Local buckling behaviour of steel plate elements supported by a plastic foam material

Sandwich panels comprising steel facings and a polystyrene foam core are increasingly used as roof and wall claddings in buildings in Australia. When they are subjected to loads causing bending and/or axial compression, the steel plate elements of their profiled facing are susceptible to local buckling. However, when compared to panels with no foam core, they demonstrate significantly improved local buckling behaviour because they are supported by foam. In order to quantify such improvements and to validate the use of available design buckling stress formulae, an investigation using finite element analyses and laboratory experiments was carried out on steel plates that are commonly used in Australia of varying yield stress and thickness supported by a polystyrene foam core. This paper presents the details of this investigation, the buckling results and their comparison with available design buckling formulae.

Local buckling studies of cold-formed steel compression members at elevated temperatures

Cold-formed steel members have been widely used in residential and commercial buildings as primary load bearing structural elements. They are often made of thin steel sheets and hence they are more susceptible to local buckling. The buckling behaviour of cold-formed steel compression members under fire conditions is not fully investigated yet and hence there is a lack of knowledge on the fire performance of cold-formed steel compression members. Current cold-formed steel design standards do not provide adequate design guidelines for the fire design of cold-formed steel compression members. Therefore a research project based on extensive experimental and numerical studies was undertaken to investigate the local buckling behaviour of light gauge cold-formed steel compression members under simulated fire conditions. First a series of 91 local buckling tests was conducted at ambient and uniform elevated temperatures up to 700oC on cold-formed lipped and unlipped channels. Suitable finite element models were then developed to simulate the behaviour of tested columns and were validated using test results. All the ultimate load capacity results for local buckling were compared with the predictions from the available design rules based on AS/NZS 4600, BS 5950 Part 5, Eurocode 3 Parts 1.2 and 1.3 and the direct strength method (DSM), based on which suitable recommendations have been made for the fire design of cold-formed steel compression members subject to local buckling at uniform elevated temperatures.

Longitudinal bar buckling behavior

Reinforced concrete columns might fail because of buckling of the longitudinal reinforcing bar when exposed to earthquake motions. Depending on the hoop stiffness and the length-over-diameter ratio, the instability can be local (in between two subsequent hoops) or global (the buckling length comprises several hoop spacings). To get insight into the topic, an extensive literary research of 19 existing models has been carried out including different approaches and assumptions which yield different results. Finite element fiberanalysis was carried out to study the local buckling behavior with varying length-over-diameter and initial imperfection-over-diameter ratios. The comparison of the analytical results with some experimental results shows good agreement before the post buckling behavior undergoes large deformation. Furthermore, different global buckling analysis cases were run considering the influence of different parameters; for certain hoop stiffnesses and length-over-diameter ratios local buckling was encountered. A parametric study yields an adimensional critical stress in function of a stiffness ratio characterized by the reinforcement configuration. Colonne in cemento armato possono collassare per via dell’instabilità dell’armatura longitudinale se sottoposte all’azione di un sisma. In funzione della rigidezza dei ferri trasversali e del rapporto lunghezza d’inflessione-diametro, l’instabilità può essere locale (fra due staffe adiacenti) o globale (la lunghezza d’instabilità comprende alcune staffe). Per introdurre alla materia, è proposta un’esauriente ricerca bibliografica di 19 modelli esistenti che include approcci e ipotesi differenti che portano a risultati distinti. Tramite un’analisi a fibre e elementi finiti si è studiata l’instabilità locale con vari rapporti lunghezza d’inflessione-diametro e imperfezione iniziale-diametro. Il confronto dei risultati analitici con quelli sperimentali mostra una buona coincidenza fino al raggiungimento di grandi spostamenti. Inoltre, il caso d’instabilità globale è stato simulato valutando l’influenza di vari parametri; per certe configurazioni di rigidezza delle staffe e lunghezza d’inflessione-diametro si hanno ottenuto casi di instabilità locale. Uno studio parametrico ha permesso di ottenere un carico critico adimensionale in funzione del rapporto di rigidezza dato dalle caratteristiche dell’armatura.

