Plastic Deformations of Impulsively Loaded, Rigid-Plastic Beams

Rigid body dynamics is used to determine the deformation of a fixed-end, rigid-plastic beam subjected to uniformly distributed impulsive loading. The proposed solution methodology allows calculations of deformations at plastic hinges and can be used to establish rigid-plastic fracture criteria for rigid-plastic beams. Unlike previous solutions to this problem, rotary inertia and the shear deformations at the support are considered. The solution for beam deformations is described in three phases: shear, bending, and membrane. Each phase ends when the corresponding component of the strain rate vector vanishes. The initial shear phase is completed when the transverse shear velocity at the support vanishes. The beam then undergoes only rigid body rotation and axial stretching at plastic hinges in the bending phase. The bending phase ends when the angular velocity vanishes. In the membrane phase, the beam acts like a string until the transverse velocity vanishes. It has been found that beams subjected to low impulse velocity attain permanent deformation in the bending phase, while beams subjected to high impulse velocity reach permanent deformation in the membrane phase. The predictions of the beam deflections using the proposed methodology are within 15% of the experimental results.     

Long-Term Behavior of Prestressed Old-New Concrete Composite Beams

This paper presents both theoretical and experimental studies of the long-term behavior of prestressed old-new concrete composite beams under sustained loads. General differential equations governing the relationship between the incremental deflection and incremental internal forces of the composite beams were deduced in the theoretical study. Closed-form solutions for simply supported composite beams were obtained and validated using test results reported in previous literature on steel-concrete composite beams. The experimental investigation consisted of static long-term load tests carried out on four prestressed old-new concrete composite beams. The behavior of the old-to-new concrete interface, time-dependent deflections, concrete strains, and prestress losses was carefully observed over 260?days. The long-term test program showed that the midspan deflections and concrete strains increased with time because of creep and shrinkage of the new prestressed concrete. The slip strains at the old-to-new concrete interface were found to be relatively small, indicating that the interface bond was sound enough to prevent slip and that the prestressing loads were effectively transferred to the old concrete. The proposed theoretical models predicted the long-term behavior of the prestressed old-new concrete composite beams with an acceptable degree of accuracy.     

Mixed-Mode Fracture of Hybrid Material Bonded Interfaces under Four-Point Bending

A combined analytical and experimental approach is presented to characterize mixed-mode fracture of hybrid material bonded interfaces under four-point bending load, and closed-form solutions of compliance and energy release rate (ERR) of the mixed-mode fracture specimens are provided. The transverse shear deformations in each sublayer of bimaterial bonded beams are included by modeling the specimen as individual Timoshenko beams, and the effect of interface crack-tip deformation on the compliance and ERR are taken into account by applying the interface deformable bilayer beam theory (flexible-joint model). The higher accuracy of the present analytical solutions for both the compliance and ERR of mixed-mode fracture specimens is manifested by comparing them with the solutions predicted by the conventional beam theory (CBT) and finite-element analysis (FEA). As an application example, the fracture of wood–fiber-reinforced plastic (FRP) bonded interface is experimentally evaluated by using mixed-mode fracture specimens [i.e., four-point asymmetric end-notched flexure (4-AENF) and four-point mixed-mode bending (4-MMB)], and the corresponding values of critical ERRs are obtained. Comparisons of the compliance rate change and the resulting critical ERR based on the CBT, the present theoretical model, and FEA demonstrate that the crack-tip deformation plays an important role in accurately characterizing the mixed-mode fracture toughness of hybrid material bonded interfaces under four-point bending load.     

Buckling of Long Orthotropic Plates Including Higher-Order Transverse Shear

The problem of buckling of long orthotropic plates under combined in-plane loading is considered. An approximate analytical solution is presented. The concept of a mixed Rayleigh-Ritz method is used considering higher-order shear deformations. The achieved load function of the half-buckling wavelength and the inclination of the nodal lines are minimized via a simplex search method. For low transverse shear stiffnesses the model predicts buckling coefficients under in-plane shear load that are of the same order of magnitude as those resulting from a uniaxial compressive load. For a thin plate, the critical shear load is larger by 42% compared to the uniaxial case. The model also suggests that for highly anisotropic materials, such as paper, the critical load solution is still influenced by the shear deformation effect at width-to-thickness ratios above 100.     

