A cell-Based smoothed discrete shear gap method (cs-fem-dsg3) for dynamic response of laminated composite plate subjected to blast loading

-DSG3 DSG3 Exponential 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -0.4 -0.2 0 0.2 0.4 0.6 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Sine 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) Ti e (sec) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Step 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 triangular . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) Time (sec) Time (sec) Time (sec) 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -36 -18 0 18 36 Khdeir and Reddy FSDT CS-FEM-DSG3 Step 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -32 -16 0 16 32 Khdeir and Reddy FSDT CS-FEM-DSG3 triangular 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -20 -10 0 10 20 Khdeir and Reddy FSDT CS-FEM-DSG3 Sine Exponential 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -30 -15 0 15 30 Khdeir and Reddy FSDT CS-FEM-DSG3 Time (sec) Time (sec) Time (sec) Time (sec) x x  x  x  x  0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -36 -18 0 18 36 Khdeir and Reddy FSDT CS-FEM-DSG3 Step 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -32 -16 0 16 32 Khdeir and Reddy FSDT CS-FEM-DSG3 triangular 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -20 -10 0 10 20 Khdeir and Reddy FSDT CS-FEM-DSG3 Sine Exponential 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -30 -15 0 15 30 Khdeir and Reddy FSDT CS-FEM-DSG3 Time (sec) Time (sec) Time (sec) Time (sec) x x  x  x  x  ( ) b) d) i . . t r flection as a function of time for various pulses loading: a) sine; ) step; c) exponential; d) triangle. tr l eflection as a function of time corresponding to differ nt pulse lo i , S 3 ( eshing 12122 triangular elements) and previous p li t t the results by the CS-FEM-DSG3 agre well with those from ir l tical solution based on the FSDT theory, and outperform those b t t cell-based gradient smo thing technique of the CS-FEM-DSG3 hel s f t e S 3 as shown in Ref [1 ], and hence make the results by the - r te than those by the DSG3. 0 . . 0.005 0.006 0.0 7 0.0 8 -1 -0.5 0 0.5 1 ( ) C en tr al d ef le ct io n ( in ) ti l 0 . . 0.005 0.006 0.0 7 0.0 8 -0.4 -0.2 0 0.2 0.4 0.6 (s c) C en tr al d ef le ct io n ( in ) 0 .001 .002 .003 0.004 .005 .006 .007 . 08 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) Ti e (sec) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Step 0 .001 .002 .003 0.004 .005 .006 .007 . 08 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 triangular . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) ) Time (sec) ) 0 0.001 0.002 0.003 .004 .005 .006 .007 .008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Red y FSDT CS-FEM-DSG3 DSG3 Exponential 0 0.001 0.002 0.003 .004 .005 .006 .007 0.008 -0.4 -0.2 0 0.2 0.4 0.6 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Red y FSDT CS-FEM DSG3 DSG3 Sine 0 0. 01 0. 02 0. 03 0.004 0.005 0.0 6 0.0 7 0.0 8 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) Ti e (sec) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Step 0 0. 01 0. 02 0. 03 0.004 0.005 0.0 6 0.0 7 0.0 8 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n (i n) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 triangular . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) Time (s c) Time (sec) Time (s c) 0 . . 0.005 0.006 0.0 7 0.0 8 -1 -0.5 0 0.5 1 ( c) C en tr al d ef le ct io n ( in ) ti l 0 . . 0.005 0.006 0.0 7 0.0 8 -0.4 -0.2 0 0.2 0.4 0.6 (s c) C en tr al d ef le ct io n ( in ) 0 .001 .002 .003 0.004 .005 .006 .007 . 08 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) Ti e (sec) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Step 0 .001 .002 0.003 0.004 .005 .006 .007 . 08 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 triangular . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) ) Time (sec) ) 0 0.001 0.002 0.003 .004 .005 .006 .007 .008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Red y FSDT CS-FEM-DSG3 DSG3 Exponential 0 0.001 0.002 0.003 .004 .005 .006 .007 0.008 -0.4 -0.2 0 0.2 0.4 0.6 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Red y FSDT CS-FEM DSG3 DSG3 Sine 0 0. 