Dynamic Modal Analysis and Stability of Cantilever Shear Buildings: Importance of Moment Equilibrium

The dynamic modal analysis (i.e., the natural frequencies, modes of vibration, generalized masses, and modal participation factors) and static stability (i.e., critical loads and buckling modes) of two-dimensional (2D) cantilever shear buildings with semirigid flexural restraint and lateral bracing at the base support as well as lumped masses at both ends and subjected to a linearly distributed axial load along its span are presented using an approach that fulfills both the lateral and moment equilibrium conditions along the member. The proposed model includes the simultaneous effects and couplings of shear deformations, translational and rotational inertias of all masses considered, a linearly applied axial load along the span, the shear force component induced by the applied axial force as the member deforms and the cross section rotates, and the rotational and lateral restraints at the base support. The proposed model shows that the stability and dynamic behavior of 2D cantilever shear buildings are highly sensitive to the coupling effects just mentioned, particularly in members with limited rotational restraint and lateral bracing at the base support. Analytical results indicate that except for members with a perfectly clamped base (i.e., zero rotation of the cross sections), the stability and dynamic behavior of shear buildings are governed by the flexural moment equation, rather than the second-order differential equation of transverse equilibrium or shear-wave equation. This equation is formulated in the technical literature by simply applying transverse equilibrium “ignoring” the flexural moment equilibrium equation. This causes erroneous results in the stability and dynamic analyses of shear buildings with base support that is not perfectly clamped. The proposed equations reproduce, as special cases: (1) the nonclassical vibration modes of shear buildings including the inversion of modes of vibration when higher modes cross lower modes in shear buildings with soft conditions at the base, and the phenomena of double frequencies at certain values of beam slenderness (L/r); and (2) the phenomena of tension buckling in shear buildings. These phenomena have been discussed recently by the writer (2005) in columns made of elastomeric materials.     

Approximate Derivation of Critical Buckling Load

This paper proposes an approximate derivation for the critical buckling load of a column, based on the application of a uniformly loaded beam's midspan moment and deflection to the buckled column's rotational equilibrium. The curvature of a pin-ended member, when it buckles under axial load, is similar to the curvature assumed by the same member when it deflects under a uniformly distributed load applied transversely along its entire length. Euler's famous equation for critical buckling load is based, of course, on the former assumption, in which the deflected column assumes the shape of a sine curve. However, dividing a uniformly loaded beam's midspan moment by its deflection provides a conservative result for the critical buckling load, within 3% of Euler's value, that can be derived solely on the basis of these commonly used beam equations.     

Static Stability Formulas of a Weakened Timoshenko Column: Effects of Shear Deformations

The static stability analysis of two-dimensional Timoshenko columns weakened at an arbitrary section is derived in a classic manner. The effects of shear deformations along the column, influenced by the additional shear force induced by the applied axial load as the member deforms according to the modified shear equation proposed by Haringx, are presented and studied in detail. The proposed model also captures: (1) the influence on the buckling load capacity of the column when an arbitrary weakened section is formed at any location; (2) the tension buckling phenomenon due to the low shear stiffness of columns made of composite materials or elastomeric rubbers; and (3) the beneficial effects of an additional lateral bracing located at the weakened section to alleviate the buckling load reduction of the column. Seven classical and nonclassical cases of columns mostly used in civil and mechanical engineering are summarized in condensed formulas which allow the straightforward determination of buckling loads and shapes.     

Buckling and Postbuckling of Antisymmetrically Laminated Cross-ply Shear-Deformable Cylindrical Shells under Axial Compression

The generalized Donnell-type equations governing large deflection of antisymmetrically laminated cross-ply cylindrical shells counting for transverse shear deformations are derived and presented. An asymptotic series solution is constructed by regular perturbation technique for postbuckling behaviors of the cylindrical shells with simply supported edges subjected to axial compression. Boundary layer influence at both ends of the shells on overall buckling and postbuckling are considered, and for consistency of the boundary valued problem, the boundary layer solutions are also designed to match the out-of-plane edge conditions by singular perturbation approach. Effects of transverse shear deformation, Batdorf’s parameter, elastic moduli ratio, and initial geometric imperfection on buckling and postbuckling performance of the shells are examined. Some numerical examples are taken for comparison of the present results of buckling loads and load–deflection curves of the shells with corresponding theoretical predictions to show effectiveness and accuracy of the present asymptotic perturbation solution.     

