Evaluation of Robustness of Structure
Date 30th August 2018
Abstract: Robust structures ensure to prevent the progressive collapse of the structure when only parts of the structure are destroyed caused by extreme events or human errors. To establish the better understanding of the aspects of robust structure, robustness evaluation becomes crucial for the existing or upcoming structures. To achieve this aims the effectiveness of the different countermeasures of the robustness indicators on the three spans continuous truss bridge are investigated based on the structural type consideration of the single span truss bridge. The totally damage state of the internally indeterminacy as well as the externally indeterminacy of the structural system are imposed. The different scenarios of the damage member combination are identified based on the possibilities of the critical condition of the system collapse according to the sizes, location and arrangement of the structural members. The dimensionless robustness indicators are adopted from the recent researchers and the indicators are determined with respect to the structural response associated with the damage system and that of the intact system. The 2D linear elastic static analysis of the eight panel single span for two structural system types as well as the eight panel three span continuous truss of the selected structural system type are done by using the OpenSees software for evaluation of the effectiveness of the different countermeasures of the robustness indexes.
- INTRODUCTION
The failures caused by not only the exceptional loadings such as winds, earthquakes, impacts or explosions but also the gravity loadings may lead the progressive collapse that caused the disastrous impact to the safety of the structures. Sufficiently robust structures can reduce the consequences significantly. Robustness is defined as many definitions as various researchers attitudes. Robustness is a desirable property of structural systems which mitigate their susceptibility to progressive or disproportionate collapse. It is defined as the insensitivity of a structure to local failure (Starossek 2006, 2009). Robustness is defined as the ability of a structure to withstand events like fire, explosions, impact or the consequences of human error, without being damaged to an extent disproportionate to the original cause (EN 1991-1-7, 2006). Many significant theoretical and technological advances are published over the recent years. Mainly qualitative and hardly quantitative recommendations are provided for robust structure demands. To examine a structure in terms of robustness, a quantitative description by means of a measure would be useful. The measure could be used for evaluation, optimization and regulation of robustness. Various approaches for the quantification of robustness or related characteristics have been published (Uwe Starossek and Marco Haberland, 2011).
Strength and ductility, as well as other performance indicators of the ultimate conditions under nonlinear behavior, may result of great significance in robustness evaluations associated with damage induced by severe loadings, like explosions or impacts (Frangopol and Curley 1987, Biondini et al. 2008, F. Biondini & S. Restelli, 2008). However, performance indicators of the serviceability conditions under linear behavior, like elastic stiffness and first yielding, may become of major importance in lifecycle robustness evaluations associated with aging of structures (Restelli 2007, F. Biondini & S. Restelli, 2008). In this study, the simple formulations of stiffness- and displacement-based measures of robustness indicators are adopted and selected to illustrate the structural robustness. Robustness is defined in (Starossek 2006, 2009) as the insensitivity of a structure to local failure. The local failure might originate from the damage of the internally indeterminacy as well as from the damage of the externally indeterminacy of the structure. As to this basics, the failures of the structure is considered from the damage of externally indeterminacy such as the failure of the external bearings and the damage of the internally indeterminacy such as the failure of the interior structural members. In addition, the local failure may exist a part or parts of the structure which lead to the progressive or disproportionate collapse. Regarding to this, the possible different damage scenarios of the single member or the combinations of the members are examined. The deterioration effect of the three span continuous truss bridge is inspected by using the different countermeasures of the robustness indicators related to the structural system properties and loading condition of the system. The analysis of the system is assessed using 2D linear elastic static analysis in OpenSees software. To conduct the robustness of the three span continuous truss bridge, the eight panel single span truss is analyzed first considering the two types of the structural systems, the frame system analysis in which, the joint connections of the members are assumed as the fixed, the bending and axial effect of the members of the system are taken into account and the truss system in which, the joints connections of the members are assumed as the pinned, only the axial effect of the members are measured. Then, the more effective system type to the robustness indexes is selected for conducting the analysis of the three spans continuous truss bridge to quantify the robustness of structure.
- ROBUSTNESS INDICATORS
The structural performance indicators under linear elastic behaviors are adopted from F. Biodini and S. Restelli, 2008. These indicators are first investigated by S. Restelli, 2007. The performance indicators of the serviceability conditions under linear behavior, like elastic stiffness and first yielding, may become of major importance in lifecycle robustness evaluations associated with aging of structures (S. Restelli, 2007). The performance indicators are used as the state indicators to identify the structural system in the original state in which the fully intact and the deteriorated state in which the damage condition is imposed.
