29

thickness of the assumed thin-walled cross section. This term has been

previously discussed in Sec. 2.6.1 of Report 248-3. In the Swiss Code

"be" is taken as d e /6 for a solid cross section. For a hollow cross

section be = t, where t represents the wall thickness of the cross

• I

de

The term de is defined in Fig. 2.9 •

, de

t

Ao = hatched area

Fig. 2.9 Definition of the term de in the effective wall thickness be (from Ref. 10)

To avoid failures due to web crushing an upper limit for the

nominal shear stress due to torsion vn must not exceed the value of

vmax ' which is a function of the concrete strength and the maximum

stirrup spacing s.

vmax = 5vcu for Srnax = de /2 but s < 12 in.

vmax = 6vcu for smax = d e /3 but s < 8 in.

30

In the above expressions for s, for small solid cross sections

(rectangle, T-section) with side ratios greater than 3:1 "de" can be

replaced by 2*de•

A comparison of these upper limits and the limit of 12 ~

suggested in the ACI Code (2) and AASHTO Standard Specs. (1) for the

case of pure torsion is shown in Fig. 2.10. Again, the Swiss limits

allow higher torsional stresses.

2000

1000

V max (pin

6Vcu

~V!U I

12Jfc

0.00 2 3 4 5

Fig. 2.10 Comparison between the upper limits of the shear stress in the case of pure torsion

f~ (pin

For the case of combined actions the nominal shear stress due to

shear and torsion must not exceed the prescribed values of vmax•

v(V + T) = v(V) + veT) < vmax (2.22)

The concrete contribution in the transition zone Vc is to be evaluated

for the simultaneous action of shear and torsion and then is to be

31

distributed in accordance with the respective levels of shear and

tor s ion so t hat v c ( V ) = (v ( V ) / v (V + T» v can d v c ( T) = (v ( T ) / v (V + T»v c •

The stirrup reinforcement required for shear and torsion are to be

determined separately and then superimposed. The longitudinal

reinforcement for shear and torsion must be determined separately and

then added to the reinforcement for bend ing. If at a cross section the

tensile force due to shear or torsion is counteracted by a compression

force due to bending, the longitudinal reinforcement required will only

be that required for the remaining tensile force.

The reinforcement for shear and torsion must meet the following

requirements. The minimum area of shear reinforcement must be equal or

greater to

(2.23)

for the case of shear, and

(2.24)

in the case of torsion. The stirrup reinforcement is to be continued on

past the design region for at least the distance of the stirrup spacing.

Stirrups must enclose the longitudinal reinforcement, and be properly

anchored so that their required strength is effective over the depth z.

The additional longitudinal reinforcement required for shear

and/or torsion is to be placed uniformly around the perimeter "u" formed

by the stirrups. Furthermore, the longitudinal steel at the corners

should be arranged so as to prevent pushing out of the concrete

32

compression field. Proper detailing also calls for sufficient anchorage

of the longitudinal reinforcement particularly at the support regions.

2.2.3 Proposed Canadian Code--General Method. The General

Method design procedure proposed in the Canad ian Code Draft of August

1982 (23) is based on the compression field theory developed by Collins

and Mitchell (17) and uses equilibrium relations from the truss model.

The General Method is applicable to both reinforced and prestressed

concrete members subjected to shear and/or torsion. Collins and

Mitchell further developed the truss model in the compression field

theory by introducing a compatibility condition for the strains of the

transverse and longitudinal steel members and the diagonal concrete

compression strut. This condition was derived only for a constant

strain profile over the section such as in the case of pure torsion,

leading to the equation

(2.25 )

whereais the angle of inclination of the diagonal strut, E:ds is the

compressive strain in the diagonal strut, E:l is the longitudinal tensile

strain, and E:s is the transverse tensile strain. Eq. 2.25 allows the

evaluation of the incl ination of the diagonal compression struts for a

given state of strain in the shear field element. Using Eq. 2.25, the

stress-strain relationships of the concrete and the steel, and the

equilibrium equations of the truss model, the compression field theory

attempts to predict the full behavioral response of reinforced and

prestressed concrete members subjected to torsion or shear.