Mathcad - Fulldevelopmentmatrix17dofhfl
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Plate bending transverse displacement, w, depends on surface coordinates of position of the point under study, a generic point located through normalized coordinates of position ξ=x/a, and η=y/a. The deflected shape of the elements is described by a Pascal cubic polynomial with the powers of the terms in ξ and η:{w}= |f(ξ,η)|*{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; 8 curvatures, two in each of the four corner nodes; and one additional thirteenth dof representing the average displacement across all the area, making a total of 13 dof.
Transformation expression: Substituting variables ξ and η by their nodal 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 extreme positions +1 and -1 for both ξ and η This {a} coefficients are applied to the curvature equations to calculate the plate generic strains.
ξ x
(
≡
1 a
f ( ξ , η) := 1 ξ η ξ
η
d ξ = dx
2
ξ⋅ η η
(
2
ξ
3
≡
y
2
ξ η ξ⋅ η
2
η
3
ξ
4
3
1
d η = dy
b
2
ξ ⋅η ξ ⋅η
2
ξ⋅ η
3
η
4
4
ξ ⋅ η ξ⋅ η
A := a1 a2 a3 a4 a5 a6 a7 a8 a8 a10 a11 a12 a13 a14 a15 a16 a17
)
T
w( ξ , η) := f ( ξ , η) ⋅ A
⎛ a1 ⎞ ⎜ ⎟ ⎜ a2 ⎟ ⎜ ⎟ ⎜ a3 ⎟ ⎜a ⎟ ⎜ 4⎟ ⎜ a5 ⎟ ⎜ ⎟ ⎜ a6 ⎟ ⎜a ⎟ ⎜ 7⎟ ⎜ a8 ⎟ ⎜ ⎟ ( a1 a2 a3 a4 a5 a6 a7 a8 a8 a10 a11 a12 a13 a14 a15 a16 a17 )T → ⎜ a8 ⎟ ⎜a ⎟ ⎜ 10 ⎟ ⎜ a11 ⎟ ⎜ ⎟ ⎜ a12 ⎟ ⎜a ⎟ ⎜ 13 ⎟ ⎜ a14 ⎟ ⎜ ⎟ ⎜ a15 ⎟ ⎜ ⎟ ⎜ a16 ⎟ ⎜a ⎟ ⎝ 17 ⎠
4
)
Here we are multiplying f ⋅ AT
(
⎛ a1 ⎞ ⎜ ⎟ ⎜ a2 ⎟ ⎜ ⎟ ⎜ a3 ⎟ ⎜a ⎟ ⎜ 4⎟ ⎜ a5 ⎟ ⎜ ⎟ ⎜ a6 ⎟ ⎜a ⎟ ⎜ 7⎟ ⎜ a8 ⎟ ⎜ ⎟ 2 2 3 2 2 3 4 3 2 2 3 4 4 4 1 ξ η ξ ξ ⋅ η η ξ ξ η ξ ⋅ η η ξ ξ ⋅ η ξ ⋅ η ξ ⋅ η η ξ ⋅ η ξ ⋅ η ⋅ ⎜ a8 ⎟ ⎜a ⎟ ⎜ 10 ⎟ ⎜ a11 ⎟ ⎜ ⎟ ⎜ a12 ⎟ ⎜a ⎟ ⎜ 13 ⎟ ⎜ a14 ⎟ ⎜ ⎟ ⎜ a15 ⎟ ⎜ ⎟ ⎜ a16 ⎟ ⎜a ⎟ ⎝ 17 ⎠
)
4
4
3
3
2
2
2
2
4
3
a16⋅ ξ ⋅ η + a11⋅ ξ + a12⋅ ξ ⋅ η + a7 ⋅ ξ + a13⋅ ξ ⋅ η + a8 ⋅ ξ ⋅ η + a4 ⋅ ξ + a17⋅ ξ ⋅ η + a14⋅ ξ ⋅ η + + ⎛ a8 ⋅ ξ⋅ η + a5 ⋅ ξ⋅ η + a2 ⋅ ξ + a15⋅ η + a10⋅ η + a6 ⋅ η + a3 ⋅ η + a1⎞
⎝
2
4
3
2
The latter is the Pascal Polinomial w(ξ,η)
⎠
...
f ( 1 , 1) → ( 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 )
q1
f ( 1 , −1 ) → ( 1 1 −1 1 −1 1 1 −1 1 −1 1 −1 1 −1 1 −1 1 ) q2 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 we have done it before, but we will first take the "x" derivative as the paper does .
(
2 2 3 2 2 3 4 3 d f ( ξ , η) → 0 0 1 0 ξ 2 ⋅ η 0 ξ 2 ⋅ ξ⋅ η 3 ⋅ η 0 ξ 2 ⋅ ξ ⋅ η 3 ⋅ ξ⋅ η 4 ⋅ η ξ 4 ⋅ ξ ⋅ η dη
Ξ( ξ , η) :=
1 d f ( ξ , η) ⋅ b dη
Ξ( 1 , 1 ) → ⎛⎜ 0 0
1
Ξ( 1 , −1 ) → ⎛⎜ 0 0
1
Ξ( −1 , 1 ) → ⎛⎜ 0 0
1
⎝
⎝
⎝
Ξ( −1 , −1 ) → ⎛⎜ 0 0
⎝
b
0
0
b
1
2
b
b
1
−
b
0 −
b 1 b
0
2 b
1
2
b
b
0 −
1 b
−
0
0
2 b
1
2
3
b
b
b
1 b 1 b
0
0
−
2
3
b
b
−
2
3
b
b
⎞ ⎟ b b b b b b⎠ 1
0
2
2
3
b
b
b
4
1
2
3
b
b
−
1
2
b
b
−
1
−
b
0 −
1
3
0 −
1 b
−
2 b
4
4
1
b
b
3
4
1
b
b
b
−
3 b
−
⎞ ⎟ b⎠
−
4
−
4
⎞ ⎟ b⎠
⎞ ⎟ b b b⎠ 4
1
4
)
(
2 2 3 2 2 3 3 4 d f ( ξ , η) → 0 1 0 2 ⋅ ξ η 0 3 ⋅ ξ 2 ⋅ ξ⋅ η η 0 4 ⋅ ξ 3 ⋅ ξ ⋅ η 2 ⋅ ξ⋅ η η 0 4 ⋅ ξ ⋅ η η dξ
1 d f ( ξ , η) ⋅ a dξ
Θ( ξ , η) :=
Θ( 1 , 1 ) → ⎛⎜ 0
1
Θ( 1 , −1 ) → ⎛⎜ 0
1
Θ( −1 , 1 ) → ⎛⎜ 0
1
Θ( −1 , −1 ) → ⎛⎜ 0
1
⎝
⎝
⎝
⎝
⌠ ⎮ ⌡
a
a
a
2
−
a
0
1 a 1
a
a
0 −
2
−
1
Π( 1 ) →
1
2
⎝
f ( ξ , η ) dξ ⋅
−1
2
a
3
2
1
a
a
a
0
0
1
2
0
−
2
1
a
a
−
2
1
a
a
3
2
1
a
a
a
3 a 3 a
0
a
0 2⋅ η
3
2⋅ η
0
3
4
3
2
1
a
a
a
a
3
2
a
a
−
0
4
−
a
0 −
4
3
a
a
0 −
4
−
0 2⋅ η
a
3 2
5
3 a
0
0
−
a
a
a 1
a
a 2
−
a
2⋅ η
1
1
2
−
4
⎞ ⎟ ⎠
0 −
4
1
a
a
0 −
4
1
a
a
1 a
0
2
0 2⋅ η
3
⎞ ⎟ ⎠ ⎞ ⎟ ⎠
4
1
a
a
⎞ ⎟ ⎠
4 2⋅ η
5
1 2aa
⎛
⎛ 1 0 −1 1 0 1 0 − 1 0 −1 1 0 1 0 1 − 1 0⎞ ⎜ ⎟ 5⋅ a a 3⋅ a a 3⋅ a a 5⋅ a 3⋅ a a ⎝a ⎠
f ( ξ , η ) dη → ⎜ 2 2 ⋅ ξ 0 2 ⋅ ξ
⎝
2
0
2 3
2⋅ ξ
3
0
2⋅ ξ 3
0 2⋅ ξ
4
0
2⋅ ξ 3
2
0
2 5
0
⎞ ⎟ 5 ⎠
2⋅ ξ
⎞
0⎟
⎛ 1 0 1 1 0 1 0 1 0 1 1 0 1 0 1 1 0⎞ ⎜ ⎟ a 3⋅ a a 3⋅ a a 5⋅ a 3⋅ a a 5⋅ a ⎝a ⎠
Π( −1 ) →
−1
a
2
0 −
f ( ξ , η ) dξ → ⎜ 2 0 2 ⋅ η
⌠ Π( η) := ⎮ ⌡
1
a
0
⎛
1
−1
⌠ ⎮ ⌡
a
0
⎠
)
Ψ( ξ) :=
⎛⎜ ⌠ 1 ⎞⎟ ⎮ f ( ξ , η ) dη ⎟ 2bb ⎜ ⌡− 1 ⎝ ⎠ 1
⌠ ⎮ ⌡
1
f ( ξ , η ) dη
−1
Ψ( ξ) :=
2⋅ b
Ψ( 1 ) →
Ψ( −1 ) →
⌠ ⎮ ⌡
1
⌠ ⎮ ⌡
⎛1 1 0 1 0 1 1 0 1 0 1 0 1 0 1 0 1 ⎞ ⎜ ⎟ b 3⋅ b b 3⋅ b b 3⋅ b 5⋅ b 5⋅ b ⎠ ⎝b b
⎛ 1 −1 0 1 0 1 −1 0 − 1 0 1 0 1 0 1 0 − 1 ⎞ ⎜ ⎟ b 3⋅ b 3⋅ b b 3⋅ b 5⋅ b 5⋅ b ⎠ b ⎝b b
1
f ( ξ , η ) dξ dη → ⎛⎜ 1 0 0
−1 −1
⎝
4
1 3
0
1 3
0 0 0 0
1 5
1
0
9
0
1 5
0 0 ⎞⎟
⎠
To find later the strains, first we find: −
d
2
dξ
−
d
2
2
dη
(
f ( ξ , η) → 0 0 0 −2 0 0 −6 ⋅ ξ −2 ⋅ η 0 0 −12⋅ ξ
2
(
2
−6 ⋅ ξ⋅ η −2 ⋅ η
f ( ξ , η) → 0 0 0 0 0 −2 0 0 −2 ⋅ ξ −6 ⋅ η 0 0 −2 ⋅ ξ
(
2
2
2
0 0 −12⋅ ξ ⋅ η 0
−6 ⋅ ξ ⋅ η −12⋅ η
2
0 −12⋅ ξ⋅ η
⎛ d d f ( ξ , η) ⎞ → 0 0 0 0 2 0 0 4 ⋅ ξ 4 ⋅ η 0 0 6 ⋅ ξ2 8 ⋅ ξ ⋅ η 6 ⋅ η2 0 8 ⋅ ξ3 8 ⋅ η3 ⎟ ⎝ dξ dη ⎠
2⎜
)
2
) )
Constructing M. (Coordinates Transformation Expression) The [M] matrix is constructed from {q} = [M]*{a} with the four values f (q1...q4); four Θ (q5...q8); four Ξ (q9... q12); two Π (q13,q14); two Ψ (q15,q16); and q17. All of them are qi vectors or 1x17 matrixes that we will use for the transformation: ⎛1 1 1 ⎜ ⎜ 1 1 −1 ⎜ 1 −1 1 ⎜ 1 −1 −1 ⎜ ⎜0 1 0 ⎜ a ⎜ 1 0 ⎜0 a ⎜ ⎜ 1 ⎜0 a 0 ⎜ ⎜0 1 0 a ⎜ ⎜ 1 ⎜0 0 b ⎜ ⎜0 0 1 ⎜ b ⎜ ⎜0 0 1 b ⎜ ⎜ 1 ⎜0 0 b ⎜ ⎜1 0 1 ⎜a a ⎜1 1 0 − ⎜ a ⎜a ⎜1 1 ⎜b b 0 ⎜ ⎜ 1 −1 0 ⎜b b ⎜ ⎜1 0 0 ⎝
1
1
1
1
1
1
1
1
1
1
1
1
1
1
−1
1
1
−1
1
−1
1
−1
1
−1
1
−1
1
−1
1
−1
1
−1
1
1
−1
1
−1
1
1
1
1
1
−1
−1
−1
−1
1
1
1
1
1
−1
2
1
3
2
1
4
3
2
1
a
a
a
a
a
a
a
a
a
2
−1
3
−2
1
4
−3
2
−1
a
a
a
a
a
a
a
a
a
2
1
2
1
a
a
−
a
a
3
2
1
2
a
a
a
−
−
1
2
3
b
b
b
−
2
3
b
b
−
2
3
b
b
1
2
3
b
b
b
−
2
1
a
a
−
2
−
a
0 0
1 a
0 0 0 0
1
2
b
b
1 b
−
2 b
0
−
1
2
b
b
0
−
1
1 3a 1 3a 1 b 1 b 1 3
b
0 0 0 0 0
−
2 b
1 a 1 a
3 a
0
b 1
0
b
0 0 0 1
3⋅ b
b
1
−1
3⋅ b
b
3
1
0
1
1
−
0
1
0
3a −
1
0
3a 0 0 0
1 3⋅ b −
1 3⋅ b 0
0 0 0
−
4
3
a
a
0
−
4
−
a
0 0
0
b 1 b 1 5
4
0
a
b
b
b
b
1 b
−
2
3
b
b
1
1
a
b
−
5a
4
1
0
a
−
4
b
1
a
3
b
1
a
0
−4
2
2
5a
1
0
a
1
1
a
0
a
−
1
0
a
0
1
−
3
4
0
b
0 0 0 0 0
−
2 b
1 3a 1 3a 1 3⋅ b 1 3⋅ b 1 9
−
4
1
b
b
−
3
4
1
b
b
b
−
3 b
0 0 0 0 0
−
4
1
b
b
1
1
a
5a
1 a 1 5⋅ b 1 5⋅ b 1 5
−
1 5a 0 0 0
⎞ ⎟ ⎟ −1 ⎟ −1 ⎟ ⎟ 1 ⎟ a ⎟ 1 ⎟ ⎟ a ⎟ 1 ⎟ a ⎟ ⎟ 1 ⎟ a ⎟ 4 ⎟ ⎟ b ⎟ 4 ⎟ − b ⎟ ⎟ 4 − ⎟ b ⎟ 4 ⎟ b ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 1 ⎟ 5⋅ b ⎟ ⎟ 1 ⎟ − 5⋅ b ⎟ ⎟ 0 ⎟ ⎠ 1 1
Mathcad is ordered to create the inverse of the former matrix M: ⎛1 1 1 ⎜ ⎜ 1 1 −1 ⎜ 1 −1 1 ⎜ 1 −1 −1 ⎜ ⎜0 1 0 ⎜ a ⎜ 1 0 ⎜0 a ⎜ ⎜ 1 ⎜0 a 0 ⎜ ⎜0 1 0 a ⎜ ⎜ 1 ⎜0 0 b ⎜ ⎜0 0 1 ⎜ b ⎜ ⎜0 0 1 b ⎜ ⎜ 1 ⎜0 0 b ⎜ ⎜1 0 1 ⎜a a ⎜1 1 0 − ⎜ a a ⎜ ⎜1 1 ⎜b b 0 ⎜ ⎜ 1 −1 0 ⎜b b ⎜ ⎜1 0 0 ⎝
1
1
1
1
1
1
1
1
1
1
1
1
1
1
−1
1
1
−1
1
−1
1
−1
1
−1
1
−1
1
−1
1
−1
1
−1
1
1
−1
1
−1
1
1
1
1
1
−1
−1
−1
−1
1
1
1
1
1
−1
2
1
3
2
1
4
3
2
1
a
a
a
a
a
a
a
a
a
2
−1
3
−2
1
4
−3
2
−1
a
a
a
a
a
a
a
a
a
2
1
1
a
−
2
a
a
a
3
2
1
2
a
a
a
−
−
1
2
3
b
b
b
−
2
3
b
b
−
2
3
b
b
1
2
3
b
b
b
−
2
1
a
a
−
2
−
a
0 0
1 a
0 0 0 0
1
2
b
b
1 b
−
2 b
0
−
1
2
b
b
0
−
1
1 3a 1 3a 1 b 1 b 1 3
b
0 0 0 0 0
−
2 b
1 a 1 a
3 a
0
b 1
0
b
0 0 0
1
1 b
1
−1
3⋅ b
b
3
1
0
3⋅ b
1
−
0
1
0
3a −
1
0
3a 0 0 0
1 3⋅ b −
1 3⋅ b 0
0 0 0
−
4
3
a
a
0
−
4
−
a
0 0
0
b 1 b 1 5
4
0
a
b
b
b
b
1 b
−
2
3
b
b
1
1
a
b
−
1
4
1
0
5a
−
4
b
a
a
3
b
1
a
0
−4
2
2
5a
1
0
a
1
1
a
0
a
−
1
0
a
0
1
−
3
4
0
b
0 0 0 0 0
−
2 b
1 3a 1 3a 1 3⋅ b 1 3⋅ b 1 9
−
4
1
b
b
−
3
4
1
b
b
b
−
3 b
0 0 0 0 0
−
4
1
b
b
1
1
a
5a
1 a 1 5⋅ b 1 5⋅ b 1 5
−
1 5a 0 0 0
⎞ ⎟ 1 ⎟ −1 ⎟ −1 ⎟ ⎟ 1 ⎟ a ⎟ 1 ⎟ ⎟ a ⎟ 1 ⎟ a ⎟ ⎟ 1 ⎟ a ⎟ 4 ⎟ ⎟ b ⎟ 4 ⎟ − b ⎟ ⎟ 4 − ⎟ b ⎟ 4 ⎟ b ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 1 ⎟ 5⋅ b ⎟ ⎟ 1 ⎟ − 5⋅ b ⎟ ⎟ 0 ⎟ ⎠ 1
−1
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
−
3
3
3
32
32
32
3
3
a
32
32
32
3
3
3
8
8
8
8
3 32
−
2
−
8
−
1
a
a
8
32
32
1
1
2
2
3
3
8
8
8
1
1
8
8
15
15
16
16
15 16 1 8 −
−
1
3
− −
1
−
15 32
−
15 16
− −
3⋅ a
3⋅ a
3⋅ a
16
16
16
a
a
a
a
8
8
−
8
8
16
8
8
8
8
3⋅ a
3⋅ a
3⋅ a
16
16
16
0
0
0
0
0
0
0
0
15
5⋅ a
5⋅ a
32
32
32
1
a
8
8
−
−
15
16 16 1 8
9
16
16
16
1
1
8
8
−
−
1 8
−
0
0
0
0
32
32
8
0
32 15
8
a
0
32
32
8
−
0
5⋅ a
15
a
32
0
15
32
a
5⋅ a
0
15
15
32
−
0
−
5⋅ a 32 0
−
5⋅ a
5⋅ a
32
32
0
0
b 8
− −
− −
−
b 8
b
b
32
32
b 32
−
0
16
0
−
32
5⋅ a
−
3⋅ a
0
15
15
−
−
0
−
32
−
8
9
−
32
a
9
32
32
32
a
8
15
a
a
8
32
32
a
−
a
15
−
b
8
−
8
1
1
15
a
a
a
8
0
1
−
−
b 32
0
8
32
a
a 32
−
8
3⋅ a
−
−
−
0
−
−
−
a
a 32
0
1
15
−
−
b 8
−
−
b 32
b
b
32
32
b
−
−
8
16
0
b
b
b
8
8
8
3⋅ b
0
0
0
3⋅ b
3⋅ b
3⋅ b
16
16
16
b
b
b
b
8
8
8
8
0
0
0
0
0
0
0
0
0
0
0
0
b
b
8
8
−
b 8
−
b 8
5⋅ b
32
32
0
0
0
5⋅ b
5⋅ b
32
32
32
−
15⋅ a
15⋅ a
8
8
0
0
0
0
15⋅ a
15⋅ a
16
16
0
0
16
− −
−
3⋅ b
5⋅ b
5⋅ b
8
0
5⋅ b 32
−
9⋅ a 8
−
9⋅ a 8
0
0
0
0
15⋅ a 16 0
−
15⋅ a 16 0
− −
0
4
8
0
16
9⋅ b
8
0
16
3⋅ a
16 0
−
−
3⋅ b
15⋅ b
16
16 0
−
15⋅ a
9⋅ a
0
32
−
9⋅ a
0
5⋅ b
4
0
3⋅ b
16
0
3⋅ a
16
0
16
32
−
3⋅ a
0
3⋅ b
−
−
0
8
3⋅ b
5⋅ b
16
15⋅ a
0
−
3⋅ a
b
0 −
−
0 −
3⋅ b 4
−
0 0 −
15⋅ b 8 0 0 0
−
9⋅ b 8
−
0 15⋅ b 16 0 15⋅ b 16
−
T
−1 Mathcad is ordered to create M
: ⎛ −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 ⎜
−
1
−
3
3
3
32
32
32
3
3
a
32
32
32
3
3
3
8
8
8
8
3 32
−
2
−
8
−
1
a
a
8
32
32
1
1
2
2
3
3
8
8
8
1
1
8
8
15
15
16
16
15 16 1 8 −
−
1
3
− −
1
−
15 32
−
15 16
− −
32
32
3⋅ a
3⋅ a
3⋅ a
16
16
16
a
a
a
a
8
8
−
8
8
16
−
32
a
a
8
8
8
3⋅ a
3⋅ a
3⋅ a
16
16
16
0
0
0
0
0
0
0
0
15
5⋅ a
5⋅ a
32
32
32
1
a
8
8
−
−
15
16 16 8
9
9
9
16
16
16
1
1
8
8
−
32
32
a
a
8
a
8
15
8
a
−
8
−
8
15
−
1 8
15
−
5⋅ a 32
a
a
8
8
−
5⋅ a 32
−
a 8
0
0
0
0
0
0
0
0
0
15
15
5⋅ a
32
32
32
−
5⋅ a 32
−
5⋅ a
5⋅ a
32
32
b 8
− −
− −
−
b 8
b
b
32
32
b 32
−
0
16
0
−
−
3⋅ a
0
15
32
−
−
0
−
−
b
−
8
a
a
8
32
a
a
−
8
8
32
32
1
1
15
32
0
1
−
b
0
1
32
a
−
a
3⋅ a
−
32
−
0
−
−
−
a
0
1
15
−
−
b 8
−
−
b 32
b
b
32
32
b
−
−
8
16
0
b
b
b
8
8
8
3⋅ b
4
0 −
−
8
8
8
0
0
0
0
0
0
3⋅ b
3⋅ b
3⋅ b
16
16
16
b
b
b
b
8
8
8
8
0
0
0
0
0
0
0
0
0
0
0
0
b
b
8
8
−
b 8
5⋅ b
5⋅ b
32
32
0
0
−
15⋅ a
15⋅ a
8
8
3⋅ b
0
0
0
0
15⋅ a
15⋅ a
16
16
0
0
16
− −
−
b 8
5⋅ b 32 0
−
9⋅ a 8
−
9⋅ a 8
0
0
0
0
15⋅ a 16
−
15⋅ a 16
− −
0
4
16 0
16
9⋅ b
16 0
16
3⋅ a
9⋅ a
0
3⋅ b
15⋅ b
16
9⋅ a
0
−
15⋅ a
3⋅ b
16
0
3⋅ a
16
0
16
−
−
3⋅ a
0
3⋅ b
32
−
0
8
3⋅ b
5⋅ b
16
15⋅ a
0
−
3⋅ a
b
0 −
−
0 −
3⋅ b 4
−
0 0 −
15⋅ b 8 0 0 0
−
9⋅ b 8 0
15⋅ b 16 0
−
⎜ 15 15 15 − ⎜− ⎝ 32 32 32
