Four plastic hinges are needed for a ring to collapse under compression between two flat plates, and two common modes are shown in Fig. 4.6 (Burton and Craig, 1963; de Runtz and Hodge, 1963). The first mode has four stationary plastic hinges and is more appropriate for mild steel, which has an upper and lower yield point. The second mode involves straighten- ing of the ring at the moving contact point. Both modes have the same force diagram for the undeformed segment and hence lead to the same force–deflection curve. The initial collapse load is the same as for the point loading case (Eq. [4.4]). From equilibrium
[4.28]
and from geometry
[4.29]
d =2Rsinq 1
2PRcosq=2Mp C
BC
B B
BC C
A
(a) (b)
Line of action of resultant force
B C
C
A'
B' C'
P' P'
R P
P
R
R d
Tangent at B''
Line of action of resultant force
Limit of mode 1
A A
A
C
C B
B
B1
B1 B1
B''
C'' P'
P'
B'' X
X
C
'
B1' Line of action of
resultant force
Hinge travelling towards
Straight Hinge travelling towards
(c) (d)
A D
B
C
20
10
Mode 1
Mode 2 Mode 3 Mode 4
1 2
Triangular form
R d PR
Mp
(e) (f)
4.5 A built-in semi-circular arch under an inwards acting point load.
(a) Initial configuration (solid line) and final configuration (dashed line); (b)–(e) collapse modes 1–4 (see text for details); (f) non- dimensional load–deflection curve. (Gill, 1976)
Combining Eq. [4.28] with [4.29], and noting Eq. [4.4], we have
[4.30]
or
[4.31]
where L is the width of the ring or tube. This demonstrates that load increases with deflection (Fig. 4.7). Note that the analysis of rings presented so far is equally applicable to tubes under similar loadings, provided the appropriate value is taken for yield stress. Thus, when the length is not greater than a few thicknesses, a short tube can be taken as a ring;Y in Eq. [4.31] is then equal to the yield stress from a uniaxial tensile test. When the length is larger than the diameter of the tube,Yis taken as mul- tiplied by the yield stress in simple tension in order to account for the plane straincondition.
It can be seen from Fig. 4.7 that this force prediction is lower than exper- imental results. This discrepancy can be accounted for by the strain- hardening effect, which has two implications here. First, the plastic bending moment resistance increases as deformation proceeds (Eqs [2.2] and [2.3]).
Second, plastic deformation takes place over a zone instead of being con- fined within a localised plastic hinge – i.e. the hinge has a certain length.
The latter effect leads to a change in load through a small but significant change in geometry and hence moment arm length. A simple way of esti- mating the strain-hardening effect is to evaluate the average strain involved
2/ 3 P Yh L
D D
=
-( )
[ ]
2 1
2 2 1 2
d
P P
D
= o
-( )
[
1 d 2 1 2]
P
2 P
2
2 P
2 P
2 P
2 P
R q Mp Mp
Mp
2 P
2 P 2
P 2
P 2 d
d2 d2
H H H H H
C
C V C V C V
V C V
(a) (b) (c)
4.6 Collapse mechanisms proposed by: (a) de Runtz and Hodge (1963) and (b) Burton and Craig (1963); (c) also shown are forces on a deforming segment.
in the deformation zone and then incorporate this into an enhancement of bending moment resistance. Assuming that the total hinge length is lh, which does not change during deformation, the average curvature is then k=q/lh.
For an assumed linear hardening relationship for bending moment [4.32]
where I is the second moment of area and Ep is the strain-hardening modulus of the assumed rigid-linear hardening material (see Eq. [2.5]).
From Eq. [4.29], q= sin-1(d/D). Replacing Mp in Eq. [4.28] with M from Eq. [4.32], we obtain
[4.33]
This equation was proposed by Redwood (1964). The value of lwas found to be 5 by measuring the plastic region in the experiment and this value is used in plotting Fig. 4.7.
Clearly Eq. [4.33] gives a better predication than Eq. [4.30], but it is still lower than the experimental results, especially when deflection is large. This is because the plastic hinges were still treated as being very localised and so the geometry is basically the same as previously assumed in Fig. 4.6. Reid and Reddy (1978) investigated this problem and proposed a plasticatheory, which replaced the concentrated hinges with an arc whose length varies
P
P D
E
Y D
o
= p
-( )
[ ]
+Ê Ë
ˆ
¯ È
ÎÍ
˘
˚˙
1 -
1
1 3
2 1 2
1
d l
sin d
M M E I M E
p p p Y
= + = Ê + p
ËÁ ˆ
¯˜
k q
1 l 3
4
3
2
1
0
0 5 10 15 20 25
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
10 20
20
30 30
40 40
50 50
60 70 d (mm) 400
300 200 100
Redwood (1964)
Redwood (1964) Reid and Reddy (1978)
de Runtz and Hodge (1963) expt.
P (kN) s (MPa)
P/Po
e %
d / D
Ep = 1.27 GPa
4.7 Non-dimensional load–displacement curves from experiments and theories (h/R=0.108, R=42.16 mm, L=101.6 mm). (Reid, 1983)
with deflection d. This essentially revealed that the effective moment arm length reduces with deflection, in addition to the enhancement of bending moment.
One quadrant of a tube HV is shown in Fig. 4.8(a), with an enlarged view of the plastic deformation zone HB shown in Fig. 4.8(b). This is similar to Fig. 4.6 except that hinge H is replaced with plastic zone HB. A linear strain- hardening relationship is assumed for the bending moment. The top moving hinge V is assumed to be a concentrated hinge as before. Plastic deforma- tion occurs within zone HB and the moment at B is the initial plastic bending moment Mp. Segment BV remains rigid and rotates during defor- mation. The governing equation for HB is (Frisch-Fay, 1962)
[4.34]
At H, MH = Mp + Pb/2. The two governing equations for the system (Fig. 4.8(a)) are
[4.35]
and
[4.36]
where g,band care defined as shown in Fig. 4.8(b). The solution procedure for these three equations was given by Reid and Reddy (1978). In particu-
d g
2=Rsin -c
P M
R
= 4 p
cosg E I s
P
p
d d
2
2 2
q = - sinq
P 2
P 2
P 2
B B
B B
V V C
H
H P
2 d
2 P
R g g
Mp
Mp Mp Mp
Mp MH
MH 2
b s c q
p / 2 – g
(a) (b)
4.8 Tube with strain-hardening material analysed using plasticatheory – Reid and Reddy (1978): (a) forces on a quadrant of a tube HV;
(b) deformation of plastic region HB.
lar, they identified the following dimensionless parameter, which governs the shape of the load–deflection curve
[4.37]
High values of mRcorrespond to a relatively flat load–deflection curve, as predicated by de Runtz and Hodge (1963) for a rigid-perfectly-plastic material. Small values of mRcause the curve to rise up significantly. Their theoretical prediction is in better agreement with experimental results than previous theories, as shown in Fig. 4.7. It may also be noted that, by choos- ing materials with different strain-hardening effects, the force–deflection curve may be adjusted to be close to the ideal rectangular force–deflection relationship of an energy absorber as described in Chapter 1.