PSI- Issue 9

I. Shardakov et al. / Procedia Structural Integrity 9 (2018) 207–214 Author name / Structural Integrity Procedia 00 (2018) 000–000

211

5

   k



1 2

T

k   L F F I is the deformation tensor k 

where k ε is the deformation tensor in the moving coordinate system;

in the original coordinate system; I is the unit tensor. – Physical equations, which take into account plastic deformations and are given in increments   , k k k k pl d d d   σ D ε ε ,

(4)

k D is the tensor of elastic constants;

k d σ is the stress tensor increment;

k d ε the deformation tensor increment;

where

( , ) σ

Q w



ε

, k pl

d





is the plastic deformation tensor increment, Q is yield surface (this form corresponds to the

σ



associated plastic flow law); w is the work of plastic deformations. – C ontact conditions on the adjacent (beam-column, bolt-beam, and bolt-column) surfaces are specified in terms of the dry friction model and are given by two inequalities, of which one expresses the condition of non-penetration of surfaces

2 k k k     U U n 1 1 2 ( ) k

0

(5)

and the other describes a limitation to the tangential force on the contact area

fr K P    ,

(6)

where 1 k U and 2 k U are the displacement vectors at the contact points of adjacent elements with numbers k 1 and k 2, respectively; 1 2 k k  n is the normal vector to the contact surface calculated with respect to the body k 1 in the contact area of bodies k 1 and k 2, 1 k σ is the stress tensor on the contact surface of the body k 2, 1 1 2 1 2 k k k k k P      σ n n is the normal stress at the contact point; 1 1 2 1 2 k k k k k P       σ n n  is the tangential stress at the contact point; fr K is the friction coefficient. - Boundary conditions

,

(7)

k X U

k

f



u

k

S



u

,

(8)

k

k

f







k

X

S





where  is the stress vector distribution function on the surface under the action of external forces are applied (for the body number k ). The mathematical model described above is solved numerically by the finite element method. Fig. 4 shows the finite element mesh. The elements are based on the quadratic approximation of displacements. The number of nodes in the design scheme is 52491. The size of the element in the contact area is 2 centimeters. 4. Comparison of calculations and experimental results Five samples of flange connections were studied. The results obtained during the first experiments allowed us to make the developed mathematical model more accurate. In particular, the conditions for contact interaction were specified, and corrections to the parameters of constitutive equations were made. The measured and calculated deformation parameters are given below. These data were obtained for the jacks under the action of the external force   F t varying according to the law illustrated in Fig. 3. Fig. 5 shows the time variation of the averaged value of vertical displacements at points U1 and U3 on the horizontal beams relative to the displacement of the vertical column at point U2 (Fig. 2): k u f is the displacement distribution function on the contact surface (for the body number k ); k f