15
0
32
0
0
5⋅ b
0
−
32
5⋅ b
−
32
5⋅ b
5⋅ b
32
32
0
15⋅ b
0
−
16
T
−1 This is the M :
3 ⎛ −1 − 3 − ⎜ 32 32 ⎜ 8 3 ⎜ −1 − 3 ⎜ 8 32 32 ⎜ 3 3 ⎜ −1 − 32 32 ⎜ 8 ⎜ 1 3 3 ⎜ −8 32 32 ⎜ a a ⎜ a − ⎜ 32 8 32 ⎜ a a a − − ⎜ 32 32 8 ⎜ ⎜ a a a − ⎜ − 32 − 8 32 ⎜ a a ⎜−a − 8 32 ⎜ 32 ⎜ b b b − ⎜ 8 32 ⎜ 32 b ⎜−b −b − ⎜ 32 32 8 ⎜ b b ⎜ b − − 32 8 ⎜ 32 ⎜ b b b − ⎜ − 32 32 8 ⎜ 15⋅ a ⎜ − 3⋅ a 0 ⎜ 16 16 ⎜ 3⋅ a 15⋅ a − 0 ⎜− 16 16 ⎜ ⎜ 3 ⋅ b 15⋅ b 0 ⎜ − 16 16 ⎜ ⎜ − 3 ⋅ b − 15⋅ b 0 16 ⎜ 16 ⎜ 9 0 0 ⎜ ⎝ 4
3
1
3
8
2
8
3 8 3 8
− −
15
15
8
16
16
−
1
−
1
3
2
8
−
1
3
1
15
2
8
8
16
8
1
3
1
8
2
8
8
3⋅ a 16
−
a
a
16
8 −
0
8
3⋅ a
16
0
a
0
8
3⋅ a
a
16
8
0 3⋅ b
0
−
b
0
−
b
3⋅ b
8
16
0 0
−
1
3
3⋅ a
−
−
3⋅ a 4 3⋅ a 4
9⋅ b 8 9⋅ b 8 −
9 4
8
b 8
−
−
16
3⋅ b 16
b
3⋅ b
8
16
0 0
9⋅ a 8 9⋅ a 8
0
−
0
−
0
3⋅ b 4 3⋅ b
−
4 9 4
a 8
−
− −
1
16
16
8
15 16
16
a
3⋅ a
8
16
0
0
0
0
0
0
0 0
−
0
0
8
−
15
1
9
1
32
8
16
8
−
15
0
0
0
−
0
0
−
3⋅ b
b
16
8
3⋅ b
b
16
8
3⋅ b
b
16
8
3⋅ b
b
16
8
0
0
0
0
−
−
8
0
16
0
15⋅ a
0
8
0
8
0
32
0
15⋅ a
0
1
8
16 0
9
16
3⋅ a
0
1
1
16
−
15⋅ b 8
15⋅ b 8 0
16
15
15
8
8
1
−
−
16
1
9
15
15
−
1
−
−
3⋅ a
3⋅ a
8
8
15
8
−
1
15
a
a
−
32
32
−
−
1
9
8
16
5⋅ a
a
32
8
5⋅ a 32
−
a 8
5⋅ a
a
32
8
5⋅ a 32
−
a 8
−
−
8
1 8
15
−
15
15
32
32
−
15
−
15
15
32
32
32
32
0
0
0
−
0
0
0
−
0
0
0
b
5⋅ b
8
32
0
0
0
0
0
0
0
−
b
5⋅ b
8
32
0
0
0
−
b
16 15⋅ a 16
−
0
−
0
0
0
−
0
0
0
−
0
0
0
9⋅ a 8 9⋅ a 8 9⋅ b 8 9⋅ b 8 9 4
8
8
−
−
0 0 0
15⋅ b 16 15⋅ b 16 0
32 5⋅ a 32
0 0 0 0
32
0
5⋅ a
32
5⋅ b
0
32
5⋅ a
32
0
15
32
5⋅ b
0
32
5⋅ a
0
0
0
−
0
0
15⋅ a
−
0
b
15
−
15⋅ a 16 −
15⋅ a 16 0 0 0
Shape functions of the plate bending elements Nj are obtained multiplying the column Nij of the former matrix by f(ξ,η) transpose. We must construct |f| transpose |f| is a 1x17 row matrix defined before, whose transpose is this 17X1 column matrix: ⎛ 1 ⎞ ⎜ ξ ⎟ ⎜ ⎟ ⎜ η ⎟ ⎜ 2 ⎟ ⎜ ξ ⎟ ⎜ ξ⋅ η ⎟ ⎜ 2 ⎟ ⎜ η ⎟ ⎜ 3 ⎟ ⎜ ξ ⎟ ⎜ 2 ⎟ ⎜ ξ ⋅η ⎟ ⎜ 2⎟ ξ⋅ η ⎟ T f ( ξ , η) → ⎜ ⎜ 3 ⎟ ⎜ η ⎟ ⎜ ξ4 ⎟ ⎜ ⎟ ⎜ ξ3⋅ η ⎟ ⎜ ⎟ ⎜ ξ2⋅ η2 ⎟ ⎜ 3⎟ ⎜ ξ⋅ η ⎟ ⎜ 4 ⎟ ⎜ η ⎟ ⎜ 4 ⎟ ⎜ ξ ⋅η ⎟ ⎜ 4⎟ ⎝ ξ⋅ η ⎠
We will find shape functions with the former matrix multiplying it by f transpose: 3 ⎛ −1 − 3 − ⎜ 32 32 ⎜ 8 3 ⎜ −1 − 3 ⎜ 8 32 32 ⎜ 3 3 ⎜ −1 − 32 8 32 ⎜ ⎜ 1 3 3 ⎜ −8 32 32 ⎜ a a ⎜ a − ⎜ 32 8 32 ⎜ a a a − − ⎜ 32 8 ⎜ 32 ⎜ a a a − ⎜ − 32 − 8 32 ⎜ a a ⎜−a − 8 32 ⎜ 32 ⎜ b b b − ⎜ 8 32 ⎜ 32 b ⎜−b −b − ⎜ 32 32 8 ⎜ b b ⎜ b − − 32 32 8 ⎜ ⎜ b b b − ⎜ − 32 32 8 ⎜ 15⋅ a ⎜ − 3⋅ a 0 ⎜ 16 16 ⎜ 3⋅ a 15⋅ a − 0 ⎜− 16 ⎜ 16 ⎜ 3 ⋅ b 15⋅ b 0 ⎜ − 16 16 ⎜ ⎜ − 3 ⋅ b − 15⋅ b 0 16 ⎜ 16 ⎜ 9 0 0 ⎜ ⎝ 4
3
1
3
8
2
8
3 8 3 8
− −
15
15
8
16
16
−
1
−
1
3
2
8
−
1
3
1
15
2
8
8
16
8
1
3
1
8
2
8
8
3⋅ a 16
−
a
a
16
8 −
0
8
3⋅ a
16
0
a
0
8
3⋅ a
a
16
8
0 3⋅ b
0
−
b
0
−
b
3⋅ b
8
16
0 0
−
1
3
3⋅ a
−
−
3⋅ a 4 3⋅ a 4
9⋅ b 8 9⋅ b 8 −
9 4
8
b 8
−
−
16
3⋅ b 16
b
3⋅ b
8
16
0 0
9⋅ a 8 9⋅ a 8
0
−
0
−
0
3⋅ b 4 3⋅ b
−
4 9 4
a 8
−
− −
1
16
16
8
15 16
16
a
3⋅ a
8
16
0
0
0
0
0
0
0 0
−
0
0
8
−
15
1
9
1
32
8
16
8
−
15
0
0
0
−
0
0
−
3⋅ b
b
16
8
3⋅ b
b
16
8
3⋅ b
b
16
8
3⋅ b
b
16
8
0
0
0
0
−
−
8
0
16
0
15⋅ a
0
8
0
8
0
32
0
15⋅ a
0
1
8
16 0
9
1
3⋅ a
0
1
16
−
15⋅ b 8
15⋅ b 8 0
8
16
15
15
16
1
8
1
−
−
−
9
15
15 16
1
−
−
3⋅ a
8
8
8
15
3⋅ a
−
1
15
a
a
−
32
32
−
−
1
9
8
16
5⋅ a
a
32
8
5⋅ a 32
−
a 8
5⋅ a
a
32
8
5⋅ a 32
−
a 8
−
−
8
1 8
15
−
15
15
32
32
−
15
−
15
15
32
32
32
32
0
0
0
−
0
0
0
−
0
0
0
b
5⋅ b
8
32
0
0
0
0
0
0
0
−
b
5⋅ b
8
32
0
0
0
−
b
16 15⋅ a 16
−
0
−
0
0
0
−
0
0
0
−
0
0
0
9⋅ a 8 9⋅ a 8 9⋅ b 8 9⋅ b 8 9 4
8
8
−
−
0 0 0
15⋅ b 16 15⋅ b 16 0
32 5⋅ a 32
0 0 0 0
32
0
5⋅ a
32
5⋅ b
0
32
5⋅ a
32
0
15
32
5⋅ b
0
32
5⋅ a
0
0
0
−
0
0
15⋅ a
−
0
b
15
−
15⋅ a 16 −
15⋅ 16 0 0 0
Shape functions from the first to the sevententh degrees of fredom: ⎛ 9⋅ ξ2⋅ η2 15⋅ ξ4 ξ3⋅ η ⎜ − − 32 8 ⎜ 16 ⎜ 4 4 3 ⎜ 15⋅ ξ ⋅ η − 15⋅ ξ + ξ ⋅ η 32 8 ⎜ 32 ⎜ 3 4 4 ⎜ ξ ⋅ η − 15⋅ ξ − 15⋅ ξ ⋅ η ⎜ 8 32 32 ⎜ 4 4 3 ⎜ 15⋅ ξ ⋅ η − 15⋅ ξ − ξ ⋅ η ⎜ 32 32 8 ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜
−
ξ
−
ξ
+
ξ
+
ξ
3
4
15⋅ ξ ⋅ η
−
8
32
3
2
9⋅ ξ ⋅ η
+
8
2
9⋅ ξ ⋅ η
+
8
2
9⋅ ξ ⋅ η
+
8 a 32 a
3 ⋅ a⋅ ξ
−
32
2
+
16
3 ⋅ a⋅ ξ
2
16 3 ⋅ a⋅ ξ
a
−
32
2
a
−
16
32
+
+
2
b 32
3⋅ η ⋅ b
−
16
2
3⋅ η ⋅ b 16
32
+
2
b 32
3⋅ η ⋅ b
−
16
2
3⋅ η ⋅ b 16
32
+
4
15⋅ a⋅ ξ ⋅ η 16 15⋅ a⋅ ξ
4
9⋅ b⋅ ξ 8
2
+
8 a⋅ ξ
3
+
8 a⋅ ξ
3
−
8 a⋅ ξ
3
−
8 η ⋅b 8 η ⋅b 8 η ⋅b 8
+
η ⋅b 8
+
5 ⋅ a⋅ ξ
3⋅ ξ
3⋅ ξ 4
− 4
− 4
− 4
−
32 5⋅ η ⋅ b
+
32 5⋅ η ⋅ b
−
32 5⋅ η ⋅ b
−
32 4
−
5⋅ η ⋅ b
2
+
2
+ 2
+
4
+ 4
+
32 15⋅ ξ ⋅ η
4
−
32 +
−
−
8 a⋅ ξ
+
8 ξ⋅ b
−
32 ξ⋅ b
−
32 ξ⋅ b
−
32 ξ⋅ b
−
32
8
16
15⋅ ξ ⋅ η
a⋅ ξ
9 ⋅ a⋅ ξ ⋅ η
−
32
8
2
a⋅ η 32 a⋅ η 32 a⋅ η 32 a⋅ η 32 η⋅ b 8 η⋅ b 8 η⋅ b 8 η⋅ b 8
ξ⋅ η
2
ξ⋅ η
ξ⋅ η
4
−
ξ ⋅ η⋅ b 8 ξ ⋅ η⋅ b 8 ξ ⋅ η⋅ b 8 ξ ⋅ η⋅ b
15⋅ a⋅ ξ ⋅ η 8 15⋅ a⋅ ξ ⋅ η 8
8
−
8
+
8 −
−
2
+
2
ξ⋅ η
+
3 ⋅ a⋅ ξ ⋅ η 16 3 ⋅ a⋅ ξ ⋅ η 16 3 ⋅ a⋅ ξ ⋅ η 16 3 ⋅ a⋅ ξ ⋅ η 16
3
a⋅ ξ ⋅ η
+
8 3
a⋅ ξ ⋅ η
−
8 3
a⋅ ξ ⋅ η
+
8 3
a⋅ ξ ⋅ η
−
8
3 ⋅ ξ⋅ η ⋅ b 16
3
+
2
+
3 ⋅ ξ⋅ η ⋅ b 16 3 ⋅ ξ⋅ η ⋅ b 16 3 ⋅ ξ⋅ η ⋅ b
3 ⋅ a⋅ ξ
16 2
+
4 3 ⋅ a⋅ ξ
2
4 15⋅ b ⋅ ξ 16
+
+
ξ⋅ η ⋅ 8 ξ⋅ η ⋅ 8 3
−
9 ⋅ a⋅ η
9 ⋅ a⋅ η
ξ⋅ η ⋅ 8
2
8
+
15
−
15
−
3⋅
2
8 15⋅ b ⋅ η 16
8
3
−
2
−
ξ⋅ η ⋅ 3
+
2
+
+
2
2
−
+
2
2
a⋅ ξ ⋅ η
+
ξ⋅ η
−
2
+
8
−
15⋅ b ⋅ ξ⋅ η
+
a⋅ ξ ⋅ η
−
2
16
−
2
2
8
+
ξ⋅ η
−
2
−
a⋅ ξ ⋅ η
−
15⋅ ξ⋅ η
−
−
2
2
16
3
8
+
15⋅ ξ⋅ η
−
ξ⋅ η
+
16
3
a⋅ ξ ⋅ η
−
15⋅ ξ⋅ η
+
8
2
16
3
8 ξ⋅ η
15⋅ ξ⋅ η
+
8
2
+
3
8
2
−
8
15⋅ b ⋅ ξ ⋅ η
15⋅ ξ ⋅ η
a⋅ ξ
9 ⋅ a⋅ ξ ⋅ η
4
32
8
2
−
15⋅ ξ ⋅ η
a⋅ ξ
2
−
16
+
32
4
−
8
32 5 ⋅ a⋅ ξ
2
8
32 5 ⋅ a⋅ ξ
−
8
4
+
15⋅ a⋅ ξ ⋅ η
8
3⋅ ξ
32 5 ⋅ a⋅ ξ
2
8
4
−
16
9⋅ b⋅ ξ ⋅ η
+
3⋅ ξ
4
15⋅ a⋅ ξ
2
−
3
4
−
16
a⋅ ξ
3
b
−
16
3
+
+
2
15⋅ ξ ⋅ η
−
3
b
−
16
3
+
+
2
15⋅ ξ ⋅ η
2
+
16
16
+
2
+
2
15⋅ ξ ⋅ η
2
16 3 ⋅ a⋅ ξ
−
16
−
16
3
+
2
16
3
2
15⋅ ξ ⋅ η
4
8
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
8
9⋅ b⋅ ξ
2
8
16
2
−
9⋅ b⋅ ξ ⋅ η
2
−
8
8
15⋅ b ⋅ ξ ⋅ η
4
16 2
+
9⋅ ξ ⋅ η
15⋅ b ⋅ ξ⋅ η
−
8
2
−
4
2
9⋅ ξ
2
16
15⋅ b ⋅ ξ
15⋅ b ⋅ η
16
9⋅ η
−
4
16
2
+
4
+
4
−
16
3⋅
9 4
In order to find later the strains, we use what we have already found: −1 ⎛⎜ d 2 a
2 2⎜ ⎝ dξ
−1 ⎛⎜ d 2
⎞⎟ ⎟ ⎠
⎛
2
⎜ ⎝
a
⎞⎟ ⎟ ⎠
⎛
2
⎜ ⎝
b
f ( ξ , η) → ⎜ 0 0 0 −
2 2⎜ b ⎝ dη
2
0 0 −
f ( ξ , η) → ⎜ 0 0 0 0 0 −
6⋅ ξ a
2
2
−
0 0 −
2⋅ η a
2
2⋅ ξ b
2
0 0 −
12⋅ ξ a
−
6⋅ η b
2
2
2
0 0 −
−
6 ⋅ ξ⋅ η a 2
2⋅ ξ b
2
2
−
−
2⋅ η a
6 ⋅ ξ⋅ η b
2
2
2
0 0 −
2
12⋅ ξ ⋅ η a
−
12⋅ η b
2
2
0 −
2
12⋅ ξ ⋅ b
2
2 2 3 3 ⎛ d d f ( ξ , η) ⎞ → ⎛⎜ 0 0 0 0 2 0 0 4⋅ ξ 4 ⋅ η 0 0 6 ⋅ ξ 8⋅ ξ⋅ η 6 ⋅ η 0 8 ⋅ ξ 8⋅ η ⎞ ⎜ ⎟ ( a ⋅ b ) ⎝ dξ dη a⋅ b a⋅ b a⋅ b a⋅ b a⋅ b ⎠ a⋅ b a⋅ b a⋅ b ⎠ ⎝
2
Strain vector: {ε}= [Ba]*{A}
2 2 ⎛⎜ −2 −6 ⋅ ξ −2 ⋅ η −12⋅ ξ −6 ⋅ ξ ⋅ η −2 ⋅ η − 0 0 0 0 0 0 0 0 0 ⎜ 2 2 2 2 2 2 a a a a a a ⎜ ⎜ 2 2 −2 ⋅ ξ −2 −2 ⋅ ξ −6 ⋅ η −6 ⋅ ξ⋅ η −12⋅ η ε ≡ z⋅ ⎜ 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 ⎜ b b b b b b ⎜ 2 2 ⎜ 2 4⋅ η 8⋅ ξ⋅ η 6⋅ η 4⋅ ξ 6⋅ ξ 0 0 0 0 0 0 0 0 0 ⎜⎝ a⋅ b a⋅ b a⋅ b a⋅ b a⋅ b a⋅ b
T
Mathcad is ready to find Ba
divided by the variable z
2 2 2 ⎛⎜ −2 −6 ⋅ ξ −2 ⋅ η −12⋅ ξ −6 ⋅ ξ⋅ η −2 ⋅ η −12⋅ ξ 0 0 0 0 0 0 0 0 0 ⎜ 2 2 2 2 2 2 2 a a a a a a a ⎜ ⎜ 2 2 −2 ⋅ ξ −2 ⋅ ξ −6 ⋅ η −6 ⋅ ξ ⋅ η −12⋅ η ⎜ 0 0 0 0 0 −2 0 0 0 0 0 2 2 2 2 2 2 ⎜ b b b b b b ⎜ 2 2 3 ⎜ 2 4⋅ η 8 ⋅ ξ⋅ η 6⋅ η 4⋅ ξ 6⋅ ξ 8⋅ ξ 0 0 0 0 0 0 0 0 0 ⎜⎝ a⋅ b a⋅ b a⋅ b a⋅ b a⋅ b a⋅ b a⋅ b
Ths is the transpose of Ba divided by z
0 ⎛ ⎜ 0 ⎜ 0 ⎜ ⎜ 2 ⎜ − 2 a ⎜ ⎜ 0 ⎜ ⎜ ⎜ 0 ⎜ ⎜ ⎜ 6⋅ ξ ⎜ − 2 a ⎜ ⎜ 2⋅ η ⎜ − 2 a ⎜ ⎜ 0 ⎜ ⎜ ⎜ ⎜ 0 ⎜ ⎜ ⎜ 12⋅ ξ2 ⎜ − 2 a ⎜ ⎜ ⎜ − 6 ⋅ ξ⋅ η ⎜ 2 a ⎜ ⎜ 2 ⋅ η2 ⎜ − 2 ⎜ a ⎜ ⎜ 0 ⎜ ⎜ ⎜ ⎜ 0 ⎜ ⎜ ⎜ 12⋅ ξ2⋅ η ⎜− 2
0
0 0 0
0 −
2 b
2
0
0
−
2⋅ ξ b
−
2
6⋅ η b
2
0
0
−
−
2
6 ⋅ ξ⋅ η b
−
2
2⋅ ξ b
2
12⋅ η b 0
⎞ ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 2 ⎟ ⎟ a⋅ b ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 4⋅ ξ ⎟ ⎟ a⋅ b ⎟ 4⋅ η ⎟ ⎟ a⋅ b ⎟ ⎟ 0 ⎟ ⎟ ⎟ ⎟ 0 ⎟ ⎟ 2 ⎟ 6⋅ ξ ⎟ a⋅ b ⎟ ⎟ 8⋅ ξ⋅ η ⎟ ⎟ a⋅ b ⎟ ⎟ 2 6⋅ η ⎟ a⋅ b ⎟ ⎟ ⎟ 0 ⎟ ⎟ ⎟ 3 8⋅ ξ ⎟ ⎟ b 0
0
2
2
⎜ ⎜ ⎜ ⎜ ⎜ ⎝
a 0
T
Mathcad is ready to find Ba ⋅ D divided by z
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜
−
0
0
0
0
0
0
2 a
0
0 −
0
2 b
−
6⋅ ξ a
−
0
2
−
0
2⋅ ξ b
−
0
12⋅ ξ a
−
−
0
2
a
2
0
0
2
2⋅ η
2
2
6 ⋅ ξ⋅ η a
2
6⋅ η b
−
2
0
2
2⋅ η a
0
0
2
2
−
−
2
2⋅ ξ b
2
6 ⋅ ξ⋅ η b
⎞ ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 2 ⎟ ⎟ a⋅ b ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 4⋅ ξ ⎟ ⎟ a⋅ b ⎟ 4⋅ η ⎟ ⎟ a⋅ b ⎟ ⎟ 0 ⎟ ⎛ Dx D1 0 ⎞ ⎟ ⎟⎜ ⎟ ⋅ ⎜ D1 Dy 0 ⎟ ⎟⎜ ⎟ 0 ⎟ ⎝ 0 0 Dxy ⎠ ⎟ 2 ⎟ 6⋅ ξ ⎟ a⋅ b ⎟ ⎟ 8⋅ ξ⋅ η ⎟ ⎟ a⋅ b ⎟ ⎟ 2 6⋅ η ⎟ a⋅ b ⎟ ⎟ ⎟ 0
2 2
⎟ ⎟ 3 ⎟ 8⋅ η ⎟ a⋅ b ⎟ ⎠ a⋅ b
2
−
12⋅ ξ⋅ η b
2
2
⎜ 2 12⋅ η ⎜ 0 0 − 2 ⎜ b ⎜ 3 ⎜ 12⋅ ξ2⋅ η 8⋅ ξ 0 ⎜− 2 a⋅ b a ⎜ ⎜ 2 3 12⋅ ξ⋅ η 8⋅ η ⎜ 0 − ⎜ 2 a⋅ b b ⎝
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜
−
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
0
0
0
0
0
0
0
0
0
2 ⋅ Dx a
−
2
a
0 −
2 ⋅ D1
−
2
−
−
4 ⋅ Dxy⋅ ξ
2
a⋅ b
2 ⋅ Dy ⋅ ξ
4 ⋅ Dxy⋅ η
2
a⋅ b
6 ⋅ Dy ⋅ η b
2
2
6 ⋅ Dx ⋅ ξ ⋅ η
−
0
2
12⋅ D1 ⋅ ξ a
−
0
2
2 ⋅ D1 ⋅ η
b
2
2
−
0
2
6 ⋅ D1 ⋅ ξ
a
2
12⋅ Dx ⋅ ξ
a⋅ b
2 ⋅ Dy
a
6 ⋅ D1 ⋅ η
a −
−
2
b −
2 ⋅ Dxy
b
2 ⋅ D1 ⋅ ξ b
−
2
2 ⋅ Dx ⋅ η a
−
−
6 ⋅ Dx ⋅ ξ a
0
2
0
b −
2 ⋅ D1
2
2
0
6 ⋅ D1 ⋅ ξ⋅ η
6 ⋅ Dxy⋅ ξ
2
a⋅ b
2
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟
T
This is Ba ⋅ D divided by z
⎜ a a ⎜ ⎜ 2 ⋅ Dx ⋅ η2 2⋅ D1⋅ ξ2 2⋅ D1⋅ η2 2 ⋅ Dy ⋅ ξ2 − − − ⎜− 2 2 2 2 ⎜ a b a b ⎜ 6 ⋅ D1 ⋅ ξ ⋅ η 6 ⋅ Dy ⋅ ξ⋅ η ⎜ − − ⎜ 2 2 ⎜ b b ⎜ 2 2 12⋅ D1 ⋅ η 12⋅ Dy ⋅ η ⎜ − − ⎜ 2 2 b b ⎜ ⎜ 2 2 12⋅ Dx ⋅ ξ ⋅ η 12⋅ D1 ⋅ ξ ⋅ η ⎜ − − ⎜ 2 2 a a ⎜ ⎜ 2 2 12⋅ D1 ⋅ ξ ⋅ η 12⋅ Dy ⋅ ξ⋅ η ⎜ − − ⎜ 2 2 b b ⎝
T
⎟ ⎟ 8 ⋅ Dxy⋅ ξ ⋅ η ⎟ ⎟ a⋅ b ⎟ ⎟ 2 6 ⋅ Dxy⋅ η ⎟ ⎟ a⋅ b ⎟ ⎟ ⎟ 0 ⎟ ⎟ 3 ⎟ 8 ⋅ Dxy⋅ ξ ⎟ ⎟ a⋅ b ⎟ 3 ⎟ 8 ⋅ Dxy⋅ η ⎟ ⎟ a⋅ b ⎠
Mathcad is ready to find Ba ⋅ D⋅ Ba divided by z
2
0 0 ⎛ ⎜ 0 0 ⎜ 0 0 ⎜ ⎜ 2 ⋅ D1 2 ⋅ Dx − − ⎜ 2 2 ⎜ a a ⎜ ⎜ 0 0 ⎜ ⎜ 2 ⋅ D1 2 ⋅ Dy ⎜ − − ⎜ 2 2 b b ⎜ 6 ⋅ Dx ⋅ ξ 6 ⋅ D1 ⋅ ξ ⎜ − − ⎜ 2 2 a a ⎜ ⎜ 2 ⋅ Dx ⋅ η 2 ⋅ D1 ⋅ η ⎜ − − 2 2 ⎜ a a ⎜ 2 ⋅ D1 ⋅ ξ 2 ⋅ Dy ⋅ ξ ⎜ − − ⎜ 2 2 b b ⎜ ⎜ 6 ⋅ D1 ⋅ η 6 ⋅ Dy ⋅ η ⎜ − − 2 2 ⎜ b b ⎜ 2 2 ⎜ 12⋅ Dx ⋅ ξ 12⋅ D1 ⋅ ξ ⎜ − − 2 2 ⎜ a a ⎜ ⎜ 6 ⋅ Dx ⋅ ξ ⋅ η 6 ⋅ D1 ⋅ ξ⋅ η − − ⎜ 2 2 ⎜ a a ⎜ 2 2 2 2 2 ⋅ D1 ⋅ η ⎜ 2 ⋅ Dx ⋅ η 2 ⋅ D1 ⋅ ξ 2 ⋅ Dy ⋅ ξ − − − ⎜− 2 2 2 2 ⎜ a b a b ⎜ 6 ⋅ D1 ⋅ ξ ⋅ η 6 ⋅ Dy ⋅ ξ⋅ η ⎜ − − ⎜ 2 2 ⎜ b b ⎜ 2 2 12⋅ D1 ⋅ η 12⋅ Dy ⋅ η ⎜ − − ⎜ 2 2 b b ⎜ ⎜ 2 2 12⋅ Dx ⋅ ξ ⋅ η 12⋅ D1 ⋅ ξ ⋅ η ⎜ − − ⎜ 2 2 a a ⎜ ⎜