2.1 Parameters for the structural behavior
The following structural performance indicators are used for the analysis.
(1)
(2)
where c is the conditioning number of the stiffness matrix K and T is the first vibration period associated with the mass matrix M and λi(K) denotes the ith eigenvalue of the matrix K and s is the displacement vector, f is the applied load vector, and ‖ . ‖ denotes the euclidean scalar norm (F. Biodini and S. Restelli, 2008). These two indicators associated with the conditioning of the stiffness matrix and the vibration period is related with the properties of the structural system only. The conditioning number and the period of the structure are obtained according to eigen value analysis of the structure that is important for identifying the performance of the structural system. The displacement performance indicator is related with both the system properties and the loading conditions. The behavior of the structure may differ depends on the different structural system and the different loading condition.
2.2 Structural Robustness Indices and its Effectiveness
The robustness indices are measured on the basic of the structural performance of the system of its original state, in which the structure is fully intact, and the deteriorated state, in which the damage condition is applied. The robustness of the system is measured directly from the robustness indices. The robustness value ‘0’ should identify the totally damage of the system by assuming at failure and ‘1’ should identify the full robustness of the system in which the system could reserve the damage. The robustness indices related with the system properties and the loading condition are expressed as follows used in this study.
(3)
(4)
where the scripts ‘0’ refers to the original intact state and ‘1’ refers to the damage state of the system. The indices ρc and ρT are related to the properties of the structural system only and the index ρs is related to both the properties of the structural system and the loading condition.
2.2.1 Evaluation on Eight Panel Single Span Truss
To discuss the effectiveness of these indices, the eight panels single span truss bridge shown in Figure 1 is evaluated under the damage of the single member and the different damage scenerios of the structural system first. The members of the single span truss are defined as O1 to O8 are top chords members, U1 to U8 are bottom chord members, V0 to V8 are vertical members, D1 to D8 and D1 to D8 are the different diagonal members for clarity of the member combination. The intensity of the damage member is considered as overall deteriorated of the members which are the internally inderterminacy. The damage scenerios are identified as the groups that may cause the more severe to the system collapse and the possible member combinations needed to be considered in the design.
Figure 1. The eight panel single span truss with member explanation
The 2D linear elastic static analysis of the eight panel single span truss is carried out imposing the total damage of the member and member combination by using the OpenSees software. On the one hand, the truss bridge is considered as the frame system taking into account the bending effect of the members due to the joint rigidity in addition to the effect of the axial force of the system. On the other hand, the analysis is carried out by considering the truss bridge as the truss system in which the connections at the joints are assumed as the pin connections regardless of the bending stresses of the members. The weights of the members are applied and distributed at each node of the structure. The results of the different robustness indices of the structure on the different damage scenarios considering the frame system and truss system of the eight panel single span truss are shown in Figure 2 to 4. Based on the results from the eight panel single span truss, the effective structural system type on the robustness indicators is identified and the robustness evaluation of three span continuous truss bridge of the selected system is conducted to check the effectiveness of the different countermeasures of the robustness indexes.
(a) (frame system) (b) (truss system)
Figure 2. Robustness Vs damage member combination for conditioning of the stiffness matrix of structure
(a) (frame system) (b) (truss system)
Figure 3. Robustness Vs damage member combination for period of structure
(a) (frame system) (b) (truss system)
Figure 4. Robustness Vs damage member combination for displacement of structure
The results from the analysis of the truss system using the truss element are different from the analysis of the frame system using the elastic beam column element of the same structure. According to the results using the elastic beam column element when considering the secondary stress effect for analysis, the indicators on the conditioning of the stiffness matrix of the structure and displacement are more effective than that of the period of the structure. The effectiveness of the robustness are different based on the different indicators. According to these three robustness indices, the central bottom and top chord member are the critical members that have the great influence on the safety of the structure and system collapse. The damage of the combination of one diagonal member in each panel is not so substantial to the collapse of the entire truss system and its robustness is about 60%. The combination of a set of central top and bottom chords directs the totally failure of the structure and the most severe of the system collapse. The scenario on damage of all vertical members has no influence on the system failure and in high robustness value. The scenario on damage of all top chords and the scenario on damage of all bottom chords are critical to lead the collapse of the system.
