Exact Bernoulli-Euler static stiffness matrix for a range of tapered beam-columns

Bernoulli-Euler theory and Bessel functions are used to obtain explicit expressions for the exact static stiffnesses for axial, torsional and flexural deformation of an axially loaded beam which is tapered such that A varies as y^l, GJ as y^m+2, and I as yⁿ⁺², where A, GJ and I have their usual meanings, y = (cx/L) + 1, c is a constant such that c > ? 1, L is the length of the beam, x is the distance from one end of the beam, l and m can have any value and n is 1,2 or ? 1. The work complements the similar dynamic stiffness derivations of Reference 2. Numerical results for a beam with substantial taper (c = 1.0) give better than seven figure agreement with the stiffnesses obtained by extrapolation from stepped beams with 400 and 500 uniform elements. A procedure is given for calculating the number of critical buckling loads of a clamped/clamped member that are exceeded by any trial load so that an existing algorithm can be used to obtain the exact critical buckling loads of structures which contain tapered members.     

An analytical approach for local buckling analysis of initially delaminated composite thin-walled columns with open and closed sections

Local buckling of intact thin-walled columns is generally performed by modeling the wall segments as long plates and by assuming that edges common to two or more plates remain straight. Thus, the buckling load can be determined by considering the wall segments as individual plates rotationally restrained by the adjacent wall segments. This technique is combined with plate theories as a new analytical method to predict the buckling load of an initially delaminated column with any arbitrary sections (open or closed). First, moments at the rotationally restrained edges of delaminated segment (web or flange) are obtained from the curvature and stiffness of the adjacent laminates. Then, the strain energy of this delaminated segment with distributed moment at edges is calculated based on the first-order shear deformation theory. Using the principal of minimum potential energy, the governing equations are obtained and solved by the Rayleigh–Ritz approximation technique. Results of the present approach are compared with three-dimensional finite-element results obtained from eigenvalue buckling analysis in ANSYS software for both box- and channel-section columns with cross-ply and angle-ply stacking sequences. Finally, the effects of delamination size and location are investigated on the buckling loads.     

Experimental investigations on buckling of cylindrical shells under axial compression and transverse shear

This paper presents experimental studies on buckling of cylindrical shell models under axial and transverse shear loads. Tests are carried out using an experimental facility specially designed, fabricated and installed, with provision forin-situ measurement of the initial geometric imperfections. The shell models are made by rolling and seam welding process and hence are expected to have imperfections more or less of a kind similar to that of real shell structures. The present work thus differs from most of the earlier investigations. The measured maximum imperfections δ_max are of the order of ±3t (t = thickness). The buckling loads obtained experimentally are compared with the numerical buckling values obtained through finite element method (FEM). In the case of axial buckling, the imperfect geometry is obtained in four ways and in the case of transverse shear buckling, the FE modelling of imperfect geometry is done in two ways. The initial geometric imperfections affect the load carrying capacity. The load reduction is considerable in the case of axial compression and is marginal in the case of transverse shear buckling. Comparisons between experimental buckling loads under axial compression, reveal that the extent of imperfection, rather than its maximum value, in a specimen influences the failure load. Buckling tests under transverse shear are conducted with and without axial constraints. While differences in experimental loads are seen to exist between the two conditions, the numerical values are almost equal. The buckling modes are different, and the experimentally observed and numerically predicted values are in complete disagreement.     

Creep buckling of nonlinear viscoelastic columns

Summary Creep buckling of columns is analyzed for materials obeying nonlinear single memory integral constitutive laws. There have been two basic methods to analyze creep buckling of columns; the Rabotnov and Shesterikov's linearized dynamical approach and Hoff's quasistatic nonlinear approach, the former predicts critical buckling loads and the latter yields critical buckling times. The two theories are generally believed to be independent of each other. Through investigation of creep phenomena other than steady creep, the two different creep stability criteria are found to be correlated and thus the linear treatment is justified by a nonlinear approach. It is also shown that the Hoff's approach can be used to derive critical creep buckling loads. The missing link between the two theorees is found in this paper.Nomenclature A cross sectional area of column - E Young's modulus - E ₀=q ₁/p ₁ instantaneous modulus of elasticity - E =q ₀ long term modulus of elasticity - H width of column - I second moment of inertia - l length of column - M bending moment - m _i transient creep powers - N number of nonlinear Kelvin elements - n steady creep power - P, P ₀ axial compressive force - P _c creep buckling load - P _E Euler load - p ₁,q ₀,q ₁ material constants - t time - t ₀ reference time - t _cr critical time of buckling - V lateral displacement of column - x, y Cartesian coordinates - strain - _i retardation times - stress - ₀ initial constant stress - steady creep parameter - _i transient creep parameters - sgn () the signum function - (^) small increment With 5 Figures     

