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Axial compression of metallic spherical shells between rigid plates
Thin-Walled Structures 34 (1999) 21–41 www.elsevier.com/locate/tws
Axial compression of metallic spherical shells between rigid plates N.K. Gupta*, G.L. Easwara Prasad, S.K. Gupta Applied Mechanics Department, Indian Institute of Technology, Delhi, New Delhi 110016, India
Abstract Aluminium spherical shells of R/t values between 15 and 240, were axially compressed in an INSTRON machine between flat plates. The modes of their collapse, load-compression and energy-compression curves, and mean collapse loads are presented. A simple analytical model has been developed for the prediction of load-compression and energy-compression curves for the metallic spherical shells, by using the concepts of stationary and rolling plastic hinges. The results thus obtained match well with the experimental results. These results have also been compared with the solutions proposed in earlier studies. Behaviour of these shells is compared with the response of spherical shells (aluminium, mild steel and galvanised steel) of shallow depth, which were also subjected to axial compression between rigid plates. Their load-deformation curves are presented, and their energy-compression behaviour and mean collapse loads are discussed. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Spherical shells; Plastic hinges; Axial collapse; Energy absorption
Nomenclature R L Z T
Mean radius of the spherical shell Span of the spherical shell Depth of the spherical shell Thickness of the spherical shell along a parallel circle at contact with top plate
* Corresponding author. Tel: ⫹ 91-11-686-19977; fax: ⫹ 91-11-686-2037; e-mail: [email protected] 0263-8231/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 3 1 ( 9 8 ) 0 0 0 4 9 - 4
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r rp h
Mean radius of the parallel circle at any point considered Radius of the rolling or travelling plastic hinge Total axial compression of the spherical shell at any stage of compression Incremental axial compression of the spherical shell Number of stationary plastic hinges length of the stationary plastic hinges induced during a compression of dhi Load on the spherical shell at any stage of compression Plastic moment Incremental length of spherical shell along the meridian covered in an incremental stage of compression “dhi” Angle subtended at centre over the incremental length “dl” Thickness of the spherical shell at apex Thickness of the spherical shell near the rim or base Total meridional length of assumed hemispherical shell from apex to rim or base Total meridional length of spherical shell from apex to point at the contact of top plate
dhi N l P Mp dl d t1 t2 lmt lmp
1. Introduction The energy absorption characteristics and collapse behaviour of structural components under impact loads are required for the design for crash worthiness of various transport vehicles like air craft and automobiles. The load-deformation characteristics of energy absorbing devices is a measure of their energy absorption capacity. It differs from one component to the other, and depends on the mode of deformation involved and the material used. The important factors which are used as a measure of the efficiency of performance of energy absorbers and their selection criteria is discussed in detail by Ezra and Fay [1]. The behaviour of thin walled tubes of circular or non circular sections under static and/or dynamic loading conditions has been studied and reported by various researchers [2]. Attention has also been given to the study of failure mechanisms and energy absorbing capabilities of rotationally symmetric conical or spherical shells. In the present work, we have considered the collapse mechanisms and energy absorbing characteristics of metallic spherical shells, when subjected to axial loading. The large deformation of a rigid-plastic spherical shell compressed between rigid plates was first studied by Updike [3]. An expression (Eq. (1)) relating the axial crushing force on the spherical shell to its deformation was proposed, predictions of which are restricted to overall compression of a few thicknesses to about one tenth of the shell radius. The load-deformation behaviour in this analysis is shown to be independent of the radius of the spherical shell. h⫽
冋 册
T P 48 2Mp
2
(1)
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The quasi-static loading of a hemispherical shell along its axis of symmetry was studied, with particular reference to large deformations and buckling, by Kitching et al. [4]. Deformation patterns which are symmetric and non-symmetric about the axis were studied analytically and experimentally for spherical shells of R/t ranging between 36 and 460. Three stages of deformation were considered. Stage I deformations were considered elastic, wherein a local flattening of the shell in contact with the plate took place. In stage II axi-symmetric inward dimple was assumed to form. In stage III non-axisymmetric integral number of lobes were considered to be formed. In stages II and III, the load-deformation relationship given by Eq. (1) was used for analytical prediction of load-deformation behaviour for a perfectly plastic material. The quasi-static crushing of a spherical shell between rigid plates was also analyzed by De Oliveira and Wierzbicki [5] as part of their studies on the crushing analysis of rotationally symmetric plastic shells. They compared the results (Eq. (2)) for load versus deformation with those of Updike [3]. They concluded that their solution would be valid for large deformation until the crush distance reached the spherical shell radius, even though there is only a marginal difference in their solution and the solutions of Updike [3]. It may be observed that the load-deformation relation [5] given by Eq. (2), is also independent of the radius of the spherical shell. They have also given an expression (Eq. (3)) for determining the radius of the rolling plastic hinge in contact with plate, viz,
冋
h⫽
T P ⫹2 43 2Mp
rp ⫽
√3TR 2 sin
册
2
⫺
冋 册
3T P 32 2Mp
(2) (3)
An analysis and quasi-static testing of spherical shells has been carried out by Kinkead et al. [6] on hemispherical shells of R/t ranging between 8 to 32. They assumed the deformations to occur in two phases. In the first phase a local flattening of the shell in contact with the rigid plate occurs, while in the second, an axi-symmetric inward dimpling of the previously flattened portion takes place. In the analytical solution proposed for the load-deformation prediction, cusps are formed at the intersection of solutions proposed for the two phases of deformations. These cusps are not borne out in the experimental data. In their static analysis the strain hardening is also considered. They have compared the analytical solutions obtained for loaddeformations with the solutions proposed by the other authors [3,5], and a large difference is reported. They have also mentioned that the analytical solution proposed by them is somewhat lengthy and cumbersome and is applicable only for the spherical shells of low R/t values. Here we consider the axial compression of aluminium spherical shells to study their collapse mechanisms and energy absorbing capabilities under quasi-static loading. R/t values of these shells are 15 to 240. It is seen that actual spun specimens
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Fig. 1.
Schematic diagram of the spherical shell.
have variation in wall thickness, which need to be considered in the analysis. In developing the analytical model to predict the load-deformation behaviour, the concepts of stationary and rolling plastic hinges are used and variation in wall thickness is considered. The results thus obtained compare with experiments very well.
2. Experimental work A detailed experimental investigation has been carried out on the axial crushing of spherical shells between rigid plates under quasi-static loading in an INSTRON machine (model 1197) of 50 T capacity. The aluminium spherical shells with R/t values ranging from 15 to 240, and the spherical shell specimens of shallow depth made of aluminium, mild steel and galvanised steel were tested. The specimens were compressed up to 50% of their depth with a cross head speed of 2 mm/min. Their Table 1 Yield stresses of the spherical shell materials Sl. No.
