Tangent Modulus Theory Research Articles

Simplified design approach for cold-formed stainless steel compression members subjected to flexural buckling

The design criteria of stainless steel compression member are more complicated than those of carbon steels due to the nonlinear stress strain behavior of the material. In general, the tangent modulus theory is used for the design of cold-formed stainless steel columns. The modified Ramberg–Osgood equation given in the ASCE Standard can be used to determine the tangent modulus at specified level of stresses. However, it is often tedious and time-consuming to determine the column buckling stress because several iterations are usually needed in the calculation. This paper presents new formulations to simplify the determination of flexural buckling stress without iterative process. Taylor series expansion theory is utilized in the study for numerical approximations. The proposed design formulas are presented herein and can be alternatively used to calculate the flexural buckling stress for austenitic type of cold-formed stainless steel columns. It is shown that the column strengths determined by using the proposed design formulas have good agreement with those calculated by using the ASCE Standard Specification. A design example is also included in the paper for cold-formed stainless steel column designed by using the ASCE Standard equations and the proposed design formulas.

Development mechanism of local plastic buckling in bars subjected to axial impact

In order to clarify the developmental mechanism of the local plastic buckling and the interaction between axial wave and buckling deformation in an axially impacted slender-bar, the non-linear dynamic equations in the incremental form are derived and solved by use of the finite difference method, with the axial wave front treated as a moving boundary. The initial local-buckling deflection given by the characteristic-value analysis is used as the initial condition of the solution of the equations, instead of the initial imperfection that is assumed in literatures. It is found that the initial buckling deflection with one half-wave, occurring near the impacted end, develops into the higher post-buckling mode with several half-waves, as the axial compression waves propagate forward. The numerical results show that no strain reversal occurs at the early stage of post-buckling process, and the solution corresponding to the tangent-modulus theory is valid for the dynamic plastic post-buckling response of the bar at this stage. The theoretical results are in good agreement with the experimental results reported in the literature.

435 円筒シェルの連続塑性座屈モードと吸収エネルギー

Experiments were performed to study the behavior of axially crushed aluminum round tubes. The mean radius to thickness (R/h) ratios of these varied from 15 to 75. The modes of collapse and the mean collapse loads obtained in these tests are presented. A simple estimation on the energy-absorption efficiency is presented by using the buckling stress expression based on the tangent modulus theory and the expression of mean collapse load considering the formation of plastic hinges. The estimation results are compared with the experiments.

Prediction of the buckling stress of intermediate wooden columns using the secant modulus

We made several buckling tests of wooden columns with intermediate slenderness ratios (λ) and examined the empirical formulas. On the basis of the examination, we formulated an equation for predicting the buckling stress (σ) of an intermediate wooden column. Sitka spruce (Picea sitchensis Carr.) and buna (Japanese beech, Fagus crenata BI.) were used for the studies. A compressive load was applied on the specimen supported with pin ends, and the buckling stress was predicted by the tangent modulus theory and two empirical equations: those of Tetmajer and Newlin-Gahagan. The predicted σ-λ relations were compared with the test results, and the applicability of these predictions were examined. Based on the comparisons, we formulated an equation that can predict the σ-λ relations of materials with various stress-strain characters in the plastic strain range.

A finite element study on Shanley's plastic buckling theory of short columns

The continuum modeling of finite element method is used to study the reliability of Shanley's plastic buckling theory which generalizes Engesser's tangent-modulus theory and Kármán's reduced-modulus theory. The study includes the deformation behavior that is the basis of the development of classical theories, the change of deformation behavior following the variation of the slenderness ratio of the column model, the adequacy of the classical approach of simply using certain effective moduli in Euler's formula to define the buckling load, and the difference of factors which affect Shanley's buckling load and the buckling load of finite element results. Buckling loads of different column models are obtained by the converged finite element solutions. They are compared with Shanley's buckling loads. In solving the nonlinear finite element systems, the incremental iterative procedure is used to update the response history. And an accelerated iteration method based on improving a modified Newton-Raphson scheme is used for obtaining converged solutions of the discretized nonlinear algebraic systems.

