6.1 Circular tubes
6.1.2 Theoretical models
Alexander model for ring mode
Alexander (1960) was the first to provide a theoretical model for axial crushing of a circular tube for the ring mode. The model is shown in Fig.
6.4. During formation of a single fold, three circumferential plastic hinges
(a) (b) (c)
6.1 Collapse modes for circular tubes under axial loading: (a) ring mode; (b) diamond mode; and (c) mixed mode (reproduced with kind permission of Elsevier).
101 102
D/h 0
2 4 6 8 10 12
L/D
Non-symmetric Euler
Mixed
Axisym.
Key to symbols : – Axisymmetric – Non-symmetric – Mixed mode – Euler – Other Other
2 3 4 5 6 7 8 2 3 4 5
6.2 Mode classification chart for circular aluminium tubes (reproduced with kind permission of Elsevier).
occur. Assuming that the fold goes completely outwards, all the material between the hinges experiences circumferential tensile strain. The external work done is dissipated by plastic bending of the three hinges and circum- ferential stretching of the materials in between.
In the following analysis, the material is assumed to be rigid, perfectly plastic. Further, there is no interaction between bending and stretching in
0 0
20 40 60
60
80 100 120
Displacement (mm)
Pm – average post-buckling load PMAX. – maximum or peak load – 1st fold
50
40
30
20
10
Force (kN)
6.3 Typical force–displacement curve (reproduced with kind permission of Elsevier).
Hinges P
D
H
P H
h
s q
6.4 A simple theoretical model for axisymmetric collapse.
the yielding criterion; hence, the material yields either by bending only or stretching only. For a complete collapse of a single fold, the plastic bending energy is
or
[6.1]
where His the half-length of the foldand Dis the tube diameter.Mois the fully plastic bending moment per unit width as before.
The corresponding stretching energy is
where Yis the yield stress.
When q=p/2
[6.2]
This equation can be obtained also by considering the change of the area between the three hinges, {=2[p(D+2H)2/4 -pD2/4] -2pDH=2pH2}, and then multiplying it by Yh, the yielding membrane force per unit length.
From the energy balance, the external work has to be dissipated by plastic energies in bending and stretching. Consequently
[6.3]
where Pmis the average external force over a complete collapse of the fold.
Substituting Eqs [6.1] and [6.2] into Eq. [6.3], we have
[6.4]
The unknown length His determined by invoking the idea that the value of His such that the external force Pmis minimum. Hence, let ∂Pm/∂H=0 to give
[6.5]
Substituting Eq. [6.5] in Eq. [6.4]
[6.6]
P
Ym ª6h Dh+1 8. h2
H = Ê Dh Dh
Ë ˆ
¯ ª
p
2 3 0 95. P
Y
h D
H Hh
m = Ê +
Ë
ˆ
¯+
p p
p
2
3 2 1
Pm2h W= b+Ws Wsª2pYhH2
Ws Y Dh D s D s
H
=2
Ú [
( +2 )]
0
p ln sinq d
Wb=2pMo(pD+2H)
Wb=2M Do 2+2Mo
Ú
(D+2H )0 2
p p
p q q
p
sin d
Remember that in the above analysis the material is assumed to deform completely outwards. If the material deforms inwards, a similar analysis leads to
[6.7]
In practice, as Alexander argued, the material deforms partially inwards and partially outwards. Hence, an average of Eqs [6.6] and [6.7] can be taken:
[6.8]
This completes the Alexander analysis of axisymmetric collapse of circular tubes, developed in 1960. The model is extremely simple, but it does capture most of the main features observed in experiments. Several modifications of this model have been presented. Johnson (1972) modified the expression for the stretching energy on the grounds that the circumferential strain varies along s.
It was recognised that the deforming tube wall bends in the meridian direction instead of the straight line (Abramowicz, 1983; Abramowicz and Jones, 1984b and 1986). In their modified model, two arcs join together to represent the deformed tube wall. This leads to an effective crush lengthde
which is smaller than 2H
[6.9]
Consequently, a slightly higher average force than Eq. [6.8] is obtained after assuming that Hremains the same
[6.10]
Grzebieta (1990) further modified the meridian profile, but adopted an equilibrium approach. Thus, the force–displacement curve can be worked out, not just the average force. To account for the fact that the tube wall deforms both inwards and outwards, Wierzbicki et al. (1992) introduced a parameter known as the eccentricity factor, which defines the outward portion over the whole length H. The value of this parameter is about 0.65 based on experiments. In this way, the occurrence and position of a second peak within each fold can be predicted. This work has been further refined by Singace et al. (1995) and Singace and El-Sokby (1996).
Effects of strain-rate and inertia
In the dynamic case, the strain-rate effect can be approximately taken into account as follows. As discussed in Section 2.4.2, this effect plays a role by
P Yh Dh h
m = Ê - D
ËÁ ˆ
¯˜ 8 91. 1 0 61. de
H
h D 2 0 86 0 52
1 2
= - Ê
Ë ˆ . . ¯ Pm ª6Yh Dh P
Ym ª6h Dh-1 8. h2
enhancing the yield stress of the material. Based on the Cowper-Symonds relation, Eq. [2.73], Eq. [6.8] can be rewritten as
[6.11]
where Band qare constants for the tube material and is the strain rate.
Their typical values are given in Table 2.1.
The key is to estimate the strain-rate over the dynamic collapse process.