Post-buckling analysis of non-uniform elastic columns

A numerical method for the second-order analysis of elastic beam-columns is described. For nominated values of axial load and end moments the deformed states and stress distribution can be determined by an automatic iterative procedure. For centrally loaded columns of non-uniform section, the method can be used to predict the critical loads from the grossly deformed equilibrium states above the first critical load. Axial and shear strains may be accounted for as well as flexural strains. Solutions for the shape of the elastica can be obtained for non-uniform columns. The method requires a small digital computer and is demonstrated for several plane, uniform and stepped elastic members. It can be readily extended to deal with transverse and distributed axial loading as well as flexural-torsional behaviour.     

Uniaxial local buckling strength of periodic lattice composites

To reveal the local buckling strength of periodic lattice composites, an important factor in optimal material design, analytical method based on the classical beam-column theory was applied. Buckling modes were decided according to the condition that the curvature of the strut columns is the smallest. Characteristic equations were built according to the equilibrium equations. The buckling strengths and constraint factors of various grids under uniaxial compression and tension were achieved. The strut network supports stronger rotation restrictions than pin-jointed nodes but weaker than the built-in ends. With more stacks of struts and connectivity at nodes, the restriction must be stronger and the buckling load is greater. Commonly, the constraint factors of isogrids and mixed triangle grids are greater than Kagome grids. The regular honeycomb and square grids possesses smaller buckling loads.     

Buckling Characteristics of Symmetrically and Antisymmetrically Laminated Composite Plates with Central Cutout

A numerical and experimental study was carried out to determine the effects of anti-symmetric laminate configuration, cutout and length/thickness ratio on the buckling behavior of E/glass-epoxy composite plates. The buckling loads were presented for symmetrically and anti-symmetrically laminated plates subjected to axial compression load. The study included two different laminate configurations ([90/45/−45/0]_as and [90/45/−45/0]_s), two different cutout shapes (circular and semi-circular), two different length/thickness ratios (L/t = 75 and 37.5) and three boundary conditions (clamped–clamped [CC], clamped–pinned [CP] and pinned–pinned [PP]). Firstly, the buckling loads of eight-ply E/glass-epoxy rectangular plates were determined experimentally. Then, the buckling loads of the laminated composites were calculated by ANSYS finite-element computer code. The changing in buckling load of the composites due to the presence of cutout and changing of length/thickness ratio was calculated. Finally, the experimental test results were compared to the buckling loads of plates obtained from the finite element analysis.     

圆管拱结构平面外弹塑性二次分岔屈曲性能

该文使用一种新的数值跟踪策略对无平面外支撑和有顶点平面外支撑的弹塑性圆管拱的平面外二次分岔屈曲的荷载-位移曲线的全过程进行跟踪分析,得到了跨中集中荷载和全跨均布荷载作用下,相同截面不同矢跨比的拱的平面外弹塑性二次分岔屈曲荷载,将其与平面内二次分岔屈曲荷载比较。计算结果表明:对于具有相同截面的无平面外支撑弹塑性圆管拱,在跨中集中荷载作用下,平面内二次分岔屈曲荷载小于平面外二次分岔屈曲荷载,平面内二次分岔屈曲会先于平面外屈曲发生,矢跨比0.2 时平面外二次分岔屈曲荷载最大。全跨均布荷载作用下的拱平面外二次分岔屈曲会先于平面内二次分岔屈曲发生;矢跨比0.3 时平面外二次分岔屈曲荷载最大。矢跨比为0.1 时,拱的平面内与平面外屈曲荷载相差最大。对于有顶点平面外支撑拱,在集中荷载作用下,支撑对分岔屈曲荷载作用不明显,只是改变了屈曲形式。全跨均布荷载作用下,支撑可以大幅提高全跨均布荷载作用下拱的平面外屈曲荷载,尤其是对矢跨比小的拱的影响更大一些,平面外二次分岔屈曲会先于平面屈曲发生。     

