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Optimization as a generator of structural instability
Int. J. mech. 8el. Pergamon Press. 1972. Vol. 14, pp. 627-629. Printed in Great Britain
LETTER TO THE EDITOR Optimization as a generator of structural instability
(Received 28 March 1972) THE PURPOSE of this note is to demonstrate how a process of structural optimization leads to designs which are liable to exhibit the dangerous failure characteristics well known in the buckling of thin shells. That is to say an idealized optimum structure is liable to fail b y a n explosive instability, while similar real structures, containing inevitable imperfections, will fail at scattered loads which m a y be considerably less t h a n that of the idealization. A hint at this phenomenon can be gained from the following simple argument. When an optimum structure deflects, the movement will carry the structure into an adjacent configuration which can carry less load. We might thus expect the ultimate failure of the structure to be quite dramatic. Moreover, since a n y assembled structure will inevitably deviate slightly from:the required (optimum) design, its load-carrylng capacity will he lower t h a n expected; that is to say an optimum structure is b y its very nature imperfectlon-sensitive. So far there can be little disagreement with our simple argument, and the issue hinges on the severity of the phenomenon under discussion. A mildly optimum structure with not too dramatic failure characteristics and modest imperfection-sensitivity might be quite safe and acceptable; a sharply optimum design with explosive failure and severe imperfection-sensitivity might be dangerous and unacceptable. To see that optimization will often lead to severe failure characteristics, we appeal now to the old adage that for effÉcient design, structural forces should be carried as far as possible b y direct membrane stresses rather than b y bending action. Some such membrane stresses will be compressive, and it is of course precisely direct compressive stresses which give rise to structural buckling and instability. Thus for a family of perfect and imperfect struts, the straight bifurcating strut represents an optimum design as is easily demonstrated b y the well-known Perry first-yield analysis. Optimization thus leads to buckling instabilities, but worse is to come. I n a structure of any appreciable complexity optimization seems to call for simultaneity of failure loads, 1 and this is becoming a significant feature in the design of large composite structures. Unfortunately the coupling of even apparently unrelated and harmless failures can give rise to severe shell-type instability, as was predicted b y Koiter and Skaloud in 1962. 2 Here the loss of stiffness in one mode precipitates an explosive failure in another. This phenomenon, unpleasant in itself, contains an additional danger because the coupling action m a y be of a non-linear post-buckling character, so that it would not be detected b y a conventional linear buckling analysis of either a perfect or an imperfect structure. Conventional design methods may thus lead to unsafe design, which could explain the recent failures of box-girder bridges since (contrary to some established thinking) plate assemblages are particularly prone to this dramatic coupling action. 8, 4 An example of this tendency is supplied b y the buckling of a complete spherical shell under uniform external pressure. Such a shell would seem intuitively to be a highly efficient design for carrying a uniform external pressure. However, due in part to the simultaneity of m a n y buck]ing modes, the shell exhibits a most violent failure characteristic with an acute imperfection-sensitivity which can lower the failure load of a carefully mamffactured shell b y a factor of 3. Clearly the severity of failure of a structure increases with the degree of optimization achieved, and this point is nicely made b y the minimum-weight Miehell truss s to support a vertical force mid-way between two abutments at the same level. Although a superb design for carrying the vertical load, this pin-jointed frame is actually a mechanism for 627
628
L e t t e r to t h e E d i t o r
horizontal m o v e m e n t of the centre pin. The design is thus violently unstable, and has effectively an infinitely severe imperfection-sensitivity since it offers no resistance at all to a horizontal p e r t u r b a t i o n . W e see now t h a t inevitable m a n u f a c t u r i n g imperfections will erode the benefits of a n o m i n a l l y o p t i m u m design, and a q u a n t i t a t i v e s t u d y of this has recently been presented b y T h o m p s o n and Lewis. 1 This s t u d y was based on the calculations of Van der N e u t a and K o i t e r and K u i k e n 4 for the local and overall buckling of a thin-walled compression m e m b e r , and t h e findings are s u m m a r i z e d in Fig. 1. H e r e t h e h e a v y curves give the load
p01
1.0 Design poromeler,
FIG. l.
