Kinematic Theorem Of Plastic Collapse Research Articles

Probabilistic design procedure for steel moment resisting frames equipped with FREEDAM connections

In this work, the Theory of Plastic Mechanism Control (TPMC) is combined with a probabilistic method to account for the influence of random material variability. Reference is made to steel Moment Resisting Frames (MRFs) equipped with FREEDAM connections. FREEDAM connections are beam-to-column connections equipped with friction dampers to dissipate the seismic input energy. TPMC is used to guarantee that in case of destructive seismic events the structural members such as beams and columns remain undamaged. To this scope, the structure is designed to assure a collapse mechanism characterized by the activation of all the friction dampers of the beam ends and the formation of plastic hinges at the base of the first storey columns only. From the probabilistic point of view, the random uncertainties are given by the static friction coefficient of the contact surfaces and the preloading of the bolts of the friction dampers as well as the yielding resistance of the steel members. The failure domain is related to all the possible failure events, where the term “failure” concerns the development of an undesired mechanism different from the global one. Generally, the design conditions to prevent undesired collapse mechanisms are stochastic events within the framework of the kinematic theorem of plastic collapse. The limit state function corresponding to each event can be represented by a hyperplane in the space of random variables. Consequently, the failure domain is a surface resulting from the intersection of the hyperplanes corresponding to the limit states of each single failure event. Since dissipative zones (member ends or friction dampers) in the frame members are common to many different mechanisms, the single limit state functions are correlated. Therefore, the probability of failure can be evaluated by means of the Bimodal or Ditlevsen bounds by assuming that the failure events are located in series. The output of the work is a simple relationship which provides the overstrength factor of FREEDAM connections to be considered in the column design phase to account for random material variability thus assuring a given level of reliability in the application of TPMC.

Theory of plastic mechanism control: A new approach for the optimization of seismic resistant steel frames

AbstractThis paper presents an important improvement in seismic design of structures based on the application of the Theory of Plastic Mechanism Control (TPMC). TPMC was originally proposed in the ‘90s requiring an iterative algorithm and upgraded in 2015 leading to the so‐called closed form solution. In the recent years, TPMC has shown its value as an effective design procedure which can be applied to any structural typology. Moreover, it has been used as a code competitor design tool in important RFCS founded research projects confirming its effectiveness, simplicity, and suitability to any structural typology. The aim of the TPMC is the design of structures able to exhibit at collapse a global type mechanism. The TPMC was originally based on the extension of the kinematic theorem of plastic collapse to the concept of mechanism equilibrium curve, working in the plane (being α the multiplier of the seismic forces and δ the top sway displacement). The novelty of this work is improvement of the design procedure by working in the plane, being θ the plastic rotation. This new TPMC approach, namely TPMC(θ), has demonstrated to be as reliable as the former approach, thus leading to structures able to develop a collapse mechanism of global type but with the optimization of the structure using lighter columns. The effectiveness of the new procedure has been confirmed by both pushover and dynamic non‐linear analyses.

Consideration of second-order effects on plastic design of steel moment resisting frames

This work mainly aims to propose a new design procedure combining the benefits of the Performance-Based Plastic Design approach (PBPD) with a rigorous accounting of second-order effects. In fact, by exploiting the kinematic theorem of plastic collapse, second-order effects can be accounted for employing the concept of collapse mechanism equilibrium curve. The same tool constitutes the base of the Theory of Plastic Mechanism Control (TPMC) design approach. Besides, the paper reports a critical comparison between TPMC and PBPD, both having the scope to design structures exhibiting a collapse mechanism of global type. These two approaches are also compared with the refined PBPD where second-order effects are accounted for by the kinematic approach. Many steel moment resisting frames are designed according to PBPD, TPMC and refined PBPD and their performances have been compared on the bases of push-over analyses.

Rigid plastic analysis for the seismic performance evaluation of steel storage racks