Elastic-Plastic Model of Pinned Beams Subjected to Impulsive Loading

An elastic-plastic dynamic analysis of simply supported beams with end membrane restraints subjected to impulsive loading is developed. The beam is elastically curved. The model takes into account elastic and plastic deformations and their effect on the stretch force and the distribution of inertia forces. Instantaneous plastification of the midspan zone due to bending and axial actions is considered. The effect of the length of the plastic zone is discussed. Equations of motion are derived by virtual work. Using the yield condition, the number of these equations is reduced by relating the variations of the displacement variables. This analysis is compared to the standard form of Lagrange's equation of motion showing that the latter is not energy conservative for this case where the deflection shape is nonlinearly dependent on the generalized displacements. Test results using a spring-powered apparatus are presented. Strain rate sensitivity is accounted for as plastic damping. The model's results are compared to test and rigid-segment model results. The comparison shows that the tests are sensitive to the curvature of the axis.     

Bond Slip Analysis of Fiber-Reinforced Polymer-Strengthened Beams

The second order differential equation of interface shear is formulated for fiber-reinforced polymer-strengthened beams using beam theory with a shear deformable adhesive layer. The solution of the boundary value problem is obtained in closed form and is used to derive deflection expressions for different loading conditions. The solution is also extended to analyze partially plated beams. The results converge to the extreme cases of very poorly and perfectly bonded plates and they help identify values of the adhesive shear modulus for effective stiffening. Furthermore, the solution of partially plated beams aids in defining anchorage lengths needed to develop the full or the highest possible composite action at midspan.     

Nonlinear Quasi-Viscoelastic Behavior of Composite Beams Curved In-Plan

A numerical formulation for the nonlinear quasi-viscoelastic (creep and shrinkage) analysis of steel-concrete composite beams that are curved in their plan is developed. The creep behavior of the concrete is considered by using the viscoelastic Maxwell-Weichert?model, in which the aging effect of the concrete is taken into account. Geometric nonlinearities and the partial shear interaction that exist at the deck-girder interface in the tangential (or longitudinal) direction and in the radial (or horizontal) direction owing to the flexibility of the shear connectors are considered in the strain-displacement relationship. The modeling based on the developed formulation is validated by comparisons with available results reported in the literature. The effects of initial curvature, partial interaction, and geometric nonlinearity on the time-dependent behavior of curved composite beams are illustrated in selected examples.     

Torsional Stiffness of Prestressing Tendons in Double-T Beams

When a prestressed double-T beam is subjected to torsion, a pair of prestressing tendons resists torsional rotation because of the restoring action of the displaced prestressing tendons. A comprehensive formulation to account for the torsional restoring action of double-T beams is presented, based on Vlasov’s hypothesis of considering warping displacement in an open-section. The deformation energies of prestressing tendons and reinforcing bars are calculated based on the deformed geometry to obtain the total potential energy. A two-noded beam element with seven degrees of freedom per node approximates an axial displacement, two translations, two flexural, and one torsional rotations, and a warping displacement to derive the finite-element equilibrium equations by minimizing the potential energy function. The role of prestressing forces of the tendons on the torsional resistance and the limitations of the traditional transformed section approach are addressed when it is applied to torsional problems. As a numerical example, an existing three-span continuous double-T beam is analyzed, and the bimoment and angle of twist are compared to those calculated using conventional three-dimensional finite-element analysis and the analytical solution of governing differential equations.     