2.0
2.45
x
The erosion of an o p t i m u m design by m a n u f a c t u r i n g imperfections.
carrying c a p a c i t y P as a function of a design p a r a m e t e r x for a perfect structure which t h u s exhibits a n o p t i m u m design at A for x -- 1. The light curves show how this load is progressively lowered b y increasing flange imperfections, which in t h e v i c i n i t y of the u n s t a b l e o p t i m u m A yield a quite severe imperfection-sensitivity due to non-linear coupling action b e t w e e n t h e local a n d overall buckling modes. F o r a given initial imperfection t h e o p t i m u m value of x is seen to shift slightly to the left, while t h e o p t i m u m itself is r o u n d e d off and for large imperfections almost eliminated. The i m p a c t of these recent findings on design is n o t i m m e d i a t e l y a p p a r e n t , but it will clearly v a r y from field to field since t h e philosophies of aerospace structures differ quite radically f r o m those of civil engineering structures. Thus a weight-conscious aircraft designer m a y be t e m p t e d to accept t h e u n p l e a s a n t failure characteristics and continue to seek t h e highest possible optimization allowing as best he can for t h e r a n d o m m a n u facturing tolerances. A civil engineer on the other h a n d m a y prefer to a b a n d o n a high degree of optimization a n d a i m to build structures w i t h mild failure characteristics, in t h e same w a y as he m i g h t prefer t h e ductility and energy absorption of a mild steel as against t h e brittle fracture of a high s t r e n g t h steel.
Department of Civil and Municipal Engineering University CoU~ge, London
J.M.T.
THOMPSON
REFERENCES 1. J . M. T. THOMPSON and G. M. LEwis, J. Mech. Phys. Solids, 20, 101 (1972). 2. W. T. KOITE~ a n d M. SK.~OUD, I n t e r v e n t i o n s . I n Comportement postcritique des Plaques utilisdes en Construction mdtaUique. Colloque international t e n u k l'Universit~ de Liege, 1962. Mdmoires de la Socidtd Royale des Sciences de Liege, 5 me s4rie, t o m e V I I I , fase. 5, pp. 64-68, 103-104.
Letter to the Editor
629
3. A. VAN DER NEUT, The interaction of local buckling and column failure of thin-walled compression members. I n Proc. Twelfth Int. Congr. Appl. Mech., Stanford University, 1968. Springer-Verlag (1969}. 4. W. T. KorrsR and G. D. L. KUIKE~, The interaction between local buckling and overall buckling on the behaviour of built-up columns. Report No. 447, May 1971, Laboratory of Engineering Mechanics, Technische Hogeschool, Delft, Holland. 5. A. G. M. M~cH~.~, Phil. Mag. 8, 589 (1904).
LETTER TO THE EDITOR Optimization as a generator of structural instability
(Received 28 March 1972) THE PURPOSE of this note is to demonstrate how a process of structural optimization leads to designs which are liable to exhibit the dangerous failure characteristics well known in the buckling of thin shells. That is to say an idealized optimum structure is liable to fail b y a n explosive instability, while similar real structures, containing inevitable imperfections, will fail at scattered loads which m a y be considerably less t h a n that of the idealization. A hint at this phenomenon can be gained from the following simple argument. When an optimum structure deflects, the movement will carry the structure into an adjacent configuration which can carry less load. We might thus expect the ultimate failure of the structure to be quite dramatic. Moreover, since a n y assembled structure will inevitably deviate slightly from:the required (optimum) design, its load-carrylng capacity will he lower t h a n expected; that is to say an optimum structure is b y its very nature imperfectlon-sensitive. So far there can be little disagreement with our simple argument, and the issue hinges on the severity of the phenomenon under discussion. A mildly optimum structure with not too dramatic failure characteristics and modest imperfection-sensitivity might be quite safe and acceptable; a sharply optimum design with explosive failure and severe imperfection-sensitivity might be dangerous and unacceptable. To see that optimization will often lead to severe failure characteristics, we appeal now to the old adage that for effÉcient design, structural forces should be carried as far as possible b y direct membrane stresses rather than b y bending action. Some such membrane stresses will be compressive, and it is of course precisely direct compressive stresses which give rise to structural buckling and instability. Thus for a family of perfect and imperfect struts, the straight bifurcating strut represents an optimum design as is easily demonstrated b y the well-known Perry first-yield analysis. Optimization thus leads to buckling instabilities, but worse is to come. I n a structure of any appreciable complexity optimization seems to call for simultaneity of failure loads, 1 and this is becoming a significant feature in the design of large composite structures. Unfortunately the coupling of even apparently unrelated and harmless failures can give rise to severe shell-type instability, as was predicted b y Koiter and Skaloud in 1962. 