Material
Heat treatment
1 2 3 4 5 6 7 8 9 10 11 12
Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Mild Steel Mild Steel Galvanised Steel Galvanised Steel
A A A A A A A A AR A AR A
A: Annealed, AR: As received
Thickness t (mm) 0.50 0.74 1.03 1.29 1.50 2.00 2.50 3.00 0.81 0.92 0.75 0.78
Yield Stress (Mpa) 74.6 75.0 78.2 80.6 91.4 92.2 98.5 116.3 307.1 185.0 330.6 179.0
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Table 2 Dimensions of aluminium spherical shells with variation in R/t values Sp. No.
Radius R (mm)
Depth Z (mm)
t1 (mm)
t2 (mm)
t (mm)
R/t
A1 A2 A3 A4 A5 A6
42.83 39.30 78.40 52.41 78.00 78.20 39.11 56.42 78.10 51.33 39.38 75.81 62.70 51.20 78.00 100.35 125.10 102.30 126.80 77.41 103.60
34.04 35.10 69.68 45.14 70.27 72.22 35.74 47.75 70.42 47.28 33.90 71.80 59.68 45.75 72.80 92.00 110.50 93.34 107.85 73.68 93.44
2.80 2.05 3.00 2.05 2.50 2.00 1.03 1.29 1.50 1.03 0.74 1.29 1.03 0.74 1.03 1.29 1.29 1.03 1.03 0.50 0.50
2.80 1.75 2.62 1.56 2.18 1.58 0.73 0.98 1.22 0.73 0.52 0.93 0.78 0.55 0.78 0.93 0.98 0.79 0.75 0.36 0.36
2.80 1.90 2.81 1.81 2.34 1.79 0.88 1.14 1.36 0.88 0.63 1.11 0.91 0.65 0.91 1.11 1.14 0.91 0.89 0.42 0.43
15.3 20.7 27.9 29.2 33.4 43.7 44.4 49.5 57.4 58.3 61.5 68.3 69.3 78.8 85.7 90.4 109.7 112.5 143.5 184.3 240.9
A7 A8 A9 A10 A11 A12 A13 A14 A15 A16 A17 A18 A19
load-compression curves were obtained on the machine chart recorder. Repeatability of the results was ensured by repeating the experiments on at least three identical specimens. The schematic diagram of the spherical shell is shown in Fig. 1. 2.1. Specimens The aluminium sheets of standard thicknesses i.e., 0.5 mm, 1 mm, 1.5 mm, 2 mm, 2.5 mm, and 3 mm were obtained commercially and then the spherical shell specimens used in the experiments were obtained by spinning process. These shells were Table 3 Dimensions of aluminium spherical shells of same radius and thickness, and of different depths Sp. No.
Span L (mm)
Depth Z (mm)
Radius R (mm)
t1 (mm)
t2 (mm)
t (mm)
R/t
B1 B2 B3 B4
127.50 141.92 152.40 154.56
30.30 46.67 67.31 72.80
78.20 77.30 76.80 78.00
1.03 1.03 1.03 1.03
0.93 0.71 0.77 0.78
0.98 0.87 0.90 0.91
89.8 88.8 85.3 85.7
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Table 4 Dimensions of Spherical shells of shallow depths Sp. No.
Material
Span L (mm) Depth Z (mm)
Specimens C1 C2 C3 C4 Specimens C5 C6 C7 C8 C9 C10
tested as received (without annealing) MS 146.98 50.36 GI 143.82 47.29 GI 124.76 37.28 GI 103.90 32.56 tested after annealing MS 148.42 51.14 GI 147.12 47.29 GI 122.06 37.71 GI 104.98 30.90 Alu. 141.92 46.67 Alu. 127.50 30.30
Radius R (mm)
t (mm)
R/t
78.80 78.32 70.83 57.72
0.81 0.73 0.97 0.74
96.9 107.3 70.8 78.0
79.41 80.56 68.24 60.03 77.30 78.20
0.92 0.78 0.96 0.78 0.87 0.98
86.3 103.7 71.1 77.0 88.8 89.8
M S: Mild Steel, G I: Galvanised Steel, Alu.: Aluminium
annealed before testing. To get the yield stress of the material, the tension specimens were cut from the same sheets and annealed along with the respective shell specimens before testing. Table 1 shows the yield stress of the material of the shell specimens. Table 2 gives the dimensions of the aluminium spherical shells with R/t values ranging from 15 to 240. Table 3 shows the dimensions of the shells which are of same radius and of different depths. Table 4 gives the details of the shells of shallow depth and made of different materials. The spherical shells obtained from the process of spinning show a variation in their wall thickness in the meridian direction. The specimens tested under axial compression were cut into two equal halves, see Fig. 2(a) and the thickness was measured along the meridians. The trend in the variation of wall thickness along the meridian direction of a spun spherical shell is found to match with that reported by Blachut and Galletly [7], and a typical variation is shown in the Fig. 2(b). It is observed that the thickness is constant along the meridian over a short length near the apex. Further up to about 50% of the meridional length the thickness is found to vary almost linearly. For the remaining 50% of the meridional length thickness is again constant. This variation in wall thickness has been considered in the present analysis. For calculations of R/t values of the shells, t is taken as the average of t1 and t2, see Fig. 1. 2.2. Modes of collapse It is observed that in the initial stage of compression the spherical shells deformed with a flat contact region against the rigid plate. This is called stage I deformations. This deformation mode is clearly observed in spherical shells of low R/t values [6]. As the R/t value increases, the range of compression over which the stage I deformations occur decreases. With the progress of deformations the spherical shells show an axi-symmetric
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Fig. 2. (a) Sectional view of the tested spherical shell (A2); (b) Typical variation of shell thickness along meridian.
inward dimple, inside a ring of contact with the rigid plate; this is stage II deformations. The axi-symmetric inward dimple expands moving the rolling plastic hinge at the contact of the plate outwards from the axis of the shell, as shown in Fig. 3. The collapse of spherical shell due to axi-symmetric inward dimpling is shown in Fig. 4(a) and Fig. 4(b). In case of spherical shells with low values of R/t the deformations of stage II is observed over a large range of compression. All specimens from Sp. No. A1 to A6 have shown stage II deformations over almost the entire range of compression. Later stages of deformation show buckling with non-symmetric shape, consisting of an integral number of lobes and stationary plastic hinges formed between the consecutive lobes, particularly in spherical shells with high R/t values. This is stage III deformations, and Fig. 5(a) and Fig. 5(b) show the tested specimen No. A17, deformed in this mode. The stage of compression, at which buckling with non-symmetric shape is initiated depends on the R/t values of the spherical shell. Fig. 6 shows
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Fig. 3. The movement of rolling plastic hinge with the progress of axial compression (schematic diagram).
