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.     

Identification of Cracking in Beam Structures Using Timoshenko and Euler Formulations

Timoshenko and Euler beam formulations, using energy approach, have been used to estimate the influence of crack size and location on the natural frequencies of cracked beams. Fracture mechanics approach has been used to consider the effect of cracking on the dynamic response of the beam. Galerkin’s approach has been used to solve the problem numerically. It is shown that for slender beams the deep beam influence is felt only when the [(basic?bending?length)/h] ratio of the fundamental sinusoid of a beam becomes very small for higher modes. When the (l/h) ratio becomes small (<10), the influence of shear rotation and rotary inertia effects become dominant; the inclusion of these effects makes the beam less stiff than a Euler beam. The crack influence on Euler and Timoshenko beams are similar for beams with l/h>10; but when l/h<10, the results of cracked Euler and Timoshenko beams slowly become different and diverge. The frequency contour method identifies the crack size and location properly, using the lower order frequencies. When structural symmetry gives an ambiguity regarding the crack location, the vibration behavior of the same beam with an asymmetrically placed mass, in conjunction with the frequency contour method, would uniquely identify the crack size and location.     

Modeling of an Unbonded CFRP Strap Shear Retrofitting System for Reinforced Concrete Beams

A retrofitting technique has been developed that uses carbon fiber-reinforced polymer (CFRP) straps to increase the shear capacity of reinforced concrete beams. The vertical straps are not bonded to the beam but are instead anchored against the beam, which makes this technique potentially more effective than bonded FRP retrofitting techniques. However, it also means that models for bonded FRPs are not appropriate for use with the straps. Instead, a model based on a shear friction approach has been developed where the strain in the straps is calculated based on a term that accounts for the effects of prestress and additional strain in the strap due to shear crack opening. The model can either consider the shear reinforcement to be smeared along the length of the beam or discrete elements. The “smeared” model was checked against an experimental database consisting of rectangular, T-, and deep beams, both in terms of predicted capacity and predicted strain in the straps. Overall the smeared model predicted the capacity of the specimens and, with some adjustments, the strains quite accurately. There were, however, cases when it was more appropriate to use the “discrete” model such as when the transverse reinforcement ratio was low or when the transverse reinforcement spacing was high. Further experimental data are required to fully validate the models and to determine appropriate limits on the use of the smeared model and the discrete model. However, the initial results are promising.     

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.     

Behavior of RC Beams Shear Strengthened with Bonded or Unbonded FRP Wraps

Reinforced concrete (RC) beams shear-strengthened with fiber-reinforced polymer (FRP) fully wrapped around the member usually fail due to rupture of FRP, commonly preceded by gradual debonding of the FRP from the beam sides. To gain a better understanding of the shear resistance mechanism of such beams, particularly the interaction between the FRP, concrete, and internal steel stirrups, nine beams were tested in the present study: three as control specimens, three with bonded FRP full wraps, and three with FRP full wraps left unbonded to the beam sides. The use of unbonded wraps was aimed at a reliable estimation of the FRP contribution to shear resistance of the beam and how bonding affects this contribution. The test results show that the unbonded FRP wraps have a slightly higher shear strength contribution than the bonded FRP wraps, and that for both types of FRP wraps, the strain distributions along the critical shear crack are close to parabolic at the ultimate state. FRP rupture of the strengthened beams occurred at a value of maximum FRP strain considerably lower than the rupture strain found from tensile tests of flat coupons, which may be attributed to the effects of the dynamic debonding process and deformation of the FRP wraps due to the relative movements between the two sides of the critical shear crack. Test results also suggest that while the internal steel stirrups are fully used at beam shear failure by FRP rupture, the contribution of the concrete to the shear capacity may be adversely affected at high values of tensile strain in FRP wraps.     

