A multilayer-reflection technique for three-dimensional photoelastic studies of perforated plates in bending

A photoelastic investigation was conducted to determine the stress-concentration factors around a large, symmetrically reinforced central hole in a square plate under 1∶1 and 2∶1 biaxial bending. Tapered-edge rings served as the reinforcement, and a major objective was to determine the ring proportions such that the maximum stress at the hole would be equal to the value which would be present in an unperforated plate under the same nominal stress. Because the stress distribution at the periphery of a hole in such a plate structure varies in the radial, tangential and thickness direction, it was necessary to employ a three-dimensional photoelastic technique. There were a number of serious disadvantages in the use of any of the standard procedures and a new three-dimensional technique for room-temperature use was developed which is particularly suitable for the determination of boundary stresses around holes in bending experiments. With the technique in its present state of development, the three-dimensional isochromatic distribution in the plate can be determined from a single model and, from this, the boundary value of stress. The new technique utilized a laminated-plate model. Selective aluminizing of the laminations allowed for the determination of fringe-order distributions in the thickness direction as well as in the radial and circumferential directions at the boundary of the hole in flat models. Uniaxial maximum fringe orders were determined and, from these, the biaxial values were obtained by superposition.     

Alleviation of the stress concentration with analogue reinforcement

In order to reduce the stress concentration around a hole in a plate, new, “analogue” reinforcements instead of reinforcing rings were used in this investigation. In two of these specimens, reinforcements with different volume fractions were arranged to coincide with the stress trajectories for an infinite plate with a hole under uniaxial tension. Two other specimens containing straight rectangular-grid-type reinforcements were made by using a photofabrication method. Specimens were then prepared by sandwiching these reinforcements between two epoxy-resin plates. Plane specimens, i.e., without reinforcement, were also made of the same epoxy resin for comparison. The stress concentrations at the edge of the hole under uniaxial tension were determined by photoelastic techniques. The measured stress-concentration factors were compared with well-known values for an infinite, isotropic, homogeneous plate containing a hole. Results were also compared with published data on [90/0/90/0]_s 7-ply laminated composite plates, and on plates strengthened with reinforcing rings. A definite reduction in stress concentration was observed on specimens containing analogue reinforcement.     

The concentration of stress and strain in finite thickness elastic plate containing a circular hole

The elastic stress and strain fields of finite thickness large plate containing a hole are systematically investigated using 3D finite element method. It is found that the stress and strain concentration factors of the finite thickness plate are different even if the plate is in elasticity state except at notch root of plate surface. The maximum stress and strain do not always occur on the mid plane of plate. They occur on the mid plane only in thin plate. The maximum stress and strain concentration factors are not on mid plane and the locations of maximum stress and strain concentration factors are different in thick plate. The maximum stress and strain concentration factors of notch root increase from their plane stress value to their peak values, then decrease gradually with increasing thickness and tend to each constant related to Poisson’s ratio of plate, respectively. The stress and strain concentration factors at notch root of plate surface are the same and are the monotonic descent functions of thickness. Their values decrease rapidly and tend to each constant related to Poisson’s ratio with plate thickness increasing. The difference between maximum and surface value of stress concentration factor is a monotonic ascent function of thickness. The thicker the plate is or the larger the Poisson’s ratio is, the larger the difference is. The corresponding difference of strain concentration factor is similar to the one of stress concentration factor. But the difference magnitude of stress concentration factor is larger than that of strain concentration factor in same plate.     

Reinforcement of a plate with a circular hole by an eccentric circular patch

We study the reinforcement of an infinite elastic plate with a circular hole by a larger eccentric circular patch completely covering the hole and rigidly adjusted to the plate along the entire boundary of itself. We assume that the plate and the patch are in a generalized plane stress state generated by the action of some given loads applied to the plate at infinity and on the boundary of the hole. We use the power series method combined with the conformal mapping method to find the Muskhelishvili complex potentials and study the stress state on the hole boundary and on the adhesion line. We consider several examples, study how the stresses depend on the geometric and elastic parameters, and compare the problem under study with the case of a plate with a circular hole without a patch. In scientific literature, numerous methods for reinforcing plates with holes, in particular, with circular holes, have been studied. In the monographs [1, 2], the problem of reinforcing the hole edges by stiffening ribs is solved. Methods for reinforcing a circular hole by using two-dimensional patches pasted to the entire plate surface are studied in [3, 4]. The case of a plate with a circular cut reinforced by a concentric circular patch adjusted to the plate along the boundary of itself or along some other circle was studied in [5, 6]. The reinforcement of an elliptic hole by a confocal elliptic patch was considered in [7].     

