2.5 Energy method
2.5.2 Energy method used in case of large deformation
Load-carrying capacity of a structure in its large deformation In the case of large deformation, the analysis is aimed at examining the vari- ation of the load-carrying capacity of a structure during its deformation
(a) (b)
L
L
F F
H
(c) F
2.22 (a) A rigid-plastic frame subjected to a force F; (b) a ‘local’
collapse mechanism; (c) a ‘global’ collapse mechanism.
process, instead of merely determining its incipient collapse load. In other words, now the objective is to find the limit load as a function of the deformation
[2.81]
where Dis a representative displacement in the structure during its large deformation process. When load P is a concentrated force, it would be straightforward and convenient to take the displacement at the loading point as D, with the same direction as P.
By assuming a large deformation mechanism of the structure under load P, the energy balance directly leads to
[2.82]
where Dfis the final displacement in the process, and the internal energy dissipation Dis calculated as an integration of the incremental plastic dis- sipation dDduring the deformation process.
Differentiation of Eq. [2.82] gives
[2.83]
If there is no unloading during the assumed large deformation process, by neglecting the influence of the plastic deformation history of the structure, the internal energy dissipation Dcan be calculated from the final configu- ration of the structure.
To show an example of using the energy method in the case of large deformation of structure, re-examine the beam pressed by a round-headed indenter, as shown in Fig. 2.12(a). With the increase in the deflection, the deformed configuration of the beam is as sketched in Fig. 2.12(b), where angle bserves as a process parameter. The internal energy dissipation is
[2.84]
where DA is the dissipation at plastic hinge A, and DBB¢is that along the bent arc BB¢.
The force Pis found from Eq. [2.83] as
[2.85]
Using the geometric relation between Dand bgiven in Eq. [2.46], the fol- lowing is obtained
[2.86]
where Ps = 4Mp/L is the initial collapse force as defined in Eq. [2.43].
Together with Eq. [2.47], Eq. [2.86] gives the variation of the load-carrying capacity of the beam during its large deflection.
P
Ps R L
= -( ) cos
sin
2
1
b b
P( )D =dD dD=4Mp (dD db)
D D D M M
R R M
A BB p p p
=2 + ¢ =2 + 1 =
2 4
b b b
P( )D =dD dD
Ein P D D
f f
=
Ú
0D ( )D Dd = =Ú
0D dP=P( )D
By comparing Eq. [2.86] with Eq. [2.45] it is found that the load-carrying capacity obtained from the energy method (Eq. [2.86]) is slightly smaller than that obtained from the equilibrium method (Eq. [2.45]). It is observed, therefore, that although the energy method leads to upper bounds for the initial collapse load of the structure whilst the equilibrium leads to lower bounds, this does not promise that the former will provide a higher esti- mate than the latter for the load-carrying capacity of the same structure during its large deformation.
Other forms of energy dissipation
For transversely loaded beams and plates, the incipient collapse mecha- nisms involve only bending deformation. However, during their large defor- mation, other forms of energy dissipation may become important or even dominant, so that the internal energy dissipation may be typically written as
[2.87]
where Db,Dm,Dfriand Dfracdenote the energy dissipation by bending, mem- brane deformation, friction and fracture, respectively.
Still taking the beam shown in Fig. 2.12(a) as an example, if a blank- holding force Fis applied vertically at the clamped end, then when the beam slides along the clamp the friction force is mF with mbeing the coefficient of friction between the beam and the clamp. Since the sliding distance at each end of the beam is (refer to see Eq. [2.49] for the geometric relation) [2.88]
the work done by the friction (i.e. the energy dissipation due to friction) is [2.89]
If the beam’s ends are axially constrained, as shown in Fig. 2.13, then the axial strain and axial force are induced by the large deflection of the beam, as given by Eqs [2.49] and [2.51], respectively. Accordingly, the right-hand side of the expression of energy dissipations (Eq. [2.87]) should have one more non-zero term
[2.90]
where the axial strain is calculated from Eq. [2.49] as a function of para- meter b. Obviously, this term proportionally increases with deflection D, so when the deflection is large enough, the axial (membrane) deformation will dominate the energy dissipation in the beam.
Dm =2NA¥eL=8NpeD Dfri =2mF¥dL
dL=(1cosb-1)L-(tanb-b)R D=Db+Dm+Dfri+Dfrac+. . .
Selecting deformation character by minimising the external load When a cylindrical tube subjected to an axial load P collapses into an axisymmetric folding pattern (see Section 6.1), its large deformation load- carrying capacity varies with the displacement periodically, but the energy dissipation in a load cycle can be written in the following form
[2.91]
where lis the half length of the fold and Aand Bare coefficients depend- ing on material and geometry. On the other hand, the work done by the external load is
[2.92]
with C being another coefficient. Therefore, equating Eq. [2.91] and Eq. [2.92] gives
[2.93]
with A¢ =A/Cand B¢ =B/C. Since Pgiven in Eq. [2.93] is an upper bound, minimising it in terms of lwill result in
[2.94]
This example shows how the character length in a large deformation mech- anism can be determined by minimising the required load.
To take another example, when a cylindrical tube with longitudinal pre- cracks is compressed in the axial direction, the large deformation mecha- nism involves both bending energy and fracture energy. The latter is proportional to the number of fractures which occur, whilst the former is inversely proportional to this number. Again, an optimum number for the fractures occurring in the tube can be obtained from the energy balance argument with the external load minimised.