Distortional buckling of I-beams by finite element method

Distortional buckling, unlike the usual lateral-torsional buckling in which the cross-section remains rigid in its own plane, involves distortion of web in the cross-section. This type of buckling typically occurs in beams with slender web and stocky flanges. Most of the published studies assume the web to deform with a cubic shape function. As this assumption may limit the accuracy of the results, a fifth order polynomial is chosen here for the web displacements. The general line-type finite element model used here has two nodes and a maximum of twelve degrees of freedom per node. The model not only can predict the correct coupled mode but also is capable of handling the local buckling of the web.

Behaviour and design of cold-formed steel compression members at elevated temperatures

Cold-formed steel members have been widely used in residential, industrial and commercial buildings as primary load bearing structural elements and non-load bearing structural elements (partitions) due to their advantages such as higher strength to weight ratio over the other structural materials such as hot-rolled steel, timber and concrete. Cold-formed steel members are often made from thin steel sheets and hence they are more susceptible to various buckling modes. Generally short columns are susceptible to local or distortional buckling while long columns to flexural or flexural-torsional buckling. Fire safety design of building structures is an essential requirement as fire events can cause loss of property and lives. Therefore it is essential to understand the fire performance of light gauge cold-formed steel structures under fire conditions. The buckling behaviour of cold-formed steel compression members under fire conditions is not well investigated yet and hence there is a lack of knowledge on the fire performance of cold-formed steel compression members. Current cold-formed steel design standards do not provide adequate design guidelines for the fire design of cold-formed steel compression members. Therefore a research project based on extensive experimental and numerical studies was undertaken at the Queensland University of Technology to investigate the buckling behaviour of light gauge cold-formed steel compression members under simulated fire conditions. As the first phase of this research, a detailed review was undertaken on the mechanical properties of light gauge cold-formed steels at elevated temperatures and the most reliable predictive models for mechanical properties and stress-strain models based on detailed experimental investigations were identified. Their accuracy was verified experimentally by carrying out a series of tensile coupon tests at ambient and elevated temperatures. As the second phase of this research, local buckling behaviour was investigated based on the experimental and numerical investigations at ambient and elevated temperatures. First a series of 91 local buckling tests was carried out at ambient and elevated temperatures on lipped and unlipped channels made of G250-0.95, G550-0.95, G250-1.95 and G450-1.90 cold-formed steels. Suitable finite element models were then developed to simulate the experimental conditions. These models were converted to ideal finite element models to undertake detailed parametric study. Finally all the ultimate load capacity results for local buckling were compared with the available design methods based on AS/NZS 4600, BS 5950 Part 5, Eurocode 3 Part 1.2 and the direct strength method (DSM), and suitable recommendations were made for the fire design of cold-formed steel compression members subject to local buckling. As the third phase of this research, flexural-torsional buckling behaviour was investigated experimentally and numerically. Two series of 39 flexural-torsional buckling tests were undertaken at ambient and elevated temperatures. The first series consisted 2800 mm long columns of G550-0.95, G250-1.95 and G450-1.90 cold-formed steel lipped channel columns while the second series contained 1800 mm long lipped channel columns of the same steel thickness and strength grades. All the experimental tests were simulated using a suitable finite element model, and the same model was used in a detailed parametric study following validation. Based on the comparison of results from the experimental and parametric studies with the available design methods, suitable design recommendations were made. This thesis presents a detailed description of the experimental and numerical studies undertaken on the mechanical properties and the local and flexural-torsional bucking behaviour of cold-formed steel compression member at ambient and elevated temperatures. It also describes the currently available ambient temperature design methods and their accuracy when used for fire design with appropriately reduced mechanical properties at elevated temperatures. Available fire design methods are also included and their accuracy in predicting the ultimate load capacity at elevated temperatures was investigated. This research has shown that the current ambient temperature design methods are capable of predicting the local and flexural-torsional buckling capacities of cold-formed steel compression members at elevated temperatures with the use of reduced mechanical properties. However, the elevated temperature design method in Eurocode 3 Part 1.2 is overly conservative and hence unsuitable, particularly in the case of flexural-torsional buckling at elevated temperatures.