01 0. 02 0. 03 0.004 0.005 0.0 6 0.0 7 0.0 8 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) Ti e (sec) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Step 0 0. 01 0. 02 0. 03 0.004 0.005 0.0 6 0.0 7 0.0 8 -1 -0.5 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 triangular . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) Time (s c) Time (sec) Time (s c) 0 0.001 .002 .003 .004 . 05 0. 06 0. 07 0. 08 -36 -18 0 18 36 Khdeir and Reddy FSDT CS-FEM-DSG3 Step 0 0.001 .002 .003 .004 . 05 0. 06 0. 07 0. 08 -32 -16 0 16 32 Khdeir and Reddy FSDT CS-FEM-DSG3 triangular 0 . . 0.005 0.006 0.0 7 0.0 8 -20 -10 0 10 20 ti l 0 . . 0.005 0.006 0.0 7 0.0 8 -30 -15 0 15 30 Time (sec) ) Time (sec) ) x x  x  x  x  0 0. 01 0. 02 0. 03 0.004 0.0 5 0.0 6 0.0 7 0.0 8 -36 -18 0 18 36 Khdeir and Reddy FSDT CS-FEM-DSG3 Step 0 0. 01 0. 02 0. 03 0.004 0.0 5 0.0 6 0.0 7 0.0 8 -32 -16 0 16 32 Khdeir and Reddy FSDT CS-FEM-DSG3 triangular 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 .008 -20 -10 0 10 20 Khdeir and Reddy FSDT CS-FEM-DSG3 Sine Exponential 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 .008 -30 -15 0 15 30 Khdeir and Reddy FSDT CS-FEM-DSG3 Time (sec) Time (s c) Time (sec) Time (s c) x x  x  x  x  ( a) b) c) d) Fig. 3. Variation of the center deflection as a function of time for various pulses loading: a) sine; b) step; c) exponen al; d) riangle. Fig. 3 shows the central deflection as a function of time corresponding to different pulse loadings by the CS-FEM-DSG3, DSG3 (meshing 12122 triangular elements) and previous publ shed results. It is seen that the results by the CS-FEM-DSG3 gr w ll with those fr m Khdeir and Reddy [1] u ing an lytical o ution based on the FSDT theo y, and outperf rm those by the DSG3. This is because the cell-based gradient smoothing technique of the CS-FEM-DSG3 helps soften the over-stiffness of th DSG3 as shown in Ref [11], a d h nce make the results by t CS-FEM-DSG3 mor a curate than those by the DSG3. 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Exponential 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -0.4 -0.2 0 0.2 0.4 0.6 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Sin 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) Ti e (sec) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Step 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 triangular . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) Time (sec) Time (sec) Time (sec) 0 0.001 0.002 0.00 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Exponential 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -0.4 -0.2 0 0.2 0.4 0.6 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Sin 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) Ti e (sec) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Step 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n (i n) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 triangular . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) Time (sec) Time (sec) Time (sec) 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Exponential 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -0.4 -0.2 0 0.2 0.4 0.6 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Sin 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) Ti e (sec) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 St p 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 triangular . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) Time (sec) Time (sec) Time (sec) 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Ti e (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Exponential 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -0.4 -0.2 0 0.2 0.4 0.6 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Sin 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) Ti e (sec) Khdeir and Re y FSDT CS-FEM-DSG3 DSG3 Step 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 triangular . