1. spymaster007
01/08/2013

2. 2

3. 3

4. 4
w
z
y
xo d
b
G
t
Shear center

5. 5

6. 6
Mathcad implementation
Importing from AISC 14th Database for unequal leg angles
(partially shown)
LS
1 2 3 4 5 6 7
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
"AISC Label" "d" "b" "t" "x" "y" "tan(a)"
"L8X6X1" 6 8 1 1.65 2.65 0.542
"L8X6X7/8" 6 8 0.875 1.6 2.6 0.546
"L8X6X3/4" 6 8 0.75 1.56 2.55 0.55
"L8X6X5/8" 6 8 0.625 1.51 2.5 0.554
"L8X6X9/16" 6 8 0.563 1.49 2.48 0.556
"L8X6X1/2" 6 8 0.5 1.46 2.46 0.557
"L8X6X7/16" 6 8 0.438 1.44 2.43 0.559
"L8X4X1" 4 8 1 1.04 3.03 0.247
"L8X4X7/8" 4 8 0.875 0.997 2.99 0.252
"L8X4X3/4" 4 8 0.75 0.949 2.94 0.257
"L8X4X5/8" 4 8 0.625 0.902 2.89 0.262
"L8X4X9/16" 4 8 0.563 0.878 2.86 0.264
"L8X4X1/2" 4 8 0.5 0.854 2.84 0.266
"L8X4X7/16" 4 8 0.438 0.829 2.81 0.268
"L7X4X3/4" 4 7 0.75 1 2.5 0.324
"L7X4X5/8" 4 7 0.625 0.958 2.45 0.329
"L7X4X1/2" 4 7 0.5 0.91 2.4 0.334
"L7X4X7/16" 4 7 0.438 0.886 2.38 ...
row rows LS( ) 77 <--- There are 76 unequal angles in total
Extracting data from the table
i 2 row
di 1 LSi 2 bi 1 LSi 3 ti 1 LSi 4 xGi 1
LSi 5
yGi 1
LSi 6 i 1 atan LSi 7 Iwi 1
LSi 8

7. 7
d
1
1
2
3
4
5
6
7
8
9
10
11
12
13
6
6
6
6
6
6
6
4
4
4
4
4
...
b
1
1
2
3
4
5
6
7
8
9
10
11
12
13
8
8
8
8
8
8
8
8
8
8
8
8
...
t
1
1
2
3
4
5
6
7
8
9
10
11
12
13
1
0.875
0.75
0.625
0.563
0.5
0.438
1
0.875
0.75
0.625
0.563
...
xG
1
1
2
3
4
5
6
7
8
9
10
11
12
13
1.65
1.6
1.56
1.51
1.49
1.46
1.44
1.04
0.997
0.949
0.902
0.878
...
yG
1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
2.65
2.6
2.55
2.5
2.48
2.46
2.43
3.03
2.99
2.94
2.89
2.86
2.84
2.81
...
1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0.497
0.5
0.503
0.506
0.507
0.508
0.51
0.242
0.247
0.252
0.256
0.258
0.26
0.262
...
Iw
1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
98.4
88.4
77.8
66.5
60.7
54.6
48.4
73.4
66.1
58.2
49.9
45.6
41
36.4
...
Generic function to calculate ßw
myfund b t xG yG Iw w x y( ) x xG cos( ) y yG sin( )
z x y( ) x xG sin( ) y yG cos( )
f x y( ) z x y( ) w x y( )
2
z x y( )
2
INT1
0
t
y
0
d
xf x y( ) d d
INT2
t
b
y
0
t
xf x y( ) d d
INT INT1 INT2
xS
yS
t
2
t
2
wS
zS
cos( )
sin( )
sin( )
cos( )
xS xG
yS yG
zo zS
out
1
Iw
INT 2zo
out
Use of FOR loop to calculate ßwfor all angles
w
outj myfundj bj tj xGj
yGj
j Iwj
j 1 76for
out

8. 8
OK, so here are the ßw values for all the unequal angles in the
table:
w
1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
3.288
3.296
3.309
3.316
3.316
3.314
3.325
5.414
5.436
5.467
5.494
5.512
5.522
5.54
4.339
...
w
1
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
4.365
4.39
4.393
4.409
3.095
3.109
3.117
3.129
3.135
3.148
3.148
3.157
3.681
3.701
3.707
...
w
1
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
2.38
2.394
2.397
2.413
2.422
2.423
2.975
2.977
2.994
2.996
3.01
0.874
0.868
0.876
0.881
...
and belowis a typical calculation of the mean of ßwfor the L8X6X
series (the first 7 sections)
k 1 7 wk
3.288
3.296
3.309
3.316
3.316
3.314
3.325
LSk 1 1
"L8X6X1"
"L8X6X7/8"
"L8X6X3/4"
"L8X6X5/8"
"L8X6X9/16"
"L8X6X1/2"
"L8X6X7/16"
MEAN mean submatrix w 1 7 1 1
------> MEAN 3.309
This is compared to ßw = 3.31 as listed in Table C.F10-1 for the L8X6 .
Angle
Shape
βw Mean
calculated
AISC Table
values
L8X6 3.309 3.31
L8X4 5.484 5.48
L7X4 4.379 4.37
L6X4 3.130 3.14
L6X3.5 3.696 3.69
L5X3.5 2.405 2.40
L5X3 2.990 2.99
L4X3.5 0.875 0.87
L4X3 1.653 1.65
L3.5X3 0.870 0.87
L3.5X2.5 1.619 1.62
L3X2.5 0.862 0.86
L3X2 1.568 1.56
L2.5X2 0.852 0.85
Summary of calculations