In Sections 2.2.3 and 2.2.4 the statically admissible stress field and the kine- matically admissible velocity/displacement field were described. It must be noted that these fields are constructed based on the original (i.e. unde- formed) configuration of the structure. For instance, the equilibrium in the beam (reflected in its bending moment diagram) shown in Fig. 2.10 was based on the straight (undeformed) configuration of the beam; and the geo- metric relation between deflection D* and rotation angle q* (see Fig. 2.11) was formulated by assuming that both of them are small. In these two exam- ples, the elongation of segments AB and AC was neglected because the for-
2 91. Mp L=Po£Ps£P*=3Mp L P*=3Mp L P≥ s
P* *D =MpqA+MpqB =3MpD* L
A
B
C
L L
P*
D*
q* qC q*
qA
2.11 A collapse mechanism for the beam shown in Fig. 2.10(a).
mulation was based on the straight configuration. As a matter of fact, the classical limit analysis and bound theorems can only be used to determine the incipient collapse mechanism, the initial limit loador its bounds.
However, when a structure or a material is designed or used for the purpose of energy absorption, it is usually expected to experience large plastic deformation under external loading. Therefore, the effects of large deformation should be taken into account when the theory of limit analy- sis is applied to analysing energy-absorbing structures.
2.3.2 Analysis of an illustrative example
To demonstrate how large deformation will affect the classical limit analy- sis, in this section we will take the following problem as an example.
Suppose initially a round-headed rigid indenter of radius R is in contact with the middle point of a straight rigid-plastic beam of length 2L; the beam is clamped at its ends (A and A¢), but allowed to have axial motions along the clamps, as shown in Fig. 2.12(a). Assume that all the contact surfaces are frictionless.
When the indenter is pressed down transversely to the beam by force P, it is easily seen that the incipient collapse mechanism of this beam contains two plastic hinges at the clamped ends A and A¢, as well as a plastic hinge at the middle section C of the beam. Following the procedure given in Section 2.2.4 and calculated from this incipient collapse mechanism (that is a kinematically admissible displacement field), the initial collapse force is found to be
[2.43]
with Mpbeing the fully plastic bending moment of the beam.
With the increase of the deflection, the deformed configuration of the beam is as sketched in Fig. 2.12(b), where a central portion of the beam, BB¢, wraps the round head of the indenter. Since the beam is supposed to be rigid, perfectly plastic, the circular segment BB¢ must be in its plastic pure bending state. Hence, in addition to the plastic hinges at clamped ends A and A¢, at which MA= -Mp, plastic hinges must form at cross-sections B and B¢, at which MB=Mpholds.
The free body diagram of the large deformation mechanism is sketched in Fig. 2.12(c), from which it is evident that with the increase of the inden- ter’s displacement, both the location and the direction of the indenter’s force will change. Note that the variation in bending moment from plastic hinge A to plastic hinge B is 2Mp, so that
[2.44]
P L R Mp 2( - sinb)=2 Ps=4Mp L
where bdenotes the angle formed by arc BC. Thus, the load-carrying capac- ity of the beam in large deformation is
[2.45]
where Psis the initial limit force given by Eq. [2.43].
On the other hand, the maximum deflection of the beam at mid-point C, D, can be expressed by angle bas
[2.46]
or in non-dimensional form
[2.47]
Equations [2.45] and [2.47] provide a relation between force P and maximum deflection Dthrough geometric parameter b. By taking R /L = 1/2, this relation is shown in Fig. 2.12(d) by the solid line.
D L
R
= +LÊ -
Ë
ˆ tan ¯
b cos
1 1b
D=(L R- sinb)tanb+R(1-cosb)=Ltanb+R R- cosb
P M
L R
P R L
p s
= - =
-( ) 4
1
sinb sinb
A
A C
(a)
(c)
A¢
L L
L R P
R
A
B
B B
C
C
(b)
(d)
A¢ b
b b
B¢
L L
P
R
0 0.1 0.2 0.3
1
L D Ps
P
F Mp
Mp
Mp Mp QA = P/2
2.12 A beam clamped at both ends without axial constraint: (a) the initial configuration when it is pressed by a round-headed indenter; (b) deformed configuration; (c) free-body diagrams of segment AB and arc BC; (d) the variation of the load-carrying capacity with central deflection.
If parameter bis small and Ris in the same order as L, (1 - 1/cosb) ª -b2/2, so that Eq. [2.47] indicates that bªD/L. Substituting it into Eq. [2.45]
results in
[2.48]
which provides an approximate linear relation between Pand D, as shown in Fig. 2.12(d) by the broken line in case of R /L =1/2.
The above analysis is based on the assumption that axial motion is allowed at the clamped ends of the beam. If axial constraints are applied to the clamped ends, as shown in Fig. 2.13(a), then the deflection of the beam will be accompanied by the elongation of the beam’s axis. Using the geometry shown in Fig. 2.12(c), the axial strain of the beam’s axis is found to be
[2.49]
If parameter bis small and Ris in the same order as L, it can be approxi- mated by eª b2/2, so that the incremental strain is deª bdb. Noting that the incremental rotation angle at plastic hinge A is dq=dband using Eq.