On the tangent modulus theory of inelastic buckling of built‐up columns

Abstract Engesser‐Shanley's Tangent Modulus Theory of inelastic buckling of a solid Euler column is extended to a three‐connection built‐up column consisting of two slightly separated identical parallel elements that are joined together at their ends and middle by rigid connections of negligible axial extent. By first assuming the validity of tangent‐modulus inelastic buckling, the instantaneous center of each member of such column at the verge of inelastic buckling is shown to be located at or beyond the bottom of the cross‐section of the convex side in the center portion of the member. Thus consistency is reached and the tangent modulus theory is shown to be valid for the inelastic buckling of the built‐up columns.

Steel column strength models

Abstract This paper uses a simple model to unify the two traditional approaches to steel column strength based either on the buckling of straight columns with residual stresses or on the first yield of crooked columns. The inelastic buckling and post-buckling loads of elastic-plastic-strain hardening steel columns with residual stresses are determined using the tangent, reduced, and Shanley theories. These are compared with the maximum load capacities of crooked columns which fail by inelastic bending. It is concluded that the strengths of real steel columns should be analysed by considering the inelastic bending of crooked elastic-plastic-strain hardening columns with residual stresses. It is suggested that this will be most conveniently done by using a tangent modulus theory (which assumes that the loading and unloading paths coincide) to determine the maximum load capacities.

Basic Studies on the Crashworthiness of Structural Elements (Part 5)

The mean crushing loads, plastic buckling loads and crushing strokes in the axisymmetric collapse mode for circular cylinders of various radius-to-thickness ratios were analytically and experimentally investigated. The finite element analysis (using axisymmetric solid elements) and the crush tests were conducted for 8 mild steel cylinders with different R/t ratios. The results obtained are summarized as follows : (1) The plastic buckling loads due to Gerard's theory agree well with the maximum loads obtained in the experiments with R/t>20. But if R/t is below 20, Gerard's solutions underestimate the loads.(2) The mean crushing loads which the present finite element code predicted, agree well with the experimental results and are improved solutions in comparision with the existing rigid-plastic solutions and the empirical formula. The above numerical results confirmed that the thin-walled assumption can not be maintained for relatively thick cylinders which are considered approximately thin in the linear problem.(3) For the mean crushing load in case of axisymmetric collapse the modified Alexander's solution is in better agreement with experimental results with R/t>30 than the other existing theories and the empirical formula. When the axisymmetric and the non-axisymmetric collapse are both considered, Magee's empirical formula is sufficiently accurate for the thin to thick walled range.(4) The crushing half wavelengths and crushing strokes are in good agreement with the solutions of the tangent modulus theory and the modified Alexander's theory respectively, confirming the results of Ref. (2).

Inelastic post buckling behavior of very short column

Abstract The main objective of this paper is to discuss and explain the post‐yielding bucking behavior of very short columns using equivalent modulus theory with emphasis on the behavior of columns made of ideal elastic‐plastic material. In general, results obtained from the proposed method are between those calculated from tangent modulus theory and double modulus theory. When the slenderness ratio l/r approaches to zero, the critical load increases sharply because of strain hardening effects. The results are predicted by the proposed method and are in a close agreement with the experimental results. For illustration and future reference purposes, column failure can be classified into 5 failure modes.

Buckling of Reinforced and Prestressed Concrete Members - Simplified Calculation Method

S tructural instability considerations provide a special group of design problems for reinforced and prestressed concrete members wherein axial or in-plane forces play an important role in structural performance. The nonlinear behavior of a member due to the actual stressstrain relationship, effects of cracking, creep and current deformations must be duly accounted for in order to carry out realistic strength predictions. On the other hand, simplified methods are needed which are practical to use in actual design situations and which can allow for the nonlinear effects in a sufficiently accurate manner. This paper describes the basic features of structural instability in reinforced and prestressed concrete members such as columns, plates and shells. Simplified methods are discussed, which are based on the tangent modulus theory and ultimate limit state considerations. Several studies have been made by comparing theoretical results with buckling tests on plate and shell specimens. In general, the design methods proposed show good agreement with available experimental results. Emphasis is put on the significance and adequate representation of construction imperfections, specifically geometry and materials. A simplified but powerful design method is given, by which geometric out-of-shape imperfections can

Optimum Design of Elastic-plastic Column

A method of optimum design for maximizing the buckling load of an elastic-plastic column of given volume, length, and material is presented. The column material is assumed to obey Ramberg-Osgood's deformation law and the buckling load is computed by means of the finite element method taking account of the tangent modulus theory. Rosen's gradient projection method is used for determining the optimum shape. Numerical results obtained for a simply supported column with similar cross section are shown to be in reasonably good agreement with the analytical solution proposed in a previous paper. As for new treated examples of columns with other cross sections and boundary conditions, it also seems that reasonable solutions are obtainable by the proposed method. Furthermore, we treat the problems including the constraint on a lower bound of the cross sectional areas of columns.

