PSI - Issue 6

A.D. Lovtsov et al. / Procedia Structural Integrity 6 (2017) 122–127 Author name / Structural Integrity Procedia 00 (2017) 000–000

125

4





int

g

 F F Mz 

Dz 

Mz 

Dz 

0,

g

1 n   











1

1

1

1

1

1

1

n

n

n

n

n

n

n















where g are the “static out-of-balance forces” and g the equivalent dynamic out-of-balance forces, int  F internal nodal forces. Axial forces N in the connecting elements are computed applying engineering strain measure. Newton-Raphson iterations are continued until residual vector 1 n  g equals required convergence tolerance. Nodal coordinates of the masses change as the system experiences deformation and current length of the elements can be calculated. Inequalities (2) and (3) are analyzed to define contact region when the convergence criterion is satisfied . If contact region at step 1 n  differs from step n static stiffness matrix should be updated and “predictor-corrector step” should be repeated. Otherwise we move to the next step. 3. Numerical examples and experimental results The model of blasting mat for numerical simulation was taken from the experiment carried out on 12.04.07 on the site of federal highway #58 "Amur" near Teploozersk village. Total of 32 blast holes were made on site with 3 meter increments (Fig. 2, a, b). Each blast hole was covered by car tire (rows 1-3). Two additional rows 4 and 5 (20 tires) were added to ensure safety on the village side. Explosions were made row by row with 0.02 seconds (s) delay starting with row 1. The mass of a single tire was 280 kg, tire internal radios 0.3 m, initial velocity 8.86 m/s. Initial velocity of a single tire after explosive detonation was calculated by Leschinskiy, 2009. The tires were bound together using 6 mm cables. According to the requirements the flight height of blasting mat should not exceed 4 m.

Fig. 2. (a) View of blasting mat; (b) Fragment of the model.

Blasting charges in the row explode simultaneously. Shortly after explosion all tires in this row move straight along the vertical axis of the well with the same acceleration . In this case, the tensile stresses arise only in the transverse elements which connect tires of adjacent rows. This allows us to make an assumption that we can implement planar model instead of spatial. Tires’ movement in a transverse row (marked with a dotted line on Fig. 2, b) is analyzed using planar model . Damping forces are not taken into consideration. Initial velocity of blasting rock and blasting mat are considered to be the same and, as assumed, they are moving upwards together. So, the allowance is made: the impact of the blasting rock on the mat is neglected. Tires movement after the moment when the maximum flight height was reached has no practical interest since flyrock is possible only when the blasted mass moves along with blasting mat upward. The numerical results depend on the time step t  . The convergence of numerical solution for this algorithm with the increase in number of incremental steps and, accordingly, decreasing t  is shown in Table 1. If 2 10 3 t     s, the maximum flight height of the tire in row 1 is 3.908 m, and maximum axial force is 12.82 kN. If 2 10 4 t     s the maximum flight height is 3.892 m, while axial force is 15.05 kN.