the tested specimens No. A7, A10, A14 and A15. For the spherical shell specimens shown in Fig. 6 the stage of compression at which the stage III deformations are initiated is shown in Table 5. It is found that generally 5 to 8 stationary hinges are induced with an equal number of integral lobes. At the instance of initiation of stationary hinges, all the hinges are not initiated at the same time, but with the progress of compression they reach the maximum number. In the same way even the rotation of the stationary hinges reach the maximum with progress of compression, from its zero value at the instance of initiation of stationary hinges, see Fig. 6. From Table 5 it can be noticed that the stage III deformation is initiated at an early stage of compression and it occurs over a large range of compression, in the case of spherical shells of high R/t values. 2.3. Rolling plastic hinge As discussed in Section 2.2, at some stage of compression, the axi-symmetric inward dimple expands outwards from the axis of the shell due to the rolling of the plastic hinge at the contact with the top plate. It is observed that the radius of the rolling plastic hinge rp at the contact of the plate varies with the progress of compression. The typical variation of rp with deformation is shown in Fig. 7. The values
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Fig. 4. Axi-symmetric inward dimpling mode of collapse of specimen no. A5; top view (a), and bottom view (b).
of rp were measured at different stages of compression by interrupting the axial compression of specimens in INSTRON machine. An equation for computing rp at any stage of deformation is obtained as: rp ⫽
K hc
(4)
Here K and c are the constants obtained from the experimental data. Results of Eq. (4) are shown in Fig. 7 along with the results obtained from Eq. (3) for comparison. 2.4. Results from experiments The typical load-deformation curves of spherical shell specimens obtained from the chart recorder of INSTRON machine are shown in Fig. 8. The shape of the load-
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Fig. 5. Buckling of specimen no. A17 showing integral number of lobes and stationary hinges; top view (a), and bottom view (b).
deformation curve is an important characteristic for an energy absorber, as it is a measure of the effectiveness of the structure in absorbing energy. An ideal energy absorber has a square load-deformation curve, i.e., once the crush mode starts, (at peak load) the collapse continues under the same load until the absorber is entirely consumed [1,2,8]. From Fig. 8 it can be observed that the load-deformation curves resemble that of an ideal energy absorber, with negligible serrations. In case of spherical shells of large thicknesses (A3, A4, A5) the load-deformation behaviour deviates from an ideal profile (see Fig. 9(b)). The linear variation of thickness of spherical shells along the meridian direction is responsible for the ideal load-deformation profile of the spherical shells. The load-deformation curves of a group of spherical shells with different radii and constant thickness and vice versa obtained from the experiments are shown in Fig. 9(a) and Fig. 9(b). The load-deformation profiles obtained for the spherical shells of shallow depth show clear variations from those of the spherical shells listed in Table 2. In these
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Fig. 6.
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Tested spherical shell specimens A7, A10, A14 and A15 showing stage III deformations.
Table 5 Compression at which stationary hinges are initiated Sp. No.
R/t
A7 A10 A14 A15
49.5 68.3 90.4 109.7
Depth Z (mm) 47.75 71.80 92.00 110.5
h (mm)
Z⬘%
22.0 18.0 16.0 12
46.1 25.1 17.4 10.9
Z⬘ Deformation of shell as a percentage of total depth (Z) at which stationary hinges are initiated
Fig. 7. Typical variation of radius of rolling plastic hinge.
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Fig. 8.
Typical load-deformation curves (exp.).
cases it is because the thickness variation along the meridian is hardly noticeable and the load-deformation does not follow the ideal profile. Fig. 10 shows the loaddeformation curves of some of the shallow spherical shells tested under quasistatic loading.
3. Analysis The concepts of rolling plastic hinges and stationary plastic hinges are used in arriving at an analytical model for predicting the load-deformation and energy-compression behaviour of the spherical shells. The mean radius of the spherical shell is given by R⫽
[[L/2]2 ⫹ Z2] 2Z
(5)
where L and Z are defined in Fig. 1. Plastic moment Mp per unit length of the hinge is given by Mp ⫽
0T 2 4
(6)
Here o is the yield stress of the material and T is the thickness of the shell at the parallel circle in contact with the plate. Plastic work dWr dissipated by the rolling plastic hinge [9–11] in the elemental area dA traversed by it, during a compression of dhi is given by, dWr ⫽
2[dA]Mp rp
(7)
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Fig. 9. Load-deformation curves of shells; (a) different radii and constant thickness, (b) different thicknesses and constant radius.
where rp is the radius of the rolling plastic hinge. The mean radius “r” of the parallel circle in contact with the plate for a compression h, is given by r ⫽ √h[2R ⫺ h]
(8)
By calculating the value of r at any stage of compression from Eq. (8), it is possible to get the value of dA for an incremental compression of dhi, as: dA ⫽ [2r]R[d]
(9)
Further to get the value of rp at any stage of compression, the experimentally
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Fig. 10.
Load-deformation curves of spherical shells of shallow depth.
measured values of rp at different stages of compression were plotted. By curve fitting, an equation for calculating rp at any stage of compression is obtained in the form rp ⫽
K dhc
(10)
The plastic work dissipated in the N stationary hinges, dWs during a compression of dhi is given by
冋 册
dWs ⫽ NMp
2 l N
(11)
where l is the length of stationary hinge. The plastic work dissipated due to the strain in the meridian direction [12] dWm during a compression of dhi is given by, dWm ⫽ 0e[dV]
(12)
Here e is the meridional strain. The strain in the meridian direction e1 corresponding to the local flattening of the shell over an incremental compression of dhi over an elemental length of the shell “dl” (see Fig. 1) is given by e1 ⫽
[dl ⫺ dl cos ] dl
e1 ⫽ [1 ⫺ cos ]
(13) (14)
The local flattening of the spherical shell can be observed only in the initial stages
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of compression, and occurs only over a shallow range of compression. Further, as the R/t value of the spherical shell increases, the range of compression over which the stage I deformation is observed, decreases. When once the deformation changes over to stage II, i.e., axi-symmetric inward dimpling, then the meridional strain will be lesser than that given by Eq. (14). As such it is assumed that the meridional strain will vary linearly with deformation. Based on this, it is assumed that the meridional strain will be maximum at the initiation of compression of the shell, and it is zero at maximum compression equal to the radius of the spherical shell, i.e.,
冋
e ⫽ e1 1 ⫺
1mp 1mt
册
(15)
Fig. 11. Comparison of experimental and computed results; (a) load-deformation curves, (b) energycompression curves.
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Fig. 12. Experimental and computed load-deformation curves of spherical shell of shallow depth.
The load on the spherical shell at any stage of compression is given by p⫽
[dWr ⫹ dWs ⫹ dWm] dhi
(16)
Thus the load-deformation curve is obtained from the above, and the energy-com-
Fig. 13. Experimental load-deformation curve of spherical shell and one computed by assuming constant meridional thickness.
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Table 6 Experimental and analytical mean collapse loads of the aluminium spherical shells Sp. No.