Inelastic stability of laterally unsupported i-beams

A computer method to study the inelastic stability of laterally unsupported steel I-beams and based on a general non-linear theory is presented. Traditionally, the problem of flexural-torsional stability of beams is treated as a lateral buckling problem. Some of the draw-backs of these earlier studies are given below: The classical theory assumes that the deformations are small. In addition the deformation field is linearized. This theory is therefore valid only when the major axis flexural rigidity is much greater than its minor axis rigidity, so that deformations before the onset of lateral buckling are negligible. The lateral buckling theory is valid for straight beams, with loads applied rigorously in the plane of symmetry. Practical beams have initial imperfections and unavoidable load eccentricities. So the true behavior is better described by the stability phenomenon. For beams of intermediate length for which buckling occurs in the inelastic range, the tangent modulus theory is generally used. For ideally straight beams the tangent modulus theory provides an estimate for the collapse load which is slightly conservative. However, for practical beams with initial deformations, this need not be the case. In the majority of existing studies on inelastic lateral buckling, the differential equations for beams under uniform moment are used without modification for beams under moment gradient. In the later case the shear center line is inclined to the centroidal and geometrical axes. The differential equations for beams under uniform moment should therefore be modified by adding additional terms. The majority of the existing studies are limited to the behavior of isolated beams with simple end-conditions and so the beneficial effect of adjacent members on the beam collapse load cannot be studied accurately. A general non-linear theory to describe the spatial behavior of beams and that doesn't have the deficiencies mentioned above, is developed in the present paper. The paper also presents a computer method of solving these non-linear equations using the method of finite differences. Several numerical examples presented and comparison with the existing theoretical and experimental results show the applicability of the theory to a wide range of problems.

Computer simulation of the scatter in steel member strength

A Monte Carlo method for digital computer simulation of the strength of (steel) members and structures is presented and is applied to rolled steel beams and columns, and thin-walled cylinders. Input data are cumulative distribution functions (histograms) for the geometric and strength variables. The output (i.e. the scatter in structural strength) is printed as histograms and is statistically analysed.Each output histogram is compared with the Gaussian normal distribution. Using the nonparametric test of homogeneity a number of histograms may then be compared.The case studies presented deal with the plastic strength of steel beams and the maximum load of axially loaded steel columns and thin-walled cylinders. Mathematical models for beams subject to pure bending moment, moment and axial force, moment and shear, or uniform torsion are presented. For the initially straight, centrally loaded column a tangent modulus theory which considers residual stresses is used.The simulations have been carried out for one HEA beam, four HEB beams and three IPE beams. Comparison of the simulation results show that the scatter in load carrying capacity of the simulated beams and columns can be regarded as normally distributed, that the load carrying capacity of beams and columns of the same group (HEB or IPE) and beams and columns of the groups HEA and HEB have distributions which differ very little from each other, and that the scatter in simulated beam strength, and in simulated column strength for short and medium length columns, is much more affected by the variation in yield strength of the material than by the variation in cross sectional data. This conclusion holds for ordinary distributions in yield strength of structural carbon steel.Comparisons of simulation results and test results show good agreement for the beams. The agreement is not so good for the columns mainly because in the tangent modulus theory it is assumed that the columns are initially straight. For the cylinders excellent agreement was achieved.The experience gained with the simulation system presented here shows that a medium size computer can be economically used to simulate a relatively large number of plays.

Dynamic Plastic Pulse Buckling Beyond Strain-Rate Reversal

A large deflection, elastic-plastic numerical theory (SABOR/DRASTIC 6) is used to investigate dynamic buckling motion beyond strain-rate reversal in cylindrical shells under radial impulse. It is found that the mode number of the most amplified harmonic is smaller than predicted by the simple analytic tangent modulus theory because of the increased stiffness after strain-rate reversal, neglected in the simple theory. Nevertheless, the simple theory gives reasonable estimates for threshold buckling impulses. In buckling well beyond threshold (about three times the threshold impulse in a thin shell, radius-to-thickness ratio of 120) plastic deformation becomes more localized at the buckle crests, but compressive strain continues to dominate so that a simple plastic hinge postbuckling analysis would be inadequate. Accuracy of the numerical theory is confirmed by comparison with experiment (asymmetric loading below buckling threshold and symmetric loading above) and with the tangent modulus theory prior to strain-rate reversal.

