# SOLUTION: MAE 456 West Virginia University Lambda Matrix and Beam Matrix Formulation HW

Pr. 2 Consider the Ray Matrix formulation presented in systematize.
Use the specification of Matrix Inverse to invent the inverse of the “Lambda Matrix” for a ray limited element
given below: (pretence the composition by laborer in point)

[Λ]
−1 = [
1 0 0 0
0 1 0 0
1 L L
2 L
3
0 1 2L 3L
2
]
−1
= [
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
]

Once you invent the []
-1
, then you are required to invent Matrix [B] as follows: (pretence the composition)
[B] = (0 0 2 6x) [

1 0 0 0
0 1 0 0
1 L L
2 L
3
0 1 2L 3L
2
]
−1
= (B1 B2 B3 B4)

Pr. 3 The you deficiency to trace the Ray Stiffness Matrix as follows: (pretence the composition)
[K] = EI ∫ [

B1B1 B1B2 B1B3 B1B4
B2B1 B2B2 B2B3 B2B4
B3B1 B3B2 B3B3 B3B4
B4B1 B4B2 B4B3 B4B4
] dx L
0

= [
K11 K12 K13 K14
K21 K22 K23 K24
K31 K32 K33 K34
K41 K42 K43 K44
]

Where Ki,n = EI ∫ Bi
L
0
Bndx

Notice that since the matrix is symmetric, you barely deficiency to complete the Upper triangle of the matrix