Effective Flexural Rigidities Research Articles

Multi-objective topology optimization and structural analysis of periodic spaceframe structures

Reduction of structural weight provides significant benefits in many engineering applications. While methods to optimise structural shape and topology of both continuous solids and discrete frame structures have existed for a while, the advent of additive layer manufacturing processes has enabled more complex geometries to be feasible. In this paper, a periodic spaceframe structure is designed for minimum mass and maximum effective flexural and torsional rigidities. A method of parametrising the spaceframe through its constituent unit cells is proposed, and Genetic Algorithm (GA) multi-objective optimisation is used to optimise its topology, size and geometry as a generic structure. The superior performance of the topology optimised periodic spaceframe is highlighted in terms of structural rigidity, large deformation capability, buckling and vibrational modal analysis in compare to equivalent beam structures of identical weight and comparable domain. The results show that the proposed method can effectively generate lightweight substitute structures of great mechanical performance in many beam structures applications, such as: aircraft wing spars. The periodic spaceframe is applied into a conventional aircraft wing structure to demonstrate the possibilities of promoting weight saving in the design of civil aircraft wings.

Effective flexural rigidities for RC beams and columns with steel fiber

Influences of different variables that affect the effective flexural rigidity of reinforced concrete (RC) members are not considered in the most seismic codes. Furthermore, in the last decades, the application of steel fibers in concrete matrix designs has been increased, requiring development of an accurate analytical procedure to calculate the effective flexural rigidity of steel fiber reinforced concrete (SFRC) members. In this paper, first, a nonlinear analytical procedure is proposed to calculate the SFRC members\' effective flexural rigidity. The proposed model\'s accuracy is confirmed by comparing the results obtained from nonlinear analysis with those recorded from the experimental testing. Then a parametric study is conducted to investigate the effects of different parameters such as varying axial load and steel fiber are then investigated through moment-curvature analysis of various SFRC (normal-strength concrete) sections. The obtained results show that increasing the steel fiber volume percentage increases the effective flexural rigidity. Also it\'s been indicated that the varying axial load affects the effective flexural rigidity. Lastly, proper equations are developed to estimate the effective flexural rigidity of SFRC members.

Influence of between-joists bridging elements on static and dynamic response of wood joisted floors

Excessive vertical vibration of lightweight floors caused by walking or similar activities can be annoying to building occupants. Previous studies have shown that the vibrational performance of floors having wood joists is enhanced by installing one or more lines of bridging elements that form transverse spines in the across-span direction. This paper presents an experimental study that defined relationships between the effective flexural rigidity of such spines and response characteristics of wood joisted floors. Behaviors of spines having both normal and special types of bridging elements were investigated, with the special types of elements able to create a broad range of spine flexural rigidities. It was found that all types of transverse bridging spines have a small influence on fundamental natural frequency but can increase higher order modal frequencies considerably. Relatively stiff types of transverse bridging spines reduced static deflections caused by concentrated gravity forces considerably. An experimental method is presented for quantifying effective flexural rigidities of transverse bridging spines. Suggestions are made concerning application of the research findings in vibration serviceability design of lightweight wood joisted floors.

Time‐dependent serviceability behavior of reinforced concrete beams: Partial interaction tension stiffening mechanics

The partial interaction (PI) mechanics of tension stiffening governs crack widening and the consequential deflection of reinforced concrete (RC) beams and, therefore, is essential in the serviceability design of RC structures. Both behaviors are dependent on the PI global load slip behavior of the tensile reinforcement relative to the surrounding concrete at a crack face, which in turn is dependent on the local PI material bond‐stress/slip relationship between the reinforcement and adjacent concrete. Due to the complexity of PI theory, it has been difficult to directly incorporate PI mechanics into design procedures. Consequently, empirical approaches to quantify effective flexural rigidities and crack widths are commonplace. While empirical approaches can be derived relatively simply for short‐term loading, the influence of concrete shrinkage and creep is difficult to incorporate. In this paper, pure mechanics‐based PI solutions to describe the crack spacing and tension stiffening behavior of RC beams subjected to long‐term loads are derived. They only require material properties and clearly show the interaction among shrinkage, creep and bond properties. They are simple enough to use directly in design and will help in the experimental derivation of the material properties. The results of these PI analyses are used in a companion paper to develop design rules for the time‐dependent serviceability deflection of RC beams and compared with test results.

