Tangent Modulus Theory Research Articles

Behaviour of short and long columns made from bamboo scrimber subjected to uniaxial compression

Engineered bamboo has been getting recognition as a forerunner in terms of desirable mechanical properties and a sustainable construction material. One of the most widespread variants is bamboo scrimber, also known as Parallel Bamboo Strand Lumber (PBSL). This paper presents a holistic investigation into the uniaxial compressive capacity of short bamboo scrimber columns in terms of failure mode, size effect, aspect ratio and failure strain. The studies in focus have been collated from published articles in literature. Furthermore, intermediately slender to long columns have been investigated with respect to the most common international timber standards and, elastic and inelastic theories. It is shown that the dominant failure mode affects the ultimate strength, especially a failure mode called kinking/local buckling. A size effect was observed and discussed with respect to the failure modes. In addition, a tangent modulus theory was proposed from a constitutive model called the Full-range Ramberg Osgood model that could better predict the critical buckling stress of intermediate-long columns over the conventional theory. Moreover, a modified AS1720 design approach was proposed to accommodate the inherent column behaviour of bamboo scrimber. Finally, a modified Newlin-Gahagan approach was proposed which produced consistent predictions for the experimental buckling stress for columns with varying slenderness across four different studies.

Experimental studies on glubam columns under axial compression

Glubam is a new type of structural material or bamboo based glulam. Understanding the compressive performance of glubam columns under axial compression is crucial for its application in construction field. Experimental and analytical investigations were conducted to study the mechanical performance of 45 columns fabricated by two types of glubam with different slenderness ratios. The failure modes in terms of various slenderness ratios were reported and evaluated: strength failure was the main failure mode of the short columns while buckling was typically observed for long and slender columns. For the intermediately slender columns, buckling was also accompanied by the material damage. The experimental results were compared with the column design curves specified in Chinese, American and European timber structural design codes for glued laminated timber, showing all the codes can provide adequately conservative prediction to the test results. Euler's equation, Newlin-Gahagan approach, Yoshihara et al.’s approach and tangent modulus theory were adopted to analyze the critical buckling of the columns under both elastic and inelastic conditions. A model is proposed to explain the buckling of intermediate glubam columns with partial splitting.

Bearing Capacity of Recycled Self-Compacting Concrete-Filled Circular Steel Tubular Long Columns Subjected to Axial Load

To investigate the ultimate bearing capacity and deformation of the recycled self-compacting concrete-filled circular steel tubular (RSCCFCST) long columns subjected to axial load, nine specimens with different recycled self-compacting concrete (RSCC) strength grades and slenderness ratios are tested. The experimental results indicate that the lateral deflection dominates the buckling failure of the specimens. The ultimate bearing capacity of the specimens is enhanced gradually as the RSCC strength grade increases but decreases as the slenderness ratio rises. The load-strain curves are linear and basically coincide at the elastic stage. The decrease in the slenderness ratio or increase in the RSCC strength grade contributes to the improvement of the stiffness and ultimate circumferential and axial strains of the columns gradually. Based on the combined tangent modulus theory and bearing capacity of the RSCCFCST short columns, two estimation models are presented to predict the ultimate bearing capacity of the RSCCFCST long columns. Additionally, comparisons between the calculation results of the ultimate strength demonstrate that the prediction models established in this study are more accurate than the other specifications mentioned.

Buckling developed in 3D printed PLA cuboidal samples under compression: Analytical, numerical and experimental investigations

