Critical Limit State Research Articles

Evaluating the Structural Integrity of Cellular Steel Beams with Web Enhancements

Abstract This study investigates the analytical approach of web reinforcement techniques, including high-strength concrete and laced reinforcement, to enrich improvement for the structural behavior of these beams. The analysis compares an unmodified cellular beam (LB1) with a web-reinforced beam (LB2), including the improvements in load-carrying capacity. The study considers the effects of reinforcement on critical design limit states, such as flexural strength, Vierendeel bending, web post-buckling, and shear resistance. The outcomes reveal significant enhancements in LB2’s structural behavior, with over a 100% increase in flexural strength and local axial force capacity and a 165% increase in web post-buckling strength. This research validates the effectiveness of web reinforcement techniques in concentrating on the limitations of cellular beams and maximizing their potential in structural applications.

Fatigue life evaluation on wire rope suspenders of the suspension bridge based on a time-varying degradation process

Suspenders are critical force transmission components of a suspension bridge and may be more prone to fatigue damage due to environmental aggression and repetitive live loads. To evaluate the safety of the suspender more accurately, a fatigue life evaluation framework of the rope suspender was proposed, where a time-varying fretting-corrosion-fatigue degradation process of the rope wire and probabilistic critical limit states based on failure area were integrated. A case study on the rope suspenders of a prototype suspension bridge was performed under the random vehicle and wind loads, and the reliability indices and fatigue life of the rope suspender were obtained. Results show that the fretting wear had an obvious effect on the degradation process of the wire rope in conjunction with corrosion fatigue, and the fatigue life of the rope suspender would be less than its design life. The proposed method can comprehensively evaluate the service life of wire rope suspenders and provide replacement suggestions more timely, ensuring the safety of the bridge operation.

Seismic fragility assessment of load‐bearing soft‐brick unreinforced masonry piers

Unreinforced masonry (URM) made with soft bricks comprises a large percentage of the building stock in developing countries. However, the poor performance of URM piers during earthquakes has led to renewed interest in understanding their behavior under lateral loads. Little experimental data is available on the seismic response, analysis, and design of URMs made of soft bricks. In this study, the micro-modeling technique is used to simulate the in-plane behavior of load-bearing, soft-brick URM piers. The parameters required in the constitutive models are obtained from material tests and used to develop a calibrated numerical model of the URM piers. Piers with various aspect ratios subjected to various axial stresses are numerically modeled to obtain monotonic and cyclic responses, and their critical displacement limit states are identified. Changes in the failure modes of masonry piers with variations in the aspect ratio and axial stress are established. Load-bearing piers exhibit three distinct failure modes: bed sliding, diagonal shear cracking, and flexure, depending on the aspect ratio and axial stress. The seismic fragility of each pier failure type is examined using nonlinear time history analyses. The results show that bed-sliding piers collapse at extremely low PGA levels. Piers failing through diagonal shear cracking also fail at low PGA levels. Flexural piers can resist seismic forces up to a slightly higher PGA level and thus are the last to collapse. The results also indicate that the effect of uncertainty in ground motions is more significant than the effect of variability in the masonry pier capacities.

Bond–Slip Law Between Steel Bar and Different Cement-Based Materials Considering Anchorage Position Function

The bond performance between steel bar and cement-based materials was the prerequisite for the two materials to work together, and previous studies showed that the bond behavior of the steel bars and cement-based materials will vary with the kinds of cement-based materials. For this reason, this paper adopted 12 direct pullout test specimens including three types of concrete and two types of steel bars. The strain of the steel bar at six measuring points was measured with a strain gauge. Based on the measured strain and free end slip of the steel bars, the distribution of steel stress, bond stress, and relative slip and the bond slip relation along the anchorage length were obtained and analyzed for different concrete and different steel bars. Based on these test results of steel strain and relative slip at six measuring points, the anchorage position function could be established in consideration of anchorage position, which was conducive to the establishment of an accurate bond–slip relationship. In addition, the anchorage length of the steel bar in Engineered Cementitious Composites (ECC) calculated from the equilibrium equation of critical limit state is only half of the anchorage length calculated in the current Code for Design of Concrete Structures (GB 50010-2010) in China. It is suggested to establish the critical anchorage length formula suitable for ECC in future studies.

