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In reply to some remarks by N. Challamel on our paper “Instability of elastic bodies”
MECHANICS RESEARCH COMMUNICATIONS
Mechanics Research Communications 32 (2005) 613–615 www.elsevier.com/locate/mechrescom
Reply
In reply to some remarks by N. Challamel on our paper ‘‘Instability of elastic bodies’’ Elie Absi a, Jean Lerbet b
b,*
a Formally Professor at ECP Paris, 44 Rue de Cronstadt, 75 015 Paris, France Institut Navier, Ecole Nationale des Ponts et Chausse´es, Lami, 6, 8 avenue Blaise Pascal, Cite´ Descartes Champs-sur-Marne, 77455 Marne La Valle´e Cedex 2, France
Available online 10 March 2005
1. General considerations In the paper (Absi and Lerbet, 2004), a new approach concerning the statics of conservative as non-conservative systems was presented in a classical example. The results dealt with the mass distribution in the system and the comparison between the static instability (divergence) and the dynamic one (flutter). We never claimed that the results concerning the example were general but it was proposed to illustrate the new static approach and its relation with the dynamic instability. Giving another example (the lengths of the bars not being the same) for which the new static limit of stability is greater than the dynamic one, some readers may think that the new proposed approach does not bring new elements as regards the stability of non-conservative elastic systems. These examples prove only that the relationship between static stability (from classic or new viewpoint) is not as simple as is commonly accepted. In the paper the main objective was to propose a new definition of static stability which generalizes the classical one. The chosen example that one finds in every book concerning nonconservative problems was given to illustrate our arguments: the spectral analysis of a system does not always provide the best limit of stability (the lowest one) as compared to the new limit of static stability. Sometimes, like in the chosen example, the new static approach is safer than the dynamic one. Obviously, changing the system and, for example, the lengths of bars leads to another conclusion. Probably a wiser attitude should consist in carrying both static and dynamic stability analyses. More precisely, the analysis of stability of a given elastic structure may be examined following three distinct approaches: the static classical one which defines a domain D0 of stability, the dynamic classical one which defines a domain D1 of stability, and the new static approach which defines a domain D2 of stability. For conservative systems D0 = D1(=D2) and the situation is clear. For non-conservative systems, D1 D0 and this is the reason why the stability is classically analysed through the spectral analysis and the domain *
Corresponding author. E-mail address: [email protected] (J. Lerbet).
0093-6413/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2005.02.008
614
E. Absi, J. Lerbet / Mechanics Research Communications 32 (2005) 613–615
S D0
A E
C
D2 Γ2
D1 Γ1
B Γ0
Fig. 1. General positions of stability domains.
D0 =S
D2
D1
Fig. 2. Stability domains in our example (l = l 0 ).
of stability identified with D1. The new concept of static stability proposed in our paper defines a domain D2 in the space of loads. Generally, relative positions of D0, D1 and D2 are like in Fig. 1 (in a symbolic two-dimensional load space S) without forgetting that D1 depends on mass distribution. In the considered classical example (same length for the bars), we have proved that for every mass distribution, D2 D1 D0(=S!!) (see Fig. 2). Obviously changing the system, another situation may occur like D1 D2 D0 (example proposed by Challamel (2004)). The conclusion of these considerations is that to go in the sense of the safety of structures, the stability domain should be D1 \ D2 and not D1. This was the general message of our paper. 2. The example of Beck’s column The example used for illustration was the well-known double bar system 0A1A2 also called BeckÕs column OA1 = l, A1A2 = l 0 (see Fig. 3). Calculations given for bars of the same length (l = l 0 ) (that is, the most frequently studied configuration) prove that our static approach is better than the classical dynamic approach for every mass distribution. According to Fig. 1, it means that for each mass distribution there exists a dynamic critical load (Fd) corresponding to a point C on the ‘‘arc’’ ACB. This load is out of the domain D2 and is instable according to the new static criterion (Fd > Fs).
E. Absi, J. Lerbet / Mechanics Research Communications 32 (2005) 613–615
615
F
y
A2 θ2
l'
k
A1
θ1 l x
k O Fig. 3. BeckÕs column.
In the example proposed by Challamel (2004), the envisaged system is no more the same because the author supposes that l 0 > l, only one mass distribution being envisaged (ponctual mass m in A1 and A2). For this new system, the critical load of dynamic stability Fd is better in the sense of safety than the one of static stability Fs (Fd < Fs). In Fig. 1, this situation corresponds to point E. Remark that with the same mass distribution but with l = l 0 , the points are A or B and then Fd = Fs. 3. Conclusion The previous results confirm our position and our analysis. For a given structure, we cannot know a priori which one of the two approaches (static and dynamic) is the best one insofar as safety is concerned. Future works shall then concern the study of the evolution of D1 with the mass distribution and the study of possible correlation between Fd and Fs or D1 and D2. Some new results have already been found in this way.
Acknowledgement We thank Challamel for his constructive remark that helped us to specify the domain of validity of our analysis.
References Absi, E., Lerbet, J., 2004. Instability of elastic bodies. Mechanics Research Communications 31 (1), 39–44. Challamel, N. 2004. Comments on the paper ‘‘Instability of elastic bodies’’ written by E. Absi and J. Lerbet, Mechanics Research Communications 31 (1), 39–44.
