Slenderness Limits Research Articles

Classification of aluminium alloy cross-sections

Cross-section classification is one of the key concepts in the design of metallic structures. Current design specifications for aluminium alloys, such as Eurocode 9 (EC9), provide clear definitions and discrete design capacities for four different classes of cross-section. On the basis of substantial, recently generated experimental and numerical data on aluminium alloy cross-sections collected from the literature, the purpose of the present study is to re-evaluate the slenderness limits that define these classes. A total of approximately 900 relevant data points have been gathered, covering stub columns, simply supported beams and continuous beams; the cross-section types include square and rectangular hollow sections (SHS/RHS) with and without internal stiffeners, I-sections, channels and angles. The members were extruded from a variety of aluminium alloy tempers with a wide range of yield and ultimate strengths. Following analysis of the available data, the slenderness limits in EC9 have been re-assessed, and new slenderness limits in the EC9 framework are proposed. In addition, the full cross-section slenderness allowing for element interaction, which is utilised in the direct strength method (DSM) and the continuous strength method (CSM) has been considered as the slenderness parameter in a new classification framework. Corresponding slenderness limits, together with a compatible effective thickness formula for Class 4 sections, are proposed. The suitability of the proposed limits has been demonstrated for the conventional design methods in EC9, as well as the alternative methods in Annexes F and H of EC9.

Slenderness limits for deflection control: A new formulation for flexural reinforced concrete elements

The first step in the design of a structure is the definition of the geometry. This process includes the definition of the depth of slabs and beams. The depth of a flexural member is often determined by control of deflections, which can only be checked in detail at an advanced stage of the project. In order to optimize the design process, it is therefore very important to choose well the span-to-depth ratio at the beginning. In order to achieve this task in an easy manner, a lower limit to the slenderness of the beams in terms of span divided by the effective depth is proposed in most major codes. However, current proposals are rather coarse and are not necessarily on the safe side. In this paper, a new formulation for the slenderness limits, based on the physics of the problem, is presented. This formulation includes the effect of the composition of the load (live load to total load ratio) as well as the possibility of using different limits to maximum deflection and considering different, more general, support conditions. It is therefore more complete and has a larger application field than current proposals.

Material properties and compressive local buckling response of high strength steel square and rectangular hollow sections

An experimental investigation into the structural performance of compressed high strength steel (HSS) square and rectangular hollow sections is described in this paper. Both hot-rolled and cold-formed HSS sections were examined. In total six S460NH and five S690QH hot-rolled section sizes and three S500MC, two S700MC and four S960QC cold-formed section sizes were tested. The experimental programme comprised tensile coupon tests on flat and corner material, measurements of geometric imperfections, full cross-section tensile tests and stub column tests. The results of the experiments presented in this paper have been combined with other available test data on high strength steel sections, and used to assess the existing design guidelines for high strength steels given in Eurocode 3. The focus has been on the material ductility requirements, the Class 3 slenderness limit for internal elements in compression and the effective width formula for Class 4 internal elements in compression.Reliability assessments of the Class 3 slenderness limit (both the current value of 42 and a proposed value of 38) and the effective width formula for Class 4 internal elements in compression were carried out. The analysis indicated that, based on the assembled test data considered in this study, and the assumptions made regarding the statistical distributions of material and geometric properties, a partial safety factor greater than unity is required for HSS. Similar findings have also recently been presented for ordinary strength steels.