Reliability Assessment of Internal Stability Limit States for Two As-Built Geosynthetic MSE Walls

This paper reports the results of calculations for the margins of safety for internal stability limit states for two as-built geosynthetic MSE walls constructed in 2006 in Washington State, U.S.A. The walls are unique because a) the internal stability design for the reinforcement was based on a working stress design method which includes the stiffness of the reinforcement as a key parameter, and b) the walls were instrumented and monitored during and after construction. Margins of safety are quantified deterministically using factors of safety, and probabilistically using reliability indices. The calculations are performed using the properties of the as-built materials as opposed to the properties assumed at the time of original design. The results show that the margins of safety for tensile and pullout strength are huge and thus these limit states are not of practical concern for these structures. The critical limit state is the soil failure limit state. Margins of safety for this limit state are shown to be closer to recommended minimums but nevertheless adequate for all reinforcement layers in both walls.

The slab effect of composite frames: From modified fiber modeling to seismic fragility assessment

The slab spatial composite effect (SSCE) is of considerable importance in the field of engineering as it greatly influences the vertical load-bearing capacity and lateral load resistance of composite frame structures. In this study, the existing uniaxial material-constitutive models of concrete (Concrete02) and steel (Steel02) in the Open System for Earthquake Engineering Simulation (OpenSees) were modified based on three effective flange widths of composite beams under two critical limit states. An improved fiber model was developed to capture the SSCE of composite frames by combining the conventional fiber models and new material-constitutive models. Through three validation studies, the ability of the proposed fiber model to trace the SSCE accurately and efficiently was established based on modeling and comparison of the representative composite frames available in literature. Moreover, seismic fragility assessment was conducted for a 10-story spatial composite frame subjected to bidirectional horizontal seismic excitation based on an incremental dynamic analysis. The comprehensive numerical simulation results indicate that the conventional fiber model without constitutive model modifications leads to significant overestimation of the probability of the structure being a specified limit state under different ground motion intensities and site classes. The conventional fiber model results are at most 47.1%, 72.0%, 53.5%, and 49.6% higher than those obtained using the improved fiber model in terms of the structure being slightly, moderately, extensively, and completely damaged, respectively. This means that the potential economic advantages of composite frames can be explored by reasonably considering the SSCE.

Deterministic and probabilistic assessment of margins of safety for internal stability of as-built PET strap reinforced soil walls

The paper demonstrates deterministic and reliability-based assessment of strength limit states (tensile resistance and pullout) and the service limit state for soil failure for mechanically stabilized earth (MSE) walls constructed with polyester (PET) strap reinforcement. The general approach considers the accuracy of the load and resistance models that appear in each limit state equation plus uncertainty in the estimate of nominal load and resistance values at time of design. Reliability index is computed using a closed-form solution that is easily implemented in a spreadsheet. Three PET strap MSE wall case studies are used to demonstrate the reliability-based assessment approach and to compare margins of safety using different load and resistance model combinations. In some walls using the Coherent Gravity Method to compute loads, the recommended nominal factors of safety for tensile strength and pullout limit states were not satisfied. However, reliability analyses showed that the walls satisfy recommended minimum target reliability index values for the limit states investigated, usually by large amounts. The most critical limit state is the soil failure limit state which is used in the Simplified Stiffness Method to keep the reinforced soil zone at working stress conditions assumed for geosynthetic MSE walls under operational conditions.

Wind fragility analysis of RC chimney with temperature effects by dual response surface method

Wind fragility analysis (WFA) of concrete chimney is often executed disregarding temperature effects. But combined wind and temperature effect is the most critical limit state to define the safety of a chimney. Hence, in this study, WFA of a 70 m tall RC chimney for combined wind and temperature effects is explored. The wind force time-history is generated by spectral representation method. The safety of chimney is assessed considering limit states of stress failure in concrete and steel. A moving-least-squares method based dual response surface method (DRSM) procedure is proposed in WFA to alleviate huge computational time requirement by the conventional direct Monte Carlo simulation (MCS) approach. The DRSM captures the record-to-record variation of wind force time-histories and uncertainty in system parameters. The proposed DRSM approach yields fragility curves which are in close conformity with the most accurate direct MCS approach within substantially less computational time. In this regard, the error by the single-level RSM and least-squares method based DRSM can be easily noted. The WFA results indicate that over temperature difference of 150°C, the temperature stress is so pronounced that the probability of failure is very high even at 30 m/s wind speed. However, below 100°C, wind governs the design.

