# Robustness Assessment of Redundant Structural Systems Based on Design Provisions and Probabilistic Damage Analyses

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Robustness Assessment Concepts and Design Provisions

#### 2.2. Design Concept in Current Developments of the Fib Model Code 2020

_{tot}from accidental/exceptional events is calculated with the following equation [11], accounting for risk associated to local (direct) damage and systemic (follow-up) damage, which can also form the basis of a respective risk management exercise:

- P[H
_{i}], the probability of occurrence of exposure to hazard H_{i}, - P[D
_{j}|H_{i}], the probability of (direct) damage D_{j}conditional on exposure to hazard H_{i}, - P[S
_{k}|D_{j}∩ H_{i}], the probability of systemic damage S_{k}conditional on the damage D_{j}and the exposure to hazard H_{i}, - C
_{dir}, the direct consequences and - C
_{ind}, the indirect consequences

_{i}], the direct damage probabilities P[D

_{j}|H

_{i}], the systemic damage probabilities P[S

_{k}|D

_{j}∩ H

_{i}], and the direct C

_{dir}and indirect C

_{ind}consequences.

#### 2.2.1. Design Scenarios

#### 2.2.2. Design Strategies

_{ind}and/or P[S|D ∩ H]) resulting from the local damages D. Measures associated with the consequence reduction strategy can include the structural segmentation or compartmentalization, and changing the context of the structure.

#### 2.3. Quantitative Robustness Assessment and Redundancy Performance Indicators

- Risk-based quantification based on a complete risk analysis in which the consequences are divided into direct and indirect consequences
- Reliability based quantification derived from e.g., the reliability of the structural system in the damaged and in the undamaged situation;
- Deterministic quantification based on structural parameters such as the load-bearing capacity, stiffness, damage energy, etc.

_{d}(A = 0, ΔE

_{d,col}) ≤ R

_{d}

_{d}, refer to situations after an accidental event. In this case, an explicit accidental action not accounted for, and the accidental action may be assumed to apply on the initially affected components and leave the rest of the structure unaffected (hence A = 0). However, dynamic effects due to sudden element collapse must be taken into account when redistributing (e.g., by means of catenary action) internal forces and moments to the remaining part of the structure, ΔE

_{d,col}.

## 3. Implemented Methodology—Redundancy Concepts

_{damaged}) and det(K

_{intact}) are the determinants of the stiffness matrices of the damaged and the intact structure respectively, and r

_{ii}the ratio between them. In this approach, damage results in a reduction of structural capacity. In this case, the change in the structural property cumulative “system stiffness” is measured. The ratio of the determinants of the stiffness matrices in the damaged and intact state can be conceived as the redundancy part of the failed component. The index takes values between zero and one, and it increases for higher degrees of robustness, i.e., if the system stiffness remains at its full level after a failure, the failed component has no influence on the system behavior.

_{f}

_{(intact)}and the damaged p

_{f}

_{(damaged)}structure [33], as proposed in Equation (6). Values between zero and infinity result, whereby lower values result in a higher degree of robustness. By definition, robustness is characterized by the fact that initial and local damage does not significantly affect the system structural reliability. To determine this index, a probabilistic analysis can be performed, with the system’s input parameters such as material strength, cross-section and loads translated to stochastic variables, in order to account for the influence of variabilities and uncertainties on the damage scenaria. Specifically the loads can be random sets for different design situations. The failure probabilities can be alternatively derived through expert judgement elicitations, e.g., in the form of an expert panel Delphi process or a risk management workshop, of course with the respective compromise in the accuracy of quantifiable results.

_{dir}and the total risk R = R

_{dir}+ R

_{ind}. A higher degree of robustness is represented by an increasing index. Should the index have a value of one, would imply that the support structure in question is absolutely robust, and no damage propagation takes place. In this case, no indirect risk Rind is generated from local damage (e.g., failure of a strut in a truss system). If, however, the indirect risk Rind increases disproportionately in relation to the initial damage, the index tends towards zero. A risk is formed from the product of the probability of damage and the resulting consequences, such as economic losses. Direct risk and the risks are calculated through Equations (8) and (9), in accordance with the formulation of Equation (1) (see also [26]).

_{dir}.

_{ind}or the support system is able to withstand the damage progress F. In this case, the damage is limited to the direct consequences that are directly related to the exposure.

