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8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
1/54
Plate bending transverse displacement, w, depends on surfacecoordinates of position of the point under study, a generic pointlocated through normalized coordinates of position =x/a, and =y/a.The deflected shape of the elements is described by a Pascal cubicpolynomial with the powers of the terms in and :{w}= |f(,)|*{A}
The row vector |f| lists the powers of the coordinates of position,using Pascal triangle and the {A} vector lists the respective
coefficients {w}=|f| {a}
The element degrees of freedom are {q}, 4 vertical displacements; 8curvatures, two in each of the four corner nodes; and one additionalthirteenth dof representing the average displacement across all thearea, making a total of 13 dof.
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
2/54
Transformation expression: Substituting variables and by theirnodal values (+1 and -1) we construct [M] from {q}=[M]{A}, Such as{q} represents the element degrees of freedom.
After finding [M]^(-1), identification of coefficients {A}={ai} is achieved,finding {A} =([M]^(-1))*{q} in the given combinations of extremepositions +1 and -1 for both and
This {a} coefficients are applied to the curvature equations tocalculate the plate generic strains.
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
3/54
x
1
a
x x
d
d=
y
1
b
y y
d
d=
f ,( ) 1 2
2
3
2
2
3
4
3
2
2
3
4
4
4
( ):=
A a1 a2 a3 a4 a5 a6 a7 a8 a8 a10 a11 a12 a13 a14 a15 a16 a17( ):= a1
w ,( ) f ,( ) AT:= A
a1 a2 a3 a4 a5 a6 a7 a8 a8 a10 a11 a12 a13 a14 a15 a16 a17( )T
a1
a2
a3
a4
a5
a6
a7
a8
a8
a10
a11
a12
a13
a14
a15
a16
a17
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
4/54
Here we are multiplying f AT
1 2
2
3
2
2
3
4
3
2
2
3
4
4
4
( )
a1
a2
a3
a4
a5
a6
a7
a8
a8
a10
a11
a12
a13
a14
a15
a16
a17
a16 4
a11 4
+ a12 3
+ a7 3
+ a13 2
2
+ a8 2
+ a4 2
+ a17 4
+ a14 3
+ +
a8 2 a5 + a2 + a15
4+ a10 3+ a6
2+ a3 + a1++
...
The latter is the Pascal Polinomial w(,)
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
5/54
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
6/54
q1 f 1 1,( ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1( )
q2
f 1 1,( ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1( )
q3 f 1 1,( ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1( )
q4 f 1 1,( ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1( )
We will use a different positive sign convention for rotation as wehave done it before, but we will first take the "x" derivative as thepaper does .
f ,( )d
d0 0 1 0 2 0
22 3
2 0
32
2 3
2 4
3
44
3( )
,( )
f ,( )d
d
1
b:=
b
1 1,( ) 0 01
b0
1
b
2
b0
1
b
2
b
3
b0
1
b
2
b
3
b
4
b
1
b
4
b
1 1,( ) 0 01
b0
1
b
2
b 0
1
b
2
b
3
b0
1
b
2
b
3
b
4
b
1
b
4
b
1 1,( ) 0 01
b0
1
b
2
b0
1
b
2
b
3
b0
1
b
2
b
3
b
4
b
1
b
4
b
1 1,( ) 0 01
b0
1
b2
b0
1
b
2
b
3
b0
1
b2
b3
b4
b1
b
4
b
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
7/54
f ,( )d
d0 1 0 2 0 3
2 2
20 4
3 3
2 2
2
30 4
3
4( )
,( )
f ,( )d
d
1
a:=
a
1 1,( ) 01
a0
2
a
1
a0
3
a
2
a
1
a0
4
a
3
a
2
a
1
a0
4
a
1
a
1 1,( ) 01
a0
2
a
1
a 0
3
a
2
a
1
a0
4
a
3
a
2
a
1
a 0
4
a
1
a
1 1,( ) 01
a
02
a
1
a
03
a
2
a
1
a
04