The Basic Principles of Assessment of the Structural Integrity and Lifetime of the BN-Type Fast Neutron Reactor Components Considering Material Degradation

The paper presents an overview of the basic principles of extending the lifetime of BN-600 fast reactors. These principles based on analysis of the main in-service mechanisms of material embrittlement and damage underlie the normative documents of the Rosatom State Corporation and were used by the authors in their developments for justification of the design lifetime of the BN-800 and BN-1200 fast reactors. The so-called critical events and limit states that determine the structural integrity and lifetime of fast reactor components have been formulated. On the basis of the results of this work, the repost “Basic Principles for Lifetime and Structural Integrity Assessment of BN-600 and BN-800 Fast Reactor Components with Regard to Material Degradation” was made at the International Conference on Fast Reactors and Related Fuel Cycles: Next Generation Nuclear Systems for Sustainable Development (FR17) held by the IAEA in Yekaterinburg in 2017.

On Initial Design and Modelling of a 10 MW Medium Speed Drivetrain for Offshore Wind Turbines

This paper presents the initial design and modelling of a medium speed drivetrain for the DTU 10 MW offshore wind turbine. Design basis and criteria of the drivetrain are described. Minimum weight and volume of the drivetrain are chosen as the main design targets. A four-point support, two main bearings and two torque arms, drivetrain configuration is selected in this study. One conventional gearbox layout, two planetary stages and one parallel stage, which is commonly used in large offshore wind turbines is adopted. Four gearbox layout options are provided and compared, and one optimal option is finally selected with holistic considerations of volume and weight as well as load sharing performance principles. Then, a high fidelity drivetrain numerical model for the 10 MW reference turbine is established using multi-body system (MBS) approach. In addition, model comparison is conducted, which shows that drivetrain eigenfrequencies in torsional mode obtained from the numerical model agree well with the values in the DTU report. Finally, future work relating to dynamic response analysis, critical limit states check for the drivetrain model is proposed.

Design of timber-concrete composite (TCC) bridges with under-deck stay cables

Timber-concrete composite (TCC) bridges represent an attractive structural system due to the synergistic use of its wood and reinforced-concrete constituent components. However, their relatively large flexibility limits their application for larger spans. This paper presents an alternative solution for TCC bridges involving the implementation of post-tensioned under-deck tendons. Based on a series of design and numerical studies, the advantages of the newly proposed system for 30-m, 60-m and 90-m spans are evaluated. This paper shows that the incorporation of under-deck post-tensioning changes the critical limit states governing the design of TCC bridges, and allows for a significant increase in their slendernesses at medium and long spans. Timber’s shear-deformation contribution to the vertical deflection of TCC bridges is significant and should be accounted for, especially when the span/depth ratio (l/h) is less than 20. However, this additional deformation can be neglected when stay cables are implemented, especially for bridges with medium and long spans. In order to achieve a more efficient structure, it is proposed that concrete-to-timber shear connections with an efficiency coefficient (γ) greater than 0.8 be used. Finally, the best practical eccentricity of the under-deck tendons and the best location of the deviators are determined on the basis of parametric analyses.

Seismic Fragility Analysis of Steel Liquid Storage Tanks Using Earthquake Ground Motions Recorded in Korea

Liquid‐containing storage tanks are important structures in industrial complexes. Because earthquake damages to liquid storage tanks can cause structural collapse, fires, and hazardous material leaks, there have been continuous efforts to mitigate earthquake damages using seismic fragility analysis. In this regard, this study focuses on the seismic responses and fragility of liquid storage tanks. First, the characteristics of earthquake ground motions are a critical factor influencing the seismic fragility of structures; thus, this study employs real earthquake records observed in the target area, southeastern Korea, with the earthquake characteristics estimated based on the ratio of peak ground acceleration to peak ground velocity. When a liquid storage tank oscillates during an earthquake, additional forces can impact the tank wall owing to hydrodynamic pressures. Therefore, this study presents a sophisticated finite element (FE) model that reflects the hydrodynamic effect of an oscillating liquid. Another advantage of such an FE model is that detailed structural responses of the entire wall shells can be estimated; this is not possible in simplified lumped mass or surrogate models. Lastly, probabilistic seismic demand models are derived for three critical limit states: elastic buckling, elephant’s foot buckling, and steel yielding. Using the real earthquake ground motion records, constructed FE model, and limit states, a seismic fragility analysis is performed for a typical anchored steel liquid storage tank in Korea. In addition, for comparison purposes, a ring‐stiffened model is investigated to derive a seismic fragility curve. The results of the seismic fragility assessment show that elastic buckling is the most vulnerable damage state. In contrast, elephant’s foot buckling and steel yielding indicate relatively severe damage levels. Furthermore, it is observed that ring stiffeners decrease the elastic buckling damage, although there is no practical effect on elephant’s foot buckling and steel yielding in all ground motion intensities.