⎞ ⎟ 0 ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ ⎟ 2 ⋅ Dxy ⎟ ⎟ a⋅ b ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 4 ⋅ Dxy⋅ ξ ⎟ ⎟ a⋅ b ⎟ ⎟ 4 ⋅ Dxy⋅ η ⎟ ⎟ a⋅ b ⎟ ⎛⎜ 0 0 0 −2 0 0 −6⋅ ξ −2⋅ η 0 ⎟⎜ 2 2 2 a a a ⎟⎜ 0 ⎟⎜ −2 −2 ⎟ ⋅⎜ 0 0 0 0 0 0 0 2 ⎟⎜ b b 0 ⎟⎜ ⎟⎜ 2 4⋅ ξ 4 0 ⎟ ⎜ 0 0 0 0 a⋅ b 0 a⋅ b a ⎝ 2 6 ⋅ Dxy⋅ ξ ⎟ ⎟ a⋅ b ⎟ ⎟ 8 ⋅ Dxy⋅ ξ ⋅ η ⎟ ⎟ a⋅ b ⎟ ⎟ 2 6 ⋅ Dxy⋅ η ⎟ ⎟ a⋅ b ⎟ ⎟ ⎟ 0 ⎟ ⎟ 3 ⎟ 8 ⋅ Dxy⋅ ξ ⎟ ⎟ a⋅ b ⎟ ⎟ 0
⎜ ⎜ ⎜ ⎝
−
12⋅ D1 ⋅ ξ ⋅ η b
2
−
2
12⋅ Dy ⋅ ξ⋅ η b
T
This is Ba ⋅ D⋅ Ba divided by z
⎡0 ⎢0 ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢
2
8 ⋅ Dxy⋅ η a⋅ b
2
3
⎟ ⎟ ⎟ ⎠
2
0 0
0
0
0
0
0 0
0
0
0
0
0 0
0
0
0
0
4 ⋅ D1
12⋅ Dx ⋅ ξ
0 0
4 ⋅ Dx a
0 0
0
4
2 2
a
0
0
4 ⋅ Dy
12⋅ D1 ⋅ ξ
4 ⋅ Dxy
0
4
a ⋅b
2 2
a ⋅b 0 0
0 0
4 ⋅ D1 a ⋅b
2 2
b
4
a ⋅b
12⋅ Dx ⋅ ξ
12⋅ D1 ⋅ ξ
36⋅ Dx ⋅ ξ
a 0 0
0
4
2 2
a ⋅b
2 2
a ⋅b
a
4 ⋅ D1 ⋅ ξ
8 ⋅ Dxy⋅ η
4 ⋅ Dy ⋅ ξ
12⋅ D1 ⋅ ξ
2 2
a ⋅b
4
a ⋅b
12⋅ D1 ⋅ η
2 2
b
b
24⋅ Dx ⋅ ξ
2
0
12⋅ Dxy⋅ ξ
4
a ⋅b
a
⎛ 2 ⋅ D ⋅ η2 ⎜ x
2 2
a ⋅b 2
2 2
2 ⋅ D1 ⋅ ξ
2⎞
⎟
2
3
72⋅ Dx ⋅ ξ
2 2
12⋅ Dx ⋅ ξ⋅ η
2
36⋅ D1 ⋅ ξ⋅ η
4
24⋅ D1 ⋅ ξ
4
4
2 2
12⋅ Dy ⋅ η
0
2 2
a 0 0
12⋅ Dx ⋅ ξ⋅ η
4
a ⋅b 0 0
4
a
4 ⋅ D1 ⋅ η
2 2
2
a ⋅b 8 ⋅ Dxy⋅ ξ
a ⋅b 0 0
2 2
4 ⋅ Dx ⋅ η a
0 0
0
4
a ⋅b
a
12⋅ D1 ⋅ ξ⋅ η
36⋅ Dx ⋅ ξ ⋅ η
2
2 2
a ⋅b
⎛ 2 ⋅ D ⋅ η2 ⎜ 1
a 2 ⋅ Dy ⋅ ξ
2⎞
⎟
4
⎛ 2 ⋅ D ⋅ η2 ⎜ x
2 ⋅ D1 ⋅ ξ
2
⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎣
2⋅ 0 0
⎜ ⎜ ⎝
xη a
+
2
b a
1
1ξ
2
2
⎟ ⎜ ⎟ 16⋅ D ⋅ ξ⋅ η 2⋅ ⎜ xy ⎠ ⎝
1η a
2
2 2
a ⋅b
yξ
+
b b
2
⎟ 6 ⋅ ξ⋅ ⎜ ⎟ ⎜ ⎠ ⎝
xη a
2
2
b a
0 ⋅ D ⋅ ξ⋅ η 12 y 0 4 b 0 2
24⋅ Dy ⋅ η 0 b
2
36⋅ D1 ⋅ ξ ⋅ η
0 0
2 2
a ⋅b
0
72⋅ D1 ⋅ ξ ⋅ η
4 ⋅ D1
4
2 2
2
2 2 2 a2 ⋅ b
3
72⋅ Dx ⋅ ξ ⋅ η 0 a
a ⋅b
24⋅ Dy ⋅ ξ ⋅ η 0 b
2
2
a ⋅b
2 2
a ⋅b
24⋅ D ⋅ ξ ⋅ η 4 ⋅ Dxy1
2
2
1
⌠ ⌠ 0 ⋅D 0 ⋅0ξ⋅ η 0 ⋅ D ⋅ η2 ⎮ ⎮ ⎡12 12 1 xy 0 0⎮ ⎮ ⎢0 0 0 0 2 2 2 2 ⎮ ⎮ ⎢ a ⋅b a ⋅b 0 ⎮ ⎮ ⎢0 0 0 ⎮ ⎮ ⎢ 2 ⎮ ⎮ ⎢24⋅ D1 ⋅ η 4 ⋅ Dx 0 0⎮ ⎮ 0 0 02 02 ⎢ 4 a ⋅b ⎮ ⎮ a ⎮ ⎮ ⎢ 2 3 ⎮ ⎮ ⎢24⋅ Dx ⋅ ξ ⋅ η 16⋅ Dxy⋅ ξ 0 0⎮ ⎮ ⎢ 0 2 2 ⎮ ⎮ ⎢0 0a4 0 a ⋅b ⎮ ⎮ ⎢ ⎮ ⎮ ⎢ 2 3 16⋅ D ⋅ η ⎮ ⎮ 24⋅ D1 ⋅ ξ ⋅ η 4 ⋅ D1 xy 0 0⎮ ⎮ ⎢ 0 02 02 2 2 2 a2 ⋅ b ⎮ ⎮ ⎢ a ⋅b ⋅ b a ⎮ ⎮ ⎢ ⎮ ⎮ ⎢ 12⋅ Dx ⋅ ξ ⎮ ⎮ ⎢ 0 0 0 ⎮ ⎮ ⎢ 4 a ⎮ ⎮ ⎢ 2 T The former ⎮ ⎮matrix is Ba ⋅ D⋅ Ba divided by z ⎮ ⎮ ⎢ 4 ⋅ Dx ⋅ η ⎮ ⎮ ⎢0 0 0 4 ⎮ ⎮ ⎢ a ⎮ ⎮ ⎢ ⎮ ⎮ ⎢ 4 ⋅ D1 ⋅ ξ ⎮ ⎮ ⎢ 0 0 0 ⎮ ⎮ ⎢ 2 2 a ⋅b ⎮ ⎮ ⎢ ⎮ ⎮ ⎮ ⎮ ⎢ 12⋅ D1 ⋅ η ⎮ ⎮ ⎢0 0 0 2 2 ⎮ ⎮ ⎢ a ⋅b ⎮ ⎮ ⎢ ⎮ ⎮ ⎢ 2 24⋅ Dx ⋅ ξ ⎮ ⎮ ⎢ 0 0 0 ⎮ ⎮ 4 ⎢ a ⎮ ⎮ ⎮ ⎮ ⎢ ⎮ ⎮ ⎢ 12⋅ Dx ⋅ ξ⋅ η ⎮ ⎮ ⎢0 0 0 4 ⎮ ⎮ ⎢ a ⎮ ⎮ ⎢ ⎮ ⎮ ⎢ ⎛ 2 ⋅ D ⋅ η2 2 ⋅ D ⋅ ξ2 ⎞ 1 ⎟ ⎮ ⎮ ⎜ x 2⋅ + ⎮ ⎮ ⎢ ⎜ 2 2 ⎟ a b ⎮ ⎮ ⎢ ⎝ ⎠ ⎮ ⎮ ⎢0 0 0 2 a ⎮ ⎮ ⎢ ⎮ ⎮ ⎢ ⎮ ⎮ ⎢ 12⋅ D1 ⋅ ξ⋅ η ⎮ ⎮ ⎢0 0 0
1ξ
+
2 2
72⋅ D1 ⋅ ξ ⋅ η
4 ⋅ Dy
4
b
4
2 2
a ⋅b
4
12⋅ D1 ⋅ ξ
0
2 2
a ⋅b 8 ⋅ Dxy⋅ ξ
4 ⋅ D1 ⋅ η
a ⋅b
2 2
a ⋅b
8 ⋅ Dxy⋅ η
4 ⋅ Dy ⋅ ξ
2 2
2 2
a ⋅b
b
4
12⋅ Dy ⋅ η
0
b
4
24⋅ D1 ⋅ ξ
0
2
2 2
a ⋅b 12⋅ Dxy⋅ ξ
2
2 2
12⋅ D1 ⋅ ξ⋅ η 2 2
a ⋅b
a ⋅b
⎛ 2 ⋅ D ⋅ η2 2 ⋅ D ⋅ ξ2 ⎞ ⎛ 2⋅ y ⎟ 1 ⎜ ⎜ 6 ⋅ ξ⋅ 2⋅ + ⎜ ⎟ ⎜ 2 2 16⋅ Dxy⋅ ξ⋅ η b ⎝ a ⎠ ⎝ 2 2
a ⋅b
12⋅ Dxy⋅ η
b 2
2
12⋅ Dy ⋅ ξ⋅ η
⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡
⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡
⎡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
−1 −1
0 0
⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎣
0 0
2 2
a ⋅b
0 0
b
2
24⋅ D1 ⋅ η
0
a ⋅b
b
2
0 0
24⋅ Dx ⋅ ξ ⋅ η
16⋅ Dxy⋅ ξ
4
a ⋅b
a 0 0
2
2 2
4
0
16⋅ Dxy⋅ η
16⋅ D1 2 2
4 2
2 2
a ⋅b 3
24⋅ Dy ⋅ ξ ⋅ η
a ⋅b
0
2
24⋅ D1 ⋅ ξ ⋅ η
2 2
a ⋅b
16⋅ Dx
3
2 2
24⋅ D1 ⋅ ξ ⋅ η
4
24⋅ Dy ⋅ η
2 2
a 0 0
2 2
a ⋅b
b
2
7
4
a ⋅b 16⋅ Dxy
0
2 2
a ⋅b 0 0
16⋅ D1 2 2
0
a ⋅b 0 0
0
16⋅ Dy b
0
4
0
48⋅ Dx a
0
4
16⋅ ⎛ 4 ⋅ Dxy⋅ a + Dx ⋅ b ⎝ 2
0 0
0
0
0
0
4 2
3⋅ a ⋅ b 0 0
0
0
0
16⋅ D1
0
2 2
a ⋅b 0 0
0
0
0
0
16⋅ D1 2 2
a ⋅b 0 0
32⋅ Dx 4
0
32⋅ D1 2 2
0
0
⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎣
a 0 0
4
2 2
a ⋅b 16⋅ Dxy
0
0
0
0
0
0
0
0
0
0
2 2
a ⋅b 16⋅ ⎛ D1 ⋅ a + Dx ⋅ b ⎝ 2
0 0
4 2
2⎞
⎠
2
0
3⋅ a ⋅ b 0 0
0
16⋅ ⎛ Dy ⋅ a + D1 ⋅ b ⎝ 2 4
2⎞
⎠
3⋅ a ⋅ b 16⋅ Dxy
0
2 2
a ⋅b 0 0
32⋅ D1 2 2
0
a ⋅b 0 0
0
32⋅ Dy b
0
4
0
0
32⋅ ⎛ 12⋅ Dxy⋅ a + 5 ⋅ Dx ⋅
⎝
2
4 2
15⋅ a ⋅ b 0 0
0
0
0
32⋅ D1 2 2
a ⋅b
0
− −
3⋅ b 16
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
⎞ ⎟ ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 9⎟ − ⎟ 4 ⎟ 0 ⎟ ⎟ 9⎟ − ⎟ 4⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 9 ⎟ 4 ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎠ 9 4
− −
3⋅ b 16
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
⎞ ⎟ 4 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 9⎟ − ⎟ 4 ⎟ 0 ⎟ ⎟ 9⎟ − ⎟ 4⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 9 ⎟ 4 ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ ⎟ 9
T
−
15⋅ b 16
0
⎞ ⎟ ⎟ 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⋅ b ⎟ 16 ⎟ ⎟ 15⋅ b ⎟ − 16 ⎟ ⎟ 0 ⎟ ⎠ −
15 32
⎟ ⎟ ⎠
5 2
5 2
a 2 a 2
a a 6
⎞ ⎟ 32 ⎟ 15 ⎟ − 32 ⎟ ⎟ 15 ⎟ 32 ⎟ 1 ⎞ 15 ⎟ ⎛⎜ ⎟ 32 ⎟ ⎜ ξ ⎟ ⎟⎜ η ⎟ 0 ⎟⎜ ⎟ ⎜ ξ2 ⎟⎟ ⎟⎜ ⎟ 0 ⎟ ξ⋅ η ⎜ ⎟ ⎜ 2 ⎟⎟ ⎟⎜ η ⎟ 0 ⎟ ⎜ ξ3 ⎟ ⎟⎜ 2 ⎟ 0 ⎟ ⎜ ξ ⋅η ⎟ ⎟⎜ 2 ⎟ 5⋅ b ⎟ ⎜ ξ⋅ η ⎟ ⎟ ⋅⎜ 3 ⎟ 32 ⎟ η ⎟ 5⋅ b ⎟ ⎜ − ⎜ 4 ⎟ 32 ⎟ ⎜ ξ ⎟ ⎟ 5 ⋅ b ⎜ ξ 3⋅ η ⎟ ⎟ − 32 ⎟ ⎜ 2 2 ⎟ ⎜ ξ ⋅η ⎟ 5⋅ b ⎟ ⎜ ⎟ 32 ⎟ ⎜ ξ ⋅ η3 ⎟ ⎟⎜ 4 ⎟ 0 ⎟⎜ η ⎟ ⎟⎜ ⎟ ⎜ ξ4⋅ η ⎟⎟ 0 ⎟ ⎟ ⎜⎝ ξ⋅ η4 ⎟⎠ 15⋅ b ⎟ 16 ⎟ ⎟ 15⋅ b ⎟ − 16 ⎟ ⎟ 0 ⎟ ⎠ −
15
⎞ ⎟ 8 32 32 8 32 8⎟ ⎟ 4 3 2 3⋅ ξ 15⋅ η η 3⋅ η 3⋅ η 1⎟ − − + + + − 8 32 32 8 32 8⎟ ⎟ 4 3 2 3⋅ ξ 15⋅ η η 3⋅ η 3⋅ η 1⎟ + − − + − − 8 32 32 8 32 8⎟ ⎟ 4 3 2 3⋅ ξ 15⋅ η η 3⋅ η 3⋅ η 1⎟ + − + + + − 8 32 32 8 32 8⎟ ⎟ 4 ⎟ η 5 ⋅ a⋅ ξ ⋅ η + ⎟ 32 ⎟ 4 ⎟ η 5 ⋅ a⋅ ξ ⋅ η − ⎟ 32 ⎟ 4 ⎟ η 5 ⋅ a⋅ ξ ⋅ η − ⎟ 32 ⎟ 4 ⎟ η 5 ⋅ a⋅ ξ ⋅ η + ⎟ 32 ⎟ ⎟ 4 b 5⋅ ξ⋅ η ⋅ b ⎟ + 32 ⎟ ⎟ 4 b 5⋅ ξ⋅ η ⋅ b ⎟ − 32 ⎟ ⎟ 4 b 5⋅ ξ⋅ η ⋅ b ⎟ − ⎟ 32 ⎟ 4 b 5⋅ ξ⋅ η ⋅ b ⎟ + ⎟ 32 ⎟ 5⋅ a⋅ η 3⋅ a ⎟ − ⎟ 16 16 ⎟ ⎟ 5⋅ a⋅ η 3⋅ a − ⎟ 16 16 ⎟ 2 ⎟ b⋅ η 3⋅ b − ⎟ 4 16 ⎟ −
3⋅ ξ
−
15⋅ η
4
−
η
3
+
3⋅ η
2
−
3⋅ η
−
1
4
16
b⋅ η
2
4
⎞
0⎟
⎟ ⎠
η
⎞ ⎟ ⎠
2
⎞ ⎟ ⎟ ⎠
−
3⋅ b 16
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
⎛ a1 ⎞ ⎜ ⎟ ⎜ a2 ⎟ ⎜ ⎟ ⎜ a3 ⎟ ⎜a ⎟ ⎜ 4⎟ ⎜ a5 ⎟ ⎜ ⎟ ⎜ a6 ⎟ 2 ⎞⎟ ⎜ ⎟ −12⋅ ξ ⋅ η a 0 ⎟⎜ 7⎟ 2 a ⎟ ⎜ a8 ⎟ 2⎟ ⎜ ⎟ −12⋅ ξ ⋅ η ⋅ a ⎟⎜ 8⎟ 0 2 ⎟ ⎜a ⎟ b ⎟ ⎜ 10 ⎟ 3 3 ⎟ ⎜ a11 ⎟ 8⋅ η 8⋅ ξ ⎟ ⎜ a⋅ b a⋅ b ⎠ ⎜ a ⎟⎟ 12 ⎜a ⎟ ⎜ 13 ⎟ ⎜ a14 ⎟ ⎜ ⎟ ⎜ a15 ⎟ ⎜ ⎟ ⎜ a16 ⎟ ⎜a ⎟ ⎝ 17 ⎠
2
⋅η
3
⎞⎟ 0 ⎟ ⎟ 2⎟ −12⋅ ξ⋅ η ⎟ 2 ⎟ b ⎟ 3 ⎟ 8⋅ η ⎟⎠ a⋅ b
T
0
0
−12⋅ ξ a
2⋅ ξ b
2
⋅η ⋅b
−6 ⋅ η b
2
0
2
−6 ⋅ ξ ⋅ η a
2
a
b 0
6⋅ ξ
2
a⋅ b
2
2
0
−12⋅ ξ ⋅ η
0
2
−2 ⋅ ξ
0
2
0
−2 ⋅ η
a 2
2
−6 ⋅ ξ⋅ η b
2
−12⋅ η b
6⋅ η
a⋅ b
a⋅ b
2
0
2
2
8⋅ ξ⋅ η
2
0
8⋅ ξ
3
a⋅ b
⎞⎟ ⎟ ⎟ 2⎟ −12⋅ ξ ⋅ η ⎟ 2 ⎟ b ⎟ 3 ⎟ 8⋅ η ⎟⎠ a⋅ b 0
0
0
0
0
0
0
4 ⋅ Dx ⋅ η
4 ⋅ D1 ⋅ ξ
4
a ⋅b
a
2 2
8 ⋅ Dxy⋅ ξ
8 ⋅ Dxy⋅ η
a ⋅b
2 2
a ⋅b
4 ⋅ D1 ⋅ η
4 ⋅ Dy ⋅ ξ
2 2
2 2
b
12⋅ Dx ⋅ ξ ⋅ η
12⋅ D1 ⋅ ξ
4
a ⋅b
a
a
2
4
2
4 ⋅ D1 ⋅ ξ ⋅ η
2 2
a ⋅b
+
2 2
16⋅ Dxy⋅ ξ ⋅ η
4 ⋅ Dy ⋅ ξ
2 2
a ⋅b
a ⋅b
b
2⎞
⎟
⎛ 2 ⋅ D ⋅ η2 ⎜ x
4
2 2
12⋅ Dy ⋅ ξ ⋅ η b
2
36
4
24⋅ D1 ⋅ ξ
4
a ⋅b
+
12
a ⋅b
24⋅ Dx ⋅ ξ ⋅ η
2
2
16⋅ Dxy⋅ η
+
2 2
a
2 2
2
a ⋅b
12⋅ Dx ⋅ ξ⋅ η
12
a ⋅b
4
2
12⋅ D1 ⋅ η
a
36
16⋅ Dxy⋅ ξ ⋅ η
+
2 2
a ⋅b
4 ⋅ D1 ⋅ ξ ⋅ η
2
2 2
16⋅ Dxy⋅ ξ
+
1
4
a ⋅b
4 ⋅ Dx ⋅ η
1
3
72⋅
2 2
24⋅ Dxy⋅ ξ 2 2
a ⋅b 2 ⋅ D1 ⋅ ξ
2⎞
⎟
3
2
12⋅ D1 ⋅ ξ ⋅ η 2 2
2
+
24⋅ Dxy⋅ ξ ⋅ η
a ⋅b
⎛ 2 ⋅ D ⋅ η2 ⎜ 1
36⋅
2 2
a ⋅b 2 ⋅ Dy ⋅ ξ
2⎞
⎟
⎛ 2⋅ D ⎜ 1
⎟ 2 ⋅ η⋅ ⎜ ⎟ ⎜ ⎠ ⎝
xη a
2
b a
0
2
+
2 2
2 2
a ⋅b
2 2
+
4
24⋅ D1 ⋅ ξ⋅ η
a ⋅b
2
a 32⋅ Dxy⋅ ξ 2 2
3
+
2 2
a ⋅b
2
12⋅ D1 ⋅ ξ 2 2
16⋅ Dxy⋅ ξ
+
4
2 2
2
+
2
24⋅ Dy ⋅ ξ ⋅ η b
+
2 2
b
4 ⋅ D1 ⋅ ξ ⋅ η 2 2
4 ⋅ Dy ⋅ ξ
2 2
a ⋅b
b
2
+
4
2
+
2 ⋅ D1 ⋅ ξ b
2
2
2
2 2
2
12⋅ Dy ⋅ ξ ⋅ η b
4
+
3
2 2
24⋅ Dxy⋅ ξ
3
2
12⋅ D1 ⋅ ξ ⋅ η
2 2
2 2
a ⋅b
2
+
a ⋅b
24⋅ Dxy⋅ ξ ⋅ η 2 2
a ⋅b
2 2⎞ ⎛ ⎛ 2 ⋅ D ⋅ η2 2 ⋅ D ⋅ ξ 2 ⎞ y ⎟ ⎟ 2 ⋅ η⋅ ⎜ 2 ⋅ Dx ⋅ η + 2⋅ D1⋅ ξ ⎟ ⎜ 1 2⋅ ξ⋅ + 2 ⎟ ⎜ ⎟ ⎜ 2 2 2 2 ⎟ b b ⎠ ⎝ a ⎠ + 32⋅ Dxy⋅ ξ ⋅ η ⎝ a ⎠ + 32⋅ Dxy⋅ ξ
2⎞
a
36⋅ D1 ⋅ ξ ⋅ η
2
2
a ⋅b
a ⋅b
4
a
16⋅ Dxy⋅ η
4
12⋅ Dx ⋅ ξ⋅ η
4
2 2
24⋅ D1 ⋅ ξ
a 2
16⋅ Dxy⋅ ξ ⋅ η
24⋅ Dx ⋅ ξ ⋅ η
4
2
a ⋅b
2
36⋅ Dx ⋅ ξ ⋅ η
4
+
2 2
3
72⋅
4 ⋅ Dy ⋅ ξ
4
16⋅ Dxy⋅ ξ ⋅ η
a ⋅b
72⋅ Dx ⋅ ξ
a
4
72⋅
a ⋅b
a ⋅b
a ⋅b
2
2 2
a ⋅b
2 2
2 2
a
2
a ⋅b
a ⋅b
Dx ⋅ η
+
32⋅ Dxy⋅ ξ ⋅ η 8 ⋅ Dxy⋅ η
2 2
12⋅ D1 ⋅ η
a
2 2
a ⋅b
a ⋅b
2
72
4 ⋅ D1 ⋅ ξ
4
36⋅ D1 ⋅ ξ⋅ η
a
0
2
12⋅ D1 ⋅ ξ
4 ⋅ D1 ⋅ ξ ⋅ η
a ⋅b
2 2
12⋅ Dx ⋅ ξ ⋅ η
4 ⋅ Dx ⋅ η a
2 2
a ⋅b
2 2
a ⋅b
a
4
32⋅ Dxy⋅ η
4 ⋅ D1 ⋅ η
4
12⋅ Dx ⋅ ξ⋅ η
2 2
36⋅
3
a ⋅b
3
2
0
4
3
24⋅ D1 ⋅ ξ ⋅ η
2 2
32⋅ Dxy⋅ ξ⋅ η
2
36⋅ Dx ⋅ ξ
b
a ⋅b
a
0
a ⋅b
4
8 ⋅ Dxy⋅ ξ
3
24⋅ Dxy⋅ η
24⋅ Dy ⋅ ξ⋅ η
4
a ⋅b
+
4
4 ⋅ Dx ⋅ η
1
2 2
2
b
⎟ ⎜ 6 ⋅ η⋅ ⎟ 32⋅ D ⋅ ξ⋅ η2 ⎜ xy ⎠ + ⎝ a ⋅b
12⋅ Dy ⋅ ξ ⋅ η
0
3
2
2
0
2 2
12⋅ D1 ⋅ ξ
b b
0
a ⋅b
4 2
a
2
a ⋅b
24⋅ Dx ⋅ ξ ⋅ η
a
24⋅ Dxy⋅ ξ⋅ η
24⋅ D1 ⋅ η
12⋅ Dx ⋅ ξ
a
a
yξ
+
2
2 2
2 2
0
1η
a ⋅b
a ⋅b
0
a
2
2
12⋅ D1 ⋅ ξ⋅ η
0
⎟ ⎜ 2ξ ⎟ 32⋅ D ⋅ ξ2⋅ η ⋅ ⋅ ⎜ xy ⎠ + ⎝
1ξ
+
2
12⋅ D1 ⋅ ξ⋅ η
2 2
a ⋅b 2
+
24⋅ Dxy⋅ ξ⋅ η
2
b
2
2 2
a ⋅b 2
12⋅ Dy ⋅ ξ ⋅ η
+
24⋅ Dxy⋅ η
3
2 2
a ⋅b
72⋅ D1 ⋅ ξ ⋅ η
+
2 2
a ⋅b
a ⋅b
2
24⋅ D1 ⋅ η
2 2
3
3
2
24⋅ Dx ⋅ ξ ⋅ η
4
a
72⋅ D1 ⋅ ξ ⋅ η
2
2 2
+
4
24⋅ Dy ⋅ ξ⋅ η b
32⋅ Dxy⋅ ξ
4
2 2
2 2
3
2 2
a ⋅b
+
3
2 2
32⋅ Dxy⋅ η
4
a ⋅b
0
0
0
0
0
0
0
0
0
0
0
0
0
32⋅ Dx a
0
4
0 16⋅ Dxy
0
2 2
a ⋅b 0
0
32⋅ D1 2 2
0
a ⋅b 16⋅ D1
0
0
0
0
0
0
0
0
0
2 2
a ⋅b 2⎞
⎠
16⋅ D1
0
2 2
a ⋅b 16⋅ ⎛ Dy ⋅ a + 4 ⋅ Dxy⋅ b
⎝
2
2⎞
⎠
0
2 4
3⋅ a ⋅ b 0
48⋅ Dy b
0
4
0
+
576 ⋅ Dx 4
0
32⋅ Dxy⋅ ξ ⋅ η 2 2
a ⋅b
2 2
a ⋅b
0
0
3
a ⋅b
32⋅ Dxy⋅ ξ⋅ η
2
4
3
24⋅ D1 ⋅ ξ ⋅ η
a ⋅b
24⋅ D1 ⋅ ξ⋅ η
a ⋅b
2
2 2
a ⋅b
a ⋅b
72⋅ Dx ⋅ ξ ⋅ η
2
b
+
4
2 2
a ⋅b
a
2 2
2 2
+
24⋅ Dy ⋅ ξ ⋅ η b
4
5⋅ a 0
0
4
16⋅ ⎛ 9 ⋅ Dxy⋅ a + 5 ⋅ Dx ⋅ b 2
⎝
0
2⎞
⎠
4 2
5⋅ a ⋅ b 0
0
32⋅ ⎛ 9 ⋅ D1 ⋅ a + 5 ⋅ Dx ⋅ b
⎝
2
2⎞
⎠
16⋅ ⎛ 9 ⋅ Dy
⎝
0
4 2
15⋅ a ⋅ b 0
0
0
(
16⋅ D1 + Dxy 2 2
a ⋅b 0
0
64⋅ D1 2 2
0
a ⋅b b
2⎞
⎠
32⋅ D1
0
0
0
0
0
2 2
a ⋅b 32⋅ ⎛ 5 ⋅ Dy ⋅ a + 12⋅ Dxy⋅ b
⎝
2
2 4
15⋅ a ⋅ b
2⎞
⎠
0
)
0
0
0
0
0
0
0
0
0
2⋅ D1 ⋅ η
24⋅ Dx ⋅ ξ
2 2
a ⋅b
a
0
2
12⋅ Dx ⋅ ξ⋅ η
4
a
4
12⋅ Dxy⋅ ξ
0
2
2 2
a ⋅b 2⋅ Dy ⋅ η
24⋅ D1 ⋅ ξ
4
a ⋅b
6⋅ D1 ⋅ ξ⋅ η
72⋅ Dx ⋅ ξ
b
2 2
a 2
2⋅ D1 ⋅ η
2
36⋅ Dx ⋅ ξ ⋅ η
4
a
a
12⋅ Dx ⋅ ξ ⋅ η
4
a
2⋅ Dy ⋅ ξ⋅ η
24⋅ D1 ⋅ ξ
4
a ⋅b
3
2
2
2
2 2
2 2 2
a ⋅b
a 2 ⋅ Dy ⋅ ξ
2⎞
⎟
2
4
3
72⋅ Dx ⋅ ξ ⋅ η
4
a 2
36⋅ Dx ⋅ ξ ⋅ η
4
⎛ 2 ⋅ D ⋅ η2 ⎜ x
2
2 2
3
a ⋅b
2 2
a ⋅b
2 2
72⋅ Dx ⋅ ξ ⋅ η
2 2
+
a ⋅b
36⋅ D1 ⋅ ξ ⋅ η
