Analytical Method For Stress Analysis Of Double Holes

Plate structural components with different shaped holes can be found everywhere in aerospace,ship,automobile,machinery and other types of engineering,as well as tunnel problems involved in hydropower,mining and underground engineering,stress concentration will inevitably occur on the hole boundary under external loads,where the failure phenomena of brittle fracture or plastic deformation are easy to take place.Therefore,it is of great significance to accurately calculate out the stress distribution and deformation on the hole boundary for engineering practice.In this dissertation,the complex variable method of plane elasticity theory is used to study the stress solution and hole shape optimization of two adjacent non-circular holes in an infinite plate,of which the key step is mapping the infinite plane region with two holes into a ring region in the image plane.When the size of the hole is much smaller than that of the plate structure,or when the tunnel is deep buried,it can be reduced to an infinite domain problem.(1)As to the problem of an infinite plate containing two elliptical holes,a mapping function is found by which the considered region is mapped onto an annulus region,and the parameters of the mapping function is analyzed.A set of linear equations for solving the two analytical functions is established through the stress boundary conditions using complex variable method.The analytical stress solution is derived when the plate is subjected to far-field uniform stress and uniform normal stress is applied on the hole boundaries.An infinite plate with two round holes or a round hole and an elliptical hole is the special case of this dissertation.The analytical solution obtained by this method are compared with the solution obtained by the stress function method,the second-order approximate solution of Schwartz alternating method and the solution of ANSYS finite element method.Some computational examples are presented to investigate the effects of the separation distance of the two holes,stress ratio,surface traction on the hole boundaries,hole size,shear stress at infinity and rotation angle of the principal stress on the stress distribution.The practical significance of this method is that it is more convenient and efficient,and can obtain very high-precision solutions.There are always four locations on the boundary of each hole where the tangential stress is zero when subjected to uniaxial tension,and their positions remain unchanged regardless of the size of the tension.The stress concentration on the hole boundary is 5 times the size of pure shear at infinity.(2)With regard to the problem of an infinite plate with two arbitrarily shaped holes,a general form of mapping function is proposed to map the discussed region into an annulus according to the Riemann mapping theory,while each of the two holes is symmetrical about the line joining their centers.The mapping functions of the specific problems are found using the mixed penalty function optimization method according to the corresponding relationship between the boundary points in the physical plane and the image plane.Some mapping examples of two various shaped holes are given.The coefficients of the mapping function converge rapidly with the increase of the number of terms.The more terms the mapping function takes,the better the mapped holes coincide with the actual ones.Then the analytical stress solution is derived using complex variable method when the plate is subjected to uniform tension and shear stress at infinity and vertical uniform tension is applied on the boundary of the hole.The newly derived results are compared with those obtained by ANSYS finite element method for the case of two square holes,and they are in good agreement.Examples of the combination of two square holes and the combination of a square hole and an elliptical hole are presented.The effects of the separation distance of the two holes,different hole shapes and their sizes,shear stress at infinity and rotation angle of the principal stress on the stress distribution are investigated under different loads.The interaction between the stress fields of the two holes can be neglected when the separation distance is large.There will be great stress concentration on the vertices of the square hole under all kinds of load,especially for the pure shear loads,the stress concentration on the vertices can reach about 15 times the size of the shear stress.(3)Optimum design of stress state of elastic perforated plate structure is a problem of great practical significance.Achieving proper shape of hole can reduce stress concentration around the boundaries remarkably.The optimal shape of a single hole in an infinite plate under uniform stress can be obtained using complex variable method based on different optimal criteria.The complex variable method is particularly suitable for the hole shape optimization in an infinite plate,in which the continuous hole boundary can be represented by the mapping function.It can also be used to solve the shape optimization problems of two or more holes.However,because of the difficulty of finding the mapping function for doubly-connected domain,the holes are mapped into slits or mapped onto a unit circle separately.In this dissertation,the shapes of two adjacent holes of equal size in a plate under uniform load is optimized.The two symmetrical holes are mapped onto an annulus simultaneously by the newly proposed mapping function,and the characteristics of the mapping function here are expounded and proved.The coefficients of the mapping function are the design variables in the differential-evolution algorithm(DE)and the tangential stress of maximum absolute value around the boundaries is minimized to achieve the optimal hole shapes for different loads,different separation distance and different hole sizes.The obtained optimal hole shapes also satisfy the equal stress hole condition and harmonic hole condition.The optimal hole shape obtained under pure shear can make the maximum tangential stress less than 3 times the applied stress.