Long‐ and short‐term serviceability behavior of reinforced concrete beams: Mechanics models for deflections and crack widths

Analysis and design of reinforced concrete (RC) flexural members at the serviceability limit state is complicated by the localized bond slip or partial interaction (PI) between tensile reinforcement and the surrounding concrete. Localized slip is responsible for crack formation, crack widening, and tension stiffening and is further complicated by the time‐dependent nature of concrete, which results in reduced tension stiffening and widening of cracks. Analysis techniques commonly applied in design are typically formulated on the assumption of full interaction between the reinforcement and the surrounding concrete. Hence, they do not simulate the behaviors observed in practice, but rather they correlate analytical predictions with experimental observations through the use of empiricisms, such as effective flexural rigidities and statistically derived expressions for crack widths. In this paper, it is shown how PI theory can be directly incorporated into elastic analysis procedures to develop simple expressions for flexural rigidities and crack widths under both instantaneous and sustained loading. The proposed approach should simplify not only the design process but also the development of design guidelines and material models, as it provides a simple rational method for investigating the time‐dependent flexural behavior of RC members.

The dynamics of multiscale, multiphysics faults: Part I - The long-term behaviour of the lithosphere

Here we used a physics-based geomechanics approach to show that the long-term strength of the lithosphere is controlled by multiple steady states that arise as a function of significant material weakening at and above a critical value of local dissipation. Ultimately, the long-term strength of the lithosphere is governed by whether it can reach this critical value to form plate boundaries and intraplate lithosphere scale fault zones. We identify two different regimes: (1) post-yield steady state creep strength, (2) post-yield localization strength. Accordingly, in regime (1) the plate tectonic loading is insufficiently large to breach the critical local dissipation needed to reduce the overall long-term strength of the lithosphere. This regime is described in classical thermal-mechanical geodynamic models based on material values derived from laboratory experiments. The approach leads to the “jelly-sandwich” model of long term lithosphere strength. The constitutive models predict that the strongest portion of the lithosphere is the upper mantle followed by the upper crust; both can support mountain ranges for long geological times and lead to large effective elastic flexural rigidities. For regime (2) we identify two possible responses. One in which the plate strength is reduced by chemical and thermal feedback processes. And the other one where crystal plastic deformation causes micron scale Fick diffusion controlled micro-shears, that in turn release fluid to feed millimetre-metre scale hydraulic subshears (Darcy diffusion controlled) to finally drive kilometre scale Fourier diffusion controlled plate boundary shears. Under these special conditions the mantle can become extremely weak such as in the “crème brûlée” model. We conclude that both models have their validity. They reveal two different end members of dynamic thermodynamic feedbacks at different critical dynamic loadings and chemical composition.

Simulating reinforced concrete members. Part 1: partial interaction properties

Reinforced concrete members are extremely complex under loading because of localised deformations in the concrete (cracks, sliding planes) and between the reinforcement and concrete (slip). An ideal model for simulating behaviour of reinforced concrete members should incorporate both global behaviour and the localised behaviours that are seen and measured in practice; these localised behaviours directly affect the global behaviour. Most commonly used models do not directly simulate these localised behaviours that can be seen or measured in real members; instead, they overcome these limitations by using empirically or semi-empirically derived strain-based pseudo properties such as the use of effective flexural rigidities for deflection; plastic hinge lengths for strength and ductility; and energy-based approaches for both concrete softening in compression and concrete softening after tensile cracking to allow for tension stiffening. Most reinforced concrete member experimental testing is associated with deriving these pseudo properties for use in design and analysis, and this component of development is thus costly. The aim of the present research is to reduce this cost substantially. In this paper, localised material behaviours and the mechanisms they induce are described. Their incorporation into reinforced concrete member behaviour without the need for empirically derived pseudo properties is described in a companion paper.

Effective flexural rigidities for ordinary reinforced concrete columns and beams

SUMMARY Current effective flexural rigidities proposed for use in design and analyses of reinforced concrete structures have been examined. The level of accuracy in the estimation of section rigidity plays a very important role in determining realistic values for the structural stiffness and hence the seismic forces imposed. The most significant parameters influencing the effective rigidity, which reflects the effect of cracking as well as the theoretical yielding of reinforced concrete sections, have been determined through comprehensive moment–curvature analyses of various reinforced concrete sections. The geometry, axial load level, concrete strength and the amount of compression and tension reinforcements have been identified as the most important factors affecting the effective rigidity. New relationships that are believed to represent the effective stiffness of reinforced concrete members have been adequately developed. Efficiency of the code-specified and proposed values has been investigated through experimental data and parametric studies. It has been observed that the relationships developed herein have provided the most accurate results in prediction of the effective rigidity of older-type structural elements with low-strength and longitudinal reinforcement ratio. They can be used for assessment of ordinary buildings. Copyright © 2012 John Wiley & Sons, Ltd.

