PSI - Issue 83

Victor Rizov et al. / Procedia Structural Integrity 83 (2026) 105–114

107

where the horizontal and vertical projections, R x a 1 and R z a 1 , are obtained as

dt d x 2 dt d z 2

t  6

a





1

R

,

(4)

R x 1

2

t  6

a





1

R

.

(5)

R z 1

2

Fig. 1. Moving beam and coordinate system.

Equations (4) and (5) can be applied for obtaining the projections of the acceleration of any point, D , of the beam since it moves translationally (these projections are shown in point, D , in Fig. 1). Since the beam studied here is stepwise functionally graded, it is treated as a multilayered structure. The horizontal and vertical projections, Dxi q and Dzi q , of the inertia load in any point of the i -th layer are

Dxi q Dzi q

i Dx ma  , ma  , i Dz

(6) (7)

where i m is the mass per unit length in the layer, the accelerations, Dx a and Dz a , are equal to R x a 1 and R z a 1 since the beam moves translationally. From Eqs. (4), (5), (6) and (7), we have