Structural behaviour and design of cold-formed steel beams at elevated temperatures

Cold-formed steel members are extensively used in the building construction industry, especially in residential, commercial and industrial buildings. In recent times, fire safety has become important in structural design due to increased fire damage to properties and loss of lives. However, past research into the fire performance of cold-formed steel members has been limited, and was confined to compression members. Therefore a research project was undertaken to investigate the structural behaviour of compact cold-formed steel lipped channel beams subject to inelastic local buckling and yielding, and lateral-torsional buckling effects under simulated fire conditions and associated section and member moment capacities. In the first phase of this research, an experimental study based on tensile coupon tests was undertaken to obtain the mechanical properties of elastic modulus and yield strength and the stress-strain relationship of cold-formed steels at uniform ambient and elevated temperatures up to 700oC. The mechanical properties deteriorated with increasing temperature and are likely to reduce the strength of cold-formed beams under fire conditions. Predictive equations were developed for yield strength and elastic modulus reduction factors while a modification was proposed for the stressstrain model at elevated temperatures. These results were used in the numerical modelling phases investigating the section and member moment capacities. The second phase of this research involved the development and validation of two finite element models to simulate the behaviour of compact cold-formed steel lipped channel beams subject to local buckling and yielding, and lateral-torsional buckling effects. Both models were first validated for elastic buckling. Lateral-torsional buckling tests of compact lipped channel beams were conducted at ambient temperature in order to validate the finite element model in predicting the non-linear ultimate strength behaviour. The results from this experimental study did not agree well with those from the developed experimental finite element model due to some unavoidable problems with testing. However, it highlighted the importance of magnitude and direction of initial geometric imperfection as well as the failure direction, and thus led to further enhancement of the finite element model. The finite element model for lateral-torsional buckling was then validated using the available experimental and numerical ultimate moment capacity results from past research. The third phase based on the validated finite element models included detailed parametric studies of section and member moment capacities of compact lipped channel beams at ambient temperature, and provided the basis for similar studies at elevated temperatures. The results showed the existence of inelastic reserve capacity for compact cold-formed steel beams at ambient temperature. However, full plastic capacity was not achieved by the mono-symmetric cold-formed steel beams. Suitable recommendations were made in relation to the accuracy and suitability of current design rules for section moment capacity. Comparison of member capacity results from finite element analyses with current design rules showed that they do not give accurate predictions of lateral-torsional buckling capacities at ambient temperature and hence new design rules were developed. The fourth phase of this research investigated the section and member moment capacities of compact lipped channel beams at uniform elevated temperatures based on detailed parametric studies using the validated finite element models. The results showed the existence of inelastic reserve capacity at elevated temperatures. Suitable recommendations were made in relation to the accuracy and suitability of current design rules for section moment capacity in fire design codes, ambient temperature design codes as well as those proposed by other researchers. The results showed that lateral-torsional buckling capacities are dependent on the ratio of yield strength and elasticity modulus reduction factors and the level of non-linearity in the stress-strain curves at elevated temperatures in addition to the temperature. Current design rules do not include the effects of non-linear stress-strain relationship and therefore their predictions were found to be inaccurate. Therefore a new design rule that uses a nonlinearity factor, which is defined as the ratio of the limit of proportionality to the yield stress at a given temperature, was developed for cold-formed steel beams subject to lateral-torsional buckling at elevated temperatures. This thesis presents the details and results of the experimental and numerical studies conducted in this research including a comparison of results with predictions using available design rules. It also presents the recommendations made regarding the accuracy of current design rules as well as the new developed design rules for coldformed steel beams both at ambient and elevated temperatures.

