13

values for the concrete contribution in this transition zone are

presented in Fig. 2.2.

In Fig. 2.2 the values of Qrd include a material safety factor

of 1.5 as recommended by the CEB Code for the case of concrete.

Therefore, the nom inal concrete contribution in shear in the uncracked

low shear stress range is 1.5 • 2.5 • Qrd or 3.75 • Qrd. In terms of

k.ff'c for the values of Qrd given in Fig. 2.2, this lower range 3.75

Qrd yields values of k ranging from 2.4 to 3.2. These values are

between the values of 2~ and 3.5..!fb which are currently

recommended in the ACI Building Code (2) and AASHTO Standard

Specifications (1) as the simplified and maximum values respectively of

the nominal concrete contribution in shear depending on the moment to

shear ratio on a section. These values then decrease for members with

higher values of shear. Such a provision gives substantial relief in

shear design of lightly loaded members.

For the case of prestressed concrete members, the same type of

linear concrete contribution in the transition zone is suggested.

However, the values of Vc of 2.5 Qrd are increased by a factor K = 1

+ [Mo/M sdu ] ~ 2, where Msdu is the maximum design moment in the shear

region under consideration, and Mo denotes the decompression moment at

transfer related to the extreme tensile fiber, for the section where

Msdu is acting. This moment is equal to that which produces a tensile

stress that cancels the compression stresses due to the applied

prestress force and other design ax ial forces.

14

Vc= concrete contribution Vc bwd

uncracked I

~~

Transition I

~I FLlIl Trus. action

Vc=2.5Qrd

2.50Qrd

Ultimate Shear Stress

where:

f' (psi) c

1740 2320 2900 3625 4350 5075 5800 6525 7250

Qrd (psi)

26.1 31.9 37.7 43.5 49.3 55.1 60.9 66.7 72.5

7.5Qrd

Fig. 2.2 Concrete contribution in the transition range CEB-Refined Method

15

In order to avoid failures due to crushing of the web, an upper

limit on the shear resistance of

(2.6)

is required. f cd represents the design concrete compression stress,

i.e. characteristic value of the concrete compression strength divided

by a resistance safety factor. In terms of ~ the maximum shear stress

Vmax/[bwd), with a resistance safety factor of 1.5 as suggested in the

CEB Code, would become equal to 0.2 ~ sin 20'. A comparison between the

CEB upper limit and AASHTO and ACI upper limit of 10~ is shown in

Fig. 2.3 for a of 45 and 30 degrees.

The design procedure for torsion in the CEB-Refined method is

also based on the Truss model with variable angle of inclination of the

2000

1000

Vmax. Maximum shear stress (psi)

ACI, AASHTO

~:::::::=--__ ----- (IO.jfc)

2 4 6

X 10 3 (psi),

8 10 f' C

Fig. 2.3 Upper Innit of the shear stress in the section

16

diagonal compression strut. A very important differentiation is made

between the cases of equilibrium and compatibility torsion. The CEB-

refined procedure in the case of torsion neglects compatibility torsion.

In this design procedure it is assumed that since compatibility torsion

is caused by deformations of adjacent members in statically

indeterminate structures it will produce secondary effects which should

be considered in evaluating servicability, but can be neglected in the

ul timate strength design of the section. Therefore, in the ul timate

strength design of the section only the cases of equilibrium torsion are

considered.

For the same reasons given in the case of shear the limits for

the angle of inclination of the diagonal compression strut remain

controlled by the values proposed in Eq. 2.1.

In the case of torsion, the ultimate torsional moment Tu must be

equal or less than the resistance value. The resistance value Tn is

made up of the resistance Ts carried by the truss and the additional

resistance of the concrete Tc in the transition range between the

uncracked state and full truss action.

The torsion carried by the truss action is evaluated using

Eq. 2.7.

A

Ts = ~ 2 AO fwd cot~ (2.7)

Equation 2.7 follows directly from Eq. 3.31 derived in Sec. 3.4

of Report 248-2 from the equilibrium conditions in the truss model. Ao

is the area enclosed by the perimeter conecting the centers of the