Plastic and yield slenderness limits for circular concrete filled tubes subjected to static pure bending

The current slenderness limits in international design codes are often based on certain rotation capacities obtained from plastic bending tests of Concrete Filled Tubes (CFT). In the past, a plastic slenderness limit of λs=188 was obtained by the first author based on a fracture rotation limit of the steel tube. However, such limit may be questionable being brittle and insufficient for plastic design of CFT members, sub-assemblies and frames where adequate strain/deformation ductility is required. The main aims of this paper are to present (i) a new method to determine new ductile slenderness limits suitable for plastic design of structures based on the measured strains in plastic bending tests on CFT; (ii) a closed-form solution for the elastic and inelastic buckling strains of CFT under pure bending using a new simplified energy approach employing the well-known Ritz method. The critical strains obtained from such analysis were used to derive new slenderness limits for CFT; and (iii) finite element modelling of CFT and compare the experimental and numerical moment-rotation responses. The effect of concrete filling on the post-buckling strength of restrained tubes is quantified. The current design rules for unrestrained Circular Hollow Sections (CHS) in steel specifications are also compared with the restrained strength obtained from the tests. Two new compact and yield slenderness limits were derived based on the strength corresponds to the appearance of the plastic ripples during the test. The experimentally obtained and the theoretically derived slenderness limits are compared against the available limits in the design codes and standards. The newly derived compact limit of λp=79 was found in a good agreement with λp=72 specified for CFT in the ANSI/AISC 360-10 specification. However, the new yield limit of λy=150 was found considerably lower than λy=254 for CFT specified in the ANSI/AISC 360-10.

Experimental investigation of cold-formed high strength steel tubular beams

This paper presents a test program on cold-formed high strength steel tubular beams. The nominal 0.2% proof stresses of the high strength steel were 700, 900 and 1100MPa in this study. Twenty-five four-point bending tests on circular, rectangular and square hollow structural sections were conducted. Load-deformation histories and failure modes of the beams were reported. Experimental results were compared against design values calculated from European code, Australian standard and North American specification for hot-rolled and cold-formed steel structures. In addition, the test strengths were also compared with the Direct Strength Method predictions. The compactness criteria were assessed by comparing the section slenderness to the slenderness limits in codes.

Flexural behavior and ductility of hybrid high performance steel I-girders

The flexural behavior and ductility of high performance steel (HPS) girders were investigated experimentally. Thirteen simply supported I-girders of HPS 485W (nominal yield strength, fy=485MPa) flanges and webs with different steel grades including Q235 (fy=235MPa), Q345 (fy=345MPa), and HPS 485W were tested. Five additional girders considering different flange strengths of Q345, Q420 (fy=420MPa), and Q500 (fy=500MPa) were tested and compared. The test results show that the effectiveness of lateral braces is mainly influenced by the location and number, which in turn affect the expansion of plastic fields. The slenderness of element (flange or web) is the major factor affecting the flexural capacity and ductility of a girder. As the flange slenderness increases, flange local buckling occurs much earlier during the inelastic loading phase and thus reduces the girder ductility. As the web slenderness increases, the deformation in the inelastic stage decreases. Under the three-point loading, the girder appears not losing its capacity when the local element buckles. While, under the four-point loading, the flange local buckling depends on the unloading stage. When the high-strength HPS 485W steel is used for flanges, the web strength is suggested to be no lower than 345MPa (Q345 steel). Using the same geometrical dimensions, the ductility of member is decrease with the increasing of flange strength, which is mainly affected by the yield stress to ultimate stress ratio (σy/σu) of steel. Hence, more restrictive slenderness limit appears necessary for structural designs using high-strength steel.

Flexural response of aluminium alloy SHS and RHS with internal stiffeners

Tubular profiles, such as square and rectangular hollow sections (SHS and RHS) are widely used in a range of structural engineering applications. Their thin-walled nature means that local buckling is a key design concern, which is usually addressed by adhering to specified slenderness limits. An alternative approach is to employ local plate stiffeners, which can be practically achieved in extruded aluminium alloy sections through the introduction of internal cross stiffeners. The flexural behaviour and design of aluminium alloy SHS and RHS with internal stiffeners is the subject of investigation of the present paper. The primary aims of the study are to generate experimental and numerical data for these new types of cross-section in bending, as well as to assess the applicability of different approaches to their design at cross-sectional and system levels. First, an experimental investigation was performed on aluminium alloy SHS and RHS beams with internal stiffeners subjected to three-point bending, four-point bending and five-point bending (i.e. continuous beams over three supports) of three different configurations. Finite element (FE) models were also developed and validated against the presented test data. Once validated, the models were employed to generate additional structural performance data through numerical parametric studies. Comparisons are then made between the experimental/numerical capacities and the design capacities predicted using a series of international design specifications for aluminium alloy structures, as well as the continuous strength method (CSM). The design strengths predicted by the American and Australian/New Zealand specifications were found to be conservative, while improved predictions are achieved by Eurocode 9. The most accurate and consistent predictions are obtained using the CSM, which was able to capture the significant observed strain hardening at the cross-sectional level and moment redistribution at the global system level.

