PSI - Issue 6

V.A. Meleshko et al. / Procedia Structural Integrity 6 (2017) 140–145 Meleshko V. А ., Rutman Y.L. / StructuralIntegrity Procedia 00 (2017) 000–000

143

4

k

  T k x 22    T k x 22 

     

l

l

  32 





 y x

( )

dx

dx

    



0

x

0

0

T

(6)

k

l

l

 

  32 

  0

  0

y x

    0 0  x x f f

dxdx

dxdx



T

The required initial conditions: at the left restraint f 0 = 1 ,  0 = 0; at the right restraint at x = l , f x = 0,  x = 0. Then we determine k 52 , k 62 from the equilibrium equations: k 22 + k 52 = 0 (7) k 32 + k 62 – k 22  l = 0 (8) The equations system for determination of stiffness factors k 26 , k 36 , k 56 , k 66 at  6 =1:                             l l x l l x dxdx T k dxdx T k x x f f y x dx T k dx T k x y x 0 0 36 26 0 0 0 0 36 26 0 ( )        (9) The required initial conditions: at the left restraint f 0 = 0 ,  0 = 0; at the right restraint at x = l , f x = 0,  x = 1. Then we determine k 56 , k 66 from the equilibrium equations: 0, 56 26   k k (10) 0, 26 66 36    k k k l (11) , 41 11 l k EA k    (12) where A – cross-section area. After determining ij k , the matrix of rod element stiffness in the global coordinates is determined by general method at each step. Then, using the incremental form for load increment at the step, the equations of the flexibility method or the displacement method are solved for the system as a whole and cross-sections moment increments are determined ( ) M x  . Determination of rod length curvature at extreme fibers:       ,   T x M x    (13) increment of extreme fibers deformation:     . 2 x h x      (14) Determination of stresses and deformation at the step:     ( ) x x x       (15)                  , 1 , 1        x E x x x E x pl (16) Then a new local matrix of rod element stiffness is formed by solving the differential equation of the bent beam centre line taking into account change of section stiffness. And the cycle is repeated. 2. Comparison of the Obtained Results with FEA in Software Systems To check the developed mathematical scheme, two test problems were solved by the generalized flexibility method in MathCad (Fig. 4) and the finite elements method in ANSYS. The following initial data were accepted in these problems:       