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Stability of square plates with through transverse cracks
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TECHNICAL NOTE STABILITY OF SQUARE PLATES WITH THROUGH TRANSVERSE CRACKS Y. ARUN ROY, B. P. SHASTRY
and G. VENKATE~WARA RAO
Structural Design and Analysis Division, Structural Engineering Group, Vikram Sarabhai Space Centre, Trivandrum 695 022, India (Received 3 July 1989)
Abstract-Stability behaviour of square plates with transverse through cracks under uniaxial compression is studied using high precision triangular plate elements. Stability parameters for various crack lengths are presented for both simply supported and clamped square plates.
The final matrix equation governing the stability analysis is written as
INTRODUCTION The stability of square plates under various loading and boundary conditions has been the subject of studies by Timoshenko and Gere[l], Kapur and Hartz 121,Dawe[3] and others. The effect of cutouts on the stability of plates has been studied by Sabir and Chow [4]. Study of the stability of plates with abrupt material discontinuties such as through cracks is very essential in assessing their structural integrity in view of the usage of advanced materials such as very high strength steels and aluminium alloys. One such class of problems, the stability of plates with through cracks transverse to the inplane loading direction, is studied in this note using the finite element method. The high precision triangular finite element developed earlier by Cowper et al. [5] for plate bending and vibration studies is extended by Shastry et al. [6] to the stability problems of plates. This element is employed in the present study and it is found to yield very accurate upper bounds for stability parameters, even with coarse meshes. Stability parameters for various crack lengths situated at the middle line of the plate when the plate is subjected to uniaxial uniform compressive loading, when the load direction is perpendicular to the cracks, are presented for both simply supported and clamped square plates for two types of transverse through cracks.
HIGH PRECISION TRIANGULAR FINITE ELEMENT
Development of this element of stability analysis is described in detail by Shastry et al. [6] and hence only a brief description of the element is presented for the sake of completeness. Figure 1 shows a three-noded (P, , P2, and PX) arbitra~ly oriented triangular element with a centroidal node Pp (used only for stress analysis and later eliminated by static condensation). 5, n are the local coordinates and X, Y are the global coordinates. Cubic displacement distributions are assumed for the inplane displacements u and v to evaluate the inplane stiffness matrix and load vector, which are directly used to obtain the prebuckling stress distribution in the plate. The required matrices for stability analysis are obtained by assuming a quintic polynomial for the lateral displacement w. The information available from prebuckling stress analysis is used to evaluate the geometric stiffness matrix,
IWt
+Wl(S) =O,
(1)
where [K] is the assembled elastic stiffness matrix, [G] is the assembled geometric stiffness matrix, 1 is the eigenvalue (stability parameter in the present study) and {S} is the eigenvector. Equation (1) is solved by any standard algorithm to obtain the stability parameter L.
NUMERICS
RESULTS
In this note square plates, subjected to uniaxial compressive inplane loading, containing two configurations of transverse through cracks (Fig 2), as described below, are considered. C I : through cracks starting from the edges of the plate and extending towards the middle of the plate. C2: through cracks starting from the middle of the plate and extending towards the edges of the plate. A 4 x 4 finite element mesh in a quarter of the plate, by exploiting the symmetries involved, is considered in order to obtain the stability parameter. The presence of a through crack is handled by imposing appropriate boundary conditions at the p~buckling and stability analyses stages. During prebuckling analysis, to obtain stress distribution, the plate can be treated as if there is no crack, as the crack provides continuous material support for inplane transverse compressive loading. For the stability analysis, appropriate symmetric boundary conditions at the noncracked region of the plate are imposed. The nodes falling in the region of the cracks make the plate di~ontinuous for bending and this condition is simulated by not imposing any conditions for the degrees of freedom at those nodes in the stability analysis. Employing this procedure, the stability parameter I, (A = u,B*t/n*D, where Us, is the critical compressive edge stress, B is the length of the plate, t is the thickness of the plate and D is the plate flexural rigidity) is determined for both simply supported and clamped plates. The results obtained are tabulated in Tables 1 and 2 as the ratio of A/B, where A is the length of the crack in the plate, for both cases 1 and 2 referred to above.