[2.26], we have
[2.50]
If the beam has a rectangular cross-section and the effective length of the plastic hinge is taken as l=2h, then Eq. [2.50] leads to an estimate of the axial force at hinge A,NA, induced by the large deflection of the beam
[2.51]
N
N L
A p
=4bª4D 2N M2
N L
p
l p
e q
b b b b
= d = = ª
d d d
D
e b
b b
b b b
= -
Ê + Ë
ˆ
¯- ÏÌ
Ó
¸˝
˛ =Ê -
Ë
ˆ
¯- ( - )
L R R L L R
L sin
cos cos1 tan
1 P
P
R
s L
ª +1 2D
A A
B C
C
(a) (b)
A¢
A¢
B¢ R
NA NA
MA P
MA
2.13 A beam clamped at both ends with axial constraint: (a) the initial configuration when loaded by a round-headed indenter; (b) the axial forces developed in the beam during its large deflection.
It is seen from Fig. 2.13(b) that the axial force NA will enhance the load- carrying capacity of the beam by adding a term 2NAsinbª2NAbª2NAD/L.
Therefore, using Eq. [2.48] and bªD/L, we have
[2.52]
It is clear that the contribution of the axial force to the collapse load depends on the slenderness of the beam, 2L/h. For example, for a short beam of 2L/h = 16, the axial force will result in the collapse load being doubled when the central deflection reaches the beam’s thickness (D =h), which is really significant.
2.3.3 Various aspects of effects of large deformation
From the analysis of this illustrative example it is observed that large defor- mation of a structure may affect its collapse mechanism and collapse load in many ways, such as:
• all the geometric relationships and the equations of equilibrium have to be formulated according to the current (instantaneous) configuration rather than the original (undeformed) configuration;
• the loading positions and directions of the external loads may vary with the deformation process – in the case of external loads applied by tools (e.g. punch, die, indenter, etc.), the friction between the structure and the tool’s surface may also alter the equilibrium and contribute to the energy dissipation;
• the instantaneous collapse mechanism of the structure may contain moving plastic hinges (or moving plastic regions), which evolute from their stationary counterparts in the incipient collapse mechanism;
• the axial forces (in 1-D structural components) or membrane forces (in 2-D structural components) may be induced by large deformation and greatly enhance the load-carrying capacity of the structure.
Whether all or some of these effects appear in an energy-absorbing struc- ture depends on its configuration, slenderness, supporting conditions and the way of loading. For instance, as shown in the above example, the large deformation behaviour of a beam with axial constraint is severely affected by the axial forces induced by the large deflection, whilst a similar beam without axial constraint is not affected. In the former case, the small defor- mation formulation may result in big errors when the maximum deflection of the beam reaches the thickness, whilst in the latter case, the small defor- mation formulation may be applicable until the deflection reaches the same order of the beam’s length.
P M L
R
L N
L P R
L hL
p
p s
ª Ê +
Ë
ˆ
¯+ Ê Ë
ˆ
¯ = Ê + +
ËÁ ˆ
¯˜
4 1 8 1 8
2
2
2 2
D D D D
For plates and shells under double-curvature bending, e.g. a circular plate under axisymmetric bending, previous studies have revealed that membrane stress will become important and even dominant as long as the maximum deflection reaches the order of the thickness of the plate or shell, regardless of the boundary condition. For instance, Timoshenko and Woinowsky-Krieger (1959) showed that as long as the central deflection of a simply-supported circular plate is equal to the thickness of the plate, the membrane strain in the middle plane of the plate will reach the same order as the bending strains. Regarding rigid-plastic circular plates, Calladine (1968) demonstrated that membrane stresses induced by large deflection will greatly enhance the load-carrying capacity of the plate even it is simply- supported.
2.3.4 Concluding remarks
(1) For 1-D structural components without axial constraint or for plates under cylindrical bending, the effects of large deformation are mainly reflected by the geometric changes, which become important when the maximum deflection of the component is comparable to the compo- nent’s characteristic length (e.g. the length of a beam or the radius of a circular ring).
(2) For 1-D structural components with axial constraint or for 2-D com- ponents (plates and shells) under double-curvature deformation, the effects of large deformation are mainly associated with the axial/mem- brane forces induced by the large deflection, which become dominant in the component’s load-carrying capacity when the deflection exceeds the thickness of the component.
(3) Usually, slender structural components suffer more in the geometric effect relevant to their large deformation, whilst stubby ones suffer more in the effect of axial/membrane forces – which is also caused by their large deformation.
(4) Depending on the configuration of the structure and the way of loading, sometimes a large deformation may also cause changes in the location and direction of the external loads, as well as changes in the collapse mechanisms (e.g. those containing moving plastic hinges).
(5) In all these cases, the various effects of large deformation need to be carefully incorporated into theoretical modelling of energy-absorbing structures.