a
⋅ D1 ⋅ ξ ⋅ η
2
24⋅ Dxy⋅ ξ ⋅ η
72⋅ D1 ⋅ ξ ⋅ η
144 ⋅ Dx ⋅ ξ
a ⋅b
2 2
a ⋅b
a ⋅b
⋅ D1 ⋅ ξ ⋅ η
3
24⋅ Dxy⋅ ξ
+
4
2
4
4
2
12⋅ D1 ⋅ ξ ⋅ η
2 2
6⋅ Dy ⋅ η
2
3
2
a ⋅b
⋅η
2 2
a ⋅b
24⋅ Dx ⋅ ξ ⋅ η
2 2
b
12⋅ D1 ⋅ ξ⋅ η
2 2
a ⋅b
b
2
a 2 ⋅ D1 ⋅ ξ
2⎞
⎟
4
⎛ 2 ⋅ D ⋅ η2 ⎜ x
4
2
+
36⋅ Dxy⋅ ξ
4
2 2
a ⋅b 2 ⋅ D1 ⋅ ξ
2⎞
⎟
2
⎛2 ⎜
η
yξ
+
2
b b
2 2
a ⋅b
⋅ Dy ⋅ ξ ⋅ η
3
4
b
2 2
a ⋅b
4
36⋅ D1 ⋅ ξ⋅ η 2 2
a ⋅b
2
3
+
a 48⋅ Dxy⋅ ξ 2 2
2 2
a ⋅b
2
+
3
2 2
a ⋅b
2 2
a ⋅b
2
36⋅ Dx ⋅ ξ ⋅ η
4
a ⋅b
2 2
2 2
a ⋅b
36⋅ D1 ⋅ ξ ⋅ η
2
12⋅ D1 ⋅ ξ ⋅ η 2 2
2 2
a ⋅b
2
24⋅ Dxy⋅ ξ ⋅ η
+
2 2
a ⋅b
a ⋅b
72⋅ D1 ⋅ ξ ⋅ η
36⋅ D1 ⋅ ξ ⋅ η
2 2
a ⋅b
144 ⋅ Dx ⋅ ξ a
2
2 2
a ⋅b
4
3
72⋅ Dx ⋅ ξ ⋅ η
4
a 3
72⋅ Dx ⋅ ξ ⋅ η a
2
2 2
a ⋅b 2
3
24⋅ Dxy⋅ ξ
+
4
2
72⋅ D1 ⋅ ξ ⋅ η
4
2
12⋅ Dx ⋅ ξ ⋅ η a
3
2
48⋅ Dxy⋅ ξ ⋅ η 12⋅ D1 ⋅ ξ⋅ η
4
24⋅ D1 ⋅ ξ
4
12⋅ Dxy⋅ ξ
a ⋅b 3
4
5
a
12⋅ Dy ⋅ ξ⋅ η
b
12⋅ Dx ⋅ ξ⋅ η
4 2
2
0
3
3
72⋅ Dx ⋅ ξ
a
36⋅ Dy ⋅ η
2
2 2
a ⋅b
2
4
72⋅ D1 ⋅ ξ ⋅ η a ⋅b
2 2
a ⋅b
b
2
24⋅ D1 ⋅ ξ
24⋅ Dx ⋅ ξ ⋅ η
2 2
0
2 2
a ⋅b 72⋅ Dx ⋅ ξ ⋅ η
a
12⋅ D1 ⋅ η
2
2
+
2 0
a ⋅b
4
a 3
2
36⋅ Dxy⋅ ξ ⋅ η
2 2
0
144 ⋅ D1 ⋅ ξ ⋅ η
12⋅ Dy ⋅ η
2
72⋅ D1 ⋅ ξ ⋅ η
2
4
4
2 2
a ⋅b
a ⋅b
24⋅ Dx ⋅ ξ a
2
2
2 2
144 ⋅ Dx ⋅ ξ ⋅ η a
b
2
0 2
2 2
0
a
2
⎟ 2⎜ 2η ⎟ 48⋅ D ⋅ ξ3⋅ η ⋅ ⋅ ⎜ xy ⎠ + ⎝
1ξ
+
36⋅ D1 ⋅ ξ ⋅ η
0 0
144 ⋅ D1 ⋅ ξ ⋅ η
2
xη
a
2
2 2
2 2
2
⎟ 6 ⋅ ξ⋅ η⋅ ⎜ ⎟ ⎜ ⎠ ⎝
2
a ⋅b
a ⋅b
a ⋅b
2
b
2 2
12⋅ D1 ⋅ η
4 2
ξ⋅ η
2
3
0 3
D1 ⋅ ξ ⋅ η
b
a
1ξ
+
72⋅ D1 ⋅ ξ ⋅ η
0 0
4
2⋅ Dy ⋅ η b
xη
a
⋅ Dy ⋅ ξ ⋅ η b
2
⎟ 12⋅ ξ2⋅ ⎜ ⎟ ⎜ ⎠ ⎝
2
36⋅ Dx ⋅ ξ ⋅ η
4
4
2
+
a
4
36⋅ Dxy⋅ ξ
4
2 2
a ⋅b
2 2 ⎛ 2 ⋅ D ⋅ η2 2 ⋅ D ⋅ ξ2 ⎞ ⎛ ⎛ 2 ⋅ D ⋅ η2 2 ⋅ D ⋅ ξ2 ⎞ 2 ⋅ D1 ⋅ ξ ⎞⎟ y ⎟ 1 ⎟ 1 ⎜ 2 ⎜ 2 ⋅ Dx ⋅ η ⎜ x 6 ⋅ ξ⋅ η⋅ 6 ⋅ η⋅ + 12⋅ ξ ⋅ + + ⎜ ⎟ ⎜ ⎟ ⎜ 2 2 2 2 2 2 ⎟ b b b ⎝ a ⎠ ⎝ a ⎠ ⎝ a ⎠ + 48⋅ Dx
b
2
36⋅ Dy ⋅ ξ ⋅ η
a 2
2
a 3
72⋅ D1 ⋅ ξ ⋅ η
2
2
a ⋅ 2
36⋅ D1 ⋅ ξ ⋅ η
2
2
+
36⋅ Dxy⋅ ξ ⋅ η
2
b
4
2 2
a ⋅b
72⋅ Dy ⋅ η b
3
2
72⋅ D1 ⋅ ξ ⋅ η
2 2
2
2
2 2
4
2 2
72⋅ Dy ⋅ ξ ⋅ η
a 3
3
72⋅ Dx ⋅ ξ ⋅ η
4
a 3
144 ⋅ D1 ⋅ ξ ⋅ η
4
2
2
2 2
+
4
2 2
3
+
0
0
0
0
0
0
0
4 2
3⋅ a ⋅ b
3
2 2
a ⋅b
0
⎠
48⋅ Dxy⋅ ξ ⋅ η
a ⋅b
2⎞
5
2 2 2
0
2
48⋅ Dxy⋅ ξ a ⋅b
72⋅ D1 ⋅ ξ ⋅ η
a ⋅b
2
0
16⋅ ⎛ D1 ⋅ a + Dx ⋅ b ⎝
3
a ⋅b
144 ⋅ Dx ⋅ ξ ⋅ η
a ⋅b
b
2
a ⋅b
a ⋅b
72⋅ D1 ⋅ ξ ⋅ η
2 2
a ⋅b
144 ⋅ D1 ⋅ ξ ⋅ η
4
+
2 2
32⋅ D1 2 2
a ⋅b 16⋅ Dxy
0
0
2 2
a ⋅b 16⋅ ⎛ Dy ⋅ a + D1 ⋅ b ⎝ 2
2⎞
⎠
0
2 4
3⋅ a ⋅ b
32⋅ Dy b
4
0
0
0
0
0
0
0
0
0
0
0
0
32
32⋅ ⎛ 9 ⋅ D1 ⋅ a + 5 ⋅ Dx ⋅ b
⎝
2
4 2
2⎞
⎠
0
64⋅ D1 2 2
4 2
2 2
15⋅ a ⋅ b
a ⋅b
(
16⋅ D1 + Dxy
0
)
0
2 2
a ⋅b 4
4
2 2⎞
2 2
y ⋅ a + 9 ⋅ Dx ⋅ b + 10⋅ D1 ⋅ a ⋅ b + 80⋅ Dxy⋅ a ⋅ b
⎠
32⋅ ⎛ 5 ⋅ Dy ⋅ a + 9 ⋅ D1 ⋅ b
⎝
0
4 4
2
2⎞
⎠
2 4
45⋅ a ⋅ b
15⋅ a ⋅ b 16⋅ ⎛ 5 ⋅ Dy ⋅ a + 9 ⋅ Dxy⋅ b
⎝
0
2
2⎞
⎠
0
2 4
5⋅ a ⋅ b 32⋅ ⎛ 5 ⋅ Dy ⋅ a + 9 ⋅ D1 ⋅ b
⎝
2
2 4
2⎞
⎠
0
15⋅ a ⋅ b
576 ⋅ Dy 5⋅ b
4
64⋅ 0
0
0
0
0
0
4 ⋅ Dx ⋅ η a
0
0
0
0
0
0
2
4 ⋅ D1 ⋅ ξ
+
4
2
12⋅ D1 ⋅ ξ⋅ η
2 2
2 2
a ⋅b
a ⋅b
16⋅ Dxy⋅ ξ ⋅ η
12⋅ Dxy⋅ η
2 2
a ⋅b
2 2
a ⋅b 2
4 ⋅ Dy ⋅ ξ b
4
4 ⋅ Dx ⋅ η a
2
+
4
2 2 2
+
2 2
b
4 2
a 12⋅ Dx ⋅ ξ⋅ η
3
4
2 ⋅ D1 ⋅ ξ
12⋅ D1 ⋅ ξ ⋅ η
2 2
a ⋅b
32⋅ Dxy⋅ ξ⋅ η
+
2
4
+
⎟
b
4
2 2
2
b 4
3
2 2
2
4 3
72⋅ D1 ⋅ ξ ⋅ η 2 2
a ⋅b 3
⎛ 2 ⋅ D ⋅ η2 ⎜ 1
+
36⋅ Dy ⋅ ξ ⋅ η
2 2
+
24⋅ Dxy⋅ η a ⋅b
a ⋅b
12⋅ D1 ⋅ ξ ⋅ η
2 2
2
12⋅ Dy ⋅ ξ ⋅ η
3
24⋅ D1 ⋅ ξ
24⋅ Dxy⋅ ξ ⋅ η a ⋅b
2 2
a ⋅b 2⎞
2
2 2
3
+
+
2 2
a ⋅b 12⋅ D1 ⋅ η
2
a ⋅b
24⋅ Dx ⋅ ξ ⋅ η
2
32⋅ Dxy⋅ ξ ⋅ η
a ⋅b 2
⋅ Dx ⋅ η
2 2
a ⋅b
a ⋅b
4 ⋅ D1 ⋅ ξ⋅ η
+
4
2
2
+
4
36⋅ D1 ⋅ ξ ⋅ η
2 2
a ⋅b 3
3
a ⋅b
12⋅ Dy ⋅ ξ ⋅ η
a
b
12⋅ D1 ⋅ ξ
4 ⋅ D1 ⋅ ξ ⋅ η
+
12⋅ Dy ⋅ ξ⋅ η
2 2
2
4
4 ⋅ Dy ⋅ ξ b
3
2
a ⋅b
12⋅ Dx ⋅ ξ⋅ η a
4 ⋅ D1 ⋅ η
+
2
48⋅ Dxy⋅ ξ ⋅ η 2 2
2 2
a ⋅b
2 ⋅ Dy ⋅ ξ
2
36⋅ D1 ⋅ ξ ⋅ η
2
2
+
36⋅ Dxy⋅ ξ ⋅ η
a ⋅b 2⎞
⎟
⎛ 2 ⋅ D ⋅ η2 ⎜ 1
2 2
a ⋅b 2 ⋅ Dy ⋅ ξ
2⎞
⎟
xη a
2
b a
⎟ 2 ⋅ ξ2⋅ ⎜ ⎟ ⎜ ⎠ + ⎝
1ξ
+
2
2
1η a
2
b
3
b
12⋅ D1 ⋅ ξ⋅ η
+
4
3
a ⋅b
2 2
4
2
a
b
a
2
+
4
12⋅ Dx ⋅ ξ⋅ η
4 ⋅ Dx ⋅ η a
3
72⋅ D1 ⋅ ξ ⋅ η
48⋅ Dxy⋅ η
12⋅ D1 ⋅ ξ
+
4
4
+
2
+
2 2
4 2
4
2
+
3
72⋅ Dy ⋅ ξ ⋅ η 1 b
4
2 2
a ⋅b
32⋅ Dxy⋅ ξ⋅ η
2 2
2
12⋅ Dy ⋅ ξ
2 2
12⋅ D1 ⋅ η
+
b
4
3
3
2 2
2
4
+
24⋅ D1 ⋅ ξ
4
7
2 2
a ⋅b 3
+
5
a ⋅b
24⋅ Dx ⋅ ξ ⋅ η
3
2 2
a ⋅b
12⋅ D1 ⋅ ξ ⋅ η
a ⋅b
2
12⋅ Dx ⋅ ξ⋅ η
+
32⋅ Dxy⋅ ξ ⋅ η
a ⋅b
a
3
48⋅ Dxy⋅ ξ ⋅ η 1
3
a ⋅b
4 ⋅ D1 ⋅ ξ⋅ η
b
1
3
2
2 2
+
3
2 2
a ⋅b 3
4
a ⋅b
4 ⋅ D1 ⋅ ξ ⋅ η
+
2 2
a ⋅b
2
4
2
a ⋅b
2
12⋅ Dy ⋅ ξ ⋅ η
a
b
2 2
2
4
4 ⋅ Dy ⋅ ξ b
3
b
2 2
2 2 a ⋅2b 2
a
+
a ⋅b
4
2 2
2
36⋅ Dy ⋅ ξ ⋅ η
72⋅ Dy ⋅ ξ ⋅ η
2
2 64⋅ D 2 ⋅ ξ ⋅ η 4 ⋅ D ⋅ ξxy 4 ⋅ D1 ⋅ η + y 2 +2 4 a ⋅b 2 2 b a ⋅b
a ⋅b
4
a ⋅b
a ⋅b
24⋅ D1 ⋅ ξ ⋅ η
2
2 2
64⋅ Dxy⋅ ξ ⋅ η 16⋅ Dxy⋅ ξ ⋅ η
+
2 2
2
b b
4
24⋅ D1 ⋅ ξ ⋅ η
2
⎟ ⎟ 48 ⎠ +
yξ
+
36⋅ Dxy⋅ η
4 ⋅ D1 ⋅ ξ
+
4
a ⋅b 3
4
2
4
4
+
4
24⋅ Dy ⋅ ξ ⋅ η b
3
2 2
24⋅ Dy ⋅ ξ ⋅ η 2 4 ⋅ Dx ⋅ η
a ⋅b 24⋅ Dx ⋅ ξ ⋅ η
a
a ⋅b
2
+
1η
3 48⋅ Dxy⋅ 0ξ⋅ η + 0 2 2 a ⋅b 0
2 2
24⋅ D1 ⋅ η
2
2
b 12⋅ Dy ⋅ ξ ⋅ η
⎟ ⎜ 6ξ ⎟ 64⋅ D ⋅ ξ2⋅ η2 ⋅ ⋅ η⋅ ⎜ xy ⎠ + ⎝
yξ
+
12⋅ D1 ⋅ ξ ⋅ η 2 2
3
+
2
48⋅ Dxy⋅ ξ ⋅ η
36⋅ D1 ⋅ ξ ⋅ η
2 2
a ⋅b
a ⋅b
2 2
a ⋅b
2 2 ⎛ 2 ⋅ D ⋅ η2 2 ⋅ D ⋅ ξ 2 ⎞ ⎛ ⎛ 2 ⋅ D ⋅ η2 2 ⋅ Dy ⋅ ξ ⎞⎟ 1 ⎟ x ⎜ 2 ⎜ 2 ⋅ D1 ⋅ η ⎜ 1 2⋅ η ⋅ + 2⋅ ξ ⋅ + 6 ⋅ ξ⋅ η⋅ + 3 2 2 ⎜ ⎟ ⎜ ⎟ ⎜ 2 2 2 2 2 64⋅ Dxy⋅ ξ ⋅ η xy⋅ ξ ⋅ η a b a b a ⎝ ⎠ + ⎝ ⎠ + ⎝ 2
⋅b
2
a
2
b 3
12⋅ Dy ⋅ ξ ⋅ η
+
12⋅ D1 ⋅ ξ⋅ η
3
+
2
2 2
a ⋅b 48⋅ Dxy⋅ ξ⋅ η
3
b
2
36⋅ Dxy⋅ η
b
+
4
+
2 2
24⋅ D1 ⋅ η 2 2
4
2
+
2
24⋅ Dx ⋅ ξ ⋅ η
24⋅ Dy ⋅ ξ ⋅ η
0 0
0
0
0
⎝
2
4 2
15⋅ a ⋅ b 0 32⋅ D1 2 2
a ⋅b 0
+
2⎞
⎠
2 2
24⋅ D1 ⋅ ξ ⋅ η 2 2
2
7
4
+
4
+
⎤ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ 32⋅ D1 ⎥ ⎥ 2 2 a ⋅b ⎥ ⎥ ⎥ 0 ⎥ ⎥ 2 2⎞ ⎥ ⎛ 32⋅ 5 ⋅ Dy ⋅ a + 12⋅ Dxy⋅ b ⎝ ⎠⎥ ⎥ 2 4 15⋅ a ⋅ b ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥
3
64⋅ Dxy⋅ ξ ⋅ η
72⋅ D1 ⋅ ξ ⋅ η
2 2
a ⋅b
2 2
a ⋅b
a ⋅b
0
0
2⋅ ⎛ 12⋅ Dxy⋅ a + 5 ⋅ Dx ⋅ b
2
0
0
24⋅ D1 ⋅ ξ ⋅ η
a ⋅b
4
a ⋅b 3
4
b 4
+
4
24⋅ Dy ⋅ ξ ⋅ η b
3
2 2
a ⋅b
a ⋅b
a
2 2
a ⋅b
64⋅ Dxy⋅ ξ ⋅ η 2 2
a ⋅b
4
48⋅ Dxy⋅ η 2 2
a ⋅b
⎥ ⎥ ⎥ 0 0 ⎥ ⎥ ⎥ ⎥ 0 0 ⎥ ⎥ ⎥ ⎥ 0 0 ⎥ ⎥ ⎥ ⎥ 0 0 ⎥ ⎥ ⎥ 2 2 ⎛ 20⋅ D ⋅ a + 21⋅ D ⋅ b ⎞ x ⎠ xy ⎥ ⎝ 0 ⎥ 4 2 35⋅ a ⋅ b ⎥ ⎥ 2 2 64⋅ ⎛ 21⋅ Dy ⋅ a + 20⋅ Dxy⋅ b ⎞ ⎝ ⎠ ⎥ 0 ⎥ 2 4 35⋅ a ⋅ b ⎦
0
0
0
0
0
0 2
24⋅ D1 ⋅ η
2
24⋅ Dx ⋅ ξ ⋅ η
2 2
a ⋅b
a
4
16⋅ Dxy⋅ ξ
0
3
2 2
a ⋅b 2
24⋅ Dy ⋅ η b
2
24⋅ D1 ⋅ ξ ⋅ η
4
2 2
a ⋅b 2
72⋅ D1 ⋅ ξ⋅ η
3
72⋅ Dx ⋅ ξ ⋅ η
2 2
a ⋅b 2
a 3
24⋅ D1 ⋅ η
2
24⋅ Dx ⋅ ξ ⋅ η
2 2
a ⋅b
a 2
24⋅ Dy ⋅ ξ⋅ η b
2 2
3
2 2 3
32⋅ Dxy⋅ ξ ⋅ η
+
32
2 2
2
72⋅ D1 ⋅ ξ ⋅ η
4
2
2 2
a ⋅b 2
4
144 ⋅ Dx ⋅ ξ ⋅ η
2 2
72⋅ D1 ⋅ ξ⋅ η
a 3
3
72⋅ Dx ⋅ ξ ⋅ η
2 2
a ⋅b
⎛ 2 ⋅ D ⋅ η2 ⎜ 1
24
a ⋅b
a ⋅b
2
4
a ⋅b
a ⋅b
2
2
32⋅ Dxy⋅ ξ
+
3
144 ⋅ D1 ⋅ ξ ⋅ η
η
2
4
24⋅ D1 ⋅ ξ ⋅ η
4
72⋅ Dy ⋅ η b
4
a 2 ⋅ Dy ⋅ ξ
2⎞
⎟
2
4
⎛ 2 ⋅ D ⋅ η2 ⎜ x
4
2
+
48⋅ Dxy⋅ ξ
5
72⋅ D
2 2
a ⋅b 2 ⋅ D1 ⋅ ξ
2⎞
⎟
2
⎛ 2⋅ D ⎜
2
8⋅ Dxy⋅ ξ ⋅ η
12⋅ η ⋅
3
⎜ ⎜ ⎝
1η a
2 2
a ⋅b 2
yξ
+
2
b b
2
0 b
0
3
4 4
2 η 12⋅ Dxy⋅ η
144 ⋅ D1 ⋅ ξ ⋅ η
2 2
a ⋅b
2
3
144 ⋅ Dy ⋅ ξ ⋅ η
12⋅ Dy ⋅ ξ⋅ η b
4
b 2
0 a
24⋅ Dxy⋅ ξ ⋅ η
2
2 2
a ⋅b
3
2 2
b
2
3
2
2
2 2
3
b
2
2
2 2
a ⋅b 2
4
144 ⋅ Dx ⋅ ξ ⋅ η a
3
3
72⋅ Dx ⋅ ξ ⋅ η
4
2
+
a
4
48⋅ Dxy⋅ ξ
5
2 2
a ⋅b
2 2 2 2 ⎛ ⎛ 2 ⋅ Dy ⋅ ξ ⎞⎟ 2 ⋅ D1 ⋅ ξ ⎞⎟ ⎟ 2 ⎜ 2 ⋅ D1 ⋅ η 2 ⎜ 2 ⋅ Dx ⋅ η 12⋅ ξ ⋅ η⋅ + + 4 ⎟ 48⋅ D ⋅ ξ⋅ η3 12⋅ η ⋅ ⎜ ⎜ 2 2 ⎟ 2 2 ⎟ xy a b a b ⎠ + ⎝ ⎠ ⎝ ⎠ + 64⋅ Dxy⋅ ξ 2
+
2 2
2⎞
2 2
4
+
2
a ⋅b
a ⋅b η
3
32⋅ Dxy⋅ ξ ⋅ η
72⋅ D1 ⋅ ξ ⋅ η
2 2
a ⋅b
2 2
a ⋅b
4
72⋅ D1 ⋅ ξ⋅ η
2 2
+
4
a ⋅b
2 2
2
32⋅ Dxy⋅ ξ
a ⋅b
a ⋅b 2
2
3
2
36⋅ Dxy⋅ ξ ⋅ η
4
4
24⋅ D1 ⋅ ξ ⋅ η
4
2 2
+
2 2
a ⋅b
24⋅ Dx ⋅ ξ ⋅ η
144 ⋅ D1 ⋅ ξ ⋅ η
a ⋅b
64⋅
3
a
72⋅ Dy ⋅ η b
2 ⋅ Dy ⋅ ξ
3
72⋅ Dx ⋅ ξ ⋅ η
3
2⋅ D1 ⋅ ξ ⋅ η
+
2 2
a ⋅b
2
24⋅ Dy ⋅ ξ⋅ η
4
2
2 2
a ⋅b
3
+
144⋅
2 64⋅ Dxy⋅ ξ ⋅ η 24⋅ D1 ⋅ ξ ⋅ η
2 2
24⋅ Dxy⋅ η
6⋅ Dy ⋅ ξ ⋅ η
η
2 2
a ⋅b
a ⋅b
a ⋅b
b
6
+
3
4
3 64⋅ Dxy⋅ ξ 16⋅ Dxy⋅ ξ
a
24⋅ D1 ⋅ η
2 2
+
2
24⋅ Dx ⋅ ξ ⋅ η
2 2
a ⋅b ⋅η
3
4
a ⋅b
2
0
3
a
2
4
144 ⋅ D1 ⋅ ξ ⋅ η
2
24⋅ Dy ⋅ η b
2 2
2 2
4
48
0
a ⋅b
144 ⋅ Dx ⋅ ξ ⋅ η
4
2 0
a ⋅b
2 2
a ⋅b +
+
144 ⋅ D1 ⋅ ξ ⋅ η
3
4
3
48⋅ Dxy⋅ ξ ⋅ η
2
2
72⋅ D1 ⋅ ξ⋅ η
2
2
2 2
6⋅ D1 ⋅ ξ ⋅ η
η
2 2
a ⋅b
2 2
2 2
a ⋅b
2
a ⋅b 2
2
a ⋅b
24⋅ D1 ⋅ η
4
b
b
3
0
2 2
a ⋅b
a
2
⎟ 2⎜ 12⋅ ξ ⋅ η ⋅ ⎟ 64⋅ D ⋅ ξ4⋅ η ⎜ xy ⎠ + ⎝
1ξ
+
72⋅ D1 ⋅ ξ ⋅ η
0 0
144 ⋅ Dy ⋅ η
12⋅ D1 ⋅ ξ⋅ η
xη
a
72⋅ Dy ⋅ ξ⋅ η
0
2
⎟ 12⋅ ξ2⋅ η⋅ ⎜ ⎟ ⎜ ⎠ ⎝
36⋅ Dy ⋅ ξ ⋅ η
2
b
2
72⋅ Dy ⋅ ξ⋅ η
a 3
2
2 2
a ⋅b 3
72⋅ D1 ⋅ ξ ⋅ η
2
3 2
+
48⋅ Dxy⋅ ξ ⋅ η
+ b 2⋅ Dy ⋅ ξ ⋅ η b η
4
b
3
b 3
+
48⋅ Dxy⋅ ξ ⋅ η 2 2
5
2
+
72⋅ Dy ⋅ ξ ⋅ η b
4
4
2
4
2 2
2
144 ⋅ Dy ⋅ ξ ⋅ η 4
3
a ⋅b
144 ⋅ D1 ⋅ ξ ⋅ η
b
a ⋅b
144 ⋅ D1 ⋅ ξ ⋅ η
3
a ⋅b 3
2 2
a ⋅b
2 2
a ⋅b η
2
+
2 2
144 ⋅ Dy ⋅ η
4
2
4
4
144 ⋅ Dx ⋅ ξ ⋅ η a
4
+
4 3
144 ⋅ D1 ⋅ ξ ⋅ η 2 2
a ⋅b
2
64⋅ Dxy⋅ ξ
6
2 2
a ⋅b 3
3 3
+
64⋅ Dxy⋅ ξ ⋅ η 2 2
a ⋅b
0 0 0 24⋅ D1 ⋅ ξ⋅ η
2
2 2
a ⋅b
16⋅ Dxy⋅ η
3
2 2
a ⋅b
24⋅ Dy ⋅ ξ⋅ η b
2
4 2
72⋅ D1 ⋅ ξ ⋅ η
2
2 2
a ⋅b ⋅ D1 ⋅ ξ⋅ η
3
32⋅ Dxy⋅ ξ⋅ η
+
2 2
2 2
a ⋅b
2⋅ Dxy⋅ η
3
a ⋅b 4
2
24⋅ Dy ⋅ ξ ⋅ η
+
2 2
a ⋅b
b
72⋅ Dy ⋅ ξ⋅ η b
2
4
3
4 3
144 ⋅ D1 ⋅ ξ ⋅ η
2
2 2
a ⋅b 2
D1 ⋅ ξ ⋅ η 2 2
3
2
+
48⋅ Dxy⋅ ξ ⋅ η
a ⋅b
D1 ⋅ η
2
2 2
a ⋅b 2 ⋅ Dy ⋅ ξ
2⎞
⎟
3
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
⎟ ⎟ 64⋅ D ⋅ ξ⋅ η4 ⎥⎥ xy a b ⎠ + ⎥ 2 2 2 b a ⋅b ⎥ ⎥ 5 2 3 0 8⋅ Dxy⋅ η 72⋅ Dy ⋅ ξ ⋅ η ⎥ + 0 ⎥ 2 2 4 a ⋅b b ⎥ 0 ⎥ 4 2 ⎥ 144 ⋅ Dy ⋅ ξ⋅ η 24⋅ D1 ⋅ ξ⋅ η ⎥ 4 2 2 b ⎥ a ⋅b ⎥ 3 3 3 3 3 ⎥ D1 ⋅ ξ ⋅ η 64⋅ Dxy⋅ ξ ⋅ η 16⋅ Dxy⋅ η + ⎥ 2 2 2 2 2 2 a ⋅b a ⋅b ⎥ a ⋅b ⎥ 6 2 4 2 ⋅ Dxy⋅ η 144 ⋅ Dy ⋅ ξ ⋅ η 24⋅ Dy ⋅ ξ⋅ η ⎥ + ⎥ 2 2 4 4 a ⋅b b ⎦ b 1η 2
+
yξ 2
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 2 2 72⋅ D1 ⋅ ξ ⋅ η ⎥ ⎥ 2 2 a ⋅b ⎥ 3 3 ⎥ 24⋅ D1 ⋅ ξ⋅ η 32⋅ Dxy⋅ ξ⋅ η ⎥ + ⎥ 2 2 2 2 a ⋅b a ⋅b ⎥ 4 2 2 ⎥ 32⋅ Dxy⋅ η 24⋅ Dy ⋅ ξ ⋅ η ⎥ + 2 2 4 ⎥ a ⋅b b ⎥ 3 ⎥ 72⋅ Dy ⋅ ξ⋅ η ⎥ 4 ⎥ dξ dη b ⎥ 3 2 ⎥ 144 ⋅ D1 ⋅ ξ ⋅ η ⎥ 2 2 ⎥ a ⋅b ⎥ 2 3 2 3 ⎥ 72⋅ D1 ⋅ ξ ⋅ η 48⋅ Dxy⋅ ξ ⋅ η ⎥ + 2 2 2 2 ⎥ a ⋅b a ⋅b ⎥ ⎛ 2 ⋅ D ⋅ η2 2 ⋅ D ⋅ ξ 2 ⎞ ⎥ y ⎟ 1 2⎜ 12 ⋅ ξ η ⋅ ⋅ + 4 4⎥ ⎜ 2 2 ⎟ ⋅η 64⋅ Dxy⋅ ξ⋅ η ⎥ a b ⎝ ⎠ + ⎥ 2 2 2 b a ⋅b ⎥ ⎥ 5 2 3 72⋅ Dy ⋅ ξ ⋅ η 48⋅ Dxy⋅ η ⎥ + ⎥
+
2 2
a ⋅b
b
144 ⋅ Dy ⋅ ξ⋅ η b 3
144 ⋅ D1 ⋅ ξ ⋅ η
64⋅ Dxy⋅ η 2 2
a ⋅b
3
+
2 2
4
4
3
a ⋅b
4
64⋅ Dxy⋅ ξ ⋅ η 2 2
a ⋅b 6
2
+
144 ⋅ Dy ⋅ ξ ⋅ η b
4
4
3
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
Transformation expression: Substituting variables ξ and η by their nodal 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 extreme positions +1 and -1 for both ξ and η This {a} coefficients are applied to the curvature equations to calculate the plate generic strains.