Buckling of Inelastic I-Beams Under Moment Gradient

A model developed for the inelastic buckling of beams under uniform moment, which accounts for strain-hardening residual stresses and monosymmetry of the yielded cross section, is extended to the case of moment gradient by allowing for the effects of nonuniform yielding. A tangent modulus theory of buckling is used, and the governing differential equations are adopted from those which govern the elastic flexural-torsional buckling of tapered monosymmetric I-beams of constant depth. The critical central concentrated loads of simply supported I-beams are obtained by solving the governing differential equations numerically by using the method of finite integrals in an iterative process. The influence of the residual stresses, height of point of application of the load, and in-plane moment distribution on inelastic buckling are investigated, and the theoretical solutions are compared with design rules.

Buckling of Composite Beams in Negative Bending

Local flange buckling and lateral buckling are studied theoretically and experimentally to determine the ultimate strength of composite beams in negative moment regions. The effect of the strain-hardening of steel on ultimate strength is included in the analysis. Based on analysis and results of tests, the ultimate moment capacity of composite beams with compact wide-flange sections is determined by local flange buckling, unless the compression flange is stiffened by cover plates. Local flange buckling and lateral buckling analysis based on tangent modulus theory provides good agreement with experimental results.

Buckling of Inelastic I-Beams under Uniform Moment

Many different theoretical models have been proposed for the flexural-torsional buckling of inelastic steel beams under uniform moment. These models are considered and what is argued to be a more accurate model is proposed, which is based on the tangent modulus theory of buckling, which uses the strain-hardening moduli for the inelastic material, and which accounts for residual stresses. The predictions for this model for a W 8 x 31 beam are compared with those for the previous models and the relative significance of the assumptions are determined. It is found that the critical moment of the beam is not very sensitive to the assumptions concerning the moduli of the inelastic material, but that the effects of monosymmetry induced by the residual stresses should be accounted for. The most important assumptions are found to be those of the magnitude and distribution of the residual stresses, which cause significant variations in the strength of the beam.

Bifurcation phenomenon and the rate-problem in plasticity

The rate-problem associated with the bifurcation of rigid/plastic bodies is formulated in general terms and illustrated by considering the buckling of a rectangular plate with lateral restraint. The connexion between the present formulation and the uniqueness criterion of Hill is also discussed in relation to the plate problem. It is found that for a sufficiently long plate under prescribed velocity of compression, the critical load approaches the value given by the tangent modulus theory.

Tangent modulus theory for cylindrical shells: Buckling under increasing load

Analytical results are presented for axisymmetric plastic buckling of axially compressed cylindrical shells which is initiated under increasing load. The rate of load increase is determined by forcing the loading condition of the J2 incremental theory of plasticity to be satisfied everywhere in the shell. Results complement those found by using classical stability concepts. The analysis constitutes the formal generalization of tangent modulus theory for plastic column behavior to axially compressed cylindrical shells.

単一山形鋼の座屈耐力に関する実験的研究

When simple steel angles are used the sub-members of a transmission tower, they are connected eccentrically with a main-member by bolts near the center of one of their flanges. In order to investigate the behavior, of angles subjected to compression under such circumstances, columns with ten different slenderness ratioo were loaded concentrically and eccentrically along the symmetric principal axes, along the non-symmetric principal axes and along the both principal axes, and the following conclusion has been drawn. Concerning with angles loaded concentrically, the experimental values agreed very well with the theoretical value of the flexual buckling in the direction of the symmetric axes, i.e. the Euler loads, for long columns and wtih the theoretical value of the flexual-torsional buckling based on the tangent modulus theory for angles with the moderate slenderness ratios. The ultimate strengths of angles loaded eccentrically along the symmetric principal axes agreed with the values obtained by Jezek's approximative formula. The ultimate strengths of angles loaded eccentrically along the non symmetric principal axes lie in the intermediate range bounded by the approximative formula mentioned above.