4.4 Results and Analysis
87
4.4.3.2 Willam-Warnke Failure Criterion
Willam-Warnke failure criterion (Chen 1982) is a five-parameter model which has
been adopted by the popular finite element software ANSYS. The compressive
meridians equation is expressed as
τ m
f
c
=a w +a w1
σ m
f
c
− a w2
σ m
f
c
2
(4.3)
where
σ m
=
(σ 1 + σ 2 + σ 3 )/3
and
τ m
=
1
√
15
(σ 1 − σ 2 )
2
+ (σ 2 − σ 3 )
2
+ (σ 3 − σ 1 )
2
1/2 are the primary normal stress
and shear stress, respectively. σ 1 , σ 2 and σ 3 are the stresses in the three principal
directions, a w , a w1 and a w2 are the Willam-Warnke model coefficients, respectively.
Figure 4.8(a) shows the present test data from Table 4.2 as well as the fitting
curves corresponding to Eq. (4.3), the compressive meridian of UHPCC in this study
can be expressed as
τ m
f
c
=0.177+0.656
σ m
f
c
− 0.062
σ m
f
c
2
(4.4)
Figure 4.8b also gives the existing compressive test data on high-strength concrete,
including UHPCC, HPFRC and SIFCON as well as the curve of Eq. (4.4). The
comparisons indicate that, compared with the Mohr–Coulomb failure criterion,
Willam-Warnke failure criterion with the expression of Eq. (4.4) gives excellent
predictions for high-strength concrete, and the influences of strength, size, and
compositions of specimens are not pronounced.
4.4.3.3 Power-Law Failure Criterion
The Power-law failure criterion can be expressed as
σ
u
1
f
c = 1 + a p (σ 3
f
c )
b p
(4.5)
where a p and b p are the coefficients.
Figure 4.9a shows the present test data on UHPCC as well as the power-law fitted
curves with the form as
σ
u
1
f
c = 1 + 3.50(σ 3
f
c )
0.72
(4.6)
Similarly, Fig. 4.9b gives the present and existing test data on high-strength
concrete, as well as the curve of Eq. (4.6). It can be seen that, compared with the
Mohr-Coulomb failure criterion, Power-law failure criterion with the expression of