Probabilistic assessment of structures

## Probabilistic Assessment of Structures using Monte Carlo Simulation

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## Optimization Study of Beam with Sudden Profile Change

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This example deals with application of SBRA (Simulation Based Reliability Assessment) method for safety assessment of a welded steel beam with a sudden profile variation. Close attention is paid especially to the determination of the position where the profile changes.

### Assignment

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The beam is simply supported. Span is L = 4 m, material grade is steel S235. The beam is exposed to two uniformly distributed loads. The design values are: dead load g = 10 kN/m, long lasting live load q = 8 kN/m. Assess I-profiles of this steel welded beam (see Fig. 2) regarding the bending moment. Determine the position where the profile is to be changed with respect to the material savings. Lateral-torsion buckling of the beam is prevented. Do not assess the serviceability limit state. The design probability of failure is Pf,lim <= 0.00007.

#### Structure Response to the Load 'S'

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The variable value of the load is equal to the product of its maximum value and the coefficient expressing its variation by the assumptive distribution, then g = 10 kN/m ∙ gvar, q = 8 kN/m ∙ qvar. Individual distributions follow in the chart (Fig.3).

The bending moment behavior along the beam Mx = ½ ∙ (g + q) ∙ (L ∙ x - x2) is traced by AntHill [2] software. The variables taking part in this simulation are g, q.

#### Structure Resistance 'R'

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The variable value of the steel yield stress is equal to the coefficient expressing its variation by the assumptive distribution, then f = fvar. The variable values of the profile moduli are equal to the products of their nominal values and the coefficients expressing their variation by the assumptive distributions, then W1 = W1,nom ∙ Wvar, W2 = W2,nom ∙ Wvar. Individual distributions follow in the chart (Fig.5).

The nominal values of the cross-sections properties are W1,nom = 1.64E-4 m3, W2,nom = 1.25E-4 m3. The profiles 1 and 2 are displayed on Fig.2. The structure resistance is defined as the product Mr = W ∙ f, than Mr1 = W1 ∙ f, Mr2 = W2 ∙ f.

### Assessment

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Critical cross-sections, considering the bending moment, are for Profile 1 the center of the beam (x = 2 m), for Profile 2 the place of its sudden profile variation (x = ?).

### Profile 1, x = 2 m

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In the center of the beam's span is the failure probability Pf = 0.000053 < Pf,lim = 0.00007. Profile 1 satisfies the requirements, calculated by AntHill [1] software.

### Profile 2, x = ? m

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With assistance of AntHill [1] software, the probability of failures is found in individual positions near the estimated location of the sudden profile change.

The profile should be changed in position x = 1 m from the support. The failure probability in this location Pf = 0.000027 (see Fig.9) < Pf,lim = 0.00007. Profile 2 satisfies the requirements. Calculated by AntHill [1] software.

### Global Analysis of Profile 1 and 2

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### Additional Study

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### Conclusions

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This study outlines the opportunity of using SBRA method for the global reliability assessment. The used example is very simple; however, it is possible to assess more complicated statically determined structures.

### References

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[1]Optimization Study of Beam with Sudden Profile Change; author Jan Hlavacek, Prague 2001

[2]Probabilistic Assessment of Structures using Monte Carlo Simulation – 1st edition; editors: P. Marek, J. Brozzetti, M. Gustar; publisher: ITAM CAS CR, Prague 2001