a
3
a
2
a
1
a
04
a
1
a
1 1,( ) 01
a0
2
a
1
a 0
3
a
2
a
1
a0
4
a
3
a
2
a
1
a 0
4
a
1
a
1
1
f ,( )
d 2 0 2 2
30 2
2 0
2
30 2
3
2
50
2 2
30 2
4
2
50
( )
1
1
f ,( )
d1
2a:=
a
1( )1
a0
1
a
1
3 a0
1
a0
1
3 a0
1
a
1
5 a0
1
3 a0
1
a
1
5 a0
1( )1
a0
1
a1
3 a 01
a0
1
3 a 01
a1
5 a 01
3 a 01
a
1
5 a 0
1
1
f ,( )
d 2 2 0 2 2
02
32
3 0
2
30 2
4 0
2 2
30
2
50
2
5
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
8/54
( )1
2b1
1
f ,( )
d
:=
b
( )1
1f ,( )
d
2 b:=
b
1( )1
b
1
b0
1
b0
1
3 b
1
b0
1
3 b0
1
b0
1
3 b0
1
5 b0
1
5 b
1( )1
b
1
b
01
b
01
3 b
1
b
01
3 b
01
b
01
3 b
01
5 b
01
5 b
1
1
1
1
f ,( )
d
d
41 0 0
1
30
1
30 0 0 0
1
50
1
90
1
50 0
To find later the strains, first we find:
2
f ,( )d
d
2
0 0 0 2 0 0 6 2 0 0 12 2
6 2 2
0 0 12 2
0( )
2
f ,( )d
d
2
0 0 0 0 0 2 0 0 2 6 0 0 2 2
6 12 2
0 12 2
( )
2
f ,( )d
d
d
d
0 0 0 0 2 0 0 4 4 0 0 6 2
8 6 2
0 8 3
8 3
( )
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
9/54
Constructing M. (Coordinates Transformation Expression)
The [M] matrix is constructed from {q} = [M]*{a} with the four values f
(q1...q4); four (q5...q8); four (q9... q12); two (q13,q14); two (q15,q16); and q17. All of them are qi vectors or 1x17 matrixes thatwe will use for the transformation:
1
1
1
1
0
0
0
0
0
0
0
0
1
a
1
a
1
b
1
b
1
1
1
1
1
1
a
1a
1
a
1
a
0
0
0
0
0
0
1
b
1
b
0
1
1
1
1
0
0
0
0
1
b
1
b
1
b
1
b
1
a
1
a
0
0
0
1
1
1
1
2
a
2a
2
a
2
a
0
0
0
0
1
3a
1
3a
1
b
1
b
1
3
1
1
1
1
1
a
1a
1
a
1
a
1
b
1
b
1
b
1
b
0
0
0
0
0
1
1
1
1
0
0
0
0
2
b
2
b
2
b
2
b
1
a
1
a
1
3 b
1
3 b
1
3
1
1
1
1
3
a
3a
3
a
3
a
0
0
0
0
0
0
1
b
1
b
0
1
1
1
1
2
a
2a
2
a
2
a
1
b
1
b
1
b
1
b
1
3a
1
3a
0
0
0
1
1
1
1
1
a
1a
1
a
1
a
2
b
2
b
2
b
2
b
0
0
1
3 b
1
3 b
0
1
1
1
1
0
0
0
0
3
b
3
b
3
b
3
b
1
a
1
a
0
0
0
1
1
1
1
4
a
4a
4
a
4
a
0
0
0
0
1
5a
1
5a
1
b
1
b
1
5
1
1
1
1
3
a
3a
3
a
3
a
1
b
1
b
1
b
1
b
0
0
0
0
0
1
1
1
1
2
a
2a
2
a
2
a
2
b
2
b
2
b
2
b
1
3a
1
3a
1
3 b
1
3 b
1
9
1
1
1
1
1
a
1a
1
a
1
a
3
b
3
b
3
b
3
b
0
0
0
0
0
1
1
1
1
0
0
0
0
4
b
4
b
4
b
4
b
1
a
1
a
1
5 b
1
5 b
1
5
1
1
1
1
4
a
4a
4
a
4
a
1
b
1
b
1
b
1
b
1
5a
1
5a
0
0
0
1
1
1
1
1
a
1a
1
a
1
a
4
b
4
b
4
b
4
b
0
0
1
5 b
1
5 b
0
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
10/54
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
11/54
Matrix M is already inverted:
1
8
3
32
3
32
3
8
1
2
3
8
1
8
15
16
15
16
1
8
15
32
1
8
9
16
1
8
15
32
15
32
15
32
1
8
3
32
3
32
3
8
1
2
3
8
1
8
15
16
15
16
1
815
32
1
8
9
16
1
8
15
32
15
32
15
32
1
8
3
32
3
32
3
8
1
2
3
8
1
8
15
16
15
16
1
8
15
32
1
8
9
16
1
8
15
32
15
32
15
32
1
8
3
32
3
32
3
8
1
2
3
8
1
8
15
16
15
16
1
815
32
1
8
9
16
1
8
15
32
15
32
15
32
a
32
a
8
a
32
3 a
16
a
8
0
a
8
3 a
16
0
0
5 a
32
a
8
0
0
0
5 a
32
0
a
32
a
8
a
32
3 a
16
a
8
0
a
8
3 a
16
0
0
5 a
32
a
8
0
0
0
5 a
32
0
a
32
a
8
a
32
3 a
16
a
8
0
a
8
3 a
16
0
0
5 a
32
a
8
0
0
0
5 a
32
0
a
32
a
8
a
32
3 a
16
a
8
0
a
8
3 a
16
0
0
5 a
32
a
8
0
0
0
5 a
32
0
b
32
b
32
b
8
0
b
8
3 b
16
0
0
3 b
16
b
8
0
0
0
b
8
5 b
32
0
5 b
32
b
32
b
32
b
8
0
b
8
3 b
16
0
0
3 b
16
b
8
0
0
0
b
8
5 b
32
0
5 b
32
b
32
b
32
b
8
0
b
8
3 b
16
0
0
3 b
16
b
8
0
0
0
b
8
5 b
32
0
5 b
32
b
32
b
32
b
8
0
b
8
3 b
16
0
0
3 b
16
b
8
0
0
0
b
8
5 b
32
0
5 b
32
3 a16
0
15 a
16
3 a
4
0
9 a
8
0
15 a
8
0
0
15 a
16