The aim of the paper is the prediction of the seismic collapse mode of steel storage pallet racks under seismic loads. The attention paid by the researchers on the behaviour of the industrial steel storage pallets racks is increased over the years thanks to their high dead-to-live load ratio. In fact, these structures, generally made by cold-formed thin-walled profiles, present very low structural costs but can support large and expensive loads. The paper presents a prediction of the seismic collapse modes of multi-storey racks. The analysis of the possible collapse modes has been made by an approach based on the kinematic theorem of plastic collapse extended to the second order effects by means of the concept of collapse mechanism equilibrium curve. In this way, the dissipative behaviour of racks is determined with a simpler method than the pushover analysis. Parametric analyses have been performed on 24 racks, differing for the geometric layout and cross-section of the components, designed in according to the EN16618 and EN15512 requirements. The obtained results have highlighted that, in all the considered cases, the global collapse mechanism, that is the safest one, never develops, leading to a dangerous situation that must be avoided to preserve the structure during a seismic event. Although the studied racks follow all the codes prescriptions, the development of a dissipative collapse mechanism is not achieved. In addition, also the variability of load distribution has been considered, reflecting the different pallet positions assumed during the in-service life of the racks, to point out its influence on the collapse mechanism. The information carried out from the paper can be very useful for designers and manufacturers because it allows to better understand the racks behaviour in seismic load condition.

Probabilistic Theory of Plastic Mechanism Control for Steel Moment Resisting Frames

This work aims at the development of an advanced method for the seismic design of Moment Resisting Frames (MRFs) based on a target value of the failure probability in the attainment of a collapse mechanism of global type for stochastic frames (considering the aleatoric uncertainty of the material properties). Therefore, the method herein presented constitutes the probabilistic version of the Theory of Plastic Mechanism Control (TPMC) already developed for frames with deterministic material properties. With reference to MRFs whose members have random values of the yield strength, when structural collapse is of concern, the failure domain is related to all the possible collapse mechanisms. Within the probabilistic TPMC, the term “failure” does not mean the attainment of a structural collapse, but the development of a collapse mechanism different from the global one.The design requirements normally needed to prevent undesired collapse mechanisms are probabilistic events within the framework of the kinematic theorem of plastic collapse. The limit state function corresponding to each event is represented by a hyperplane in the space of random variables, so that the failure domain is a surface resulting from the intersection of the hyperplanes corresponding to the limit states representing the single failure events. Since plastic hinges in frame’s members are common to many different mechanisms, the single limit state events are correlated. Therefore, by applying the theory of binary systems and considering that the limit states are events located in series, the probability of failure can be computed by means of Ditlevsen bounds. This approach has been validated by means of Monte Carlo simulations.In order to achieve a predefined level of reliability in the attainment of the design goal, the reliability analysis is repeated for increasing values of the overstrength factor of the dissipative zones to be used in TPMC, aiming to its calibration. Finally, on the basis of the results of a parametric analysis, a simple relationship to compute the value of the overstrength factor needed to include the influence of random material variability in the application of TPMC is proposed.

11.10: Probabilistic theory of plastic mechanism control

ABSTRACTThis work aims at the development of an advanced method for the seismic design of Moment Resisting Frames (MRFs) based on the evaluation of the probability of failure in the attainment of a collapse mechanism of global type in case of stochastic frames. Therefore, the method presented constitutes the probabilistic version of the Theory of Plastic Mechanism Control (TPMC) already successfully developed for frames with deterministic material properties. With reference to MRFs whose members have random values of the yield strength, the failure domain derives from all the possible collapse mechanisms. Under the point of view of structural reliability analysis, the term “failure” means the attainment of a collapse mechanism different from the global one. The design requirements normally needed to prevent undesired collapse mechanisms are probabilistic events within the framework of the kinematic theorem of plastic collapse. At each event corresponds a limit state function representing a hyperplane in the space of the random variables, so that the failure domain is a manifold surface resulting from the intersection of the hyperplanes corresponding to the limit states of the single events. Because of plastic hinges in frame's members are common to many different mechanisms, the single limit state events are correlated. Therefore, by applying the theory of binary systems and taking into account that, from the structural reliability point of view, the limit states are events located in series, the probability of failure is computed by means of Ditlevsen bounds.

Influence of the bracing scheme on seismic performances of MRF-EBF dual systems

The work herein presented has the aim to investigate the influence of the bracing scheme on the seismic performances of Moment Resisting Frames-Eccentrically Braced Frames (MRF-EBF) dual systems, designed according to two design approaches: the first one is the Theory of Plastic Mechanism Control (TPMC) while the second one is based on Eurocode 8 (EC8) design provisions. Even though TPMC design approach is not introduced in modern seismic codes, it has earned the reputation of being a robust design approach because of its strong theoretical background, based on the kinematic theorem of plastic collapse extended to the collapse mechanism equilibrium curve to assure a collapse mechanism of global type. Conversely, the design approach based on EC8 promotes the application of the so-called beam-column hierarchy criterion which is usually able to avoid soft-storey mechanisms but does not assure the yielding of all the dissipative zones, because a collapse mechanism of global type is very rarely attained. The main purpose of the present work is to compare, given the above mentioned design approaches, the different seismic performances resulting from the use of the four different bracing scheme proposed by codes. For this reason, 5 bay buildings with 4, 6 and 8 storeys have been examined considering four bracing schemes providing a total number of 12 structural schemes analyzed. The seismic performances have been evaluated by means of Incremental Dynamic Analyses (IDA) carried out until the achievement of structural collapse and repeated for a set of ten earthquake ground motion.