Here we present a simple estimate for the average circumferential strain- rate, which is assumed to be representative of the problem. The mean strain in a completely flattened fold of the circular tube is
[6.12]
Assume that the tube starts to deform with an initial velocity and that this velocity decreases linearly with time. This corresponds to a constant decel- eration with a constant external axial load. The total time to deform one fold completely is
[6.13]
Therefore the average strain-rate is
[6.14]
Substituting this into Eq. [6.11] results in
[6.15]
Note that the contribution of the second term is not as much as one might expect because the value of qis usually large.
The inertia effect of axial collapse of tubes can be large, similar to the type II inertia-sensitive structures discussed in Section 7.2. Detailed analy- sis can be found in Karagiozova et al. (2000).
Theories for non-symmetric modes
Theoretical models for the diamond mode are less successful than those for the ring mode. Most of the models involve bending of triangularised ele- ments about hinge lines with the mid-surface being inextensional. Pugsley and Macaulay (1960) were among the first researchers to consider the diamond mode. They proposed
[6.16]
The constants were determined to best fit the experiments. Johnson et al.
(1977b) attempted to develop a theory for the diamond mode based on experiments with PVC tubes. From the actual geometry of folding, the
P Y Dh
h p =10D+0 13. Pm Yh Dh Vo D
=6
[
1+( 2 )1q]
e˙q =eq T=Vo 2D T=2H Vo
eqªH D
e˙ Pm Yh Dh B
ª6
[
1+(e˙ )1q]
arrangement of hinge lines can be worked out for a given number of lobes.
Figure 6.5 shows such an arrangement for three lobes. The external work is dissipated only by plastic bending of elements about hinge lines together with flattening of the initially curved elements. For long tubes, the calcu- lated force is
[6.17]
where n is the number of circumferential lobes. This formula requires a prior knowledge of n and there is no established method of determining this.
Further theoretical studies have been conducted by Singace (1999). An eccentricity factor was introduced in the same way as for the ring mode.
The equation developed is
[6.18]
Experiments on axial crushing of circular tubes
Experimental results on axial crushing of circular tubes are extensive. Most of the researchers who proposed theoretical models conducted experiments in an attempt to validate the models. However, the range of D/hused was usually very limited. The most recent work is by Guillow et al. (2001) and it covers a sufficiently large range of D/h and L/D in a single testing program. For aluminium tubes, the average force is plotted against D/hin Fig. 6.6.
Figure 6.6 is a double logarithmic plot. It is surprising to note that all the points, regardless of the collapse mode, fall into a single curve, which may
P
M n
n n
D h
m o
= - + Ê
Ë ˆ
¯
p p p
3 2
2
2
tan P
M n
n n n
m
2 o 1
2 2
p
p p
ª + Ê
Ë ˆ
¯+ Ê
Ë ˆ cosec cot ¯
C
C 2n p 2n
ho h1=m ho
6.5 A theoretical collapse model for non-symmetric mode; n=3 (reproduced with kind permission of the Council of the Institution of Mechanical Engineers).
be approximated using a straight line. Hence an empirical equation emerges as
[6.19]
Recall the theoretical analysis presented above for the ring mode. The value of Pm/Mois largely proportional to and dependent upon n. Experi- mental results in Fig. 6.6 clearly defy these theoretical observations. Huang and Lu (2003) have explained this by proposing a model with an effective hinge arc length, the value of which is 3hto 5h.
Structural effectiveness and solidity ratio
To facilitate the presentation of test results, two important parameters are introduced here, namely structural effectiveness and solidity ratio. The structural effectiveness is defined as
[6.20]
where Ais the net cross-sectional area of the thin-walled tube. Hence,AY represents the axial squash load and his always less than 1. For a circular tube, therefore
h= P AY
m
D h/ P
M
D h
m o
= Ê
Ë ˆ 72 3 ¯
0 3
.
.
101 102
102
D/h Pm /Mo
– Axisymmetric – Non-symmetric – Mixed mode
2 2
2 3
3 Eq. [6.19]
3 4
4
4 5
5
5 6
6
7 7
8
6.6 Dimensionless plot of average force against D/hfor aluminium tubes (reproduced with kind permission of Elsevier).
[6.21]
The solidity ratio is defined as
[6.22]
Here A1is the enclosed total area of the section and is pD2/4 for circular tubes. Obviously,f<1.
Experiments for thin-walled tubes can be summarised using these two parameters. Figure 6.7 shows a plot for circular tubes. It is clear that empiri- cally for circular tubes
[6.23]
The empirical equation [6.19] can be recast into the form
[6.24]
This has the same value of power 0.7 as in Eq. [6.23], but with a much higher value of coefficient 5.7.Wierzbicki and Abramowicz (1983) obtained a value of 5.15, which is close to the present one.
Dynamic tests of circular tubes lead to a higher average load compared with the static tests (Abramowicz and Jones, 1984b). Figure 6.8 shows the ratio of the dynamic average load,Pmd, to the static one,Pms. Equation [6.15]
is also plotted with D = 6844s-1and q= 3.91 for the mild steel’s ultimate stress.
h=5 7. f0 7. h=2f0 7. f=A A1 h=Pm pDhY
f h
0 0 0.2 0.4 0.6 0.8 1.0
0.1 0.2 0.3 0.4
Mamalis and Johnson (1983) Abramowicz and Jones (1986) Alexander (1960)
Macaulay and Redwood (1964) Eq. [6.23]
6.7 Plot of structural effectiveness against solidity ratio for circular tubes.