R/t
Mean collapse loads (kN) Experiment
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16 A17 A18 A19
15.3 20.7 27.9 29.2 33.4 43.7 44.4 49.5 57.4 58.3 61.5 68.3 69.3 78.8 85.7 90.4 109.7 112.5 143.5 184.3 240.9
13.64 10.17 27.17 10.66 22.32 15.42 2.67 4.89 8.80 2.86 1.71 5.72 3.42 2.12 4.79 7.49 7.87 5.06 6.13 1.62 1.51
Analytical
16.45 9.67 28.02 11.22 22.37 14.67 2.88 5.00 9.79 3.50 1.71 5.94 4.00 2.16 5.59 7.52 8.27 5.87 6.39 1.85 1.94
Updike
10.16 6.19 22.23 7.25 15.15 8.60 1.90 2.88 5.70 2.15 1.02 3.47 2.21 1.22 2.80 3.80 4.00 2.92 2.92 0.76 0.74
De Oliveira and Wierzbicki 9.63 5.94 21.32 6.90 14.48 8.20 1.81 2.75 5.42 2.05 0.98 3.30 2.10 1.16 2.66 3.60 3.79 2.77 2.77 0.72 0.71
pression curve is obtained from the cumulative summation of the energy absorbed over the incremental compression dhi. 4. Results and discussions Load-deformation and energy-compression curves obtained from the analytical model compare very well with the experimental observations, as shown in Fig. 11(a) and Fig. 11(b). The results obtained from the solutions proposed by the earlier researchers [3,5] have also been shown for comparison. It is seen that the results of the proposed analytical model match the experimental load-deformation and energycompression data very well. To study the effect of depth on the mode of collapse a set of experiments were conducted on spherical shells of different depths but having same radius, see Table 3. The mode of collapse in these cases does not show any noticeable changes when compared with the observations made in the case of shells listed in Table 2. But in case of spherical shells of shallow depth, see Table 4, the load-deformation profile is not following the ideal profile as observed in Figs. 8 and 9(a) and Fig. 9(b). By
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Table 7 Experimental and analytical mean collapse loads of aluminium spherical shells with same radius, thickness, and different depths Sp. No.
B1 B2 B3 B4
Span L (mm)
127.50 141.92 152.40 154.56
Depth Z (mm)
30.30 46.67 67.31 72.80
Radius R (mm)
78.20 77.30 76.80 78.00
t (mm)
0.98 0.87 0.90 0.91
R/t
89.8 88.8 85.3 85.7
Mean collapse load (kN) Exp.
Ana.
3.26 3.72 4.36 4.79
3.02 3.29 4.75 5.59
Table 8 Experimental and analytical mean collapse loads of spherical shells of shallow depths Sp. No.
Mat.
Depth Z (mm)
Specimens tested as received (without C1 MS 50.36 C2 GI 47.29 C3 GI 37.28 C4 GI 32.56 Specimens tested after annealing C5 MS 51.14 C6 GI 47.29 C7 GI 37.71 C8 GI 30.90 C9 Alu. 46.67 C10 Alu. 30.30
Radius R (mm)
t (mm)
R/t
Mean collapse load (kN) Exp.
Ana.
annealing) 78.80 78.32 70.83 57.72
0.81 0.73 0.97 0.74
96.9 107.3 70.8 78.0
12.32 9.24 11.81 6.83
13.97 9.83 11.84 7.73
79.41 80.56 68.24 60.03 77.30 78.20
0.92 0.78 0.96 0.78 0.87 0.98
86.3 103.7 71.1 77.0 88.8 89.8
8.29 7.27 7.17 4.04 3.72 3.26
9.88 7.18 7.45 4.86 3.29 3.02
Mat.: Material, M S: Mild Steel, G I: Galvanised Steel, Alu.: Aluminium Exp.: Experimental Ana.: Analytical
keeping the thickness constant the analytical model described above can be used for the prediction of the load-deformation as well as the energy-compression curves of the spherical shells of the shallow depth (see Fig. 12). From Fig. 11 it can be seen that the analytical predictions match well with the experimental observations. Fig. 13 shows the difference in the load-deformation behaviour due to the assumed uniform thickness and varying thickness of the shell. The mean collapse loads for the spherical shells have been calculated both from the experimental observations and from the analytical model. Tables 6–8 give the mean collapse loads as obtained actually from the experiments and from the analytical model. The yield stress values of different aluminium sheets show considerable variation, see Table 1. To carry out the comparative study, the mean collapse loads
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were normalised for an yield stress of 75 MPa, in case of aluminium spherical shells. Fig. 14(a) and Fig. 14(b) show the variation of mean collapse load with different radii and constant thickness, and vice versa. From Fig. 9(a), Fig. 9(b) and Fig. 14(a), Fig. 14(b) it can be seen that the thickness of the spherical shell influences the mode of collapse as well as on the load-deformation curve. Fig. 15(a) and Fig. 15(b) show the energy-compression curves of aluminium spherical shells. 5. Conclusions Aluminium spherical shells of R/t values ranging from 15 to 240 and a few spherical shells of shallow depth were tested for axial compression in an INSTRON
Fig. 14. Variation of mean collapse load with R/t values of the spherical shells; (a) different radii and constant thickness, (b) different thicknesses and constant radius.
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Fig. 15. Energy-compression curves of shells; (a) different radii and constant thickness, (b) different thicknesses and constant radius.
machine. Their typical modes of collapse and load-deformation curves are presented. A simple analytical model is also presented to predict the load-deformation and energy-compression behaviour. From Fig. 11(a) and Fig. 11(b) one can observe matching between experimental results and results predicted from analytical model. Further it can be seen from Fig. 8 and Fig. 9(a) and Fig. 9(b) that the load-deformation profiles of spherical shells resemble a square wave profile and as such the spherical shells can be ideal energy absorbers. From Fig. 13 we can see that the thickness variation in the meridian direction contributes to making the curve flat. The mode of deformation of a spherical shell is highly dependent on R/t of the shell. For its high values (greater than 50)mode III deformations prevails over a large
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41
range. For intermediate R/t values (greater than 10 up to 50)mode II deformations are predominant. The mean collapse loads for the spherical shells tested have been calculated from the load-deformation curves. The analytical and experimental mean collapse load values for spherical shells tested, match well, as seen in Tables 6–8.
References [1] Arthur A, Ezra, Fay RJ. An assessment of energy absorbing devices for prospective use in aircraft impact situations. In: Herrmann G, Perrone N, editors. Dynamic response of structures. 1972:225– 246. Poc. of a symposium held at Stanford University, California, June 28 and 29, 1971. [2] Johnson W, Reid SR. Metallic energy dissipating systems. Applied Mechanics Reviews 1978;31:277–88. [3] Updike DP. On the large deformation of a rigid-plastic spherical shell compressed by a rigid plate. J. of Engineering for Industry 1972;94:949–955. [4] Kitching R, Houlston R, Johnson W. A theoretical and experimental study of hemispherical shells subjected to axial loads between flat plates. Int. J. of Mech. Sci 1975;17:693–703. [5] De Oliveira JG, Wierzbicki T. Crushing analysis of rotationally symmetric plastic shells. J. of Strain Analysis 1982;17:229–36. [6] Kinead AN, Jennings A, Newell J, Leinster JC. Spherical shells in inelastic collision with a rigid wall-Tentative analysis and recent quasi-static testing. J. of Strain Analysis 1994;29:17–41. [7] Blachut J, Galletly GD. Buckling strength of imperfect steel hemispheres. Thin Walled Structures 1995;23:1–20. [8] Thornton PH, Edwards PJ. Energy absorption in composite tubes. J. of Compos. Mat 1982;16:521–45. [9] Gupta NK, Atul K. Lateral crushing of square and rectangular tubes by non-orthogonally placed narrow width indentors. Int. J. of Mech. Sci 1995;37:31–50. [10] Gupta NK, Sinha SK. Collapse of a laterally compressed square tube resting on a flat base. Int. J. of Sol. Struct 1990;26(5/6):601–15. [11] Meng Q, Al-Hassani STS, Soden PD. Axial crushing of square tubes. Int. J. of Mech. Sci 1983;25:747–73. [12] Gupta NK, Velmurugan R, Gupta SK, An analysis of axial crushing of composite tubes. J. of Compos. Mat. 1997. (Accepted for publication).