Size Effects for Reinforced Concrete Beams Strengthened in Shear with CFRP Strips

The principal motivation of this study is to obtain a clear understanding of size effects for fiber-reinforced polymer (FRP) shear-strengthened beams. The experimental program consists of seven beams of various sizes grouped in three test series. One beam of each series is used as a benchmark and its behavior is compared with a beam strengthened with a U-shaped carbon FRP (CFRP) jacket. The third test series includes an additional beam strengthened with completely wrapped external CFRP sheets. The experimental results show that the effective axial strains of the CFRP sheets are higher in the smaller specimens. Moreover, with a larger beam size, one can expect less strain in the FRPs. A nonlinear finite-element numerical analysis is developed to model the behavior of the CFRP shear-strengthened beams. The numerical model is able to simulate the characteristics of the shear-strengthened beams, including the interfacial behavior between the concrete and the CFRP sheets. Three prediction models available in current design guidelines for computing the CFRP effective strain and shear contribution to the shear capacity of the CFRP shear-strengthened beams are compared with the experimental results.     

Large-Deflection Stability of Slender Beam-Columns with Both Ends Partially Restrained against Rotation

The large-deflection analysis and postbuckling behavior of laterally braced or unbraced slender beam columns of symmetrical cross section subjected to end loads (forces and moments) with both ends partially restrained against rotation including the effects of out-of-plumbness is developed in a classical manner. The classical theory of the “Elastica” and the corresponding elliptical functions utilized herein are those presented previously by the senior writer. The proposed method can be used in the large-deflection elastic analysis and postbuckling behavior of slender beam columns with rigid, semirigid, and simple flexural connections and both ends. Only bending strains are considered, i.e., the effects of axial and shear strains are neglected. An example is included that shows the effects of flexible connections at both ends on the large-deflection analysis and postbuckling behavior of slender beam columns.     

Assessment of Construction Joint Effect in Full-Scale Concrete Beams by Acoustic Emission Activity

In the present paper the mechanical and acoustic emission (AE) behaviors of full-scale reinforced concrete beams are evaluated. One of the beams was constructed in two parts, which were assembled later in order to evaluate the effect of the joints in the structural behavior. The load was applied by means of a four-point-bending configuration. It is revealed that at initial stages of loading, the conventional measurements of strain and deflection, as well as pulse velocity, do not show any discrepancy, although the structural performance of the two beams is eventually proven to be quite different. On the contrary, AE parameters, even from early load steps, indicate that the damage accumulation is much faster in the assembled beam. This is confirmed by the calculated sources of AE events which are close to the construction joints. The results show that the AE technique is suitable to monitor the deterioration process of full-scale structures and yields valuable information that cannot be obtained at the early stages of damage by any other way.     

Spatial Postbuckling Analysis of Nonsymmetric Thin-Walled Frames.?II: Geometrically Nonlinear FE Procedures

In a companion paper, transformation rules of Rodriguez' finite rotations and semitangential rotations are derived, respectively, and their rotational properties are discussed. The shear deformable displacement field of nonsymmetric thin-walled space frames is introduced based on semitangential rotations and the potential energy corresponding to semitangential internal moments are consistently derived using the proposed displacement field. In this paper, for spatial postbuckling analysis of thin-walled space frames, the elastic strain energy including bending-torsion coupled terms and shear deformation effects is derived using transformations of displacements and stress resultants defined at the centroid and the centroid-shear center, respectively. Tangent stiffness matrices of the frame element are derived by using Hermitian polynomials including shear effects, and an improved corotational formulation is presented by separating rigid body motions and pure deformations from incremental displacements, calculating the corresponding generalized forces and updating direction cosines of frame elements. In addition, a scheme to evaluate load correction stiffness matrices due to the off-axis loading and conservative moments is addressed. FE solutions for the postbuckling are presented and compared with results by ABAQUS shell models.     

Lateral–Torsional Buckling of Singly Symmetric Tapered Beams: Theory and Applications

A general variational formulation to analyze the elastic lateral–torsional buckling (LTB) behavior of singly symmetric thin-walled tapered beams is presented, numerically implemented, validated and illustrated. It (1) begins with a precise geometrical definition of a tapered beam; (2) extends the kinematical assumptions traditionally adopted to study the LTB of prismatic beams; (3) includes a careful derivation of the beam total potential energy; and (4) employs Trefftz’s criterion to ensure the beam adjacent equilibrium. In order to validate and illustrate the application and capabilities of the proposed formulation, several numerical results are presented, discussed and, when possible, also compared with values reported by other authors. These results (1) are obtained by means of the Rayleigh–Ritz method, using trigonometric functions to approximate the beam critical buckling mode, and (2) concern the critical moments of doubly and singly symmetric web-tapered I-section simply supported beams and cantilevers acted by point loads. In particular, one shows that modeling a tapered beam as an assembly of prismatic beam segments is conceptually inconsistent and may lead to rather inaccurate (safe or unsafe) results. Finally, it is worth mentioning that the paper includes a state-of-the-art review concerning one-dimensional analytical formulations for the LTB behavior of tapered beams.     

