = 0.05 and 0.10). However, the observed decrements are practically negligible from a practical viewpoint and certainly so from an experimental standpoint.
Technical Note
987
Geometric scale 10 I I
Y
Iz
X Fig. 2. Finite-element net.
Table 1.
Mode 1 2 3 4 5
Natural frequency coefficients of a clamped, thin, elastic square-plate (Leissa, 1973)
~ph/D~a 2 35.992 73.413 73.413 108.27 131.64
Note: the fifth normal mode is characterized by an inner, closed curved nodal line.
One concludes, then, that for the situation under study, the stress concentration field does not alter considerably the lower natural frequency coefficients; their values being determined primarily by the value of the applied in-plane stress. This conclusion was also reached in the case of a simply supported square-plate (Sonzogni and Laura, 1996). Another interesting feature of the present problem is the fact that the fifth natural frequency remains practically constant as Sx a21D varies from 0 (Table 1) to 100 (the increment for this wide range of values is of the order of 3%). On the other hand, the fundamental frequency experiences an increment of the order of 40% as Sx a2/D increases from 0 to 100.
P.A.A. Laura and V. Sonzogni
988
Frequency coefficients ~i of a solid, clamped square-plate, as a function of Sx a21D
Table 2.
Sx a2/D Mode 1 2 3 4 5
1
5
10
20
50
36.11 73.32 73.59 108.17 131.50
36.76 73.64 74.81 108.99 131.81
37.56 74.03 76.32 110.01 132.05
39.11 74.79 79.23 112.01 132.49
43.36 77.04 87.34 117.80 133.76
100 49.51 80.59 99.28 126.79 135.81
Note: the fifth normal mode is characterizedby an inner, closed curved nodal line. Table 3.
Frequency coefficients l~i of a clamped square-plate with a central circular hole, as a function of
Sxa21D Sx a2/D Mode (a)
Ri/a = 0.05
(b)
Rila = 0.10
1
5
10
20
50
100
1 2 3 4 5
36.02 73.32 73.58 107.95 131.07
36.68 73.63 74.80 108.78 131.25
37.49 74.02 76.30 109.80 131.46
39.05 74.79 79.20 111.82 131.89
43.34 77.05 87.27 117.65 133.15
49.53 80.61 99.16 126.71 135.20
1 2 3 4 5
35.92 73.17 73.43 107.37 130.32
36.60 73.49 74.63 108.22 130.65
37.43 73.89 76.11 109.27 130.90
39.03 74.68 78.97 111.34 131.33
43.39 76.96 86.93 117.29 132.57
49.67 80.57 98.67 126.50 134.57
Note: the fifth normal mode is characterizedby an inner, closed curved nodal line.
Acknowledgements
The present study has been sponsored by CONICET (PIA 1996-1997) and by the Secretaria General de Ciencia y Tecnologia of Universidad Nacional del Sur (Program 1997-1998). REFERENCES Durelli, A. J., Phillips, E. A. and Tsao, C. H. (1958) Introduction to the Theoretical and Experimental Analysis of Stress and Strain. McGraw-Hill, New York. Laura, P. A. A., Ercoli, L. and LaMalfa, S. (1995) Dynamic stiffening of a printed circuit board. Acoustica 81, 196. Leissa, A. W. (1969) Vibration of Plates. NASA SP 160. Leissa, A. W. (1973) The free vibration of rectangular plates. Journal of Sound and Vibration 31, 257-293. SAMCEF User's Manuals, V. 5.1 (1994) Samtech and University of Liege, Belgium. Sonzogni, V. and Laura, P. A. A. (1996) An Attempt to Model the Effect of a Stress Concentration Field on the Lower Natural Frequencies of Structural Elements. Institute of Applied Mechanics, Bahia Blanca, Argentina, Publication no. 96-38. Timoshenko, S. and Goodier, J. N. (1951) Theory of Elasticity. McGraw-Hill, New York.
