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Elasto-plastic behaviour of double-curved shells under a concentrated load
Inr.,..Von-hror .Urchonrcr. Vol.22.No. 5,pp. 391-399.1987 Pnntcd I" Great Bntam.
CUO 7462 87 5300 + 0.00 f: I987 Pergamon Journals Ltd.
ELASTO-PLASTIC BEHAVIOUR OF DOUBLE-CURVED SHELLS UNDER A CONCENTRATED LOAD S.
A. LUKASIEWICZ
Dept. of Mechanical Engineering. The University of Calgary, Calgary. Alberta. Canada
and W. OPALINSKI Institute of Aeronautical Technology and Applied Mechanics, Warsaw Technical University. Warsaw, Poland (Received 9 February 1987; received
for publication
4
March 1987)
paper presents the application of the so-called geometrical elements method to the solution of the elasto-plastic behaviour of spherical shells subjected to an axisymmetrical concentrated load. The approach is based on the observation that during large deformations, the shell structure deforms in a nearly isometrical manner. The shell is sub-divided into elements of two kinds: purely-isometrically deformed elements and quasi-isometrically deformed elements. Equilibrium of the structure is defined by the stationariness of the total potential energy. The total energy is compared with Pogorelov’s result for the same strain energy. The solution obtained defines the large deformation behaviour and motion of the plastic zones on the surface of the shell. A simplified solution for similar problems of the shells with double positive Gaussian curvature is also presented. Abstract-The
1. INTRODUCTION
When performing experiments with spherical shells made of an elasto-plastic material we observe that, at a certain value of the load, the shell undergoes permanent deformation. The first zone which becomes plastic is in the vicinity of the point of application of the load. The deflection field corresponding to the action of the concentrated force in the initial linear stage is of local character and corresponds well to results obtained from the linear shell theory. However, when the force is further increased, the shell deforms isometrically with the depression caused by the load taking the form of a “reversed sphere” (mirror-like reflection of the original one, Fig. 1). Areas of plastic deformation appear at points of maximum curvature, regions referred to as “ridge zones” (see Fig. 1). The deformation mechanism in these ridge zones consists of the motion of these zones with the creation of new plastic regions and simultaneous unloading in the previously plastified areas. A region which was under the largest curvature will flatten and will finally bend in the reverse direction in order to become part of the central isometrically transformed area. The purpose of this paper is to present a solution which predicts behaviour in agreement with the above described process of shell deformation. 2. GEOMETRICAL
APPROACH
The problem of a spherical shell under the action of a central concentrated force using non-linear shell equations or energy methods has been studied by many authors such as Bieseno [l], Archer [3] and Ashwell [4]. The present paper follows the approach introduced by Pogorelov [S], who discussed isometric deformations of shells. He noticed that when a shell undergoes large deflections, it takes the form of its isometric transformation; i.e. geometrical bending plays the most important role in the mechanism of deformation of the shell, while strains in its middle surface are of minor importance. An isometric or inextensible transformation of a surface is, by definition, one in which the first metric tensor remains unchanged. A surface deformation during which the metric changes slightly without causing cracking or plastification in the mid-surface, is called a “quasi-isometric’* deformation. Such a deformation does not, therefore, differ much from an isometric deformation, a fact which 391
392
S. A. LUKASIEWICZ and W. OPALWKI
I!!L
QUASI
ISOMETRIC -
DEFORMATION
ISOMETRICAL
DEFORMATION
Fig. I.
