6.2 Square tubes
6.2.3 Details of the plastic zones
In the idealised deformation mode in Fig. 6.12, all the plastic deformation occurs within localised plastic hinges. This would be acceptable if there were no propagation of plastic deformation. However, in this case, we mentioned that the plastic hinge KC is travelling as deformation proceeds, originally from position K¢C¢. A localised hinge with infinite curvature would absorb an infinite amount of plastic energy when it travels, as will be seen later.
The same can be said about point C, which moves from its original position C¢.
We therefore need a more realistic model which is kinematically admis- sible. Figure 6.13 shows such a model, obtained by extending the plastic deformation into a plastic zone instead of concentrated hinges. In this model, plastic deformation occurs only in shaded regions. Thus, during deformation the four plane trapezoidal plates move as rigid bodies. Two cylindrical surfaces are bounded by two straight hinge lines which propa- gate in opposite directions, leading to a wider zone. Two adjacent trape- zoidal plates are connected via a conical surface bounded by two straight lines. As KC in Fig. 6.12 moves, one straight line imparts a curvature to an originally flat sheet (part of JKCA) and the other removes this curvature so that the curved sheet bends back to flat, joining KLCD. Finally, the four active deforming zones are connected by a section of a toroidal shell. This doubly curved surface has a non-zero Gaussian curvature (defined as the product of two principal curvatures), while the cylindrical segments before and after passing this toroidal shell have a zero Gaussian curvature. There is therefore a change of Gaussian curvature when the material deforms into
6.13 A more realistic kinematically admissible folding mechanism (Wierzbicki and Abramowicz, 1983).
this shell and then back to a cylindrical one, and there must be in-plane stretching (Calladine, 1983b).
Our next task is to evaluate energy dissipation for each of the four types of plastic zone. Those for the two cylindrical shells are straightforward to perform, similar to the analysis in the previous chapters. The energies dis- sipated in the two travelling conical zones and in the toroidal shell are less so, and this issue will be dealt with next.
Energy dissipation in a travelling hinge
Consider a strip shown in Fig. 6.14, with a travelling hinge defined by an arc AB of radius r. Suppose this hinge moves by a distance Dsinto a new posi- tion, but the radius remains unchanged. For the sake of convenience, we assume that Dsis sufficiently large that the whole arc AB is unbent into a straight segment A¢B¢. The energy for unbending AB is then
[6.31]
where Mpis the fully plastic bending moment. Segment BC has been first bent into an arc of radius rand then unbent to a flat strip B¢C¢. The energy required by this process is
[6.32]
Similarly, for CD
[6.33]
and for DE
[6.34]
W DE r M
DE = 1 p
W CD rM
CD= 1 p
W BC r M
BC = 1 p
2 W AB
r M M
AB= 1 p = p(p-b)
A
r r
B C D
D¢ C¢
B¢
A¢
E b
Ds
b – Db
6.14 A strip demonstrating a travelling hinge.
Therefore, the total bending energy for this plastic hinge to travel a distance Dsis
[6.35]
Equation [6.35] was also given by Meng et al. (1983). This equation demon- strates that the energy absorbed by a travelling hinge is directly propor- tional to the distance travelled and inversely proportional to the radius rof the hinge. This explains why a sharp crease (of a radius zero) could not travel – to do so would require an infinite amount of external work. The process of a travelling hinge may be understood in another way: it can be regarded as material being pushed, in an opposite direction, through an anvil of radius r. Thus, the energy is absorbed by bending and then unbend- ing a strip of distance Ds, leading immediately to Eq. [6.35].
Energy dissipation in a sheet passing over a toroidal surface For the toroidal segment shown in Fig. 6.15, a generic point within this surface may be described by two coordinates (q,f). Here q denotes the meridian coordinate (Fig. 6.15(c)) and fis along the circumferential direc- tion (Fig. 6.15(b)). The limits of qand fare
[6.36]
[6.37]
Also
[6.38]
where bis the radius in the meridian direction (Fig. 6.15). When a material is forced to pass outward over this toroidal surface, there is a circumferen- tial strain and its increment corresponding to a tangential velocity vtis
[6.39]
Here,vtsinqis the horizontal component of vt, and
˙ sin cos ˙
tan
sin
e q aa cos
y
q
f = = q
+ v
r H
b a
t
o
r=bcosq+a - £ £b f b p y q p 2- £ £ 2+y
W W W W W
AB BC CD DE r M rM AB s AB CD
CE
rM s
AB BC CD DE
p
p
p
= + + +
=
(
+ + +)
◊ ◊= + Ê - +
ËÁ ˆ
¯˜ + È
ÎÍ ˘
˚˙
=
2 1
1 2
2 21
D D
[6.40]
In the toroidal region, the main plastic flow occurs in terms of f. Though there is a non-zero curvature change in circumferential direction, the cor- responding bending energy can be shown to be zero (resulting from the fact that yielding occurs in circumferential tension with a fully yielding mem- brane force No=Yh). Hence, the plastic dissipation rate is
[6.41]
Substituting Eqs [6.38], [6.39] and [6.40] into Eq. [6.41], we have
[6.42]
Note that the angle yis assumed to increase linearly with the coordinate f from yoto p/2 as
Integrating W˙1with respect to a, we obtain
y y p y
p f
= + -
o
2 o
˙
tan
cos cos cos ˙
W N bHo
o o
o o
o
1 4
2
2
= ( - )
- Ê + -
Ë
ˆ
¯ È
ÎÍ
˘
˚˙
p
p y y
a y y p y
p b a
˙ ˙ ˙
W No s N r b
s
o
1=
Ú
efd =Ú
ef df dqe˙
v s H
t
o o
= ˙ =
tan
cos ˙ y tan
aa y
nt
(a)
2b 2b
(b) a f
q
a r b
(c)
6.15 Plastic deformation of a toroidal surface: a 3D sketch (a) and definitions of various parameters (b,c) (Wierzbicki and Abramowicz, 1983).
