3. HARMONIC END FORCING OF AN UNDAMPED 

 UNIFORM BEAM WITHOUT SHEAR WARPING 



Because of the two independent parameters q and E, in the formulas , 

 the discussion of special cases is complicated. Accordingly, the varia- 

 tion of the coefficients with w will be discussed in greater detail only 

 in the case of 5 = 0; that is, when shear warping is neglected. 



When ^ = 0, the formulas become, with two useful additions: 



V = a^^F + a^^*^ ' e = ^12^ "^ ^22*^ [lUa,b] 



sC-cS 1 sS 



11 1-cC EIq3 ' 12 21 1-cC ^^^2 



sC+cS 1 2 1+cC 1 



"22 - - 1-cC Elq ' \i'^22 " ^"12 ' 1-cC 



(EI)V 



^ = ^l^o■^^2% •' ° = ^2^0 -^ ^22«o [I5a,b] 



. = . sCicS ^,^3 ^ = . = - ^^ Elq2 

 11 1+cC 12 21 i+^Q 



sC-cS 2 i_cc , >2 1+ 



^22 = - i:^ ^^^ ^ ^1^22-^12 = IT^ (EI) q 



The added formulas are obtained with the use of Formula [lib]. It 



will be noted that when f = , a^, = a and b = b,^. 



^ 21 12 21 12 



As qj, increases , sinh qj, and cosh qj, soon become large and nearly 

 equal; for exajnple , sinh 3 = IO.OI8, cosh 3 = IO.O68. Hence, at leaist 

 from q2. = 2tt upward, the following simplified formulas may be sufficiently 

 accurate; they are obtained by dividing numerator and denominator by C 

 and then replacing S/C by unity, provided £, = 0: 



10 



