On the density of finite element matrices

Explicit formulae for the density of a class of finite element matrices are presented. The formulae are functions of the characteristics of the mesh and the element (piecewise polynomial) employed. Results are given for two-dimensional triangular and quadrilateral elements and three-dimensional tetrahedral and cuboid elements. These results are useful in the management of storage in finite element computer programmes.     

Analysis of RC structures subjected to air-blast loading accounting for strain rate effect of steel reinforcement

A new method to calculate the dynamic strength of reinforced concrete (RC) structures, with emphasis on the embedded response of mild steel reinforcement, subjected to air-blast loading is derived. The proposed method is based on a single-degree-of-freedom (SDOF) system with appropriate modification for strain-rate effect. Approximate formulae are used and explained in detail. A numerical example is presented to show how the derived formulae are used for the analysis of concrete structures subjected to air-blast loading.     

Complex hypersingular integrals and integral equations in plane elasticity

Summary Complex hypersingular (finite-part) integrals and integral equations are considered in the functional class of N. Muskhelishvili. The appropriate definition is given. Three regularization (equivalence) formulae follow from this definition. They reduce hypersingular integrals to singular ones and allow to derive hypersingular analogues for Sokhotsky-Plemelj's formulae and for conditions that are necessary and sufficient for the function to be piecewise holomorphic. Two approaches to get and investigate complex hypersingular equations follow from these results: one of them is based on the equivalence formulae; as to the other, it is based on above-mentioned conditions. As an example, authors' equation for plane elasticity is studied. The existence of a unique solution is stated and some advantages over singular equations are outlined. To solve hypersingular equations the quadrature rules are presented. The accuracy of different quadrature formulae is compared, the examples being used. They confirm the need to take into account asymptotics and to carry out a thorough analytical investigation to get safe numerical results.     

Incomplete convolutions in production and inventory models

In this paper, we study incomplete convolutions of continuous distribution functions, as they appear in the analysis of (multi-stage) production and inventory systems. Three example systems are discussed where these incomplete convolutions naturally arise. We derive explicit, nonrecursive formulae for these convolutions, for the relevant case in which the underlying distributions are (mixtures of) Erlang distributions with the same scale parameter. Numerical results for one example system, the multi-stage serial inventory system, are presented to show the effectiveness of these formulae.     

Reflectivity and Transmittivity of a Stack of Thin Films in the Soft X-ray Region

Formulae are presented for the S-polarized complex reflectivity and transmittivity of a stack of dielectric films, valid in the soft X-ray region and based on the Ewald microscopical model of a dielectric. These formulae are compared with the classical Fresnel formulae and the results of model calculations are presented.     

Alternative integration formulae for triangular finite elements

Some new formulae are presented for numerical integration over triangular regions. These formulae, which are triangularly symmetric, involve four, six, and seven integration points, respectively. They were derived by determining co-ordinates and weight factors for which polynomial functions of up to a certain degree are integrated exactly. The effectiveness of the formulae is demonstrated by applying them to the integration of five non-polynomial functions, two of which are singular within the region of integration. Results obtained with the new formulae are compared with results obtained with other, comparable formulae. It is seen that the new formulae are more accurate than some existing ones with the same numbers of points.     

A discussion about a class of stress intensity factors and its verification

In this paper, new formulae of a class of stress intensity factors for an infinite plane with two collinear semi-infinite cracks are presented. The formulae differ from those gathered in several handbooks used all over the world. Some experiments and finite element calculations have been developed to verify the new formulae and the results have shown its reliability. Finally, the new formulae and the old are listed to show the differences between them.     

Size effect in the strength of concrete structures

This paper reports on the range of applicability of the various size effect formulae available in the literature. In particular, the failure loads of three point bend (TPB) beams are analysed according to the size effect formulae of Bažant and of Karihaloo for notched beams and according to those of Bažant and of Carpinteri for unnotched beams, and the results of this analysis presented. Improvements to Karihaloo’s size effect formula are also proposed.     

斜卷边冷弯薄壁槽钢轴压柱弹性畸变屈曲计算

畸变屈曲是冷弯薄壁卷边槽钢的一种重要屈曲模式,弹性畸变屈曲临界应力是目前一些主要规范计算畸变屈曲的关键。该文首先推导出畸变屈曲模式下,构件纵向翘曲位移的显式计算公式,从而简化计算过程;然后基于经典的薄板理论,计算平面内位移,确定构件平面内的变形,在现有研究基础上得到斜卷边冷弯薄壁槽钢轴压柱弹性畸变屈曲计算公式;最后与有限条法、近似模型计算法进行对比,验证了该文计算方法的正确和有效性。该文中的弹性畸变屈曲计算公式简单,便于手算,可供工程设计人员参考。     

A comment on the penetration theory by dikshit and sundararjan

In the paper published by Dikshit and Sundararjan [1], the experimental results for the penetration of a projectile with an ogival nose into steel plates (RHA) have been investigated, and suitable analytical relations were presented. However, some formulae in their paper seem to be incorrect. In this paper along with these formula, the corrected formulae have also been presented.     

Composite materials dynamic fracture studies by generalized Shmuely difference algorithm

The generalized Shmuely Difference Algorithm (GSDA) is presented here to analyze the dynamic fracture performance of orthogonal-anisotropic composite materials, such as glass fibre reinforced phenolplast. The difference recurrence formulae and boundary condition difference extrapolation formulae are derived and programmed. The dynamic stress intensity factors (DSIF) of the isotropic and anisotropic centrally cracked plates are computed respectively using GSDA and compared with that published previously. GSDA is proved effective and reliable.     