Effect of rotationally restrained and Pasternak foundation on buckling of an orthotropic rectangular Mindlin plate

This article, based on first-order shear deformation theory, presents the buckling analysis of a rotationally restrained orthotropic rectangular Mindlin plate resting on a Pasternak elastic foundation. Thus, the Mindlin–Reissner plate theory is employed for which the governing equations are solved by the Rayleigh–Ritz method. Uniformly distributed in-plane loads are applied to two simply supported opposite edges of the plate and the other two edges have rotationally restrained conditions without loading. Finally, the effects of plate parameters, such as foundation stiffness coefficients, aspect ratios, and ratio of elastic modulus in the x to y direction on the buckling loads are presented. The results show that the buckling load would increase when the ratio of the elastic modulus in the x to y direction increases and the plate is close to isotropic. The variation of buckling load versus changing ratio of elastic modulus in the x to y direction in the state of without elastic foundation and with clamp support is more than the rest of the state.     

Weight functions for bimaterial interface cracks

An efficient approach using the analytically decoupled near-tip displacement solution for bimaterial interface cracks presented in this paper involves: (1) the calculation of the decoupled strain energy release rates G _I and G _II associated respectively with the decoupled stress intensity factors K _I and K _II and (2) the extension of Rice's displacement derivative representation of Bueckner's weight function vectors beyond the homogeneous media. It is shown that the stress intensity factors for a bimaterial interface crack predicted by the present approach agree very well with those solutions available in the literature. The computational efficiency is enhanced through the use of singular elements in the crack-tip neighborhood.As reported in the homogeneous case, the calculated weight function for a bimaterial interface crack is load-independent but depends strongly on geometry and constraint conditions. Due to the coupling nature of the stress intensity factors of a bimaterial interface crack, the invariant characteristics of the dimensionless weight function vectors are different from those of a crack in homogeneous material. In addition, the elastic constants of two constituents can significantly alter the weight function behavior for a cracked bimaterial medium.Due to the load-independent characteristic of the weight functions, the stress intensity factors for a bimaterial interface crack can be obtained accurately and inexpensively by performing the sum of worklike products between the applied loads and the weight functions for the cracked bimaterial body under any loading conditions once the weight functions are explicitly predetermined. The same calculation can also be applied for the identical cracked bimaterial medium with different constraint conditions by including the self-equilibrium forces that contain both the external loads and the reaction forces induced at the constraint locations. Moreover, the physical interpretation of the weight functions can provide a guidance for damage tolerant design application.     

Probabilistic analysis of delamination onset in linear anisotropic elastic and viscoelastic composite columns

Previously developed deterministic and stochastic combined load invariant failure criteria are used to determine the onset of delamination in elastic and viscoelastic columns. The analysis includes the effects of initial imperfections as well as offset column loads and transverse shear contributions. The delamination predictions are found to be sensitive to the magnitude of applied loads and of initial imperfections. Illustrative numerical examples are presented for elastic and viscoelastic columns with random combined failure stresses in bending, shear, compression and with normal interlaminar stresses. Probabilities of delamination onset are established for various axial loads and initial imperfections and in the viscoelastic columns additionally as a function of lifetime. Since the failure theories consider the combined effects of bending, shear, compression and normal interlaminar stresses, delamination onset is predicted at smaller axial loads than the critical buckling loads in the elastic case and at shorter viscoelastic lifetimes compared to equivalent columns with no delamination effects.     

The behaviour of FRP wrapped HSC columns under different eccentric loads

The majority of columns are subjected to a combination of an axial load and a bending moment in one or two directions. With a few exceptions, most of the research in the area of FRP wrapped columns have concentrated on the behaviour of concentrically loaded columns. This paper presents results of testing nine reinforced high strength concrete columns. The column specimens are circular in shape with 205 mm diameter and 925 mm height. Concrete compressive strength was 65 MPa. All columns were reinforced with steel. Three columns were not wrapped, three columns were wrapped with three layers of carbon FRP and three columns were wrapped with three layers of E-Glass FRP. From each of the three groups, one column was tested concentrically, one column was tested with a 25 mm eccentric load and one column was tested with a 50 mm eccentric load. Results of testing the columns have shown that the carbon FRP is most effective in increasing the strength and ductility of columns.     