Nonlinear Zigzag Theory for Buckling of Hybrid Piezoelectric Rectangular Beams under Electrothermomechanical Loads

A new efficient electromechanically coupled geometrically nonlinear (of von Karman type) zigzag theory is developed for buckling analysis of hybrid piezoelectric beams, under electrothermomechanical loads. The thermal and potential fields are approximated as piecewise linear in sublayers. The deflection is approximated as piecewise quadratic to explicitly account for the transverse normal strain due to thermal and electric fields. The longitudinal displacement is approximated as a combination of third order global variation and a layerwise linear variation. The shear continuity conditions at the layer interfaces and the shear traction-free conditions at the top and bottom are used to formulate the theory in terms of three primary displacement variables. The governing coupled nonlinear field equations and boundary conditions are derived using a variational principle. Analytical solutions for buckling of symmetrically laminated simply supported beams under electrothermal loads are obtained for comparing the results with the available exact two-dimensional (2D) piezothermoelasticity solution. The comparison establishes that the present results are in excellent agreement with the 2D solution which neglects the prebuckling transverse strain effect.     

Negative regulation of the anti-human immunodeficiency virus and chemotactic activity of human stromal cell-derived factor 1alpha by CD26/dipeptidyl peptidase IV

A method to characterize the energy distribution in the whole photon field is valuable when designing an accelerator for choosing target and flattening filter or scan pattern. Another field of application is beam characterization for treatment planning systems or other dosimetric purposes. This work is focused on the energy distribution in different 50 MV bremsstrahlung beams with different scanning of electrons on three different targets. Fluence differential in energy and angle at the exit of each target has been determined by Monte Carlo calculations for a narrow beam. Data for broad beams were obtained by convolution of the narrow beams with different scan patterns. Photon energy fluence differential in energy at SSD 100 were thus found to be rather different for the targets studied. The results are presented as mean energy profiles and narrow beam half-value layer (HVL) in water. Two different experimental setups were used to measure HVL at the central axis and at off-axis positions. The two methods gave results which differ by 5%-6% and the calculated data where within these experimental results. In conclusion, the presented method for characterization of the photon field energy distribution is well within the experimental results and can thus be used to improve accelerator design or dosimetric calculations, e.g., for treatment planning.     

Buckling and Postbuckling Behavior of Cross-Ply Composite Plate Subjected to Nonuniform In-Plane Loads

This paper presents a study of buckling and postbuckling behaviour of simply supported composite plates subjected to nonuniform in-plane loading. The mathematical model is based on higher order shear deformation theory incorporating von Kármán nonlinear strain displacement relations. Because the applied in-plane edge load is nonuniform, in the first step the plane elasticity problem is solved to evaluate the stress distribution within the prebuckling range. Using these stress distributions, the governing equations for postbuckling analysis of composite plates are obtained through the theorem of minimum potential energy. Adopting Galerkin’s approximation, the governing nonlinear partial differential equations are reduced into a set of nonlinear algebraic equations in the case of postbuckling analysis, and homogeneous linear algebraic equations in the case of buckling analysis. The critical buckling load is obtained from the solution of associated linear eigenvalue problem. Postbuckling equilibrium paths are obtained by solving nonlinear algebraic equations employing the Newton-Raphson iterative scheme. Explicit expressions for the plate in-plane stress distributions within the prebuckling range are reported for isotropic and composite plates subjected to parabolic in-plane edge loading. Buckling loads are determined for three plate aspect ratios (a/b = 0.5, 1, 1.5) and three different types of in-plane load distributions. The effect of shear deformation on the buckling loads of composite plate is reported. The present buckling results are compared with previously published results wherever possible.     