光谱共焦位移传感器的色散物镜设计

光谱共焦位移传感器具有精度高、适应性强等特点,传感器的关健部件是产生轴向色散的物镜.本文对传感器的工作原理进行了分析,推导了计算物镜光学参数的公式,并依据此公式计算结果设计物镜,得到了满足设计指标的物镜,且与理论计算相符.传感器的工作波长范围是500～700 nm,设计的色散物镜利用高色散玻璃和双胶合透镜达到了1 mm...     

圆轴扭转强度的计算及其误差

通过对三个圆轴扭转强度的计算公式的比较，给出了经们与真实声转强度公式的误差函数和曲一及误差的最大值。     

Number of non-zero blocks in the stiffness matrix of the finite element network

In Reference 1 a series of general formulae for evaluating the total number of non-zero blocks in the stiffness matrix, both of two-dimensional and three-dimensional (arbitrary polygonal/polyhedral) finite element networks, was presented. In this paper, an alternative, more direct and efficient method for deriving such formulae is presented. In addition, this new method can be used for more complex situations. In fact, it may be used for almost all kinds of finite element networks.     

冷弯薄壁卷边槽钢梁的弹性畸变屈曲荷载简化计算公式

以既有理论研究成果为基础,结合应用广义梁理论,对冷弯薄壁卷边槽钢在绕截面强轴弯矩作用下的弹性畸变屈曲荷载进行了深入细致的分析。首先基于45个截面冷弯薄壁卷边槽钢梁畸变屈曲荷载的参数分析,提出了屈曲临界半波长度的近似计算公式;接着对腹板提供给翼缘板的转动约束刚度进行了分析,提出了精度较高的线性近似计算公式;依据畸变屈曲特征方程,提出了考虑翼缘自身剪切与畸变影响的修正参数的近似计算公式;利用以上结果,最后提出了冷弯薄壁卷边槽钢在绕强轴弯矩作用下的弹性畸变屈曲荷载简化计算公式。简化公式具有足够的精度,便于手算,实用性强,可供工程设计人员和各国冷弯薄壁型钢设计规范修订相应内容时参考采用。     

Morphometry of fibre reinforced cementitious materials

A spacing and geometric efficiency concept is developed that deviates from the available formulae by its foundation on geometric probability theory. The solutions can easily be extended to cover non-random structures as well. Two-dimensional structural information is presented by means of sections and projections (X-ray radiography). Counting of features or intersections with a superimposed test-line system can in both cases provide the necessary information for the determination of the fibre spacing. The spacing formulae are also transformed for design purposes, in which case they contain only geometric parameters and the volume fraction. The spacing concept comprises formulae for the mean free spacing and the average nearest neighbour distance in two- and three-dimensions. As such they can be applied to structure-insensitive and structure-sensitive properties (cracking), respectively, of fibre reinforced cementitious materials. A brief survey of experimental results is presented.     

冷弯薄壁卷边Z形钢梁的弹性畸变屈曲荷载

该文以既有理论研究成果为基础,结合应用广义梁理论,对两端简支的冷弯薄壁卷边Z形钢梁在绕截面强轴弯矩作用下的弹性畸变屈曲荷载进行了深入细致的分析。通过对45个截面的参数分析,将原先适用于冷弯薄壁卷边槽钢梁的参数计算公式推广应用于Z形钢梁,并根据Z形钢截面的几何特性对相应参数进行了重新推导和修改。利用以上结果,最后提出了冷弯薄壁卷边Z形钢梁在绕强轴弯矩作用下的弹性畸变屈曲荷载简化计算公式。简化公式具有足够的精度、便于手算、实用性强,可供工程设计人员和各国冷弯薄壁型钢设计规范修订相应内容时参考采用。     

Winkler地基上变厚度圆（环）板的非对称自由振动

本文提出了Winkler地基上变厚度圆(环)板非对称自由振动的传递矩阵法.应用贝塞尔函数理论,求得等厚度圆板和环板单元非对称自由振动传递矩阵的正确公式.然后将Winkler地基上的变厚度圆(环)板划分成一系列的等厚度的圆板和环板单元,应用传递矩阵原理得到变厚度圆(环)板的整体传递矩阵公式.最后给出了一些数值结果,表明板厚和地基模量变化对固有频率的影响.     

An implicit curve-fitting method for fast calculation of thermal properties of pure and mixed refrigerants

Calculations of refrigerant thermal properties are desired to be very fast and stable in cases of simulation of refrigeration system, etc. The traditional method based on equation of state cannot meet such requirement because of unavoidable iterations in calculation. In this paper, a new calculation method for refrigerant thermal properties is presented. Low order implicit polynomial equations are got by using curve-fitting method at first, and then explicit formulae for calculating refrigerant thermal properties quickly are obtained by getting the analytical solution of these implicit equations. Explicit fast calculation formulae for thermal properties of R22 and R407C, covering the saturated temperature of −6080 °C and superheat of 0–65 °C, are presented as examples. The calculation speeds of the formulae of R22 are about 140 times faster than those of REFPROP 6.01 while the formulae of R407C are about 1000 times faster. The total mean relative deviations of the fast calculation formulae for R22 and R407C are less than 0.02%.     

Calculation of strain-energy release rates with higher order and singular finite elements

A general finite element procedure for obtaining strain-energy release rates for crack growth in isotropic materials is presented. The procedure is applicable to two-dimensional finite element analyses and uses the virtual crack-closure method. The procedure was applied to non-singular 4-noded (linear), 8-noded (parabolic), and 12-noded (cubic) elements and to quarter-point and cubic singularity elements. Simple formulae for strain-energy release rates were obtained with this procedure for both non-singular and singularity elements. The formulae were evaluated by applying them to two mode I and two mixed mode problems. Comparisons with results from the literature for these problems showed that the formulae give accurate strain-energy release rates.