Influence of Shear Deformation on Buckling of Pultruded Fiber Reinforced Plastic Profiles

Theoretical studies of the influence of shear deformation on the flexural, torsional, and lateral buckling of pultruded fiber reinforced plastic (FRP)-I-profiles are presented. Theoretical developments are based on the governing energy equations and full section member properties. The solution for flexural buckling is consistent with the established solution based on the governing differential equation. The new solutions for torsional and lateral buckling incorporate a reduction factor similar to that for flexural buckling. The solution for lateral buckling also incorporates the influence of prebuckling displacements. Closed form solutions for a series of simply supported, pultruded FRP I-profiles, based on experimentally determined full section flexural and torsional properties, indicate the following conclusions. For members subjected to axial compression, shear deformation can reduce the elastic flexural and torsional buckling loads by up to approximately 15% and 10%, respectively. For members subjected to bending, prebuckling displacements can increase the buckling moments by over 20% while shear deformation decreases the buckling moments by less than 5%.     

Timoshenko Beam-Column with Generalized End Conditions and Nonclassical Modes of Vibration of Shear Beams

The main objective of this publication is to derive, in a classic manner, the characteristic equations for the undamped natural frequencies and the corresponding modes of vibration of a two-dimensional (2D) Timoshenko beam–column with generalized support conditions (i.e., with semirigid flexural restraints and lateral bracings as well as lumped masses at both ends) and subjected to a constant axial load along its span. The model includes the simultaneous effects (or couplings) of bending and shear deformations, translational and rotational inertias of all masses considered. The proposed model is general, showing that the natural frequencies and the corresponding modes of vibration of 2D beam–columns are highly sensitive to the coupling effects just mentioned. This is particularly true in members with low shear stiffness and with the end flexural restraints and lateral bracing approaching those of free–free and pinned–free conditions. A second objective of this paper is to show that the obtained solution reproduce, as a special case, the nonclassical modes of shearbeams, including the inversion of modes of vibration (i.e., higher modes crossing lower modes) in shear beams with pinned–free and free–free end conditions, and the phenomena of double frequencies at certain values of beam slenderness (r/L).     

Shear Lag Analysis in Prestressed Concrete Box Girders

Most studies on shear lag effect in box girders are only concerned about concentrated loads and uniformly distributed loads. In this paper, a finite-element method based on the variational principle is presented to analyze the effect of prestressing on shear lag in box girders. The procedures and main steps are listed to demonstrate how to use the proposed FEM, which is verified by the analytical method and the numerical examples. The shear lag effect in box girders with different types of support conditions under prestressing is analyzed in detail. The shear lag effect in box girders under prestressing is more apparent than that under uniformly distributed loads or vertical concentrated loads. The values and distributions of shear lag coefficients are relate to the anchorage locations of prestressing and the distributions of internal forces along the girder under the combined uniformly distributed load and prestressing. Among the conclusions of the study is that negative shear lag under the uniformly distributed load and prestressing may occur both at the midspan of a simply supported box girder and at the fixed end of a cantilever box girder.     

Composite T-Beams Using Reduced-Scale Rectangular FRP Tubes and Concrete Slabs

A composite system consisting of rectangular glass fiber reinforced polymer (GFRP) tubes connected to concrete slabs, using GFRP dowels has been developed. Seven beam specimens have been tested, including hollow and concrete-filled GFRP tubes with and without concrete slabs. Beam–slab specimens had two different shear span-to-depth ratios and one specimen had carbon–fiber reinforced polymer (CFRP)-laminated tension flange for enhanced flexural performance. Additionally, three double-shear GFRP tube-slab assemblies have been tested to assess the shear behavior of GFRP dowels, in both hollow and concrete-filled tubes. Three compression stubs of concrete-filled tubes were also tested by loading them parallel to the cross-section plane, to study GFRP web buckling behavior. The study showed that GFRP dowels performed well in shear and that composite action is quite feasible. While hollow tubes can act compositely with concrete slabs, more slip between the tube and slab would occur, compared to a concrete-filled tube-slab system. Simplified models are proposed to predict critical web buckling load of fiber reinforced polymer (FRP) tubes. Based on the models, a critical shear span-to-depth ratio of 4 was determined, below which web buckling may occur before flexural failure.     