ξ x
(
≡
1 a
f ( ξ , η) := 1 ξ η ξ
η
d ξ = dx
2
ξ⋅ η η
(
2
ξ
3
≡
y
2
ξ η ξ⋅ η
2
η
3
ξ
4
3
1
d η = dy
b
2
ξ ⋅η ξ ⋅η
2
ξ⋅ η
3
η
4
4
ξ ⋅ η ξ⋅ η
A := a1 a2 a3 a4 a5 a6 a7 a8 a8 a10 a11 a12 a13 a14 a15 a16 a17
)
T
w( ξ , η) := f ( ξ , η) ⋅ A
⎛ a1 ⎞ ⎜ ⎟ ⎜ a2 ⎟ ⎜ ⎟ ⎜ a3 ⎟ ⎜a ⎟ ⎜ 4⎟ ⎜ a5 ⎟ ⎜ ⎟ ⎜ a6 ⎟ ⎜a ⎟ ⎜ 7⎟ ⎜ a8 ⎟ ⎜ ⎟ ( a1 a2 a3 a4 a5 a6 a7 a8 a8 a10 a11 a12 a13 a14 a15 a16 a17 )T → ⎜ a8 ⎟ ⎜a ⎟ ⎜ 10 ⎟ ⎜ a11 ⎟ ⎜ ⎟ ⎜ a12 ⎟ ⎜a ⎟ ⎜ 13 ⎟ ⎜ a14 ⎟ ⎜ ⎟ ⎜ a15 ⎟ ⎜ ⎟ ⎜ a16 ⎟ ⎜a ⎟ ⎝ 17 ⎠
4
)
Here we are multiplying f ⋅ AT
(
⎛ a1 ⎞ ⎜ ⎟ ⎜ a2 ⎟ ⎜ ⎟ ⎜ a3 ⎟ ⎜a ⎟ ⎜ 4⎟ ⎜ a5 ⎟ ⎜ ⎟ ⎜ a6 ⎟ ⎜a ⎟ ⎜ 7⎟ ⎜ a8 ⎟ ⎜ ⎟ 2 2 3 2 2 3 4 3 2 2 3 4 4 4 1 ξ η ξ ξ ⋅ η η ξ ξ η ξ ⋅ η η ξ ξ ⋅ η ξ ⋅ η ξ ⋅ η η ξ ⋅ η ξ ⋅ η ⋅ ⎜ a8 ⎟ ⎜a ⎟ ⎜ 10 ⎟ ⎜ a11 ⎟ ⎜ ⎟ ⎜ a12 ⎟ ⎜a ⎟ ⎜ 13 ⎟ ⎜ a14 ⎟ ⎜ ⎟ ⎜ a15 ⎟ ⎜ ⎟ ⎜ a16 ⎟ ⎜a ⎟ ⎝ 17 ⎠
)
4
4
3
3
2
2
2
2
4
3
a16⋅ ξ ⋅ η + a11⋅ ξ + a12⋅ ξ ⋅ η + a7 ⋅ ξ + a13⋅ ξ ⋅ η + a8 ⋅ ξ ⋅ η + a4 ⋅ ξ + a17⋅ ξ ⋅ η + a14⋅ ξ ⋅ η + + ⎛ a8 ⋅ ξ⋅ η + a5 ⋅ ξ⋅ η + a2 ⋅ ξ + a15⋅ η + a10⋅ η + a6 ⋅ η + a3 ⋅ η + a1⎞
⎝
2
4
3
2
The latter is the Pascal Polinomial w(ξ,η)
⎠
...
f ( 1 , 1) → ( 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 )
q1
f ( 1 , −1 ) → ( 1 1 −1 1 −1 1 1 −1 1 −1 1 −1 1 −1 1 −1 1 ) q2 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 we have done it before, but we will first take the "x" derivative as the paper does .
(
2 2 3 2 2 3 4 3 d f ( ξ , η) → 0 0 1 0 ξ 2 ⋅ η 0 ξ 2 ⋅ ξ⋅ η 3 ⋅ η 0 ξ 2 ⋅ ξ ⋅ η 3 ⋅ ξ⋅ η 4 ⋅ η ξ 4 ⋅ ξ ⋅ η dη
Ξ( ξ , η) :=
1 d f ( ξ , η) ⋅ b dη
Ξ( 1 , 1 ) → ⎛⎜ 0 0
1
Ξ( 1 , −1 ) → ⎛⎜ 0 0
1
Ξ( −1 , 1 ) → ⎛⎜ 0 0
1
⎝
⎝
⎝
Ξ( −1 , −1 ) → ⎛⎜ 0 0
⎝
b
0
0
b
1
2
b
b
1
−
b
0 −
b 1 b
0
2 b
1
2
b
b
0 −
1 b
−
0
0
2 b
1
2
3
b
b
b
1 b 1 b
0
0
−
2
3
b
b
−
2
3
b
b
⎞ ⎟ b b b b b b⎠ 1
0
2
2
3
b
b
b
4
1
2
3
b
b
−
1
2
b
b
−
1
−
b
0 −
1
3
0 −
1 b
−
2 b
4
4
1
b
b
3
4
1
b
b
b
−
3 b
−
⎞ ⎟ b⎠
−
4
−
4
⎞ ⎟ b⎠
⎞ ⎟ b b b⎠ 4
1
4
)
(
2 2 3 2 2 3 3 4 d f ( ξ , η) → 0 1 0 2 ⋅ ξ η 0 3 ⋅ ξ 2 ⋅ ξ⋅ η η 0 4 ⋅ ξ 3 ⋅ ξ ⋅ η 2 ⋅ ξ⋅ η η 0 4 ⋅ ξ ⋅ η η dξ
1 d f ( ξ , η) ⋅ a dξ
Θ( ξ , η) :=
Θ( 1 , 1 ) → ⎛⎜ 0
1
Θ( 1 , −1 ) → ⎛⎜ 0
1
Θ( −1 , 1 ) → ⎛⎜ 0
1
Θ( −1 , −1 ) → ⎛⎜ 0
1
⎝
⎝
⎝
⎝
⌠ ⎮ ⌡
a
a
a
2
−
a
0
1 a 1
a
a
0 −
2
−
1
Π( 1 ) →
1
2
⎝
f ( ξ , η ) dξ ⋅
−1
2
a
3
2
1
a
a
a
0
0
1
2
0
−
2
1
a
a
−
2
1
a
a
3
2
1
a
a
a
3 a 3 a
0
a
0 2⋅ η
3
2⋅ η
0
3
4
3
2
1
a
a
a
a
3
2
a
a
−
0
4
−
a
0 −
4
3
a
a
0 −
4
−
0 2⋅ η
a
3 2
5
3 a
0
0
−
a
a
a 1
a
a 2
−
a
2⋅ η
1
1
2
−
4
⎞ ⎟ ⎠
0 −
4
1
a
a
0 −
4
1
a
a
1 a
0
2
0 2⋅ η
3
⎞ ⎟ ⎠ ⎞ ⎟ ⎠
4
1
a
a
⎞ ⎟ ⎠
4 2⋅ η
5
1 2aa
⎛
⎛ 1 0 −1 1 0 1 0 − 1 0 −1 1 0 1 0 1 − 1 0⎞ ⎜ ⎟ 5⋅ a a 3⋅ a a 3⋅ a a 5⋅ a 3⋅ a a ⎝a ⎠
f ( ξ , η ) dη → ⎜ 2 2 ⋅ ξ 0 2 ⋅ ξ
⎝
2
0
2 3
2⋅ ξ
3
0
2⋅ ξ 3
0 2⋅ ξ
4
0
2⋅ ξ 3
2
0
2 5
0
⎞ ⎟ 5 ⎠
2⋅ ξ
⎞
0⎟
⎛ 1 0 1 1 0 1 0 1 0 1 1 0 1 0 1 1 0⎞ ⎜ ⎟ a 3⋅ a a 3⋅ a a 5⋅ a 3⋅ a a 5⋅ a ⎝a ⎠
Π( −1 ) →
−1
a
2
0 −
f ( ξ , η ) dξ → ⎜ 2 0 2 ⋅ η
⌠ Π( η) := ⎮ ⌡
1
a
0
⎛
1
−1
⌠ ⎮ ⌡
a
0
⎠
)
Ψ( ξ) :=
⎛⎜ ⌠ 1 ⎞⎟ ⎮ f ( ξ , η ) dη ⎟ 2bb ⎜ ⌡− 1 ⎝ ⎠ 1
⌠ ⎮ ⌡
1
f ( ξ , η ) dη
−1
Ψ( ξ) :=
2⋅ b
Ψ( 1 ) →
Ψ( −1 ) →
⌠ ⎮ ⌡
1
⌠ ⎮ ⌡
⎛1 1 0 1 0 1 1 0 1 0 1 0 1 0 1 0 1 ⎞ ⎜ ⎟ b 3⋅ b b 3⋅ b b 3⋅ b 5⋅ b 5⋅ b ⎠ ⎝b b
⎛ 1 −1 0 1 0 1 −1 0 − 1 0 1 0 1 0 1 0 − 1 ⎞ ⎜ ⎟ b 3⋅ b 3⋅ b b 3⋅ b 5⋅ b 5⋅ b ⎠ b ⎝b b
1
f ( ξ , η ) dξ dη → ⎛⎜ 1 0 0
−1 −1
⎝
4
1 3
0
1 3
0 0 0 0
1 5
1
0
9
0
1 5
0 0 ⎞⎟
⎠
To find later the strains, first we find: −
d
2
dξ
−
d
2
2
dη
(
f ( ξ , η) → 0 0 0 −2 0 0 −6 ⋅ ξ −2 ⋅ η 0 0 −12⋅ ξ
2
(
2
−6 ⋅ ξ⋅ η −2 ⋅ η
f ( ξ , η) → 0 0 0 0 0 −2 0 0 −2 ⋅ ξ −6 ⋅ η 0 0 −2 ⋅ ξ
(
2
2
2
0 0 −12⋅ ξ ⋅ η 0
−6 ⋅ ξ ⋅ η −12⋅ η
2
0 −12⋅ ξ⋅ η
⎛ d d f ( ξ , η) ⎞ → 0 0 0 0 2 0 0 4 ⋅ ξ 4 ⋅ η 0 0 6 ⋅ ξ2 8 ⋅ ξ ⋅ η 6 ⋅ η2 0 8 ⋅ ξ3 8 ⋅ η3 ⎟ ⎝ dξ dη ⎠
2⎜
)
2
) )
Constructing M. (Coordinates Transformation Expression) The [M] matrix is constructed from {q} = [M]*{a} with the four values f (q1...q4); four Θ (q5...q8); four Ξ (q9... q12); two Π (q13,q14); two Ψ (q15,q16); and q17. All of them are qi vectors or 1x17 matrixes that we will use for the transformation: ⎛1 1 1 ⎜ ⎜ 1 1 −1 ⎜ 1 −1 1 ⎜ 1 −1 −1 ⎜ ⎜0 1 0 ⎜ a ⎜ 1 0 ⎜0 a ⎜ ⎜ 1 ⎜0 a 0 ⎜ ⎜0 1 0 a ⎜ ⎜ 1 ⎜0 0 b ⎜ ⎜0 0 1 ⎜ b ⎜ ⎜0 0 1 b ⎜ ⎜ 1 ⎜0 0 b ⎜ ⎜1 0 1 ⎜a a ⎜1 1 0 − ⎜ a ⎜a ⎜1 1 ⎜b b 0 ⎜ ⎜ 1 −1 0 ⎜b b ⎜ ⎜1 0 0 ⎝
1
1
1
1
1
1
1
1
1
1
1
1
1
1
−1
1
1
−1
1
−1
1
−1
1
−1
1
−1
1
−1
1
−1
1
−1
1
1
−1
1
−1
1
1
1
1
1
−1
−1
−1
−1
1
1
1
1
1
−1
2
1
3
2
1
4
3
2
1
a
a
a
a
a
a
a
a
a
2
−1
3
−2
1
4
−3
2
−1
a
a
a
a
a
a
a
a
a
2
1
2
1
a
a
−
a
a
3
2
1
2
a
a
a
−
−
1
2
3
b
b
b
−
2
3
b
b
−
2
3
b
b
1
2
3
b
b
b
−
2
1
a
a
−
2
−
a
0 0
1 a
0 0 0 0
1
2
b
b
1 b
−
2 b
0
−
1
2
b
b
0
−
1
1 3a 1 3a 1 b 1 b 1 3
b
0 0 0 0 0
−
2 b
1 a 1 a
3 a
0
b 1
0
b
0 0 0 1
3⋅ b
b
1
−1
3⋅ b
b
3
1
0
1
1
−
0
1
0
3a −
1
0
3a 0 0 0
1 3⋅ b −
1 3⋅ b 0
0 0 0
−
4
3
a
a
0
−
4
−
a
0 0
0
b 1 b 1 5
4
0
a
b
b
b
b
1 b
−
2
3
b
b
1
1
a
b
−
5a
4
1
0
a
−
4
b
1
a
3
b
1
a
0
−4
2
2
5a
1
0
a
1
1
a
0
a
−
1
0
a
0
1
−
3
4
0
b
0 0 0 0 0
−
2 b
1 3a 1 3a 1 3⋅ b 1 3⋅ b 1 9
−
4
1
b
b
−
3
4
1
b
b
b
−
3 b
0 0 0 0 0
−
4
1
b
b
1
1
a
5a
1 a 1 5⋅ b 1 5⋅ b 1 5
−
1 5a 0 0 0
⎞ ⎟ ⎟ −1 ⎟ −1 ⎟ ⎟ 1 ⎟ a ⎟ 1 ⎟ ⎟ a ⎟ 1 ⎟ a ⎟ ⎟ 1 ⎟ a ⎟ 4 ⎟ ⎟ b ⎟ 4 ⎟ − b ⎟ ⎟ 4 − ⎟ b ⎟ 4 ⎟ b ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 1 ⎟ 5⋅ b ⎟ ⎟ 1 ⎟ − 5⋅ b ⎟ ⎟ 0 ⎟ ⎠ 1 1
Mathcad is ordered to create the inverse of the former matrix M: ⎛1 1 1 ⎜ ⎜ 1 1 −1 ⎜ 1 −1 1 ⎜ 1 −1 −1 ⎜ ⎜0 1 0 ⎜ a ⎜ 1 0 ⎜0 a ⎜ ⎜ 1 ⎜0 a 0 ⎜ ⎜0 1 0 a ⎜ ⎜ 1 ⎜0 0 b ⎜ ⎜0 0 1 ⎜ b ⎜ ⎜0 0 1 b ⎜ ⎜ 1 ⎜0 0 b ⎜ ⎜1 0 1 ⎜a a ⎜1 1 0 − ⎜ a a ⎜ ⎜1 1 ⎜b b 0 ⎜ ⎜ 1 −1 0 ⎜b b ⎜ ⎜1 0 0 ⎝
1
1
1
1
1
1
1
1
1
1
1
1
1
1
−1
1
1
−1
1
−1
1
−1
1
−1
1
−1
1
−1
1
−1
1
−1
1
1
−1
1
−1
1
1
1
1
1
−1
−1
−1
−1
1
1
1
1
1
−1
2
1
3
2
1
4
3
2
1
a
a
a
a
a
a
a
a
a
2
−1
3
−2
1
4
−3
2
−1
a
a
a
a
a
a
a
a
a
2
1
1
a
−
2
a
a
a
3
2
1
2
a
a
a
−
−
1
2
3
b
b
b
−
2
3
b
b
−
2
3
b
b
1
2
3
b
b
b
−
2
1
a
a
−
2
−
a
0 0
1 a
0 0 0 0
1
2
b
b
1 b
−
2 b
0
−
1
2
b
b
0
−
1
1 3a 1 3a 1 b 1 b 1 3
b
0 0 0 0 0
−
2 b
1 a 1 a
3 a
0
b 1
0
b
0 0 0
1
1 b
1
−1
3⋅ b
b
3
1
0
3⋅ b
1
−
0
1
0
3a −
1
0
3a 0 0 0
1 3⋅ b −
1 3⋅ b 0
0 0 0
−
4
3
a
a
0
−
4
−
a
0 0
0
b 1 b 1 5
4
0
a
b
b
b
b
1 b
−
2
3
b
b
1
1
a
b
−
1
4
1
0
5a
−
4
b
a
a
3
b
1
a
0
−4
2
2
5a
1
0
a
1
1
a
0
a
−
1
0
a
0
1
−
3
4
0
b
0 0 0 0 0
−
2 b
1 3a 1 3a 1 3⋅ b 1 3⋅ b 1 9
−
4
1
b
b
−
3
4
1
b
b
b
−
3 b
0 0 0 0 0
−
4
1
b
b
1
1
a
5a
1 a 1 5⋅ b 1 5⋅ b 1 5
−
1 5a 0 0 0
⎞ ⎟ 1 ⎟ −1 ⎟ −1 ⎟ ⎟ 1 ⎟ a ⎟ 1 ⎟ ⎟ a ⎟ 1 ⎟ a ⎟ ⎟ 1 ⎟ a ⎟ 4 ⎟ ⎟ b ⎟ 4 ⎟ − b ⎟ ⎟ 4 − ⎟ b ⎟ 4 ⎟ b ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 1 ⎟ 5⋅ b ⎟ ⎟ 1 ⎟ − 5⋅ b ⎟ ⎟ 0 ⎟ ⎠ 1
−1
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
−
3
3
3
32
32
32
3
3
a
32
32
32
3
3
3
8
8
8
8
3 32
−
2
−
8
−
1
a
a
8
32
32
1
1
2
2
3
3
8
8
8
1
1
8
8
15
15
16
16
15 16 1 8 −
−
1
3
− −
1
−
15 32
−
15 16
− −
3⋅ a
3⋅ a
3⋅ a
16
16
16
a
a
a
a
8
8
−
8
8
16
8
8
8
8
3⋅ a
3⋅ a
3⋅ a
16
16
16
0
0
0
0
0
0
0
0
15
5⋅ a
5⋅ a
32
32
32
1
a
8
8
−
−
15
16 16 1 8
9
16
16
16
1
1
8
8
−
−
1 8
−
0
0
0
0
32
32
8
0
32 15
8
a
0
32
32
8
−
0
5⋅ a
15
a
32
0
15
32
a
5⋅ a
0
15
15
32
−
0
−
5⋅ a 32 0
−
5⋅ a
5⋅ a
32
32
0
0
b 8
− −
− −
−
b 8
b
b
32
32
b 32
−
0
16
0
−
32
5⋅ a
−
3⋅ a
0
15
15
−
−
0
−
32
−
8
9
−
32
a
9
32
32
32
a
8
15
a
a
8
32
32
a
−
a
15
−
b
8
−
8
1
1
15
a
a
a
8
0
1
−
−
b 32
0
8
32
a
a 32
−
8
3⋅ a
−
−
−
0
−
−
−
a
a 32
0
1
15
−
−
b 8
−
−
b 32
b
b
32
32
b
−
−
8
16
0
b
b
b
8
8
8
3⋅ b
0
0
0
3⋅ b
3⋅ b
3⋅ b
16
16
16
b
b
b
b
8
8
8
8
0
0
0
0
0
0
0
0
0
0
0
0
b
b
8
8
−
b 8
−
b 8
5⋅ b
32
32
0
0
0
5⋅ b
5⋅ b
32
32
32
−
15⋅ a
15⋅ a
8
8
0
0
0
0
15⋅ a
15⋅ a
16
16
0
0
16
− −
−
3⋅ b
5⋅ b
5⋅ b
8
0
5⋅ b 32
−
9⋅ a 8
−
9⋅ a 8
0
0
0
0
15⋅ a 16 0
−
15⋅ a 16 0
− −
0
4
8
0
16
9⋅ b
8
0
16
3⋅ a
16 0
−
−
3⋅ b
15⋅ b
16
16 0
−
15⋅ a
9⋅ a
0
32
−
9⋅ a
0
5⋅ b
4
0
3⋅ b
16
0
3⋅ a
16
0
16
32
−
3⋅ a
0
3⋅ b
−
−
0
8
3⋅ b
5⋅ b
16
15⋅ a
0
−
3⋅ a
b
0 −
−
0 −
3⋅ b 4
−
0 0 −
15⋅ b 8 0 0 0
−
9⋅ b 8
−
0 15⋅ b 16 0 15⋅ b 16
−
T
−1 Mathcad is ordered to create M
: ⎛ −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 ⎜
−
1
−
3
3
3
32
32
32
3
3
a
32
32
32
3
3
3
8
8
8
8
3 32
−
2
−
8
−
1
a
a
8
32
32
1
1
2
2
3
3
8
8
8
1
1
8
8
15
15
16
16
15 16 1 8 −
−
1
3
− −
1
−
15 32
−
15 16
− −
32
32
3⋅ a
3⋅ a
3⋅ a
16
16
16
a
a
a
a
8
8
−
8
8
16
−
32
a
a
8
8
8
3⋅ a
3⋅ a
3⋅ a
16
16
16
0
0
0
0
0
0
0
0
15
5⋅ a
5⋅ a
32
32
32
1
a
8
8
−
−
15
16 16 8
9
9
9
16
16
16
1
1
8
8
−
32
32
a
a
8
a
8
15
8
a
−
8
−
8
15
−
1 8
15
−
5⋅ a 32
a
a
8
8
−
5⋅ a 32
−
a 8
0
0
0
0
0
0
0
0
0
15
15
5⋅ a
32
32
32
−
5⋅ a 32
−
5⋅ a
5⋅ a
32
32
b 8
− −
− −
−
b 8
b
b
32
32
b 32
−
0
16
0
−
−
3⋅ a
0
15
32
−
−
0
−
−
b
−
8
a
a
8
32
a
a
−
8
8
32
32
1
1
15
32
0
1
−
b
0
1
32
a
−
a
3⋅ a
−
32
−
0
−
−
−
a
0
1
15
−
−
b 8
−
−
b 32
b
b
32
32
b
−
−
8
16
0
b
b
b
8
8
8
3⋅ b
4
0 −
−
8
8
8
0
0
0
0
0
0
3⋅ b
3⋅ b
3⋅ b
16
16
16
b
b
b
b
8
8
8
8
0
0
0
0
0
0
0
0
0
0
0
0
b
b
8
8
−
b 8
5⋅ b
5⋅ b
32
32
0
0
−
15⋅ a
15⋅ a
8
8
3⋅ b
0
0
0
0
15⋅ a
15⋅ a
16
16
0
0
16
− −
−
b 8
5⋅ b 32 0
−
9⋅ a 8
−
9⋅ a 8
0
0
0
0
15⋅ a 16
−
15⋅ a 16
− −
0
4
16 0
16
9⋅ b
16 0
16
3⋅ a
9⋅ a
0
3⋅ b
15⋅ b
16
9⋅ a
0
−
15⋅ a
3⋅ b
16
0
3⋅ a
16
0
16
−
−
3⋅ a
0
3⋅ b
32
−
0
8
3⋅ b
5⋅ b
16
15⋅ a
0
−
3⋅ a
b
0 −
−
0 −
3⋅ b 4
−
0 0 −
15⋅ b 8 0 0 0
−
9⋅ b 8 0
15⋅ b 16 0
−
⎜ 15 15 15 − ⎜− ⎝ 32 32 32
15
0
32
0
0
5⋅ b
0
−
32
5⋅ b
−
32
5⋅ b
5⋅ b
32
32
0
15⋅ b
0
−
16
T
−1 This is the M :
3 ⎛ −1 − 3 − ⎜ 32 32 ⎜ 8 3 ⎜ −1 − 3 ⎜ 8 32 32 ⎜ 3 3 ⎜ −1 − 32 32 ⎜ 8 ⎜ 1 3 3 ⎜ −8 32 32 ⎜ a a ⎜ a − ⎜ 32 8 32 ⎜ a a a − − ⎜ 32 32 8 ⎜ ⎜ a a a − ⎜ − 32 − 8 32 ⎜ a a ⎜−a − 8 32 ⎜ 32 ⎜ b b b − ⎜ 8 32 ⎜ 32 b ⎜−b −b − ⎜ 32 32 8 ⎜ b b ⎜ b − − 32 8 ⎜ 32 ⎜ b b b − ⎜ − 32 32 8 ⎜ 15⋅ a ⎜ − 3⋅ a 0 ⎜ 16 16 ⎜ 3⋅ a 15⋅ a − 0 ⎜− 16 16 ⎜ ⎜ 3 ⋅ b 15⋅ b 0 ⎜ − 16 16 ⎜ ⎜ − 3 ⋅ b − 15⋅ b 0 16 ⎜ 16 ⎜ 9 0 0 ⎜ ⎝ 4
3
1
3
8
2
8
3 8 3 8
− −
15
15
8
16
16
−
1
−
1
3
2
8
−
1
3
1
15
2
8
8
16
8
1
3
1
8
2
8
8
3⋅ a 16
−
a
a
16
8 −
0
8
3⋅ a
16
0
a
0
8
3⋅ a
a
16
8
0 3⋅ b
0
−