makes it possible to predict the deflected shape of the shell, at least approximately, by looking for the deformed configuration among its isometric transformations. Considering a spherical shell, we come to the conclusion that the simplest isometric transfo~ation corresponds to a mirror-like reflection of the original surface. It can be assumed, therefore, that the deflection produced by a concentrated force has the approximate shape shown in Fig. 1 and consists of the dimple B-B, (region I), together with the ridge A-C-B, (region II), the latter being the region of large bending moments and curvatures. Region I, except for a small circular area around the point of load application, can be considered as a purely isomet~~ally transfo~ed area. The ridge area is the region where isometric transformation of the surface without discontinuity is impossible. Therefore, in this region, bending as well as stretching of the mid-surface occurs and the strain energy is the sum of the bending and membrane energy. If the deflections of the shell are defined, the problem can be solved using the principle of stationary value of the total potential energy 6T = 0. The energy of the system can be calculated as a function of only one geometrical parameter, namely the depth of the dimple. Pogoreiov calculated the energy in the central region and in the ridge area using additional conditions for the components of the displacement vector resulting from the assumption that the strains in the ridge, in the meridional direction, are very close to zero. Moreover, he assumed that the displacements in the ridge are of local character, and vanish rapidly at a certain distance from the ridge. The ridge was very narrow (,‘a” close to “b” in Fig. I). The experiments do not confirm these assumptions. Finally, Pogorelov obtained a very simple expression for the energy due to the change in the shape of the shell U = 2acE(2f)3/2hS’2R- ‘,
(1)
where 2f is the total deflection at the point of load application, c is a constant value, c z 0.19; E, h and R are Young’s modulus, thickness and radius of the she& respectively. Knowing equation (I), only one additional step is required to obtain the load-deflection relation, namely the calculation of the work of the external load, W = P2j’. Equating the variation of the tota potential energy to zero results in the following simple formula for the deflection of the shell p = 3mEh5’= R vm
(21
Surprisingly, this simple result is in good agreement with experimental results and results obtained by other authors. The comparison with Penning’s [13’J experimental results can
Elasto-plastic
behaviour
of double-curved
shells under a concentrated
load
393
be found in [12]. However, in the case of simultaneous appii~ation of an external concentrated load and internal pressure, some discrepancies were observed [ 1i]. These errors could be produced, first of all, by the assumption that the strain energy can be calculated for the unit length of the ridge, not taking into account the real dimensions of that area. However, the more precise calculations performed by the present authors confirmed Pogoreiov’s result. 3. ELASTIC
STATE
Consider a spherical shell of the radius R and thickness h. We can distinguish three areas in the large deformed shell: isometricai
O
quasi-isometrical
a
undeformed
b < r,
where a and b are the radii of the ridge area, a and b are unknowns to be defined {Fig. 1). The strain energy in the isometrical area is given by the formula
u = $(l
where D =
+ v)7m2,
Eh3
12(1 - v2)’
where v is Poisson’s ratio. The energy in the ridge area can be obtained assuming the deflection function w. This function should satisfy the geometrical boundary conditions of the continuity of displacements between the purely isometrically deformed inner area and the external undeformed area. The following relation was used. 2
dw dr=
i--) 1-i
1-i
(4)
where A, B are constants which can be found from the boundary conditions for r = a$
= i,andp
d2w
= 24 = arccosi.
(5)
The respective conditions for r = b are satisfied identically. The constant C is subject to variation. The membrane energy in the ridge area can be obtained by solving the shallow shell equation, which in the case of axisymmetry is reduced to the equation
(6) where 4 is the stress function. The solution for 4 can be found in the form of the series 4 = i$I airi+’ + Crr + C2i* The constants a; can be obtained comparing both sides of equation (6). C,, C2 result from the boundary conditions for the membrane state of stress which requires that for r = a,
S. A. LUKASEWCZ
394
and W. OPALINSKI
6, = 0. 0, = 0. The bending and membrane energy can be obtained from
The solution of the problem was obtained numerically minimising the total potential energy of the system with respect to parameters a, b and C only. In this way. we could check Pogorelov’s result calculating the coefficient c from the equation
where u is the total strain energy associated with the dimple on the shell surface. It was found that the value of c is almost constant and independent of h/R. The following values for R/h = 1200 were obtained for P* = 2
c=O,I74
for P* = 9
c = 0,184,
where P* = f!!_
Eh”
The above values are close to Pogorelov’s value for the coefficient c = 0,19. 4. ELASTO-PLASTIC
STATE
The efasto-plastic behaviour of a spherical shell loaded by a concentrated force has not been studied by many authors. In 1967, Leckie [9] obtained a limit load corresponding to the snap-through behaviour of a spherical shell loaded at its apex with a concentrated load applied through a rigid boss. This study was based on a so-called two-moment limited interaction criterion which allows the moment and force resultants to assume their respective ultimate values separately. However, this approach is not fully justified physically. Recently, Padovan and Tovichakchaikul (1983) using F.E.M. obtained results for the elasto-plastic collapse of a spherical cap under a central point load. The results presented by them seem to be wrong or very inaccurate [lo], Using the described geometrical method, the problem can be solved in the following way. The strain energy in the plastic state can be obtained from the assumed displacement field. However, the position of the limit surface between the elastic and plastic state must be determined first. The stress intensity is the function of the curvature changes x,, K,. (f. = +h.;
-t K:)(l
-
Vf
V’) + (4V -
1 -
VZ)K,Ktj1'2,
(11)
where t is the coordinate measured across the thickness. If we know K,, and K,, we can calculate czOacross the thickness of the shell. Then, we can check if it is smaller than the yield stress cY and define the limit value of z = zr, for c0 = rrv. The bending energy in the isometrically deformed central area, I, can be obtained very easily. The principal changes of curvatures are K, = K, = -2/R and the strains are E,
=
Es =
-
14 R
for2 < tp.