In this work, the behaviour of short-length PLA 3D printed elements under compression is investigated, with a specific focus on buckling. An extensive experimental campaign is conducted on square polymeric columns produced via Fused Filament Fabrication (FFF), with the longitudinal axis oriented in the out-of-plane direction in 3D printing coordinate system. Compressive and tensile mechanical properties of FFF 3D printed PLA are determined; an asymmetry in out-of-plane tensile-compressive behaviour is discussed. The compression of the specimens is monitored using a single camera Digital Image Correlation system, supported by a preliminary validation of its set-up. The slenderness ratio at which the elements start to buckle is identified. The experimental results are compared with three analytical models for buckling prediction in isotropic materials and with linear and non-linear Finite Elements (FE) models. For a broad range of slenderness ratio, the tangent modulus theory gave a good, if conservative, estimation of the critical loads when used with the compressive mechanical properties. Furthermore, the predictions of the non-linear FE models were found to be consistent with the experimental results. • Compressive and tensile mechanical properties in FFF PLA are compared. • Buckling in FFF PLA short columns is experimentally studied. • DIC is used for buckling identification and monitoring. • Analytical and numerical models are compared with experimental results.

Analytical and numerical studies on reducing lateral restraints in conventional & all steel Buckling Restrained Braces

A typical Buckling Restrained Brace (BRB) consists of a steel core and a mortar-filled tube to prevent buckling of the core. In BRBs, the mortar used in a casing considerably increases their weight. In another type of such braces, to reduce the weight, a steel casing is used to avoid buckling of the steel core. In this paper, discontinuous restraint is investigated analytically and numerically to manage buckling behavior of the core along the steel tube.In the analytical section, equations are proposed to determine a suitable distance between the restraints in rectangular and cruciform sections using inelastic buckling, tangent and double modulus theories and their combination with the Romberg-Osgood behavioral model. In the numerical section, first the cyclic behavior of 9 Conventional BRB models with continuous and discontinuous mortar casing is evaluated using the Abaqus software. Then, the appropriate distance between restraints is determined for the numerical model and compared to those from analytical equations. Results show that the tangent modulus theory is more compatible with the finite element model. Moreover, in cruciform cores, it is possible to reduce the amount of mortar up to 64% without any changes in the cyclic behavior of bracing. Finally, two all-steel BRB models are exposed to cyclic loading where steel plates prevent core bucking. Those results demonstrate that the cumulative ductility exceeds the minimum value defined by AISC.

Mechanical properties of wood columns with rectangular hollow cross section

Abstract A rectangular hollow wood column made of Spruce-Pine-Fir (SPF) lumbers was developed, for the purpose of improving the utilization efficiency of timber. The present work conducted experimental investigations of the columns under centrically and eccentrically compression. A wide range of slenderness ratios of columns and eccentricities of loads was considered to carefully examine the failure modes in columns. The results indicated that combined material failures and buckling failures were observed for the intermediately slender columns under axial compression, while buckling failures were observed for the long columns. The ultimate strength of columns under axial compression significantly decreased with the increase of slenderness ratios. The predicted results by the tangent modulus theory closed to those of experiments. The bending failure, which underwent a pronounced nonlinear process, was the major characteristic of the intermediately slender column with an eccentric compressive load. Typically, the load carrying capacity of columns showed a decreasing trend with an increasing in eccentric ratios. Both the secondary bending and material nonlinearity were important factors that impacted the nonlinear response of the column. An analytical model was proposed for evaluating the load carrying capacity of the rectangular hollow columns under the interaction between compression and bending. There was good agreement between the results of the experiments and calculations.

Improvement of Dynamic Buckling Behavior of Intermediate Aluminized Stainless Steel Columns

This paper experimentally investigated the dynamic buckling behavior of AISI 303 stainless steel aluminized and as received intermediate columns. Twenty seven specimens without aluminizing (type 1) and 75 specimens with hot-dip aluminizing at different aluminizing conditions of dipping temperature and dipping time (type 2), were tested under dynamic compression loading (compression and torsion), dynamic bending loading (bending and torsion), and under dynamic combined loading (compression, bending, and torsion) by using a rotating buckling test machine. The experimental results werecompared with tangent modulus theory, reduced modulus theory, and Perry Robertson interaction formula. Reduced modulus was formulated to circular cross-section for the specimens of type (1).The experimental results obtained showed an advantageous influence of hot-dip aluminizing treatment on the dynamic buckling behavior of AISI 303 stainless steel intermediate columns. The improvements based on the average value of critical stress were19.4 % for intermediate columns type (2) compared with columns type (1) under dynamic compression loading, 8.7 % for intermediate columns type (2) compared with columns type (1) under dynamic bending loading, and 16.5 % for intermediate columns type (2) compared with columns type (1) under dynamic combined loading.