0
9 a
8
0
0
15 a
16
0
3 a16
0
15 a
16
3 a
4
0
9 a
8
0
15 a
8
0
0
15 a
16
0
9 a
8
0
0
15 a
16
0
3 b16
15 b
16
0
9 b
8
0
3 b
4
0
0
15 b
8
0
0
0
9 b
8
0
15 b
16
0
15 b
16
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
12/54
Mathcad is ordered to create M1 T
:
1
8
3
32
3
32
3
8
1
2
3
8
1
8
15
16
1516
1
8
15
32
1
8
9
16
1
8
15
32
15
32
1
8
3
32
3
32
3
8
1
2
3
8
1
8
15
16
1516
1
8
15
32
1
8
9
16
1
8
15
32
15
32
1
8
3
32
3
32
3
8
1
2
3
8
1
8
15
16
1516
1
8
15
32
1
8
9
16
1
8
15
32
15
32
1
8
3
32
3
32
3
8
1
2
3
8
1
8
15
16
1516
1
8
15
32
1
8
9
16
1
8
15
32
15
32
a
32
a
8
a
32
3 a
16
a
8
0
a
8
3 a
16
0
0
5 a
32
a
8
0
0
0
5 a
32
a
32
a
8
a
32
3 a
16
a
8
0
a
8
3 a
16
0
0
5 a
32
a
8
0
0
0
5 a
32
a
32
a
8
a
32
3 a
16
a
8
0
a
8
3 a
16
0
0
5 a
32
a
8
0
0
0
5 a
32
a
32
a
8
a
32
3 a
16
a
8
0
a
8
3 a
16
0
0
5 a
32
a
8
0
0
0
5 a
32
b
32
b
32
b
8
0
b
8
3 b
16
0
0
3 b16
b
8
0
0
0
b
8
5 b
32
0
b
32
b
32
b
8
0
b
8
3 b
16
0
0
3 b16
b
8
0
0
0
b
8
5 b
32
0
b
32
b
32
b
8
0
b
8
3 b
16
0
0
3 b16
b
8
0
0
0
b
8
5 b
32
0
b
32
b
32
b
8
0
b
8
3 b
16
0
0
3 b16
b
8
0
0
0
b
8
5 b
32
0
3 a
16
0
15 a
16
3 a
4
0
9 a
8
0
15 a
8
0
0
15 a
16
0
9 a
8
0
0
15 a
16
3 a
16
0
15 a
16
3 a
4
0
9 a
8
0
15 a
8
0
0
15 a
16
0
9 a
8
0
0
15 a
16
3 b
16
15 b
16
0
9 b
8
0
3 b
4
0
0
15 b8
0
0
0
9 b
8
0
15 b
16
0
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
13/54
15
32
15
32
15
32
15
320 0 0 0
5 b
32
5 b
32
5 b
32
5 b
320 0
15 b
16
This is the M1
T
:
1
8
1
8
1
8
1
8
a
32
a
32
a
32
a
32
b
32
b
32
b
32
b
32
3 a
16
3 a
16
3 b
16
3 b
16
9
4
3
32
3
32
3
32
3
32
a
8
a
8
a
8
a
8
b
32
b
32
b
32
b
32
0
0
15 b
16
15 b
16
0
3
32
3
32
3
32
3
32
a
32
a
32
a
32
a
32
b
8
b
8
b
8
b
8
15 a
16
15 a
16
0
0
0
3
8
3
8
3
8
3
8
3 a16
3 a
16
3 a
16
3 a
16
0
0
0
0
3 a
4
3 a
4
9 b
8
9 b
8
9
4
1
2
1
2
1
2
1
2
a
8
a
8
a
8
a
8
b
8
b
8
b
8
b
8
0
0
0
0
0
3
8
3
8
3
8
3
8
0
0
0
0
3 b
16
3 b16
3 b
16
3 b
16
9 a
8
9 a
8
3 b
4
3 b
4
9
4
1
8
1
8
1
8
1
8
a
8
a
8
a
8
a
8
0
0
0
0
0
0
0
0
0
15
16
15
16
15
16
15
16
3 a16
3 a
16
3 a
16
3 a
16
0
0
0
0
15 a
8
15 a
8
0
0
0
15
16
15
16
15
16
15
16
0
0
0
0
3 b
16
3 b16
3 b
16
3 b
16
0
0
15 b
8
15 b
8
0
1
8
1
8
1
8
1
8
0
0
0
0
b
8
b
8
b
8
b
8
0
0
0
0
0
15
32
15
32
15
32
15
32
5 a32
5 a
32
5 a
32
5 a
32
0
0
0
0
15 a
16
15 a
16
0
0
0
1
8
1
8
1
8
1
8
a
8
a
8
a
8
a
8
0
0
0
0
0
0
0
0
0
9
16
9
16
9
16
9
16
0
0
0
0
0
0
0
0
9 a
8
9 a
8
9 b
8
9 b
8
9
4
1
8
1
8
1
8
1
8
0
0
0
0
b
8
b
8
b
8
b
8
0
0
0
0
0
15
32
15
32
15
32
15
32
0
0
0
0
5 b
32
5 b32
5 b
32
5 b
32
0
0
15 b
16
15 b
16
0
15
32
15
32
15
32
15
32
5 a32
5 a
32
5 a
32
5 a
32
0
0
0
0
15 a
16
15 a
16
0
0
0
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
14/54
Shape functions of the plate bendingelements Nj are obtained multiplying thecolumn Nij of the former matrix by f(,)transpose. We must construct |f| transpose
|f| is a 1x17 row matrix definedbefore, whose transpose is this17X1 column matrix:
f ,( )T
1
2
2
3
2
2
3
4
3
2
2
3
4
4
4
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
15/54
We will find shape functions with the former matrix multiplying it by f transpose:
1
8
1
8
1
8
1
8
a
32
a
32
a
32
a
32
b
32
b
32
b
32
b
32
3 a
16
3 a
16
3 b16
3 b
16
9
4
3
32
3
32
3
32
3
32
a
8
a
8
a
8
a
8
b
32
b
32
b