387
Technical Note
388 Y
Table 1. Stability parameter A for simply supported plates with transverse through cracks
t
A/B
Case 1
Case 2
0.0 0.25 0.5 0.75
4.000 3.997 3.949 3.687
4.000 3.379 3.030 2.329
Table 2. Stability parameter E, for clamped plates with transverse through cracks
P,f Xl,YlI
AIB
Case 1
Case 2
0.0 0.25 0.5 0.75
10.07 9.692 9.623 8.709
10.07 8.689 5.780 4.429
Fig. 1. An arbitrarily oriented triangular plate element. It can be seen from Table 1 that for a simply supported plate, for case 1 cracks, the stability parameter is almost insensitive to the crack ratio A/S. The stability parameter. which is 4.0 for plates with no cracks, varies from 3.997 to 3.687 for case 1 when the crack ratio is increased from 0.25 to 0.75. However, for the plates with case 2 cracks, the stability parameter varies from 3.379 to 2.329, showing considerable change in the buckling strength. and reducing to almost half of that of the plate
without a crack for A/B = 0.75. For the clamped plate (Table 2) a similar observation can, in general, be made. The stability parameter varies from 9.692 to 8.709 for case I cracks and 8.689 to 4.429 for case 2 cracks, when the crack ratio is varied from 0.25 to 0.75. From these results it can be seen that, as far as the stability is concerned, the presence of central cracks is more serious than that of edge cracks for the problem considered in the present study. CONCLUDING
CRACKLENGTH
CASE.l. CRACKS EXTEN~NG FROM EDGE OF THE PLATE TOWARDS THE HMDLEOFTHE PLATE
REMARKS
The stability of plates with transverse through cracks under uniaxial uniform compressive edge load is studied using the finite element method. The effect of the crack, situated on one line of symmetry in the present study, is taken care of by applying appropriate symmetric conditions on the noncracked portion of the line of symmetry. The present study shows that edge cracks are much less serious than centre cracks, from the buckling strength point of view, for both simply supported and clamped plates.
REFERENCES
1. S. P. Timoshenko and G, M. Gere, Theory of Eimic Stability. McGraw-Hill, New York (1961). 2. K. Kapur and B. Hartz. Stability of plates using the finite element method. J. Engng Mech. Dill., ASCE 92, 177-19.5 (1966).
3. D. J. Dawe, Application of the discrete element method to the buckling analysis of rectangular plates under arbitrary membrane loading. Aeronaut. Quart. 20, 124~12g (1969).
CASE.2. CRACKS EXTENDING FROM MULE OF THE PLATE TOWARDS THE EDGE OF THE PLATE
Fig. 2. Square plate with transverse through cracks.
4. A. B. Sabir and F. Y. Chow, Finite element analysis for buckling of plates with eccentrically located -square holes. In FEICOM-85, pp. 371-382. Pergamon, Oxford. 5 G. R. Cowper, E. Kosko, G. M. Lindberg and M. D. Olson, A high precision triangular plate bending element. Aeronautical report LR-514, National Research Council of Canada (1968). 6 B. P. Shastry, G. Venkateswara Rao and M. N. Reddy, Stability of stiffened plates using high precision finite elements. Nucl. Engng Des. 36, 91-95 (1976).
TECHNICAL NOTE STABILITY OF SQUARE PLATES WITH THROUGH TRANSVERSE CRACKS Y. ARUN ROY, B. P. SHASTRY
and G. VENKATE~WARA RAO
Structural Design and Analysis Division, Structural Engineering Group, Vikram Sarabhai Space Centre, Trivandrum 695 022, India (Received 3 July 1989)
Abstract-Stability behaviour of square plates with transverse through cracks under uniaxial compression is studied using high precision triangular plate elements. Stability parameters for various crack lengths are presented for both simply supported and clamped square plates.
The final matrix equation governing the stability analysis is written as
INTRODUCTION The stability of square plates under various loading and boundary conditions has been the subject of studies by Timoshenko and Gere[l], Kapur and Hartz 121,Dawe[3] and others. The effect of cutouts on the stability of plates has been studied by Sabir and Chow [4]. Study of the stability of plates with abrupt material discontinuties such as through cracks is very essential in assessing their structural integrity in view of the usage of advanced materials such as very high strength steels and aluminium alloys. One such class of problems, the stability of plates with through cracks transverse to the inplane loading direction, is studied in this note using the finite element method. The high precision triangular finite element developed earlier by Cowper et al. [5] for plate bending and vibration studies is extended by Shastry et al. [6] to the stability problems of plates. This element is employed in the present study and it is found to yield very accurate upper bounds for stability parameters, even with coarse meshes. Stability parameters for various crack lengths situated at the middle line of the plate when the plate is subjected to uniaxial uniform compressive loading, when the load direction is perpendicular to the cracks, are presented for both simply supported and clamped square plates for two types of transverse through cracks.
HIGH PRECISION TRIANGULAR FINITE ELEMENT
Development of this element of stability analysis is described in detail by Shastry et al. [6] and hence only a brief description of the element is presented for the sake of completeness. Figure 1 shows a three-noded (P, , P2, and PX) arbitra~ly oriented triangular element with a centroidal node Pp (used only for stress analysis and later eliminated by static condensation). 5, n are the local coordinates and X, Y are the global coordinates. Cubic displacement distributions are assumed for the inplane displacements u and v to evaluate the inplane stiffness matrix and load vector, which are directly used to obtain the prebuckling stress distribution in the plate. The required matrices for stability analysis are obtained by assuming a quintic polynomial for the lateral displacement w. The information available from prebuckling stress analysis is used to evaluate the geometric stiffness matrix,
IWt
+Wl(S) =O,
(1)
where [K] is the assembled elastic stiffness matrix, [G] is the assembled geometric stiffness matrix, 1 is the eigenvalue (stability parameter in the present study) and {S} is the eigenvector. Equation (1) is solved by any standard algorithm to obtain the stability parameter L.