[6.43]
where
[6.44]
Since bis a function of a, I1(yo) can be evaluated, e.g. I1(p/4) =0.58 for yo
=p/4, and I1(p/6) =1.05 for yo=p/6.
Energy dissipation in plastic hinges
The plastic energy dissipated in the fixed horizontal hinges AC and CD is [6.45]
or
[6.46]
Finally, the inclined hinges have a total length
[6.47]
Hence
[6.48]
[6.49]
where
Hence, I3(p/4) =1.11 and I3(p/6) =2.39. The rate of external work done is [6.50]
or
[6.51]
The work balance therefore requires
[6.52]
2P Hm =W1+W2+W3
W˙ext =2PH
˙ ˙ sin ˙
Wext =Pd =2PH aa
I3 d
0
1 2
y y
a g a
p o
o
( )=tan
Ú
cos sin W3=4MoI3( )yo H2 b
˙
tan cos W M Lv sin
b M H
o b
t o
o 3
2
2 4 1
= =
y a g
L H
= 2 sing
W2 M co M co 0
2 2
=
Ú
da =pp
˙ ˙
W2=2M co a I
d
1 0
2
2
2 1 2
y p
p y y a
y p y
p b y p y
p b a
p o
o o
o
o
o
o
( )=
( - )
Ê - Ë
ˆ
¯ + - Ê -
Ë
ˆ
¯ È
ÎÍ
˘
˚˙
ÏÌ Ó
¸˝
˛ tan
Ú
cossin sin cos cos
W N bH M Hb
o o o h o
1=4 I1( )y =16 I1( )y
where Pmis the average load. It can be seen that, on substituting the expres- sions for the three energy components (Eqs [6.43], [6.46] and [6.49]), the average load is of the following form
[6.53]
where A1,A2and A3are appropriate functions. The only two unknown para- meters, the radius b and the height of half a fold H, can be determined by letting
[6.54]
This leads to
[6.55]
[6.56]
Substituting them back into Eq. [6.53], we have
[6.57]
This indicates that total plastic energy comprises equal contributions from all the three major mechanisms of energy dissipation.
For a square or rectangular section of c1¥d, we take . I1= 0.58, I3=1.11. Also, because the top and bottom horizontal hinges (JK, KL, FG and GH) occur,W2has to be doubled. The corresponding energy balance gives
[6.58]
Hence,A1=32I1=18.56,A2=4p,A3=8I3=8.91. Consequently
[6.59]
and
[6.60]
For a square tube,c1=d=cand we have
[6.61]
The fact that Pm is proportional to h
5–
3is a reflection of the energy con- tribution of bending and stretching, in this case, 2 : 1. We know that for a
Pm =9 56Yh c
5 3
1
. 3
P M
c h
m o
=38 27. 3
H =0 983. 3 hc2, b=0 687. 3 h c2
2 64 1 8 16 3
2
HP M bH
h c H
m = oÊ + + b
ËÁ ˆ
¯˜
I p I
c=1(c +d) 2 1 P
Mm A A A c h
o
=33 1 2 33
H = A22 A A c h
1 3
3 3 2
b=3 A A A1 3 123ch2
∂
∂
∂
∂ P
H
P b
m m
=0, =0 P
M A b h A c
H A H b
m o
= 1 + 2 + 3
bending-only deformation, the force is proportional to h2while it is pro- portional to hfor a membrane-only deformation. When both are present, the force is proportional to hraised to a power between 1 and 2.
Using the structural effectiveness h and solidity ratio f introduced earlier, we have, for square sections
[6.62]
In the above model, only the circumference c plays a role in the calculation of Pmand H. The aspect ratio is immaterial. This has been partially verified by experiment: the observed fold length is indeed independent of the aspect ratio (Aya and Takahashi, 1974).