b
0
−
b
3⋅ b
8
16
0 0
−
1
3
3⋅ a
−
−
3⋅ a 4 3⋅ a 4
9⋅ b 8 9⋅ b 8 −
9 4
8
b 8
−
−
16
3⋅ b 16
b
3⋅ b
8
16
0 0
9⋅ a 8 9⋅ a 8
0
−
0
−
0
3⋅ b 4 3⋅ b
−
4 9 4
a 8
−
− −
1
16
16
8
15 16
16
a
3⋅ a
8
16
0
0
0
0
0
0
0 0
−
0
0
8
−
15
1
9
1
32
8
16
8
−
15
0
0
0
−
0
0
−
3⋅ b
b
16
8
3⋅ b
b
16
8
3⋅ b
b
16
8
3⋅ b
b
16
8
0
0
0
0
−
−
8
0
16
0
15⋅ a
0
8
0
8
0
32
0
15⋅ a
0
1
8
16 0
9
16
3⋅ a
0
1
1
16
−
15⋅ b 8
15⋅ b 8 0
16
15
15
8
8
1
−
−
16
1
9
15
15
−
1
−
−
3⋅ a
3⋅ a
8
8
15
8
−
1
15
a
a
−
32
32
−
−
1
9
8
16
5⋅ a
a
32
8
5⋅ a 32
−
a 8
5⋅ a
a
32
8
5⋅ a 32
−
a 8
−
−
8
1 8
15
−
15
15
32
32
−
15
−
15
15
32
32
32
32
0
0
0
−
0
0
0
−
0
0
0
b
5⋅ b
8
32
0
0
0
0
0
0
0
−
b
5⋅ b
8
32
0
0
0
−
b
16 15⋅ a 16
−
0
−
0
0
0
−
0
0
0
−
0
0
0
9⋅ a 8 9⋅ a 8 9⋅ b 8 9⋅ b 8 9 4
8
8
−
−
0 0 0
15⋅ b 16 15⋅ b 16 0
32 5⋅ a 32
0 0 0 0
32
0
5⋅ a
32
5⋅ b
0
32
5⋅ a
32
0
15
32
5⋅ b
0
32
5⋅ a
0
0
0
−
0
0
15⋅ a
−
0
b
15
−
15⋅ a 16 −
15⋅ a 16 0 0 0
Shape functions of the plate bending elements Nj are obtained multiplying the column Nij of the former matrix by f(ξ,η) transpose. We must construct |f| transpose |f| is a 1x17 row matrix defined before, whose transpose is this 17X1 column matrix: ⎛ 1 ⎞ ⎜ ξ ⎟ ⎜ ⎟ ⎜ η ⎟ ⎜ 2 ⎟ ⎜ ξ ⎟ ⎜ ξ⋅ η ⎟ ⎜ 2 ⎟ ⎜ η ⎟ ⎜ 3 ⎟ ⎜ ξ ⎟ ⎜ 2 ⎟ ⎜ ξ ⋅η ⎟ ⎜ 2⎟ ξ⋅ η ⎟ T f ( ξ , η) → ⎜ ⎜ 3 ⎟ ⎜ η ⎟ ⎜ ξ4 ⎟ ⎜ ⎟ ⎜ ξ3⋅ η ⎟ ⎜ ⎟ ⎜ ξ2⋅ η2 ⎟ ⎜ 3⎟ ⎜ ξ⋅ η ⎟ ⎜ 4 ⎟ ⎜ η ⎟ ⎜ 4 ⎟ ⎜ ξ ⋅η ⎟ ⎜ 4⎟ ⎝ ξ⋅ η ⎠
We will find shape functions with the former matrix multiplying it by f transpose: 3 ⎛ −1 − 3 − ⎜ 32 32 ⎜ 8 3 ⎜ −1 − 3 ⎜ 8 32 32 ⎜ 3 3 ⎜ −1 − 32 8 32 ⎜ ⎜ 1 3 3 ⎜ −8 32 32 ⎜ a a ⎜ a − ⎜ 32 8 32 ⎜ a a a − − ⎜ 32 8 ⎜ 32 ⎜ a a a − ⎜ − 32 − 8 32 ⎜ a a ⎜−a − 8 32 ⎜ 32 ⎜ b b b − ⎜ 8 32 ⎜ 32 b ⎜−b −b − ⎜ 32 32 8 ⎜ b b ⎜ b − − 32 32 8 ⎜ ⎜ b b b − ⎜ − 32 32 8 ⎜ 15⋅ a ⎜ − 3⋅ a 0 ⎜ 16 16 ⎜ 3⋅ a 15⋅ a − 0 ⎜− 16 ⎜ 16 ⎜ 3 ⋅ b 15⋅ b 0 ⎜ − 16 16 ⎜ ⎜ − 3 ⋅ b − 15⋅ b 0 16 ⎜ 16 ⎜ 9 0 0 ⎜ ⎝ 4
3
1
3
8
2
8
3 8 3 8
− −
15
15
8
16
16
−
1
−
1
3
2
8
−
1
3
1
15
2
8
8
16
8
1
3
1
8
2
8
8
3⋅ a 16
−
a
a
16
8 −
0
8
3⋅ a
16
0
a
0
8
3⋅ a
a
16
8
0 3⋅ b
0
−
b
0
−
b
3⋅ b
8
16
0 0
−
1
3
3⋅ a
−
−
3⋅ a 4 3⋅ a 4
9⋅ b 8 9⋅ b 8 −
9 4
8
b 8
−
−
16
3⋅ b 16
b
3⋅ b
8
16
0 0
9⋅ a 8 9⋅ a 8
0
−
0
−
0
3⋅ b 4 3⋅ b
−
4 9 4
a 8
−
− −
1
16
16
8
15 16
16
a
3⋅ a
8
16
0
0
0
0
0
0
0 0
−
0
0
8
−
15
1
9
1
32
8
16
8
−
15
0
0
0
−
0
0
−
3⋅ b
b
16
8
3⋅ b
b
16
8
3⋅ b
b
16
8
3⋅ b
b
16
8
0
0
0
0
−
−
8
0
16
0
15⋅ a
0
8
0
8
0
32
0
15⋅ a
0
1
8
16 0
9
1
3⋅ a
0
1
16
−
15⋅ b 8
15⋅ b 8 0
8
16
15
15
16
1
8
1
−
−
−
9
15
15 16
1
−
−
3⋅ a
8
8
8
15
3⋅ a
−
1
15
a
a
−
32
32
−
−
1
9
8
16
5⋅ a
a
32
8
5⋅ a 32
−
a 8
5⋅ a
a
32
8
5⋅ a 32
−
a 8
−
−
8
1 8
15
−
15
15
32
32
−
15
−
15
15
32
32
32
32
0
0
0
−
0
0
0
−
0
0
0
b
5⋅ b
8
32
0
0
0
0
0
0
0
−
b
5⋅ b
8
32
0
0
0
−
b
16 15⋅ a 16
−
0
−
0
0
0
−
0
0
0
−
0
0
0
9⋅ a 8 9⋅ a 8 9⋅ b 8 9⋅ b 8 9 4
8
8
−
−
0 0 0
15⋅ b 16 15⋅ b 16 0
32 5⋅ a 32
0 0 0 0
32
0
5⋅ a
32
5⋅ b
0
32
5⋅ a
32
0
15
32
5⋅ b
0
32
5⋅ a
0
0
0
−
0
0
15⋅ a
−
0
b
15
−
15⋅ a 16 −
15⋅ 16 0 0 0
Shape functions from the first to the sevententh degrees of fredom: ⎛ 9⋅ ξ2⋅ η2 15⋅ ξ4 ξ3⋅ η ⎜ − − 32 8 ⎜ 16 ⎜ 4 4 3 ⎜ 15⋅ ξ ⋅ η − 15⋅ ξ + ξ ⋅ η 32 8 ⎜ 32 ⎜ 3 4 4 ⎜ ξ ⋅ η − 15⋅ ξ − 15⋅ ξ ⋅ η ⎜ 8 32 32 ⎜ 4 4 3 ⎜ 15⋅ ξ ⋅ η − 15⋅ ξ − ξ ⋅ η ⎜ 32 32 8 ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜
−
ξ
−
ξ
+
ξ
+
ξ
3
4
15⋅ ξ ⋅ η
−
8
32
3
2
9⋅ ξ ⋅ η
+
8
2
9⋅ ξ ⋅ η
+
8
2
9⋅ ξ ⋅ η
+
8 a 32 a
3 ⋅ a⋅ ξ
−
32
2
+
16
3 ⋅ a⋅ ξ
2
16 3 ⋅ a⋅ ξ
a
−
32
2
a
−
16
32
+
+
2
b 32
3⋅ η ⋅ b
−
16
2
3⋅ η ⋅ b 16
32
+
2
b 32
3⋅ η ⋅ b
−
16
2
3⋅ η ⋅ b 16
32
+
4
15⋅ a⋅ ξ ⋅ η 16 15⋅ a⋅ ξ
4
9⋅ b⋅ ξ 8
2
+
8 a⋅ ξ
3
+
8 a⋅ ξ
3
−
8 a⋅ ξ
3
−
8 η ⋅b 8 η ⋅b 8 η ⋅b 8
+
η ⋅b 8
+
5 ⋅ a⋅ ξ
3⋅ ξ
3⋅ ξ 4
− 4
− 4
− 4
−
32 5⋅ η ⋅ b
+
32 5⋅ η ⋅ b
−
32 5⋅ η ⋅ b
−
32 4
−
5⋅ η ⋅ b
2
+
2
+ 2
+
4
+ 4
+
32 15⋅ ξ ⋅ η
4
−
32 +
−
−
8 a⋅ ξ
+
8 ξ⋅ b
−
32 ξ⋅ b
−
32 ξ⋅ b
−
32 ξ⋅ b
−
32
8
16
15⋅ ξ ⋅ η
a⋅ ξ
9 ⋅ a⋅ ξ ⋅ η
−
32
8
2
a⋅ η 32 a⋅ η 32 a⋅ η 32 a⋅ η 32 η⋅ b 8 η⋅ b 8 η⋅ b 8 η⋅ b 8
ξ⋅ η
2
ξ⋅ η
ξ⋅ η
4
−
ξ ⋅ η⋅ b 8 ξ ⋅ η⋅ b 8 ξ ⋅ η⋅ b 8 ξ ⋅ η⋅ b
15⋅ a⋅ ξ ⋅ η 8 15⋅ a⋅ ξ ⋅ η 8
8
−
8
+
8 −
−
2
+
2
ξ⋅ η
+
3 ⋅ a⋅ ξ ⋅ η 16 3 ⋅ a⋅ ξ ⋅ η 16 3 ⋅ a⋅ ξ ⋅ η 16 3 ⋅ a⋅ ξ ⋅ η 16
3
a⋅ ξ ⋅ η
+
8 3
a⋅ ξ ⋅ η
−
8 3
a⋅ ξ ⋅ η
+
8 3
a⋅ ξ ⋅ η
−
8
3 ⋅ ξ⋅ η ⋅ b 16
3
+
2
+
3 ⋅ ξ⋅ η ⋅ b 16 3 ⋅ ξ⋅ η ⋅ b 16 3 ⋅ ξ⋅ η ⋅ b
3 ⋅ a⋅ ξ
16 2
+
4 3 ⋅ a⋅ ξ
2
4 15⋅ b ⋅ ξ 16
+
+
ξ⋅ η ⋅ 8 ξ⋅ η ⋅ 8 3
−
9 ⋅ a⋅ η
9 ⋅ a⋅ η
ξ⋅ η ⋅ 8
2
8
+
15
−
15
−
3⋅
2
8 15⋅ b ⋅ η 16
8
3
−
2
−
ξ⋅ η ⋅ 3
+
2
+
+
2
2
−
+
2
2
a⋅ ξ ⋅ η
+
ξ⋅ η
−
2
+
8
−
15⋅ b ⋅ ξ⋅ η
+
a⋅ ξ ⋅ η
−
2
16
−
2
2
8
+
ξ⋅ η
−
2
−
a⋅ ξ ⋅ η
−
15⋅ ξ⋅ η
−
−
2
2
16
3
8
+
15⋅ ξ⋅ η
−
ξ⋅ η
+
16
3
a⋅ ξ ⋅ η
−
15⋅ ξ⋅ η
+
8
2
16
3
8 ξ⋅ η
15⋅ ξ⋅ η
+
8
2
+
3
8
2
−
8
15⋅ b ⋅ ξ ⋅ η
15⋅ ξ ⋅ η
a⋅ ξ
9 ⋅ a⋅ ξ ⋅ η
4
32
8
2
−
15⋅ ξ ⋅ η
a⋅ ξ
2
−
16
+
32
4
−
8
32 5 ⋅ a⋅ ξ
2
8
32 5 ⋅ a⋅ ξ
−
8
4
+
15⋅ a⋅ ξ ⋅ η
8
3⋅ ξ
32 5 ⋅ a⋅ ξ
2
8
4
−
16
9⋅ b⋅ ξ ⋅ η
+
3⋅ ξ
4
15⋅ a⋅ ξ
2
−
3
4
−
16
a⋅ ξ
3
b
−
16
3
+
+
2
15⋅ ξ ⋅ η
−
3
b
−
16
3
+
+
2
15⋅ ξ ⋅ η
2
+
16
16
+
2
+
2
15⋅ ξ ⋅ η
2
16 3 ⋅ a⋅ ξ
−
16
−
16
3
+
2
16
3
2
15⋅ ξ ⋅ η
4
8
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
8
9⋅ b⋅ ξ
2
8
16
2
−
9⋅ b⋅ ξ ⋅ η
2
−
8
8
15⋅ b ⋅ ξ ⋅ η
4
16 2
+
9⋅ ξ ⋅ η
15⋅ b ⋅ ξ⋅ η
−
8
2
−
4
2
9⋅ ξ
2
16
15⋅ b ⋅ ξ
15⋅ b ⋅ η
16
9⋅ η
−
4
16
2
+
4
+
4
−
16
3⋅
9 4
In order to find later the strains, we use what we have already found: −1 ⎛⎜ d 2 a
2 2⎜ ⎝ dξ
−1 ⎛⎜ d 2
⎞⎟ ⎟ ⎠
⎛
2
⎜ ⎝
a
⎞⎟ ⎟ ⎠
⎛
2
⎜ ⎝
b
f ( ξ , η) → ⎜ 0 0 0 −
2 2⎜ b ⎝ dη
2
0 0 −
f ( ξ , η) → ⎜ 0 0 0 0 0 −
6⋅ ξ a
2
2
−
0 0 −
2⋅ η a
2
2⋅ ξ b
2
0 0 −
12⋅ ξ a
−
6⋅ η b
2
2
2
0 0 −
−
6 ⋅ ξ⋅ η a 2
2⋅ ξ b
2
2
−
−
2⋅ η a
6 ⋅ ξ⋅ η b
2
2
2
0 0 −
2
12⋅ ξ ⋅ η a
−
12⋅ η b
2
2
0 −
2
12⋅ ξ ⋅ b
2
2 2 3 3 ⎛ d d f ( ξ , η) ⎞ → ⎛⎜ 0 0 0 0 2 0 0 4⋅ ξ 4 ⋅ η 0 0 6 ⋅ ξ 8⋅ ξ⋅ η 6 ⋅ η 0 8 ⋅ ξ 8⋅ η ⎞ ⎜ ⎟ ( a ⋅ b ) ⎝ dξ dη a⋅ b a⋅ b a⋅ b a⋅ b a⋅ b ⎠ a⋅ b a⋅ b a⋅ b ⎠ ⎝
2
Strain vector: {ε}= [Ba]*{A}
2 2 ⎛⎜ −2 −6 ⋅ ξ −2 ⋅ η −12⋅ ξ −6 ⋅ ξ ⋅ η −2 ⋅ η − 0 0 0 0 0 0 0 0 0 ⎜ 2 2 2 2 2 2 a a a a a a ⎜ ⎜ 2 2 −2 ⋅ ξ −2 −2 ⋅ ξ −6 ⋅ η −6 ⋅ ξ⋅ η −12⋅ η ε ≡ z⋅ ⎜ 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 ⎜ b b b b b b ⎜ 2 2 ⎜ 2 4⋅ η 8⋅ ξ⋅ η 6⋅ η 4⋅ ξ 6⋅ ξ 0 0 0 0 0 0 0 0 0 ⎜⎝ a⋅ b a⋅ b a⋅ b a⋅ b a⋅ b a⋅ b
T
Mathcad is ready to find Ba
divided by the variable z
2 2 2 ⎛⎜ −2 −6 ⋅ ξ −2 ⋅ η −12⋅ ξ −6 ⋅ ξ⋅ η −2 ⋅ η −12⋅ ξ 0 0 0 0 0 0 0 0 0 ⎜ 2 2 2 2 2 2 2 a a a a a a a ⎜ ⎜ 2 2 −2 ⋅ ξ −2 ⋅ ξ −6 ⋅ η −6 ⋅ ξ ⋅ η −12⋅ η ⎜ 0 0 0 0 0 −2 0 0 0 0 0 2 2 2 2 2 2 ⎜ b b b b b b ⎜ 2 2 3 ⎜ 2 4⋅ η 8 ⋅ ξ⋅ η 6⋅ η 4⋅ ξ 6⋅ ξ 8⋅ ξ 0 0 0 0 0 0 0 0 0 ⎜⎝ a⋅ b a⋅ b a⋅ b a⋅ b a⋅ b a⋅ b a⋅ b
Ths is the transpose of Ba divided by z
0 ⎛ ⎜ 0 ⎜ 0 ⎜ ⎜ 2 ⎜ − 2 a ⎜ ⎜ 0 ⎜ ⎜ ⎜ 0 ⎜ ⎜ ⎜ 6⋅ ξ ⎜ − 2 a ⎜ ⎜ 2⋅ η ⎜ − 2 a ⎜ ⎜ 0 ⎜ ⎜ ⎜ ⎜ 0 ⎜ ⎜ ⎜ 12⋅ ξ2 ⎜ − 2 a ⎜ ⎜ ⎜ − 6 ⋅ ξ⋅ η ⎜ 2 a ⎜ ⎜ 2 ⋅ η2 ⎜ − 2 ⎜ a ⎜ ⎜ 0 ⎜ ⎜ ⎜ ⎜ 0 ⎜ ⎜ ⎜ 12⋅ ξ2⋅ η ⎜− 2
0
0 0 0
0 −
2 b
2
0
0
−
2⋅ ξ b
−
2
6⋅ η b
2
0
0
−
−
2
6 ⋅ ξ⋅ η b
−
2
2⋅ ξ b
2
12⋅ η b 0
⎞ ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 2 ⎟ ⎟ a⋅ b ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 4⋅ ξ ⎟ ⎟ a⋅ b ⎟ 4⋅ η ⎟ ⎟ a⋅ b ⎟ ⎟ 0 ⎟ ⎟ ⎟ ⎟ 0 ⎟ ⎟ 2 ⎟ 6⋅ ξ ⎟ a⋅ b ⎟ ⎟ 8⋅ ξ⋅ η ⎟ ⎟ a⋅ b ⎟ ⎟ 2 6⋅ η ⎟ a⋅ b ⎟ ⎟ ⎟ 0 ⎟ ⎟ ⎟ 3 8⋅ ξ ⎟ ⎟ b 0
0
2
2
⎜ ⎜ ⎜ ⎜ ⎜ ⎝
a 0
T
Mathcad is ready to find Ba ⋅ D divided by z
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜
−
0
0
0
0
0
0
2 a
0
0 −
0
2 b
−
6⋅ ξ a
−
0
2
−
0
2⋅ ξ b
−
0
12⋅ ξ a
−
−
0
2
a
2
0
0
2
2⋅ η
2
2
6 ⋅ ξ⋅ η a
2
6⋅ η b
−
2
0
2
2⋅ η a
0
0
2
2
−
−
2
2⋅ ξ b
2
6 ⋅ ξ⋅ η b
⎞ ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 2 ⎟ ⎟ a⋅ b ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 4⋅ ξ ⎟ ⎟ a⋅ b ⎟ 4⋅ η ⎟ ⎟ a⋅ b ⎟ ⎟ 0 ⎟ ⎛ Dx D1 0 ⎞ ⎟ ⎟⎜ ⎟ ⋅ ⎜ D1 Dy 0 ⎟ ⎟⎜ ⎟ 0 ⎟ ⎝ 0 0 Dxy ⎠ ⎟ 2 ⎟ 6⋅ ξ ⎟ a⋅ b ⎟ ⎟ 8⋅ ξ⋅ η ⎟ ⎟ a⋅ b ⎟ ⎟ 2 6⋅ η ⎟ a⋅ b ⎟ ⎟ ⎟ 0
2 2
⎟ ⎟ 3 ⎟ 8⋅ η ⎟ a⋅ b ⎟ ⎠ a⋅ b
2
−
12⋅ ξ⋅ η b
2
2
⎜ 2 12⋅ η ⎜ 0 0 − 2 ⎜ b ⎜ 3 ⎜ 12⋅ ξ2⋅ η 8⋅ ξ 0 ⎜− 2 a⋅ b a ⎜ ⎜ 2 3 12⋅ ξ⋅ η 8⋅ η ⎜ 0 − ⎜ 2 a⋅ b b ⎝
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜
−
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
0
0
0
0
0
0
0
0
0
2 ⋅ Dx a
−
2
a
0 −
2 ⋅ D1
−
2
−
−
4 ⋅ Dxy⋅ ξ
2
a⋅ b
2 ⋅ Dy ⋅ ξ
4 ⋅ Dxy⋅ η
2
a⋅ b
6 ⋅ Dy ⋅ η b
2
2
6 ⋅ Dx ⋅ ξ ⋅ η
−
0
2
12⋅ D1 ⋅ ξ a
−
0
2
2 ⋅ D1 ⋅ η
b
2
2
−
0
2
6 ⋅ D1 ⋅ ξ
a
2
12⋅ Dx ⋅ ξ
a⋅ b
2 ⋅ Dy
a
6 ⋅ D1 ⋅ η
a −
−
2
b −
2 ⋅ Dxy
b
2 ⋅ D1 ⋅ ξ b
−
2
2 ⋅ Dx ⋅ η a
−
−
6 ⋅ Dx ⋅ ξ a
0
2
0
b −
2 ⋅ D1
2
2
0
6 ⋅ D1 ⋅ ξ⋅ η
6 ⋅ Dxy⋅ ξ
2
a⋅ b
2
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟
T
This is Ba ⋅ D divided by z
⎜ a a ⎜ ⎜ 2 ⋅ Dx ⋅ η2 2⋅ D1⋅ ξ2 2⋅ D1⋅ η2 2 ⋅ Dy ⋅ ξ2 − − − ⎜− 2 2 2 2 ⎜ a b a b ⎜ 6 ⋅ D1 ⋅ ξ ⋅ η 6 ⋅ Dy ⋅ ξ⋅ η ⎜ − − ⎜ 2 2 ⎜ b b ⎜ 2 2 12⋅ D1 ⋅ η 12⋅ Dy ⋅ η ⎜ − − ⎜ 2 2 b b ⎜ ⎜ 2 2 12⋅ Dx ⋅ ξ ⋅ η 12⋅ D1 ⋅ ξ ⋅ η ⎜ − − ⎜ 2 2 a a ⎜ ⎜ 2 2 12⋅ D1 ⋅ ξ ⋅ η 12⋅ Dy ⋅ ξ⋅ η ⎜ − − ⎜ 2 2 b b ⎝
T
⎟ ⎟ 8 ⋅ Dxy⋅ ξ ⋅ η ⎟ ⎟ a⋅ b ⎟ ⎟ 2 6 ⋅ Dxy⋅ η ⎟ ⎟ a⋅ b ⎟ ⎟ ⎟ 0 ⎟ ⎟ 3 ⎟ 8 ⋅ Dxy⋅ ξ ⎟ ⎟ a⋅ b ⎟ 3 ⎟ 8 ⋅ Dxy⋅ η ⎟ ⎟ a⋅ b ⎠
Mathcad is ready to find Ba ⋅ D⋅ Ba divided by z
2
0 0 ⎛ ⎜ 0 0 ⎜ 0 0 ⎜ ⎜ 2 ⋅ D1 2 ⋅ Dx − − ⎜ 2 2 ⎜ a a ⎜ ⎜ 0 0 ⎜ ⎜ 2 ⋅ D1 2 ⋅ Dy ⎜ − − ⎜ 2 2 b b ⎜ 6 ⋅ Dx ⋅ ξ 6 ⋅ D1 ⋅ ξ ⎜ − − ⎜ 2 2 a a ⎜ ⎜ 2 ⋅ Dx ⋅ η 2 ⋅ D1 ⋅ η ⎜ − − 2 2 ⎜ a a ⎜ 2 ⋅ D1 ⋅ ξ 2 ⋅ Dy ⋅ ξ ⎜ − − ⎜ 2 2 b b ⎜ ⎜ 6 ⋅ D1 ⋅ η 6 ⋅ Dy ⋅ η ⎜ − − 2 2 ⎜ b b ⎜ 2 2 ⎜ 12⋅ Dx ⋅ ξ 12⋅ D1 ⋅ ξ ⎜ − − 2 2 ⎜ a a ⎜ ⎜ 6 ⋅ Dx ⋅ ξ ⋅ η 6 ⋅ D1 ⋅ ξ⋅ η − − ⎜ 2 2 ⎜ a a ⎜ 2 2 2 2 2 ⋅ D1 ⋅ η ⎜ 2 ⋅ Dx ⋅ η 2 ⋅ D1 ⋅ ξ 2 ⋅ Dy ⋅ ξ − − − ⎜− 2 2 2 2 ⎜ a b a b ⎜ 6 ⋅ D1 ⋅ ξ ⋅ η 6 ⋅ Dy ⋅ ξ⋅ η ⎜ − − ⎜ 2 2 ⎜ b b ⎜ 2 2 12⋅ D1 ⋅ η 12⋅ Dy ⋅ η ⎜ − − ⎜ 2 2 b b ⎜ ⎜ 2 2 12⋅ Dx ⋅ ξ ⋅ η 12⋅ D1 ⋅ ξ ⋅ η ⎜ − − ⎜ 2 2 a a ⎜ ⎜
⎞ ⎟ 0 ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ ⎟ 2 ⋅ Dxy ⎟ ⎟ a⋅ b ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 4 ⋅ Dxy⋅ ξ ⎟ ⎟ a⋅ b ⎟ ⎟ 4 ⋅ Dxy⋅ η ⎟ ⎟ a⋅ b ⎟ ⎛⎜ 0 0 0 −2 0 0 −6⋅ ξ −2⋅ η 0 ⎟⎜ 2 2 2 a a a ⎟⎜ 0 ⎟⎜ −2 −2 ⎟ ⋅⎜ 0 0 0 0 0 0 0 2 ⎟⎜ b b 0 ⎟⎜ ⎟⎜ 2 4⋅ ξ 4 0 ⎟ ⎜ 0 0 0 0 a⋅ b 0 a⋅ b a ⎝ 2 6 ⋅ Dxy⋅ ξ ⎟ ⎟ a⋅ b ⎟ ⎟ 8 ⋅ Dxy⋅ ξ ⋅ η ⎟ ⎟ a⋅ b ⎟ ⎟ 2 6 ⋅ Dxy⋅ η ⎟ ⎟ a⋅ b ⎟ ⎟ ⎟ 0 ⎟ ⎟ 3 ⎟ 8 ⋅ Dxy⋅ ξ ⎟ ⎟ a⋅ b ⎟ ⎟ 0