32
b
32
0
0
15 b16
15 b
16
0
3
32
3
32
3
32
3
32
a
32
a
32
a
32
a
32
b
8
b
8
b
8
b
8
15 a
16
15 a
16
0
0
0
3
8
3
8
3
8
3
8
3 a
16
3 a
16
3 a
16
3 a
16
0
0
0
0
3 a
4
3 a
4
9 b8
9 b
8
9
4
1
2
1
2
1
2
1
2
a
8
a
8
a
8
a
8
b
8
b
8
b
8
b
8
0
0
0
0
0
3
8
3
8
3
8
3
8
0
0
0
0
3 b
16
3 b
16
3 b
16
3 b
16
9 a
8
9 a
8
3 b4
3 b
4
9
4
1
8
1
8
1
8
1
8
a
8
a
8
a
8
a
8
0
0
0
0
0
0
0
0
0
15
16
15
16
15
16
15
16
3 a
16
3 a
16
3 a
16
3 a
16
0
0
0
0
15 a
8
15 a
8
0
0
0
15
16
15
16
15
16
15
16
0
0
0
0
3 b
16
3 b
16
3 b
16
3 b
16
0
0
15 b8
15 b
8
0
1
8
1
8
1
8
1
8
0
0
0
0
b
8
b
8
b
8
b
8
0
0
0
0
0
15
32
15
32
15
32
15
32
5 a
32
5 a
32
5 a
32
5 a
32
0
0
0
0
15 a
16
15 a
16
0
0
0
1
8
1
8
1
8
1
8
a
8
a
8
a
8
a
8
0
0
0
0
0
0
0
0
0
9
16
9
16
9
16
9
16
0
0
0
0
0
0
0
0
9 a
8
9 a
8
9 b8
9 b
8
9
4
1
8
1
8
1
8
1
8
0
0
0
0
b
8
b
8
b
8
b
8
0
0
0
0
0
15
32
15
32
15
32
15
32
0
0
0
0
5 b
32
5 b
32
5 b
32
5 b
32
0
0
15 b16
15 b
16
0
15
3
15
32
15
3
15
32
5 a
32
5
3
5
3
5 a
32
0
0
0
0
15
16
15
16
0
0
0
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
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Shape functions from the first to the sevententh degrees of fredom:
9 2
2
16
15 4
32
3
8
3
8
15 4
32
15 2
16+
3 2
8+
15 4
32
3
8
15 2
16+
2+
15 4
32
15 4
32
3
8+
3
8
9 2
2
16+
15 2
16
3 2
8+
15 4
32
3
8+
15 2
16+
2
3
8
15 4
32
15 4
32
3
8+
9 2
2
16+
15 2
16+
3 2
8+
15 4
32+
3
8+
15 2
16
2 +
15 4
32
15 4
32
3
8
3
8+
9 2
2
16+
15 2
16
3 2
8+
15 4
32+
3
8
15 2
16
2+ +
a
32
3 a 2
16
a 3
8+
5 a 4
32+
a
8
a
32+
a
8
3 a 2
16
a 3
8+
a
32
3 a 2
16
a 3
8+
5 a 4
32+
a
8
a
32
a
8+
3 a 2
16+
a 3
8
3 a 2
16
a
32
a 3
8+
5 a 4
32
a
8
a
32
a
8
3 a 2
16+
a 3
8+
3 a 2
16
a
32
a 3
8+
5 a 4
32
a
8
a
32+
a
8+
3 a 2
16
a 3
8
b
32
3 2 b16
3 b
8+
5 4 b32
+ b32
+ b
8
b8
3 2 b
16
3 8
+
3 2
b
16
b
32
3
b
8+
5 4
b
32
b
32
b
8
b
8
3 2
b
16+
3
8+
b
32
3 2
b
16
3
b
8+
5 4
b
32+
b
32
b
8
b
8+
3 2
b
16+
3
8
3 2
b
16
b
32
3
b
8+
5 4
b
32
b
32+
b
8
b
8+
3 2
b
16
3
8
15 a 4 16
15 a 416
+ 9 a 2 28
15 a 2 8
3 a 24
9 a 28
+ 15+
15 a 4
16
15 a 4
16
9 a 2
2
8
15 a 2
8+
3 a 2
4
9 a 2
8+
15
9 b 2
8
9 b 2
2
8
15 b 4
16+
15 b 2
8
15 b
16+
15 b 4
16+
3
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
17/54
9 b 2
8
9 b 2
2
8
15 b 4
16
15 b 2
8+
15 b
16
15 b 4
16+
3
9 2
2
4
9 2
4
9 2
4
9
4+
In order to find later the strains, we use what we have already found:
1
a2 2
f ,( )d
d
2
0 0 0
2
a2
0 06
a2
2
a2
0 012
2
a2
6
a2
2
2
a2
0 012
2
a2
1
b2 2
f ,( )d
d
2
0 0 0 0 0
2
b2
0 02
b2
6
b2
0 02
2
b2
6
b2
12
2
b2
012
b2
2
a b( ) f ,( )d
d
d
d
0 0 0 02
a b0 0
4
a b
4
a b0 0
6 2
a b
8
a b
6 2
a b0
8 3
a b
8 3
a b
Strain vector: {}= [Ba]*{A}
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
18/54
z
0
0
0
0
0
0
0
0
0
2
a2
0
0
0
0
2
a b
0
2
b2
0
6
a2
0
0
2
a2
0
4
a b
0
2
b2
4
a b
0
6
b2
0
12 2
a2
0
0
6
a2
0
6 2
a b
2 2
a2
2 2
b2
8
a b
0
6
b2
6 2
a b
0
12 2
b2
0
z
Mathcad is ready to find BaT
divided by the variable z
0
0
0
0
0
0
0
0
0
2
a2
0
0
0
0
2
a b
0
2
b2
0
6
a2
0
0
2
a2
0
4
a b
0
2
b2
4
a b
0
6
b2
0
12 2
a2
0
0
6
a2
0
6 2
a b
2 2
a2
2 2
b2
8
a b
0
6
b2
6 2
a b
0
12 2
b2
0
12
a2
0
8
a b
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
19/54
Ths is the transpose of Ba
divided by z
0
0
0
2
a2
0
