Fig. 1

Individual FGM beam model (color online)"

Fig. 2

General FGM stepped beam model (color online)"

Fig. 3

Deformations of neutral plane: (a) before bending; (b) after bending (color online)"

Fig. 4

Two different three-level FGM stepped beam models (color online)"

Table 1

Geometric parameters for Model 1 and Model 2"

Table 2

Natural frequencies ωi and errors for Model 1 with C-C ends under different power-law indices"

Table 3

Natural frequencies ωi and errors for Model 2 with C-C ends under different power-law indices"

Fig. 5

Mode shapes of FGM stepped beam with C-C ends when power-law index is 0.3: (a) Model 1 by TMM; (b) Model 1 by ABAQUS; (c) Model 2 by TMM; (d) Model 2 by ABAQUS (color online)"

Fig. 6

Natural frequencies ωi of Model 1 with different boundary conditions versus power-law index (color online)"

Table 4

Bending stiffness of Level 1 for Model 1 with C-C ends under different power-law indices"

Table 5

Natural frequencies ωi for Model 1 with C-C ends under different cross-sectional sizes"

Table 6

Natural frequencies ωi for Model 1 with S-S ends under different cross-sectional sizes"

Table 7

Natural frequencies ωi for Model 1 with C-S ends under different cross-sectional sizes"

Table 8

Natural frequencies ωi for Model 1 with C-F ends under different cross-sectional sizes"

Fig. 7

Natural frequencies ωi of Model 1 with different boundary conditions versus cross-sectional sizes of Level 1 when power-law index is 0.3 (color online)"

Fig. 8

Natural frequencies ωi of Model 1 with different boundary conditions versus step locations when power-law index is 0.3 (color online)"

Fig. 9

Steady response amplitudes of Model 1 with different boundary conditions versus power-law index (color online)"

Fig. 10

Steady response amplitudes of Model 1 with different boundary conditions versus cross-sectional sizes of Level 1 when power-law index is 0.3 (color online)"

Fig. 11

Steady response amplitudes of Model 1 with different boundary conditions versus step locations when power-law index is 0.3 (color online)"

Fig. 12

Two different five-level FGM stepped beam models (color online)"

Table 9

Geometric parameters for Model A and Model B"

Table 10

Natural frequencies ωi and errors for Model A with C-C ends under different power-law indices"

Table 11

Natural frequencies ωi and errors for Model B with C-C ends under different power-law indices"

Fig. 13

Mode shapes of FGM stepped beam with C-C ends when power-law index is 0.3: (a) Model A by TMM; (b) Model A by ABAQUS; (c) Model B by TMM; (d) Model B by ABAQUS (color online)"