CFRP阶梯加固钢板端部胶层应力的理论模型研究

CFRP阶梯加固可以有效减小CFRP端部胶层应力,防止胶层过早剥离而导致CFRP加固失效．文中通过理论推导建立CFRP阶梯加固钢板的端部胶层剪应力和正应力的理论模型,并采用有限元模型验证了理论模型,然后利用理论模型研究了加固参数对胶层应力的影响．研究结果表明：理论模型可以有效地计算端部胶层应力,且当阶梯端部长度超过“最小端部长度”时,理论模型可以准确计算端部胶层的最大应力值及其发生位置;利用理论模型便于分析各种参数对端部胶层应力的影响,其中胶层厚度、阶梯数量和CFRP厚度对端部胶层应力影响较大．     

Fatigue of fillet welded joints

The Albrecht method was used to compute stress intensity factors in non-load carrying fillet welds subjected to tension and cantilever bending for cracks growing from the tip of the weld toe through the plate thickness. The stresses were obtained using a two dimensional finite element programme with eight node isoparametric elements with two degrees of freedom in each node. A detailed study of the influence of weld and plate geometry on both the stress concentration and stress intensity factor values was carried out.The results in bending showed stress intensity values 20 to 30% lower than in tension. The weld affected the stress intensity and stress concentration results only for small crack length values less than 0.1 to 0.12 of the thickness of the main plate. The results also indicated that neither the lack of penetration nor the plate length affected the stress intensity values significantly. Weld angle and plate thickness were the main parameters influencing the stress intensity values. The results in tension were compared with similar results published in the literature and good agreement was found.     

孔口缝合补强对含孔复合材料层板应力集中影响的数值模拟

提出的一种缝合线计算模型,通过数值模拟计算与实验结果比较,得到缝合线计算模型中相关的弹性参数.对复合材料开口缝合补强结构进行有限元模拟计算,分析了孔边及邻近区域应变、应力的分布规律,得到不同缝合参数、孔边不同位置以及不同载荷条件下的应变、应力集中系数,并给出合理的孔口缝合参数设计方法及相关结论.研究结果表明:含孔拉伸试件在孔边θ=0°处,切向拉伸应力最大;在θ=90°处,切向压缩应力最大;在孔口0°和90°之间存在拉应力与压应力的转换点,缝合补强后,此转换点大约在θ=56°左右.     

Photoelastic study of axially loaded thick-notched bars

The objective of this investigation was to study the stress distribution and concentration at the root of Ushaped notches in a thick bar subjected to uniform tension. The method of stress freezing and slicing techniques was used. Six models of PLM-4B plastic with various grooved parameters were employed. The variation of the cross stress normal to the bar thickness and at the root of the groove was investigated. The maximum axial tangential stress at the root of the fillet was found to occur at the middle plane of the bar. The effect of the bar thickness and the fillet parameters on the stress concentration was studied and the experimental data ware correlated with the existing two-dimensional solutions.     

Bending of a stretched plate containing an eccentrically plate reinforced hole of arbitrary shape

In this paper we consider the problem of a stretched plate containing a hole of arbitrary shape which is reinforced by thickening the plate, on one side only, in a region surrounding the hole. Due to the eccentricity of the reinforcement a bending boundary layer occurs in the neighbourhood of the junction between the plate and the reinforcement. The equations for the moments at the junction are found to be identical to those for the circular hole in Ref. [1]. The boundary layer occurring at a clamped edge of arbitrary shape is also discussed.     

Bending of a highly stretched plate containing an eccentrically plate-reinforced circular hole

In this paper, we consider the problem of finding the stress distribution in a highly stretched plate containing a circular hole that is eccentrically reinforced by thickening the plate, on one side only, in an annular region concentric with the hole. A solution of the nonlinear Kármán plate equations is obtained that is asymptotically valid for large membrane stresses. We show that, except for a narrow bending boundary layer in the neighbourhood of the boundary between the reinforced area and the rest of the plate, a state of plane stress prevails and the reinforced area undergoes a transverse deflection that brings its middle surface into the plane of the middle surface of the plate.     