Elasto-plastic
behaviour
of double-curved
shells under a concentrated
load
39s
Using the Deformation Theory of Plasticity, we have E, = &IE)+ &!P)= ; (0, - vg,) + & E, = &i”’+ &iP)= ;(c,
(a, - s),
- vg,) + &(%
(12)
- 9,
where s = :(c, + a,). Applying a model of idea1 elasto-plastic material, we have the following condition for the yield surface a,’ 5 a,’ - a,a, + a:.
(13)
If we assume the displacements field for the shell, the state of strain is known, then the set of equations (12) can be solved with respect to the stress components. We have a, = E*(E, + v*EJ, a, = I?*(&,+ v*E,),
(14)
where E* = E
v+k v* = 1 +k’
l+k
k=!E
2 E’p”
The function k(r) can be obtained in the following way. First we define the value of zp, which gives the border between the elastic and plastic zone. At z = z,, the strains are still elastic and k = 0. In the plastic region z > zp, k # 0 and increases when coming to the external surfaces of the shell. Using a simple iterative technique, we can find its value from equations (13) and (14). The stress distribution across the thickness is linear in the elastic state. However, in the plastic region, this distribution is non-linear as we see in Fig. 2 which presents the distribution of stress across the thickness of the shell with the assumed changes of curvatures. )
Z/-
h 2
I.0 I
lY= t
0.8 -cl PLASTIC
-2-l
(a)
0
I I
I 2
I 3
I1) 4 5 &J-l00
2
(P) Fig. 2.
40
ZONE
80
120
160
200
396
S. A. LUKASIE~ICZ and W. OP~LINSKI
Knowing the state of strain and stress in the shell, we calculate the strain energy from (15)
where .sLE’,&I”’are elastic components and sip’, .s~” are plastic components of the strain tensor, and c,, or are components of the stress tensor. As we observed by performing experiments, some regions of the she11 can become unloaded, being previously in the plastic state. In this case we have to define the new elastic components of the strain and stress states. The plastic strains remain unchanged until the yield surface is reached again at its other point. The plastic strains at this point are the sum of the actual strain and residual strain from the previous step of loading, .stPRJ. E(P)= &g;,,, + &(PR’.
5. NUMERICAL
(16)
EXAMPLE
The above derived equations (12-15) were used in the analysis of the elasto-plastic behaviour of the spherical-shell under central concentrated force. The minimum of potential energy was obtained using an optimisation routine which minimised the total potential energy with respect to the three parameters a, b and C only. The routine used a nongradient method of penalty function. The results of the calculations are presented in Fig. 3 for the following parameters: radius of the sphere R = 5OOmm, thickness h = 0.5 mm, Young’s
R = 500mm h = 0.5 E = 0.8 Y = 0.35
Eh3 Fig. 3.
mm IO5
Elasto-plastic
behaviour
of double-curved
shells under a concentrated
load
397
1
v 2.12
3.18
4.24
5.30 c/h
h=0.3(mm) l
P
_
6.36
= 1.8
PR
-Eh3 Fig. 3.
modulus E = 0.8 lo5 MPa, Poisson’s ratio v = 0.35, yield stress by = 200 MPa, radius of the loaded area c = 0.54mm. These data give R/h = 1000, ay/E = 0.0025. We observe that the ridge region moves outwards as the load increases. The first plastic zone appears near the point of the application of the load for
Then the plastic zone spreads more and more as the load increases, but this zone is always very localized, (Fig.4). At the load value P* = 5.5, the plastic zone at the ridge can be observed. With the increase of the load, this zone spreads outwards. As the ridge area moves, there remains the unloaded area with residual strains. The load deflection diagram is presented in Fig. 4. We observe that the deflection of the shell in an elasto-plastic state does not deviate much from the curve given by the elastic solution, equation (2). This behaviour can be explained and justified by the fact that the shell still has a relatively large elastic core (half the thickness of the shell) in the elasto-plastic zone, which controls the mechanics of the deformations. However, the difference in stress for the same value of deflection, in the case of an elasto-plastic material and a hypothetical elastic material, is considerable. Stresses in the elastic case are 3 to 4 times larger than in the elasto-plastic analysis.