_{i}) is estimated and combined with their redundancy components (r

_{ii}). The reliability indices (β

_{i}) participate though their function Φ, which is the standard normal probability function, and the expression Φ(−β

_{i}) can be assumed to express the element’s failure probability. This approach implicitly accounts for the influence of the degree of indeterminacy on the assessment of structural robustness, i.e., by assigning a contribution to each component according to their redundancy. The definition of static indeterminacy is given in Equation (11), where a is the number of external reactions, p the number of solid components, and k the number of nodes (it is noted that this definition is valid for “classic” trussed structures only, i.e., trusses with hinges at every nodes, without continuous chords). The sum of the individual redundancy contributions sums up to the total system indeterminacy f. Based on this index, the influences from the correlation between the elements (non-diagonal entries of a redundancy matrix e.g., in a truss system) remain dissociated, although, for a sensible evaluation of the robustness, these influences should be regarded. On the example of a truss system, it is clear that the structural system consists of a certain number of interrelated linear components, whereas the non-diagonal entries of the redundancy matrix reflect the influence of a localized failure on the adjacent components:

## 4. Case Studies of Computations of Robustness Performance Indicators

_{Var.1}= 8. In Variant 2, one node is omitted and additional diagonal elements are arranged in the upper field area. This change increases the degree of external static indeterminacy by one degree (f

_{Var.2}= 9). In both variants, all bars are assumed to consist of a uniform cross-section QRO 80 × 6.3 and material S355 (EA = 176,400 kN, I

_{y}= I

_{z}= 62.3 cm

^{4}). The elements are assumed without flexural stiffness, and stability for compression members is taken into account according to EN 1993-1-1 [36,37]. Some further case studies for deterministic and probabilistic case studies in the context of the investigations presented herein can be found in [38].

_{stiffness,Var1}= 0.028 < ρ

_{stiffness,Var2}= 0.076

^{−2}. Based on this assumption and assuming independent failure probabilities for each member, the probabilistic redundancy indices for Variants 1 and 2 are calculated below. Final system failure occurs as soon as the instability limit is reached, based on the static indeterminacy calculation (i.e., at loss of one element after reaching static determinacy and transformation of the system to a mechanism). The possible failure paths for the intact and damaged systems are plotted below in Figure 7 for variant 1, and Figure 8 for variant 2.

_{f(intact)}= 2 × (10

^{−2})

^{2}+ 6 × (10

^{−2})

^{6}+ 12 × (10

^{−2})

^{7}+ 6 × (10

^{−2})

^{8}= 2.00 × 10

^{−4}

_{f(damage)}= 2 × (10

^{−2})

^{1}+ 4 × (10

^{−2})

^{5}+ 12 × (10

^{−2})

^{6}+ 7 × (10

^{−2})

^{7}= 2.00 × 10

^{−2}

_{f(intact)}= 6 × (10

^{−2})

^{7}+ 8 × (10

^{−2})

^{8}+ 11 × (10

^{−2})

^{9}= 6.08 × 10

^{−14}

_{f(damage)}= 6 × (10

^{−2})

^{6}+ 7 × (10

^{−2})

^{8}+ 11 × (10

^{−2})

^{9}= 6.00 × 10

^{−12}

_{ii}of each individual truss member (the ratio between the determinants of stiffness matrices and a respective reliability index are needed, as seen from Equations (21)–(26)). These values are plotted in Table 2 and Table 3 for each truss member.

_{i})) ≈ 1 and this term can be neglected in the calculation per Equations (23) and (26). As it can be seen from the computations, the robustness advantage of variant 2 is reflected by the respective robustness indicator:

_{redundancy,Var1}= 8/18 = 0.44

_{redundancy,Var2}= 9/16 = 0.56

## 5. Probabilistic Elaborations

_{EA,Var1}= 1.764 × 10

^{5}kN, with a range of possible coefficients of variations (5%, 10%, and 15%—see also Figure 9), which constitutes the stochastic model’s random variables. The standard deviations (σEA) in particular are:

- σ
_{EA,Var1(5%)}= 0.05 × 176,400 = 8820 kN (Coefficient of variation = 5%) - σ
_{EA,Var1(10%)}= 0.10 × 176,400 = 17,640 kN (Coefficient of variation = 10%) - σ
_{EA,Var1(15%)}= 0.20 × 176,400 = 35,280 kN (Coefficient of variation = 20%)