0
6
a2
2
a2
0
0
12 2
a2
6
a2
2 2
a2
0
0
12 2
0
0
0
0
0
2
b
2
0
0
2
b2
6
b2
0
0
2 2
b2
6
b2
12 2
b2
0
0
0
0
0
2
a b
0
0
4
a b
4
a b
0
0
6 2
a b
8
a b
6 2
a b
0
8 3
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
20/54
a
012
2
b2
a
8 3
a b
Mathcad is ready to find BaT
D divided by z
0
0
0
2
a2
0
0
6
a2
2
a2
0
0
12 2
a2
6
a2
2 2
a2
0
0
0
0
0
0
2
b2
0
0
2
b2
6
b2
0
0
2 2
b2
6
b2
0
0
0
0
2
a b
0
0
4
a b
4 a b
0
0
6 2
a b
8
a b
6 2
a b
Dx
D1
0
D1
Dy
0
0
0
Dxy
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
21/54
0
12 2
a2
0
12 2
b2
0
12 2
b2
0
8 3
a b
8 3
a b
This is BaT
D divided by z
0
0
0
2 Dx
a2
0
2 D1
b2
6 Dx
a2
2 Dx
a2
2 D1
b2
6 D1
b2
12 Dx 2
a2
6 Dx
2
0
0
0
2 D1
a2
0
2 Dy
b2
6 D1
a2
2 D1
a2
2 Dy
b2
6 Dy
b2
12 D1 2
a2
6 D1
2
0
0
0
0
2 Dxy
a b
0
0
4 Dxy
a b
4 Dxy
a b
0
0
6 Dxy 2
a b
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
22/54
a
2 Dx 2
a2
2 D1
2
b2
6 D1
b2
12 D1 2
b2
12 Dx 2
a2
12 D1
2
b2
a
2 D1 2
a2
2 Dy
2
b2
6 Dy
b2
12 Dy 2
b2
12 D1 2
a2
12 Dy
2
b2
8 Dxy
a b
6 Dxy 2
a b
0
8 Dxy 3
a b
8 Dxy
3
a b
Mathcad is ready to find BaT
D Ba divided by z2
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
23/54
0
0
0
2 Dx
a2
0
2 D1
b2
6 Dx
a2
2 Dx
a2
2 D1
b2
6 D1
b2
12 Dx 2
a
2
6 Dx
a2
2 Dx 2
a2
2 D1
2
b2
6 D1
b2
12 D1 2
b2
12 Dx 2
a2
0
0
0
2 D1
a2
0
2 Dy
b2
6 D1
a2
2 D1
a2
2 Dy
b2
6 Dy
b2
12 D1 2
a
2
6 D1
a2
2 D1 2
a2
2 Dy
2
b2
6 Dy
b2
12 Dy 2
b2
12 D1 2
a2
0
0
0
0
2 Dxy
a b
0
0
4 Dxy a b
4 Dxy
a b
0
0
6 Dxy 2
a b
8 Dxy
a b
6 Dxy 2
a b
0
8 Dxy 3
a b
0
0
0
0
0
0
0
0
0
2
a2
0
0
0
0
2a b
0
2
b2
0
6
a2
0
0
2
a2
0
4 a b
4a
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
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12 D1 2
b2
12 Dy
2
b2
8 Dxy
3
a b
This is BaT
D Ba divided by z2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4 Dx
a4
0
4 D1
a2
b2
12 Dx
a4
4 Dx
a4
4 D1
a2
b2
12 D1
a2
b2
24 Dx
2
a4
12 Dx
a4
2 D 2
2 D 2
0
0
0
0
4 Dxy
a2
b2
0
0
8 Dxy
a2
b2
8 Dxy
a2
b2
0
0
12 Dxy 2
a2
b2
0
0
0
4 D1
a2
b2
0
4 Dy
b4
12 D1
a2
b2
4 D1
a2
b2
4 Dy
b4
12 Dy
b4
24 D1
2
a2
b2
12 D1
a2
b2
2 D 2
2 D 2
0
0
0
12 Dx
a4
0
12 D1
a2
b2
36 Dx 2
a4
12 Dx
a4
12 D1 2
a2
b2
36 D1
a2
b2
72 Dx
3
a4
36 Dx 2
a4
2 D 2
2 D
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
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0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
a2
b2
+
a2
12 D1
a2
b2
24 D1 2
a2
b2
24 Dx 2
a4
24 D1 2
a2
b2
16 Dxy
a2
b2
12 Dxy 2
a2
b2
0
16 Dxy 3
a2
b2
16 Dxy 3
a2
b2
2
a2
b2
+
b2
12 Dy
b4
24 Dy 2
b4
24 D1 2
a2
b2
24 Dy 2
b4
6
a2
b2
+
a2
36 D1 2
a2
b2
72 D1 2
a2
b2
72 Dx 3
a4
72 D1 2
2
a2
b2
The former matrix is BaT
D Ba divided by z2
1 10
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4 Dx
a4
0
4 D
1
a2
b2
12 Dx
a4
4 Dx
a4
4 D1
a2 b2
12 D1
a2
b2
24 Dx 2
a4
12 Dx
a4
22 Dx
2
a2
2 D1 2
b2
+
a2
12 D1
0
0
0
0
4 Dxy
a2
b2
0
0
8 Dxy
a2
b2
8 Dxy
a2 b2
0
0
12 Dxy 2
a2
b2
16 Dxy
a2
b2
12 Dxy 2
0
0
0
4 D1
a2
b2
0
4 D
y
b4
12 D1
a2
b2
4 D1
a2
b2
4 Dy
b4
12 Dy
b4
24 D1 2
a2
b2
12 D1
a2
b2
22 D1
2
a2
2 Dy 2
b2
+
b2
12 Dy
6 2
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
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1 1
0
0
0
0
0
0
0
0
0
a2
b2
24 D1 2
a
2
b
2
24 Dx
2
a4
24 D1 2