PR
I
Eh3 10.0
Fig. 5.
398
S. A. LUKASIEWKZ
and
Fig.
6. ELASTO.PLASTIC
BE~AVI~UR
W. OPALWSKI
6.
OF A DOUBLE
CURVED
SHELL
A simpli~ed solution for the eIasto-plastic case of a shetl of positive Gaussian curvature can be obtained using the assumption that the energy stored in the sheI can be cakulated for the unit length of the ridge, based on the local configuration of this area, Fig. 5. This assumption is the result of an observation made for the sphere where the boundaries of the zones were calculated as a function of the applied load and the vaiues of curvature changes at the ridge. Evaluating the total potential energy contained in the efastic and plastic zones, it was found that the toad deflection reiation for the eIa$to-plastic case was close to the corresponding elastic curve.
The resuIts obtained in this way are presented in Fig. 7a and b, which also show the plastic zones and the untoaded zones. These figures present the results for two values of applied loads. We observe that the ridge area moves in an outward direction. The plastified zones move forward leaving behind the unloaded zones with residual stresses.
PLASflC
ZONE NON -DEFORMED
RESIDUAL ELASTIC
ISOMETRICAUY OEFORMED AREA
AREA
STRAJNS UNLOADED
NON- 1SDMETRtCALLY OEfORMEO AREA Fig.
7%
ZONE
Elasto-plastic behaviour of double-curved shells under a concentrated load
399
Y
(mm)
PR :6 Eh3
~=“.a
t
I
PLASTIC
ZONE
ESIDUAL
STRAIN
Fig. 7b.
7. CONCLUSIONS
This paper presents the application of the geometrical approach to the analysis of the large, elasto-plastic deformation of spherical shells. The method appears to be effective and gives better results than F.E.M. The problem was reduced to the evaluation of a small number of unknown parameters only. REFERENCES 1. C. B. Biezeno, Uber die Bestimmung der Durchschlagkraft einer schwachgekrummten kreisformigen Platte. 2. Angew Murh. Met. 15, 10 (1935); C. B. Biezeno and R. Grammel, Engineering Dynamics, p. 484. Blackie, London (1965). 2. Chien Wei-Zang and Hu Hai-Chang, On the snapping of a thin spherical cap, 9th lnt. Congress Appl. Mech., Vol. 6. p. 309, University of Brussels (1957). 3. D. G. Ashwell, On large deflection of a spherical shell with an inward point load. In Proceedings of IUTAM Symposium Theory Thin Elastic Shells Delft, 1959, pp. 43-63. North Holland, Amsterdam (1960). 4. R. R. Archer, On the numerical solutions of the non-linear equations for shells of revolution. J. Muth. Phys. 41, 165-178 (1962). 5. A. V. Pogorelov, Geometric Shell Stability Theory. Izd. Nauka, Moscow (1966). English translation U.S. Air Force FTD-ID(RS)T-2219-78. 6. S. A. Lukasiewicz and W. Szyszkowski, Geometrical methods in the non-linear theory of shells. In Proceedings Symposium Shell Strucrures, Theory and Applications, PAN, Krakow, Poland, pp. 25-27, April (1974) in Polish. 7. S. Lukasiewicz. Geometrical elements method for the solution of non-linear shell problems. J. Aeronaut. Sot. India 37, 297-312 (1985). 8. S. Lukasiewicz and W, Szyszkowski, Geometrical analysis of large elastic deflections of axially compressed cylindrical and conical shells. Int. J. Non-linear Mech. 14, 273-284 (1979). 9. F. A. Leckie, Plastic instability of a spherical shell. In Proc. IUTAM Symposium on Theory of Thin Shells, Copenhagen (1967). 10. T. Padovan and S. Tovichakchaikul, On the solution of elastic-plastic and dynamic postbuckling collapse of general structure. Advances and Trends in Structural and Solid Mechanics, Proceedings of the Symposium, Washington, DC, 1982. Pergamon Press, Oxford. 11. S. Lukasiewicz and P. G. Glockner, Collapse by ponding of shells. lnt. J. Solids Struct. 19, 251-261 (1983). 12. S. Lukasiewin, Local Loads in Plates and Shells. SijthofI & Noordhoff, Netherlands PWN, Warsaw (1979). 13. F. A. Penning, Experimental buckling modes of clamped shallow shells under concentrated loads. J. appk Mech. 33, 297-304 (1966).