NUMERICS
RESULTS
In this note square plates, subjected to uniaxial compressive inplane loading, containing two configurations of transverse through cracks (Fig 2), as described below, are considered. C I : through cracks starting from the edges of the plate and extending towards the middle of the plate. C2: through cracks starting from the middle of the plate and extending towards the edges of the plate. A 4 x 4 finite element mesh in a quarter of the plate, by exploiting the symmetries involved, is considered in order to obtain the stability parameter. The presence of a through crack is handled by imposing appropriate boundary conditions at the p~buckling and stability analyses stages. During prebuckling analysis, to obtain stress distribution, the plate can be treated as if there is no crack, as the crack provides continuous material support for inplane transverse compressive loading. For the stability analysis, appropriate symmetric boundary conditions at the noncracked region of the plate are imposed. The nodes falling in the region of the cracks make the plate di~ontinuous for bending and this condition is simulated by not imposing any conditions for the degrees of freedom at those nodes in the stability analysis. Employing this procedure, the stability parameter I, (A = u,B*t/n*D, where Us, is the critical compressive edge stress, B is the length of the plate, t is the thickness of the plate and D is the plate flexural rigidity) is determined for both simply supported and clamped plates. The results obtained are tabulated in Tables 1 and 2 as the ratio of A/B, where A is the length of the crack in the plate, for both cases 1 and 2 referred to above.
387
Technical Note
388 Y
Table 1. Stability parameter A for simply supported plates with transverse through cracks
t
A/B
Case 1
Case 2
0.0 0.25 0.5 0.75
4.000 3.997 3.949 3.687
4.000 3.379 3.030 2.329
Table 2. Stability parameter E, for clamped plates with transverse through cracks
P,f Xl,YlI
AIB
Case 1
Case 2
0.0 0.25 0.5 0.75
10.07 9.692 9.623 8.709
10.07 8.689 5.780 4.429
Fig. 1. An arbitrarily oriented triangular plate element. It can be seen from Table 1 that for a simply supported plate, for case 1 cracks, the stability parameter is almost insensitive to the crack ratio A/S. The stability parameter. which is 4.0 for plates with no cracks, varies from 3.997 to 3.687 for case 1 when the crack ratio is increased from 0.25 to 0.75. However, for the plates with case 2 cracks, the stability parameter varies from 3.379 to 2.329, showing considerable change in the buckling strength. and reducing to almost half of that of the plate
without a crack for A/B = 0.75. For the clamped plate (Table 2) a similar observation can, in general, be made. The stability parameter varies from 9.692 to 8.709 for case I cracks and 8.689 to 4.429 for case 2 cracks, when the crack ratio is varied from 0.25 to 0.75. From these results it can be seen that, as far as the stability is concerned, the presence of central cracks is more serious than that of edge cracks for the problem considered in the present study. CONCLUDING
CRACKLENGTH
CASE.l. CRACKS EXTEN~NG FROM EDGE OF THE PLATE TOWARDS THE HMDLEOFTHE PLATE
REMARKS
The stability of plates with transverse through cracks under uniaxial uniform compressive edge load is studied using the finite element method. The effect of the crack, situated on one line of symmetry in the present study, is taken care of by applying appropriate symmetric conditions on the noncracked portion of the line of symmetry. The present study shows that edge cracks are much less serious than centre cracks, from the buckling strength point of view, for both simply supported and clamped plates.
REFERENCES
1. S. P. Timoshenko and G, M. Gere, Theory of Eimic Stability. McGraw-Hill, New York (1961). 2. K. Kapur and B. Hartz. Stability of plates using the finite element method. J. Engng Mech. Dill., ASCE 92, 177-19.5 (1966).
3. D. J. Dawe, Application of the discrete element method to the buckling analysis of rectangular plates under arbitrary membrane loading. Aeronaut. Quart. 20, 124~12g (1969).
CASE.2. CRACKS EXTENDING FROM MULE OF THE PLATE TOWARDS THE EDGE OF THE PLATE
Fig. 2. Square plate with transverse through cracks.
4. A. B. Sabir and F. Y. Chow, Finite element analysis for buckling of plates with eccentrically located -square holes. In FEICOM-85, pp. 371-382. Pergamon, Oxford. 5 G. R. Cowper, E. Kosko, G. M. Lindberg and M. D. Olson, A high precision triangular plate bending element. Aeronautical report LR-514, National Research Council of Canada (1968). 6 B. P. Shastry, G. Venkateswara Rao and M. N. Reddy, Stability of stiffened plates using high precision finite elements. Nucl. Engng Des. 36, 91-95 (1976).