**Figure 9.**Probabilistic distributions of the strain stiffness EA with the variability 5% (

**a**), 10% (

**b**) and 20% (

**c**). The red line represents the best-fit theoretical distribution function.

## 6. Discussion and Conclusions

- Robustness is a multifaceted discipline. Although basic principles can apply globally, the redundancy aspect of robustness is de facto crucial for multi-component systems. Without excluding a different possibility, this study has managed to treat truss systems with tools and methodologies that are customized for only these certain types of structures. The definition of individual members and the assessment with the stiffness- and the redundancy-based robustness indices presented in Equations (3) and (7), can become more intricate in case of monolithic or composite structures, as it requires separate treatment of the connection details.
- Quantification of the robustness can be translated to performance indicators, which can be computed and measured for individual structures. These can then be used in a life-cycle engineering approach by relating these performance indicators to performance criteria for progressive collapse. Furthermore, it is feasible to link these performance indicators with actual risks, with which structure stakeholders are typically concerned, as opposed to prescriptive structural characteristics. These performance indicators can in turn be associated with the condition assessment at section or member level. In the case of the probabilistic robustness index, the failure probability of a single member can be directly included in the calculations, as demonstrated in Equations (11) and (12). In the case of the stiffness-based index, a recalculation of the entire system with the altered or degraded parameters in one or more members needs to be carried out, as shown in Equation (10). For the robustness index by combined reliability and redundancy of the elements, both a member failure probability and a structural reanalysis of the system is integrated (see Equations (13) and (14), and Table 2).
- All the discussed robustness performance indicators are plausible. However, they are not necessarily equally beneficial to robustness assessment. As the degree of information involved with the assessment decreases, the indicators become generic, and they may fail to reveal the order of differences in the robustness of compared solutions or systems. The most useful index out of the ones evaluated is the robustness index by combined reliability and redundancy of the elements, since it accounts for a combined contribution of individual components on redundancy, and the individual components’ safety and reliability (see Equations (13) and (14), and Table 2). This indicates that this index is a more comprehensive representation of the system’s performance and it allows for a sensible measure to the system’s global safety and reliability assessment. Furthermore, it is seen that the probabilistic robustness indices in the examined truss variants have a very small difference in numerical value (99.0 and 97.68 for Variant 1 and 2 respectively), while it is clear from the calculations that the actual probability of failure of Variant 1 is several orders of magnitude higher to that of Variant 1. Based on this example, it is evident that the probabilistic robustness index does not always facilitate a decision process.
- Variability and randomness as regards the component strength and actions on the structure can have a substantial influence on the robustness assessment. Reasonably high variabilities have been shown to dramatically affect, not only the possible dispersion in certain the performance measures (e.g., deformations), but on the prevailing failure modes of the structure. In particular, a change in the truss components axial stiffness variability from 5% to 20%, led from contained and stable performance, to a high probability of progressive collapse, under the same loading conditions. As the results in Figure 10 show, actually multiple modes (peaks) appear in the output distributions for structures with a higher input variation, which implies not only high uncertainties in the load bearing capacity and reliability of the progressively damaged system, but uncertainty on the type of critical failure type. This starts to become evident at the systems with 10% and 20% variations after loss of 2 and 3 members respectively.
- A probabilistic elaboration of the structure’s collapse characteristics has proven to be very insightful, not only due to the appreciate treatment of uncertainty and randomness mentioned above, but because they also enable an inverse identification of the critical failure paths (see Figure 7 and Figure 8) and the characterization of weak links in a multicomponent system. On the example of Variant 1 analyzed in Section 4, prioritization of maintenance would be assigned primarily to elements 25 and 26, and perhaps secondarily to element 5 and 19 (although improvement of elements 25 and 26 may alter the originally anticipated failure paths). On the example of Variant 2, key elements are the ones identified as 5 and 18 in Figure 8. In turn, this can facilitate a rational design for robustness, as well as an efficient strategy for the design of, as for example the elements mentioned above, ‘key elements” as defined in Section 2.2.2. This further allows for a robustness-based strategy for structural maintenance, including health monitoring optimization and targeted strengthening. Alternatively, elements of high significance for the survival of the structure under an unexpected event, can be specifically designed as redundant or adaptive components in innovative structural concepts.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Increase of redundancy on the example of a truss system (

**a**), by the addition of members and increase of its static indeterminacy (

**b**).

**Figure 2.**Direct and indirect consequences, adapted from [26].

**Figure 6.**Deformation patterns of the investigated systems at sequences of component losses of truss system variants 1 and 2 (FEM results).

