On Applications of Generalized Functions in the Discontinuous Beam Bending Differential Equations

This paper discusses the mathematical modeling for the mechanics of solid using the distribution theory of Schwartz to the beam bending differential Equations. This problem is solved by the use of generalized functions, among which is the well known Dirac delta function. The governing differential Equation is Euler-Bernoulli beams with jump discontinuities on displacements and rotations. Also, the governing differential Equations of a Timoshenko beam with jump discontinuities in slope, deflection, flexural stiffness, and shear stiffness are obtained in the space of generalized functions. The operator of one of the governing differential Equations changes so that for both Equations the Dirac Delta function and its first distributional derivative appear in the new force terms as we present the same in a Euler-Bernoulli beam. Examples are provided to illustrate the abstract theory. This research is useful to Mechanical Engineering, Ocean Engineering, Civil Engineering, and Aerospace Engineering.

Analysis of bending and buckling of pre-twisted beams:A bioinspired study

Twisting chirality is widely observed in artificial and natural materials and structures at different length scales. In this paper, we theoretically investigate the effect of twisting chiral morphology on the mechanical properties of elas- tic beams by using the Timoshenko beam model. Particular attention is paid to the transverse bending and axial buckling of a pre-twisted rectangular beam. The analytical solution is first derived for the deflection of a clamped-free beam under a uniformly or periodically distributed transverse force. The critical buckling condition of the beam subjected to its self- weight and an axial compressive force is further solved. The results show that the twisting morphology can significantly improve the resistance of beams to both transverse bending and axial buckling. This study helps understand some phenomena associated with twisting chirality in nature and provides inspirations for the design of novel devices and structures.

APPLICATIONS OF WAVELET GALERKIN FEM TO BENDING OF BEAM AND PLATE STRUCTURES

In this paper, an approach is proposed for taking calculations of high order differentials of scaling functions in wavelet theory in order to apply the wavelet Galerkin FEM to numerical analysis of those boundary-value problems with order higher than 2. After that, it is realized that the wavelet Galerkin FEM is used to solve mechanical problems such as bending of beams and plates. The numerical results show that this method has good precision.

TRANSVERSELY ISOTROPIC DAMAGE AROUND CONDUCTING CRACK-TIP IN FOUR-POINT BENDING PIEZOELECTRIC BEAM

A static damage constitutive model was proposed on basis of the electrical enthalpy density, and then some characteristics of transversely isotropic damage were discussed. Finally, the effects of both crack depth and applied loads on damage distributions were investigated through numerically analyzing transversely isotropic damage in a four_point bending PZT_PIC151 beam with a central conducting crack. Some conclusions were given: 1) Crack depth and mechanical loading have great influence on both mechanical and electrical damages.With their increment,the damages at crack_tip obviously increase and their region sizes also expand. 2) Effects of electrical loading on the two kinds of damages are obviously different. Electrical loading monotonously changes magnitude but region size of mechanical damage, whose effect on electrical damage is very complex.

Simulation of Fracture Process of Lightweight Aggregate Concrete Based on Digital Image Processing Technology

The mechanical properties and failure mechanism of lightweight aggregate concrete(LWAC)is a hot topic in the engineering field,and the relationship between its microstructure and macroscopic mechanical properties is also a frontier research topic in the academic field.In this study,the image processing technology is used to establish a micro-structure model of lightweight aggregate concrete.Through the information extraction and processing of the section image of actual light aggregate concrete specimens,the mesostructural model of light aggregate concrete with real aggregate characteristics is established.The numerical simulation of uniaxial tensile test,uniaxial compression test and three-point bending test of lightweight aggregate concrete are carried out using a new finite element method-the base force element method respectively.Firstly,the image processing technology is used to produce beam specimens,uniaxial compression specimens and uniaxial tensile specimens of light aggregate concrete,which can better simulate the aggregate shape and random distribution of real light aggregate concrete.Secondly,the three-point bending test is numerically simulated.Thirdly,the uniaxial compression specimen generated by image processing technology is numerically simulated.Fourth,the uniaxial tensile specimen generated by image processing technology is numerically simulated.The mechanical behavior and damage mode of the specimen during loading were analyzed.The results of numerical simulation are compared and analyzed with those of relevant experiments.The feasibility and correctness of the micromodel established in this study for analyzing the micromechanics of lightweight aggregate concrete materials are verified.Image processing technology has a broad application prospect in the field of concrete mesoscopic damage analysis.