Modified slenderness limits for bending resistance of longitudinally stiffened plate girders

In this study, the slenderness limits for flanges and webs were examined to rationally evaluate the flange local buckling (FLB) strength of longitudinally stiffened I-girders. It is known that the bending resistance of a plate girder greatly increases when the web is longitudinally stiffened. This additional strength can be attributed to the fact that the stiffened web provides improved restraint to the rotation of the compression flanges as well as web bend-buckling strength. This study conducted a series of numerical analyses and it was found that the American Association of State Highway and Transportation Officials' (AASHTO's) load and resistance factor design (LRFD) requirements provide highly conservative estimates of the FLB strength of longitudinally stiffened plate girders, especially in noncompact sections. The reason is that the buckling coefficient and corresponding slenderness limit for a noncompact flange is determined based on the bending resistance of an unstiffened girder. Therefore, this study conducted a series of eigenvalue analyses in which the slenderness ratios of flanges and webs were considered. As a result, a buckling coefficient equation that reasonably reflects the interaction of the stiffened web and flange was derived. Based on the results of the analyses, this study proposed modified slenderness limits for the noncompact web and noncompact flange of a stiffened girder. It was verified that the bending resistance of longitudinally stiffened girders could be rationally estimated on the basis of the AASHTO LRFD specifications only by replacing the slenderness limits with the proposed formulas.

Behavior of concentrically loaded geopolymer-concrete circular columns reinforced longitudinally and transversely with GFRP bars

The behavior of concentrically loaded geopolymer-concrete circular columns reinforced longitudinally and transversely with glass–fiber-reinforced-polymer (GFRP) bars was investigated. Six full-scale short columns (L/r=8) were cast: one column without transverse reinforcement; three columns with circular hoops spaced at 50mm, 100mm, and 200mm on centers; and two columns with spirals spaced at 50mm and 100mm on centers. In addition, two slender columns (L/r=16) transversely reinforced with hoops and spirals both spaced at 100mm on centers were fabricated. Based on the experimental results, the GFRP bars contributed an average of 7.6% to the overall capacity of the tested columns. The hoop- and spiral-confined slender columns failed at a load equal to 66% and 82%, respectively, of the strength of their counterpart short columns. Irrespective of the tie configuration, the columns with higher volumetric ratios showed better compressive behavior than those with lower volumetric ratios. The ductility and confinement efficiency of the spiral-confined columns were higher than that of their counterpart hoop-confined columns. The tested columns yielded relatively superior compression performance compared to OPC-based concrete columns reinforced with GFRP bars and ties. Further studies dealing with the behavior and slenderness limit in GFRP-reinforced geopolymer concrete slender columns are recommended to increase its uptake in the construction industry.

NEW PROPOSED FORMULA FOR R.C. COLUMN SLENDERNESS RATIO LINKED WITH CONCRETE COMPRESSIVE STRENGTH AND STRESSES ON COLUMNS

Many codes for reinforced concrete columns design adopted by various countries are in fact built upon empirical formulae. However, Egyptian code adopted slenderness limit 30 for braced columns and 23 for unbraced columns. The mentioned values in codes do not change with different variables such as concrete compressive strength and stresses on columns as well as increases in column dimensions. This paper aimed to study the influences of parameters that affected slenderness ratio such as concrete compressive strength and stresses on columns. Also, the present study aimed to find a formula for slenderness ratio linked with concrete compressive strength and stresses on column ( . The results of the proposed formula fit very well with ECP 203 code and showed that an increase in slenderness limits with increase compressive concrete strength as well as decrease stresses on columns. Thus, the effect of compressive concrete strength and stresses on columns should be taken into consideration in calculating the slenderness limits.