**Figure 7.**Failure paths of variant 1; intact system (

**a**); damaged system after loss of component 25 (

**b**).

**Figure 8.**Failure paths of variant 2; intact system (

**a**); damaged system after loss of component 25 (

**b**).

**Figure 10.**Distribution of system deformations of Variant 1 in the intact and different damage states, under the assumptions of variable axial stiffness (5%, 10% and 20% scatter of strain stiffness, for loss of 0 to 4 components); X-axes: vertical deformations ranging (mm); Y-axes: frequencies (-).

**Table 1.**System deformations (w) [mm] and utilization ratios (η) [–] during successive failure of the truss system components; element i.d. numbers according to Figure 4.

Number of Failing Components | Variant 1 | Variant 2 | ||||
---|---|---|---|---|---|---|

w | η | Element | w | η | Element | |

0 | 4.8 | 0.55 | 25 | 8.2 | 0.72 | 3 |

1 | 5.9 | 0.51 | 26 | 10.2 | 0.69 | 22 |

2 | 6.8 | 0.50 | 2 | 12.1 | 0.64 | 2 |

3 | 9.8 | 0.68 | 4 | 18 | 0.93 | 24 |

4 | 77.3 | 2.65 | 23 | 21.8 | 0.81 | 17 |

5 | 125.6 | 2.54 | 3 | 35.6 | 1.23 | 14 |

6 | 430.1 | 6.02 | 24 | 48.4 | 1.14 | 1 |

7 | − | − | − | 136.3 | 1.96 | 20 |

**Table 2.**Redundancy contributions (r

_{ii}) and reliability indexes (β

_{i}) for components of variant 1.

i | r_{ii} | β_{i} | i | r_{ii} | β_{i} |
---|---|---|---|---|---|

1; 21 | 0.541 | 4.75 | 8; 15 | 0.249 | 4.75 |

2; 22 | 0.528 | 4.75 | 9; 16 | 0.362 | 4.75 |

3; 23 | 0.623 | 4.75 | 10; 17 | 0.249 | 4.75 |

4; 24 | 0.311 | 4.75 | 11; 18 | 0.223 | 4.75 |

5; 19 | 0.038 | 4.75 | 12; 13 | 0.050 | 4.75 |

6; 20 | 0.028 | 4.75 | 25; 26 | 0.430 | 4.75 |

7; 14 | 0.368 | 4.75 | - | - | - |

**Table 3.**Redundancy contributions (r

_{ii}) and reliability indexes (β

_{i}) for components of variant 2.

i | r_{ii} | β_{i} | i | r_{ii} | β_{i} |
---|---|---|---|---|---|

1; 21 | 0.582 | 4.75 | 7; 20 | 0.088 | 4.75 |

2; 23 | 0.507 | 4.75 | 8; 14 | 0.435 | 4.75 |

3; 22 | 0.423 | 4.75 | 9; 15 | 0.381 | 4.75 |

4; 24 | 0.587 | 4.75 | 10; 16 | 0.400 | 4.75 |

5; 25 | 0.246 | 4.75 | 11; 17 | 0.490 | 4.75 |

6; 20 | 0.079 | 4.75 | 12; 18 | 0.246 | 4.75 |

13 | 0.076 | 4.75 | - | - | - |

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Spyridis, P.; Strauss, A.
Robustness Assessment of Redundant Structural Systems Based on Design Provisions and Probabilistic Damage Analyses. *Buildings* **2020**, *10*, 213.
https://doi.org/10.3390/buildings10120213

**AMA Style**

Spyridis P, Strauss A.
Robustness Assessment of Redundant Structural Systems Based on Design Provisions and Probabilistic Damage Analyses. *Buildings*. 2020; 10(12):213.
https://doi.org/10.3390/buildings10120213

**Chicago/Turabian Style**

Spyridis, Panagiotis, and Alfred Strauss.
2020. "Robustness Assessment of Redundant Structural Systems Based on Design Provisions and Probabilistic Damage Analyses" *Buildings* 10, no. 12: 213.
https://doi.org/10.3390/buildings10120213