Plastic design of seismic resistant reinforced concrete frame

A new method for designing moment resisting concrete frames failing in a global mode is presented in this paper. Starting from the analysis of the typical collapse mechanisms of frames subjected to horizontal forces, the method is based on the application of the kinematic theorem of plastic collapse. The beam section properties are assumed to be known quantities, because they are designed to resist vertical loads. As a consequence, the unknowns of the design problem are the column sections. They are determined by means of design conditions expressing that the kinematically admissible multiplier of the horizontal forces corresponding to the global mechanism has to be the smallest among all kinematically admissible multipliers. In addition, the proposed design method includes the influence of second-order effects. In particular, second-order effects can play an important role in the seismic design and can be accounted for by means of the mechanism equilibrium curves of the analysed collapse mechanism. The practical application of the proposed methodology is herein presented with reference to the design of a multi-storey frame whose pattern of yielding is validated by means of push-over analysis.

On the plastic analysis of concentrically braced frames with shear panel, obtaining predetermined collapse mechanism

SummaryConventional design methods do not ensure that the desired collapse mechanism is developed at target displacement. In this paper, a case study is presented to analyze concentrically braced frames with steel shear panel (CBFSP). Also, extensive investigation in the failure modes are made, to have the global yielding mode at the final state. For this purpose, each of one‐story, three‐story, six‐story and nine‐story CBFSP models were decomposed into three parts where the members' closed‐form equations of internal forces were identified and superimposed. On the basis of the kinematic theorem of plastic collapse, the possible mechanisms and the related energy equations were defined to estimate the lateral load multiplier. First, the shear panels, columns, vertical and horizontal boundary elements were designed using the values of internal forces and seismic loads. Next, sections of the beams and braces were selected by constraining, where the mechanism equilibrium curve of the desired mechanism had to be placed below the others within the admissible roof displacement. Finally, for assessment of the precision of the method, results of the pushover analysis of the finite element models were compared with the theoretical ones. The findings show that, despite more effort for design, the investigated method is reliable and satisfactory. Copyright © 2014 John Wiley & Sons, Ltd.

Theory of plastic mechanism control for MRF–CBF dual systems and its validation

In this paper the theory of plastic mechanism control is presented for moment resisting frame–concentrically braced frames dual systems, i.e. for structural systems combined by moment resisting frames and concentrically braced frames. It is aimed at the design of structures failing in global mode, i.e. whose collapse mechanism is characterised by the yielding of all the tensile diagonals and the occurrence of buckling in the compressed ones, and by plastic hinge formation at all the beam ends and at the base of first storey columns. The proposed methodology is based on the application of the kinematic theorem of plastic collapse, by imposing that the global mechanism equilibrium curve has to lie below all the other equilibrium curves corresponding to undesired mechanisms. The practical application of the design methodology is illustrated by means of a worked example. In addition, the results of a non-linear static pushover and dynamic analyses of the designed structure are also discussed in order to demonstrate the effectiveness of the proposed design procedure.

Theory of plastic mechanism control for eccentrically braced frames with inverted y-scheme

The theory of plastic mechanism control is presented with reference to eccentrically braced frames with vertical links, i.e. EB-Frames with inverted Y-scheme. Such theory has been originally developed with reference to moment resisting steel frames and gradually extended to other seismic resistant structural typologies aiming to the assure, in all cases, a collapse mechanism of global type. This result is of primary importance in earthquake resistant design, because partial and local failure modes are responsible for the worsening of the energy dissipation capacity leading to an increased risk of collapse under destructive seismic events.With reference to EB-Frames with inverted Y-scheme, the proposed method is based on the assumption that vertical link elements are preliminarily designed according to the internal actions due to the design seismic forces. Beam and diagonal sections are also assumed to be known quantities, because they need to be designed to fulfil a local hierarchy criteria assuring, at storey level, that yielding occurs in the link element only. Conversely, column sections constitute the unknowns of the design problem.The theory of plastic mechanism control includes also the influence of second order effects by means of the concept of mechanism equilibrium curve. In fact, the design requirements are derived by means of the kinematic theorem of plastic collapse extended to the concept of mechanism equilibrium curve. Column sections are obtained by imposing that the mechanism equilibrium curve corresponding to the global mechanism has to be located below those corresponding to all the undesired mechanisms within a displacement range compatible with the local ductility supply.Aiming at the evaluation of the accuracy of the presented design method, the inelastic response of EB-Frames with inverted Y-scheme designed according to the proposed method are investigated by means of push-over analyses to check the collapse mechanism actually developed.

