118 Energy absorption of structures and materials
r = 1.032
r = 0.774
r = 0.538
r = 0.258
r = 0.0645
Theoretical results
[Morris and Calladine, 1971]
Experimental results [Morris, 1971]
0 0
1 1
2 2
3 3
4 4
5 6 7 8 9
d /h
P 2pMo
5.3 Experimental and theoretical non-dimensional load–displacement curves for boss-loaded cylindrical shell with fully clamped ends.
The central cross-section is assumed to be of a circular arc closed with a straight line matching the wedge tip BD, see Fig. 5.4(c). The overall plastic zone size is defined by characteristic length xin Fig. 5.4(b) and width BD is 2l. Strictly speaking, all the travelling plastic hinges must have a (small) radius, instead of being of sharp creases. However, for the present model, this point is not investigated. The ends of the tube can be either free or fully fixed.
For the assumed central cross-section geometry, because the circumfer- ential length remains the same, we have a new radius for the current part:
2l
(a)
(b) A
D
C B
x
x
R dp
b R¢
2l
(c)
5.4 Indentation of a tube by a wedge indenter. Assumed hinge lines:
(a) and (b); and central cross-section (c).
[5.3]
The plastic dent depth
[5.4]
and the half width of the flat part is
[5.5]
The fully plastic bending moment of the central cross-section for the current configuration is straightforward to work out; it is
[5.6]
where Mt=YD2h, the fully plastic bending moment of the circular tube, and a =–12(p+b- sinb). Numerical calculations lead to an approximate equa- tion for the bending moment M
[5.7]
This indicates that the fully plastic bending moment capacity decreases linearly with plastic dent depth. This result is very similar to the behaviour of an equivalent initially square cross-section of side length a= pD/4, being deformed into a rectangular section with a height reduc- tion of dp.
The rate of external work should be equal to rate of the plastic energy dissipation, as discussed in Section 2.2. The unknown parameters xand lare obtained by minimising the external work. de Oliveira et al.(1982) found that
[5.8]
and
[5.9]
where Np =pDhYis the fully plastic axial force of the tube and Nis the axial force generated within the tube.N/Np=0 and 1 for a tube with free ends and fully fixed ends, respectively.
Wierzbicki and Suh (1988) subsequently improved the above analysis by considering a series of more realistic deformed sectional shapes of the tube,
P M D
h D
N N
t p
p
= - Ê -
ËÁ ˆ
¯˜ È
ÎÍ ˘
˚˙ ÏÌ
Ó
¸˝
˛
4 1 1
2 1
2
2 1
p d 2
x pd
= - Ê -
ËÁ ˆ
¯˜ È
ÎÍ ˘
˚˙ ÏÌ
Ó
¸˝
˛ D h
N N
p
4 1 1 p
2 1
2 1 2
M=(1-dp D M) t
M= ( - + )Mt
( - + )
p a b b b
p b b
2
2
2 2
sin sin sin cos sin
l= ¢R sinb
d b p b p b
p b b
p
=R( + - - )
- +
2sin 2 cos
sin
¢ = - + R pR
p b sinb
connected by strings in the longitudinal direction. The final equations they obtained are
[5.10]
and
[5.11]
The notation used is the same as for Eqs (5.8) and (5.9).
Tube denting experiments were performed by Reid and Goudie (1989), following preliminary tests by Thomas et al.(1976). Mild steel seamed tubes (D = 50.8 mm,h =1.6 mm) were either simply supported or fully fixed at the ends and then loaded at mid-span by means of a wedge-shaped inden- ter. In such an experiment, local denting of the tube occurs initially, with little tube global deformation. The total indenter displacement dtlis always larger than the local plastic dent depth dp.
Experimentally, it was found that dpis linearly proportional to dtl(e.g.dp
= 0.835dtlfor a fully fixed tube with length L= 305 mm). As deformation proceeds, the fully plastic bending moment at the central section reduces as a result of the change in geometry [Eq. 5.6]. When the indenter force P produces a bending moment equal to this moment capacity (i.e.PL/4 =M), structural collapse of the tube occurs, similar to that of a solid beam section under three-point loading (Fig. 2.12). For fully fixed end tubes, axial stretch- ing develops.
From the empirical relation between local plastic dent depth dpand total indenter displacementdtl, Eq. [5.9] can be re-cast in terms of dtl. Also, for a given load Pthe elastic deformation of the tubular beam can be determined using the usual beam theory and hence a ‘theoretical’ load–deflection curve can be produced. Figure 5.5 compares the results from experiments and theory using this approach (Reid and Goudie, 1989) for 457 mm span tubes with ends free and fully fixed. The curves indicate that elastic-plastic analy- sis agrees better with the experiments than rigid-plastic theory.
Jones and his co-workers (Jones et al., 1992; Jones and Shen, 1992) further modified the theoretical analysis and conducted extensive impact tests using a drop hammer on steel tubes with D/h=11~60 and L/Dª10. Fully clamped tubes were impacted by a rigid wedge indenter at mid-span, quarter span or near a support. Typical results are shown in Fig. 5.6 in terms of plastic deformation and input energy for a mild steel tube:L=600 mm, D=60 mm,h=2 mm. Yield stress is used in the theoretical calculations. The
P Yh D hR
N N
p
p
= - Ê -
ËÁ ˆ
¯˜ È
ÎÍ ˘
˚˙ ÏÌ
Ó
¸˝
˛
4 3 1 1
4 1
2
3 1
p d 2
x pd
= - Ê -
ËÁ ˆ
¯˜ È
ÎÍ ˘
˚˙ ÏÌ
Ó
¸˝
˛ D
h
N N
p
2 p
2
3 1 1
4 1
3 1 2
0 0 5 15
10 10
20 20
25
30 40
(a)
50 60
Indenter displacement dtl (mm) Simply supported
457mm span
Rigid-plastic Elastic-plastic Experimental Denting–bending Structural collapse
Load P (kN)
0 0 20
10 10
20 30
40
30 40
(b)
50 60 70
Indenter displacement dtl (mm) Fully fixed
457mm span
Rigid-plastic Elastic-plastic Experimental
Denting–bending–stretching
Load P (kN)
Membrane stretching
5.5 Theoretical and experimental load–indenter displacement curves:
(a) simply supported; (b) fully fixed (Reid and Goudie, 1989) (reproduced with kind permission of John Wiley & Sons Inc.).
deflection is slightly less if an average of yield stress and ultimate stress is taken as the flow stress.
Impact energy leading to tube material rupture was studied by Shen and Chen (1998). Denting of a tube by two wedge indenters was reported by
60
40
20
0
500 1000 1500 2000
E (Nm) (a)
d (mm)
80
40
20
0
500 1000 1500 2000
E (Nm) (b)
d (mm)
60
5.6 Comparison of tube deflection from theory (straight line) and impact experiments (dots): (a) impact at mid-span; (b) impact at one-quarter span (Jones et al., 1992) (reproduced with kind permission of the Council of the Institution of Mechanical Engineers).
Watson et al.(1976) and Lu (1993b). Empirical formulae were given by Lu (1993b) and Ong and Lu (1996). In particular, for mild steel tubes with L/D
=10,P=3.78Yd0.47h1.6D-0.07, where dis the indenter displacement. Kardaras and Lu (2000) conducted a finite element analysis of tube indentation by point loads.