The results using the truss element analysis, the robustness indices of conditioning of stiffness matrix and the period of the structure have more tendency than that of the displacement of the structure on the robustness of the system. According to the structural system properties, the robustness index of the conditioning of the stiffness of the structure, it is found that damage of all top chord members or collapse of all bottom chord members leads to the collapse of the system. Either damage of one member top or bottom chord for center span corresponds more than 80% of the system collapse. In the contrast, for the damage of one member top or bottom chord of span adjacent to the support, the system can reserve the strength without leading to the collapse of the system. The damage of all vertical members excluding two vertical members on the supports can survive the system with the high robustness index of close to 1. The damage of the combination of one diagonal member in all panel leads to 60% of the collapse of the system. The most critical members are the top and bottom chord of the central span of the system that the most influence on the behavior of the of the system collapse.
Based on the analysis of two different methods, truss system regardless of the secondary stress and frame system considering the secondary effect, the robustness indices of the fame system are smaller than the truss system due to the effect of the secondary stress. On the other hand, these two systems give the same critical members on the system although the tendencies are different according to the different robustness indices.
As a conclusion, from the comparison of the two analysis expresses the effect of the secondary stress is influence on the safety, strength and robustness of the structure. The sensitivity of the system collapse is highly dependent on the different combination of damaged members, member size and structural configuration of the system. The robustness indicators based on the system properties, the period of the structure, have the less influence than that of the conditioning of the stiffness matrix of the structure. The damage of all interior vertical members is not significant influence on the collapse of the entire system according to the different countermeasures of the robustness indices related with the structural system properties and loading conditions.
2.2.2 Evaluation on Three Spans Continuous Truss Bridge
The assessment of three spans continuous truss bridge is examined based on the information obtained from the structural system type of the single span truss bridge to establish better understanding of the aspects related to the effectiveness of the different countermeasures robustness indexes. The different scenarios of three continuous truss are identified relating with the failure of the internally indeterminacy as well as the failure of the externally indeterminacy of the system that are the conditions which are assumed to occur under construction and during the lifetime of a structure. The 2D linear elastic gravity analysis using OpenSees software is carried out for the assessment of the effectiveness of the different robustness indexes that are related with the structural system properties and loading conditions to monitor the significance of specific damage scenarios. The three spans continuous truss system is shown in Figure 5 and the members and bearings are defined as O1 to O24 are top chords members, U1 to U24 are bottom chord members, V0 to V24 are vertical members, D1 to D24 and D1 to D24 for eight panels in each span and E1 to E4 for support bearings. The tendencies of the different countermeasures on the different scenarios are shown in Figure 6 to 9.
Figure 5. Eight Panel Three Span Continuous Truss Bridge
Figure 6. Robustness Vs damage member combination for conditioning of the stiffness matrix of structure
Figure 7. Robustness Vs damage member combination for period of structure
Figure 8. Robustness Vs damage member combination for displacement of structure
Figure 9. Robustness Vs damage member combination for displacement of structure
The most critical members of three span continuous truss bridge are investigated using the linear analysis of the frame system. The robustness indices on the conditioning of the stiffness matrix and the period of the structure provide the consistent result. The index on the displacement of the structure distributes the different results depending on the location of the node of the collected response of the structure.
According to the results on the internally indeterminacy, the damage of all vertical members are not key factor on the collapse of the entire system. Also the damage of all single diagonal member of every panel does not deal the significant effect on the system collapse. The damage of all diagonals of the whole truss or the damage of all diagonals in each span contribute the total collapse of the structure according to the robustness index of the conditioning, nearly collapse based on the period index and all scenarios contribute the different solutions for the displacement index. The damage of a set of the diagonals of the center of the first or the last span provides the intermediate robustness and a pair of the diagonals of the first span close to the second support supplies the lower robustness value.
The central bottom and top chord of each span provide the intermediate robustness of the system and the top and bottom chord adjacent to the supports are in high robustness. On the other hand, the simultaneously damage of pair of the top and bottom chords of three spans lead to the total collapse of the system. The damage of all top chords and the damage of all bottom chords approach to the system failure according to the robustness index on the conditioning of the stiffness matrix.
The scenarios on the damage of the externally indeterminacy are the most critical for the system collapse and provide the lowest robustness indices of the system. Among these externally indeterminacy scenarios, the removal of the second hinge support (E2) is the most severe case as it carries the support for the horizontal and vertical direction and due to the effect of the interior support of the continuous truss system. The damage of the exterior roller supports (E1) and (E4) have almost the same tendency on the collapse of the structure. The damage of the interior roller support (E3) is the higher robustness than that of the exterior support (E1) or (E4).