Surface effects on a mode-Ⅲ reinforced nano-elliptical hole embedded in one-dimensional hexagonal piezoelectric quasicrystals

To effectively reduce the field concentration around a hole or crack, an anti-plane shear problem of a nano-elliptical hole or a nano-crack pasting a reinforcement layer in a one-dimensional(1 D) hexagonal piezoelectric quasicrystal(PQC) is investigated subject to remotely mechanical and electrical loadings. The surface effect and dielectric characteristics inside the hole are considered for actuality. By utilizing the technique of conformal mapping and the complex variable method, the phonon st...     

Semi-analytical solution for three-dimensional vibration of thick continuous grading fiber reinforced (CGFR) annular plates on Pasternak elastic foundations with arbitrary boundary conditions on their circular edges

In this paper free vibration of continuous grading fiber reinforced (CGFR) annular plates on an elastic foundation, based on the three-dimensional theory of elasticity, for different boundary conditions at the circular edges is investigated. The foundation is described by the Pasternak or two-parameter model. The CGFR annular plates have an arbitrary variation of fiber volume fraction in the thickness direction. A semi-analytical approach composed of differential quadrature method (DQM) and series solution is adopted to solve the equations of motion. The fast rate of convergence of the method is demonstrated and comparison studies are carried out to establish its very high accuracy and versatility. Some new results for the natural frequencies of the plate are prepared, which include the effects of elastic coefficients of foundation, boundary conditions, material and geometrical parameters. Besides, results for CGFR plate with arbitrary variation of fiber volume fraction in the thickness direction of the plate are compared with discrete laminated composite plate. The main contribution of this work is to present useful results for continuous grading of fiber reinforcement in the thickness direction of a plate on an elastic foundation and comparison with similar discrete laminated composite plate. The interesting and new results show that non-dimensional natural frequency parameters of a functionally graded fiber volume fraction is larger than that of a discrete laminated and close to that of a 2-layer. The new results can be taken as the benchmark solutions for those from numerical methods and future researches.     

弯曲波在含非圆孔洞无限压电薄板中的散射

基于线性压电动力学理论,采用波函数展开法、保角映射以及复变函数,对含非圆孔洞无限大压电薄板弹性波的散射及动应力集中问题进行了分析,给出了其动弯矩集中系数(DMCF)的解析表达式.为说明问题,以PZT-4为例,讨论了外加电场、椭圆孔长短半轴比、椭圆孔倾角以及入射波频率对含圆孔和椭圆孔无限大压电薄板弹性波散射的影响,并分别...     

Three-dimensional photoelastic study of stresses in rack gears

A three-dimensional photoelastic analysis using the stress-freezing and slicing techniques was employed in studying the stress distribution and concentration across the thickness of a thick rack-gear tooth. Full-size photoelastic models of about 25-mm thickness, having various teeth parameters were machined from PLM-4B plates. A special mounting fixture was designed to hold the models in the loading frame and a knife-edged line load was used in loading the models. In the analysis, the method of oblique incidence was adopted for the separation of the principal stresses. The intension was to determine the magnitude and location of the maximum stress at the tensile fillet and to establish the stress-concentration factors for various geometric tooth configurations.From this study, it was concluded that the fillet stresses depend largely upon the pressure angle, fillet radius and the position of the load. Maximum values of the tensile fillet stresses occur at the middle plane of the tooth thickness. Furthermore, the value of the stress-concentration factor increases with the increase of the tooth thickness. In general, this investigation gave values of stresses much higher than those values calculated by the simple-flexure theory.     

Moiré holographic analysis of reinforced plate models

This paper deals with an extension of moiré holography for structural analysis on reinforced plate models. It is shown that unequal optical paths, caused by lack of uniform thickness, are compensated for and do not alter the moiré fringe patterns. Moreover, a simple technique is described for replicating high-sensitivity moiré gratings on Plexiglas models. The validity of the method has been satisfactorily tested on plate models with different geometries.Paper was presented at the 1988 SEM Spring Coinference on Experimental Mechanics, held in Portland, OR on June 5–10.     