Axial compression of metallic spherical shells between rigid plates N.K. Gupta*, G.L. Easwara Prasad, S.K. Gupta Applied Mechanics Department, Indian Institute of Technology, Delhi, New Delhi 110016, India
Abstract Aluminium spherical shells of R/t values between 15 and 240, were axially compressed in an INSTRON machine between flat plates. The modes of their collapse, load-compression and energy-compression curves, and mean collapse loads are presented. A simple analytical model has been developed for the prediction of load-compression and energy-compression curves for the metallic spherical shells, by using the concepts of stationary and rolling plastic hinges. The results thus obtained match well with the experimental results. These results have also been compared with the solutions proposed in earlier studies. Behaviour of these shells is compared with the response of spherical shells (aluminium, mild steel and galvanised steel) of shallow depth, which were also subjected to axial compression between rigid plates. Their load-deformation curves are presented, and their energy-compression behaviour and mean collapse loads are discussed. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Spherical shells; Plastic hinges; Axial collapse; Energy absorption
Nomenclature R L Z T
Mean radius of the spherical shell Span of the spherical shell Depth of the spherical shell Thickness of the spherical shell along a parallel circle at contact with top plate
* Corresponding author. Tel: ⫹ 91-11-686-19977; fax: ⫹ 91-11-686-2037; e-mail: [email protected] 0263-8231/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 3 1 ( 9 8 ) 0 0 0 4 9 - 4
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r rp h
Mean radius of the parallel circle at any point considered Radius of the rolling or travelling plastic hinge Total axial compression of the spherical shell at any stage of compression Incremental axial compression of the spherical shell Number of stationary plastic hinges length of the stationary plastic hinges induced during a compression of dhi Load on the spherical shell at any stage of compression Plastic moment Incremental length of spherical shell along the meridian covered in an incremental stage of compression “dhi” Angle subtended at centre over the incremental length “dl” Thickness of the spherical shell at apex Thickness of the spherical shell near the rim or base Total meridional length of assumed hemispherical shell from apex to rim or base Total meridional length of spherical shell from apex to point at the contact of top plate
dhi N l P Mp dl d t1 t2 lmt lmp
1. Introduction The energy absorption characteristics and collapse behaviour of structural components under impact loads are required for the design for crash worthiness of various transport vehicles like air craft and automobiles. The load-deformation characteristics of energy absorbing devices is a measure of their energy absorption capacity. It differs from one component to the other, and depends on the mode of deformation involved and the material used. The important factors which are used as a measure of the efficiency of performance of energy absorbers and their selection criteria is discussed in detail by Ezra and Fay [1]. The behaviour of thin walled tubes of circular or non circular sections under static and/or dynamic loading conditions has been studied and reported by various researchers [2]. Attention has also been given to the study of failure mechanisms and energy absorbing capabilities of rotationally symmetric conical or spherical shells. In the present work, we have considered the collapse mechanisms and energy absorbing characteristics of metallic spherical shells, when subjected to axial loading. The large deformation of a rigid-plastic spherical shell compressed between rigid plates was first studied by Updike [3]. An expression (Eq. (1)) relating the axial crushing force on the spherical shell to its deformation was proposed, predictions of which are restricted to overall compression of a few thicknesses to about one tenth of the shell radius. The load-deformation behaviour in this analysis is shown to be independent of the radius of the spherical shell. h⫽
冋 册
T P 48 2Mp
2
(1)
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23
The quasi-static loading of a hemispherical shell along its axis of symmetry was studied, with particular reference to large deformations and buckling, by Kitching et al. [4]. Deformation patterns which are symmetric and non-symmetric about the axis were studied analytically and experimentally for spherical shells of R/t ranging between 36 and 460. Three stages of deformation were considered. Stage I deformations were considered elastic, wherein a local flattening of the shell in contact with the plate took place. In stage II axi-symmetric inward dimple was assumed to form. In stage III non-axisymmetric integral number of lobes were considered to be formed. In stages II and III, the load-deformation relationship given by Eq. (1) was used for analytical prediction of load-deformation behaviour for a perfectly plastic material. The quasi-static crushing of a spherical shell between rigid plates was also analyzed by De Oliveira and Wierzbicki [5] as part of their studies on the crushing analysis of rotationally symmetric plastic shells. They compared the results (Eq. (2)) for load versus deformation with those of Updike [3]. They concluded that their solution would be valid for large deformation until the crush distance reached the spherical shell radius, even though there is only a marginal difference in their solution and the solutions of Updike [3]. It may be observed that the load-deformation relation [5] given by Eq. (2), is also independent of the radius of the spherical shell. They have also given an expression (Eq. (3)) for determining the radius of the rolling plastic hinge in contact with plate, viz,
冋
h⫽
T P ⫹2 43 2Mp
rp ⫽
√3TR 2 sin
册
2
⫺
冋 册
3T P 32 2Mp
(2) (3)
An analysis and quasi-static testing of spherical shells has been carried out by Kinkead et al. [6] on hemispherical shells of R/t ranging between 8 to 32. They assumed the deformations to occur in two phases. In the first phase a local flattening of the shell in contact with the rigid plate occurs, while in the second, an axi-symmetric inward dimpling of the previously flattened portion takes place. In the analytical solution proposed for the load-deformation prediction, cusps are formed at the intersection of solutions proposed for the two phases of deformations. These cusps are not borne out in the experimental data. In their static analysis the strain hardening is also considered. They have compared the analytical solutions obtained for loaddeformations with the solutions proposed by the other authors [3,5], and a large difference is reported. They have also mentioned that the analytical solution proposed by them is somewhat lengthy and cumbersome and is applicable only for the spherical shells of low R/t values. Here we consider the axial compression of aluminium spherical shells to study their collapse mechanisms and energy absorbing capabilities under quasi-static loading. R/t values of these shells are 15 to 240. It is seen that actual spun specimens
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Fig. 1.
Schematic diagram of the spherical shell.
have variation in wall thickness, which need to be considered in the analysis. In developing the analytical model to predict the load-deformation behaviour, the concepts of stationary and rolling plastic hinges are used and variation in wall thickness is considered. The results thus obtained compare with experiments very well.
