Issue 54
T. I. J. Brito et alii, Frattura ed Integrità Strutturale, 54 (2020) 1-20; DOI: 10.3221/IGF-ESIS.54.01
b
b
sin
cos
1
b
b
R
R
sin
sin
b
b
b
b
sin
cos
B
0
b
b
b
R
R
sin
sin
b
b
cos 1 sin sin b b
cos 1 cos sin b b
b
b
cos
sin
0
b
b
(5)
sin
cos
b
b
b
b
b R b R
b
b
b
cos
cos
sin
cos
b
b
0
b
b
R
R
R b R
sin
sin
b
b
b
b
b
cos
b
b
b
b
b
b
b
sin
cos
cos
cos
sin
1
b
b
R
R
sin
sin
b
b
b
b
cos 1 sin b
cos 1 cos b
b
b
b
b
b
b
b
b
cos
sin
0
b
b
sin
sin
Then, the equilibrium relation of the structure is given by:
m
1 b P Q
(6)
E b
being { Q E } b the expanded internal forces matrix, given as follows:
E Q
0 0 0} T
1 {0 0 0 node ui wi i Q Q Q node i
uj Q Q Q wj j
(7)
b
node n
node j
Therefore, the equilibrium relation can be expressed as:
1 m T E b b b P B M
(8)
where [ B E ] b is the expanded transformation matrix, given in a similar way of { Q E } b . Kinematics of circular arch elements
Consider again the structure depicted in Fig. 1a, the node i presents two translations parallel to the global X - and Z -axes ( u i and w i , respectively) and a rotation in the XZ plane ( θ i ). Such translations and rotation are here called generalised displacements. Then, the generalised displacements of all nodes of the structure are gathered in the generalised displacements matrix, given by: 1 1 1 ... ... ... T i i i j j j n n n u w u w u w u w U (9) Now, the generalised deformations are defined as conjugate quantities of the generalised stress i.e. i , j and δ i are conjugated to M i , M j and N i , respectively. Such quantities are gathered in the generalised deformations matrix: T i j i b Φ (10)
Therefore, the mechanical power of the element is expressed as:
4
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