Theory of plastic mechanism control for the seismic design of braced frames equipped with friction dampers

An innovative approach for the design of a seismic resistant system composed by the combination of a MR-Frame and a bracing system equipped with friction dampers is presented. From a multi-scale point of view, at local scale, supplementary energy dissipation is provided by means of friction dampers, while, at global scale, the development of a global type mechanism is assured involving all the friction dampers equipping the structure. The activation of all the friction dampers requires an advanced design procedure. Toward this end, the theory of plastic mechanism control, which is based on the application of the kinematic theorem of plastic collapse is extended to the concept of mechanism equilibrium curve, is applied. The fulfillment of the design goal has been pointed out by means of both pushover and dynamic non linear analyses whose results are herein presented and discussed.

Computational Studies and Molecular Dynamics of the Potent Biochemical and Molecular Markers in Relevance to Oral Cancer

Seismic codes do not provide specific hierarchy criteria for irregular Moment Resisting Frames (MRFs). In fact, the provisions for irregular frames are exactly the same adopted for regular MRFs with the only exception of a 20% reduction of the design value of the q-factor, accounting for a presumed worsening of the frame energy dissipation capac- ity. It is well known that seismic code provisions, based on hierarchy criteria, are often unsuitable to prevent partial mecha- nisms and to assure the development of a global collapse mechanism. In particular, irregular structures are prone to de- velop partial or soft storey mechanisms in case of significant set-backs. As already showed in previous work (1-10) a ra- tional design procedure based on plastic mechanism theory can be adopted for different design typologies leading to ex- cellent results in the field of mechanism control. This procedure is herein briefly presented and applied to different study cases. It is based on the application of the kinematic theorem of plastic collapse extended to the concept of mechanism equilibrium curve in order to consider second order effects. Sending the reader back to the original work on the theory of plastic mechanism control (1) for an exhaustive presentation of the theory, referred to regular MR-Frames, the work herein presented focuses on the issues to be faced to apply such theory to the particular case of steel frames with set- backs. In order to verify the results obtained by the application of the proposed design procedure, non linear static analy- ses (11) have been carried out for all the structures considered and a comparison with the results coming from the applica- tion of Eurocode 8 design procedures (12) is also performed.

Failure Mode Control and Seismic Response of Dissipative Truss Moment Frames

AbstractIn this paper, a new design approach for dissipative truss moment frames (DTMFs) able to guarantee, under seismic forces, the development of a yield mechanism of global type is presented and applied. In particular, DTMFs consist of a truss moment frame where the energy dissipation is provided by means of special devices located at the ends of the truss girders at the bottom chord level. The proposed design methodology is based on the kinematic theorem of plastic collapse. The method is based on the assumption that sections of truss elements and the yield threshold of dissipative parts (i.e., special devices) are known, and therefore the column sections are the unknowns of the design problem, which are obtained by imposing that the equilibrium curve corresponding to the global mechanism has to lie below those corresponding to all the possible mechanisms within a displacement range compatible with local ductility supply of dissipative elements. The design methodology has been implemented in a comput...