⎜ ⎜ ⎜ ⎝
−
12⋅ D1 ⋅ ξ ⋅ η b
2
−
2
12⋅ Dy ⋅ ξ⋅ η b
T
This is Ba ⋅ D⋅ Ba divided by z
⎡0 ⎢0 ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢
2
8 ⋅ Dxy⋅ η a⋅ b
2
3
⎟ ⎟ ⎟ ⎠
2
0 0
0
0
0
0
0 0
0
0
0
0
0 0
0
0
0
0
4 ⋅ D1
12⋅ Dx ⋅ ξ
0 0
4 ⋅ Dx a
0 0
0
4
2 2
a
0
0
4 ⋅ Dy
12⋅ D1 ⋅ ξ
4 ⋅ Dxy
0
4
a ⋅b
2 2
a ⋅b 0 0
0 0
4 ⋅ D1 a ⋅b
2 2
b
4
a ⋅b
12⋅ Dx ⋅ ξ
12⋅ D1 ⋅ ξ
36⋅ Dx ⋅ ξ
a 0 0
0
4
2 2
a ⋅b
2 2
a ⋅b
a
4 ⋅ D1 ⋅ ξ
8 ⋅ Dxy⋅ η
4 ⋅ Dy ⋅ ξ
12⋅ D1 ⋅ ξ
2 2
a ⋅b
4
a ⋅b
12⋅ D1 ⋅ η
2 2
b
b
24⋅ Dx ⋅ ξ
2
0
12⋅ Dxy⋅ ξ
4
a ⋅b
a
⎛ 2 ⋅ D ⋅ η2 ⎜ x
2 2
a ⋅b 2
2 2
2 ⋅ D1 ⋅ ξ
2⎞
⎟
2
3
72⋅ Dx ⋅ ξ
2 2
12⋅ Dx ⋅ ξ⋅ η
2
36⋅ D1 ⋅ ξ⋅ η
4
24⋅ D1 ⋅ ξ
4
4
2 2
12⋅ Dy ⋅ η
0
2 2
a 0 0
12⋅ Dx ⋅ ξ⋅ η
4
a ⋅b 0 0
4
a
4 ⋅ D1 ⋅ η
2 2
2
a ⋅b 8 ⋅ Dxy⋅ ξ
a ⋅b 0 0
2 2
4 ⋅ Dx ⋅ η a
0 0
0
4
a ⋅b
a
12⋅ D1 ⋅ ξ⋅ η
36⋅ Dx ⋅ ξ ⋅ η
2
2 2
a ⋅b
⎛ 2 ⋅ D ⋅ η2 ⎜ 1
a 2 ⋅ Dy ⋅ ξ
2⎞
⎟
4
⎛ 2 ⋅ D ⋅ η2 ⎜ x
2 ⋅ D1 ⋅ ξ
2
⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎣
2⋅ 0 0
⎜ ⎜ ⎝
xη a
+
2
b a
1
1ξ
2
2
⎟ ⎜ ⎟ 16⋅ D ⋅ ξ⋅ η 2⋅ ⎜ xy ⎠ ⎝
1η a
2
2 2
a ⋅b
yξ
+
b b
2
⎟ 6 ⋅ ξ⋅ ⎜ ⎟ ⎜ ⎠ ⎝
xη a
2
2
b a
0 ⋅ D ⋅ ξ⋅ η 12 y 0 4 b 0 2
24⋅ Dy ⋅ η 0 b
2
36⋅ D1 ⋅ ξ ⋅ η
0 0
2 2
a ⋅b
0
72⋅ D1 ⋅ ξ ⋅ η
4 ⋅ D1
4
2 2
2
2 2 2 a2 ⋅ b
3
72⋅ Dx ⋅ ξ ⋅ η 0 a
a ⋅b
24⋅ Dy ⋅ ξ ⋅ η 0 b
2
2
a ⋅b
2 2
a ⋅b
24⋅ D ⋅ ξ ⋅ η 4 ⋅ Dxy1
2
2
1
⌠ ⌠ 0 ⋅D 0 ⋅0ξ⋅ η 0 ⋅ D ⋅ η2 ⎮ ⎮ ⎡12 12 1 xy 0 0⎮ ⎮ ⎢0 0 0 0 2 2 2 2 ⎮ ⎮ ⎢ a ⋅b a ⋅b 0 ⎮ ⎮ ⎢0 0 0 ⎮ ⎮ ⎢ 2 ⎮ ⎮ ⎢24⋅ D1 ⋅ η 4 ⋅ Dx 0 0⎮ ⎮ 0 0 02 02 ⎢ 4 a ⋅b ⎮ ⎮ a ⎮ ⎮ ⎢ 2 3 ⎮ ⎮ ⎢24⋅ Dx ⋅ ξ ⋅ η 16⋅ Dxy⋅ ξ 0 0⎮ ⎮ ⎢ 0 2 2 ⎮ ⎮ ⎢0 0a4 0 a ⋅b ⎮ ⎮ ⎢ ⎮ ⎮ ⎢ 2 3 16⋅ D ⋅ η ⎮ ⎮ 24⋅ D1 ⋅ ξ ⋅ η 4 ⋅ D1 xy 0 0⎮ ⎮ ⎢ 0 02 02 2 2 2 a2 ⋅ b ⎮ ⎮ ⎢ a ⋅b ⋅ b a ⎮ ⎮ ⎢ ⎮ ⎮ ⎢ 12⋅ Dx ⋅ ξ ⎮ ⎮ ⎢ 0 0 0 ⎮ ⎮ ⎢ 4 a ⎮ ⎮ ⎢ 2 T The former ⎮ ⎮matrix is Ba ⋅ D⋅ Ba divided by z ⎮ ⎮ ⎢ 4 ⋅ Dx ⋅ η ⎮ ⎮ ⎢0 0 0 4 ⎮ ⎮ ⎢ a ⎮ ⎮ ⎢ ⎮ ⎮ ⎢ 4 ⋅ D1 ⋅ ξ ⎮ ⎮ ⎢ 0 0 0 ⎮ ⎮ ⎢ 2 2 a ⋅b ⎮ ⎮ ⎢ ⎮ ⎮ ⎮ ⎮ ⎢ 12⋅ D1 ⋅ η ⎮ ⎮ ⎢0 0 0 2 2 ⎮ ⎮ ⎢ a ⋅b ⎮ ⎮ ⎢ ⎮ ⎮ ⎢ 2 24⋅ Dx ⋅ ξ ⎮ ⎮ ⎢ 0 0 0 ⎮ ⎮ 4 ⎢ a ⎮ ⎮ ⎮ ⎮ ⎢ ⎮ ⎮ ⎢ 12⋅ Dx ⋅ ξ⋅ η ⎮ ⎮ ⎢0 0 0 4 ⎮ ⎮ ⎢ a ⎮ ⎮ ⎢ ⎮ ⎮ ⎢ ⎛ 2 ⋅ D ⋅ η2 2 ⋅ D ⋅ ξ2 ⎞ 1 ⎟ ⎮ ⎮ ⎜ x 2⋅ + ⎮ ⎮ ⎢ ⎜ 2 2 ⎟ a b ⎮ ⎮ ⎢ ⎝ ⎠ ⎮ ⎮ ⎢0 0 0 2 a ⎮ ⎮ ⎢ ⎮ ⎮ ⎢ ⎮ ⎮ ⎢ 12⋅ D1 ⋅ ξ⋅ η ⎮ ⎮ ⎢0 0 0
1ξ
+
2 2
72⋅ D1 ⋅ ξ ⋅ η
4 ⋅ Dy
4
b
4
2 2
a ⋅b
4
12⋅ D1 ⋅ ξ
0
2 2
a ⋅b 8 ⋅ Dxy⋅ ξ
4 ⋅ D1 ⋅ η
a ⋅b
2 2
a ⋅b
8 ⋅ Dxy⋅ η
4 ⋅ Dy ⋅ ξ
2 2
2 2
a ⋅b
b
4
12⋅ Dy ⋅ η
0
b
4
24⋅ D1 ⋅ ξ
0
2
2 2
a ⋅b 12⋅ Dxy⋅ ξ
2
2 2
12⋅ D1 ⋅ ξ⋅ η 2 2
a ⋅b
a ⋅b
⎛ 2 ⋅ D ⋅ η2 2 ⋅ D ⋅ ξ2 ⎞ ⎛ 2⋅ y ⎟ 1 ⎜ ⎜ 6 ⋅ ξ⋅ 2⋅ + ⎜ ⎟ ⎜ 2 2 16⋅ Dxy⋅ ξ⋅ η b ⎝ a ⎠ ⎝ 2 2
a ⋅b
12⋅ Dxy⋅ η
b 2
2
12⋅ Dy ⋅ ξ⋅ η
⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡
⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡
⎡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
−1 −1
0 0
⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎣
0 0
2 2
a ⋅b
0 0
b
2
24⋅ D1 ⋅ η
0
a ⋅b
b
2
0 0
24⋅ Dx ⋅ ξ ⋅ η
16⋅ Dxy⋅ ξ
4
a ⋅b
a 0 0
2
2 2
4
0
16⋅ Dxy⋅ η
16⋅ D1 2 2
4 2
2 2
a ⋅b 3
24⋅ Dy ⋅ ξ ⋅ η
a ⋅b
0
2
24⋅ D1 ⋅ ξ ⋅ η
2 2
a ⋅b
16⋅ Dx
3
2 2
24⋅ D1 ⋅ ξ ⋅ η
4
24⋅ Dy ⋅ η
2 2
a 0 0
2 2
a ⋅b
b
2
7
4
a ⋅b 16⋅ Dxy
0
2 2
a ⋅b 0 0
16⋅ D1 2 2
0
a ⋅b 0 0
0
16⋅ Dy b
0
4
0
48⋅ Dx a
0
4
16⋅ ⎛ 4 ⋅ Dxy⋅ a + Dx ⋅ b ⎝ 2
0 0
0
0
0
0
4 2
3⋅ a ⋅ b 0 0
0
0
0
16⋅ D1
0
2 2
a ⋅b 0 0
0
0
0
0
16⋅ D1 2 2
a ⋅b 0 0
32⋅ Dx 4
0
32⋅ D1 2 2
0
0
⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎣
a 0 0
4
2 2
a ⋅b 16⋅ Dxy
0
0
0
0
0
0
0
0
0
0
2 2
a ⋅b 16⋅ ⎛ D1 ⋅ a + Dx ⋅ b ⎝ 2
0 0
4 2
2⎞
⎠
2
0
3⋅ a ⋅ b 0 0
0
16⋅ ⎛ Dy ⋅ a + D1 ⋅ b ⎝ 2 4
2⎞
⎠
3⋅ a ⋅ b 16⋅ Dxy
0
2 2
a ⋅b 0 0
32⋅ D1 2 2
0
a ⋅b 0 0
0
32⋅ Dy b
0
4
0
0
32⋅ ⎛ 12⋅ Dxy⋅ a + 5 ⋅ Dx ⋅
⎝
2
4 2
15⋅ a ⋅ b 0 0
0
0
0
32⋅ D1 2 2
a ⋅b
0
− −
3⋅ b 16
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
⎞ ⎟ ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 9⎟ − ⎟ 4 ⎟ 0 ⎟ ⎟ 9⎟ − ⎟ 4⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 9 ⎟ 4 ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎠ 9 4
− −
3⋅ b 16
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
⎞ ⎟ 4 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 9⎟ − ⎟ 4 ⎟ 0 ⎟ ⎟ 9⎟ − ⎟ 4⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 9 ⎟ 4 ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ ⎟ 9
T
−
15⋅ b 16
0
⎞ ⎟ ⎟ 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⋅ b ⎟ 16 ⎟ ⎟ 15⋅ b ⎟ − 16 ⎟ ⎟ 0 ⎟ ⎠ −
15 32
⎟ ⎟ ⎠
5 2
5 2
a 2 a 2
a a 6
⎞ ⎟ 32 ⎟ 15 ⎟ − 32 ⎟ ⎟ 15 ⎟ 32 ⎟ 1 ⎞ 15 ⎟ ⎛⎜ ⎟ 32 ⎟ ⎜ ξ ⎟ ⎟⎜ η ⎟ 0 ⎟⎜ ⎟ ⎜ ξ2 ⎟⎟ ⎟⎜ ⎟ 0 ⎟ ξ⋅ η ⎜ ⎟ ⎜ 2 ⎟⎟ ⎟⎜ η ⎟ 0 ⎟ ⎜ ξ3 ⎟ ⎟⎜ 2 ⎟ 0 ⎟ ⎜ ξ ⋅η ⎟ ⎟⎜ 2 ⎟ 5⋅ b ⎟ ⎜ ξ⋅ η ⎟ ⎟ ⋅⎜ 3 ⎟ 32 ⎟ η ⎟ 5⋅ b ⎟ ⎜ − ⎜ 4 ⎟ 32 ⎟ ⎜ ξ ⎟ ⎟ 5 ⋅ b ⎜ ξ 3⋅ η ⎟ ⎟ − 32 ⎟ ⎜ 2 2 ⎟ ⎜ ξ ⋅η ⎟ 5⋅ b ⎟ ⎜ ⎟ 32 ⎟ ⎜ ξ ⋅ η3 ⎟ ⎟⎜ 4 ⎟ 0 ⎟⎜ η ⎟ ⎟⎜ ⎟ ⎜ ξ4⋅ η ⎟⎟ 0 ⎟ ⎟ ⎜⎝ ξ⋅ η4 ⎟⎠ 15⋅ b ⎟ 16 ⎟ ⎟ 15⋅ b ⎟ − 16 ⎟ ⎟ 0 ⎟ ⎠ −
15
⎞ ⎟ 8 32 32 8 32 8⎟ ⎟ 4 3 2 3⋅ ξ 15⋅ η η 3⋅ η 3⋅ η 1⎟ − − + + + − 8 32 32 8 32 8⎟ ⎟ 4 3 2 3⋅ ξ 15⋅ η η 3⋅ η 3⋅ η 1⎟ + − − + − − 8 32 32 8 32 8⎟ ⎟ 4 3 2 3⋅ ξ 15⋅ η η 3⋅ η 3⋅ η 1⎟ + − + + + − 8 32 32 8 32 8⎟ ⎟ 4 ⎟ η 5 ⋅ a⋅ ξ ⋅ η + ⎟ 32 ⎟ 4 ⎟ η 5 ⋅ a⋅ ξ ⋅ η − ⎟ 32 ⎟ 4 ⎟ η 5 ⋅ a⋅ ξ ⋅ η − ⎟ 32 ⎟ 4 ⎟ η 5 ⋅ a⋅ ξ ⋅ η + ⎟ 32 ⎟ ⎟ 4 b 5⋅ ξ⋅ η ⋅ b ⎟ + 32 ⎟ ⎟ 4 b 5⋅ ξ⋅ η ⋅ b ⎟ − 32 ⎟ ⎟ 4 b 5⋅ ξ⋅ η ⋅ b ⎟ − ⎟ 32 ⎟ 4 b 5⋅ ξ⋅ η ⋅ b ⎟ + ⎟ 32 ⎟ 5⋅ a⋅ η 3⋅ a ⎟ − ⎟ 16 16 ⎟ ⎟ 5⋅ a⋅ η 3⋅ a − ⎟ 16 16 ⎟ 2 ⎟ b⋅ η 3⋅ b − ⎟ 4 16 ⎟ −
3⋅ ξ
−
15⋅ η
4
−
η
3
+
3⋅ η
2
−
3⋅ η
−
1
4
16
b⋅ η
2
4
⎞
0⎟
⎟ ⎠
η
⎞ ⎟ ⎠
2
⎞ ⎟ ⎟ ⎠
−
3⋅ b 16
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
⎛ a1 ⎞ ⎜ ⎟ ⎜ a2 ⎟ ⎜ ⎟ ⎜ a3 ⎟ ⎜a ⎟ ⎜ 4⎟ ⎜ a5 ⎟ ⎜ ⎟ ⎜ a6 ⎟ 2 ⎞⎟ ⎜ ⎟ −12⋅ ξ ⋅ η a 0 ⎟⎜ 7⎟ 2 a ⎟ ⎜ a8 ⎟ 2⎟ ⎜ ⎟ −12⋅ ξ ⋅ η ⋅ a ⎟⎜ 8⎟ 0 2 ⎟ ⎜a ⎟ b ⎟ ⎜ 10 ⎟ 3 3 ⎟ ⎜ a11 ⎟ 8⋅ η 8⋅ ξ ⎟ ⎜ a⋅ b a⋅ b ⎠ ⎜ a ⎟⎟ 12 ⎜a ⎟ ⎜ 13 ⎟ ⎜ a14 ⎟ ⎜ ⎟ ⎜ a15 ⎟ ⎜ ⎟ ⎜ a16 ⎟ ⎜a ⎟ ⎝ 17 ⎠
2
⋅η
3
⎞⎟ 0 ⎟ ⎟ 2⎟ −12⋅ ξ⋅ η ⎟ 2 ⎟ b ⎟ 3 ⎟ 8⋅ η ⎟⎠ a⋅ b
T
0
0
−12⋅ ξ a
2⋅ ξ b
2
⋅η ⋅b
−6 ⋅ η b
2
0
2
−6 ⋅ ξ ⋅ η a
2
a
b 0
6⋅ ξ
2
a⋅ b
2
2
0
−12⋅ ξ ⋅ η
0
2
−2 ⋅ ξ
0
2
0
−2 ⋅ η
a 2
2
−6 ⋅ ξ⋅ η b
2
−12⋅ η b
6⋅ η
a⋅ b
a⋅ b
2
0
2
2
8⋅ ξ⋅ η
2
0
8⋅ ξ
3
a⋅ b
⎞⎟ ⎟ ⎟ 2⎟ −12⋅ ξ ⋅ η ⎟ 2 ⎟ b ⎟ 3 ⎟ 8⋅ η ⎟⎠ a⋅ b 0
0
0
0
0
0
0
4 ⋅ Dx ⋅ η
4 ⋅ D1 ⋅ ξ
4
a ⋅b
a
2 2
8 ⋅ Dxy⋅ ξ
8 ⋅ Dxy⋅ η
a ⋅b
2 2
a ⋅b
4 ⋅ D1 ⋅ η
4 ⋅ Dy ⋅ ξ
2 2
2 2
b
12⋅ Dx ⋅ ξ ⋅ η
12⋅ D1 ⋅ ξ
4
a ⋅b
a
a
2
4
2
4 ⋅ D1 ⋅ ξ ⋅ η
2 2
a ⋅b
+
2 2
16⋅ Dxy⋅ ξ ⋅ η
4 ⋅ Dy ⋅ ξ
2 2
a ⋅b
a ⋅b
b
2⎞
⎟
⎛ 2 ⋅ D ⋅ η2 ⎜ x
4
2 2
12⋅ Dy ⋅ ξ ⋅ η b
2
36
4
24⋅ D1 ⋅ ξ
4
a ⋅b
+
12
a ⋅b
24⋅ Dx ⋅ ξ ⋅ η
2
2
16⋅ Dxy⋅ η
+
2 2
a
2 2
2
a ⋅b
12⋅ Dx ⋅ ξ⋅ η
12
a ⋅b
4
2
12⋅ D1 ⋅ η
a
36
16⋅ Dxy⋅ ξ ⋅ η
+
2 2
a ⋅b
4 ⋅ D1 ⋅ ξ ⋅ η
2
2 2
16⋅ Dxy⋅ ξ
+
1
4
a ⋅b
4 ⋅ Dx ⋅ η
1
3
72⋅
2 2
24⋅ Dxy⋅ ξ 2 2
a ⋅b 2 ⋅ D1 ⋅ ξ
2⎞
⎟
3
2
12⋅ D1 ⋅ ξ ⋅ η 2 2
2
+
24⋅ Dxy⋅ ξ ⋅ η
a ⋅b
⎛ 2 ⋅ D ⋅ η2 ⎜ 1
36⋅
2 2
a ⋅b 2 ⋅ Dy ⋅ ξ
2⎞
⎟
⎛ 2⋅ D ⎜ 1
⎟ 2 ⋅ η⋅ ⎜ ⎟ ⎜ ⎠ ⎝
xη a
2
b a
0
2
+
2 2
2 2
a ⋅b
2 2
+
4
24⋅ D1 ⋅ ξ⋅ η
a ⋅b
2
a 32⋅ Dxy⋅ ξ 2 2
3
+
2 2
a ⋅b
2
12⋅ D1 ⋅ ξ 2 2
16⋅ Dxy⋅ ξ
+
4
2 2
2
+
2
24⋅ Dy ⋅ ξ ⋅ η b
+
2 2
b
4 ⋅ D1 ⋅ ξ ⋅ η 2 2
4 ⋅ Dy ⋅ ξ
2 2
a ⋅b
b
2
+
4
2
+
2 ⋅ D1 ⋅ ξ b
2
2
2
2 2
2
12⋅ Dy ⋅ ξ ⋅ η b
4
+
3
2 2
24⋅ Dxy⋅ ξ
3
2
12⋅ D1 ⋅ ξ ⋅ η
2 2
2 2
a ⋅b
2
+
a ⋅b
24⋅ Dxy⋅ ξ ⋅ η 2 2
a ⋅b
2 2⎞ ⎛ ⎛ 2 ⋅ D ⋅ η2 2 ⋅ D ⋅ ξ 2 ⎞ y ⎟ ⎟ 2 ⋅ η⋅ ⎜ 2 ⋅ Dx ⋅ η + 2⋅ D1⋅ ξ ⎟ ⎜ 1 2⋅ ξ⋅ + 2 ⎟ ⎜ ⎟ ⎜ 2 2 2 2 ⎟ b b ⎠ ⎝ a ⎠ + 32⋅ Dxy⋅ ξ ⋅ η ⎝ a ⎠ + 32⋅ Dxy⋅ ξ
2⎞
a
36⋅ D1 ⋅ ξ ⋅ η
2
2
a ⋅b
a ⋅b
4
a
16⋅ Dxy⋅ η
4
12⋅ Dx ⋅ ξ⋅ η
4
2 2
24⋅ D1 ⋅ ξ
a 2
16⋅ Dxy⋅ ξ ⋅ η
24⋅ Dx ⋅ ξ ⋅ η
4
2
a ⋅b
2
36⋅ Dx ⋅ ξ ⋅ η
4
+
2 2
3
72⋅
4 ⋅ Dy ⋅ ξ
4
16⋅ Dxy⋅ ξ ⋅ η
a ⋅b
72⋅ Dx ⋅ ξ
a
4
72⋅
a ⋅b
a ⋅b
a ⋅b
2
2 2
a ⋅b
2 2
2 2
a
2
a ⋅b
a ⋅b
Dx ⋅ η
+
32⋅ Dxy⋅ ξ ⋅ η 8 ⋅ Dxy⋅ η
2 2
12⋅ D1 ⋅ η
a
2 2
a ⋅b
a ⋅b
2
72
4 ⋅ D1 ⋅ ξ
4
36⋅ D1 ⋅ ξ⋅ η
a
0
2
12⋅ D1 ⋅ ξ
4 ⋅ D1 ⋅ ξ ⋅ η
a ⋅b
2 2
12⋅ Dx ⋅ ξ ⋅ η
4 ⋅ Dx ⋅ η a
2 2
a ⋅b
2 2
a ⋅b
a
4
32⋅ Dxy⋅ η
4 ⋅ D1 ⋅ η
4
12⋅ Dx ⋅ ξ⋅ η
2 2
36⋅
3
a ⋅b
3
2
0
4
3
24⋅ D1 ⋅ ξ ⋅ η
2 2
32⋅ Dxy⋅ ξ⋅ η
2
36⋅ Dx ⋅ ξ
b
a ⋅b
a
0
a ⋅b
4
8 ⋅ Dxy⋅ ξ
3
24⋅ Dxy⋅ η
24⋅ Dy ⋅ ξ⋅ η
4
a ⋅b
+
4
4 ⋅ Dx ⋅ η
1
2 2
2
b
⎟ ⎜ 6 ⋅ η⋅ ⎟ 32⋅ D ⋅ ξ⋅ η2 ⎜ xy ⎠ + ⎝ a ⋅b
12⋅ Dy ⋅ ξ ⋅ η
0
3
2
2
0
2 2
12⋅ D1 ⋅ ξ
b b
0
a ⋅b
4 2
a
2
a ⋅b
24⋅ Dx ⋅ ξ ⋅ η
a
24⋅ Dxy⋅ ξ⋅ η
24⋅ D1 ⋅ η
12⋅ Dx ⋅ ξ
a
a
yξ
+
2
2 2
2 2
0
1η
a ⋅b
a ⋅b
0
a
2
2
12⋅ D1 ⋅ ξ⋅ η
0
⎟ ⎜ 2ξ ⎟ 32⋅ D ⋅ ξ2⋅ η ⋅ ⋅ ⎜ xy ⎠ + ⎝
1ξ
+
2
12⋅ D1 ⋅ ξ⋅ η
2 2
a ⋅b 2
+
24⋅ Dxy⋅ ξ⋅ η
2
b
2
2 2
a ⋅b 2
12⋅ Dy ⋅ ξ ⋅ η
+
24⋅ Dxy⋅ η
3
2 2
a ⋅b
72⋅ D1 ⋅ ξ ⋅ η
+
2 2
a ⋅b
a ⋅b
2
24⋅ D1 ⋅ η
2 2
3
3
2
24⋅ Dx ⋅ ξ ⋅ η
4
a
72⋅ D1 ⋅ ξ ⋅ η
2
2 2
+
4
24⋅ Dy ⋅ ξ⋅ η b
32⋅ Dxy⋅ ξ
4
2 2
2 2
3
2 2
a ⋅b
+
3
2 2
32⋅ Dxy⋅ η
4
a ⋅b
0
0
0
0
0
0
0
0
0
0
0
0
0
32⋅ Dx a
0
4
0 16⋅ Dxy
0
2 2
a ⋅b 0
0
32⋅ D1 2 2
0
a ⋅b 16⋅ D1
0
0
0
0
0
0
0
0
0
2 2
a ⋅b 2⎞
⎠
16⋅ D1
0
2 2
a ⋅b 16⋅ ⎛ Dy ⋅ a + 4 ⋅ Dxy⋅ b
⎝
2
2⎞
⎠
0
2 4
3⋅ a ⋅ b 0
48⋅ Dy b
0
4
0
+
576 ⋅ Dx 4
0
32⋅ Dxy⋅ ξ ⋅ η 2 2
a ⋅b
2 2
a ⋅b
0
0
3
a ⋅b
32⋅ Dxy⋅ ξ⋅ η
2
4
3
24⋅ D1 ⋅ ξ ⋅ η
a ⋅b
24⋅ D1 ⋅ ξ⋅ η
a ⋅b
2
2 2
a ⋅b
a ⋅b
72⋅ Dx ⋅ ξ ⋅ η
2
b
+
4
2 2
a ⋅b
a
2 2
2 2
+
24⋅ Dy ⋅ ξ ⋅ η b
4
5⋅ a 0
0
4
16⋅ ⎛ 9 ⋅ Dxy⋅ a + 5 ⋅ Dx ⋅ b 2
⎝
0
2⎞
⎠
4 2
5⋅ a ⋅ b 0
0
32⋅ ⎛ 9 ⋅ D1 ⋅ a + 5 ⋅ Dx ⋅ b
⎝
2
2⎞
⎠
16⋅ ⎛ 9 ⋅ Dy
⎝
0
4 2
15⋅ a ⋅ b 0
0
0
(
16⋅ D1 + Dxy 2 2
a ⋅b 0
0
64⋅ D1 2 2
0
a ⋅b b
2⎞
⎠
32⋅ D1
0
0
0
0
0
2 2
a ⋅b 32⋅ ⎛ 5 ⋅ Dy ⋅ a + 12⋅ Dxy⋅ b
⎝
2
2 4
15⋅ a ⋅ b
2⎞
⎠
0
)
0
0
0
0
0
0
0
0
0
2⋅ D1 ⋅ η
24⋅ Dx ⋅ ξ
2 2
a ⋅b
a
0
2
12⋅ Dx ⋅ ξ⋅ η
4
a
4
12⋅ Dxy⋅ ξ
0
2
2 2
a ⋅b 2⋅ Dy ⋅ η
24⋅ D1 ⋅ ξ
4
a ⋅b
6⋅ D1 ⋅ ξ⋅ η
72⋅ Dx ⋅ ξ
b
2 2
a 2
2⋅ D1 ⋅ η
2
36⋅ Dx ⋅ ξ ⋅ η
4
a
a
12⋅ Dx ⋅ ξ ⋅ η
4
a
2⋅ Dy ⋅ ξ⋅ η
24⋅ D1 ⋅ ξ
4
a ⋅b
3
2
2
2
2 2
2 2 2
a ⋅b
a 2 ⋅ Dy ⋅ ξ
2⎞
⎟
2
4
3
72⋅ Dx ⋅ ξ ⋅ η
4
a 2
36⋅ Dx ⋅ ξ ⋅ η
4
⎛ 2 ⋅ D ⋅ η2 ⎜ x
2
2 2
3
a ⋅b
2 2
a ⋅b
2 2
72⋅ Dx ⋅ ξ ⋅ η
2 2
+
a ⋅b
36⋅ D1 ⋅ ξ ⋅ η
a
⋅ D1 ⋅ ξ ⋅ η
2
24⋅ Dxy⋅ ξ ⋅ η
72⋅ D1 ⋅ ξ ⋅ η
144 ⋅ Dx ⋅ ξ