Planar Bending of Sandwich Beams with Transverse Loads off the Centroidal Axis

Conventional analysis methods for beams do not distinguish between transverse loads that are applied at the beam centroidal axis and those acting either above or below the centroidal axis. In contrast, this paper formulates a sandwich beam finite element solution which models the effect of load height relative to the centroidal axis. Towards this goal, the governing equilibrium equations and associated boundary conditions are derived based on a Timoshenko beam formulation for the core material. Special shape functions satisfying the homogeneous form of the equilibrium equations are derived and subsequently used to formulate exact stiffness matrices. By omitting the stiffness terms related to the faces, the formulation for a homogeneous Timoshenko beam can be recovered. Also, the Euler–Bernouilli counterpart of the formulation is recovered as a limiting case of the current Timoshenko beam formulation. Effects of load height relative to the centroid are observed to have similarities with those induced by axial forces in beam-columns. For a simply supported beam, downward acting loads located below the centroidal axis are found to induce a stiffening effect while those acting above the centroidal axis are found to induce a softening effect, resulting in higher transverse displacements.     

Recurrent Single Delaminated Beam Model for Vibration Analysis of Multidelaminated Beams

Free vibration analysis of a through-width multidelaminated beam is performed in the present study. Multiple delaminations are assumed to spread from the top through the thickness direction of the beam. The natural frequencies of the multidelaminated beams are obtained from a recurrent single delaminated beam (RSDB) model, which is the subsingle delaminated beam from the top surface of a global beam. Each frequency equation for the RSDB with unknown boundary conditions is obtained through continuity conditions. Then this result is updated to the next one. With these sequential operations, the final frequency equation of the multidelaminated beams is obtained for both end boundary conditions of the global beam. The numerical results for the beams are compared with those of finite element analysis to give the reliance on the proposed model and to investigate the effects of the shape, number, and size of multidelaminations on the natural frequency. It was shown that the variations in the natural frequency for the multidelaminated beams were significantly affected by the delamination length.     

Interaction between Steel Stirrups and Shear-Strengthening FRP Strips in RC Beams

RC beams shear strengthened with either fiber-reinforced polymer (FRP) U-jackets/U-strips or side strips commonly fail due to debonding of the bonded FRP shear reinforcement. As such debonding occurs in a brittle manner at relatively small shear crack widths, some of the internal steel stirrups may not have reached yielding. Consequently, the yield strength of internal steel stirrups in such a strengthened RC beam cannot be fully used. In this paper, a computational model for shear interaction between FRP strips and steel stirrups is first presented, in which a general parabolic crack shape function is employed to represent the widening process of a single major shear crack in an RC beam. In addition, appropriate bond-slip relationships are adopted to accurately depict the bond behavior of FRP strips and steel stirrups. Numerical results obtained using this computational model show that a substantial adverse effect of shear interaction generally exists between steel stirrups and FRP strips for RC beams shear strengthened with FRP side strips. For RC beams shear strengthened with FRP U-strips, shear interaction can still have a significant adverse effect when FRP strips with a high axial stiffness are used. Therefore, for accurate evaluation of the shear resistance of RC beams shear strengthened with FRP strips, this adverse effect of shear interaction should be properly considered in design.     

Spatial Stability of Nonsymmetric Thin-Walled Curved Beams. I: Analytical Approach

An improved formulation for spatial stability of thin-walled curved beams with nonsymmetric cross sections is presented based on the displacement field considering both constant curvature effects and the second-order terms of finite-semitangential rotations. By introducing Vlasov's assumptions and invoking the inextensibility condition, the total potential energy is derived from the principle of linearized virtual work for a continuum. In this formulation, all displacement parameters and the warping function are defined at the centroid axis so that the coupled terms of bending and torsion are added to the elastic strain energy. Also, the potential energy due to initial stress resultants is consistently derived corresponding to the semitangential rotation and moment. Analytical solutions are newly derived for in-plane and lateral-torsional buckling of monosymmetric thin-walled curved beams subjected to pure bending or uniform compression with simply supported conditions. In a companion paper, finite-element procedures for spatial buckling analysis of thin-walled circular curved beams under arbitrary boundary conditions are developed by using thin-walled straight and curved beam elements with nonsymmetric sections. Numerical examples are presented to demonstrate the accuracy and the practical usefulness of the analytical and numerical solutions.     