Optimal Design of Fibre-reinforced Laminated Plates Accounting for Manufacturing Uncertainty

A procedure to design symmetrically laminated plates under buckling loads for minimum weight with manufacturing uncertainty (tolerance) in the ply angle and plate thickness, which are the design variables, is described. A minimum buckling load capacity is the design constraint implemented. It is assumed that the probability of any tolerance value occurring within the tolerance band, compared with any other, is equal, and thus the approach is a worst case scenario approach. The effects of bending–twisting coupling are neglected in implementing the procedure, and the Downhill Simplex method is used as the search technique, but the methodology is flexible and allows any appropriate problem formulation and search algorithm to be substituted. Two different tolerance scenarios are used for the purposes of illustrating the methodology, and plates with varying aspect ratios and loading ratios are optimally designed and compared. The results demonstrate the importance of carrying out design optimisation of composite structures with the effects of manufacturing tolerances included.     

Repair and strengthening of reinforced concrete square columns using ferrocement jackets

A series of 10 one-third scale square reinforced concrete column specimens were cast; preloaded under axial compression up to various fractions (0%, 60%, 80%, and 100%) of its ultimate load; repaired using ferrocement jackets containing two layers of Welded Wire Mesh (WWM) encapsulated in high strength mortar; and then retested to failure. The overall response of the specimens was investigated in terms of load carrying capacity, axial displacement, axial stress and strain, lateral displacement, and ductility. The test results indicated that jacketing reinforced concrete square columns with this form of ferrocement provided about 33% and 26% increases in axial load capacity and axial stiffness, respectively, compared to the control columns. The test results also indicated that repairing similar reinforced concrete columns (after preloading them to failure) with the same ferrocement jacket almost restored their original load capacity and stiffness. Furthermore, the repaired columns failed in a ductile manner compared to the brittle failure exhibited by the control columns.     

轴向时变冲击载荷作用下直杆弹性动力屈曲研究

基于应力波理论,用半解析半数值方法对轴向时变冲击载荷作用下的直杆进行研究,给出了一种利用压应力波前附加约束条件求解轴向时变载荷作用下直杆弹性动力屈曲问题的方法。以三角脉冲载荷作用下的直杆为例,对其临界屈曲长度、初始屈曲模态和动力特征参数进行了求解,探讨了脉冲载荷峰值和载荷持续时间对临界屈曲长度和屈曲模态的影响。总结了三角脉冲载荷作用下直杆弹性动力屈曲的规律,并与阶跃载荷作用下的情况进行对比分析,结果与之前文献研究结果吻合良好。     

Effect of loading rate on dynamic fracture initiation toughness of brittle materials

Numerical simulation is carried out to investigate the effect of loading rate on dynamic fracture initiation toughness including the crack-tip constraint. Finite element analyses are performed for a single edge cracked plate whose crack surface is subjected to uniform pressure with various loading rate. The first three terms in the Williams’ asymptotic series solution is utilized to characterize the crack-tip stress field under dynamic loads. The coefficient of the third term in Williams’ solution, A ₃, was utilized as a crack tip constraint parameter. Numerical results demonstrate that (a) the dynamic crack tip opening stress field is well represented by the three term solution at various loading rate, (b) the loading rate can be reflected by the constraint, and (c) the constraint A ₃ decreases with increasing loading rate. To predict the dynamic fracture initiation toughness, a failure criterion based on the attainment of a critical opening stress at a critical distance ahead of the crack tip is assumed. Using this failure criterion with the constraint parameter, A ₃, fracture initiation toughness is determined and in agreement with available experimental data for Homalite-100 material at various loading rate.     

