Single Degree Of Freedom Analysis Research Articles

Prediction of annealed glass window response to blast loading

Several software packages that are capable of predicting the response of annealed glass windows to blast loads are evaluated, by comparison to experimental data that has been obtained from full-scale arena blast tests in the field. A total of 34 instrumented, monolithic, glass window panes were subjected to explosive blast waves of varying intensity. A series of small-scale material tests and full-scale pane tests was also performed in the laboratory to obtain mechanical properties for use in the predictive models. The utility of various software tools, based either on single degree of freedom analysis or explicit finite element analysis, is hence assessed, for use in the analysis or design of glazing under blast loading.

Static resistance function for steel-plate composite (SC) walls subject to impactive loading

Steel-plate composite (SC) walls consist of a plain concrete core reinforced with two steel faceplates on the surfaces. Modules (consisting of steel faceplates, shear connectors and tie-bars) can be shop-fabricated and shipped to the site for erection and concrete casting, which expedites construction schedule and thus economy. SC structures have recently been used in nuclear power plant designs and are being considered for the next generation of small modular reactors. Design for impactive and impulsive loading is an important consideration for SC walls in safety-related nuclear facilities. The authors have previously developed design methods to prevent local failure (perforation) of SC walls due to missile impact. This paper presents the development of static resistance functions for use in single-degree-of-freedom (SDOF) analyses to predict the maximum displacement response of SC walls subjected to missile impact and designed to resist local failure (perforation). The static resistance function for SC walls is developed using results of numerical analyses and parametric studies conducted using benchmarked 3D finite element (FE) models. The influence of various design parameters are discussed and used to develop idealized bilinear resistance functions for SC walls with fixed edges and simply supported edges. Results from dynamic non-linear FE analysis of SC panels subjected to rigid missile impact are compared with the maximum displacements predicted by SDOF analyses using the bilinear resistance function.

강재압축재의 방폭성능에 대한 중력하중효과의 해석적 연구

Equivalent Single-Degree-of-Freedom(SDOF) analysis, most used for blast-resistant design, does not consider the effects of gravity load on the performance evaluation of blast resistance of structural members. However, since there exists gravity load on columns and walls of structures, the blast resistance of structural members should be evaluated considering gravity load on them. In this paper, an approach to reflect the gravity load effects on the equivalent SDOF analysis for dynamic blast response of structural members is proposed. For this purpose, the parametric studies using finite element analysis were performed by varying maximum blast load, blast load duration, and gravity load with constant the resistance and natural period of a structural member. The finite element analysis results were compared with the equivalent SDOF analysis results and the blast response of the structure member was estimated by conducting finite element analyses for various gravity loads. Finally, a graphical solution for ductility of a structural member with the variables of blast load, gravity load and structural member properties was developed. The blast response of structural members under gravity load could be estimated reasonably and easily by using this graphical solution.

Experimental study of laminated glass window responses under impulsive and blast loading

Laminated glass panes are widely adopted as blast-resistant glass windows to mitigate the hazard from ejecting fractured glass fragments. The response of laminated glass windows under blast loads is often predicted by equivalent static analysis or simplified equivalent single degree of freedom (SDOF) analysis. The equivalent SDOF and equivalent static analyses are also respectively adopted in UFC and ASTM design guide for glass window designs. Owing to the inherent problems, the SDOF analysis can only predict the global responses of glass windows and the predictions are not necessarily always satisfactory. Therefore the accuracy and applicability of the SDOF analysis is sometimes questioned. Often numerical simulations and/or experimental tests have to be carried out for reliable predictions of laminated glass window responses to blast loads. In this study, experimental tests on laminated glass windows subjected to impact and blast loads were carried out to evaluate the accuracy of available analyses and design methods. Pendulum impact tests were conducted first on laminated panes of various thicknesses. Full-scale field blast tests were performed on laminated glass windows of dimension 1.5 m × 1.2 m. Glass pane deflections were monitored by mechanical linear voltage displacement transducer (LVDT) and high-speed cameras. The responses of the tested windows are compared with the estimations of SDOF models and design standards in this paper. Available blast testing data by other researchers are also included together with the current testing data to evaluate the accuracy of the SDOF and equivalent static analyses defined in the design guides. The adequacy of these simplified approaches in predicting laminated glass window responses to blast loads is discussed.