a ⋅b
2 2
a ⋅b
a ⋅b
⋅ D1 ⋅ ξ ⋅ η
3
24⋅ Dxy⋅ ξ
+
4
2
4
4
2
12⋅ D1 ⋅ ξ ⋅ η
2 2
6⋅ Dy ⋅ η
2
3
2
a ⋅b
⋅η
2 2
a ⋅b
24⋅ Dx ⋅ ξ ⋅ η
2 2
b
12⋅ D1 ⋅ ξ⋅ η
2 2
a ⋅b
b
2
a 2 ⋅ D1 ⋅ ξ
2⎞
⎟
4
⎛ 2 ⋅ D ⋅ η2 ⎜ x
4
2
+
36⋅ Dxy⋅ ξ
4
2 2
a ⋅b 2 ⋅ D1 ⋅ ξ
2⎞
⎟
2
⎛2 ⎜
η
yξ
+
2
b b
2 2
a ⋅b
⋅ Dy ⋅ ξ ⋅ η
3
4
b
2 2
a ⋅b
4
36⋅ D1 ⋅ ξ⋅ η 2 2
a ⋅b
2
3
+
a 48⋅ Dxy⋅ ξ 2 2
2 2
a ⋅b
2
+
3
2 2
a ⋅b
2 2
a ⋅b
2
36⋅ Dx ⋅ ξ ⋅ η
4
a ⋅b
2 2
2 2
a ⋅b
36⋅ D1 ⋅ ξ ⋅ η
2
12⋅ D1 ⋅ ξ ⋅ η 2 2
2 2
a ⋅b
2
24⋅ Dxy⋅ ξ ⋅ η
+
2 2
a ⋅b
a ⋅b
72⋅ D1 ⋅ ξ ⋅ η
36⋅ D1 ⋅ ξ ⋅ η
2 2
a ⋅b
144 ⋅ Dx ⋅ ξ a
2
2 2
a ⋅b
4
3
72⋅ Dx ⋅ ξ ⋅ η
4
a 3
72⋅ Dx ⋅ ξ ⋅ η a
2
2 2
a ⋅b 2
3
24⋅ Dxy⋅ ξ
+
4
2
72⋅ D1 ⋅ ξ ⋅ η
4
2
12⋅ Dx ⋅ ξ ⋅ η a
3
2
48⋅ Dxy⋅ ξ ⋅ η 12⋅ D1 ⋅ ξ⋅ η
4
24⋅ D1 ⋅ ξ
4
12⋅ Dxy⋅ ξ
a ⋅b 3
4
5
a
12⋅ Dy ⋅ ξ⋅ η
b
12⋅ Dx ⋅ ξ⋅ η
4 2
2
0
3
3
72⋅ Dx ⋅ ξ
a
36⋅ Dy ⋅ η
2
2 2
a ⋅b
2
4
72⋅ D1 ⋅ ξ ⋅ η a ⋅b
2 2
a ⋅b
b
2
24⋅ D1 ⋅ ξ
24⋅ Dx ⋅ ξ ⋅ η
2 2
0
2 2
a ⋅b 72⋅ Dx ⋅ ξ ⋅ η
a
12⋅ D1 ⋅ η
2
2
+
2 0
a ⋅b
4
a 3
2
36⋅ Dxy⋅ ξ ⋅ η
2 2
0
144 ⋅ D1 ⋅ ξ ⋅ η
12⋅ Dy ⋅ η
2
72⋅ D1 ⋅ ξ ⋅ η
2
4
4
2 2
a ⋅b
a ⋅b
24⋅ Dx ⋅ ξ a
2
2
2 2
144 ⋅ Dx ⋅ ξ ⋅ η a
b
2
0 2
2 2
0
a
2
⎟ 2⎜ 2η ⎟ 48⋅ D ⋅ ξ3⋅ η ⋅ ⋅ ⎜ xy ⎠ + ⎝
1ξ
+
36⋅ D1 ⋅ ξ ⋅ η
0 0
144 ⋅ D1 ⋅ ξ ⋅ η
2
xη
a
2
2 2
2 2
2
⎟ 6 ⋅ ξ⋅ η⋅ ⎜ ⎟ ⎜ ⎠ ⎝
2
a ⋅b
a ⋅b
a ⋅b
2
b
2 2
12⋅ D1 ⋅ η
4 2
ξ⋅ η
2
3
0 3
D1 ⋅ ξ ⋅ η
b
a
1ξ
+
72⋅ D1 ⋅ ξ ⋅ η
0 0
4
2⋅ Dy ⋅ η b
xη
a
⋅ Dy ⋅ ξ ⋅ η b
2
⎟ 12⋅ ξ2⋅ ⎜ ⎟ ⎜ ⎠ ⎝
2
36⋅ Dx ⋅ ξ ⋅ η
4
4
2
+
a
4
36⋅ Dxy⋅ ξ
4
2 2
a ⋅b
2 2 ⎛ 2 ⋅ D ⋅ η2 2 ⋅ D ⋅ ξ2 ⎞ ⎛ ⎛ 2 ⋅ D ⋅ η2 2 ⋅ D ⋅ ξ2 ⎞ 2 ⋅ D1 ⋅ ξ ⎞⎟ y ⎟ 1 ⎟ 1 ⎜ 2 ⎜ 2 ⋅ Dx ⋅ η ⎜ x 6 ⋅ ξ⋅ η⋅ 6 ⋅ η⋅ + 12⋅ ξ ⋅ + + ⎜ ⎟ ⎜ ⎟ ⎜ 2 2 2 2 2 2 ⎟ b b b ⎝ a ⎠ ⎝ a ⎠ ⎝ a ⎠ + 48⋅ Dx
b
2
36⋅ Dy ⋅ ξ ⋅ η
a 2
2
a 3
72⋅ D1 ⋅ ξ ⋅ η
2
2
a ⋅ 2
36⋅ D1 ⋅ ξ ⋅ η
2
2
+
36⋅ Dxy⋅ ξ ⋅ η
2
b
4
2 2
a ⋅b
72⋅ Dy ⋅ η b
3
2
72⋅ D1 ⋅ ξ ⋅ η
2 2
2
2
2 2
4
2 2
72⋅ Dy ⋅ ξ ⋅ η
a 3
3
72⋅ Dx ⋅ ξ ⋅ η
4
a 3
144 ⋅ D1 ⋅ ξ ⋅ η
4
2
2
2 2
+
4
2 2
3
+
0
0
0
0
0
0
0
4 2
3⋅ a ⋅ b
3
2 2
a ⋅b
0
⎠
48⋅ Dxy⋅ ξ ⋅ η
a ⋅b
2⎞
5
2 2 2
0
2
48⋅ Dxy⋅ ξ a ⋅b
72⋅ D1 ⋅ ξ ⋅ η
a ⋅b
2
0
16⋅ ⎛ D1 ⋅ a + Dx ⋅ b ⎝
3
a ⋅b
144 ⋅ Dx ⋅ ξ ⋅ η
a ⋅b
b
2
a ⋅b
a ⋅b
72⋅ D1 ⋅ ξ ⋅ η
2 2
a ⋅b
144 ⋅ D1 ⋅ ξ ⋅ η
4
+
2 2
32⋅ D1 2 2
a ⋅b 16⋅ Dxy
0
0
2 2
a ⋅b 16⋅ ⎛ Dy ⋅ a + D1 ⋅ b ⎝ 2
2⎞
⎠
0
2 4
3⋅ a ⋅ b
32⋅ Dy b
4
0
0
0
0
0
0
0
0
0
0
0
0
32
32⋅ ⎛ 9 ⋅ D1 ⋅ a + 5 ⋅ Dx ⋅ b
⎝
2
4 2
2⎞
⎠
0
64⋅ D1 2 2
4 2
2 2
15⋅ a ⋅ b
a ⋅b
(
16⋅ D1 + Dxy
0
)
0
2 2
a ⋅b 4
4
2 2⎞
2 2
y ⋅ a + 9 ⋅ Dx ⋅ b + 10⋅ D1 ⋅ a ⋅ b + 80⋅ Dxy⋅ a ⋅ b
⎠
32⋅ ⎛ 5 ⋅ Dy ⋅ a + 9 ⋅ D1 ⋅ b
⎝
0
4 4
2
2⎞
⎠
2 4
45⋅ a ⋅ b
15⋅ a ⋅ b 16⋅ ⎛ 5 ⋅ Dy ⋅ a + 9 ⋅ Dxy⋅ b
⎝
0
2
2⎞
⎠
0
2 4
5⋅ a ⋅ b 32⋅ ⎛ 5 ⋅ Dy ⋅ a + 9 ⋅ D1 ⋅ b
⎝
2
2 4
2⎞
⎠
0
15⋅ a ⋅ b
576 ⋅ Dy 5⋅ b
4
64⋅ 0
0
0
0
0
0
4 ⋅ Dx ⋅ η a
0
0
0
0
0
0
2
4 ⋅ D1 ⋅ ξ
+
4
2
12⋅ D1 ⋅ ξ⋅ η
2 2
2 2
a ⋅b
a ⋅b
16⋅ Dxy⋅ ξ ⋅ η
12⋅ Dxy⋅ η
2 2
a ⋅b
2 2
a ⋅b 2
4 ⋅ Dy ⋅ ξ b
4
4 ⋅ Dx ⋅ η a
2
+
4
2 2 2
+
2 2
b
4 2
a 12⋅ Dx ⋅ ξ⋅ η
3
4
2 ⋅ D1 ⋅ ξ
12⋅ D1 ⋅ ξ ⋅ η
2 2
a ⋅b
32⋅ Dxy⋅ ξ⋅ η
+
2
4
+
⎟
b
4
2 2
2
b 4
3
2 2
2
4 3
72⋅ D1 ⋅ ξ ⋅ η 2 2
a ⋅b 3
⎛ 2 ⋅ D ⋅ η2 ⎜ 1
+
36⋅ Dy ⋅ ξ ⋅ η
2 2
+
24⋅ Dxy⋅ η a ⋅b
a ⋅b
12⋅ D1 ⋅ ξ ⋅ η
2 2
2
12⋅ Dy ⋅ ξ ⋅ η
3
24⋅ D1 ⋅ ξ
24⋅ Dxy⋅ ξ ⋅ η a ⋅b
2 2
a ⋅b 2⎞
2
2 2
3
+
+
2 2
a ⋅b 12⋅ D1 ⋅ η
2
a ⋅b
24⋅ Dx ⋅ ξ ⋅ η
2
32⋅ Dxy⋅ ξ ⋅ η
a ⋅b 2
⋅ Dx ⋅ η
2 2
a ⋅b
a ⋅b
4 ⋅ D1 ⋅ ξ⋅ η
+
4
2
2
+
4
36⋅ D1 ⋅ ξ ⋅ η
2 2
a ⋅b 3
3
a ⋅b
12⋅ Dy ⋅ ξ ⋅ η
a
b
12⋅ D1 ⋅ ξ
4 ⋅ D1 ⋅ ξ ⋅ η
+
12⋅ Dy ⋅ ξ⋅ η
2 2
2
4
4 ⋅ Dy ⋅ ξ b
3
2
a ⋅b
12⋅ Dx ⋅ ξ⋅ η a
4 ⋅ D1 ⋅ η
+
2
48⋅ Dxy⋅ ξ ⋅ η 2 2
2 2
a ⋅b
2 ⋅ Dy ⋅ ξ
2
36⋅ D1 ⋅ ξ ⋅ η
2
2
+
36⋅ Dxy⋅ ξ ⋅ η
a ⋅b 2⎞
⎟
⎛ 2 ⋅ D ⋅ η2 ⎜ 1
2 2
a ⋅b 2 ⋅ Dy ⋅ ξ
2⎞
⎟
xη a
2
b a
⎟ 2 ⋅ ξ2⋅ ⎜ ⎟ ⎜ ⎠ + ⎝
1ξ
+
2
2
1η a
2
b
3
b
12⋅ D1 ⋅ ξ⋅ η
+
4
3
a ⋅b
2 2
4
2
a
b
a
2
+
4
12⋅ Dx ⋅ ξ⋅ η
4 ⋅ Dx ⋅ η a
3
72⋅ D1 ⋅ ξ ⋅ η
48⋅ Dxy⋅ η
12⋅ D1 ⋅ ξ
+
4
4
+
2
+
2 2
4 2
4
2
+
3
72⋅ Dy ⋅ ξ ⋅ η 1 b
4
2 2
a ⋅b
32⋅ Dxy⋅ ξ⋅ η
2 2
2
12⋅ Dy ⋅ ξ
2 2
12⋅ D1 ⋅ η
+
b
4
3
3
2 2
2
4
+
24⋅ D1 ⋅ ξ
4
7
2 2
a ⋅b 3
+
5
a ⋅b
24⋅ Dx ⋅ ξ ⋅ η
3
2 2
a ⋅b
12⋅ D1 ⋅ ξ ⋅ η
a ⋅b
2
12⋅ Dx ⋅ ξ⋅ η
+
32⋅ Dxy⋅ ξ ⋅ η
a ⋅b
a
3
48⋅ Dxy⋅ ξ ⋅ η 1
3
a ⋅b
4 ⋅ D1 ⋅ ξ⋅ η
b
1
3
2
2 2
+
3
2 2
a ⋅b 3
4
a ⋅b
4 ⋅ D1 ⋅ ξ ⋅ η
+
2 2
a ⋅b
2
4
2
a ⋅b
2
12⋅ Dy ⋅ ξ ⋅ η
a
b
2 2
2
4
4 ⋅ Dy ⋅ ξ b
3
b
2 2
2 2 a ⋅2b 2
a
+
a ⋅b
4
2 2
2
36⋅ Dy ⋅ ξ ⋅ η
72⋅ Dy ⋅ ξ ⋅ η
2
2 64⋅ D 2 ⋅ ξ ⋅ η 4 ⋅ D ⋅ ξxy 4 ⋅ D1 ⋅ η + y 2 +2 4 a ⋅b 2 2 b a ⋅b
a ⋅b
4
a ⋅b
a ⋅b
24⋅ D1 ⋅ ξ ⋅ η
2
2 2
64⋅ Dxy⋅ ξ ⋅ η 16⋅ Dxy⋅ ξ ⋅ η
+
2 2
2
b b
4
24⋅ D1 ⋅ ξ ⋅ η
2
⎟ ⎟ 48 ⎠ +
yξ
+
36⋅ Dxy⋅ η
4 ⋅ D1 ⋅ ξ
+
4
a ⋅b 3
4
2
4
4
+
4
24⋅ Dy ⋅ ξ ⋅ η b
3
2 2
24⋅ Dy ⋅ ξ ⋅ η 2 4 ⋅ Dx ⋅ η
a ⋅b 24⋅ Dx ⋅ ξ ⋅ η
a
a ⋅b
2
+
1η
3 48⋅ Dxy⋅ 0ξ⋅ η + 0 2 2 a ⋅b 0
2 2
24⋅ D1 ⋅ η
2
2
b 12⋅ Dy ⋅ ξ ⋅ η
⎟ ⎜ 6ξ ⎟ 64⋅ D ⋅ ξ2⋅ η2 ⋅ ⋅ η⋅ ⎜ xy ⎠ + ⎝
yξ
+
12⋅ D1 ⋅ ξ ⋅ η 2 2
3
+
2
48⋅ Dxy⋅ ξ ⋅ η
36⋅ D1 ⋅ ξ ⋅ η
2 2
a ⋅b
a ⋅b
2 2
a ⋅b
2 2 ⎛ 2 ⋅ D ⋅ η2 2 ⋅ D ⋅ ξ 2 ⎞ ⎛ ⎛ 2 ⋅ D ⋅ η2 2 ⋅ Dy ⋅ ξ ⎞⎟ 1 ⎟ x ⎜ 2 ⎜ 2 ⋅ D1 ⋅ η ⎜ 1 2⋅ η ⋅ + 2⋅ ξ ⋅ + 6 ⋅ ξ⋅ η⋅ + 3 2 2 ⎜ ⎟ ⎜ ⎟ ⎜ 2 2 2 2 2 64⋅ Dxy⋅ ξ ⋅ η xy⋅ ξ ⋅ η a b a b a ⎝ ⎠ + ⎝ ⎠ + ⎝ 2
⋅b
2
a
2
b 3
12⋅ Dy ⋅ ξ ⋅ η
+
12⋅ D1 ⋅ ξ⋅ η
3
+
2
2 2
a ⋅b 48⋅ Dxy⋅ ξ⋅ η
3
b
2
36⋅ Dxy⋅ η
b
+
4
+
2 2
24⋅ D1 ⋅ η 2 2
4
2
+
2
24⋅ Dx ⋅ ξ ⋅ η
24⋅ Dy ⋅ ξ ⋅ η
0 0
0
0
0
⎝
2
4 2
15⋅ a ⋅ b 0 32⋅ D1 2 2
a ⋅b 0
+
2⎞
⎠
2 2
24⋅ D1 ⋅ ξ ⋅ η 2 2
2
7
4
+
4
+
⎤ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ 32⋅ D1 ⎥ ⎥ 2 2 a ⋅b ⎥ ⎥ ⎥ 0 ⎥ ⎥ 2 2⎞ ⎥ ⎛ 32⋅ 5 ⋅ Dy ⋅ a + 12⋅ Dxy⋅ b ⎝ ⎠⎥ ⎥ 2 4 15⋅ a ⋅ b ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥
3
64⋅ Dxy⋅ ξ ⋅ η
72⋅ D1 ⋅ ξ ⋅ η
2 2
a ⋅b
2 2
a ⋅b
a ⋅b
0
0
2⋅ ⎛ 12⋅ Dxy⋅ a + 5 ⋅ Dx ⋅ b
2
0
0
24⋅ D1 ⋅ ξ ⋅ η
a ⋅b
4
a ⋅b 3
4
b 4
+
4
24⋅ Dy ⋅ ξ ⋅ η b
3
2 2
a ⋅b
a ⋅b
a
2 2
a ⋅b
64⋅ Dxy⋅ ξ ⋅ η 2 2
a ⋅b
4
48⋅ Dxy⋅ η 2 2
a ⋅b
⎥ ⎥ ⎥ 0 0 ⎥ ⎥ ⎥ ⎥ 0 0 ⎥ ⎥ ⎥ ⎥ 0 0 ⎥ ⎥ ⎥ ⎥ 0 0 ⎥ ⎥ ⎥ 2 2 ⎛ 20⋅ D ⋅ a + 21⋅ D ⋅ b ⎞ x ⎠ xy ⎥ ⎝ 0 ⎥ 4 2 35⋅ a ⋅ b ⎥ ⎥ 2 2 64⋅ ⎛ 21⋅ Dy ⋅ a + 20⋅ Dxy⋅ b ⎞ ⎝ ⎠ ⎥ 0 ⎥ 2 4 35⋅ a ⋅ b ⎦
0
0
0
0
0
0 2
24⋅ D1 ⋅ η
2
24⋅ Dx ⋅ ξ ⋅ η
2 2
a ⋅b
a
4
16⋅ Dxy⋅ ξ
0
3
2 2
a ⋅b 2
24⋅ Dy ⋅ η b
2
24⋅ D1 ⋅ ξ ⋅ η
4
2 2
a ⋅b 2
72⋅ D1 ⋅ ξ⋅ η
3
72⋅ Dx ⋅ ξ ⋅ η
2 2
a ⋅b 2
a 3
24⋅ D1 ⋅ η
2
24⋅ Dx ⋅ ξ ⋅ η
2 2
a ⋅b
a 2
24⋅ Dy ⋅ ξ⋅ η b
2 2
3
2 2 3
32⋅ Dxy⋅ ξ ⋅ η
+
32
2 2
2
72⋅ D1 ⋅ ξ ⋅ η
4
2
2 2
a ⋅b 2
4
144 ⋅ Dx ⋅ ξ ⋅ η
2 2
72⋅ D1 ⋅ ξ⋅ η
a 3
3
72⋅ Dx ⋅ ξ ⋅ η
2 2
a ⋅b
⎛ 2 ⋅ D ⋅ η2 ⎜ 1
24
a ⋅b
a ⋅b
2
4
a ⋅b
a ⋅b
2
2
32⋅ Dxy⋅ ξ
+
3
144 ⋅ D1 ⋅ ξ ⋅ η
η
2
4
24⋅ D1 ⋅ ξ ⋅ η
4
72⋅ Dy ⋅ η b
4
a 2 ⋅ Dy ⋅ ξ
2⎞
⎟
2
4
⎛ 2 ⋅ D ⋅ η2 ⎜ x
4
2
+
48⋅ Dxy⋅ ξ
5
72⋅ D
2 2
a ⋅b 2 ⋅ D1 ⋅ ξ
2⎞
⎟
2
⎛ 2⋅ D ⎜
2
8⋅ Dxy⋅ ξ ⋅ η
12⋅ η ⋅
3
⎜ ⎜ ⎝
1η a
2 2
a ⋅b 2
yξ
+
2
b b
2
0 b
0
3
4 4
2 η 12⋅ Dxy⋅ η
144 ⋅ D1 ⋅ ξ ⋅ η
2 2
a ⋅b
2
3
144 ⋅ Dy ⋅ ξ ⋅ η
12⋅ Dy ⋅ ξ⋅ η b
4
b 2
0 a
24⋅ Dxy⋅ ξ ⋅ η
2
2 2
a ⋅b
3
2 2
b
2
3
2
2
2 2
3
b
2
2
2 2
a ⋅b 2
4
144 ⋅ Dx ⋅ ξ ⋅ η a
3
3
72⋅ Dx ⋅ ξ ⋅ η
4
2
+
a
4
48⋅ Dxy⋅ ξ
5
2 2
a ⋅b
2 2 2 2 ⎛ ⎛ 2 ⋅ Dy ⋅ ξ ⎞⎟ 2 ⋅ D1 ⋅ ξ ⎞⎟ ⎟ 2 ⎜ 2 ⋅ D1 ⋅ η 2 ⎜ 2 ⋅ Dx ⋅ η 12⋅ ξ ⋅ η⋅ + + 4 ⎟ 48⋅ D ⋅ ξ⋅ η3 12⋅ η ⋅ ⎜ ⎜ 2 2 ⎟ 2 2 ⎟ xy a b a b ⎠ + ⎝ ⎠ ⎝ ⎠ + 64⋅ Dxy⋅ ξ 2
+
2 2
2⎞
2 2
4
+
2
a ⋅b
a ⋅b η
3
32⋅ Dxy⋅ ξ ⋅ η
72⋅ D1 ⋅ ξ ⋅ η
2 2
a ⋅b
2 2
a ⋅b
4
72⋅ D1 ⋅ ξ⋅ η
2 2
+
4
a ⋅b
2 2
2
32⋅ Dxy⋅ ξ
a ⋅b
a ⋅b 2
2
3
2
36⋅ Dxy⋅ ξ ⋅ η
4
4
24⋅ D1 ⋅ ξ ⋅ η
4
2 2
+
2 2
a ⋅b
24⋅ Dx ⋅ ξ ⋅ η
144 ⋅ D1 ⋅ ξ ⋅ η
a ⋅b
64⋅
3
a
72⋅ Dy ⋅ η b
2 ⋅ Dy ⋅ ξ
3
72⋅ Dx ⋅ ξ ⋅ η
3
2⋅ D1 ⋅ ξ ⋅ η
+
2 2
a ⋅b
2
24⋅ Dy ⋅ ξ⋅ η
4
2
2 2
a ⋅b
3
+
144⋅
2 64⋅ Dxy⋅ ξ ⋅ η 24⋅ D1 ⋅ ξ ⋅ η
2 2
24⋅ Dxy⋅ η
6⋅ Dy ⋅ ξ ⋅ η
η
2 2
a ⋅b
a ⋅b
a ⋅b
b
6
+
3
4
3 64⋅ Dxy⋅ ξ 16⋅ Dxy⋅ ξ
a
24⋅ D1 ⋅ η
2 2
+
2
24⋅ Dx ⋅ ξ ⋅ η
2 2
a ⋅b ⋅η
3
4
a ⋅b
2
0
3
a
2
4
144 ⋅ D1 ⋅ ξ ⋅ η
2
24⋅ Dy ⋅ η b
2 2
2 2
4
48
0
a ⋅b
144 ⋅ Dx ⋅ ξ ⋅ η
4
2 0
a ⋅b
2 2
a ⋅b +
+
144 ⋅ D1 ⋅ ξ ⋅ η
3
4
3
48⋅ Dxy⋅ ξ ⋅ η
2
2
72⋅ D1 ⋅ ξ⋅ η
2
2
2 2
6⋅ D1 ⋅ ξ ⋅ η
η
2 2
a ⋅b
2 2
2 2
a ⋅b
2
a ⋅b 2
2
a ⋅b
24⋅ D1 ⋅ η
4
b
b
3
0
2 2
a ⋅b
a
2
⎟ 2⎜ 12⋅ ξ ⋅ η ⋅ ⎟ 64⋅ D ⋅ ξ4⋅ η ⎜ xy ⎠ + ⎝
1ξ
+
72⋅ D1 ⋅ ξ ⋅ η
0 0
144 ⋅ Dy ⋅ η
12⋅ D1 ⋅ ξ⋅ η
xη
a
72⋅ Dy ⋅ ξ⋅ η
0
2
⎟ 12⋅ ξ2⋅ η⋅ ⎜ ⎟ ⎜ ⎠ ⎝
36⋅ Dy ⋅ ξ ⋅ η
2
b
2
72⋅ Dy ⋅ ξ⋅ η
a 3
2
2 2
a ⋅b 3
72⋅ D1 ⋅ ξ ⋅ η
2
3 2
+
48⋅ Dxy⋅ ξ ⋅ η
+ b 2⋅ Dy ⋅ ξ ⋅ η b η
4
b
3
b 3
+
48⋅ Dxy⋅ ξ ⋅ η 2 2
5
2
+
72⋅ Dy ⋅ ξ ⋅ η b
4
4
2
4
2 2
2
144 ⋅ Dy ⋅ ξ ⋅ η 4
3
a ⋅b
144 ⋅ D1 ⋅ ξ ⋅ η
b
a ⋅b
144 ⋅ D1 ⋅ ξ ⋅ η
3
a ⋅b 3
2 2
a ⋅b
2 2
a ⋅b η
2
+
2 2
144 ⋅ Dy ⋅ η
4
2
4
4
144 ⋅ Dx ⋅ ξ ⋅ η a
4
+
4 3
144 ⋅ D1 ⋅ ξ ⋅ η 2 2
a ⋅b
2
64⋅ Dxy⋅ ξ
6
2 2
a ⋅b 3
3 3
+
64⋅ Dxy⋅ ξ ⋅ η 2 2
a ⋅b
0 0 0 24⋅ D1 ⋅ ξ⋅ η
2
2 2
a ⋅b
16⋅ Dxy⋅ η
3
2 2
a ⋅b
24⋅ Dy ⋅ ξ⋅ η b
2
4 2
72⋅ D1 ⋅ ξ ⋅ η
2
2 2
a ⋅b ⋅ D1 ⋅ ξ⋅ η
3
32⋅ Dxy⋅ ξ⋅ η
+
2 2
2 2
a ⋅b
2⋅ Dxy⋅ η
3
a ⋅b 4
2
24⋅ Dy ⋅ ξ ⋅ η
+
2 2
a ⋅b
b
72⋅ Dy ⋅ ξ⋅ η b
2
4
3
4 3
144 ⋅ D1 ⋅ ξ ⋅ η
2
2 2
a ⋅b 2
D1 ⋅ ξ ⋅ η 2 2
3
2
+
48⋅ Dxy⋅ ξ ⋅ η
a ⋅b
D1 ⋅ η
2
2 2
a ⋅b 2 ⋅ Dy ⋅ ξ
2⎞
⎟
3
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
⎟ ⎟ 64⋅ D ⋅ ξ⋅ η4 ⎥⎥ xy a b ⎠ + ⎥ 2 2 2 b a ⋅b ⎥ ⎥ 5 2 3 0 8⋅ Dxy⋅ η 72⋅ Dy ⋅ ξ ⋅ η ⎥ + 0 ⎥ 2 2 4 a ⋅b b ⎥ 0 ⎥ 4 2 ⎥ 144 ⋅ Dy ⋅ ξ⋅ η 24⋅ D1 ⋅ ξ⋅ η ⎥ 4 2 2 b ⎥ a ⋅b ⎥ 3 3 3 3 3 ⎥ D1 ⋅ ξ ⋅ η 64⋅ Dxy⋅ ξ ⋅ η 16⋅ Dxy⋅ η + ⎥ 2 2 2 2 2 2 a ⋅b a ⋅b ⎥ a ⋅b ⎥ 6 2 4 2 ⋅ Dxy⋅ η 144 ⋅ Dy ⋅ ξ ⋅ η 24⋅ Dy ⋅ ξ⋅ η ⎥ + ⎥ 2 2 4 4 a ⋅b b ⎦ b 1η 2
+
yξ 2
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 2 2 72⋅ D1 ⋅ ξ ⋅ η ⎥ ⎥ 2 2 a ⋅b ⎥ 3 3 ⎥ 24⋅ D1 ⋅ ξ⋅ η 32⋅ Dxy⋅ ξ⋅ η ⎥ + ⎥ 2 2 2 2 a ⋅b a ⋅b ⎥ 4 2 2 ⎥ 32⋅ Dxy⋅ η 24⋅ Dy ⋅ ξ ⋅ η ⎥ + 2 2 4 ⎥ a ⋅b b ⎥ 3 ⎥ 72⋅ Dy ⋅ ξ⋅ η ⎥ 4 ⎥ dξ dη b ⎥ 3 2 ⎥ 144 ⋅ D1 ⋅ ξ ⋅ η ⎥ 2 2 ⎥ a ⋅b ⎥ 2 3 2 3 ⎥ 72⋅ D1 ⋅ ξ ⋅ η 48⋅ Dxy⋅ ξ ⋅ η ⎥ + 2 2 2 2 ⎥ a ⋅b a ⋅b ⎥ ⎛ 2 ⋅ D ⋅ η2 2 ⋅ D ⋅ ξ 2 ⎞ ⎥ y ⎟ 1 2⎜ 12 ⋅ ξ η ⋅ ⋅ + 4 4⎥ ⎜ 2 2 ⎟ ⋅η 64⋅ Dxy⋅ ξ⋅ η ⎥ a b ⎝ ⎠ + ⎥ 2 2 2 b a ⋅b ⎥ ⎥ 5 2 3 72⋅ Dy ⋅ ξ ⋅ η 48⋅ Dxy⋅ η ⎥ + ⎥
+
2 2
a ⋅b
b
144 ⋅ Dy ⋅ ξ⋅ η b 3
144 ⋅ D1 ⋅ ξ ⋅ η
64⋅ Dxy⋅ η 2 2
a ⋅b
3
+
2 2
4
4
3
a ⋅b
4
64⋅ Dxy⋅ ξ ⋅ η 2 2
a ⋅b 6
2
+
144 ⋅ Dy ⋅ ξ ⋅ η b
4
4
3
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
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