Mechanics Research Communications 32 (2005) 613–615 www.elsevier.com/locate/mechrescom
Reply
In reply to some remarks by N. Challamel on our paper ‘‘Instability of elastic bodies’’ Elie Absi a, Jean Lerbet b
b,*
a Formally Professor at ECP Paris, 44 Rue de Cronstadt, 75 015 Paris, France Institut Navier, Ecole Nationale des Ponts et Chausse´es, Lami, 6, 8 avenue Blaise Pascal, Cite´ Descartes Champs-sur-Marne, 77455 Marne La Valle´e Cedex 2, France
Available online 10 March 2005
1. General considerations In the paper (Absi and Lerbet, 2004), a new approach concerning the statics of conservative as non-conservative systems was presented in a classical example. The results dealt with the mass distribution in the system and the comparison between the static instability (divergence) and the dynamic one (flutter). We never claimed that the results concerning the example were general but it was proposed to illustrate the new static approach and its relation with the dynamic instability. Giving another example (the lengths of the bars not being the same) for which the new static limit of stability is greater than the dynamic one, some readers may think that the new proposed approach does not bring new elements as regards the stability of non-conservative elastic systems. These examples prove only that the relationship between static stability (from classic or new viewpoint) is not as simple as is commonly accepted. In the paper the main objective was to propose a new definition of static stability which generalizes the classical one. The chosen example that one finds in every book concerning nonconservative problems was given to illustrate our arguments: the spectral analysis of a system does not always provide the best limit of stability (the lowest one) as compared to the new limit of static stability. Sometimes, like in the chosen example, the new static approach is safer than the dynamic one. Obviously, changing the system and, for example, the lengths of bars leads to another conclusion. Probably a wiser attitude should consist in carrying both static and dynamic stability analyses. More precisely, the analysis of stability of a given elastic structure may be examined following three distinct approaches: the static classical one which defines a domain D0 of stability, the dynamic classical one which defines a domain D1 of stability, and the new static approach which defines a domain D2 of stability. For conservative systems D0 = D1(=D2) and the situation is clear. For non-conservative systems, D1 D0 and this is the reason why the stability is classically analysed through the spectral analysis and the domain *
Corresponding author. E-mail address: [email protected] (J. Lerbet).
0093-6413/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2005.02.008
614
E. Absi, J. Lerbet / Mechanics Research Communications 32 (2005) 613–615
S D0
A E
C
D2 Γ2
D1 Γ1
B Γ0
Fig. 1. General positions of stability domains.
D0 =S
D2
D1
Fig. 2. Stability domains in our example (l = l 0 ).
of stability identified with D1. The new concept of static stability proposed in our paper defines a domain D2 in the space of loads. Generally, relative positions of D0, D1 and D2 are like in Fig. 1 (in a symbolic two-dimensional load space S) without forgetting that D1 depends on mass distribution. In the considered classical example (same length for the bars), we have proved that for every mass distribution, D2 D1 D0(=S!!) (see Fig. 2). Obviously changing the system, another situation may occur like D1 D2 D0 (example proposed by Challamel (2004)). The conclusion of these considerations is that to go in the sense of the safety of structures, the stability domain should be D1 \ D2 and not D1. This was the general message of our paper. 2. The example of Beck’s column The example used for illustration was the well-known double bar system 0A1A2 also called BeckÕs column OA1 = l, A1A2 = l 0 (see Fig. 3). Calculations given for bars of the same length (l = l 0 ) (that is, the most frequently studied configuration) prove that our static approach is better than the classical dynamic approach for every mass distribution. According to Fig. 1, it means that for each mass distribution there exists a dynamic critical load (Fd) corresponding to a point C on the ‘‘arc’’ ACB. This load is out of the domain D2 and is instable according to the new static criterion (Fd > Fs).
E. Absi, J. Lerbet / Mechanics Research Communications 32 (2005) 613–615
615
F
y
A2 θ2
l'
k
A1
θ1 l x
k O Fig. 3. BeckÕs column.
In the example proposed by Challamel (2004), the envisaged system is no more the same because the author supposes that l 0 > l, only one mass distribution being envisaged (ponctual mass m in A1 and A2). For this new system, the critical load of dynamic stability Fd is better in the sense of safety than the one of static stability Fs (Fd < Fs). In Fig. 1, this situation corresponds to point E. Remark that with the same mass distribution but with l = l 0 , the points are A or B and then Fd = Fs. 3. Conclusion The previous results confirm our position and our analysis. For a given structure, we cannot know a priori which one of the two approaches (static and dynamic) is the best one insofar as safety is concerned. Future works shall then concern the study of the evolution of D1 with the mass distribution and the study of possible correlation between Fd and Fs or D1 and D2. Some new results have already been found in this way.
Acknowledgement We thank Challamel for his constructive remark that helped us to specify the domain of validity of our analysis.
References Absi, E., Lerbet, J., 2004. Instability of elastic bodies. Mechanics Research Communications 31 (1), 39–44. Challamel, N. 2004. Comments on the paper ‘‘Instability of elastic bodies’’ written by E. Absi and J. Lerbet, Mechanics Research Communications 31 (1), 39–44.