Element interaction of cold-formed stainless steel cross-sections subjected to major axis bending

Numerical investigation of cold-formed normal and high strength stainless steel square and rectangular hollow sections subjected to major axis bending is presented in this paper. A non-linear finite element model which includes geometric and material non-linearities was developed and verified against experimental results. The material properties of the flat and corner portions of the specimen sections were carefully incorporated. It was shown that the finite element model closely predicted the failure modes, ultimate moments and deflections of the tested specimens. The model was then used for an extensive parametric study to investigate the interaction effects of constituent plate elements on Class 3 and 2 slenderness limits and cross-section moment capacities of cold-formed stainless steel square and rectangular hollow sections in bending.The numerical strengths predicted from this study together with the experimental results were compared with the theoretical elastic and plastic bending moments. It was shown that the element interaction surely influences the flexural behavior of cold-formed stainless steel square and rectangular hollow sections especially for slender cross-sections. The design provisions on Class 3 and 2 slenderness limits and effective width equations specified in ASCE Specification, EC3 Code and proposed by Gardner and Theofanous are not suitable for square and rectangular hollow sections in bending since they did not take into consideration interaction effects of constituent plate elements. Hence, the new Class 3 and Class 2 slenderness limits and the cross-section moment capacity design equations are proposed in this study based on the whole cross-section response, which carefully consider the interaction effects of constituent plate elements.

Recommendations for verifying lateral stability of precast beams in transitory phases

Abstract This paper presents recommendations for security check of precast beams in transitory phases, compare results of parametric analyzes with national and international code recommendations and confront the formulations used for the calculation of critical load of lateral instability. In transport and lifting phases, precast beams are susceptible to loss lateral stability because the established supports provides little restriction to the element rotate on its principal axis and move laterally. To recommend limits of slenderness, parametric analysis are performed using formulations based on bifurcacional instability, including eigenvalue problems with the finite element method. The results show that the safety limits for I beams and rectangular beams are different. For the analyzed cases and with reference to beam slenderness equation used by fib Model Code [13], the limit determined for rectangular beams would be 85 and for I beams 53, which could be taken as 50, as recommended by the code. Within the analyzed cases of I beams, only the fib Model Code [13] recommendation attend the slenderness limit for transitory phases.

Cross-sectional optimization of cold-formed steel channels to Eurocode 3

Cold-formed steel structural systems are widely used in modern construction. However, identifying optimal cross section geometries for cold-formed steel elements is a complex problem, since the strength of these members is controlled by combinations of local, distortional, and global buckling. This paper presents a procedure to obtain optimized steel channel cross-sections for use in compression or bending. A simple lipped C-shape is taken as a starting point, but the optimization process allows for the addition of double-fold (return) lips, inclined lips and triangular web stiffeners. The cross-sections are optimized with respect to their structural capacity, determined according to the relevant Eurocode (EN1993-1-3), using genetic algorithms. All plate slenderness limit values and all limits on the relative dimensions of the cross-sectional components, set by the Eurocode, are thereby taken into account as constraints on the optimization problem. The optimization for compression is carried out for different column lengths and includes the effects of the shift of the effective centroid induced by local buckling. Detailed finite element models are used to confirm the relative gains in capacity obtained through the optimization process.

