PSI - Issue 25

Victor Rizov / Procedia Structural Integrity 25 (2020) 112–127

115

4

Author name / Structural Integrity Procedia 00 (2019) 000–000

where z is the vertical centroidal axis of the beam cross-section,  is the curvature of the beam, n z is the coordinate of the neutral axis. The following equations for equilibrium of the elementary forces in the beam cross-section are used in order to determine the curvature and the coordinate of the neutral axis:

A    ( )

N

dA

,

(9)

A    ( )

M

zdA

,

(10)

where N is the axial force, M is the bending moment, A is the area of the beam cross-section. If the material of the beam has different mechanical behaviour in tension and compression, the strain energy cumulated in the beam is expressed as

V ( )  c

V ( ) 

U

u dV

u dV 0 t





,

(11)

0

c

t

c u 0 and

t u 0 are, respectively, the strain energy densities in the compression and tension zones, c V and

where

t V are, respectively, the volumes of the parts of the beam loaded in compression and tension. The strain energy densities in the compression and tension zones are written as

2 1

2

c c u E 



,

(12)

2 1

2

t t u E 



,

(13)

where c E and t E are the moduli of elasticity in compression and tension, respectively. It should be mentioned that both c E and t E vary continuously in the height and length directions of the beam. When the material has different mechanical behaviour in tension, the curvature and the coordinate of neutral axis are obtained by using the equations for equilibrium which are written as

A ( ) c 

A ( ) 

N

dA

dA



c 



t 

,

(14)

t

A ( ) c 

A ( ) 

M

zdA

zdA



c 



t 

,

(15)

t

where c  and t  are, respectively, the normal stresses in the compression and tension zones of the beam cross section, c A and t A are, respectively, the areas of the compression and tension zones. The stress, c  and t  , are found as