PSI - Issue 25

Victor Rizov / Procedia Structural Integrity 25 (2020) 88–100 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

90

3

a         w i m Fi i n  

b a b M Mi i    1 

a U



b G F i 1 i

.

(2)



i

1

Fi w , and the angle of rotation,

Mi  , are obtained by applying the integrals of Maxwell-Mohr

The displacement,

i s     1 i

w

M dx Fi i 



,

(3)

Fi

l

i

i s       1 i i

w

M dx Mi i 

,

(4)

l

i

where s is the number of the beam portions, i l and i  are, respectively, the length and the curvature of the i -th beam portion, x is the longitudinal centroidal axis of the beam, Fi M and Mi M are the bending moments in the i -th beam portion induced by the unit loading for the displacement and the angle of rotation, respectively. The strain energy in the beam is written as

j f 1 1 ( )        j i s i

U

u dV ij 0



,

(5)

V

ij

ij u 0 are, respectively, the volume and the strain energy density in the

where f is the number of layers, ij V and j -th layer in the i -portion of the beam. The strain energy density is expressed as

2 1

ij u

0 

  ,

(6)

ij ij

where ij  and ij  are, respectively, the normal stress and the longitudinal strain in the j -th layer of the i -portion of the beam. By using the Hooke’s law, the strain energy density is obtained as

2 1

2 j ij ij u E   , 0

(7)

where j E is the modulus of elasticity in the j -th layer of the beam. The distribution of the strain along the height of the beam cross-section is treated in accordance to the Bernoulli’s hypothesis for plane sections since beams of high length to height ratio are un der consideration in the present study. Thus, the distribution of ij  is written as   n i ij z z     , (8) where z is the vertical centroidal axis of the beam cross-section, n z is the coordinate of the neutral axis. The curvature and the coordinate of the neutral axis are determined by using the following equations for