In summary, the different tendencies are obtained based on the different robustness indices. On the other hand, all the indices gives the same information that the damage of all top chords, the damage of all bottom chords, the simultaneously damage of a pair of central top and bottom chords of all spans, the damage of all diagonals lead to the collapse of the whole system. The effect of the robustness indices depends on the location of the members, the size of the members and the system configuration. The damage of the externally indeterminacy are the most effective and severe cases compared to the damage of the other internally indeterminacy. Even the damage of one or two externally indeterminacy can cause the collapse of the system. The effect of the damage of one externally indeterminacy are close to the effect of the damage of all top chords or the damage of all bottom chords or the damage of all diagonals or the damage of the a set of central upper and lower chords in three spans of the whole truss. The failures of externally indeterminacy are the most effective of the system.
- SUMMARY AND CONCLUSIONS
The failure of some parts of the structural members may cause the successive collapse of the structural system and subsequently lead to the total system collapse. The damage of the parts that leads to system collapse are the critical members. As a consequence, the evaluation of the robustness of the system is crucial to find out those critical members whose damage will highlight the system collapse and safety. Such members should receive additional quality assurance, inspection, and maintenance compared to members (or group of members) with lesser importance (Dan M, Frangopol, M. ASCE, and James P. Curley, 1987). The failure of single member has the vast impact to become the progressive collapse which forces the totally damage of the structure as well as the damage of the combination of the members have more chance to cause excessive collapse finally to the totally collapse of the system. Regarding to this, the different countermeasures of the robustness indexes, in which the structural system properties and the loading condition are taking into consideration, are adopted for checking of the important members of system and the effectiveness of these different robustness indexes are evaluated.
To discuss the effectiveness of the different countermeasures of the robustness indexes, the different scenarios of the damage member combinations are identified relating with the possibility of the critical condition to the system collapse according to the member sizes, location and system configuration and the possibility of the damage that seem need to know whether it may damage or not. The damage condition of the system is defined as the total failure of the internally indeterminacy of the system and combined the failure of these internally indeterminacies for evaluation process.
Two structural systems of the eight panel single span truss bridge are used to evaluate the effectiveness of the different countermeasures of the robustness indexes assuming as the frame system in which the joint connections are assumed as fixed, both the bending action of the members and the member axial forces are included, and the truss system in which the joint connections are assumed as pined, no bending is considered and only the member axial forces are included. The weights of the structural members are assigned at each node of the system. The linear elastic gravity analysis of the eight panel single span truss is done by using OpenSees software to check the performance of two structural systems. The results obtained from the analysis of two systems are compared to recognize the different performance of two systems on the different countermeasures of the robustness indexes.
The robustness indexes from frame analysis are smaller than those obtained from truss analysis due to the bending effect of the system and, but the critical members and member combinations are same for both system. For frame system and truss system, the central top and bottom chord members are the most critical and the influence of the damage of that members are severe to the collapse of the system according to the robustness indexes of conditioning of the stiffness matrix and period of the structure. The combination of these two members directs the totally failure of the structure and the most severe of the system collapse. The scenario on damage of all vertical members has no influence on the system failure and high robustness value. The scenario on damage of all top chords and the scenario on damage of all bottom chords are critical to lead the collapse of the system. The different tendencies of the robustness indexes are obtained based on the different countermeasures of the robustness indexes for two different structural systems. The frame system is more influence on the system collapse than the truss system.
By considering the frame structure, the most critical members of three spans continuous truss bridge are investigated using the linear analysis of the frame system. The totally damage of the members and the external supports are imposed in different member combination to check the safety of the system using the different robustness indices on conditioning of the stiffness matrix, period and displacement of the structure. The externally indeterminacy as well as the internally indeterminacy are taken into account to select the different scenarios for the damage of the structure. The different scenarios are considered based on that the damage of the member combination may cause the possibilities of the critical condition to the system collapse and depend on the member location necessary to know whether it may effects or not according to their member sizes and location and configuration of the system.
The damage of the externally indeterminacy are the most effective and severe cases compared to the damage of the other internally indeterminacy. Even the damage of one or two externally indeterminacy can cause the collapse of the system. The effect of the damage of one externally indeterminacy are close to the effect of the damage of all top chords or the damage of all bottom chords or the damage of all diagonals or the damage of a set of central upper and lower chords in three spans of the whole truss. The externally indeterminacy are the most effective of the system.
By conducting this analysis, the three different robustness indexes can be regarded as a very effective measure in the structural robustness evaluation. It can be identified the important members or member combinations as those whose failures (or severe damage) has a great influence on the system collapse which should receive additional quality assurance, inspection, and maintenance.
References
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