A new model for wrinkling and collapse analysis of membrane inflated beam

A new model is proposed to accurately predict the wrinkling and collapse loads of a membrane inflated beam. In this model, the pressure effects are considered and a modified factor is introduced to obtain an accurate prediction. The former is achieved by modifying the pressure-related structural parameters based on elastic small strain considerations, and the modified factor is determined by our test data. Compared with previous models and our test data, the present model, named as shell-membrane model, can accurately predict the wrinkling and collapse loads of membrane inflated beams.

On plane Λ-fractional linear elasticity theory

Non-local plane elasticity problems are discussed in the context of Λ-fractional linear elasticity theory. Adapting the Λ-fractional derivative along with the Λ-fractional space, where geometry and mechanics are valid in the conventional way, non-local plane elasticity problems are solved with the help of biharmonic functions. Then, the results are transferred into the initial plane.Applications are presented to homogeneous and the fractional beam bending problem.

A Mechanics Model for Stamping a Sheet on Elastic Die with Large Deformation

A semi analytical method was proposed to solve the mechanics problem of stamping a sheet on elastic die. The sheet was divided into four parts according to its deformation and contact with the punch and elastic die. Analytical solutions were derived individually for each part by using elastic large deflection and plastic large deformation. Solutions were found out with MATLAB by developing a numerical algorithm. Interface forces were obtained by iteration under the compatibility conditions between the neighboring parts of the sheet. Computation shows the method is efficient.

Peridynamic Modeling and Simulation of Fracture Process in Fiber-Reinforced Concrete

In this study,a peridynamic fiber-reinforced concrete model is developed based on the bond-based peridynamic model with rotation effect(BBPDR).The fibers are modelled by a semi-discrete method and distributed with random locations and angles in the concrete specimen,since the fiber content is low,and its scale is smaller than the concrete matrix.The interactions between fibers and concrete matrix are investigated by the improvement of the bond’s strength and stiffness.Also,the frictional effect between the fibers and the concrete matrix is considered,which is divided into static friction and slip friction.To validate the proposed model,several examples are simulated,including the tensile test and the three-point bending beam test.And the numerical results of the proposed model are compared with the experiments and other numerical models.The comparisons show that the proposed model is capable of simulating the fracture behavior of the fiber-reinforced concrete.After adding the fibers,the tensile strength,bending strength,and toughness of the fiber-reinforced concrete specimens are improved.Besides,the fibers distribution has an impact on the crack path,especially in the three-point bending beam test.

Elasto-Plastic Test of Q235 Steel Bending Beam With Cracking Resistance

More than 30 bending beams with rectangular cross-section and different thicknesses and heights were pre- pared from Q295 steel. The specimen dimensions were about 240 mm （length） × 60 mm （height） ×70 mm （thick ness）. Flaws were cut along its middle line with a wire cutter, with lengths ranging from 6 to 35 mm. Each specimen was tested with three-point bend loading, and a process curve was obtained between load and the displacement of the loading point, in order to analyze the fracture process when opening the crack. A deformation near the prefabricated crack was observed in the testing period, and the variation of the fracture characteristic parameters was analyzed for different sizes. For a comprehensive understanding of carbon steel fracture resistance behavior, its elasticity and plas- ticity were established by determining its Young＇s modulus and Poisson＇s ratio with an optical strain gauge. This gauge was also used for the loading process test. It was found that the fracture toughness varied with the dimensions, and the toughness of the elastic limit loading was almost constant. Using the relationship of crack resistance stress intensity factor and fracture criterion, the bearing capacity of the material structure could be estimated, which shows a good agreement with the experimental test data.

Evaluation of reclaimed asphalt pavement binder stiffness without extraction and recovery

A new testing procedure to estimate the low-temperature stiffness of the reclaimed asphalt pavement （RAP） binder was developed. In the testing procedure, the SuperpaveTM Bending Beam Rheometer （BBR） with special modifications and binder blending charts by Asphalt Institute were utilized. Modifications involved the development of a new kind of sample mold and different testing parameters were made to BBR testing procedure to capture the theological properties of bitumen mortars produced by mixing fresh binder with fine RAP materials or RAP aggregate. The stiffness relationship between binder and bitumen mortar was established based on the BBR test results. The blended binder stiffness in bitumen RAP mortar was estimated from the RAP mortar stiffness based on the binder-mortar relationship. And finally, the RAP binder stiffness was estimated from the blended binder and fresh binder stiffness based on the blending charts by Asphalt Institute. The results indicate that the new procedure can capture the rheological properties of bitumen mortar and can be used to estimate the low temperature stiffness of RAP binder without binder extraction and/or any chemical treatments.