BEHAVIOR OF SLENDER RC COLUMNS WITH INITIAL DEFORMATION IN FIRE

Reinforced concrete columns formed with high-strength concrete have been developed for use as axial force carrying members. They have a reduced cross section compared with conventional “short” columns and are known as “slender” columns. These columns realize larger building interior spaces and offer architectural space with good visibility. However, under fire conditions, bending stiffness of RC columns decreases considerably due to thermal degradation of the cover concrete. It is anticipated that slender RC columns would be more vulnerable to buckling failure than to cross sectional failure. The authors have proposed an evaluation method based on tangent modulus theory for the buckling strength of straight RC columns subjected to centric axial load in fire, and shown that this proposed method offered a good approximation of both experimental fire-resistance time and failure mode. However, existing members have initial deformation formed in manufacturing process. For this reason, it is anticipated that slender RC columns with initial deformation fails earlier than straight ones. This paper presents the main results from an analytical study of slender RC columns with initial deformation shaped sine wave under fire conditions, leading to a discussion of fire performance, which includes deformation behavior and fire-resistance time. All analytical models consist of concrete with a compressive strength of approximately 120N/mm2 and have a square section where the length of one side is 250mm and 5000mm in height. From thermal stress analysis in consideration of geometrical nonlinearity, the followings are obtained: 1. Fire-resistance time of slender RC columns whose the initial deformation α/h was less than approximately 1/103 decreased sharply. 2. There are two failure modes of slender RC columns with initial deformation which are buckling and three-hinged mode. 3. The time, while columns resist an axial force after they start to bend at the time evaluated by the analytical method based on tangent modulus theory, is approximately 10~15 minutes. 4. The larger transient strain was, the shorter fire-resistance time was because the large additional moment occurred with bending deformation increases in fire. 5. The directions of initial deformation have little influence on fire-resistance time. 6. The mechanical boundary conditions at edge of columns have large impact on fire-resistance time.

Analytical solution for assessing continuum buckling in sedimentary rock slopes based on the tangent-modulus theory

• A formula to calculate the length of the passive segment is provided. • Know whether the mode of failure is brittle, elastic or inelastic determines stability. • Nine real collapses were satisfactorily resolved using the proposed method.

Cement-stabilised rammed-earth square column in compression

The paper presents a novel experimental investigation, comprising tests on cement-stabilised rammed-earth (CSRE) prisms and columns of varying slenderness ratios, focusing on the effects of concentric axial loading and variation of the moisture content along the height of the column, and of the slenderness ratio on capacity/stress reduction factors. An attempt was also made to determine the ultimate compressive strength (σu) of columns using tangent modulus theory and the validity of using masonry design rules for the design of CSRE columns. The results showed that the load-carrying capacity of columns decreased with increasing slenderness ratio. The location/zone in the column where the difference in the moisture content was relatively higher (i.e. 3·86% and above) at the time of testing was found to be the weakest zone leading to untimely failure. Tangent modulus theory underestimated σu. The experimental reduction factors were found to be more conservative than the masonry code provisions. The characteristic strength obtained for CSRE columns yielded a relatively higher safety factor and showed that it is possible to construct a single-storey load-bearing house when designed properly.