Advanced analysis of steel frame structures comprising non-compact sections

During the past decade, a significant amount of research has been conducted internationally with the aim of developing, implementing, and verifying "advanced analysis" methods suitable for non-linear analysis and design of steel frame structures. Application of these methods permits comprehensive assessment of the actual failure modes and ultimate strengths of structural systems in practical design situations, without resort to simplified elastic methods of analysis and semi-empirical specification equations. Advanced analysis has the potential to extend the creativity of structural engineers and simplify the design process, while ensuring greater economy and more uniform safety with respect to the ultimate limit state. The application of advanced analysis methods has previously been restricted to steel frames comprising only members with compact cross-sections that are not subject to the effects of local buckling. This precluded the use of advanced analysis from the design of steel frames comprising a significant proportion of the most commonly used Australian sections, which are non-compact and subject to the effects of local buckling. This thesis contains a detailed description of research conducted over the past three years in an attempt to extend the scope of advanced analysis by developing methods that include the effects of local buckling in a non-linear analysis formulation, suitable for practical design of steel frames comprising non-compact sections. Two alternative concentrated plasticity formulations are presented in this thesis: the refined plastic hinge method and the pseudo plastic zone method. Both methods implicitly account for the effects of gradual cross-sectional yielding, longitudinal spread of plasticity, initial geometric imperfections, residual stresses, and local buckling. The accuracy and precision of the methods for the analysis of steel frames comprising non-compact sections has been established by comparison with a comprehensive range of analytical benchmark frame solutions. Both the refined plastic hinge and pseudo plastic zone methods are more accurate and precise than the conventional individual member design methods based on elastic analysis and specification equations. For example, the pseudo plastic zone method predicts the ultimate strength of the analytical benchmark frames with an average conservative error of less than one percent, and has an acceptable maximum unconservati_ve error of less than five percent. The pseudo plastic zone model can allow the design capacity to be increased by up to 30 percent for simple frames, mainly due to the consideration of inelastic redistribution. The benefits may be even more significant for complex frames with significant redundancy, which provides greater scope for inelastic redistribution. The analytical benchmark frame solutions were obtained using a distributed plasticity shell finite element model. A detailed description of this model and the results of all the 120 benchmark analyses are provided. The model explicitly accounts for the effects of gradual cross-sectional yielding, longitudinal spread of plasticity, initial geometric imperfections, residual stresses, and local buckling. Its accuracy was verified by comparison with a variety of analytical solutions and the results of three large-scale experimental tests of steel frames comprising non-compact sections. A description of the experimental method and test results is also provided.

Bolted beam-column moment connections between cold-formed steel members

ABSTRACT Twelve beam-to-column connections between cold-formed steel sections consisting of three beam depths and four connection types were tested in isolation to investigate their behavior based on strength, stiffness and ductility. Resulting moment-rotation curves indicate that the tested connections are efficient moment connections where moment capacities ranged from about 65% to 100% of the connected beam capac-ity. With a moment capacity of greater than 80% of connected beam member capacity, some of the connec-tions can be regarded as full strength connections. Connections also possessed sufficient ductility with rota-tions of 20 mRad at failure although some connections were too ductile with rotations in excess of 30 mRad. Generally, most of the connections possess the strength and ductility to be considered as partial strength con-nections. The ultimate failures of almost all of the connections were due to local buckling of the compression web and flange elements of the beam closest to the connection.

Structural behaviour and design of cold-formed steel wall systems under fire conditions

