PSI - Issue 41

Victor Rizov et al. / Procedia Structural Integrity 41 (2022) 115–124 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

117 3

where F v and M v are parameters which control the variation of F and M , respectively. Apparently, the beam in Fig. 1 is twice statically undetermined. The bending moment, 1 H M , and the axial force, 1 H N , in the axial double rod are treated as redundant unknowns. The time-dependent mechanical behaviour of the beam under consideration is treated by using the non-linear viscoelastic model depicted in Fig. 2.

Fig. 2. Non-linear viscoelastic model.

The model has equal number of springs and dashpots assembled as depicted in Fig. 2. In the i -th layer of the beam, the stress-strain relationship of the j -th spring of the model is written as ij ij j r ij iE iE E    , (3) where j iE  is the stress, ij iE  is the strain, ij E is the modulus of elasticity, ij r is a material property. The stress in the j -th dashpot in the i -th layer of the beam is expressed as ij ij j s ij i i        , (4) where j i   is the stress, ij i    is the first derivative of the strain with respect to time, ij  is the coefficient of viscosity, ij s is a material property. The model in Fig. 2 is under stress,  , which varies with time according to the following law: v t    , (5) where  v is a parameter controlling the variation of the stress. Apparently,    j iE , (6)     j i . (7) By using (3) and (6), the strain in the j -th spring is found as

1

ij E v t 

  

r   

ij





.

(8)

iE

ij

From (4) and (7), one derives

s v t 1       ij

   

ij   i 



.

(9)

ij

The differential equation (9) is solved with respect to strain. The result is

s



ij

1 1

s

   

s      ij

1 s t s ij

v

ij

C



i 





,

(10)

1



ij 

ij

ij

1 C is the integration constant. Since

where