Effect of loading types and reinforcement ratio on an effective moment of inertia and deflection of a reinforced concrete beam

In the design of reinforced concrete structures, a designer must satisfy not only the strength requirements but also the serviceability requirements, and therefore the control of the deformation becomes more important. To ensure serviceability criterion, it is necessary to accurately predict the cracking and deflection of reinforced concrete structures under service loads. For accurate determination of the member deflections, cracked members in the reinforced concrete structures need to be identified and their effective flexural and shear rigidities determined. The effect of concrete cracking on the stiffness of a flexural member is largely dependent on both the magnitude and shape of the moment diagram, which is related to the type of applied loading. In the present study, the effects of the loading types and the reinforcement ratio on the flexural stiffness of beams has been investigated by using the computer program developed for the analysis of reinforced concrete frames with members in cracked state. In the program, the variation of the flexural stiffness of a cracked member has been obtained by using ACI, CEB and probability-based effective stiffness model. Shear deformation effect is also taken into account in the analysis and the variation of shear stiffness in the cracked regions of members has been considered by employing reduced shear stiffness model available in the literature. Comparisons of the different models for the effective moment of inertia have been made with the reinforced concrete test beams. The effect of shear deformation on the total deflection of reinforced concrete beams has also been investigated, and the contribution of shear deformation to the total deflection of beam have been theoretically obtained in the case of various loading case by using the developed computer program. The applicability of the proposed analytical procedure to the beams under different loading conditions has been tested by a comparison of the analytical and experimental results, and the analytical results have been found in good agreement with the test results.

Moment resistance and flexural rigidity of lapped connections in multi-span cold-formed Z purlin systems

This paper reports an experimental investigation on the structural behavior of lapped connections over the internal supports in multi-span cold-formed steel Z purlin systems. The moment resistance and effective flexural rigidity of two typical connection configurations, i.e. web bolts plus self-drilling screws at both flanges or at the top flange only, are considered. In the moment resistance test, a total of 30 specimens with different connection configurations and connection lengths are tested, from which, after careful calibration, the assessment of moment resistance of such lapped connections as well as several useful design suggestions are proposed. In the effective flexural rigidity test, full-scale two-spanned purlins, with two external pin ends and one lapped connection over the internal support, are adopted. Lapped connections with different configurations and two practical connection lengths of 10% and 20% of span are concerned. Based on test results, the effective flexural rigidities of lapped connections studied in this paper are proposed, respectively, for the stress analysis and deflection analysis.

Effective characteristics of corrugated plates

Corrugated plates are widely used in modern constructions and structures, because they, in contrast to plane plates, possess greater rigidity. In many cases, such a plate can be modeled by a homogeneous anisotropic plate with certain effective flexural and tensional rigidities. Depending on the geometry of corrugations and their location, the equivalent homogeneous plate can also have rigidities of mutual influence. These rigidities allow one to take into account the influence of bending moments on the strain in the midplane and, conversely, the influence of longitudinal strains on the plate bending [1]. The behavior of the corrugated plate under the action of a load normal to the midsurface is described by equations of the theory of flexible plates with initial deflection. These equations form a coupled system of nonlinear partial differential equations with variable coefficients [2]. The dependence of the coefficients on the coordinates is determined by the corrugation geometry. In the case of a plate with periodic corrugation, the coefficients significantly vary within one typical element and depend on the values of local variables determined in each of the typical elements. There is a connection between the local and global variables, and therefore, the functions of local coordinates are simultaneously functions of global coordinates, which are sometimes called rapidly oscillating functions [3].

Three dimensional analysis of reinforced concrete frames with cracked beam and column elements

In the design process of reinforced concrete buildings, the serviceability limit state for lateral drift is an important design criterion that must be satisfied to prevent large second-order P –delta effects. Deformation control is also important to ensure the serviceability requirements. For accurate determination of the deflections, cracked members in the reinforced concrete structure need to be identified along with their effective flexural and shear rigidities. In this study, a computer program has been developed using the rigid diaphragm model for the three dimensional analysis of reinforced concrete frames with cracked beam and column elements. ACI, CEB and probability-based effective stiffness models are used for the effective moment of inertia of the cracked members. In the analysis, shear deformations, which can be large following crack developments, are taken into account and the variation of the shear rigidity due to cracking is considered by reduced shear stiffness models. The computer program is based on an iterative procedure which is subsequently verified experimentally through a series of reinforced concrete frame tests. A parametric study is also carried out on a four-story, three-dimensional reinforced concrete frame. The iterative analytical procedure can provide an accurate and efficient prediction of deflections of reinforced concrete structures due to cracking under service loads. The most significant feature of the proposed procedure is that the variations in the flexural stiffness of beams and columns can be observed explicitly. The procedure is efficient from the viewpoint of computational time and convergence rate, and is also more direct than the finite element method.