CUO 7462 87 5300 + 0.00 f: I987 Pergamon Journals Ltd.
ELASTO-PLASTIC BEHAVIOUR OF DOUBLE-CURVED SHELLS UNDER A CONCENTRATED LOAD S.
A. LUKASIEWICZ
Dept. of Mechanical Engineering. The University of Calgary, Calgary. Alberta. Canada
and W. OPALINSKI Institute of Aeronautical Technology and Applied Mechanics, Warsaw Technical University. Warsaw, Poland (Received 9 February 1987; received
for publication
4
March 1987)
paper presents the application of the so-called geometrical elements method to the solution of the elasto-plastic behaviour of spherical shells subjected to an axisymmetrical concentrated load. The approach is based on the observation that during large deformations, the shell structure deforms in a nearly isometrical manner. The shell is sub-divided into elements of two kinds: purely-isometrically deformed elements and quasi-isometrically deformed elements. Equilibrium of the structure is defined by the stationariness of the total potential energy. The total energy is compared with Pogorelov’s result for the same strain energy. The solution obtained defines the large deformation behaviour and motion of the plastic zones on the surface of the shell. A simplified solution for similar problems of the shells with double positive Gaussian curvature is also presented. Abstract-The
1. INTRODUCTION
When performing experiments with spherical shells made of an elasto-plastic material we observe that, at a certain value of the load, the shell undergoes permanent deformation. The first zone which becomes plastic is in the vicinity of the point of application of the load. The deflection field corresponding to the action of the concentrated force in the initial linear stage is of local character and corresponds well to results obtained from the linear shell theory. However, when the force is further increased, the shell deforms isometrically with the depression caused by the load taking the form of a “reversed sphere” (mirror-like reflection of the original one, Fig. 1). Areas of plastic deformation appear at points of maximum curvature, regions referred to as “ridge zones” (see Fig. 1). The deformation mechanism in these ridge zones consists of the motion of these zones with the creation of new plastic regions and simultaneous unloading in the previously plastified areas. A region which was under the largest curvature will flatten and will finally bend in the reverse direction in order to become part of the central isometrically transformed area. The purpose of this paper is to present a solution which predicts behaviour in agreement with the above described process of shell deformation. 2. GEOMETRICAL
APPROACH
The problem of a spherical shell under the action of a central concentrated force using non-linear shell equations or energy methods has been studied by many authors such as Bieseno [l], Archer [3] and Ashwell [4]. The present paper follows the approach introduced by Pogorelov [S], who discussed isometric deformations of shells. He noticed that when a shell undergoes large deflections, it takes the form of its isometric transformation; i.e. geometrical bending plays the most important role in the mechanism of deformation of the shell, while strains in its middle surface are of minor importance. An isometric or inextensible transformation of a surface is, by definition, one in which the first metric tensor remains unchanged. A surface deformation during which the metric changes slightly without causing cracking or plastification in the mid-surface, is called a “quasi-isometric’* deformation. Such a deformation does not, therefore, differ much from an isometric deformation, a fact which 391
392
S. A. LUKASIEWICZ and W. OPALWKI
I!!L
QUASI
ISOMETRIC -
DEFORMATION
ISOMETRICAL
DEFORMATION
Fig. I.
