F. K. Kong MA, MSc, PhD, CEng, FICE, FIStructE, R. H. Evans CBE, DSc, D ès Sc, DTech, PhD, CEng, FICE, FIMechE, FIStructE (auth.)-Reinforced and Prestressed Concrete-Springer US (1987)

100 Reinforced concrete beams-the ultimate limit state
SOLUTION
Try xld = 0.7. From eqns (4.5-5) and (4.5-6),
fs = e
~,:'d)(0.0035) = 0.0015
e~ = (x/d ;;/' /d)(0.0035) = 0.00275
From Fig. 4.5-1 (for /y = 460 N/mm 2 ),
fs = 300 N/mm 2 ; f~ = 400 N/mm 2
substituting into eqn (4.5-7) (with k 1 from Fig. 4.4-4),
ktfcu(~) = e!s - e'f~
LHS = (0.396)( 40)(0. 7) = 11.09
RHS = (0.03)(300) - (0.01)(400) = 5 < LHS
Therefore the assumed x/d is too high.
Try x/d = 0.4. Repeat the above process;
whence
fs = e
0.~.4)(0.0035) = 0.00525
e~ = ( 0·4 Q. 4 °· 15 )(o.0035) = 0.00245
fs = 400 N/mm 2 ; f~ = 400 N/mm 2
LHS of eqn (4.5-7) = (0.396)(40)(0.4) = 6.34
RHS = (0.03)(400) - (0.01)(400) = 8.00 > LHS
Therefore the assumed xl d is too low. By inspection the correct xl d value
must be between 0.4 and 0.7.
Try xld = 0.5. Repeat the above process:
whence
f 5 = ( 1 0.5°·5) (0.0035) = 0.0035
e~ = ( 0·5 Q. 5 °· 15 ) (0.0035) = 0.00245
fs = 400 N/mm 2 ; f~ = 400 N/mm 2
LHS of eqn (4.5-7) = (0.396)(40)(0.5) = 7.92
RHS = (0.03)(400) - (0.01)(400) = 8.00 * LHS
Hence x/ d = 0.5 is sufficiently accurate.
From eqn (4.5-8),