2. Experimental work A detailed experimental investigation has been carried out on the axial crushing of spherical shells between rigid plates under quasi-static loading in an INSTRON machine (model 1197) of 50 T capacity. The aluminium spherical shells with R/t values ranging from 15 to 240, and the spherical shell specimens of shallow depth made of aluminium, mild steel and galvanised steel were tested. The specimens were compressed up to 50% of their depth with a cross head speed of 2 mm/min. Their Table 1 Yield stresses of the spherical shell materials Sl. No.
Material
Heat treatment
1 2 3 4 5 6 7 8 9 10 11 12
Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Mild Steel Mild Steel Galvanised Steel Galvanised Steel
A A A A A A A A AR A AR A
A: Annealed, AR: As received
Thickness t (mm) 0.50 0.74 1.03 1.29 1.50 2.00 2.50 3.00 0.81 0.92 0.75 0.78
Yield Stress (Mpa) 74.6 75.0 78.2 80.6 91.4 92.2 98.5 116.3 307.1 185.0 330.6 179.0
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25
Table 2 Dimensions of aluminium spherical shells with variation in R/t values Sp. No.
Radius R (mm)
Depth Z (mm)
t1 (mm)
t2 (mm)
t (mm)
R/t
A1 A2 A3 A4 A5 A6
42.83 39.30 78.40 52.41 78.00 78.20 39.11 56.42 78.10 51.33 39.38 75.81 62.70 51.20 78.00 100.35 125.10 102.30 126.80 77.41 103.60
34.04 35.10 69.68 45.14 70.27 72.22 35.74 47.75 70.42 47.28 33.90 71.80 59.68 45.75 72.80 92.00 110.50 93.34 107.85 73.68 93.44
2.80 2.05 3.00 2.05 2.50 2.00 1.03 1.29 1.50 1.03 0.74 1.29 1.03 0.74 1.03 1.29 1.29 1.03 1.03 0.50 0.50
2.80 1.75 2.62 1.56 2.18 1.58 0.73 0.98 1.22 0.73 0.52 0.93 0.78 0.55 0.78 0.93 0.98 0.79 0.75 0.36 0.36
2.80 1.90 2.81 1.81 2.34 1.79 0.88 1.14 1.36 0.88 0.63 1.11 0.91 0.65 0.91 1.11 1.14 0.91 0.89 0.42 0.43
15.3 20.7 27.9 29.2 33.4 43.7 44.4 49.5 57.4 58.3 61.5 68.3 69.3 78.8 85.7 90.4 109.7 112.5 143.5 184.3 240.9
A7 A8 A9 A10 A11 A12 A13 A14 A15 A16 A17 A18 A19
load-compression curves were obtained on the machine chart recorder. Repeatability of the results was ensured by repeating the experiments on at least three identical specimens. The schematic diagram of the spherical shell is shown in Fig. 1. 2.1. Specimens The aluminium sheets of standard thicknesses i.e., 0.5 mm, 1 mm, 1.5 mm, 2 mm, 2.5 mm, and 3 mm were obtained commercially and then the spherical shell specimens used in the experiments were obtained by spinning process. These shells were Table 3 Dimensions of aluminium spherical shells of same radius and thickness, and of different depths Sp. No.
Span L (mm)
Depth Z (mm)
Radius R (mm)
t1 (mm)
t2 (mm)
t (mm)
R/t
B1 B2 B3 B4
127.50 141.92 152.40 154.56
30.30 46.67 67.31 72.80
78.20 77.30 76.80 78.00
1.03 1.03 1.03 1.03
0.93 0.71 0.77 0.78
0.98 0.87 0.90 0.91
89.8 88.8 85.3 85.7
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Table 4 Dimensions of Spherical shells of shallow depths Sp. No.
Material
Span L (mm) Depth Z (mm)
Specimens C1 C2 C3 C4 Specimens C5 C6 C7 C8 C9 C10
tested as received (without annealing) MS 146.98 50.36 GI 143.82 47.29 GI 124.76 37.28 GI 103.90 32.56 tested after annealing MS 148.42 51.14 GI 147.12 47.29 GI 122.06 37.71 GI 104.98 30.90 Alu. 141.92 46.67 Alu. 127.50 30.30
Radius R (mm)
t (mm)
R/t
78.80 78.32 70.83 57.72
0.81 0.73 0.97 0.74
96.9 107.3 70.8 78.0
79.41 80.56 68.24 60.03 77.30 78.20
0.92 0.78 0.96 0.78 0.87 0.98
86.3 103.7 71.1 77.0 88.8 89.8
M S: Mild Steel, G I: Galvanised Steel, Alu.: Aluminium
annealed before testing. To get the yield stress of the material, the tension specimens were cut from the same sheets and annealed along with the respective shell specimens before testing. Table 1 shows the yield stress of the material of the shell specimens. Table 2 gives the dimensions of the aluminium spherical shells with R/t values ranging from 15 to 240. Table 3 shows the dimensions of the shells which are of same radius and of different depths. Table 4 gives the details of the shells of shallow depth and made of different materials. The spherical shells obtained from the process of spinning show a variation in their wall thickness in the meridian direction. The specimens tested under axial compression were cut into two equal halves, see Fig. 2(a) and the thickness was measured along the meridians. The trend in the variation of wall thickness along the meridian direction of a spun spherical shell is found to match with that reported by Blachut and Galletly [7], and a typical variation is shown in the Fig. 2(b). It is observed that the thickness is constant along the meridian over a short length near the apex. Further up to about 50% of the meridional length the thickness is found to vary almost linearly. For the remaining 50% of the meridional length thickness is again constant. This variation in wall thickness has been considered in the present analysis. For calculations of R/t values of the shells, t is taken as the average of t1 and t2, see Fig. 1. 2.2. Modes of collapse It is observed that in the initial stage of compression the spherical shells deformed with a flat contact region against the rigid plate. This is called stage I deformations. This deformation mode is clearly observed in spherical shells of low R/t values [6]. As the R/t value increases, the range of compression over which the stage I deformations occur decreases. With the progress of deformations the spherical shells show an axi-symmetric
N.K. Gupta et al. / Thin-Walled Structures 34 (1999) 21–41
27
Fig. 2. (a) Sectional view of the tested spherical shell (A2); (b) Typical variation of shell thickness along meridian.
inward dimple, inside a ring of contact with the rigid plate; this is stage II deformations. The axi-symmetric inward dimple expands moving the rolling plastic hinge at the contact of the plate outwards from the axis of the shell, as shown in Fig. 3. The collapse of spherical shell due to axi-symmetric inward dimpling is shown in Fig. 4(a) and Fig. 4(b). In case of spherical shells with low values of R/t the deformations of stage II is observed over a large range of compression. All specimens from Sp. No. A1 to A6 have shown stage II deformations over almost the entire range of compression. Later stages of deformation show buckling with non-symmetric shape, consisting of an integral number of lobes and stationary plastic hinges formed between the consecutive lobes, particularly in spherical shells with high R/t values. This is stage III deformations, and Fig. 5(a) and Fig. 5(b) show the tested specimen No. A17, deformed in this mode. The stage of compression, at which buckling with non-symmetric shape is initiated depends on the R/t values of the spherical shell. Fig. 6 shows
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N.K. Gupta et al. / Thin-Walled Structures 34 (1999) 21–41
Fig. 3. The movement of rolling plastic hinge with the progress of axial compression (schematic diagram).
