Issue 54

T. I. J. Brito et alii, Frattura ed Integrità Strutturale, 54 (2020) 1-20; DOI: 10.3221/IGF-ESIS.54.01

        T T E  Φ  

 



(11)

M U Q M

b

b

b

b

Assuming small displacements and small deformations, the matrix of generalised deformations is given by:       E b b  Φ B U (12) Elasticity of circular arch elements Consider an elastic circular arch element with its internal forces at edges in the local coordinate system as the one depicted in Fig. 2a. Thus, the strain energy of the element is expressed by:     2 2 b M N      where N ( ψ ) and M ( ψ ) are the normal force and bending moment for a cross section ψ , respectively, AE is the axial stiffness and EI is the flexural stiffness of the element. Note that N ( ψ ) and M ( ψ ) can be obtained by equilibrium (Fig. 2b): ( ) ( 1 cos ) ( ) (1 cos ) sin sin sin ( cos ) ( ) cos sin i j i b b i i b b i i b i b b j i b b M M n R M M N R M N R N R M N N R                       (14) 0  2 2 b b R d U EI AE          (13)

Figure 2: Elastic circular arch element: (a) internal forces and (b) stresses at any cross section.

where V ( ψ ) is the shear force for a cross section ψ . By applying Castigliano’s Theorem, the elastic generalised deformations are given by:

      

         

2

2

2







U

U

U

  

U

b

b

b

b

e    i e j e i i U       b          j             

2   M M M N M M    

i

j

i

i

i

i M         N j i b   

2

2

2







U

U

U M

  e Φ

    0 F M

b

b

b

 





(15)

2

b

b

M M M   

 

M N



M

b

i

j

j

i

j

U

2

2

2

b U N M N M N N                b b i U U 

b

2



i

i

j

i

i

5