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)

230 Shear, bond and torsion
Similarly, on a vertical face of the member, each crack will intersect XI/Sv
vertical legs.
Assume that the member is torsionally under-reinforced, so that at
failure the links intersected by the diagonal cracks yield in tension. First
consider the horiZontal legs of the links (Fig. 6.9-1):
Tension in a horizontal leg = !Asvlyv
Moment about axis of member = !Asv/yv x ~l
From eqn (6.9-2), each crack on a horizontal face intersects YI/Sv
horizontal legs, and the torsional moment due to the tension in these
legs is
1 Xl YI
'1Asv/yv x -2 x - Sv
Considering the two horizontal forces, the total torsional moment is
twice this amount, Le.
Similarly,
T (horizontal legs) = !AsvlyvXtYt
Sv
Adding,
T (vertical legs) = !AsvlyvYtXl
Sv
(6.9-3)
where the notation is as explained at the beginning of this section. Of
course, the validity of eqn (6.9-3) is conditional upon eqn (6.9-1) being
satisfied.
Equation (6.9-3) is a powerful tool in the hands of designers with a good
understanding of structural behaviour. Note the following statements:
(a) The Lampert and Collins study shows that the torsional strength of
a properly reinforced beam is independent of the concrete strength,
provided the beam is torsionally under-reinforced; that is, the
longitudinal reinforcement and the links reach yield before the
ultimate torque is reached; in fact, their equation applies only to
under-reinforced beams. This restriction does not diminish the usefulness
of eqn (6.9-3), because over-reinforced sections are avoided
in design in any case. Not only are torsionally over-reinforced beams
uneconomical, but they do not have the necessary ductility when
subjected to overload. We shall see in Section 6.11 how overreinforcement
is guarded against in practice.
(b) The torsional strength T is proportional to the area X t Y t enclosed by
the links; within reasonable limits, the ratio Xt/Yl does not seem to
be important.
(c) The torsional strength of a box beam is sensibly the same as a solid
beam (in fact eqn 6.9-3 does not differentiate between the two types