Finite Element Assessment of Methods for Incorporating Axial Load Effects into Blast Design SDOF Analyses of Precast Wall Panels

AbstractInnovations in the U.S. construction and precast concrete manufacturing industries have resulted in efficient methods of constructing multistory buildings that are entirely supported by precast concrete walls (i.e., no structural frame). Simultaneously, the U.S. government has mandated overarching requirements and incentives that promote energy efficiency in all government buildings and facilities. The combination of construction efficiencies and relatively recent requirements for improving energy efficiency makes total precast construction an attractive option for constructing government, diplomatic and U.S. Department of Defense (DoD) buildings and facilities. However, to design load bearing wall panels for DoD blast response criteria, engineers need a relatively simple method for analyzing dynamic response under combined lateral and axial load effects. This paper presents methods for incorporating the P-delta and P-M interaction effects into the single-degree-of-freedom (SDOF) framework that is...

Single Degree of Freedom Analysis of Steel Beams under Blast Loading

The paper deals with the analysis of steel beams subjected to blast load approximated as a one degree system of freedom (SDOF). It requires knowing the parameters of blast pressure wave, its effect on structure and the tools for the solution of dynamic analysis. The blast wave is estimated with linear decay and exponential decay using positive and negative phase. The results of SDOF model are compared with the corresponding experimental accelerations and strain time-histories. There is a described dynamic analysis for such structure.

Single-Degree-of-Freedom Based Pressure-Impulse Diagrams for Blast Damage Assessment

Pressure-impulse (P-I) diagrams, which relates damage with both impulse and pressure, are widely used in the design and damage assessment of structural elements under blast loading. Among many methods of deriving P-I diagrams, single degree of freedom (SDOF) models are widely used to develop P-I diagrams for damage assessment of structural members exposed to blast loading. The popularity of the SDOF method in structural response calculation in its simplicity and cost-effective approach that requires limited input data and less computational effort. The SDOF model gives reasonably good results if the response mode shape is representative of the real behaviour. Pressure-impulse diagrams based on SDOF models are derived based on idealised structural resistance functions and the effect of few of the parameters related to structural response and blast loading are ignored. Effects of idealisation of resistance function, inclusion of damping and load rise time on P-I diagrams constructed from SDOF models have been investigated in this study. In idealisation of load, the negative phase of the blast pressure pulse is ignored in SDOF analysis. The effect of this simplification has also been explored. Matrix Laboratory (MATLAB) codes were developed for response calculation of the SDOF system and for repeated analyses of the SDOF models to construct the P-I diagrams. Resistance functions were found to have significant effect on the P-I diagrams were observed. Inclusion of negative phase was found to have notable impact of the shape of P-I diagrams in the dynamic zone.

Study of Pressure-Impulse Diagrams for Reinforced Concrete Columns Using Finite Element Analysis

The failure of reinforced concrete columns when subjected to blast events can be catastrophic to the building as a whole. The choice of the type of lateral confinement may influence the response of columns subjected to blast load much more when compared to the application of traditional loads. The objective of the research reported in this paper is to study the non-linear response of a typical reinforced concrete column with four types of lateral reinforcement subjected to uniform blast loading, using finite element (FE) and single degree of freedom (SDOF) methods. An advanced FE modeling tool, LS-DYNA and a widely used SDOF analysis tool – SBEDS – are used for this purpose. Pressure-Impulse (PI) diagrams for different levels of protection (LOP) are developed using both the methods. The mid height displacement is used to represent the levels of protection. A parametric analysis is used to develop PI diagrams in order to study the effect of confinement on the reduction of displacement at impulsive, dynamic, and quasi-static loading conditions. PI curves developed from the FE methods are compared with the curves from the SDOF methodology. Regression equations are also developed using statistical analysis, to characterize PI diagrams generated from the FE method. Recommendations for analysis, design and detailing of reinforced concrete columns for blast protection based on this research are presented.

A new SDOF method of one-way reinforced concrete slab under non-uniform blast loading

A new effective model for calculation of the equivalent uniform blast load for non-uniform blast load such as close-in explosion of a one-way square and rectangle reinforced concrete slab is proposed in this paper. The model is then validated using single degree of freedom (SDOF) system with the experiments and blast tests for square slabs and rectangle slabs. Test results showed that the model is accurate in predicting the damage level on the tested RC slabs under the given explosive charge weight and stand-off distance especially for close-in blast load. The results are also compared with those obtained by conventional SDOF analysis and finite element (FE) analysis using solid elements. It is shown that the new model is more accurate than the conventional SDOF analysis and is running faster than the FE analysis.