Shear Lag Effect in Simply Supported Prestressed Concrete Box Girder

In this paper, derivation and computed formulas are provided for the shear lag coefficient in a simply supported prestressed concrete box girder under dead load. In the case of prestressed tendons having parabolic configurations, formulas to compute the shear lag effect are also developed. The magnitude of upward loading intensity caused by prestress as well as the relationship between the height of the box girder and the sag of prestressed tendons have been fully treated. Conclusions are drawn that the shear lag effect caused by dead load and prestress force is equivalent to dead load acting alone, provided that the prestressed tendon is set up with a parabolic profile. Shear lag effect caused by movable load is also analyzed according to the eccentricity of the load to the half-width ratio of the box girder. Charts were prepared to predict the shear lag coefficient for live load. Finally, having considered the shear deformation of flanges, the deflection of box girders is studied for both uniformly distributed load and concentrated load. Examples are given for illustrative purpose.     

Contribution to Shear Wrinkling of GFRP Webs in Cell-Core Sandwiches

Glass-fiber-reinforced polymer (GFRP) cell-core sandwiches are composed of outer GFRP face sheets, a foam core, and a grid of GFRP webs integrated into the core to reinforce the shear load capacity. One of the critical failure modes of cell-core sandwich structures is shear wrinkling, a local buckling failure in the sandwich webs because of shear loading. The shear wrinkling behavior of GFRP laminates with different laminate sequences, stabilized by a polyurethane foam core, was experimentally and numerically investigated. Shear wrinkling was simulated by a biaxial compression–tension setup. The results show that an increasing transverse tension load significantly decreases the wrinkling load. The decreasing effect of tension is explained by the lateral contraction because of Poisson’s effect, which causes an increase in the initial imperfections and subsequent accelerated bending.     

Static Stability of Beam-Columns under Combined Conservative and Nonconservative End Forces: Effects of Semirigid Connections

Stability criteria that evaluate the effects of combined conservative and nonconservative end axial forces on the elastic divergence buckling load of prismatic beam-columns with semirigid connections is presented using the classic static equilibrium method. The proposed method and stability equations follow the same format and classification of ideal beam-columns under gravity loads presented previously by Aristizabal-Ochoa in 1996 and 1997. Criterion is also given to determine the minimum lateral bracing required by beam-columns with semirigid connections to achieve “braced” buckling (i.e., with sidesway inhibited). Analytical results obtained from three cases of cantilever columns presented in this paper indicate that: (1) the proposed method captures the limit on the range of applicability of the Euler’s method in the stability analysis of beam-columns subjected to simultaneous combinations of conservative and nonconservative loads. The static method as proposed herein can give the correct solution to the stability of beam-columns within a wide range of combinations of conservative and nonconservative axial loads without the need to investigate their small oscillation behavior about the equilibrium position; and (2) dynamic instability or flutter starts to take place when the static critical loads corresponding to the first and second mode of buckling of the column become identical to each other. “Flutter” in these examples is caused by the presence of nonconservative axial forces (tension or compression) and the softening of both the flexural restraints and the lateral bracing. In addition, the “transition” from static instability (with sidesway and critical zero frequency) to dynamic instability (with no sidesway or purely imaginary sidesway frequencies) was determined using static equilibrium. It was found also that the static critical load under braced conditions (i.e., with sidesway inhibited) is the upper bound of the dynamic buckling load of a cantilever column under nonconservative compressive forces. Analytical studies indicate the buckling load of a beam-column is not only a function of the degrees of fixity (ρa and ρb), but also of the types and relative intensities of the applied end forces (Pci and Pfj), their application parameters (ci, ηj, and ξj), and the lateral bracing provided by other members (SΔ).     