Pergamon PII: S0029-8018(97)00051-0
TECHNICAL NOTE KIRSCH PROBLEM AND THE LOWER NATURAL FREQUENCIES OF A CLAMPED SQUARE-PLATE P. A. A. Laura* and V. Sonzognit *Department of Engineering, UniversidadNacional del Sur and Institute of Applied Mechanics (CONICET), 8000 Bahia Blanca, Argentina tlNTEC (CONICET-UNL)and Facultad Regional Santa Fe (UTN), 3000 Santa Fe, Argentina (Received 10 October 1996; accepted in final form 6 November 1996)
Abstraet--A clamped square-plateis subjectedto a uniform state of stress in the x-direction. The present study deals with the basic question of determining the variation of the lower natural frequencies of transverse vibration of a plate as a small diameter, central hole is made in it, introducing a stress concentrationfield. © 1997 Elsevier Science Ltd.
1. INTRODUCTION Consider a thin plate of infinite width and length that has a circular hole at its center. The plate is subjected to a uniform tensile stress Sx' in the x-direction, see Fig. 1. It was shown in a classical paper by G. Kirsch (Timoshenko and Goodier, 1951) that the components of the plane-stress tensor are given by Sx'(
O'r~- ~ -
1-
Oo = ~ -
1+
~ t2)
/
"l-~-
Sx'(
4Ri2 T ) 1-~-+--cos20
-~-
1+3
cos 20
Ux
~t
Fig. 1. Kirsch problem. 985
(1)
.~s~
986
P. A. A. Laura and V. Sonzogni ~'r0 =
2
~ - 3
sin 20.
At the edge of the hole, r --- Ri, one has tr0 = Sx'(1 - 2 cos 20);O'r = ~'r0= 0
(2)
and, consequently, at 0 = ¢r/2 or 37r/2 the tangential stress tr0 is three times the uniform stress applied at the ends of the plate. At 0 = 0 or ¢r one has o-0 = - Sx', a compressive stress equal in magnitude to the applied tensile stress (Durelli et al., 1958). In the case of a plate of finite size with a very small, central, circular hole the basic conclusions are essentially the same. Consider now the situation in which a square plate is clamped at its four edges and subjected to a uniform stress, applied in the x-direction, at the ends of the plate. Determination of the natural frequencies of transverse vibration is a straightforward task. If, owing to service conditions (passage of ducts, cables, etc.) one makes a hole at the plate center, one has a non uniform stress field. From a designer's viewpoint it is a matter of basic structural dynamics to know, especially when the plate or slab is subjected to dynamic loads, if the natural frequencies change with respect to those encountered when the structure is simply connected. Following a previous study (Sonzogni and Laura, 1996) use will be made of a very efficient finite-element code (SAMCEF, 1994). 2. FINITE ELEMENT DETERMINATIONS The numerical results were obtained using a SAMCEF finite-element code using hybrid elements of triangular and rectangular shape (dements type 55 and 56 of the SAMCEF library). All calculations were performed with a finite-element net containing 1694 elements and 1468 nodes (Fig. 2). It is important to note that some verifications were performed using 2718 elements and 2344 nodes. The eigenvalues determined using this net practically agreed with the results obtained using the previously mentioned arrangement. Poisson's ratio was taken as equal to 0.30 in all calculations. 3. NUMERICAL RESULTS AND CONCLUSIONS Table 1 depicts values of the natural frequency coefficients of a clamped square-plate of side a determined by means of the Rayleigh-Ritz method and solving a 36 × 36 determinantal equation (Leissa, 1973). These eigenvalues are quite accurate from an engineering viewpoint (less than 1% higher than the exact values). If a small, central hole is made in the plate, the fundamental frequency decreases its value by a small amount, but starts increasing as the hole diameter becomes larger (Leissa, 1969). This is the so-called 'dynamic stiffening' effect (Laura et al., 1995). Table 2 shows values of ~'~i in the case of a solid, clamped square-plate subjected to a uniform, tensile stress resultant Sx = Sx'.h, while Table 3 deals with two cases of plates with central circular holes: (a) Ri/a = 0.05 and (b) RJa = 0.10. From the analysis of Table 2 and Table 3 one concludes that for a particular mode shape and a particular value of the applied stress parameter Sx a21D, the corresponding frequency values decrease as a hole is made in the plate (Ri/a = 0.05 and 0.10). However, the observed decrements are practically negligible from a practical viewpoint and certainly so from an experimental standpoint.
