926 



DR JOHN DOUOALL ON AN ANALYTICAL THEORY OF THE 



111 I. and II. below the rigid body rotations with coefficient t are inserted for a 

 purpose explained later (Art. 22, after (5)). 



What we call the integral stress across a section is defined by the three forces 



along the axes 



jj(2a;, zy, zz)d^, 

 and the three couples in the coordinate planes 



I I (?/z2 - zzy, zzx — xSc, xzy - i/zx)dS. 

 For a free solution the integral stress is independent of the section taken. 



Ux = 5^^ + z{a{x^ - if) -TO^], 

 o 



u„ = •2(TZxy, 



Mj = - z'^x + -x{x^ + 2/^)^(o" + ^~''' j'^^'*'- 



zy = fjL(2a- + l)xy, 



£ = - 2Ezx. 



Ua = 



( 1 



Uz 



= ■> -Z^ + z{(Tp^ - TOp') > COS (.), 



= < - -s !2^ + '-{'^P^ + ■'"«^) [ sin OJ, 



= "1 - z'^p -It -^ p^ - { a + - - T ]a^p \ 



■p > COS OJ. 



pp = 0, /3aJ = 0, p.' = /x( 0- + - j(p'^ - a'^) COS 0) 

 zS = /* I (^ cr - -jpH (o- + - j rt2 J sill ^_ 



J 



The integral stress is an ic-force through the origin of amount 



j jzxdS= - ^-n-Ea^ 



The deformation obtained by multiplying all the displacements here by l/7rEa* we 

 shall denote by Si; or, briefly. 



11. 



Si=(this solution) ^r^, ■ 



u^=2azxy, 



U, = lz^ + z{a(f^-x^)-ra^^}, 



' 2^y + 2 y(^^ + y'^) - ( ^ + 9 - "^Wu- 