A model for the compressive buckling of extended chain polymers

A model for the compressive buckling of an extended polymer chain is presented. The application of classical elastic instability analysis to an idealized polymer chain reveals that the bending rigidity and critical buckling loads for a chain are proportional to the force constants for valence bond angle bending and torsion. Highly oriented polymer fibres are treated as a collection of elastic chains that interact laterally. The critical stresses to buckle this collection of chains are calculated following a procedure developed to predict the compressive strengths of fibre-reinforced composites. This buckling stress is predicted to be equal to the shear modulus of the fibres and is the limiting value of compressive strength. Comparison of experimental and predicted values shows that the theory overestimates the compressive strength, but that there is a correlation of shear modulus with axial compressive strength. Consideration of flaws in both the theory and the material indicate that the compressive strength should be proportional to either the shear modulus or shear strength of the fibres.Nomenclature P axial compressive load (force) - P _cr critical buckling load (force) - M,M _i bending moments - l length of a link - p number of links - k elastic hinge constant - , _i angular rotation of hinges - L overall chain or column length - v,v _i lateral deflection of buckled chain or column - x, y, z Cartesian coordinate axes - E Young's modulus of isotropic column - I moment of inertia - a _ij matrix coefficients - A _p coefficient for exact buckling loads of chains - T energy change due to work of external load on buckled column or chain - U ₁ bending strain energy change of buckled column or chain - U ₂, U ₂ ^e , U ₂ ^s strain energy changes in elastic foundation, where e refers to extension mode buckling and s refers to shear mode buckling - E _t transverse modulus - G longitudinal shear modulus - b dimension associated with chain packing - A cross-sectional area per chain (=b ²) - f(x) curve fitted to shape of buckled chain - m,n,r integers - a _n coefficients of trigonometric series - _y normal strain iny-direction - _y normal stress iny-direction - xy shear strain inxy plane - xy shear stress inxy plane - u _x displacement inx-direction - u _y displacement iny-direction - V volume     

Effects of creep properties of materials on unified creep constraint parameter Ac for cracked pipes

Extensive finite element analyses of cracked pipes with different crack sizes and orientations have been conducted to investigate effects of creep properties of materials on the unified creep constraint parameter A_c. The results show that the constraint parameter A_c is independent on Norton’s coefficient A, and it is only affected by the creep exponent n of materials. For a given crack size, with increasing n, A_c decreases and constraint level increases. The A_c of lower constraint cracks is more sensitive to n. The unified correlation equations between A_c and n have been obtained for cracked pipes with a wide range of crack sizes and constraint levels. They may be used to estimate the constraint parameter A_c at different positions along the crack fronts in cracked pipes made of materials with different n values. The two-parameter C*-A_c approach for assessing creep life of cracked pipes has also been discussed.     

Limit loads and approximate J estimates for axial through-wall cracked pipe bends

This paper presents plastic limit loads and approximate J estimates for axial through-wall cracked pipe bends under internal pressure and in-plane bending. These loads and estimates are based on small strain finite element limit analyses using elastic-perfectly plastic materials. Geometric variables associated with the crack and pipe bend are systematically varied, and three possible crack locations (intrados, crown and extrados) are considered. Effects of the bend and crack geometries on plastic limit loads are quantified, and closed-form limit load solutions are given. Based on the proposed limit load solutions, a reference stress based the J estimation scheme for axial through-wall cracked pipe bends under internal pressure and in-plane bending is proposed.     

460MPa高强钢箱形截面轴压柱局部稳定有限元分析和设计方法研究

为研究460MPa高强度钢材等边箱形截面轴心受压柱的局部稳定受力性能和设计方法, 该文采用有限元软件ANSYS建立有限元模型, 考虑残余应力和局部初始几何缺陷的影响, 对3组共10个460MPa等边箱形截面轴心受压短柱的局部屈曲进行了有限元分析, 并将极限承载力计算值与试验实测值进行对比, 结果证明该模型能够准确分析计算460MPa高强度钢材等边箱形截面轴心受压柱的局部屈曲受力性能。利用该模型进行有限元参数分析, 与钢板的宽厚比相比, 钢板厚度、钢板长宽比、残余压应力值和局部初始几何缺陷幅值等因素对局部屈曲受力性能的影响都很小。将试验实测值和参数分析的结果与中国、美国和欧洲钢结构设计规范中箱形截面局部稳定设计曲线进行对比, 表明现行规范的设计方法不够安全。该文提出的建议设计公式适用于460MPa高强度钢材等边箱形截面轴心受压柱局部屈曲极限应力的设计计算。