Behavior of reinforced concrete columns under combined effects of axial and blast-induced transverse loads

The results of numerical studies on the dynamic response of reinforced concrete (RC) columns subjected to axial and blast-induced transverse loads utilizing an advanced single-degree-of-freedom (SDOF) model are presented in this paper. The main variables considered in this study were the level of axial force and longitudinal reinforcement ratio. This work addressed the effect of various levels of axial compressive load on the resistance function, time-history response, and load–impulse diagram when the columns were subjected to transverse loads due to blast. The blast loads were idealized as triangular pulses and the effects of flexural, diagonal shear, and tension membrane behaviors were included in the RC column response. The results from the SDOF analyses were validated using the commercial finite element (FE) program ABAQUS. The results of the parametric study indicated that the level of axial compressive load has a significant influence on the behavior of RC columns when subjected to transverse blast-induced loads.

Strength and stability of steel beam columns under blast load

A Single-Degree-of-Freedom (SDOF) model is used to determine the effect of axial load on column strength and stability during a blast event. The model, which accounts for the axial load–bending interaction (P–δ effect) and strain rate effect on the column dynamic response, is validated by comparing its results with experimental data from blast tests on full scale steel columns and with the results of the finite element software LS-DYNA. Maximum displacements and moments obtained from SDOF analysis are also compared with the results of the interaction formulas recommended by the Unified Facilities Criteria (UFC 3-340-02) design manual for steel structures. It is shown that the UFC method overestimates the column capacity for ductility ratios μ greater than one, irrespective of the axial load to Euler elastic buckling load ratio (P/Pe). Also for P/Pe>0.5, even if μ<1.0, the UFC method still overestimates the actual column capacity. For dealing with this problem in practical applications, non-dimensional beam column curves are developed to include the effects of the blast load and column properties on both its strength and stability.

A new non-uniform blast load model for SDOF method of one-way reinforced concrete slab

A new effective model for calculation of the equivalent uniform blast load for non-uniform blast load such as close-in explosion of a one-way square and rectangle reinforced concrete slab is proposed in this paper. The model is then validated using single degree of freedom (SDOF) system with the experiments and blast tests for square slabs and rectangle slabs. Test results showed that the model is accurate in predicting the explosive charge weight and stand-off distance to impose a given damage level on the tested RC slabs especially for close-in blast load. It is shown that the new model is more accurate than the conventional SDOF analysis and is running faster than the FE analysis.

Dynamic Response of Retrofitted Masonry Walls for Blast Loading

A full-scale blast test was conducted on eight masonry walls reinforced with two and four layers of carbon fibers and two types of polymer matrices. The walls were then subjected to a 0.45-kg pentolite booster suspended from the ceiling of a test structure. The pressure-time history caused by the blast and the resulting displacement response were measured during the test. This paper presents a summary of the test program and the corresponding results from a nonlinear single degree of freedom analysis. The results provide a basis for determining effective means of retrofitting existing masonry walls and designing new structures to withstand blast loads. The paper also outlines a fiber-reinforced polymer retrofit design procedure for walls subjected to blast loading.

Finite difference analysis of simply supported RC slabs for blast loadings

A finite difference procedure that can account for strain-rate effects, both shear and flexural deformations, permits variations in cross-section geometry and strength and loading over the length of a component is proposed to accurately and efficiently analyze the dynamic response of a simply supported structural member under blast loads. A section-based layered analysis model that accommodates varying strain rates across a member’s cross-section is used to derive sectional moment–curvature relationships. A formula is derived to estimate the distribution of strain rate over the depth of a cross-section along the length of the member, and the corresponding strain rate effects are incorporated into the section-based layered analysis model. The Timoshenko beam equations that include both shear deformations and rotational inertia are solved numerically using an explicit finite difference scheme. The accuracy of the proposed finite difference analysis model is part validated using results of blast testing of reinforced concrete slabs with combinations of explosive weights and standoff distances. The results are also compared with those obtained by conventional single-degree-of-freedom (SDOF) analysis and finite element (FE) analysis using solid elements. The finite difference analysis procedure is both fast-running and accurate and most suitable for design office application, combining the speed of SDOF analysis and the detail and accuracy of FE analysis.