Dynamic stability analysis for Beck’s type of inelastic Timoshenko steel columns

In this study, the dynamic stability analysis of a Beck’s type of steel column subjected to a sub-tangential follower force is presented considering the effects of gradual yielding along the length of column, the shear deformation, and the rotary inertia. The nonlinear tangent modulus-stress, tangent modulus-slenderness ratio, and compressive stress-strain relationships of an inelastic column are derived from the column strength curve given in Korea Bridge Design Code (KBDC) based on the tangent modulus theory. The equations of motion of the Timoshenko inelastic column are derived from the Hamilton’s principle. The finite element method using the Hermite cubic interpolation functions is employed to obtain the mass, inelastic stiffness, geometric stiffness, and load correction matrices, and the evaluation procedure for critical values of divergence and flutter of the non-conservative system is briefly described. The obtained results are compared with available results of other numerical or analytical methods. Finally, the effects of various structural parameters such as the material inelasticity, the shear deformation, the rotary inertia, the effective length factor, and the sub-tangentiality factor on the divergence and flutter behavior of non-conservative columns are parametrically investigated.

Axial load-capacity of rectangular cement stabilized rammed earth column

The paper presents a novel study on structural behaviour of axially loaded cement stabilized rammed earth (CSRE) prisms and columns of square and rectangular cross-sections. The effects of slenderness ratio and aspect ratio on the load-capacity of columns and stress reduction factors were assessed. Experimental results were compared with that obtained by tangent modulus theory. Experimental capacity/stress reduction factors were compared with published codal values. The result shows that the load-capacity of column decreases as the value of slenderness ratio increases and aspect ratio of column was found to influence the load-capacity. The ultimate compressive strength of column predicted by tangent modulus theory tend to converge the experimental values at higher slenderness ratios. The stress reduction factors in earthen (NZS 4297, AS HB 195) and masonry (IS 1905) standards are found to be in close agreement with the experimental values. The characteristic strength of column yields relatively higher safety factor (∼23–35) indicating a possibility of using CSRE for construction of load-bearing houses.

Behavior of cement-stabilized rammed earth circular column under axial loading

Detailed study has been carried out on the load carrying capacity of cement stabilized rammed earth (CSRE) circular columns under axial compression. Tests on CSRE cylinders and columns were performed to determine the effects of concentric axial loading and slenderness ratio; and stress reduction factors were assessed. A comparative study was made between the ultimate compressive strength (σu) of columns determined using tangent modulus theory and experimental values. Furthermore, the validity of using masonry design rules for the design of CSRE columns was also assessed. The result shows that with increasing slenderness ratio the load carrying capacity of columns decreases. The ultimate compressive strength of column predicted by tangent modulus theory tend to converge with experimental values at higher slenderness ratio and the masonry codal predictions are observed to be un-conservative as compared to experimental reduction factors. Lastly, the characteristic strength determined for columns yields relatively higher safety factor and indicates that it is possible to construct a single storey load bearing houses when designed properly.

Steel Rod Stability and Inelastic Buckling Study

Research on stability problem, has put forward the famous bar under axial pressure formula from the middle of the eighteenth Century Euler critical force, more than 200 years, the elastic range but recognize that Euler's formula is only suitable for the material, but it took more than a hundred years. Study on the inelastic critical load of column, double modulus theory and the tangent modulus theory. These theories are starting from the ideal straight rod, and practical bar called " defects exist various accidental eccentricity " ( such as physical and geometrical aspects of the initial deflection rod, pressure asymmetry in aspects of material unevenness, residual stress) is not consistent. Thus the utility pressure bar, should according to the eccentric pressing rod is considered more practical.

A Simplified Method for Calculating Ultimate Strength of Pressure Hull in Deep-Sea Manned Submersible

Based on the strain energy density theory, the connotation of tangent modulus theory is developed. At critical equilibrium state, a dimensionless number Φt of a structure under pressure is introduced in this paper. Derived from the stress-strain curve of material, a four-parameter formula that contains Φt is established. This formula is referred to as the function of strength utilization ratio that may be used for calculating the inelastic buckling of deep submersible pressure hull.Φt can be used to express the effect of initial imperfection of the shell and to guide the experimental study. Compared with the data from some experiments or other methods, it has demonstrated that the method given by this paper is more precise and convenient for predicting the failure of thin and moderately thick shell under pressure.