Behavior of FRP-Retrofitted Joints Built with Plain Bars and Low-Strength Concrete

Two series of tests on eight full-scale exterior beam-column joint subassemblages built with plain bars and low-strength concrete were conducted. No transverse reinforcement was present in the joint cores. In the first series of tests, which included three specimens, the behavior of joints before fiber-reinforced polymer (FRP) retrofitting was investigated. In the second series, which included five specimens, the behavior of the FRP-retrofitted joints was investigated. The six specimens consisted of a column, an in-plane beam, a transverse beam, and a slab part, and two specimens were plane members without transverse beams and slabs. The utilized retrofitting scheme is easily applicable for actual exterior beam-column joints, even in the presence of a transverse beam and a slab. Two types of strength limitation were observed for specimens in the first series. The strength of the specimen with beam longitudinal bars sufficiently anchored to the joint core was limited by the shear strength of the joint. The strengths of the other two specimens were limited by the slip of the beams’ longitudinal bars at their anchorages. In the second series of tests, significantly better performance was obtained both in terms of shear strength and ductility, provided that the slip of the beam bars was prevented. Furthermore, by using a simple theoretical algorithm based on truss analogy, the strength and deformability characteristics of the tested reference and FRP-retrofitted joints are predicted with reasonable accuracy. The same algorithm is used for predicting the joint shear strength of specimens tested by other researchers, and satisfactory agreement is obtained between the predictions and test results.     

Intensity modulation with electrons: calculations, measurements and clinical applications

Intensity modulation of electron beams is one step towards truly conformal therapy. This can be realized with the MM50 racetrack microtron that utilizes a scanning beam technique. By adjusting the scan pattern it is possible to obtain arbitrary fluence distributions. Since the monitor chambers in the treatment head are segmented in both x- and y-directions it is possible to verify the fluence distribution to the patient at any time during the treatment. Intensity modulated electron beams have been measured with film and a plane parallel chamber and compared with calculations. The calculations were based on a pencil beam method. An intensity distribution at the multileaf collimator (MLC) level was calculated by superposition of measured pencil beams over scan patterns. By convolving this distribution with a Gaussian pencil beam, which has propagated from the MLC to the isocentre, a fluence distribution at isocentre level was obtained. The agreement between calculations and measurements was within 2% in dose or 1 mm in distance in the penumbra zones. A standard set of intensity modulated electron beams has been developed. These beams have been implemented in a treatment planning system and are used for manual optimization. A clinical example (prostate) of such an application is presented and compared with a standard irradiation technique.     

Initial recombination in a parallel-plate ionization chamber exposed to heavy ions

For exact determination of absorbed dose in heavy-ion irradiation fields which are used in radiation therapy and biological experiments, ionization chambers have been characterized with defined heavy-ion beams and correction factors. The LET (linear energy transfer) dependence of columnar recombination in a parallel-plate ionization chamber has been examined. Using 135 MeV/u carbon and neon beams, the ion collection efficiency was measured for several gases (air, carbon dioxide, argon and tissue-equivalent gas). 95 MeV/u argon beams and 90 MeV/u iron beams were also used for measurements of columnar recombination in air. As expected by Jaffe theory, the inverse of the ratio of the ionization charge to the saturated ionization charge had a linear relationship with the inverse of the electric field strength in the region below 0.002 V(-1) cm. The gradient of the line increases as the LET of the heavy ions increases. A strong LET dependence of the gradient was observed in air and carbon dioxide. The LET dependence was not observed in tissue-equivalent gas, nitrogen or argon. The exact depth-dose distribution of the heavy-ion beam was obtained by this correction of the initial recombination effect for the collected ionization charge. The columnar recombination in air was analysed using Jaffe theory; the obtained parameter b (a track radius) should be in the range between 0.001 cm and 0.005 cm, whereas the value obtained by Jaffe is 0.00179 cm. The value of the parameter b should increase as the LET of the heavy-ion beam increases in order to reproduce the experimental values of the initial recombination.