a2
b2
a2
b2
0
16 Dxy 3
a2
b2
16 Dxy 3
a2
b2
b4
24 Dy 2
b
4
24 D1 2
a2
b2
24 Dy 2
b4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
16 Dx
a4
0
16 D1
a2
b2
0
0
0
0
32 Dx
4
0
0
0
0
16 Dxy
a2
b2
0
0
0
0
0
0
0
0
0
16 D1
a2
b2
0
16 Dy
b4
0
0
0
0
32 D1
2 2
0
0
0
0
0
0
48 Dx
a4
0
16 D1
a2
b2
0
0
0
0
0
0
0
0
0
16 4 Dxy a2
Dx b+
3 a4
b2
0
16 D1
a2
b2
0
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
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0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
a
0
16 D1 a2 Dx b
2+
3 a4
b2
0
32 D1
a2
b2
0
0
16 Dxy
a2
b2
0
16 Dxy
a2
b2
0
0
0
a b
0
16 Dy a2 D1 b
2+
3 a2
b4
0
32 Dy
b4
0
0
0
0
0
0
0
32 D1
a2
b2
0
0
0
0
32 12 Dxy a2
5 Dx +
15 a4
b2
0
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
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3 b16
15 b
16
0
9 b
8
0
3 b
4
0
0
15 b
8
0
0
0
9 b
8
0
15 b
16
0
15 b
16
9
4
0
0
9
4
0
9
4
0
0
0
0
0
0
9
4
0
0
0
0
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
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3 b
16
15 b
16
0
9 b
8
0
3 b
4
0
0
15 b8
0
0
0
9 b
8
0
15 b
16
0
9
4
0
0
9
4
0
9
4
0
0
0
0
0
0
9
4
0
0
0
T
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15 b
160
15
32
15
32
15
32
15
32
0
0
0
0
5 b
32
5 b32
5 b
32
5 b
32
0
0
15 b
16
15 b
16
0
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
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8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
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a
15
32
15
32
15
32
15
32
0
0
0
0
5 b
32
5 b
32
5 b
32
5 b
32
0
0
15 b16
15 b
16
0
1
2
2
3
2
2
3
4
3
2
2
3
4
4
4
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
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3
32
15 4
32
3
8
3 2
8+
3
32
1
8
3
32
15 4
32
3
8+
3 2
8+
3
32+
1
8
3
32
15 4
32
3
8
3 2
8+
3
32
1
8
3
32
15 4
32
3
8+
3 2
8+
3
32+
1
8
5 a 4
32+
5 a 4
32
5 a 4
32
5 a 4
32+
b 5 4 b32
+
b 5 4
b
32
b 5 4
b
32
b 5 4
b
32+
a 16
3 a16
a
16
3 a
16
b 2
4
3 b
16
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b 2
4
3 b
16
0
2
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12 2
a2
0
8 3
a b
0
12 2
b2
8 3
a b
a1
a2
a3
a4
a5
a6
a7
a8
a8
a10
a11
a12
a13
a14
a15
a16
a17
0
12 2
b2
8 3
a b
T
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
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2
b
0
6
b2
0
12 2
a2
0
0
6
a2
0
6
2
a b
2 2
a2
2 2
b2
8 a b
0
6
b2
6
2
a b
0
12 2
b2
0
12 2
a2
0
8
3
a b
0
12 2
b2
8
3
a b
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0
0
0
4 Dx
a4
8 Dxy
a2
b2
4 D1
a2
b2
12 Dx
a4
4 Dx 2
a4
16 Dxy 2
a2
b2
+
4 D1
a2
b2
16 Dxy
a2
b2
+
12 D1 2
a2
b2
24 Dx
2
a4
12 Dx 2
a4
24 Dxy 3
a2
b2
+
2 D 2
2 D 2
0
0
0
4 D1
a2
b2
8 Dxy
a2
b2
4 Dy
b4
12 D1 2
a2
b2
4 D1
a2
b2
16 Dxy
a2
b2
+
4 Dy 2
b4
16 Dxy 2
a2
b2
+
12 Dy
b4
24 D1
3
a2
b2
12 D1 2
a2
b2
24 Dxy 2
a2
b2
+
2 D 2
2 D 2
1
1
36
1
1
3
72
36
2 D
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2
a2
b2
+
a2
32 Dxy 2
a2
b2
+
12 D1 2
a2
b2
24 Dxy 2
a2
b2
+
24 D1 3
a2
b2
24 Dx 2
2
a4
32 Dxy 4
a2
b2
+
24 D1 3
a2
b2
32 Dxy 3
a2
b2
+
2
a2
b2
+
b2
32 Dxy 2
a2
b2
+
12 Dy 2
b4
24 Dxy 3
a2
b2
+
24 Dy 2
b4
24 D1 3
a2
b2
32 Dxy 3
a2
b2
+
32 Dxy 4
a2
b2
24 Dy 2
2
b4+
6