makes it possible to predict the deflected shape of the shell, at least approximately, by looking for the deformed configuration among its isometric transformations. Considering a spherical shell, we come to the conclusion that the simplest isometric transfo~ation corresponds to a mirror-like reflection of the original surface. It can be assumed, therefore, that the deflection produced by a concentrated force has the approximate shape shown in Fig. 1 and consists of the dimple B-B, (region I), together with the ridge A-C-B, (region II), the latter being the region of large bending moments and curvatures. Region I, except for a small circular area around the point of load application, can be considered as a purely isomet~~ally transfo~ed area. The ridge area is the region where isometric transformation of the surface without discontinuity is impossible. Therefore, in this region, bending as well as stretching of the mid-surface occurs and the strain energy is the sum of the bending and membrane energy. If the deflections of the shell are defined, the problem can be solved using the principle of stationary value of the total potential energy 6T = 0. The energy of the system can be calculated as a function of only one geometrical parameter, namely the depth of the dimple. Pogoreiov calculated the energy in the central region and in the ridge area using additional conditions for the components of the displacement vector resulting from the assumption that the strains in the ridge, in the meridional direction, are very close to zero. Moreover, he assumed that the displacements in the ridge are of local character, and vanish rapidly at a certain distance from the ridge. The ridge was very narrow (,‘a” close to “b” in Fig. I). The experiments do not confirm these assumptions. Finally, Pogorelov obtained a very simple expression for the energy due to the change in the shape of the shell U = 2acE(2f)3/2hS’2R- ‘,
(1)
where 2f is the total deflection at the point of load application, c is a constant value, c z 0.19; E, h and R are Young’s modulus, thickness and radius of the she& respectively. Knowing equation (I), only one additional step is required to obtain the load-deflection relation, namely the calculation of the work of the external load, W = P2j’. Equating the variation of the tota potential energy to zero results in the following simple formula for the deflection of the shell p = 3mEh5’= R vm
(21
Surprisingly, this simple result is in good agreement with experimental results and results obtained by other authors. The comparison with Penning’s [13’J experimental results can
Elasto-plastic
behaviour
of double-curved
shells under a concentrated
load
393
be found in [12]. However, in the case of simultaneous appii~ation of an external concentrated load and internal pressure, some discrepancies were observed [ 1i]. These errors could be produced, first of all, by the assumption that the strain energy can be calculated for the unit length of the ridge, not taking into account the real dimensions of that area. However, the more precise calculations performed by the present authors confirmed Pogoreiov’s result. 3. ELASTIC
STATE
Consider a spherical shell of the radius R and thickness h. We can distinguish three areas in the large deformed shell: isometricai
O
quasi-isometrical
a
undeformed
b < r,
where a and b are the radii of the ridge area, a and b are unknowns to be defined {Fig. 1). The strain energy in the isometrical area is given by the formula
u = $(l
where D =
+ v)7m2,
Eh3
12(1 - v2)’
where v is Poisson’s ratio. The energy in the ridge area can be obtained assuming the deflection function w. This function should satisfy the geometrical boundary conditions of the continuity of displacements between the purely isometrically deformed inner area and the external undeformed area. The following relation was used. 2
dw dr=
i--) 1-i
1-i
(4)
where A, B are constants which can be found from the boundary conditions for r = a$
= i,andp
d2w
= 24 = arccosi.
(5)
The respective conditions for r = b are satisfied identically. The constant C is subject to variation. The membrane energy in the ridge area can be obtained by solving the shallow shell equation, which in the case of axisymmetry is reduced to the equation
(6) where 4 is the stress function. The solution for 4 can be found in the form of the series 4 = i$I airi+’ + Crr + C2i* The constants a; can be obtained comparing both sides of equation (6). C,, C2 result from the boundary conditions for the membrane state of stress which requires that for r = a,
S. A. LUKASEWCZ
394
and W. OPALINSKI
6, = 0. 0, = 0. The bending and membrane energy can be obtained from
The solution of the problem was obtained numerically minimising the total potential energy of the system with respect to parameters a, b and C only. In this way. we could check Pogorelov’s result calculating the coefficient c from the equation
where u is the total strain energy associated with the dimple on the shell surface. It was found that the value of c is almost constant and independent of h/R. The following values for R/h = 1200 were obtained for P* = 2
c=O,I74
for P* = 9
c = 0,184,
where P* = f!!_
Eh”
The above values are close to Pogorelov’s value for the coefficient c = 0,19. 4. ELASTO-PLASTIC
STATE
The efasto-plastic behaviour of a spherical shell loaded by a concentrated force has not been studied by many authors. In 1967, Leckie [9] obtained a limit load corresponding to the snap-through behaviour of a spherical shell loaded at its apex with a concentrated load applied through a rigid boss. This study was based on a so-called two-moment limited interaction criterion which allows the moment and force resultants to assume their respective ultimate values separately. However, this approach is not fully justified physically. Recently, Padovan and Tovichakchaikul (1983) using F.E.M. obtained results for the elasto-plastic collapse of a spherical cap under a central point load. The results presented by them seem to be wrong or very inaccurate [lo], Using the described geometrical method, the problem can be solved in the following way. The strain energy in the plastic state can be obtained from the assumed displacement field. However, the position of the limit surface between the elastic and plastic state must be determined first. The stress intensity is the function of the curvature changes x,, K,. (f. = +h.;
-t K:)(l
-
Vf
V’) + (4V -
1 -
VZ)K,Ktj1'2,
(11)
where t is the coordinate measured across the thickness. If we know K,, and K,, we can calculate czOacross the thickness of the shell. Then, we can check if it is smaller than the yield stress cY and define the limit value of z = zr, for c0 = rrv. The bending energy in the isometrically deformed central area, I, can be obtained very easily. The principal changes of curvatures are K, = K, = -2/R and the strains are E,
=
Es =
-
14 R
for2 < tp.