the tested specimens No. A7, A10, A14 and A15. For the spherical shell specimens shown in Fig. 6 the stage of compression at which the stage III deformations are initiated is shown in Table 5. It is found that generally 5 to 8 stationary hinges are induced with an equal number of integral lobes. At the instance of initiation of stationary hinges, all the hinges are not initiated at the same time, but with the progress of compression they reach the maximum number. In the same way even the rotation of the stationary hinges reach the maximum with progress of compression, from its zero value at the instance of initiation of stationary hinges, see Fig. 6. From Table 5 it can be noticed that the stage III deformation is initiated at an early stage of compression and it occurs over a large range of compression, in the case of spherical shells of high R/t values. 2.3. Rolling plastic hinge As discussed in Section 2.2, at some stage of compression, the axi-symmetric inward dimple expands outwards from the axis of the shell due to the rolling of the plastic hinge at the contact with the top plate. It is observed that the radius of the rolling plastic hinge rp at the contact of the plate varies with the progress of compression. The typical variation of rp with deformation is shown in Fig. 7. The values
N.K. Gupta et al. / Thin-Walled Structures 34 (1999) 21–41
29
Fig. 4. Axi-symmetric inward dimpling mode of collapse of specimen no. A5; top view (a), and bottom view (b).
of rp were measured at different stages of compression by interrupting the axial compression of specimens in INSTRON machine. An equation for computing rp at any stage of deformation is obtained as: rp ⫽
K hc
(4)
Here K and c are the constants obtained from the experimental data. Results of Eq. (4) are shown in Fig. 7 along with the results obtained from Eq. (3) for comparison. 2.4. Results from experiments The typical load-deformation curves of spherical shell specimens obtained from the chart recorder of INSTRON machine are shown in Fig. 8. The shape of the load-
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N.K. Gupta et al. / Thin-Walled Structures 34 (1999) 21–41
Fig. 5. Buckling of specimen no. A17 showing integral number of lobes and stationary hinges; top view (a), and bottom view (b).
deformation curve is an important characteristic for an energy absorber, as it is a measure of the effectiveness of the structure in absorbing energy. An ideal energy absorber has a square load-deformation curve, i.e., once the crush mode starts, (at peak load) the collapse continues under the same load until the absorber is entirely consumed [1,2,8]. From Fig. 8 it can be observed that the load-deformation curves resemble that of an ideal energy absorber, with negligible serrations. In case of spherical shells of large thicknesses (A3, A4, A5) the load-deformation behaviour deviates from an ideal profile (see Fig. 9(b)). The linear variation of thickness of spherical shells along the meridian direction is responsible for the ideal load-deformation profile of the spherical shells. The load-deformation curves of a group of spherical shells with different radii and constant thickness and vice versa obtained from the experiments are shown in Fig. 9(a) and Fig. 9(b). The load-deformation profiles obtained for the spherical shells of shallow depth show clear variations from those of the spherical shells listed in Table 2. In these
N.K. Gupta et al. / Thin-Walled Structures 34 (1999) 21–41
Fig. 6.
31
Tested spherical shell specimens A7, A10, A14 and A15 showing stage III deformations.
Table 5 Compression at which stationary hinges are initiated Sp. No.
R/t
A7 A10 A14 A15
49.5 68.3 90.4 109.7
Depth Z (mm) 47.75 71.80 92.00 110.5
h (mm)
Z⬘%
22.0 18.0 16.0 12
46.1 25.1 17.4 10.9
Z⬘ Deformation of shell as a percentage of total depth (Z) at which stationary hinges are initiated
Fig. 7. Typical variation of radius of rolling plastic hinge.
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N.K. Gupta et al. / Thin-Walled Structures 34 (1999) 21–41
Fig. 8.
Typical load-deformation curves (exp.).
cases it is because the thickness variation along the meridian is hardly noticeable and the load-deformation does not follow the ideal profile. Fig. 10 shows the loaddeformation curves of some of the shallow spherical shells tested under quasistatic loading.
3. Analysis The concepts of rolling plastic hinges and stationary plastic hinges are used in arriving at an analytical model for predicting the load-deformation and energy-compression behaviour of the spherical shells. The mean radius of the spherical shell is given by R⫽
[[L/2]2 ⫹ Z2] 2Z
(5)
where L and Z are defined in Fig. 1. Plastic moment Mp per unit length of the hinge is given by Mp ⫽
0T 2 4
(6)
Here o is the yield stress of the material and T is the thickness of the shell at the parallel circle in contact with the plate. Plastic work dWr dissipated by the rolling plastic hinge [9–11] in the elemental area dA traversed by it, during a compression of dhi is given by, dWr ⫽
2[dA]Mp rp
(7)
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33
Fig. 9. Load-deformation curves of shells; (a) different radii and constant thickness, (b) different thicknesses and constant radius.
where rp is the radius of the rolling plastic hinge. The mean radius “r” of the parallel circle in contact with the plate for a compression h, is given by r ⫽ √h[2R ⫺ h]
(8)
By calculating the value of r at any stage of compression from Eq. (8), it is possible to get the value of dA for an incremental compression of dhi, as: dA ⫽ [2r]R[d]
(9)
Further to get the value of rp at any stage of compression, the experimentally
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N.K. Gupta et al. / Thin-Walled Structures 34 (1999) 21–41
Fig. 10.
Load-deformation curves of spherical shells of shallow depth.
measured values of rp at different stages of compression were plotted. By curve fitting, an equation for calculating rp at any stage of compression is obtained in the form rp ⫽
K dhc
(10)
The plastic work dissipated in the N stationary hinges, dWs during a compression of dhi is given by
冋 册
dWs ⫽ NMp
2 l N
(11)
where l is the length of stationary hinge. The plastic work dissipated due to the strain in the meridian direction [12] dWm during a compression of dhi is given by, dWm ⫽ 0e[dV]
(12)
Here e is the meridional strain. The strain in the meridian direction e1 corresponding to the local flattening of the shell over an incremental compression of dhi over an elemental length of the shell “dl” (see Fig. 1) is given by e1 ⫽
[dl ⫺ dl cos ] dl
e1 ⫽ [1 ⫺ cos ]
(13) (14)
The local flattening of the spherical shell can be observed only in the initial stages
N.K. Gupta et al. / Thin-Walled Structures 34 (1999) 21–41
35
of compression, and occurs only over a shallow range of compression. Further, as the R/t value of the spherical shell increases, the range of compression over which the stage I deformation is observed, decreases. When once the deformation changes over to stage II, i.e., axi-symmetric inward dimpling, then the meridional strain will be lesser than that given by Eq. (14). As such it is assumed that the meridional strain will vary linearly with deformation. Based on this, it is assumed that the meridional strain will be maximum at the initiation of compression of the shell, and it is zero at maximum compression equal to the radius of the spherical shell, i.e.,
冋
e ⫽ e1 1 ⫺
1mp 1mt
册
(15)
Fig. 11. Comparison of experimental and computed results; (a) load-deformation curves, (b) energycompression curves.
