Optimisation of structures based on reliability.

Tierean, Mircea ; Baltes, Liana ; Eftimie, Lucian 等

1. INTRODUCTION

In the case of deterministic optimisation, the input values of the problem (admitted stress and strains, geometrical dimensions, forces, strain, and boarding conditions) are considered univocally as known. The no statistic approaches to those values impose the choice of a safety factor, which covers every possible breaking case. The choosing of this safety factor has a great influence regarding the cost and structure mass, sometimes bigger than the results of the mathematics program. In the case of probabilities optimisation, these problem variables are considered as random values, here existing the advantage of considering some accidental strains, which can predictably load the structure, but they have a small appearance probability.

The purpose of the optimisation based on reliability is to find the geometrical dimensions of structure elements, by imposing the failure probability.

2. HYPOTHESIS

1. The probability concept is used not just with a relative frequency, but as a measure for a certain hypothesis trust grade, which has been used in computing without being confirmed by experiments.

2. The structures loading and bearing capacity are random values, with normal distribution.

3. The uncertainty that affects the safety is considered in calculus by scattering parameters of the random variables.

4. The structure's geometry and loading positions are considered as deterministic.

5. There are neglected the shearing forces and torsion moments affect regarding truss and frame structures.

6. The loading action is independent.

7. The bars' cross sections are constants, related to their longitudinal axes; they are completely characterised by a single parameter.

8. The loading is statically applied.

9. The temperature and the structure's environment do not influence the resistance.

10. The destruction of every element brings about the destruction of the structure (Tierean et al., 2006).

3. THE PROBLEM'S GENERAL WORDING

There is considered a structure consisting of "n" bars, traction-compression strained for the first case (truss), and bending-traction-compression strained for the second case (frame). The structure is loaded by "l" forces. There are considered as initial data: the geometrical dimensions, material and loading conditions. The shape of the bars' cross section is specified, so it is considered that only one dimension (g) defines univocally all other geometrical characteristics that may appear into calculus. The optimisation problem can be phrased this way: determine the geometrical characteristics (g), so that the elementary failure probability [F.sub.i](g) is equal with the admitted value (Thoft-Christensen et al., 1986):

[F.sub.i]([bar.g])= P[[L.sub.i]([bar.g]) [less than or equal to] 0] = [F.sun.ai], (i = 1,2, ..., n). (1)

For the truss structure, the safety limit that corresponds to the "m" breaking case is:

[L.sub.j] = [n.summation over (j=1])[a.sub.ij][T.sub.j] - [l.summation over (j=1)][b.sub.ij][S.sub.j], (i=1, 2,..., m), (2)

where:

1 = the number of loading cases;

[T.sub.j] = the admissible loading of the "j" element;

[S.sub.j] = the loading in the "j" case;

[a.sub.ij] = the "j" element resistance factor;

[b.sub.ij] = the "j" element loading factor, for the "i" breaking type.

In the case of frame structure, the "i" element safety limit is:

[L.sub.i] = [X.sup.2.sub.1i][A.sub.j][W.sub.j] - ([M.sub.j][A.sub.j] + [N.sub.j][W.sub.j])[X.sub.1i], (3)

where:

[X.sub.1i] = the admissible strength of the "i" bar material;

[M.sub.j] = the bending moment, for the critical section "j";

[N.sub.j] = the axial force, for the critical section "j";

m = the number of critical sections;

[W.sub.j] =W([g.sub.j]) = the strength modulus of the "j" section;

[A.sub.j] = A([g.sub.j]) = the critical section "j" area.

For a Gaussian random design variable's distribution, the elementary reliability is:

[F.sub.i] = P[[L.sub.i] [less than or equal to] 0] = [phi](0-[[bar.L].sub.i]/[[sigma].sub.Li]) = [phi](- [[bar.L].sub.i]/[[sigma].sub.Li]) [less than or equal to] [F.sub.ai], (4)

where: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

[L.sub.i] = the safety limit mean value;

[[sigma].sub.Li] = the safety limit mean square error.

Let [F.sub.ai] = [phi](-[f.sub.i]) [??] [f.sub.i] = -[[phi].sup.-1] ([F.sub.ai]) be the optimisation problem restriction; it results:

[phi] (-[[bar.L].sub.i]/[[sigma].sub.Li]) = [sigma](-[f.sub.i]) [??] [f.sub.i] = [[bar.L].sub.i]/[[sigma].sub.Li] (5)

or to avoid the square root: [f.sup.2.sub.i] = [([[bar.L].sub.i]/ [[sigma].sub.Li]).sup.2] (6)

In each iteration (q) the Eq. (6) is solved and the obtained vector is the optimum solution for that iteration. The algorithm has the following steps:

Step 1. There are established the elementary failure probabilities ([F.sub.ai]), the elementary diameters' initial values [g.sub.i.sup.(0)] =([g.sub.1.sup.(0)], [g.sub.2.sup.(0)]), ..., [g.sub.n.sup.(0)]), q=0 and the admitted error limit e.