a
36
7
72
72
0
0
0
12 Dx
a4
0
12 D
1
a2
b2
36 Dx 2
a4
12 Dx
a4
12 D1 2
a2 b2
36 D1
a2
b2
72 Dx 3
a4
36 Dx 2
a4
Dx 2
a2
2 D1 2
b2
+
a2
36 D1 2
0
0
0
4 Dx
a4
8 Dxy
a2
b2
4 D
1
a2
b2
12 Dx
a4
4 Dx 2
a4
16 Dxy 2
a2
b2
+
4 D1
a2 b2
16 Dxy
a2 b2
+
12 D1 2
a2
b2
24 Dx 2
a4
12 Dx 2
a4
24 Dxy 3
a2
b2
+
2 2 Dx
2
a2
2 D1 2
b2
+
a2
32 Dxy 2
a2
b2
+
12 D1 2
24 Dxy 2
0
0
0
4 D1
a2
b2
8 Dxy
a2
b2
4 D
y
b4
12 D1 2
a2
b2
4 D1
a2
b2
16 Dxy
a2
b2
+
4 Dy 2
b4
16 Dxy 2
a2 b2
+
12 Dy
b4
24 D1 3
a2
b2
12 D1 2
a2
b2
24 Dxy 2
a2
b2
+
2 2 D1
2
a2
2 Dy 2
b2
+
b2
32 Dxy
a2
b
+
12 Dy 2
24 Dxy 3
8/14/2019 Mathcad - fullDevelopmentMatrix17dofHFL
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a2
b2
72 D1 2
a
2
b
2
72 Dx
3
a4
2 D1 2
2
a2
b2
a2
b2
a2
b2
24 D1 3
a
2
b
2
24 Dx
2
2
a4
32 Dxy 4
a2
b2
+
24 D1 3
a2
b2
32 Dxy 3
a2
b2
+
b4
a2
b2
24 Dy 2
b
4
24 D1 3
a2
b2
32 Dxy 3
a2
b2
+
32 Dxy 4
a2
b2
24 Dy 2
2
b4
+
2
0
0
0
0
0
0
16 D1
a2
b2
0
16 Dy a2
4 Dxy b2
+
3 a2
b4
0
0
0
0
0
0
0
0
0
16 D1
a2
b2
0
48 Dy
b4
0
0
0
0
32 Dx
a4
0
32 D1
a2
b2
0
0
0
0
576 Dx
4
0
0
0
0
16 Dxy
a2
b2
0
0
0
0
0
0
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b
2
0
0
0
0
0
32 5 Dy a2
12 Dxy b2
+
15 a2
b4
0
0
0
0
32 D1
a2
b2
0
5 a
0
32 9 D1 a2 5 Dx b
2+
15 a4
b2
0
64 D1
a2
b2
0
0
16 9 Dxy a2
5 Dx b2
+
5 a4
b2
0
16 D1 Dxy+( )
a2
b2
0
0
0
16 9 D
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0
0
0
2 D1
a2
b2
0
2 Dy
b4
D1
a2
b2
D1 2
a2
b2
Dy
b4
Dy 2
b4
D1
2
a2
b2
D1 2
a2
b2
2
2 D 2
0
0
0
24 Dx 2
a4
0
24 D1 2
a2
b2
72 Dx 3
a4
24 Dx 2
a4
24 D1 3
a2
b2
72 D1 2
a2
b2
144 Dx
4
a4
72 Dx 3
a4
2 D 2
2 D 2
0
0
0
12 Dx
a4
12 Dxy 2
a2
b2
12 D1
a2
b2
36 Dx 2
a4
12 Dx 2
a4
24 Dxy 3
a2
b2
+
12 D1 2
a2
b2
24 Dxy 2
a2
b2
+
36 D1 2
a2
b2
72 Dx
3
a4
36 Dx 2
2
a4
36 Dxy 4
a2
b2
+
2 D 2
2 D 2
2
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b4
72 Dy 3
b
4
72 D1 2
2
a2
b2
72 Dy 3
b4
a2
b2
144 D1 2
2
a
2
b
2
144 Dx
4
a4
144 D1 3
2
a2
b2
a2
b2
a2
b2
72 D1 3
a
2
b
2
72 Dx
3
2
a4
48 Dxy 5
a2
b2
+
72 D1 2
3
a2
b2
48 Dxy 2
3
a2
b2
+
0
0
0
16 D1 a2
Dx b2
+
3 a4
b2
0
16 Dy a2 D1 b2+
3 a2
b4
0
0
0
0
32 9 D1 a2
5 Dx b2
+
4 2
0
0
0
0
16 Dxy
a2
b2
0
0
0
0
0
0
0
0
0
32 D1
a2
b2
0
32 Dy
b4
0
0
0
0
64 D1
2 2
32
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15 a b
0
a4 9 Dx b4+ 10 D1 a
2 b2+ 80 Dxy a2 b2+
45 a4
b4
0
32 5 Dy a2
9 D1 b2
+
15 a2
b4
0
0
16 D1 Dxy+( )
a2
b2
0
16 5 Dy a2
9 Dxy b2
+
5 a2
b4
0
0
0
a b
0
32 5 Dy a2 9 D1 b
2+
15 a2
b4
0
576 Dy
5 b4
0
0
64
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0
0
0
4 Dx 2
a4
4 D1 2
a2
b2
+
16 Dxy
a2
b2
4 Dy 2
b4
4 D1 2
a2
b2
+
12 Dx 2
a4
12 D1 3
a2
b2
+
4 Dx 3
a4
4 D1 2
a2
b2
+32 Dxy
2
a2
b2
+
4 Dy 3
b4
4 D1 2
a2
b2
+32 Dxy
2
a2
b2
+
12 Dy 2
b4
12 D1 3
a2
b2
+
24 Dx
2
2
a4
24 D1
4
a2
b2
+
12 Dx 3
a4
12 D1 3
a2
b2
+48 Dxy
3
a2
b2
+
D 2
2 D 2
2 D 2 2 D 2
0
0
0
12 D1
a2
b2
12 Dxy 2
a2
b2
12 Dy
b4
36 D1 2
a2
b2
12 D1 2
a2
b2
24 Dxy
a2
b2
+
12 Dy 2
b4
24 Dxy 3
a2
b2
+
36 Dy 2
b4
72 D1
3
a2
b2
36 D1 2
2
a2
b2
36 Dxy 2
a2
b2
+
2 D 2
2 D 2
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a2
b2
+
a2
2
a2
b2
+
b2
+64 Dxy
2
2
a2
b2
+
12 Dy 3
b4
12 D1 3
a2
b2
+ 48 Dxy 3
a2
b2
+
24 D1 4
a2
b2
24 Dy 2
2
b4
+
24 Dx 2
3
a4
24 D1 4
a2
b2
+64 Dxy