Inelastic two-degree-of-freedom model for roof frame under airblast loading

When a roof frame is subjected to the airblast loading, the conventional way to analyze the damage of the frame or design the frame is to use single degree of freedom (SDOF) model. Although a roof frame consists of beams and girders, a typical SDOF analysis can be conducted only separately for each component. Thus, the rigid body motion of beams by deflections of supporting girders can not be easily considered. Neglecting the beam-girder interaction in the SDOF analysis may cause serious inaccuracies in the response values in both Pressure-Impulse curve (P-I) and Charge Weight-Standoff Diagrams (CWSD). In this paper, an inelastic two degrees of freedom (TDOF) model is developed, based on force equilibrium equations, to consider beam-girder interaction, and to assess if the modified SDOF analysis can be a reasonable design approach.

Modeling of transmission of shaped sonic booms into residential structures.

The indoor perception of shaped minimized sonic booms is a concern for the acceptability of overland supersonic flight. An investigation of the transmission of shaped booms into residential structures has been conducted. Two classical methods were used. The first is single degree of freedom analysis. The second is modal analysis, modeling a wall as a simply supported rectangular panel and the interior as a rectangular cavity. Time domain responses were computed for both methods. Both methods allowed arbitrary waveforms, and the modal model allows arbitrary wave incidence angle. Sound pressures inside a room and acceleration of the wall were calculated and compared to measured data collected during the NASA sonic boom tests conducted at the Edwards Air Force Base in 2007. The effects of wave signature parameters on transmission were studied to evaluate the advantages of various kinds of low boom shapes, and the influence of thickened shock structures. Future extension of this research will be to employ finite element modeling. [This work was funded by NASA Langley Research Center, Contract No. NNL07AD96T.]

Inelastic Dynamic Response of Curtainwall Systems to Blast Loading

Abstract Blast-resistant curtainwall systems require laminated glass that is adequately adhered to the mullions to develop the membrane capacity of the glass, as well as mullions that are capable of transferring the collected forces to the structural slabs. Dynamic inelastic analyses are required to determine the response of curtainwalls to highly impulsive blast loading. The most economical way to minimize the hazard to occupants is to allow the glass to undergo large deformations prior to dismemberment and fragmentation. Curtainwall performance is often evaluated using a sequence of single-degree-of-freedom (SDOF) analyses. The glass is represented as an SDOF to determine the reaction forces, which are then applied to SDOF analyses of the mullions. These individual calculations represent the dynamic inelastic response of the glass and mullions, but often neglect their interaction and phasing. As a result, SDOF methods tend to overpredict the response of the curtainwall components, resulting in an uneconomical design. Dynamic inelastic finite element analyses that account for the interaction of the glass and mullions and accurately represent the potential failure mechanisms provide the most accurate evaluation, leading to the most economical design for blast-resistant curtainwall systems. In addition to material nonlinearities, the finite element analyses also account for geometric stiffening of the components, therefore producing more accurate anchorage forces to be developed at the slabs.

Post-buckling behavior during earthquakes and seismic margin of FBR main vessels

Shaking-table tests of cylindrical shells were performed in order to examine the buckling and post-buckling characteristics of FBR main vessels under seismic shear loads. Static buckling tests were also performed, and it is confirmed that there is no significant difference between the static and dynamic load-displacement relations. Based on the test results, hysteresis rules of restoring force were formulated for both elastic and plastic shear-buckling. Non-linear dynamic-response analyses of the single-degree-of-freedom (SDOF) system were then carried out by using the hysteresis rules. The analyses were able to simulate the dynamic test results, especially energy-absorption capacity due to hysteresis behavior. Finally, the non-linear SDOF analysis was applied to the FBR main-vessel cylinder. It is pointed out that the hysteresis behavior could absorb a considerable amount of energy input from seismic motion, which would contribute to the seismic margin of FBR main vessels.

Modified SDOF Analysis of RC Box‐Type Structures

An improved single‐degree‐of‐freedom (SDOF) approach is presented for the analysis of shallow‐buried reinforced concrete box‐type structures under the effect of blast and shock environments. The method is based on rational models for structural resistance functions that describe the combined effects of flexure, shear, and in‐plane compression on the system response. The approach is demonstrated by employing it for the analysis of seven different events for which experimental information is available, and the accuracy of the method is evaluated by comparing numerical and experimental results.