Elasto-plastic
behaviour
of double-curved
shells under a concentrated
load
39s
Using the Deformation Theory of Plasticity, we have E, = &IE)+ &!P)= ; (0, - vg,) + & E, = &i”’+ &iP)= ;(c,
(a, - s),
- vg,) + &(%
(12)
- 9,
where s = :(c, + a,). Applying a model of idea1 elasto-plastic material, we have the following condition for the yield surface a,’ 5 a,’ - a,a, + a:.
(13)
If we assume the displacements field for the shell, the state of strain is known, then the set of equations (12) can be solved with respect to the stress components. We have a, = E*(E, + v*EJ, a, = I?*(&,+ v*E,),
(14)
where E* = E
v+k v* = 1 +k’
l+k
k=!E
2 E’p”
The function k(r) can be obtained in the following way. First we define the value of zp, which gives the border between the elastic and plastic zone. At z = z,, the strains are still elastic and k = 0. In the plastic region z > zp, k # 0 and increases when coming to the external surfaces of the shell. Using a simple iterative technique, we can find its value from equations (13) and (14). The stress distribution across the thickness is linear in the elastic state. However, in the plastic region, this distribution is non-linear as we see in Fig. 2 which presents the distribution of stress across the thickness of the shell with the assumed changes of curvatures. )
Z/-
h 2
I.0 I
lY= t
0.8 -cl PLASTIC
-2-l
(a)
0
I I
I 2
I 3
I1) 4 5 &J-l00
2
(P) Fig. 2.
40
ZONE
80
120
160
200
396
S. A. LUKASIE~ICZ and W. OP~LINSKI
Knowing the state of strain and stress in the shell, we calculate the strain energy from (15)
where .sLE’,&I”’are elastic components and sip’, .s~” are plastic components of the strain tensor, and c,, or are components of the stress tensor. As we observed by performing experiments, some regions of the she11 can become unloaded, being previously in the plastic state. In this case we have to define the new elastic components of the strain and stress states. The plastic strains remain unchanged until the yield surface is reached again at its other point. The plastic strains at this point are the sum of the actual strain and residual strain from the previous step of loading, .stPRJ. E(P)= &g;,,, + &(PR’.
5. NUMERICAL
(16)
EXAMPLE
The above derived equations (12-15) were used in the analysis of the elasto-plastic behaviour of the spherical-shell under central concentrated force. The minimum of potential energy was obtained using an optimisation routine which minimised the total potential energy with respect to the three parameters a, b and C only. The routine used a nongradient method of penalty function. The results of the calculations are presented in Fig. 3 for the following parameters: radius of the sphere R = 5OOmm, thickness h = 0.5 mm, Young’s
R = 500mm h = 0.5 E = 0.8 Y = 0.35
Eh3 Fig. 3.
mm IO5
Elasto-plastic
behaviour
of double-curved
shells under a concentrated
load
397
1
v 2.12
3.18
4.24
5.30 c/h
h=0.3(mm) l
P
_
6.36
= 1.8
PR
-Eh3 Fig. 3.
modulus E = 0.8 lo5 MPa, Poisson’s ratio v = 0.35, yield stress by = 200 MPa, radius of the loaded area c = 0.54mm. These data give R/h = 1000, ay/E = 0.0025. We observe that the ridge region moves outwards as the load increases. The first plastic zone appears near the point of the application of the load for
Then the plastic zone spreads more and more as the load increases, but this zone is always very localized, (Fig.4). At the load value P* = 5.5, the plastic zone at the ridge can be observed. With the increase of the load, this zone spreads outwards. As the ridge area moves, there remains the unloaded area with residual strains. The load deflection diagram is presented in Fig. 4. We observe that the deflection of the shell in an elasto-plastic state does not deviate much from the curve given by the elastic solution, equation (2). This behaviour can be explained and justified by the fact that the shell still has a relatively large elastic core (half the thickness of the shell) in the elasto-plastic zone, which controls the mechanics of the deformations. However, the difference in stress for the same value of deflection, in the case of an elasto-plastic material and a hypothetical elastic material, is considerable. Stresses in the elastic case are 3 to 4 times larger than in the elasto-plastic analysis.