Experiments on cold-formed ferritic stainless steel slender sections

The usage of stainless steel in construction has been increasing owing to its corrosion resistance, aesthetic appearance and favourable mechanical properties. The most common stainless steel grades used for structural applications are austenitic steels. The main drawback of these grades relies on their nickel content (around 8–10%), resulting in a relatively high initial material cost. Other stainless steel grades with lower nickel content such as the ferritic steels offer the benefits of stainless steels in terms of functional qualities and design but within a limited cost frame. Hence, ferritic stainless steels may be a viable alternative for structural applications. Given the fact that little experimental information on ferritic stainless steels is currently available, the purpose of this investigation is to report a series of material and cross-section tests on ferritic grade EN 1.4003 (similar to 3Cr12) stainless steel square and rectangular hollow sections to enable a better understanding of their material response and structural performance. Four different cross-section geometries have been tested under pure compression and in-plane bending. Measurements of geometric imperfections and material properties are also presented. The obtained test results are used to assess the adequacy of the slenderness limits and effective width formula given in EN 1993-1-4 to ferritic stainless steels, those proposed by Gardner and Theofanous and Zhou et al. design approach.

Material and local buckling response of ferritic stainless steel sections

An investigation into the material response and local buckling behaviour of ferritic stainless steel structural cross-sections is presented in this paper. Particular attention is given to the strain hardening characteristics and ductility since these differ most markedly from the more common austenitic and duplex stainless steel grades. Based on collated stress-strain data on ferritic stainless steel, key aspects of the material model given in Annex C of EN 1993-1-4 [1] were evaluated and found to require adjustment. Proposed modifications are presented herein.The local buckling behaviour of ferritic stainless steel sections in compression and bending was examined numerically, using the finite element (FE) package ABAQUS. The studied section types were cold-formed square hollow sections (SHS), rectangular hollow sections (RHS) and channels, as well as welded I-sections. The models were first validated against experimental data collected from the literature, after which parametric studies were performed to generate data over a wide range of section geometries and slendernesses. The obtained numerical results, together with existing experimental data from the literature were used to assess the applicability of the slenderness limits and effective width formulae set out in EN 1993-1-4 [1] to ferritic stainless steel sections.The comparisons of the generated FE results for ferritic stainless steel with the design provisions of EN 1993-1-4 [1], highlighted, in line with other stainless steel grades, the inherent conservatism associated with the use of the 0.2% proof stress as the limiting design stress. To overcome this, the continuous strength method (CSM) was developed as an alternative design approach to exploit the deformation capacity and strain hardening potential of stocky cross-sections. An extension of the method to ferritic stainless steels, including the specification of a revised strain hardening slope for the CSM material model, is proposed herein. Comparisons with test and FE data showed that the CSM predictions are more accurate and consistent than existing provisions thus leading to significant material savings and hence more efficient structural design.

Cross-section classification for cold-formed and built-up high strength carbon and stainless steel tubes under compression

This paper presents an assessment on the current yield slenderness limits under axial compression for cold-formed and built-up high strength carbon and stainless steel hollow sections. The definition of high strength steel in this paper refers to the materials with nominal yield strength or 0.2% proof stress which is greater than 700MPa (Eurocode 3: Part 1-12) for carbon steel and greater than 480MPa (Eurocode 3: Part 1-4) for stainless steel. A total of 147 stub column experimental results was compiled and assessed. Companion 46 finite element simulations were performed in this study and the results concurred with the experimental observations. The structural performance data showed that the current slenderness limits adopted in Eurocode 3, American and Australian Standards for both carbon and stainless steel under axial compression can be safely adopted for the high strength steel counterparts. New slenderness parameters and limits were also proposed to improve the design efficiency.

Theoretical and Experimental Study of Slender Concrete Columns Reinforced with GFRP Bars

This paper studies slender concrete columns reinforced with GFRP bars. For this target an experimental work was performed utilizing five such columns. Based on the experimental work the slenderness limit of 22 for unbraced steel-reinforced concrete columns as given by the ACI318-11 code was suggested to be little changed to 23.3 for GFRP reinforced concrete columns. The provisions of the ACI318-11 code for slender concrete columns reinforced with steel bars are reviewed. New expression from the literature for the effective flexural stiffness EI of slender concrete columns reinforced with GFRP bars is used to calculate the moment magnification factor and a procedure for plotting the interaction diagrams of the columns of this research is described.