In recent times, light gauge steel framed (LSF) structures, such as cold-formed steel wall systems, are increasingly used, but without a full understanding of their fire performance. Traditionally the fire resistance rating of these load-bearing LSF wall systems is based on approximate prescriptive methods developed based on limited fire tests. Very often they are limited to standard wall configurations used by the industry. Increased fire rating is provided simply by adding more plasterboards to these walls. This is not an acceptable situation as it not only inhibits innovation and structural and cost efficiencies but also casts doubt over the fire safety of these wall systems. Hence a detailed fire research study into the performance of LSF wall systems was undertaken using full scale fire tests and extensive numerical studies. A new composite wall panel developed at QUT was also considered in this study, where the insulation was used externally between the plasterboards on both sides of the steel wall frame instead of locating it in the cavity. Three full scale fire tests of LSF wall systems built using the new composite panel system were undertaken at a higher load ratio using a gas furnace designed to deliver heat in accordance with the standard time temperature curve in AS 1530.4 (SA, 2005). Fire tests included the measurements of load-deformation characteristics of LSF walls until failure as well as associated time-temperature measurements across the thickness and along the length of all the specimens. Tests of LSF walls under axial compression load have shown the improvement to their fire performance and fire resistance rating when the new composite panel was used. Hence this research recommends the use of the new composite panel system for cold-formed LSF walls. The numerical study was undertaken using a finite element program ABAQUS. The finite element analyses were conducted under both steady state and transient state conditions using the measured hot and cold flange temperature distributions from the fire tests. The elevated temperature reduction factors for mechanical properties were based on the equations proposed by Dolamune Kankanamge and Mahendran (2011). These finite element models were first validated by comparing their results with experimental test results from this study and Kolarkar (2010). The developed finite element models were able to predict the failure times within 5 minutes. The validated model was then used in a detailed numerical study into the strength of cold-formed thin-walled steel channels used in both the conventional and the new composite panel systems to increase the understanding of their behaviour under nonuniform elevated temperature conditions and to develop fire design rules. The measured time-temperature distributions obtained from the fire tests were used. Since the fire tests showed that the plasterboards provided sufficient lateral restraint until the failure of LSF wall panels, this assumption was also used in the analyses and was further validated by comparison with experimental results. Hence in this study of LSF wall studs, only the flexural buckling about the major axis and local buckling were considered. A new fire design method was proposed using AS/NZS 4600 (SA, 2005), NAS (AISI, 2007) and Eurocode 3 Part 1.3 (ECS, 2006). The importance of considering thermal bowing, magnified thermal bowing and neutral axis shift in the fire design was also investigated. A spread sheet based design tool was developed based on the above design codes to predict the failure load ratio versus time and temperature for varying LSF wall configurations including insulations. Idealised time-temperature profiles were developed based on the measured temperature values of the studs. This was used in a detailed numerical study to fully understand the structural behaviour of LSF wall panels. Appropriate equations were proposed to find the critical temperatures for different composite panels, varying in steel thickness, steel grade and screw spacing for any load ratio. Hence useful and simple design rules were proposed based on the current cold-formed steel structures and fire design standards, and their accuracy and advantages were discussed. The results were also used to validate the fire design rules developed based on AS/NZS 4600 (SA, 2005) and Eurocode Part 1.3 (ECS, 2006). This demonstrated the significant improvements to the design method when compared to the currently used prescriptive design methods for LSF wall systems under fire conditions. In summary, this research has developed comprehensive experimental and numerical thermal and structural performance data for both the conventional and the proposed new load bearing LSF wall systems under standard fire conditions. Finite element models were developed to predict the failure times of LSF walls accurately. Idealized hot flange temperature profiles were developed for non-insulated, cavity and externally insulated load bearing wall systems. Suitable fire design rules and spread sheet based design tools were developed based on the existing standards to predict the ultimate failure load, failure times and failure temperatures of LSF wall studs. Simplified equations were proposed to find the critical temperatures for varying wall panel configurations and load ratios. The results from this research are useful to both structural and fire engineers and researchers. Most importantly, this research has significantly improved the knowledge and understanding of cold-formed LSF loadbearing walls under standard fire conditions.

Numerical and experimental studies of cold-formed steel floor systems under standard fire conditions

Light gauge cold-formed steel frame (LSF) structures are increasingly used in industrial, commercial and residential buildings because of their non-combustibility, dimensional stability, and ease of installation. A floor-ceiling system is an example of its applications. LSF floor-ceiling systems must be designed to serve as fire compartment boundaries and provide adequate fire resistance. Fire rated floor-ceiling assemblies formed with new materials and construction methodologies have been increasingly used in buildings. However, limited research has been undertaken in the past and hence a thorough understanding of their fire resistance behaviour is not available. Recently a new composite panel in which an external insulation layer is used between two plasterboards has been developed at QUT to provide a higher fire rating to LSF floors under standard fire conditions. But its increased fire rating could not be determined using the currently available design methods. Research on LSF floor systems under fire conditions is relatively recent and the behaviour of floor joists and other components in the systems is not fully understood. The present design methods thus require the use of expensive fire protection materials to protect them from excessive heat increase during a fire. This leads to uneconomical and conservative designs. Fire rating of these floor systems is provided simply by adding more plasterboard sheets to the steel joists and such an approach is totally inefficient. Hence a detailed fire research study was undertaken into the structural and thermal performance of LSF floor systems including those protected by the new composite panel system using full scale fire tests and extensive numerical studies. Experimental study included both the conventional and the new steel floor-ceiling systems under structural and fire loads using a gas furnace designed to deliver heat in accordance with the standard time- temperature curve in AS 1530.4 (SA, 2005). Fire tests included the behavioural and deflection characteristics of LSF floor joists until failure as well as related time-temperature measurements across the section and along the length of all the specimens. Full scale fire tests have shown that the structural and thermal performance of externally insulated LSF floor system was superior than traditional LSF floors with or without cavity insulation. Therefore this research recommends the use of the new composite panel system for cold-formed LSF floor-ceiling systems. The numerical analyses of LSF floor joists were undertaken using the finite element program ABAQUS based on the measured time-temperature profiles obtained from fire tests under both steady state and transient state conditions. Mechanical properties at elevated temperatures were considered based on the equations proposed by Dolamune Kankanamge and Mahendran (2011). Finite element models were calibrated using the full scale test results and used to further provide a detailed understanding of the structural fire behaviour of the LSF floor-ceiling systems. The models also confirmed the superior performance of the new composite panel system. The validated model was then used in a detailed parametric study. Fire tests and the numerical studies showed that plasterboards provided sufficient lateral restraint to LSF floor joists until their failure. Hence only the section moment capacity of LSF floor joists subjected to local buckling effects was considered in this research. To predict the section moment capacity at elevated temperatures, the effective section modulus of joists at ambient temperature is generally considered adequate. However, this research has shown that it leads to considerable over- estimation of the local buckling capacity of joist subject to non-uniform temperature distributions under fire conditions. Therefore new simplified fire design rules were proposed for LSF floor joist to determine the section moment capacity at elevated temperature based on AS/NZS 4600 (SA, 2005), NAS (AISI, 2007) and Eurocode 3 Part 1.3 (ECS, 2006). The accuracy of the proposed fire design rules was verified with finite element analysis results. A spread sheet based design tool was also developed based on these design rules to predict the failure load ratio versus time, moment capacity versus time and temperature for various LSF floor configurations. Idealised time-temperature profiles of LSF floor joists were developed based on fire test measurements. They were used in the detailed parametric study to fully understand the structural and fire behaviour of LSF floor panels. Simple design rules were also proposed to predict both critical average joist temperatures and failure times (fire rating) of LSF floor systems with various floor configurations and structural parameters under any given load ratio. Findings from this research have led to a comprehensive understanding of the structural and fire behaviour of LSF floor systems including those protected by the new composite panel, and simple design methods. These design rules were proposed within the guidelines of the Australian/New Zealand, American and European cold- formed steel structures standard codes of practice. These may also lead to further improvements to fire resistance through suitable modifications to the current composite panel system.