Technical Note
987
Geometric scale 10 I I
Y
Iz
X Fig. 2. Finite-element net.
Table 1.
Mode 1 2 3 4 5
Natural frequency coefficients of a clamped, thin, elastic square-plate (Leissa, 1973)
~ph/D~a 2 35.992 73.413 73.413 108.27 131.64
Note: the fifth normal mode is characterized by an inner, closed curved nodal line.
One concludes, then, that for the situation under study, the stress concentration field does not alter considerably the lower natural frequency coefficients; their values being determined primarily by the value of the applied in-plane stress. This conclusion was also reached in the case of a simply supported square-plate (Sonzogni and Laura, 1996). Another interesting feature of the present problem is the fact that the fifth natural frequency remains practically constant as Sx a21D varies from 0 (Table 1) to 100 (the increment for this wide range of values is of the order of 3%). On the other hand, the fundamental frequency experiences an increment of the order of 40% as Sx a2/D increases from 0 to 100.
P.A.A. Laura and V. Sonzogni
988
Frequency coefficients ~i of a solid, clamped square-plate, as a function of Sx a21D
Table 2.
Sx a2/D Mode 1 2 3 4 5
1
5
10
20
50
36.11 73.32 73.59 108.17 131.50
36.76 73.64 74.81 108.99 131.81
37.56 74.03 76.32 110.01 132.05
39.11 74.79 79.23 112.01 132.49
43.36 77.04 87.34 117.80 133.76
100 49.51 80.59 99.28 126.79 135.81
Note: the fifth normal mode is characterizedby an inner, closed curved nodal line. Table 3.
Frequency coefficients l~i of a clamped square-plate with a central circular hole, as a function of
Sxa21D Sx a2/D Mode (a)
Ri/a = 0.05
(b)
Rila = 0.10
1
5
10
20
50
100
1 2 3 4 5
36.02 73.32 73.58 107.95 131.07
36.68 73.63 74.80 108.78 131.25
37.49 74.02 76.30 109.80 131.46
39.05 74.79 79.20 111.82 131.89
43.34 77.05 87.27 117.65 133.15
49.53 80.61 99.16 126.71 135.20
1 2 3 4 5
35.92 73.17 73.43 107.37 130.32
36.60 73.49 74.63 108.22 130.65
37.43 73.89 76.11 109.27 130.90
39.03 74.68 78.97 111.34 131.33
43.39 76.96 86.93 117.29 132.57
49.67 80.57 98.67 126.50 134.57
Note: the fifth normal mode is characterizedby an inner, closed curved nodal line.
Acknowledgements
The present study has been sponsored by CONICET (PIA 1996-1997) and by the Secretaria General de Ciencia y Tecnologia of Universidad Nacional del Sur (Program 1997-1998). REFERENCES Durelli, A. J., Phillips, E. A. and Tsao, C. H. (1958) Introduction to the Theoretical and Experimental Analysis of Stress and Strain. McGraw-Hill, New York. Laura, P. A. A., Ercoli, L. and LaMalfa, S. (1995) Dynamic stiffening of a printed circuit board. Acoustica 81, 196. Leissa, A. W. (1969) Vibration of Plates. NASA SP 160. Leissa, A. W. (1973) The free vibration of rectangular plates. Journal of Sound and Vibration 31, 257-293. SAMCEF User's Manuals, V. 5.1 (1994) Samtech and University of Liege, Belgium. Sonzogni, V. and Laura, P. A. A. (1996) An Attempt to Model the Effect of a Stress Concentration Field on the Lower Natural Frequencies of Structural Elements. Institute of Applied Mechanics, Bahia Blanca, Argentina, Publication no. 96-38. Timoshenko, S. and Goodier, J. N. (1951) Theory of Elasticity. McGraw-Hill, New York.