Road Performances of Mesoporous Nano-silica Modified Asphalt Binders

The objective of this paper was to find new modifier to improve the aging resistance and low temperature cracking resistance of asphalt. To investigate the aging resistance of modified asphalt binders, mesoporous nano-silica（doping Ti^（4＋）） was used as a asphalt modifier. Some physical properties including penetration, ductility, and softening point of asphalt were analyzed with RTFO（Rotating thin film oven） aging and ultraviolet aging. Moreover, the performances of high and low temperature of modified asphalt binders with pressure aging were tested by dynamic shear rheometer（DSR） test and bending beam rheometer（BBR） test. These results showed that the penetration decreased, low temperature ductility, and softening point increased when adding mesoporous nano-silica to base asphalt. After ultraviolet radiation aging, the penetration loss and ductility loss of modified asphalt decreased than that of original asphalt, the increase of softening point was also significantly reduced than that of base asphalt. Furthermore, The test results of DSR and BBR showed that the G＊sinδ and creep modulus‘s＇ of pressure aged asphalt decreased, but the creep rate ‘m＇ increased. It can be concluded that the aging resistance and cracking resistance of modified asphalt are improved by adding mesoporous nano-silica, especially the doping of Ti^（4＋） could improve the aging resistance obviously.

Estimation of Rheological Properties of RAP Binder

The low-temperature rheological properties of binders in the recycled asphalt pavement （RAP） material without the damaging effects of solvent extractions were analyzed. The developed procedure is based on testing of bitumen-RAP mortars produced by mixing the fresh （virgin） binder with RAP material smaller than # 8 sieve. Different mortars were prepared, containing RAP material passing the #8 sieve and 15% by weight of fresh binder. Low temperature properties of these mortars were investigated by using the bending beam rheometer （BBR） test procedure that is specially modified for testing of the RAP mortars. The modification involved the development of a new kind of mold and different testing parameters. The RAP material used in the experimental study were both extracted from real reclaimed pavement and prepared in the laboratory, by aging binder through repeated PAV cycles. The results indicate the new procedure can capture the effect of aged properties of bitumen in the RAP and can be used to estimate the PG low temperature grade of the blended binder. Although data is limited in size, the modification of the BBR appears to be simple and provide repeatable data.

Effect of asphalt thin beams mixed with three nominal maximum aggregate sizes in the bending beam rheometer on the prediction of thermal properties of bituminous material

The objective of this research is to evaluate an impact of asphalt mixture beams with varying sizes of aggregate in the Bending Beam Rheometer （BBR） for testing/predicting thermal cracking properties of asphalt pavements. The BBR test has following benefits： the equipment is cheaper, it uses smaller specimens, faster conditioning, easier availability for quality control, easier to manage, etc. However some concerns have been raised： some consider that the size of the aggregate may affect the test＇s results; the other concern is that such small beams cannot represent the whole properties of the asphalt pavement. To address these criticisms, imaging techniques, statistical analysis, and viscoelastic modeling are used. Asphalt thin beams prepared with three different nominal maximum aggregate size （NMAS） （12.5mm, 9.5 mm, and 4.75 mm） were tested at three different temperatures （-18℃, -24℃, and -30℃）. Based on results from statistical analyses and viscoelasticity, the ratio of asphalt binders and voids and stiffness differences among the three NMAS specimens are not significant, meaning that the impact of asphalt thin beams prepared with the three NMAS on the prediction of thermal cracking is minimal and can be neglected.

A HIGH CONVERGENT PRECISION EXACT ANALYTIC METHOD FOR DIFFERENTIAL EQUATION WITH VARIABLE COEFFICIENTS

The exact analytic method was given by [1] . It can be used for arbitrary variable coefficient differential equations and the solution obtained can have the second order convergent precision. In this paper, a new high precision algorithm is given based on [1], through a bending problem of variable cross-section beams. It can have the fourth convergent precision without increasing computation work. The present computation method is not only simple but also fast. The numerical examples are given at the end of this paper which indicate that the high convergent precision can be obtained using only a few elements. The correctness of the theory in this paper is confirmed.

Location estimating of load for bending beam

We propose a method by which location of load for bending beam can be approxi- mately retrieved by matching the measured and theoretically forecasted displacement distribu- tion. To show the method validity, a principal experiment is performed.

Positive Solution for a Singular Fourth-Order Differential System

In this paper, we investigate the existence of positive solutions for the singular fourth-order differential system u(4) = φu + f (t, u, u”, φ), 0 < t < 1, -φ” = μg (t, u, u”), 0 < t < 1, u (0) = u (1) = u”(0) = u”(1) = 0, φ (0) = φ (1) = 0;where μ > 0 is a constant, and the nonlinear terms f, g may be singular with respect to both the time and space variables. The results obtained herein generalize and improve some known results including singular and non-singular cases.