Design Optimization of Hybrid FRP/RC Bridge

The hybrid bridge consists of a Reinforced Concrete (RC) slab supported by U-shaped Fiber Reinforced Polymer (FRP) girders. Previous studies on similar hybrid bridges constructed in the United States and Europe seem to substantiate these hybrid designs for lightweight, high strength, and durable highway bridge construction. In the current study, computational and optimization analyses were carried out to investigate six composite material systems consisting of E-glass and carbon fibers. Optimization constraints are determined by stress, deflection and manufacturing requirements. Finite Element Analysis (FEA) and optimization software were utilized, and a framework was developed to run the complete analyses in an automated fashion. Prior to that, FEA validation of previous studies on similar U-shaped FRP girders that were constructed in Poland and Texas is presented. A finer optimization analysis is performed for the case of the Texas hybrid bridge. The optimization outcome of the hybrid FRP/RC bridge shows the appropriate composite material selection and cross-section geometry that satisfies all the applicable Limit States (LS) and, at the same time, results in the lightest design. Critical limit states show that shear stress criteria determine the optimum design for bridge spans less than 15.24 m and deflection criteria controls for longer spans. Increased side wall thickness can reduce maximum observed shear stresses, but leads to a high weight penalty. A taller cross-section and a thicker girder base can efficiently lower the observed deflections and normal stresses. Finally, substantial weight savings can be achieved by the optimization framework if base and side-wall thickness are treated as independent variables.

Effects of Changing Double Angles Configuration Used as Braces in Seismic Resisting Systems

Seismic resisting systems consisting of double angles are used in many parts of the world. Generally, these double angles are arranged in the shape of a T, with a very small distance between them. However, sometimes these angles are distanced and faced in order to improve their mechanical characteristics about the axis of symmetry. In the past, their design was made in the same way as the double angles arranged in a T shape, that is, considering the limit states of flexural buckling and buckling by flexural-torsional, but ignoring the properties of the connectors and their effect on the modified slenderness ratio, as well as the fact that in this case the warping constant is not negligible. These parameters are taken into account in this research in order to study the effects of increasing the distance between the connectors and their possible use as braces in seismic resisting systems. The theoretical results were compared with the experimental results of fifty-seven specimens tested in the laboratory of structures of the Universidad Nacional de Colombia – Sede Manizales. The models were classified according to the main angles, the connectors, the total lengths, and the width of separation. All of them were subjected to axial compressive stress, with free rotation at both ends. Three identical specimens of each model were constructed. The flexural buckling length about x-axis was limited to two meters in all specimens tested whereas the flexural bucking length about y-axis and flexural-torsional buckling length were not limited, i.e. these lengths are equivalent to the total length of each specimen tested. This in order that the critical limit state was to be the flexural-torsional buckling as a function of the torsional buckling term in Z, except in the models of class 2 in which this induced condition was not reached. This was proposed to better evaluate the torsional buckling term in Z. The experimental results show that the nominal compressive strength for the flexural-torsional buckling limit state, when it is governed by torsion, is undervalued. A new methodology is proposed for the calculation of the nominal compressive strength for the flexural-torsional buckling limit state, when it is governed by torsion.

Reliability analysis of a large-scale landslide using SOED-based RSM

A design matrix in response surface method (RSM) that satisfies the orthogonality is very useful because the mean square error can be minimized, so that the response surface is more precise. But the orthogonality of a second-order design matrix in conventional RSM cannot be satisfied. In this paper, a second-order orthogonal experimental design (SOED)-based RSM is proposed by considering the orthogonality of high-order design matrix. The SOED is constructed by changing the length of star points, and the main characteristic of SOED is that the design matrix is diagonal. When the high-order terms are considered in the SOED-based RSM, a globe optimal solution can be found. As the regression equation is determined, the reliability index can be analyzed by the normalized distance between the mean value of the performance function and the critical limit state of the safety factor. A practical large-scale landslide with two slip surfaces is taken to verify the applicability and precision of the proposed method in detail. It is found that the SOED-based response surface is more rigorous than the conventional RSM.

Reliability index and parameter importance for bridge traffic loading definition changes

With the continued evolution of traffic loading specifications, safety classifications of bridge structures are subject to change, independent of the actual condition of the structures at that point in time. As investment decisions are often based on these safety classifications, a reclassification of safety level due to changing of traffic load definitions can lead to misinterpretation of the actual state of the structure, and thus lead to a misallocation of resources. Should a reclassification of safety occur after a change in traffic load specification, the question as to whether modern design codes are producing more or less robust bridges than previous design codes is raised. To investigate this, three bridge structures were assessed for evolving definitions of traffic load. Using deterministic and probabilistic methods, critical limit states were assessed and the associated reliability indices and parametric sensitivity factors were determined and compared across various code specifications. This comparison allowed for an evaluation of how the evolution of traffic load over time influences the computed safety of bridge structures.

