Description
Geometry - Volume
Simple Volume Equation
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AJs are available with square or round ends on the arms (Figures 1 & 2). All other
geometric aspects of square and round end AJs are the same. The volume for square
end AJs is well approximated by
V = 3t3L (square end AJs)
in which V is the volume, t is the arm
thickness, and L is the length. The volume for round end AJs is well
approximated by
V = 3t3L- 0.6438t3 (round end AJs)
The volume of a round end AJ is approximately 4% less than the corresponding square end unit.
Geometry Definition
The primary dimensions of AJs are characterized by the
overall length, L, the arm thickness, t, and the fillet length,
f. These are defined in Figure 3. The ratio of the arm thickness to the
overall length is the waist ratio, r = t / L. The ratio
of the length of the fillet to the thickness of the arm is the fillet ratio,
s = f / t.
The use of standard values for the coefficients
allows the AJ volume to be fully described by just the length. The standard
value for the fillet ratio is s = 1/2. AJs used in open channel
applications have a waist ratio r = 1/6.0 or r = 1/6.5. In
coastal applications, more robust units are used with r = 1/5.0 or r
= 1/5.2.
Complete Volume Equation
A slightly more accurate equation for the volume may be developed which includes the
influence of having rounded corners at the junction between the fillet and arm and
rounded corners on the ends of the arms.
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Figure 1 Square end AJ
Figure 2 Round end AJ
Figure 3 AJ geometry definitions.
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The fillets have quarter rounds where they join with the arms. The round corner
eliminates the stress concentration which would develop if there was a square
corner. The radius of this quarter round is related to the fillet length as
pfrL.
There may also be round corners on the ends of the arms of square end AJs.
These slightly rounded corners improve the appearance and also
eliminate the sharp square corner which is easily chipped. The
radii of the rounded ends are proportional to the arm thickness, brL.
Note that round end AJs just correspond to the case where b = 1/2.
Including the corner rounds, the volume of both square and round end AJs
is given by
V= {3r2 + r3[-2 + 8s2 +
(s2p2 - b2)(12 - 3
&pi)]}L3 = aL3
in which a is the volume coefficient.
For a typical square end coastal AJs unit, the geometric coefficients are
r= 1/5.2, s = 1/2, b= 1/10, p = 1/4.
For these values, the approximate volume equation has an error with respect
to the full equation of -0.2%.
For a typical square end river AJs unit, the geometric coefficients are
r= 1/6.0, s = 1/2, b= 1/50, p = 1/10.
For this case, the approximation has an error of -0.04%. For most applications,
the approximate volume equations provide sufficient accuracy.
Standard Length AJs Volumes
Table 1 provides volumes for typical configurations. The waist ratio of
r
= 1/4 corresponds to “stubby” units. These are typically used for underlayer or
repair of armour layers on structures with other types of armour. Waist ratios
of
r = 1/5 and 1/5.2 are typical coastal units. Waist ratios of
r
= 1/6 and 1/6.5 are typical river units. The more slender river units are only
recommended in the smaller sizes.
The web site provides a volume calculation tool which solves the full equation and
is applicable to all AJs configurations. It is located at:
www.a-jacks.com/Tools/Geometry/Geometry.aspx.
| AJs volume as a function of length and waist ratio. |
| (s=1/2, b=1/10, p=1/4) |
| |
| r = |
1/4.0 |
1/5.0 |
1/5.2 |
1/6.0 |
1/6.5 |
| a = |
0.18803 |
0.12027 |
0.11119 |
0.08349 |
0.07113 |
| |
| L(m) |
V(m3) |
V(m3) |
V(m3) |
V(m3) |
V(m3) |
| 0.6 |
0.0406 |
0.0260 |
0.0240 |
0.0180 |
0.0154 |
| 1.0 |
0.1880 |
0.1203 |
0.1112 |
0.0835 |
0.0711 |
| 1.5 |
0.6346 |
0.4059 |
0.3753 |
0.2818 |
0.2401 |
| 2.0 |
1.5042 |
0.9622 |
0.8895 |
0.6679 |
|
| 2.5 |
2.9379 |
1.8792 |
1.7373 |
|
|
| 3.0 |
5.0768 |
3.2473 |
3.0021 |
|
|
| 3.5 |
8.0617 |
5.1566 |
4.7671 |
|
|
| 4.0 |
12.0338 |
7.6973 |
7.1160 |
|
|
| 4.5 |
17.1341 |
10.9596 |
10.1319 |
|
|
| 5.0 |
23.5035 |
15.0338 |
13.8984 |
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