Experimental investigation of slender circular RC columns strengthened with FRP composites

The relevant design code provisions for Fiber Reinforced Polymer (FRP) strengthened RC columns are restricted to the short RC columns strengthened with FRP jackets. These design provisions are thus strictly not applicable to those long RC columns where second-order/slenderness effect is substantial. In the present study, the effectiveness of hoop and longitudinal Carbon FRP (CFRP) wraps in reducing the lateral deflections and improving the strength of slender circular RC columns has been studied experimentally. A total of 12 small-scale circular RC columns of 150mm diameter were cast in three groups, each group containing 4 columns of the same height. The columns of the first group belonged to short columns of 600mm height, whereas the columns of second and third groups of 900 and 1200mm heights respectively represented slender columns. Columns of each group had one control and 3 strengthened columns. The strengthened columns were prepared using three different strengthening schemes. In the first strengthening scheme, the columns were wrapped using a single layer of hoop CFRP sheet, whereas other strengthening schemes employed 2 and 4 longitudinal CFRP sheets in addition to one layer of hoop CFRP wrap. The columns were tested under monotonic compression with initial eccentricity of 25mm. In general, CFRP-strengthening improves the strength and ductility of slender RC columns substantially. The test results indicate that the CFRP hoop wraps provide confinement to concrete and lateral support to the longitudinal fibers and thus increase the strength of the RC columns. In slender columns, the effect of longitudinal FRP fibers in carrying the load in post-yielding stage is more significant than hoop FRP fibers. The existing ACI expression of slenderness limit for RC columns was extended to propose a simple analytical equation for the slenderness limit of FRP-strengthened RC columns. The proposed expression for slenderness limit is valid for both RC and FRP-confined columns and matches well with the experimentally observed slenderness limit.

Slenderness limit for SSTT-confined HSC column

Due to the confinement effects, Steel-Straps Tensioning Technique (SSTT) can significantly enhance the strength and ductility of high-strength concrete (HSC) members (Moghaddam et al. 2008). However, the enhancement especially in strength may result in slender member and more susceptible to instability (Jiang and Teng 2012a). This instability is particularly significant in HSC member as it inherent the brittle nature of the material (Galano et al. 2008). The current slenderness limit expression used in the design is mainly derived from the experiment and analysis results based on Normal strength concrete (NSC) column and therefore the direct application of these slenderness limit expressions to the HSC column is being questioned. Besides, a particular slenderness limit for the SSTT-confined HSC column which incorporated the pre-tensioned force and multilayers effects is not yet available. Hence, an analytical study was carried out in the view of developing a simple equation in order to determine the slenderness limit for HSC column confined with SSTT. Based on the analytical results, it was concluded that the existing slenderness limit expressions used in the design are appropriate for neither HSC columns nor SSTTconfined HSC columns. In this paper, a slenderness limit expression which has incorporated the SSTTconfinement effects is proposed. The proposed expression can also be applied to unconfined HSC columns.

A closed-form solution for elastic buckling of thin-walled unstiffened circular cylinders in pure flexure

To date, despite the significant development in the field of structural mechanics, there still remains a paradox in the solutions available for a classical shell buckling problem. The difference in strength between a cylindrical shell under uniform axial compression and that under pure bending is not quite well investigated. This lack of research is reflected in the wide variations in the elastic bending strength and the slenderness limits given in current international design standards. The discrepancies in the available classical solutions and hence the design rules have initiated the current research. The main aim of this paper is to present a closed-form solution for the elastic buckling strength of unstiffened circular cylinders under pure bending using a new simplified energy approach employing the well-known Ritz method. Two types of analyses are presented for cylinders with large (D/t>200) and medium (100<D/t<200) diameter-to-thickness ratios. A unique testing rig was used to experimentally verify the new theory using a Moiré fringe film. The theoretical results are compared against the available and present test results and the existing classical solutions. The current design rules for thin-cylinders in international steel specifications are also compared, and the newly derived design curve is proposed which was found in a good agreement with the available test results.