Simple General Solution for Interfacial Stresses in Plated Beams

The flexural strength of a reinforced concrete, metallic or timber beam can be increased by bonding a thin plate, made of steel or fiber-reinforced polymer, to its tension face. A main failure mode of such plated beams involves debonding of the plate end from the beam and such plate-end debonding depends strongly on the interfacial stresses between the beam and the plate. Consequently, many analytical solutions have been developed for the interfacial stresses of specific plated beam problems, with almost all of them being for simply supported plated straight beams of constant section subjected to simple loadings. The existing analytical solutions are therefore neither general enough nor simple enough for direct exploitation in assessing the risk of plate-end debonding failure. This paper corrects this deficiency by presenting a simple, accurate yet general solution for interfacial stresses. The solution is applicable to plated beams of all geometric (e.g., curved beams), sectional (e.g., tapered beams), loading (e.g., a linearly varying distributed load), and boundary conditions (e.g., continuous beams). The accuracy of the solution is demonstrated through comparisons with finite element results. The paper also presents simple and accurate approximations for the peak values of interfacial shear and normal stresses at the plate end. In these approximate expressions, only the sectional forces and properties of the plate end section are involved, which greatly facilitates their direct exploitation in predicting debonding failure.     

In-Plane Free Vibration of Symmetric Cross-Ply Laminated Circular Bars

A parametric study is performed to investigate influences of the opening angles, the slenderness ratios, the material types, the boundary conditions, and the thickness-to-width ratios of the cross section on the in-plane natural frequencies of symmetric cross-ply laminated circular composite beams. Governing equations are obtained based on the classical beam theory. The transfer matrix method is successfully applied to calculate exact natural frequencies with the help of an effective numerical algorithm, which was previously used for isotropic materials. The effects of the shear deformation, the axial deformation, and the rotary inertia are included in the formulation based on the first-order shear deformation theory. The physical system is considered as a continuous system. To verify the present theory, two examples are worked out for straight beams. A quite good agreement is observed with the reported results.     

Spatial Stability of Shear Deformable Nonsymmetric Thin-Walled Curved Beams: A Centroid-Shear Center Formulation

An improved shear deformable curved beam theory to overcome the drawback of currently available beam theories is newly proposed for the spatially coupled stability analysis of thin-walled curved beams with nonsymmetric cross sections. For this, the displacement field is introduced considering the second order terms of semitangential rotations. Next the elastic strain energy is newly derived by using transformation equations of displacement parameters and stress resultants and considering shear deformation effects due to shear forces and restrained warping torsion. Then the potential energy due to initial stress resultants is consistently derived with accurate calculation of the Wagner effect. Finally, equilibrium equations and force–deformation relations are obtained using a stationary condition of total potential energy. The closed-form solutions for in-plane and out-of-plane buckling of curved beams subjected to uniform compression and pure bending are newly derived. Additionally, finite-element procedures are developed by using curved beam elements with arbitrary thin-walled sections. In order to illustrate the accuracy and the practical usefulness of this study, closed-form and numerical solutions for spatial buckling are compared with results by available references and ABAQUS’ shell elements.     

Uncertainty Analysis of Creep and Shrinkage Effects in Long-Span Continuous Rigid Frame of Sutong Bridge

The long-term behavior of long-span prestressed concrete continuous rigid-frame bridges is significantly sensitive to creep and shrinkage. Therefore, it is important to accurately estimate creep and shrinkage effects. This paper presents modified prediction models that are based on the creep and shrinkage models in the existing bridge code. These modified prediction models match well with the test results of the high-strength concrete used in the continuous rigid frame of the Sutong Bridge in China. Results indicate that the accuracy in predicting creep and shrinkage can be enhanced greatly by measuring short-term creep and shrinkage on the given concrete and by modifying the prediction model parameters accordingly. Subsequently, the probabilistic analysis method of structural creep and shrinkage effects was studied. Uncertainty analysis of time-dependent effects in the given bridge was performed using the modified model, and results were compared with field-test data. Two approaches for mitigating deflections that were used in the continuous rigid frame of the Sutong Bridge are introduced. Finally, the time-dependent deflection at the midspan attributable to creep and shrinkage was analyzed.     

