Fixed Angle Shear Theories - Unified Theory of Concrete Structures - page 231

β

Figure 6.10 Function of deviation angle, f 3 ( )

/ f c ( MPa ) is greater than 0.9. The
constant of 0.9 governs and Eq. ( 6.51) degenerates into Eq. (6.50). When f c is greater than
42 MPa, however, 5

When concrete strength f c is less than 42 MPa, 5 . 8

/ f c ( MPa ) is less than 0.9 and this term will dominate.

calculated by
Equation (6.51) will be less than that calculated by Equation (6.50). Equation (6.51) provides
a simple concept: the higher the concrete strength f c , the lower the softened coefficient

. 8 ζ

ζ .

6.1.7.3 Function of Deviation Angle f 3 ( β

)
In the fixed angle theory, the softening coefficient ζ is also a function of the deviation angle β ,

ε 1 and f c . The angle in addition to the variables β is defined as: β = α 1 − α r , the difference

between the angle α 1 of the applied principal stresses in the 1-2 coordinate and the angle α r of

the principal concrete stresses in the r − d coordinate. Angle β has been discussed in Sections

4.3.1 and 4.3.5, and in Figure 4.14.
Figure 6.10 shows that f 3 ( β ) is a linear function of the deviation angle β , and can be

24 ◦ . This straight-line expression was
determined by regression analysis (Wang, 2006) based on three sets of test data:

expressed by a simple formula, f 3 ( β ) = 1 − | β | /

1. Pang and Hsu (1995) tested ten panels (A series and B series) made of normal-strength
concrete of approximately 42 MPa and reinforced with steel percentages from 0.6 to 3.0%
in the longitudinal and transverse directions. The ratio

ρ /ρ t ranged from 1 to 5.
2. Zhang and Hsu (1998) tested nine panels (VA series and VB series) made of high-strength
concrete of approximately 100 MPa, and reinforced with steel percentages in the range
0.42-6.0%. The ratio

ρ /ρ t also varied from 1 to 5.
3. Chintrakarn (2001) tested six panels (M series) with concrete strength of approximately
45 MPa. The ratios ρ /ρ t in panels M1-M6 were much larger than those tested by Pang

and Hsu (1995) and Zhang and Hsu (1998). The four panels M2, M3, M4 and M5 had
ρ /ρ t ratios of 4.0, 9.0, 7.2, and 15.6, respectively. No steel was placed in the transverse
direction in panels M1 and M6, giving a

ρ /ρ t ratio of infinity.

0. When a
2-D shear element is reinforced with the same amounts of steel bars in the two directions

The straight-line function f 3 ( β ) has an intercept of f 3 ( β ) = 1, when β =