# Correction to: A performance measure approach for risk optimization

• André Jacomel Torii
• Rafael Holdorf Lopez
• André Téofilo Beck
Correction

Correction to: Structural and Multidisciplinary Optimization

https://doi.org/10.1007/s00158-019-02243-5

The original article unfortunately contains error in one of the equations and the results of the last example. This is also to emphasize that these corrections do not affect the conclusions of the paper.

Equation (46) in Section 5.4, concerning the cantilever beam, should be corrected to

$$C = \frac{1}{3} h b^{2} G.$$
(46)
The numerical results are correct, since this was just a typing error.
In Section 5.5, concerning the reinforced concrete arch, the computational routines for evaluation of the volume of the structure were incorrect. Thus, the correct optimal designs are slightly different, even though the conclusions remain the same. The correct deterministic design is
$$\mathbf{d} = \{66.99, 0.28, 0.66, 1.35, 1.44, 0.94, 1.63, 3.21\}.$$
The correct risk optimization design is
$$\mathbf{d} = \{81.91, 0.35, 0.72, 1.35, 1.44, 0.92, 1.61, 3.21\}.$$
The critical buckling load factor and the critical resistance factor are λcr = 1.7954 and rcr = μ/μr = 0.5673, respectively. Equation (59) should be corrected to
$$\mathbf{d}^{(0)} = \kappa \times \{81.91, 0.35, 0.72, 1.35, 1.44, 0.92, 1.61, 3.21\}.$$
(59)
Figures 8910 and 11 should be corrected to Figs. 811 presented below. Tables 1516 and 17 should be corrected to Tables 1517 presented below.
Table 15

Example 5: comparison of the results

Approaches

Initial solution

c

NFE*

β

PMA-SORA

Parabolic

14.3483

2,780 (52,820)

{3.0121, 3.4879}

PMA-SORA

Deterministic

14.3483

2,759 (52,421)

{3.0156, 3.4597}

RO-IP

Parabolic

Fails to converge at the first iteration

RO-IP

Deterministic

Fails to converge at the first iteration

RO-HLRF

Parabolic

Fails to converge at the first iteration

RO-HLRF

Deterministic

Fails to converge at the first iteration

*NFE in parenthesis consider gradient evaluation with forward finite differences

Table 16

Example 5: results obtained with PMA-SORA

Iteration

c

d

β

1

19.9035

{76.33, 0.31, 0.69, 1.35, 1.44, 0.93, 1.62, 3.21}

{2.00, 2.00}

2

15.9118

{78.81, 0.31, 0.68, 1.35, 1.43, 0.93, 1.62, 3.21}

{2.50, 2.26}

3

15.3487

{78.59, 0.34, 0.71, 1.35, 1.44, 0.92, 1.61, 3.21}

{2.41, 2.77}

4

14.5862

{80.08, 0.34, 0.71, 1.35, 1.44, 0.92, 1.61, 3.21}

{2.70, 2.91}

5

14.3882

{80.98, 0.34, 0.71, 1.35, 1.44, 0.92, 1.61, 3.21}

{2.87, 3.07}

6

14.3306

{81.62, 0.34, 0.71, 1.35, 1.44, 0.92, 1.61, 3.21}

{2.98, 3.21}

7

14.3208

{81.90, 0.35, 0.71, 1.35, 1.44, 0.92, 1.61, 3.21}

{3.03, 3.30}

8

14.3184

{82.00, 0.35, 0.72, 1.35, 1.44, 0.92, 1.61, 3.21}

{3.04, 3.35}

9

14.3167

{82.02, 0.35, 0.72, 1.35, 1.44, 0.92, 1.61, 3.21}

{3.04, 3.40}

10

14.3154

{81.99, 0.35, 0.72, 1.35, 1.44, 0.92, 1.61, 3.21}

{3.03, 3.45}

11

14.3151

{81.93, 0.35, 0.72, 1.35, 1.44, 0.92, 1.61, 3.21}

{3.02, 3.46}

12

14.3151

{81.91, 0.35, 0.72, 1.35, 1.44, 0.92, 1.61, 3.21}

{3.02, 3.46}

13

14.3151

{81.91, 0.35, 0.72, 1.35, 1.44, 0.92, 1.61, 3.21}

{3.02, 3.46}

Table 17

Example 5: NFE* for different values of κ

Approaches

κ = 0.9

κ = 1.1

PMA-SORA

2,621 (49,799)

2,583 (49,077)

RO-IP

failed

1,782 (36,100)

RO-HLRF

failed

867 (18,981)

*NFE in parenthesis consider gradient evaluation with forward finite differences

## Notes

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

## Authors and Affiliations

• André Jacomel Torii
• 1
Email author
• Rafael Holdorf Lopez
• 2
• André Téofilo Beck
• 3