Buckling of Multiwalled Carbon Nanotubes Using Timoshenko Beam Theory

A Timoshenko beam model is presented in this paper for the buckling of axially loaded multiwalled carbon nanotubes surrounded by an elastic medium. Unlike the Euler beam model, the Timoshenko beam model allows for the effect of transverse shear deformation which becomes significant for carbon nanotubes with small length-to-diameter ratios. These stocky tubes are normally encountered in applications such as nanoprobes or nanotweezers. The proposed model treats each of the nested and concentric nanotubes as individual Timoshenko beams interacting with adjacent nanotubes in the presence of van der Waals forces. In particular, the buckling of double-walled carbon nanotubes modeled as a pair of double Timoshenko beams is studied closely and an explicit expression for the critical axial stress is derived. The study clearly demonstrates a significant reduction in the buckling loads of the tubes with small length-to-diameter ratios when shear deformation is taken into consideration.     

Second-Order Axial Deflections of Imperfect 3-D Beam-Column

Beam-columns, in general, undergo axial elongation not only from the applied axial forces but also from the transverse deflections. A practical method that takes into account the effects of these transverse deflections on the total axial deformation of a beam-column δt is by multiplying the first-order axial stiffness AE∕L by the geometrically nonlinear factor s1 [i.e., δt = P∕(s1AE∕L)]. A general solution for s1 is derived for the combined effects of end moments, a uniformly distributed load, a series of concentrated loads, sidesway, and out-of-straightness. This solution requires numerical integration and is limited to 3D elastic prismatic beam-columns with doubly symmetrical cross sections or singly symmetrical 2D beam-columns under small strains. The proposed solution can be applied to the second-order and stability analyses of frames and to the evaluation of the axial load induced by transverse loads in beams built into rigid supports. These effects are particularly important in long-span structures. An example is presented to show the validity of the proposed formulation.     

Analytical Solution for Fracture Analysis of CFRP Sheet–Strengthened Cracked Concrete Beams

Fiber-reinforced polymer (FRP) composite materials have been widely used in the field of retrofitting. Theoretical analysis of FRP plate- or sheet-strengthened cracked concrete beams is necessary for estimating service reliability of the structural members. In previous studies, the effect of a perfectly bonded FRP plate or sheet was equivalent to a cohesive force acting at the bottom of crack to delay the crack propagation in concrete and reduce the crack width. However, delamination between FRP and cracked beam is inevitable due to interfacial shear stress concentration at the bottom of crack. The intention of this paper is to present an analytical solution for fracture analysis of carbon FRP (CFRP) sheet–strengthened cracked concrete beams by considering both vertical crack propagation in concrete and interfacial debonding at CFRP-concrete interface. The interfacial debonding is modeled as the interfacial shear crack propagation in this paper. Four different stages are discussed after initial cracking state of the concrete. At the first stage, only fictitious crack propagation occurs in the concrete. At the second stage, macrocrack propagates in the concrete without interfacial debonding. At the third stage, both vertical macrocrack propagation in the concrete and horizontal shear crack propagation at the CFRP-concrete interface occur in the strengthened beam. The tensile stress in the CFRP sheet and interfacial shear stress along the span are formulated based on the deformation compatibility condition at the CFRP-concrete interface at this stage. Finally, macroshear crack propagates at the interface until the CFRP sheet is completely peeled out from the beam, and then the member is fractured. The applied load is determined as a function of the referred two crack lengths at different stages. At the beginning, the applied load increases to one peak value with the full propagation of fictitious crack at the first stage. At the third stage, the applied load is improved to another peak value due to the relatively high cohesive effect of the CFRP sheet. Then the two peak values are determined by the Lagrange multiplier method. The validity of the proposed analytical solution is verified with the experimental results and numerical simulations. It can be concluded that the proposed analytical solution can predict the load-bearing capacity of CFRP sheet-strengthened cracked concrete beams with reasonable accuracy.     