2 Here the loss of stiffness in one mode precipitates an explosive failure in another. This phenomenon, unpleasant in itself, contains an additional danger because the coupling action m a y be of a non-linear post-buckling character, so that it would not be detected b y a conventional linear buckling analysis of either a perfect or an imperfect structure. Conventional design methods may thus lead to unsafe design, which could explain the recent failures of box-girder bridges since (contrary to some established thinking) plate assemblages are particularly prone to this dramatic coupling action. 8, 4 An example of this tendency is supplied b y the buckling of a complete spherical shell under uniform external pressure. Such a shell would seem intuitively to be a highly efficient design for carrying a uniform external pressure. However, due in part to the simultaneity of m a n y buck]ing modes, the shell exhibits a most violent failure characteristic with an acute imperfection-sensitivity which can lower the failure load of a carefully mamffactured shell b y a factor of 3. Clearly the severity of failure of a structure increases with the degree of optimization achieved, and this point is nicely made b y the minimum-weight Miehell truss s to support a vertical force mid-way between two abutments at the same level. Although a superb design for carrying the vertical load, this pin-jointed frame is actually a mechanism for 627
628
L e t t e r to t h e E d i t o r
horizontal m o v e m e n t of the centre pin. The design is thus violently unstable, and has effectively an infinitely severe imperfection-sensitivity since it offers no resistance at all to a horizontal p e r t u r b a t i o n . W e see now t h a t inevitable m a n u f a c t u r i n g imperfections will erode the benefits of a n o m i n a l l y o p t i m u m design, and a q u a n t i t a t i v e s t u d y of this has recently been presented b y T h o m p s o n and Lewis. 1 This s t u d y was based on the calculations of Van der N e u t a and K o i t e r and K u i k e n 4 for the local and overall buckling of a thin-walled compression m e m b e r , and t h e findings are s u m m a r i z e d in Fig. 1. H e r e t h e h e a v y curves give the load
p01
1.0 Design poromeler,
FIG. l.
2.0
2.45
x
The erosion of an o p t i m u m design by m a n u f a c t u r i n g imperfections.
carrying c a p a c i t y P as a function of a design p a r a m e t e r x for a perfect structure which t h u s exhibits a n o p t i m u m design at A for x -- 1. The light curves show how this load is progressively lowered b y increasing flange imperfections, which in t h e v i c i n i t y of the u n s t a b l e o p t i m u m A yield a quite severe imperfection-sensitivity due to non-linear coupling action b e t w e e n t h e local a n d overall buckling modes. F o r a given initial imperfection t h e o p t i m u m value of x is seen to shift slightly to the left, while t h e o p t i m u m itself is r o u n d e d off and for large imperfections almost eliminated. The i m p a c t of these recent findings on design is n o t i m m e d i a t e l y a p p a r e n t , but it will clearly v a r y from field to field since t h e philosophies of aerospace structures differ quite radically f r o m those of civil engineering structures. Thus a weight-conscious aircraft designer m a y be t e m p t e d to accept t h e u n p l e a s a n t failure characteristics and continue to seek t h e highest possible optimization allowing as best he can for t h e r a n d o m m a n u facturing tolerances. A civil engineer on the other h a n d m a y prefer to a b a n d o n a high degree of optimization a n d a i m to build structures w i t h mild failure characteristics, in t h e same w a y as he m i g h t prefer t h e ductility and energy absorption of a mild steel as against t h e brittle fracture of a high s t r e n g t h steel.
Department of Civil and Municipal Engineering University CoU~ge, London
J.M.T.
THOMPSON
REFERENCES 1. J . M. T. THOMPSON and G. M. LEwis, J. Mech. Phys. Solids, 20, 101 (1972). 2. W. T. KOITE~ a n d M. SK.~OUD, I n t e r v e n t i o n s . I n Comportement postcritique des Plaques utilisdes en Construction mdtaUique. Colloque international t e n u k l'Universit~ de Liege, 1962. Mdmoires de la Socidtd Royale des Sciences de Liege, 5 me s4rie, t o m e V I I I , fase. 5, pp. 64-68, 103-104.
Letter to the Editor
629
3. A. VAN DER NEUT, The interaction of local buckling and column failure of thin-walled compression members. I n Proc. Twelfth Int. Congr. Appl. Mech., Stanford University, 1968. Springer-Verlag (1969}. 4. W. T. KorrsR and G. D. L. KUIKE~, The interaction between local buckling and overall buckling on the behaviour of built-up columns. Report No. 447, May 1971, Laboratory of Engineering Mechanics, Technische Hogeschool, Delft, Holland. 5. A. G. M. M~cH~.~, Phil. Mag. 8, 589 (1904).