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N.K. Gupta et al. / Thin-Walled Structures 34 (1999) 21–41
Fig. 12. Experimental and computed load-deformation curves of spherical shell of shallow depth.
The load on the spherical shell at any stage of compression is given by p⫽
[dWr ⫹ dWs ⫹ dWm] dhi
(16)
Thus the load-deformation curve is obtained from the above, and the energy-com-
Fig. 13. Experimental load-deformation curve of spherical shell and one computed by assuming constant meridional thickness.
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37
Table 6 Experimental and analytical mean collapse loads of the aluminium spherical shells Sp. No.
R/t
Mean collapse loads (kN) Experiment
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16 A17 A18 A19
15.3 20.7 27.9 29.2 33.4 43.7 44.4 49.5 57.4 58.3 61.5 68.3 69.3 78.8 85.7 90.4 109.7 112.5 143.5 184.3 240.9
13.64 10.17 27.17 10.66 22.32 15.42 2.67 4.89 8.80 2.86 1.71 5.72 3.42 2.12 4.79 7.49 7.87 5.06 6.13 1.62 1.51
Analytical
16.45 9.67 28.02 11.22 22.37 14.67 2.88 5.00 9.79 3.50 1.71 5.94 4.00 2.16 5.59 7.52 8.27 5.87 6.39 1.85 1.94
Updike
10.16 6.19 22.23 7.25 15.15 8.60 1.90 2.88 5.70 2.15 1.02 3.47 2.21 1.22 2.80 3.80 4.00 2.92 2.92 0.76 0.74
De Oliveira and Wierzbicki 9.63 5.94 21.32 6.90 14.48 8.20 1.81 2.75 5.42 2.05 0.98 3.30 2.10 1.16 2.66 3.60 3.79 2.77 2.77 0.72 0.71
pression curve is obtained from the cumulative summation of the energy absorbed over the incremental compression dhi. 4. Results and discussions Load-deformation and energy-compression curves obtained from the analytical model compare very well with the experimental observations, as shown in Fig. 11(a) and Fig. 11(b). The results obtained from the solutions proposed by the earlier researchers [3,5] have also been shown for comparison. It is seen that the results of the proposed analytical model match the experimental load-deformation and energycompression data very well. To study the effect of depth on the mode of collapse a set of experiments were conducted on spherical shells of different depths but having same radius, see Table 3. The mode of collapse in these cases does not show any noticeable changes when compared with the observations made in the case of shells listed in Table 2. But in case of spherical shells of shallow depth, see Table 4, the load-deformation profile is not following the ideal profile as observed in Figs. 8 and 9(a) and Fig. 9(b). By
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Table 7 Experimental and analytical mean collapse loads of aluminium spherical shells with same radius, thickness, and different depths Sp. No.
B1 B2 B3 B4
Span L (mm)
127.50 141.92 152.40 154.56
Depth Z (mm)
30.30 46.67 67.31 72.80
Radius R (mm)
78.20 77.30 76.80 78.00
t (mm)
0.98 0.87 0.90 0.91
R/t
89.8 88.8 85.3 85.7
Mean collapse load (kN) Exp.
Ana.
3.26 3.72 4.36 4.79
3.02 3.29 4.75 5.59
Table 8 Experimental and analytical mean collapse loads of spherical shells of shallow depths Sp. No.
Mat.
Depth Z (mm)
Specimens tested as received (without C1 MS 50.36 C2 GI 47.29 C3 GI 37.28 C4 GI 32.56 Specimens tested after annealing C5 MS 51.14 C6 GI 47.29 C7 GI 37.71 C8 GI 30.90 C9 Alu. 46.67 C10 Alu. 30.30
Radius R (mm)
t (mm)
R/t
Mean collapse load (kN) Exp.
Ana.
annealing) 78.80 78.32 70.83 57.72
0.81 0.73 0.97 0.74
96.9 107.3 70.8 78.0
12.32 9.24 11.81 6.83
13.97 9.83 11.84 7.73
79.41 80.56 68.24 60.03 77.30 78.20
0.92 0.78 0.96 0.78 0.87 0.98
86.3 103.7 71.1 77.0 88.8 89.8
8.29 7.27 7.17 4.04 3.72 3.26
9.88 7.18 7.45 4.86 3.29 3.02
Mat.: Material, M S: Mild Steel, G I: Galvanised Steel, Alu.: Aluminium Exp.: Experimental Ana.: Analytical
keeping the thickness constant the analytical model described above can be used for the prediction of the load-deformation as well as the energy-compression curves of the spherical shells of the shallow depth (see Fig. 12). From Fig. 11 it can be seen that the analytical predictions match well with the experimental observations. Fig. 13 shows the difference in the load-deformation behaviour due to the assumed uniform thickness and varying thickness of the shell. The mean collapse loads for the spherical shells have been calculated both from the experimental observations and from the analytical model. Tables 6–8 give the mean collapse loads as obtained actually from the experiments and from the analytical model. The yield stress values of different aluminium sheets show considerable variation, see Table 1. To carry out the comparative study, the mean collapse loads
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were normalised for an yield stress of 75 MPa, in case of aluminium spherical shells. Fig. 14(a) and Fig. 14(b) show the variation of mean collapse load with different radii and constant thickness, and vice versa. From Fig. 9(a), Fig. 9(b) and Fig. 14(a), Fig. 14(b) it can be seen that the thickness of the spherical shell influences the mode of collapse as well as on the load-deformation curve. Fig. 15(a) and Fig. 15(b) show the energy-compression curves of aluminium spherical shells. 5. Conclusions Aluminium spherical shells of R/t values ranging from 15 to 240 and a few spherical shells of shallow depth were tested for axial compression in an INSTRON
Fig. 14. Variation of mean collapse load with R/t values of the spherical shells; (a) different radii and constant thickness, (b) different thicknesses and constant radius.
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Fig. 15. Energy-compression curves of shells; (a) different radii and constant thickness, (b) different thicknesses and constant radius.
machine. Their typical modes of collapse and load-deformation curves are presented. A simple analytical model is also presented to predict the load-deformation and energy-compression behaviour. From Fig. 11(a) and Fig. 11(b) one can observe matching between experimental results and results predicted from analytical model. Further it can be seen from Fig. 8 and Fig. 9(a) and Fig. 9(b) that the load-deformation profiles of spherical shells resemble a square wave profile and as such the spherical shells can be ideal energy absorbers. From Fig. 13 we can see that the thickness variation in the meridian direction contributes to making the curve flat. The mode of deformation of a spherical shell is highly dependent on R/t of the shell. For its high values (greater than 50)mode III deformations prevails over a large
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range. For intermediate R/t values (greater than 10 up to 50)mode II deformations are predominant. The mean collapse loads for the spherical shells tested have been calculated from the load-deformation curves. The analytical and experimental mean collapse load values for spherical shells tested, match well, as seen in Tables 6–8.
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