Structural Concrete - Hassoun

CHAPTER15
DESIGN
FOR
TORSION
Apartment building, Habitat 67, Montreal, Canada.
15.1 INTRODUCTION
Torsional stresses develop in a beam section when a moment acts on that section parallel to its
surface. Such moments, called torsional moments, cause a rotation in the structural member and
cracking on its surface, usually in the shape of a spiral. To illustrate torsional stresses, let a torque,
T, be applied on a circular cantilever beam made of elastic homogeneous material, as shown in
Fig. 15.1. The torque will cause a rotation of the beam. Point B moves to point B ′ at one end of
the beam, whereas the other end is fixed. The angle θ is called the angle of twist. The plane AO ′
OB will be distorted to the shape AO ′ OB ′ . Assuming that all longitudinal elements have the same
length, the shear strain is
γ = BB′
L
= rθ L
where L is the length of the beam and r is the radius of the circular section.
In reinforced concrete structures, members may be subjected to torsional moments when they
are curved in plan, support cantilever slabs, act as spandrel beams (end beams), or are part of a spiral
stairway.
Structural members may be subjected to pure torsion only or, as in most cases, subjected
simultaneously to shearing forces and bending moments. Example 15.1 illustrates the different
forces that may act at different sections of a cantilever beam.
Example 15.1
Calculate the forces acting at sections 1, 2, and 3 of the cantilever beam shown in Fig. 15.2. The beam is
subjected to a vertical force P 1 = 15 K, a horizontal force P 2 = 12 K acting at C, and a horizontal force
P 3 = 20 K acting at B and perpendicular to the direction of the force P 2 .
Solution
Let N be the normal force, V the shearing force, M the bending moment, and T the torsional moment.
The forces are as follows:
523