。 dT 。Here, a is the linear thermal expansion coefficient。 The relevant problem is solved through for linear problems by incremental loading method in which the differentials dT , dr and de are replaced by the increments DT , Dr and De, respectively, at each loading step。 Thus, Eqs。 (5) and (6) becomeDr ¼ CepðDe — D~eT Þþ Dr~T : ð10Þ

The finite element method may now be introduced for analysing  the problem。  It is assumed  that  the  space

domain X and a given time domain are separate as given in Eq。 (1)。 The displacement vector ut  for any element  e

412 S。 Sen, B。 Aksakal / Materials and Design 25 (2004)   407–417

and any time t may be expressed by interpolated value of the nodal displacement vector ute  of this element。 Hence,

ut ¼ Nute ; ð11Þ

and strain vector

et  ¼ Bute: ð12Þ

The stiffness matrix and the nodal thermal load vector of element e in the elastic and plastic regions are given as follows:

for the elastic region

Considering that the stress field is equivalent to  the stress of the final cooling stage and by substituting the nodal displacement vector into the geometrical Eq。 (12) and substituting this into Eqs。 (9) and (10) the transient stress field is then  found。

5。Results and discussions

In order to validate and show the reliability of the model and solutions it is important to investigate the radial, rr, and hoop, rh, stresses, along the    shrink-fitted

region of shaft–hub assembly。 In this work the varia- tions of stresses are presented with respect to the nor- malised  variable  ratios  of  y/l  and  r/R  and      various

e T interference values。 To verify the boundary    conditions

and for the plastic  regionZ

and the finite element mesh for axisymmetric solution, the results of radial stress, rr, axial stress, ry ,  hoop stress, rh  and shear stress rxy  were given in Fig。 3   along

the symmetry axis, in the centre line OC, r axis, for the ratio of l=d ¼ 0:5 and h ¼ 42 lm interference。 That kind

of results also gives us an opportunity to compare  the

By using the direct stiffness method the whole equilib- rium equation is then   obtained

Kt D~u ¼ DRt  þ Dt: ð17Þ

The equilibrium equation is non-linear for a given time and   could   be   solved   by   the   full Newton–Raphson

analytical solutions where the heat effect was neglected。 In Fig。 4 along the contact surface, line AB, as illus- trated in Fig。 2, the radial displacements on the shaft and hub were shown for l=d ¼ 0:5 and h ¼ 42 lm in- terference values。 The radial displacement values for both hub and shaft have nearly the same values and increase  with  increasing  y=l  ratio。  After  the  ratio  of

y=l ¼ 1 the increments become very small and will  cer-

method。 The total nodal displacement vector ~ut  þ Dt at

time t þ Dt is obtained as

tainly go to zero after the value of    y=l

Figs。  3  and  4  the  results  show  the

¼ 1:25。 In both ability  of  the摘要一般的应力计算、干涉配合、设计估计都要通过使用传统的方程，这使得一个部分显示弹塑性行为的过程变成了一件更加复杂的事件。在这项研究中,应力分布中的干涉安装、shaft-hub组装研究与接触length-shaft直径的比例都要使用有限元方法分析瞬态传导、传热的边界条件在系统加热和冷却过程中被认为是不断改变地。结果表明，在发生塑性变形区及其分布中过盈配合连接被评估后，目前的结论设计师在shaft-hub配件加上轮和齿轮系统传输组件等的多维度分析是肯定会获得帮助，使得可以控制冷却条件