Local Delamination Buckling of Laminated Composite Beams Using Novel Joint Deformation Models

Local delamination buckling formulas for laminated composite beams are derived based on the rigid, semirigid, and flexible joint models with respect to three bilayer beam (i.e., conventional composite, shear-deformable bilayer, and interface-deformable bilayer, respectively) theories. Two local delamination buckling modes (i.e., sublayer delamination buckling and symmetrical delamination buckling) are analyzed and their critical buckling loads based on the three joint models are obtained. A numerical finite-element simulation is carried out to validate the accuracy of the formulas, and parametric studies of delamination length ratio, the transverse shear effect, and the influence of interface compliance are conducted to demonstrate the improvement of the flexible joint model compared to the rigid and semirigid joint models. The explicit local delamination buckling solutions developed in this study facilitate the design analysis and optimization of laminated composite structures and provide simplified and improved practical design equations and guidelines for buckling analyses.     

Energy Capacity Criterion for the Design of Columns against Collapse

Current practice in buckling design is based on the ultimate strength of a member such that excessive lateral deflections do not occur. This ultimate strength is often dictated by empirically derived formulae which correspond to postulated element imperfections including residual stresses. In this paper, the postbuckling behavior of columns is considered, and the limit state is considered to be total collapse. The load versus end displacement relationship is derived for a simplified elastic–perfectly plastic column model. Using this curve, the energy absorbed and released in going from one load/deflection state to another can be easily quantified. With this, the energy absorption capacity of an already loaded column is considered to be the critical performance metric. This “energy capacity criterion” is then used to develop alternative design curves for different levels of energy absorption capacity.     

Second-Order Stiffness Matrix and Loading Vector of a Beam-Column with Semirigid Connections on an Elastic Foundation

The second-order stiffness matrix and corresponding loading vector of a prismatic beam–column subjected to a constant axial load and supported on a uniformly distributed elastic foundation (Winkler type) along its span with its ends connected to elastic supports are derived in a classical manner. The stiffness coefficients are expressed in terms of the ballast coefficient of the elastic foundation, applied axial load, support conditions, bending, and shear deformations. These individual parameters may be dropped when the appropriate effect is not considered; therefore, the proposed model captures all the different models of beams and beam–columns including those based on the theories of Bernoulli–Euler, Timoshenko, Rayleigh, and bending and shear.The expressions developed for the load vector are also general for any type or combinations of transverse loads including concentrated and partially nonuniform distributed loads. In addition, the transfer equations necessary to determine the transverse deflections, rotations, shear, and bending moments along the member are also developed and presented.     

抗弯刚度比对加劲板屈曲性能的影响

以一大型薄壁钢结构的加劲板为研究对象,采用有限元方法,考虑了13种不同的刚度比、多种不同的加劲肋布置方式以及边界条件等因素,分析了加劲板线性屈曲和非线性屈曲性能.抗弯刚度比对加劲板的屈曲性能影响显著,加劲板最佳抗弯刚度比将其线性屈曲模态划分为整体屈曲和局部屈曲,其值为10~20.加劲板非线性屈曲荷载随抗弯刚度比增大而提高.另外,在加载方向增加加劲肋布置可以提高加劲板局部屈曲荷载,在非加载方向增加加劲肋布置对加劲板的局部屈曲性能影响较小.      

Flotation to Trigger Lateral Buckles in Pipelines on a Flat Seabed

Triggering lateral buckles in subsea pipelines can be an effective way to ensure that the thermal expansion is spread over several buckles at a controlled spacing, rather than concentrating in a single buckle that may become overstressed, or causing excessive displacements at the ends of a pipeline where spool-piece connections or jumpers may become overstressed. For a given location where a buckle is to be triggered one calculates that axial load that would develop if the buckle fails to form, and ensures the buckle would be triggered at or below this axial load. This paper gives simple analytical expressions for triggering of such lateral buckles by applying buoyancy to the pipeline, e.g., by inflatable parachute-like bags, which can later be removed. The expressions apply for a single buoyancy load, two equal buoyancy loads, or distributed buoyancy over a specified length. They explicitly solve the design problem of establishing the required amount of buoyancy for a given axial load at which the buckle must be triggered. It is shown quite generally that bifurcation buckling into a lateral buckling mode occurs before vertical instability, and at moderate uplift displacements. The analytical results are based on a flat seabed, the theory of moderately large deflections, and elastic pipe behavior. An example involving a single point buoyancy load is also solved by finite-element analysis based on large deflection theory to verify the accuracy of the approach.     