Inelastic Static and Dynamic Stability Analyses of a Column Subjected to a Nonconservative Force

This study performs the inelastic static and dynamic stability analyses of a column subjected to a nonconservative force using the finite element method. Based on the tangent modulus theory, the nonlinear compressive stress–strain and tangent modulus–stress relationships of an inelastic column are derived from the column strength curves given in two design codes. The extended Hamilton principle is employed to obtain the mass, elastic stiffness, and geometric stiffness matrices. Evaluation procedures for the critical values of buckling and flutter of the nonconservative systems are briefly introduced. In numerical examples, the influence of various parameters on the inelastic static and dynamic stability of the nonconservative systems is addressed as follows: (1) the effect of the nonconservativeness parameter on the critical buckling and flutter loads of the column for the inelastic stability analysis from the characteristic curves, (2) the variation of the effective length factor (K-factor) for inelastic columns with respect to the nonconservativeness parameter.

Structural Behavior of Story-High Cement-Stabilized Rammed-Earth Walls under Compression

A rammed-earth wall is a monolithic construction made by compacting processed soil in progressive layers in a rigid formwork. There is a growing interest in using this low-embodied-carbon building material in buildings. The paper investigates the strength and structural behavior of story-high cement-stabilized rammed-earth (CSRE) walls, reviews literature on the strength of CSRE, and discusses results of the compressive strength of CSRE prisms, wallettes, and story-high walls. The strength of the story-high wall was compared with the strength of wallettes and prisms. There is a nearly 30% reduction in strength as the height-to-thickness ratio increases from about 5 to 20. The ultimate compressive strength of CSRE walls predicted using the tangent modulus theory is in close agreement with the experimental values. The shear failures noticed in the story-high walls resemble the shear failures of short-height prism and wallette specimens. The paper ends with a discussion of structural design and characteristic compressive strength of CSRE walls.

Buckling of laser-welded sandwich panels: Ultimate strength and experiments

This is the last of three companion papers which examine the elastic buckling and collapse of laser-welded sandwich panels with an adhesively bonded core and unidirectional vertical webs. By evaluation of the buckling stress in the first two papers it has been found that the buckling stress in compression parallel and normal to the webs typically reaches the proportional limit of the face plate and web material well before elastic buckling occurs. Hence, this paper presents an extension of the buckling model into the elastoplastic regime, with the aim of determining the ultimate (local) strength of the sandwich and of allowing experimental verification of the results. Using tangent modulus theory to ‘plasticize’ the elastic buckling model, the ultimate strength is evaluated for a sandwich configuration with high-strength steel face plates and a broad range of core moduli. The critical load predicted by the inelastic buckling model agrees well with non-linear finite element results and experimental values obtained from compression testing.

Effect of strain hardening on the buckling of structural members and design codes recommendations

Some aspects of buckling behaviour of structural elements made of nonlinear materials with strain hardening, for which the stress–strain law has no yield plateau (stainless steels, aluminium alloys and others) are discussed. Considering compressed bars and plates made of materials with linear and nonlinear strain hardening, on the basis of tangent modulus theory and the deformation theory, we show that there exists a “threshold” value of the hardening factor below which the hardening does not influence the limit load. But if the hardening factor exceeds the threshold value then the hardening significantly increases the limit load and this increasing is not properly accounted for in some worldwide design codes.

Effective buckling length of steel column members based on elastic/inelastic system buckling analyses

This study presents an improved method that uses the elastic and inelastic system buckling analyses for determining the K-factors of steel column members. The inelastic system buckling analysis is based on the tangent modulus theory for a single column and the application is extended to the frame structural system. The tangent modulus of an inelastic column is first derived as a function of nominal compressive stress from the column strength curve given in the design codes. The tangential stiffness matrix of a beam-column element is then formulated by using the so-called stability function or Hermitian interpolation functions. Two inelastic system buckling analysis procedures are newly proposed by utilizing nonlinear eigenvalue analysis algorithms. Finally, a practical method for determining the K-factors of individual members in a steel frame structure is proposed based on the inelastic and/or elastic system buckling analyses. The K-factors according to the proposed procedure are calculated for numerical examples and compared with other results in available references.