Modeling of macrosegregation caused by volumetric deformation in a coherent mushy zone

A two-phase volume-averaged continuum model is presented that quantifies macrosegregation formation during solidification of metallic alloys caused by deformation of the dendritic network and associated melt flow in the coherent part of the mushy zone. Also, the macrosegregation formation associated with the solidification shrinkage (inverse segregation) is taken into account. Based on experimental evidence established elsewhere, volumetric viscoplastic deformation (densification/dilatation) of the coherent dendritic network is included in the model. While the thermomechanical model previously outlined (M. M’Hamdi, A. Mo, and C.L. Martin: Metall. Mater. Trans. A, 2002, vol. 33A, pp. 2081–93) has been used to calculate the temperature and velocity fields associated with the thermally induced deformations and shrinkage driven melt flow, the solute conservation equation including both the liquid and a solid volume-averaged velocity is solved in the present study. In modeling examples, the macrosegregation formation caused by mechanically imposed as well as by thermally induced deformations has been calculated. The modeling results for an Al-4 wt pct Cu alloy indicate that even quite small volumetric strains (≈2 pct), which can be associated with thermally induced deformations, can lead to a macroscopic composition variation in the final casting comparable to that resulting from the solidification shrinkage induced melt flow. These results can be explained by the relatively large volumetric viscoplastic deformation in the coherent mush resulting from the applied constitutive model, as well as the relatively large difference in composition for the studied Al-Cu alloy in the solid and liquid phases at high solid fractions at which the deformation takes place.     

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.     

Modeling Steel Frame Buildings in Three Dimensions. II: Elastofiber Beam Element and Examples

This is the second of two papers describing a procedure for the three-dimensional nonlinear time-history analysis of steel-framed buildings. An overview of the procedure and the theory for the panel zone element and the plastic hinge beam element are presented in part I. In this paper, the theory for an efficient new element for modeling beams and columns in steel frames called the elastofiber element is presented, along with four illustrative examples. The elastofiber beam element is divided into three segments—two end nonlinear segments and an interior elastic segment. The cross sections of the end segments are subdivided into fibers. Associated with each fiber is a nonlinear hysteretic stress-strain law for axial stress and strain. This accounts for coupling of nonlinear material behavior between bending about the major and minor axes of the cross section and axial deformation. Examples presented include large deflection of an elastic cantilever beam, cyclic loading of a cantilever beam, pushover analysis of a 20-story steel moment-frame building to collapse, and strong ground motion analysis of a two-story unsymmetric steel moment-frame building.     

Beam Bending Solutions Based on Nonlocal Timoshenko Beam Theory

This paper is concerned with the bending problem of micro- and nanobeams based on the Eringen nonlocal elasticity theory and Timoshenko beam theory. In the former theory, the small-scale effect is taken into consideration while the effect of transverse shear deformation is accounted for in the latter theory. The governing equations and the boundary conditions are derived using the principle of virtual work. General solutions for the deflection, rotation, and stress resultants are presented for transversely loaded beams. In addition, specialized bending solutions are given for beams with various end conditions. These solutions account for a better representation of the bending behavior of short, stubby, micro- and nanobeams where the small-scale effect and transverse shear deformation are significant. Considering particular loading and boundary conditions, the effects of small-scale and shear deformation on the bending results may be observed because of the analytical forms of the solutions.     

Preflex Beams: A Method of Calculation of Creep and Shrinkage Effects

This paper presents a method of calculation of creep and shrinkage effects for composite beams. It is particularly applicable to Preflex and Flexstress beams, which are composed of a steel I-girder with the bottom flange encased by concrete. The concrete is prestressed by predeflection of the steel beam and the subsequent release after hardening of the concrete flange or by means of prestressing cables. The presented approach using concrete age-adjusted modular ratios allows the calculation of time-dependent stresses in the concrete flange due to creep and shrinkage, with sufficient accuracy for practical applications and without carrying out cumbersome numerical computations. The results can be extended directly to the analysis of ordinary steel–concrete composite beams. The main goal of the present paper is the calibration of the parameters which must be introduced to simplify the equations describing the system. This calibration is discussed and its sensitivity to some calculation inputs is presented. The conclusions are very encouraging and the simplified approach seems to agree very well with the results of the numerical approach.