Theory of plastic mechanism control of dissipative truss moment frames

An innovative structural typology using energy dissipating devices, such as friction devices or hysteretic devices, is proposed to design truss moment frames exhibiting high energy dissipation capacity. The configuration of the proposed structural system consists in a special Truss Moment Frame where the energy dissipation is provided by means of special devices located at the ends of truss girders at the bottom chord level and connecting the bottom chord of the truss girder to the column. For this innovative structural typology, namely Dissipative Truss Moment Frames (DTMFs), an advanced design approach is developed and proposed in this work. In particular, the proposed design approach is able to guarantee, under seismic forces, the development of a collapse mechanism of global type assuring that all the dissipative devices are involved and plastic hinges develop only at the base of first storey columns. This result is of primary importance in earthquake resistant design, because partial and local failure modes are responsible of the worsening of the energy dissipation capacity leading to an increased risk of collapse under destructive ground motions.The member sections of the truss girders and the threshold axial resistance of the special dissipative devices are assumed to be known quantities, because they are designed to withstand vertical loads, whereas the unknowns of the design problem are the column sections. According to the kinematic theorem of plastic collapse, they are obtained by imposing that the equilibrium curve corresponding to the global mechanism has to lie below those corresponding to all the undesired mechanisms within a displacement range compatible with the local ductility supply of dissipative zones and devices. In addition, the presented approach includes also the influence of second order effects and accounts for the concentrated forces acting on the truss girders, resulting from the actions transmitted by secondary beams.In addition, the design algorithm for the application of the proposed design procedure is discussed and presented in detail by means of a worked example. Finally, aiming at a preliminary validation of the proposed design methodology, the results of non-linear static analyses (push-over) are also reported.

Failure Mode and Drift Control of MRF-CBF Dual Systems

In this paper, a new method for designing moment resisting frame (MRF) - concentrically braced frame (CBF) dual systems failing in global mode is presented. Starting from the analyses of the typical collapse mechanisms of such structural typology subjected to seismic horizontal forces, the method is based on the application of the kinematic theorem of plastic collapse. Beam and diagonal sections are assumed to be known quantities, because they are designed to resist vertical loads and horizontal forces, respectively. Therefore, the column sections are the only unknowns of the design problem. Column sections are obtained by imposing that the equilibrium curve corresponding to the global mechanism has to lie below all the equilibrium curves corresponding to the undesired collapse mechanisms within a displacement range compatible with the local ductility supply of dissipative elements. Such procedure has been applied to design several MRF-CBF dual systems and has been validated by means of non-linear static analyses aimed to check the actual pattern of yielding.

Optimal plastic design of pitched roof frames for multiple loading

The paper describes the use of algebraic linear programming for the minimum weight design of steel portal frames subject to the constraints of the Kinematic Theorem of plastic collapse. Minimum weight design is a classic linear programming problem which can be solved algebraically for classes of frames with arbitrary geometric dimensions and arbitrary load magnitudes. In a recent paper, the process of algebraic linear programming was reduced to the repeated application of a number of vector formulas and a computer program was developed for the derivation of the solution charts for specific classes of frames. In this paper the method is extended to the problem of frames subjected to multiple load cases. It is shown that simple problems whose solution can normally be displayed in the form of two-dimensional charts now require three-dimensional charts or a number of two-dimensional charts.

PLASTIC DESIGN OF SEISMIC RESISTANT STEEL FRAMES

A new method for designing moment resisting steel frames failing in a global mode is presented in this paper. Starting from the analysis of the typical collapse mechanisms of frames subjected to horizontal forces, the method is based on the application of the kinematic theorem of plastic collapse. The beam section properties are assumed to be known quantities, because they are designed to resist vertical loads. As a consequence, the unknowns of the design problem are the column sections. They are determined by means of design conditions expressing that the kinematically admissible multiplier of the horizontal forces corresponding to the global mechanism has to be the smallest among all kinematically admissible multipliers. In addition, the proposed design method includes both the influence of distributed loads acting on the beams and the influence of second-order effects. In particular, the influence of second-order effects, which can play an important role in the seismic design of steel frames, is accounted for by the mechanism equilibrium curves of the analysed collapse mechanisms. Moreover, in order to show the practical application of the proposed design procedure, a worked example is presented. Finally, the inelastic behaviour of the designed frame is compared to that obtained when the simple member hierarchy criterion or a similar rule is applied. © 1997 by John Wiley & Sons, Ltd.

ALGEBRAIC LINEAR PROGRAMMING APPLIED TO OPTIMAL PLASTIC DESIGN OF STEEL PORTAL FRAMES

The paper is concerned with the minimum weight design of steel portal frames subject to the constraints of the kinematic theorem of plastic collapse. Minimum weight design is a classic linear programming problem which can be solved algebraically for classes of frames with arbitrary geometric dimensions and arbitrary load magnitudes. The solution for each class of frames is expressed in the form of a chart. This technique has been inhibited by the substantial amount of tedious algebraic manipulation necessary for each specified class of frames. In the paper, the process of algebraic linear programming is reduced to the repeated application of a number of vector formulas. A computer program is described which, using these formulas, derives the solution chart for specified classes of frames. Examples illustrate the use of the program to derive a design chart and the use of a chart to solve a specific minimum weight design problem.