In chapter 8 solutions by eigenfunction expansion to 1-dimensional problems of mechanics and 2-dimensional problems of theory of elasticity are researched.

In this paper, the strength design calculation fumula of cylindrical shell of boiler is arialyzed by the theory of elasticity shell, the analysis result is important to the boiler design calculation and safety.

STUDY ON THE RELATIONSHIPS BETWEEN THE STRUCTURE OF NETWORKS FOR PHASE STATE AND THE MECHANICAL PROPERTIES OF TPUE Ⅰ. THE MOLECULAR THEORY OF ELASTICITY FOR TPUE AT LARGE DEFORMATION

The solution of the boundary value problems of the theory of elasticity is sought in the form of expansions into series of the associated Legendre polynomials.

It is shown that when there are no gas phases and the liquid is incompressible the system of equations reduces to the general equations of the theory of elasticity of an anisotropic body with fictitious stress components.

Averaged systems of equations of the theory of elasticity in a medium with weakly compressible inclusions

The latter provides an explanation for the generation of the second shear harmonic that is observed in real solids contrary to the predictions of the nonlinear theory of elasticity, which prohibits such phenomena.

Possible descriptions of phasons in incommensurate crystal phases and quasicrystals have been reviewed in terms of the theory of elasticity and superspace symmetry.

Two methods for analyzing caisson-beams are introduced in this paper.One is the well-known method of redundant forces. The author has simplified this methed by using couples of redundant forces to set up a typical equation and pointing out the rule that the matrix of the coefficients of simultaneous linear equations which are organized from the expansion of that typical equation. This method can be easily solvd when the number of unknown redundant forces, or that of equations, is less than 3 or 4; but it will...

Two methods for analyzing caisson-beams are introduced in this paper.One is the well-known method of redundant forces. The author has simplified this methed by using couples of redundant forces to set up a typical equation and pointing out the rule that the matrix of the coefficients of simultaneous linear equations which are organized from the expansion of that typical equation. This method can be easily solvd when the number of unknown redundant forces, or that of equations, is less than 3 or 4; but it will be difficult when the number is more than that. In order to solve this difficulty the author suggests another kind of method of which the essential principle is mentioned in the following.Supposing that the distance between the beams is sufficiently short in comparing with their spans, we can set up a partial differential equation for its deffiection W, as we often do in the theory of elasticity. In this way we can solve it with its boundary conditions of simple supporting by sine series. From this we can easily get the formulas of bending moments, shears and twist moments of each beam by partially differentiating the function of deffiection. The result of the calculation proves that it quite agrees with the method of redundant forces when the distance between beams is no longer than 1/5 of their spans.There are some tables given in this raper for practical use.

In recennt years field tests have shown that the moduli of elasticity of rock foundation E_(OC) are generally not greater than those of concrete E_σ.When E_(OC)=E_σ, the coefficients of foundation restraint R are given in the following table,as shown by photoelasticity experiment. Coefficient of Foundation Restraint R (When E_σ=E_(OC)) y=height of the point in consideration. L=length of the block. When E_B≠E_(OC),R may be computed hy the following semi-empirical formula: R'=(1.82R)/(1+0.82(E_σ)/(E_(OC))) Where...

In recennt years field tests have shown that the moduli of elasticity of rock foundation E_(OC) are generally not greater than those of concrete E_σ.When E_(OC)=E_σ, the coefficients of foundation restraint R are given in the following table,as shown by photoelasticity experiment. Coefficient of Foundation Restraint R (When E_σ=E_(OC)) y=height of the point in consideration. L=length of the block. When E_B≠E_(OC),R may be computed hy the following semi-empirical formula: R'=(1.82R)/(1+0.82(E_σ)/(E_(OC))) Where R is taken from the above table. Taking into consideration the rise and drop of temperature,the change of E_σ and R with the age of concrete and the plastic flow of concrete,the allowable tem- perature difference Δt=t_p+t_r-t_f may-be computed by the following formula; t_p+k_rt_r-t_f≤(17)/(KR) Where t_p=placing temperature of concrete, t_r=rise of temperature due to heat of hydration, t_f=final stable temperature, K=factor of safety k_r=0.60～70. When E_B=E_(OC),R=0.55,K=1.30,by the above formula,Δt=t_p+t_r-t_f=25～30℃, which is much higher than 17℃,the allowable temperature difference now in use. When there is a temparature difference between the upper and lower portion of the block,the coefficients of restraint of the lower portion are given in the following table,as obtained by the theory of elasticity: Coefficient of Restraint of lower,portion. In the construction of Gootien dam.as 30% rock is placed in the concrete,there is only 110kg of pozzolan Portland cement in 1 M~3 of concrete,the adiabatic tem- perature rise is lower than 18℃;thus the heiglrt of lift may be raised to 6-10 M or greater from November to May of next year.Only in June,July and August pre- cooling and pipe-cooling may be necessary.

The main exertion of this writing is to investigate the condition of stress and deformation caused by a cylindrical inclusion of circular cross section, which possesses different coefficient of expansion as the material of the vicinal region, in a semi-infinite elastic solid, when the composite body is subjected to the same relatively high temperature. It was so arranged that the axis of the cylindrical inclusion is parallel to the free surface of the semi-infinite elastic solid; c represents the distance from...

The main exertion of this writing is to investigate the condition of stress and deformation caused by a cylindrical inclusion of circular cross section, which possesses different coefficient of expansion as the material of the vicinal region, in a semi-infinite elastic solid, when the composite body is subjected to the same relatively high temperature. It was so arranged that the axis of the cylindrical inclusion is parallel to the free surface of the semi-infinite elastic solid; c represents the distance from this axis to the free surface, while a0 is the diameter of the cylinder. In order to let the cylinder totally buried, we have to request that c > a0.The whole composite body was thought to be heated to a uniform temperature TO-On account of the different coefficients of thermal expansion of the two materials, the evoked stresses must be proportional to the difference of these coefficients.But the same stress and deformation condition will also take place when the two materials possess the same coefficient of expansion, and the inclusion is heated to the temperature T0 solely, while the rest of the composite body is kept at 2ero temperature.With the y-axis lieing in the free surface and the x-axis perpendicular to it, we have to, according to the case with two different coefficients of thermal expansion, solve the following differential equations:where η0 means the difference of the thermal expansion coefficients, v the Poisson ratio,the thermal displacement function, R =It was found that the following two thermal displacement functions do satisfy the two differential equations (1) outside und inside of the cylindrical inclusion respectively, while K represents a composite material constant.Nevertheless, the thermal displacement function has further aroused a boundary value problem of pure elasticity nature, which is solved with the usual methods presented in the theory of elasticity.When the two partial solutions are superposed, then we get the following complete solution of the announced problem: Finally, it is still necessary to remark, that some data were unfavourably chosen so as to have rendered that the resulting maximum stresses, in the numerical example, surmount already the elastic limit of the related material. We have to ask our reader to consider the material so as to have a higher elastic limit, because the author has obstinately hesitated to take new values and then to repeat a rather large scale tedious numerical calculation.