180
6 Impact Resistance of Basalt Aggregated UHP-SFRC/Fabric …
h per
d
= 3.19
x
d
− 0.718
x
d
2 for
x
d
≤ 1.35 or
h per
d
≤ 3
h per
d
= 1.32
x
d
+ 1.24
x
d
2 for 1.35 <
x
d
< 13.5 or 3 <
h per
d
< 18
(6.1c)
(ii) Kar (1978) further proposed the equations for predicting perforation limit of
steel projectile
h per −a
d
= 3.19
x
d
− 0.718
x
d
2 for
x
d
≤ 1.35
h per −a
d
= 1.32
x
d
+ 1.24
x
d
2 for 1.35 <
x
d
≤ 13.5
(6.2)
where a is half of the aggregate size in concrete and x is derived from Eq. (6.1a).
(iii) Chen et al. (2008) proposed the formula to predict the perforation limit as
follows
⎧
⎨
⎩
x
d
=
(4k/π)(1+kπ/4N )
(1/I 0 +1/N )
for
x
d
≤ k
x
d
=
2
π
ln
(1+I 0 /N )
1+kπ/4N
+ k for
x
d
> k
(6.3a)
I 0 =
M V
2
0
Sf c d 3 ; N =
M
N 1 ρd 3
(6.3b)
⎧
⎨
⎩
h per
d
=
x
d
+
√
1+
√
3S(x+kd ) tan α−1
2 tan α
for
x
d
≤ k
h per
d
=
x
d
+
√
1+
√
3S tan α−1
2 tan α
for
x
d
> k
(6.3c)
where k = 0.707 + h c /d is a dimensionless parameter and h c is the length of the
hard core nose; ρ t is the density of the target; N 1 is the nose shape factor and its
value for different nose shaped projectile can be referred in Whiffen (1943) and Wu
et al. (2012); and α = 55° is adopted as discussed in Sect. 6.4.1. Equations (6.3a–c)
are also widely used to predict the ballistic limit V BL and residual velocity V r of
the projectile impacting on the concrete target. For the relatively thick target in the
present test where x > kd, it can be obtained reversely from Eqs. (3a–c) that
V BL =
Sf c d 3 I BL
M
; I BL =
N +
kπ
4
e
π
2N
H
d −
H plug
d −k
− N
(6.3d)
V r =
V
2
0 − V
2
BL
0.5
(6.3e)
where H t is the thickness of the given target plate, and H plug is the thickness of the
rear crater (Chen et al. 2008).
(iv) In Wu et al. (2015a), we have proposed an empirical approach for predicting the
terminal ballistic parameters of projectile perforating finite-thickness concrete
panel. The residual impact function I r is expressed as follows
6 Impact Resistance of Basalt Aggregated UHP-SFRC/Fabric …
h per
d
= 3.19
x
d
− 0.718
x
d
2 for
x
d
≤ 1.35 or
h per
d
≤ 3
h per
d
= 1.32
x
d
+ 1.24
x
d
2 for 1.35 <
x
d
< 13.5 or 3 <
h per
d
< 18
(6.1c)
(ii) Kar (1978) further proposed the equations for predicting perforation limit of
steel projectile
h per −a
d
= 3.19
x
d
− 0.718
x
d
2 for
x
d
≤ 1.35
h per −a
d
= 1.32
x
d
+ 1.24
x
d
2 for 1.35 <
x
d
≤ 13.5
(6.2)
where a is half of the aggregate size in concrete and x is derived from Eq. (6.1a).
(iii) Chen et al. (2008) proposed the formula to predict the perforation limit as
follows
⎧
⎨
⎩
x
d
=
(4k/π)(1+kπ/4N )
(1/I 0 +1/N )
for
x
d
≤ k
x
d
=
2
π
ln
(1+I 0 /N )
1+kπ/4N
+ k for
x
d
> k
(6.3a)
I 0 =
M V
2
0
Sf c d 3 ; N =
M
N 1 ρd 3
(6.3b)
⎧
⎨
⎩
h per
d
=
x
d
+
√
1+
√
3S(x+kd ) tan α−1
2 tan α
for
x
d
≤ k
h per
d
=
x
d
+
√
1+
√
3S tan α−1
2 tan α
for
x
d
> k
(6.3c)
where k = 0.707 + h c /d is a dimensionless parameter and h c is the length of the
hard core nose; ρ t is the density of the target; N 1 is the nose shape factor and its
value for different nose shaped projectile can be referred in Whiffen (1943) and Wu
et al. (2012); and α = 55° is adopted as discussed in Sect. 6.4.1. Equations (6.3a–c)
are also widely used to predict the ballistic limit V BL and residual velocity V r of
the projectile impacting on the concrete target. For the relatively thick target in the
present test where x > kd, it can be obtained reversely from Eqs. (3a–c) that
V BL =
Sf c d 3 I BL
M
; I BL =
N +
kπ
4
e
π
2N
H
d −
H plug
d −k
− N
(6.3d)
V r =
V
2
0 − V
2
BL
0.5
(6.3e)
where H t is the thickness of the given target plate, and H plug is the thickness of the
rear crater (Chen et al. 2008).
(iv) In Wu et al. (2015a), we have proposed an empirical approach for predicting the
terminal ballistic parameters of projectile perforating finite-thickness concrete
panel. The residual impact function I r is expressed as follows