Tangent Stiffness Equations for Laterally Distributed Loaded Members

Tangent stiffness equations for a beam-column, which is subjected to either uniformly or sinusoidally distributed lateral loads, are presented. The equations have been derived by differentiating the slope-deflection equations under axial forces for a member. Thus, the tangent stiffness equations take into consideration axial forces, bowing effect, and laterally distributed loads. As a numerical example, elastic buckling behavior of parallel chord latticed beams with laterally distributed loads is investigated to compare the results obtained from the present method with those from the conventional matrix method in which the distributed loads are considered as a series of concentrated loads at additional intermediate nodes of a member. Furthermore, buckling tests were carried out to confirm the equations derived as well as to clarify the buckling behavior of space frame structures. In conclusion, it can be said that the new equations can provide a good efficient way of estimating the equilibrium paths and buckling loads. They can also lead to a significant savings in core storage and computing time required for the analysis of space frame structures.     

Finite Beam Element Considering Shear-Lag Effect in Box Girder

A finite-element method considering the interaction of the bending and shear-lag deformation of a box girder was established. Meanwhile a shear-lag-induced stiffness matrix was defined. The stiffness matrices considering the effect of the shear lag were deduced. At each node of the beam element, two shear-lag degrees of freedom were used as boundary conditions for the box girders. The proposed formulations were then applied to analyze the effects of the shear lag on the deflection, the internal forces, and the shear-lag coefficients in the simply supported cantilever and continuous box beams under uniformly distributed and concentrated loads. The numerical results obtained using the proposed procedure were in good agreement with those using the finite-shell-element method, the finite-stringer method, the analytical method based on the variational principle, and the model tests. The proposed method is reliable and more effective for the analysis of the shear lag in the actual box-girder structures.     

In-Plane Nonlinear Buckling Analysis of Deep Circular Arches Incorporating Transverse Stresses

Large discrepancies exist among current classical theories for the in-plane buckling of arches that are subjected to a constant-directed radial load uniformly distributed around the arch axis. Discrepancies also exist between the classical solutions and nonlinear finite-element results. A new theory is developed in this paper for the nonlinear analysis of circular arches in which the nonlinear strain-displacement relationship is based on finite displacement theory. In the resulting variational equilibrium equation, the energy terms due to both nonlinear shear and transverse stresses are included. This paper also derives a set of linearized equations for the elastic in-plane buckling of arches, and presents a detailed analysis of the buckling of deep circular arches under constant-directed uniform radial loading including the effects of shear and transverse stresses, and of the prebuckling deformations. The solutions of the new theory agree very well with nonlinear finite-element results. Various assumptions often used by other researchers, in particular the assumption of inextensibility of the arch axis, are examined. The discrepancies among the current theories are clarified in the paper.     

Buckling of Delaminated Composite Beams with Shear Deformation Effect

A general 1D model of composite delaminated beams with shear deformation effect is derived for buckling behavior. The constitutive models of composite laminated beams are derived from the classical 2D laminate theory. The present cylindrical bending models can be used—with much greater accuracy than their well-known plane-strain and plane-stress counterparts—as upper and lower bounds toward one of which the behavior tends, depending on the width-to-length ratio. The analysis is based on the first-order Timoshenko-Mindlin kinematic approach. The differential equations are solved with the aid of a specially developed, very efficient interlaced finite-difference scheme eliminating the “shear locking” phenomenon. A parametric study of the shear deformation effect associated with various constitutive models is carried out for angle-ply delaminated laminate. It was found that the most significant difference between the models is associated with the mix of local and global modes.