PR
I
Eh3 10.0
Fig. 5.
398
S. A. LUKASIEWKZ
and
Fig.
6. ELASTO.PLASTIC
BE~AVI~UR
W. OPALWSKI
6.
OF A DOUBLE
CURVED
SHELL
A simpli~ed solution for the eIasto-plastic case of a shetl of positive Gaussian curvature can be obtained using the assumption that the energy stored in the sheI can be cakulated for the unit length of the ridge, based on the local configuration of this area, Fig. 5. This assumption is the result of an observation made for the sphere where the boundaries of the zones were calculated as a function of the applied load and the vaiues of curvature changes at the ridge. Evaluating the total potential energy contained in the efastic and plastic zones, it was found that the toad deflection reiation for the eIa$to-plastic case was close to the corresponding elastic curve.
The resuIts obtained in this way are presented in Fig. 7a and b, which also show the plastic zones and the untoaded zones. These figures present the results for two values of applied loads. We observe that the ridge area moves in an outward direction. The plastified zones move forward leaving behind the unloaded zones with residual stresses.
PLASflC
ZONE NON -DEFORMED
RESIDUAL ELASTIC
ISOMETRICAUY OEFORMED AREA
AREA
STRAJNS UNLOADED
NON- 1SDMETRtCALLY OEfORMEO AREA Fig.
7%
ZONE
Elasto-plastic behaviour of double-curved shells under a concentrated load
399
Y
(mm)
PR :6 Eh3
~=“.a
t
I
PLASTIC
ZONE
ESIDUAL
STRAIN
Fig. 7b.
7. CONCLUSIONS
This paper presents the application of the geometrical approach to the analysis of the large, elasto-plastic deformation of spherical shells. The method appears to be effective and gives better results than F.E.M. The problem was reduced to the evaluation of a small number of unknown parameters only. REFERENCES 1. C. B. Biezeno, Uber die Bestimmung der Durchschlagkraft einer schwachgekrummten kreisformigen Platte. 2. Angew Murh. Met. 15, 10 (1935); C. B. Biezeno and R. Grammel, Engineering Dynamics, p. 484. Blackie, London (1965). 2. Chien Wei-Zang and Hu Hai-Chang, On the snapping of a thin spherical cap, 9th lnt. Congress Appl. Mech., Vol. 6. p. 309, University of Brussels (1957). 3. D. G. Ashwell, On large deflection of a spherical shell with an inward point load. In Proceedings of IUTAM Symposium Theory Thin Elastic Shells Delft, 1959, pp. 43-63. North Holland, Amsterdam (1960). 4. R. R. Archer, On the numerical solutions of the non-linear equations for shells of revolution. J. Muth. Phys. 41, 165-178 (1962). 5. A. V. Pogorelov, Geometric Shell Stability Theory. Izd. Nauka, Moscow (1966). English translation U.S. Air Force FTD-ID(RS)T-2219-78. 6. S. A. Lukasiewicz and W. Szyszkowski, Geometrical methods in the non-linear theory of shells. In Proceedings Symposium Shell Strucrures, Theory and Applications, PAN, Krakow, Poland, pp. 25-27, April (1974) in Polish. 7. S. Lukasiewicz. Geometrical elements method for the solution of non-linear shell problems. J. Aeronaut. Sot. India 37, 297-312 (1985). 8. S. Lukasiewicz and W, Szyszkowski, Geometrical analysis of large elastic deflections of axially compressed cylindrical and conical shells. Int. J. Non-linear Mech. 14, 273-284 (1979). 9. F. A. Leckie, Plastic instability of a spherical shell. In Proc. IUTAM Symposium on Theory of Thin Shells, Copenhagen (1967). 10. T. Padovan and S. Tovichakchaikul, On the solution of elastic-plastic and dynamic postbuckling collapse of general structure. Advances and Trends in Structural and Solid Mechanics, Proceedings of the Symposium, Washington, DC, 1982. Pergamon Press, Oxford. 11. S. Lukasiewicz and P. G. Glockner, Collapse by ponding of shells. lnt. J. Solids Struct. 19, 251-261 (1983). 12. S. Lukasiewin, Local Loads in Plates and Shells. SijthofI & Noordhoff, Netherlands PWN, Warsaw (1979). 13. F. A. Penning, Experimental buckling modes of clamped shallow shells under concentrated loads. J. appk Mech. 33, 297-304 (1966).