4
a2
b2
+
24 Dy 3
2
b4
24 D1 4
a2
b2
+
64 Dxy 4
a2
b2
+
6
a2
b2
+
b2
48+
36 Dxy 4
a2
b2
36 Dy 2
b4
+
72 Dy 3
b4
72 D1 3
2
a2
b2
48 Dxy 3
a2
b2
+
48 Dxy 5
a2
b2
72 Dy 2
b4+
y 3
b2
0
0
0
4 Dx 2
a4
4 D1 2
a2
b2
+
16 Dxy
a2
b2
4 D
y
2
b4
4 D
1
2
a2
b2
+
12 Dx 2
a4
12 D1 3
a2
b2
+
4 Dx 3
a4
4 D1 2
a2
b2
+32 Dxy
2
a2
b2
+
4 Dy 3
b4
4 D1 2
a2 b2
+32 Dxy
2
a2 b2
+
12 Dy 2
b4
12 D1 3
a2
b2
+
24 Dx 2
2
a4
24 D1 4
a2
b2
+
12 Dx 3
a4
12 D1 3
a2
b2
+48 Dxy
3
a2
b2
+
2 2
2 Dx
2
a2
2 D1 2
b2
+
a2
2 2
2 D1
2
a2
2 Dy 2
b2
+
b2
+64 Dxy
2
2
a2
b2
+
12 Dy 3
12 D1 3
48 Dxy 3
1
1
1
3
12 D1
a2
b2
12 Dy
b4
3
7
36 D1 2
a2
b2
6 2 D1
2
a2
b2
36 Dxy
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b4
a2
b2
a2
b2
24 D1 4
a
2
b
2
24 Dy 2
2
b
4+
24 Dx 2
3
a4
24 D1 4
a2
b2
+64 Dxy
4
a2
b2
+
24 Dy 3
2
b4
24 D1 4
a2
b2
+64 Dxy
4
a2
b2
+
a2
b2
7
72 D1 3
a2
b2
48 Dxy
a2
b2
0
0
0
0
0
0
0
12 Dxy a2
5 Dx b2
+
15 a4
b2
0
32 D1
a2
b2
0
0
0
0
0
0
0
32 D1
a2
b2
0
32 5 Dy a2
12 Dxy b2
+
15 a2
b4
0
0
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0
0
0
0
20 Dxy a2
21 Dx b2
+
35 a4
b2
0
0
0
0
0
0
64 21 Dy a2
20 Dxy b2
+
35 a2
b4
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2
2
0
0
0
24 D1 2
a2
b2
0
24 Dy 2
b4
72 D1 2
a2
b2
24 D1 3
a2
b2
24 Dy 2
b4
72 Dy 3
b4
144 D1
2
2
a2
b2
72 D1 3
a2
b2
2 D 2
2 D 2
0
0
0
24 Dx 2
a4
16 Dxy 3
a2
b2
24 D1 2
a2
b2
72 Dx 3
a4
24 Dx 2 2
a4
32 Dxy 4
a2
b2
+
24 D1 3
a2
b2
32 Dxy 3
a2
b2
+
72 D1 2
2
a2
b2
144 Dx
4
a4
72 Dx 3
2
a4
48 Dxy 5
a2
b2
+
2 D 2
2 D 2
24
3
72
2
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Dxy 3
a2
b2
2
12
a2
b2
+
b2
72 Dy 3
b4
144 Dy 4
b4
144 D1 2
3
a2
b2
144 Dy 4
b4
12
a2
b2
+
a2
64 Dxy 4
a2
b2
+
72 D1 3
2
a2
b2
48 Dxy 3
2
a2
b2
+
144 D1 2
3
a2
b2
144 Dx 4
2
a4
64 Dxy 6
a2
b2
+
144 D1 3
3
a2
b2
64 Dxy 3
3
a2
b2
+
12
4
144
64
0
0
0
2 D1
a2
b2
2 Dxy 2
a2
b2
2 D
y
b4
6 D1 2
a2
b2
224 Dxy
2
a2
b2
+
2 24 Dxy
3
a2 b2
+
6 Dy 2
b4
2 D1 3
a2
b2
236 Dxy
2
2
a2
b2
+
2 Dy 2
b2
48 Dxy 3
a2
b2
+
436 Dy
2
2
0
0
0
24 D1 2
a2
b2
0
24 D
y
2
b4
72 D1 2
a2
b2
24 D1 3
a2
b2
24 Dy 2
b4
72 Dy 3
b4
144 D1 2
2
a2
b2
72 D1 3
a2
b2
12 2
2 D1
2
a2
2 Dy 2
b2
+
b2
72 Dy 3
0
0
0
24 Dx 2
a4
16 Dxy 3
a2
b2
24 D
1
2
a2
b2
72 Dx 3
a4
24 Dx 2
2
a4
32 Dxy 4
a2
b2
+
24 D1 3
a2 b2
32 Dxy 3
a2 b2
+
72 D1 2
2
a2
b2
144 Dx 4
a4
72 Dx 3
2
a4
48 Dxy 5
a2
b2
+
12 2
2 Dx
2
a2
2 D1 2
b2
+
a2
64 Dxy 4
a2
b2
+
72 D1 3
2
48 Dxy 3
2
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0
0
0
24 D1 2
a2
b2
16 Dxy 3
a2
b2
24 Dy 2
b4
72 D1 2
2
a2
b2
D1 3
a2
b2
32 Dxy 3
a2
b2
+
Dxy 4
a2
b2
24 Dy 2
2
b4
+
72 Dy 3
b4
144 D1
3
2
a2
b2
1 2
3
a2
b2
48 Dxy 2
3
a2
b2
+
2
2 D 2
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a2
b2
+
b2
64 Dxy 4
a2
b2
+
Dxy 5
a2
b2
72 Dy 2
3
b4
+
144 Dy 4
b4
D1 3
3
a2
b2
64 Dxy 3
3
a2
b2
+
Dxy 6
a2
b2
144 Dy 2
4
b4+
0
0
0
24 D1 2
a2
b2
16 Dxy 3
a2
b2
24 D
y
2
b4
72 D1 2
2
a2
b2
24 D1 3
a2
b2
32 Dxy 3
a2
b2
+
32 Dxy 4
a2 b2
24 Dy 2
2
b4
+
72 Dy 3
b4
144 D1 3
2
a2
b2
72 D1 2
3
a2
b2
48 Dxy 2
3
a2
b2
+
12 2
2 D1
2
a2
2 Dy 2
b2
+
b2
64 Dxy 4
a2
b2
+
48 Dxy 5
72 Dy 2
3
d d
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a2
b2
b4
144 Dy 4
b
4
144 D1 3
3
a2
b2
64 Dxy 3
3
a2
b2
+
64 Dxy 6
a2
b2
144 Dy 2
4
b4
+