Structural behavior of lapped cold-formed steel Z sections with generic bolted configurations

In order to improve the buildability of cold-formed steel structures, a series of research and development projects have been undertaken by the authors to study the structural behavior of bolted moment connections between cold-formed steel sections. As one of the major applications of cold-formed steel sections in building construction is modern roof structures with multi-span purlin systems, an extensive experimental and theoretical investigation on the structural behavior of lapped moment connections between cold-formed steel Z sections was carried out. Both generic configurations with high structural efficiency were adopted, namely, Config. W4 and W6, in which only the webs of the sections were bolted together for easy installation. Moreover, an analysis and design method was proposed to assess both the moment resistance and the effective flexural rigidity of the generic lapped connections. The research work aims to provide understanding on the structural behavior of cold-formed steel Z sections with lapped connections, and hence, to develop a set of rational design rules for multi-span purlin systems with overlaps. In order to verify the applicability of the proposed analysis and design method for other commonly adopted connection configurations, namely, Config. W2F2 and W4F2 where bolts were installed to both section webs and flanges within the lapped connections, an experimental and theoretical investigation on a total of 12 one point load tests on lapped cold-formed steel Z sections were carried out. Among all tests, section failure under combined bending and shear at the ends of lap was found to be critical while twisting of the lapped Z sections was apparent throughout the entire deformation ranges. In general, the structural behavior of lapped connections with Config. W2F2 and W4F2 was found to be similar to those with Config. W4 and W6. Moreover, the formulation of the proposed analysis and design method was modified to accommodate the presence of the flange bolts, and back analysis of the lapped Z sections against combined bending and shear using the proposed method was performed. After careful calibration against test data, the method was shown to be structurally adequate and efficient for lapped connections with Config. W2F2 and W4F2. Moreover, simple design expressions were also proposed for the evaluation of effective flexural rigidities of the lapped connections. Comparison on the structural behavior between the two sets of connection configurations was also presented.

Analytical prediction on deformation characteristics of lapped connections between cold-formed steel Z sections

This paper presents an analytical method to predict the deformation characteristics of lapped connections between cold-formed steel Z sections. Based on the test results of typical lap shear tests on bolted fastenings between cold-formed steel strips of various strengths and thicknesses, a normalized bearing deformation curve was established. The curve was then adopted to predict the deformation characteristics of lapped connections between cold-formed steel sections where deformations due to global bending and shear actions as well as local bearing were significantly. The predicted deformation characteristics of a total of 26 lapped Z sections with different section sizes and connection configurations up to failure were then compared with the measured results, and comparison is found to be highly satisfactory. Moreover, an empirical formula was also proposed for the moment rotation relationship of lapped connections at large deformations after failure under combined bending and shear. The proposed method is considered to be very important in assessing the structural behaviour of multi-span purlin systems with lapped Z sections, in particular, in determining the effective flexural rigidities of lapped connections over internal supports. Such design data is essential for the establishment of internal force distribution along the entire lengths of the purlin members. Hence, the co-existing shear forces and moments at the ends of lap of the lapped connections are obtained accurately to check against section failure under combined bending and shear.

Stability of Anisotropic Laminated Rings and Long Cylinders Subjected to External Hydrostatic Pressure

The problem of buckling of rings under external pressure has attracted interest since the late 1950s; however, the formulations developed, to date, to obtain the critical pressure are limited to special cases of orthotropic laminated construction. In this work, analytical and numerical treatments are carried out to provide results on the buckling of thin and moderately thick anisotropic rings and long cylinders. A generalized closed-form analytical formula for the buckling of thin anisotropic laminated rings is developed. Standard energy-based formulation and classical lamination theory are used to obtain the equilibrium equations assuming an intermediate class of deformation. The constitutive equations are statically condensed, in terms of the ring’s boundary conditions, to produce the effective axial, coupling, and flexural rigidities. In addition, a three-dimensional (3D) tube finite-element model is developed for nonlinear analysis of anisotropic laminated composite rings or long cylinders. The elemen...