Experimental evaluation of longitudinal splitting of bamboo flexural components

Splitting or longitudinal shear failures in full-culm bamboo are a critical limit state when using the material for construction applications. Understanding and quantifying the behaviour must be through standardised test methods for international adoption into building practices to be successful. Current testing procedures only determine an effective flexural strength due to splitting failures. The present study investigates longitudinal shear failure in full-culm Phyllostachys pubescens (Moso) and Bambusa stenostachya (Tre Gai) bamboo flexural specimens through a number of standard and modified test methods. In particular, the paper proposes modifications to ‘standard’ culm flexural tests intended to better capture actual shear-dominated behaviour. Notched specimens were used to better establish relationships between mode I and mode II behaviours. The test results illustrate the significant degree of interaction between mode I and mode II stresses resulting from flexural tests and their associated failures, specifically the significant deterioration of mode II capacity in the presence of even a small amount of mode I stress. It is proposed that a uniform experimental approach using horizontally notched flexural specimens having different shear spans and notch locations would be appropriate to establish a relationship between mode I and II behaviour.

Lateral-torsional buckling of butterfly-shaped beams with rectangular cross section

This paper analyzes lateral-torsional buckling (LTB) of elastic beams with rectangular cross section having constant thickness, and depth symmetric with respect to the midpoint and tapered linearly in each half with smallest depth at the midpoint. This type of butterfly-shaped beam is used as a shear link in steel plates that serve as structural fuses to reduce the seismic response of buildings. LTB has been observed in previous quasi-static tests and found to be a critical limit state. The ends of the beam are prevented from twisting and can be assumed to be pinned with respect to bending in the weak direction. Numerical results are obtained for LTB of butterfly-shaped beams with the use of a shooting method assuming uniform torsion. Pure bending, reversed moments, and end moments with unequal magnitudes are investigated first. A compressive load is included next, and interaction curves for combinations of critical moment (squared) versus axial load are determined. Finally, the effect of in-plane shear deformation on LTB is examined. Plots showing the influence of various parameters are presented, along with some analytical approximations for the critical moment in terms of geometric and material quantities.

Robust simulation of buckled structures using reduced order modeling

Lightweight metallic structures are a mainstay in aerospace engineering. For these structures, stability, rather than strength, is often the critical limit state in design. For example, buckling of panels and stiffeners may occur during emergency high-g maneuvers, while in supersonic and hypersonic aircraft, it may be induced by thermal stresses. The longstanding solution to such challenges was to increase the sizing of the structural members, which is counter to the ever present need to minimize weight for reasons of efficiency and performance.In this work we present some recent results in the area of reduced order modeling of post- buckled thin beams. A thorough parametric study of the response of a beam to changing harmonic loading parameters, which is useful in exposing complex phenomena and exercising numerical models, is presented. Two error metrics that use but require no time stepping of a (computationally expensive) truth model are also introduced. The error metrics are applied to several interesting forcing parameter cases identified from the parametric study and are shown to yield useful information about the quality of a candidate reduced order model.Parametric studies, especially when considering forcing and structural geometry parameters, coupled environments, and uncertainties would be computationally intractable with finite element models. The goal is to make rapid simulation of complex nonlinear dynamic behavior possible for distributed systems via fast and accurate reduced order models. This ability is crucial in allowing designers to rigorously probe the robustness of their designs to account for variations in loading, structural imperfections, and other uncertainties.

Multiobjective optimization of post-tensioned concrete box-girder road bridges considering cost, CO2 emissions, and safety

This paper presents a multiobjective optimization of post-tensioned concrete road bridges in terms of cost, CO2 emissions, and overall safety factor. A computer tool links the optimization modulus with a set of modules for the finite-element analysis and limit states verification. This is applied for the case study of a three-span continuous post-tensioned box-girder road bridge, located in a coastal region. A multiobjective harmony search is used to automatically search a set of optimum structural solutions regarding the geometry, concrete strength, reinforcing and post-tensioned steel. Diversification strategies are combined with intensification strategies to improve solution quality. Results indicate that cost and CO2 emissions are close to each other for any safety range. A one-euro reduction, involves a 2.34kg CO2 emissions reduction. Output identifies the best variables to improve safety and the critical limit states. This tool also provides bridge managers with a set of trade-off optimum solutions, which balance their preferences most closely, and meet the requirements previously defined.