Step 2. There are calculated the forces (and moments) corresponding to the [g.sup.(q)] vector.

Step 3. There are calculated the elementary failure probabilities.

Step 4. If the relation

[absolute value of [F.sub.i]([[bar.g].sup.(q)] - [F.sub.ai]/[F.sub.ai]] [less than or equal to] [epsilon], (i = 1,2, ..., n), (7)

is accomplished, it results that the optimum solution is [g.sup.(q)]. For the opposite case it will be made the jump to step 5.

Step 5. There is modified the "g" vector, which will have the components [g.sub.i.sup.(q+1)] resulted by solving the equation (4).

Step 6. For compression strained bars it will be made the buckling check. If the buckling exists, the optimum solution is [g.sup.(q)]. For the opposite case it will be made a jump at step 2, with q=q+1.

4. CALCULUS EXAMPLE

The presented method was applied to a 10 bar structure (Fig. 1). It is considered that the bars are made of steel with [X.sub.1] = 273 MPa and [[sigma].sub.X1] = 19.492 MPa. For the calculus, there were used the loading variation factor Vq = 0.2; the admitted failure probability [F.sub.i] = [10.sup.-6] and the admitted range [epsilon] = [10.sup.-3] [2].

The loadings are [Q.sub.1]=5x[10.sup.5]N, [Q.sub.2]=[10.sup.5]N and the structure's dimensions are b=6000 mm, c=3000 mm.

The calculus program starts with the structure drawing section, valid for a 5xn bars structure generating process. In the input data block there are introduced the bar's number and the horizontal dimensions "b", respectively the vertical ones "c". In the first stage there are generated the left diagonals and the vertical beams, by the successive traversing of all junctions by the position vector "r". In the second stage there are generated the right diagonals and the inferior sole, by returning to the initial junction and in the contrary sense traversing the previously generated vertical beams. In the last stage it results the structure's final draw, by joining the second before the last junction with the second one (Tofan et al., 1995).

In the next step, by using the finite element method, there are computed the internal forces and moments. After the initial structure check, the bars are dimensioned again, by using the Eq. (6). The compressed bars must be checked at buckling and will be verified the Eq. (7) accomplishment. Finally it is graphically represented the bar's section variation into the current iteration, the total area variation factor and the structure's strain. With a view for reaching the optimum solution there are 5 iterations necessary, finally obtaining a [mu] =77.6% total area decrease for truss and 4 iterations with [mu] =57.2% for frame.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

Fig. 2 presents the bar's area variation for the 5th iteration (truss) and fig. 3 presents the bar's area variation for the 4th iteration (frame). In the case of the truss structure, in the last iteration, the bar's reliability spectrum is [[10.sup.-6]; 5.577x[10.sup.-6]]. In the case of the frame structure, in the last iteration, the bar's reliability spectrum is [9.976x[10.sup.-7]; 3.539x[10.sup.-6]].

In the case of the frame structure, for the buckling length determination, the junctions are considered as elastic-fixed junctions.

5. CONCLUSIONS

Comparing the results obtained at optimisation of the 10 bars structures it can be concluded:

* the method ensures a fast convergence to the optimum value;

* in the case of the truss structure's optimisation, a number of bars do not change their section, other bars decrease it, but there are some bars which after decrease in the first iteration, present in the following ones some increases;

* in the last step of the optimisation process, there can be very easily found the elementary reliability lowest value; this way it is diagnosed which element will break first;

* for frame structure's optimisation, almost all bars change their section during the iterations; the optimisation time is short;

* optimising the same structure, initially considered with fixed junctions, and then with joint junctions, it is established that, in the case of the frame structure, the optimisation process is performed in a less number of iterations; it is obtained a shorter area decrease than in the case of the truss structure;

* the optimisation programs are done in such a way, that for every step there can be closely watched the work parameters; there is easy service here.

* From these conclusions, it can be deduced that the optimisation on the basis of reliability is an efficient designing instrument; this way it can be ensured a low value for the bar's areas, but only respecting the reliability requirements.

6. REFERENCES

Tierean, M.H.; Baltes, L.S.; Mirza Rosca, J. & Santana Lopez, A. (2006). Reliability, a useful way for the strength structures' optimisation, Third international conference "Mechanics and Machine Elements", 2-4 Nov. 2006, Sofia, Bulgaria, ISBN 10: 954-438-587-8, pag. 97-103

Tofan, M.C.; Goia, I.; Tierean, M.H. & Ulea, M. (1995). Deformatele structurilor (Structure's strains), Editura Lux Libris, ISBN 973-96854-2-0, Brasov, Romania

Thoft-Christensen, P.; Murotsu, Y. (1986). Application of structural systems reliability theory, Springer-Verlag, Berlin

Mathsoft, Inc. (2001)MathCAD User's Guide, Cambridge, MA http://www.ptc.com/go/mathsoft/mathcad/