Analysis and design of lapped connections between cold-formed steel Z sections

This paper presents an analysis and design method for lapped connections between cold-formed steel Z sections after careful calibration against test data obtained from a total of 26 one point-load tests on lapped connections. Based on the experimental observations on the lapped connection tests where combined bending and shear is always critical in the cross-sections at the end of laps, an analysis method is proposed to evaluate all the internal forces within the lapped connections. Once the co-existing moments and shear forces in the lapped connections are evaluated, the critical sections are readily checked against combined bending and shear using codified design rules. Moreover, design expressions are also proposed for the evaluation of effective flexural rigidities of lapped connections with various bolt configurations against practical lap length to section depth ratios. Consequently, the structural behaviour of lapped sections between cold-formed steel Z sections in terms of strength and stiffness is quantified rationally for general design. The research project aims to provide understanding to the structural performance of lapped connections between cold-formed steel Z sections, and hence, to develop a set of rational design rules for multi-span purlin systems with overlaps in modern roof construction.

BUCKLING OF THIN LAMINATED ORTHOTROPIC COMPOSITE RINGs/LONG CYLINDERS UNDER EXTERNAL PRESSURE

The buckling of isotropic rings under external pressure has attracted the interest of researchers since late 1950s. The formula for critical fluid buckling pressure of thin rings is very well known. This formula was directly extended to account for homogeneous orthotropic rings as well. The buckling of orthotropic cylindrical shells was also a subject of interest since the 1960s. However, the formulations developed, to date, require numerical solutions to obtain the critical pressure. In this work, a generalized closed form analytical formula for the buckling of thin orthotropic multi-angle laminated rings/long cylinders is developed. Standard energy based formulation is used to express the kinematics and equilibrium equations. Classical lamination theory is implemented to introduce the constitutive equations of thin shells. These equations are statically condensed, in terms of the ring's boundary conditions, to produce effective axial, coupling and flexural rigidities for the cases of rings and long cylinders. The critical buckling pressure may be calculated by hand using the derived equation in terms of these effective elastic rigidities. Comparisons are made with some existing results. Parametric studies are conducted to compare the present results with those of the buckling equations implemented by design standards. Various fiber orientations and stacking sequences are considered.

The isostatic compensation of the Colorado Plateau-Basin and Range transition zone in central Utah

The western margin of the Colorado Plateau in central Utah is similar in tectonic style to the adjacent Basin and Range Province and appears to be a transition between the two provinces. It is important to understand the isostatic compensation of the topography, the source of the associated gravity anomalies, and the regional mechanical structure since these variations can be linked to the mechanism controlling the uplift of the Colorado Plateau and adjacent regions. In this paper the isostatic response function (admittance function) technique is used to study the isostatic compensation of the Colorado Plateau-Basin and Range transition zone in central Utah and determine the applicability of this technique in the study of small geologically complex areas. The observed admittance, determined from the topography and free air gravity anomaly, had small errors for the long wavelength estimates, small (nearly zero) phase angles, and high coherence. This suggests that most of the energy in the power spectrum of the free air anomaly is correlated with the gravity fields of the topography and its compensation. The residual gravity anomaly based on the observed admittance is small except in the region of the Wasatch Front where the presence of a fairly symmetric high-low-high trend suggests compensated crustal or lithospheric structures not correlated to the topography. The observed admittance function is compared to three simple isostatic models: a surface loaded continuous elastic plate, a subsurface loaded continuous elastic plate, and a Pratt model of local compensation. It was found that the best fit elastic plate models defined similar results in spite of differing assumptions as to the form of the loading. These models both had best fit effective flexural rigidities of 10 20 Nm and Moho depths of approximately 35 km. The calculated residual gravity anomalies based on these models are nearly identical to the residual anomaly of the Pratt model with a compensation depth of 65 km. The residual anomaly of all three models reproduced the trends and magnitude of the residual anomaly based on the observed admittance. It is suggested that this local compensation is due to subsurface laterally varying thermal density anomalies. These results differ from previous isostatic rebound studies of ancient Lake Bonneville, which give larger magnitude effective flexural rigidities. These differences can possibly be attributed to differing time scales of loading between Lake Bonneville and the long term thermal loading of the Basin and Range extension and associated volcanism or the overall simplicity of the considered models. Care should be exercised in the interpretation of admittance functions determined across transition zones between adjacent tectonic provinces. Depths of compensation and flexural rigidities can represent averaged values. All additional geophysical and geologic information should be considered in the interpretations.