Finite element modelling of load bearing cold-formed steel wall systems under fire conditions

Light Gauge Steel Framing (LSF) walls are made of cold-formed, thin-walled steel lipped channel studs with plasterboard linings on both sides. However, these thin-walled steel sections heat up quickly and lose their strength under fire conditions despite the protection provided by plasterboards. A new composite wall panel was recently proposed to improve the fire resistance rating of LSF walls, where an insulation layer was used externally between the plasterboards on both sides of the wall frame instead of using it in the cavity. A research study using both fire tests and numerical studies was undertaken to investigate the structural and thermal behaviour of load bearing LSF walls made of both conventional and the new composite panels under standard fire conditions and to determine their fire resistance rating. This paper presents the details of finite element models of LSF wall studs developed to simulate the structural performance of LSF wall panels under standard fire conditions. Finite element analyses were conducted under both steady and transient state conditions using the time-temperature profiles measured during the fire tests. The developed models were validated using the fire test results of 11 LSF wall panels with various plasterboard/insulation configurations and load ratios. They were able to predict the fire resistance rating within five minutes. The use of accurate numerical models allowed the inclusion of various complex structural and thermal effects such as local buckling, thermal bowing and neutral axis shift that occurred in thin-walled steel studs under non-uniform elevated temperature conditions. Finite element analyses also demonstrated the improvements offered by the new composite panel system over the conventional cavity insulated system.

Refined plastic hinge analysis of steel frame structures comprising non-compact sections : II : verification

Application of "advanced analysis" methods suitable for non-linear analysis and design of steel frame structures permits direct and accurate determination of ultimate system strengths, without resort to simplified elastic methods of analysis and semi-empirical specification equations. However, the application of advanced analysis methods has previously been restricted to steel frames comprising only compact sections that are not influenced by the effects of local buckling. A refined plastic hinge method suitable for practical advanced analysis of steel frame structures comprising non-compact sections is presented in a companion paper. The method implicitly accounts for the effects of gradual cross-sectional yielding, longitudinal spread of plasticity, initial geometric imperfections, residual stresses, and local buckling. The accuracy and precision of the method for the analysis of steel frames comprising non-compact sections is established in this paper by comparison with a comprehensive range of analytical benchmark frame solutions. The refined plastic hinge method is shown to be more accurate and precise than the conventional individual member design methods based on elastic analysis and specification equations.