Study of residual stresses in linearly varying biaxial-stress fields

A theory has been developed for the calculation of relaxation strains effected by drilling a hole in a plate with a linearly varying stress field. With this theory, a technique was developed for the measurement of residual stress at the toe of tee-fillet welds. The above technique was employed for the measurement of residual stresses at the toe of tee-fillet welds in 11/2-in. HY-80 steel with the fillet in the as-welded, ground, shot peened, ground and shot peened, and mechanically peened condition. It was found that experimental data conform to the assumed theory, and that residual stresses in aswelded tee-fillet welds in both the transverse and longitudinal directions approach the yield strength of the steel. It was also found that residual stresses are reduced approximately 25 percent by grinding, 50 percent by shot peening and 50 percent by grinding and shot peening. Mechanical peening drastically affected residual stresses by converting high tension at the toe of the fillet weld to high compression of approximately the same magnitude.     

Photoelastic stress analysis of an elliptical hole in a thick plate subjected to uniform in-plane compressive Loading

A three-dimensional photoelastic analysis was conducted to determine the stress distribution and concentration around the periphery of a centrally located elliptical hole in a plate of finite thickness. The edge of the plate was subjected to a uniformly distributed compressive uniaxial in-plane load. The principle of superposition was employed to study the effect of uniform biaxial loading.Elliptical holes with five different major/minor axis ratios () ranging from 1.0 to 2.64 were investigated. Among the results of this study, it was established that the variation of the principal stresses at the edge of the hole is not linear across the plate thickness. It was also found that in loading the plate in a direction parallel to the major axis of the ellipse, the value of the maximum tangential principal stress () occurs in a plane other than the middle plane of the plate. However, in loading the plate in a direction either parallel or perpendicular to the major axis, the maximum transverse stress ( _z ) occurs at the middle plane. In addition, the maximum value of ( _z ) was about 20 percent of the maximum value of the tangential stress for all models tested. Furthermore, the effect of the bixial loading has reduced the value of the maximum tangential stress at the periphery of the hole as compared with uniaxial loading.As a three-dimensional theoretical solution does not exist for this problem, the present findings were correlated with the well established two-dimensional solutions.     

Geometric elasticity problem of stress concentration in a plate with a circular hole

We use the geometric elasticity equations [1], which permit relating the medium stress state to the geometry of the Riemannian space generated by the stresses, to consider the plane problem of stress concentration near a circular hole in a thin unbounded plate loaded by normal and tangential stresses. The Riemannian space metric coefficient corresponding to the coordinate normal to the plate plane is treated as the variable thickness of the plate in three-dimensional Euclidean space, which determines the optimal law for the plate material distribution. We consider plates in uniaxial tension, biaxial tension, and shear. For the plate with thickness variation laws thus obtained, we construct direct numerical solutions of the corresponding classical elasticity problems and determine the stress concentration factors.     

Analysis of the stress state of a flattened conical shell of moderate thickness with a structurally weakening hole

Conclusions We determined the relationship between the nature of the stress distribution on the hole surface in a flexible plate as a function of thickness. We observed a great difference between the stress densities in flattened, thin and moderate-thickness conical shells and the stress concentrations near holes in thin cylindrical shells and thin, almost cylindrical, conical shells. The stress distribution near the hole in flattened conical shells of moderate thickness is similar to the stress distribution near the holes in flexible, thick plates. During loading of conical shells by an axial force, the lowest stress concentration factor near the holes is obtained when the axis of the hole is parallel to the shell axis. As the thickness of the shell is increased, the stress concentration factor near the holes increases.Kiev University. Ukrainian Institute of Water Management Engineers, Rovno. Translated from Prikladnaya Mekhanika, Vol. 24, No. 9, pp. 65–70, September, 1988.     

Viscoplastic deformation of reinforced plates with varying thickness under explosive loads

The problem of viscoplastic bending of reinforced plates with varying thickness is formulated. An original method for the integration of this initial-boundary-value problem is developed. The numerical solution is compared with an analytical solution obtained with a rigid-plastic model of an isotropic circular plate. The efficiency of the method is demonstrated by analyzing the inelastic dynamics of reinforced plates with constant and varying thickness. It is shown that the maximum residual deflections of plates can be reduced severalfold by means of rational profiling and reinforcement __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 85–98, February 2008.