Expanding Area of Application

As shown by G. Maier in (Maier, 1970) Quadratic Programming allows us to model a broad range of piecewise-linear structural behaviors. The constitutive laws of such behaviors are given in Fig.2. An exhaustive overview of the QP-based approach, including both continuum and discrete models, can be found in (Borkowski, 2004). Let us recall, for the sake of brevity, only two cases – a cable-strut elastic structure and a structure made of strain-hardening material.

The entries of the stress matrix s for the cable-strut structure can be split into two sub­matrices: ss represents axial forces in struts and sc represents axial forces in cables. Obviously, cables work only in tension, while struts can be either extended or compressed. Table 2 shows the set of relations governing elastic behavior of the cable-strut system. In order to simplify things, we assume purely static loading.

The adjoint variable for the axial force in cable is its slackness – the axial strain taken with negative sign. Grouping the slackness unknowns into a column matrix qc, we consider it as a matrix

of non-negative variables. Then, the entries of sc remain formally unconstrained in sign: they will become non-negative in the solution due to the constraint VLq > 0.

Applying the templates (6), (7) to the Table 2, we obtain the following energy principles for the cable-strut system:

max {- – SsE^1 Ss | CT Ss + C Sc = P0, Sc > 0}

Note that the existence of solution is not warranted: for a certain load p0 there might be no equili­brated stress state. Then the constraints of the QP-problem (19) become contradictory. On the other hand, if a solution exists, then it is unique due to the convexity of the problems (18), (19).

Table 2. Governing relations for a cable-strut structure

w

qc

ss

sc

1

VLw =

0

0

CT

s

CT

p0

= 0

=

0

0

0

I

0

> 0

v4 =

Cs

0

– E-1

0

0

= 0

<1

p

II

Cc

i

0

0

0

= 0

qc > 0, qT VLC = 0

Similar procedure can be applied to a structure made of elastic-strain hardening material. A complete set of governing relations for such a structure is shown in Table 3. The first row of this table includes linearised yield condition NTS — < k0. A positive definite (r x r) – matrix of hardening H is

responsible for a shift of the yield planes caused by the plastic strains. The equation of equilibrium that relates the given load p0 to the unknown stress s can be recognized in the second row. The last row

ensures the kinematic compatibility of strains q = qe + qp and displacements w. Elastic strains obey the Hooke’s law qe = E_1s . Plastic strains are governed by the associated flow rule qp = N X. Plastic multipliers represented by a column matrix X є Rr are supposed to be non-negative.

Table 3. Governing relations for a structure made of elastic-strain hardening material.

X

w

s

1

vL„ =

H

0

– NT

k0

> 0

VLw =

0

0

CT

– p0

= 0

VLs =

– N

C

– E-1

0

= 0

X> 0, XT VLX = 0

Applying the templates (6), (7) to the Table 2, we obtain the following energy principles for elastic – strain hardening structures:

min { 1XTH X +1 s1^1 s + – wTp01 C w – N X – E-1s = 0}

s, w,x>o 2 2

max {-1 XT H X-1 s E1 s | NT s – H X< k0, CTs = p0}

Solutions of these QP-problems exist for any p0 since the admissible domain for stresses adjusts itself

automatically to the loading. The convexity of the problems (20), (21) ensures uniqueness of the structural response. The only drawback of the model (20), (21) is its holonomic nature: a possible local unloading is not taken into account.

Assuming H = 0, we obtain structure made of elastic-perfectly plastic material. The relevant dual QP-problems

min {1 sTE_1 s + k^ – wTp01 C w – N X – E-1s = 0} (22)

s, w, X>0 2

max {1 s E 1 s | NT s < k0, CTs = p0} (23)

s 2

still retain the uniqueness of solutions but the nice property of the existence of solution for any p0 is lost. The yield surface NTs = k0 is now fixed and too high loading would cause the absence of statically admissible field of stresses.

If we would like to neglect elastic strains as well, the static energy principle (23) would loose its cost function (since there would be E-1 = 0). This shows clearly that for a structure made of the

rigid-perfectly plastic material the problem “find the response to the given load” is ill-posed. The right formulation is “find the load factor that corresponds to the state of plastic collapse”. A complete set of relations for this formulation is given in Table 4.

Table 4. Governing relations for ultimate load factor.

X

w

s

в

1

VL,=

0

0

– NT

0

k0

> 0

VLW =

0

0

CT

– p0

0

= 0

VLs =

– N

C

0

0

0

= 0

и

[>

0

H = Q.

1

0

0

1

= 0

x> 0, a, T vlx = 0

The primal problem generated by this table is

min { kT X | – N X + Cw = 0, p^w = 1} (24)

X, !Л

and the dual one has been already given as Eq. (5). All sub-matrices located at the diagonal of the Table 4 have zero values. This leads to vanishing quadratic terms in the cost functions of the dual problems. On the other hand, we can not expect the solution to be unique in terms of stresses and/or collapse mechanisms, since linear functions are not strictly convex (concave).