[工学]弹性力学双语课件习题.ppt

structural mechanics 2.13.4 When the body force components may be expresses by (1) the body forces are said to have a potential and V is called the potential function. Show that, in this case, the differential of equilibrium can be satisfied by Derive the differential equation of ? 试证明，如果体力虽然不是常量，但却是有势的力，即体力分量可以表示为： (1) 其中V是势函数，则应力分量亦可用应力函数表示成为 满足平衡微分方程。试导出相应的相容方程。 Replace ? with ?/(1-?) ,we obtain ?4?=-（1-2?）?2V /（1-?） Which is the compatibility equation for the plane strain problems. 2.Derive the solution for the stress function from assumed stress component Substituting into Eqs.(2.12.10),we get After integration, we have in which are unknown functions of x only. 3.Find the stress function from the compatibility equation Substituting Eq.(a) into compatibility equation, we get This is a first degree function of y,the compatibility equation requires that the equality should satisfy on each point in the column.So the coefficients should equal to zero,then After integration, we have The constant term in and the first degree term and constant term in has been omitted.Because they will not affect the expressions for the stress components.Substituting Eqs.(b) and (c) into Eq.(a),then we get the stress function 4.Get the components of stress from the stress function by Eq.(2.12.10) 5.Consider the boundary conditions to obtain the unknown coefficients (a)The boundary conditions on x=0 are The first is satisfied automatically,and from the second equation we get C=0. (b)The boundary conditions on x=b are The first is satisfied automatically,and from the second equation we get (c)Method 1 for the boundary conditions on y=0 : we use Saint-Venant principle only when the exact boundary conditions can not be satisfied. Boundary conditions on y=0 are But A=B=0 is in contradiction with equation