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) Time (sec) Time (sec) Time (sec) 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -36 -18 0 18 36 Khdeir and Reddy FSDT CS-FEM-DSG3 Step 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -32 -16 0 16 32 Khdeir and Reddy FSDT CS-FEM-DSG3 triangular 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -20 -10 0 10 20 Khdeir and Reddy FSDT CS-FEM-DSG3 Sine Exponential 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -30 -15 0 15 30 Khdeir and Reddy FSDT CS-FEM-DSG3 Time (sec) Time (sec) Time (sec) Time (sec) x x  x  x  x  0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -36 -18 0 18 36 Khdeir and Reddy FSDT CS-FEM-DSG3 Step 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -32 -16 0 16 32 Khdeir and Reddy FSDT CS-FEM-DSG3 triangular 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -20 -10 0 10 20 Khdeir and Reddy FSDT CS-FEM-DSG3 Sine Exponential 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -30 -15 0 15 30 Khdeir and Reddy FSDT CS-FEM-DSG3 Time (sec) Time (sec) Time (sec) Time (sec) x x  x  x  x  (c) b) d) Fig. 3. Variation of t s a function of time for various pulses loading: a) sine; ) onen al; d) r angle. Fig. 3 sho s t i as a function of time corresponding to differ nt pulse lo dings by the - ( eshing 12122 triangular elements) and pr vious published results. It i lts by the CS-FEM-DSG3 ag e well with those fr m Khdeir and Red y [ ] s l tion bas d on the FS T theory, and outperform those by the DSG3. This is radient smoo ing technique of the CS-FEM-DSG3 helps soften the over- ti as shown in Ref [1 ], and hence make the results by the CS-FE - S 3 t se by the DSG3. 0 0. 01 0. 02 0.003 0. . 7 0.008 -1 -0.5 0 0.5 1 Ti e (s C en tr al d ef le ct io n ( in ) Khdeir and Reddy FS CS-FEM-DSG3 DSG3 0 0. 01 0. 02 0.003 0. . 07 0.008 -0.4 -0.2 0 0.2 0.4 0.6 Ti e (s C en tr al d ef le ct io n ( in ) Khdeir and Reddy FS CS-FEM-DSG3 DSG3 0 0.001 0.002 0.003 0.004 0.005 .006 .007 .008 -1 -0.5 0 0.5 1 Tim (sec) C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) Ti e (sec) Khdeir an Reddy FSDT CS-FEM-DSG3 DSG3 Step 0 0.001 0.002 0.003 0.004 0.005 .006 .007 .008 -1 -0.5 0 0.5 1 Tim (sec) C en tr al d ef le ct io n ( in ) Khdeir an Reddy FSDT CS-FEM-DSG3 DSG3 triangular . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) i ( Time (sec) i 0 0.001 .002 .00 .004 0.005 0.006 .007 .008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Exponential 0 0.001 .002 .003 .004 0.005 0.006 .007 .008 -0.4 -0.2 0 0.2 0.4 0.6 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Sine 0 0.0 1 0. 02 0. 03 0. 04 0.005 0.0 6 0.0 7 0.0 8 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) Ti e (sec) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Step 0 0.0 1 0. 02 0. 03 0. 04 0.005 0.0 6 0.0 7 0.0 8 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 triangular . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) Time (sec) Time (sec) Time (sec) 0 0. 01 0. 02 0.003 0. . 07 0.008 -1 -0.5 0 0.5 1 Ti e (s C en tr al d ef le ct io n ( in ) Khdeir and Reddy FS CS-FEM-DSG3 DSG3 0 0. 01 0. 02 0.003 0. . 07 0.008 -0.4 -0.2 0 0.2 0.4 0.6 Ti e ( C en tr al d ef le ct io n ( in ) Khdeir and Reddy FS CS-FEM-DSG3 DSG3 i 0 0.001 0.002 0.003 0.004 0.005 .006 .007 .008 -1 -0.5 0 0.5 1 Tim (sec) C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) Ti e (sec) Khdeir an Reddy FSDT CS-FEM-DSG3 DSG3 St p 0 0.001 0.002 0.003 0.004 0.005 .006 .007 .008 -1 -0.5 0 0.5 1 Tim (sec) C en tr al d ef le ct io n ( in ) Khdeir an Reddy FSDT CS-FEM-DSG3 DSG3 triangular . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) i Time (sec) i 0 0.001 .002 .003 .004 0.005 0.006 .007 .008 -1 -0.5 0 0.5 1 ime (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Exponential 0 0.001 .002 .003 .004 0.005 0.006 .007 .008 -0.4 -0.2 0 0.2 0.4 0.6 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 S ne 0 0.0 1 0. 02 0. 03 0. 04 0.005 0.0 6 0.0 7 0.0 8 -1 - . 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) Ti e (sec) Khdeir and Red y FSDT CS-FEM-DSG3 DSG3 Step 0 0.0 1 0. 02 0. 03 0. 04 0.005 0.0 6 0.0 7 0.0 8 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 triangular . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) Time (sec) Time (sec) Time (sec) 0 0.001 0.002 .003 .004 .005 0.006 .007 .008 -36 -18 0 18 36 Khdeir and Reddy FSDT CS-FEM-DSG3 Step 0 0.001 0.002 .003 .004 .005 0.006 .007 .008 -32 -16 0 16 32 Khdeir and Reddy FSDT CS-FEM-DSG3 triangular 0 0. 01 0. 02 0.003 0. . 07 0.008 -20 -10 0 10 20 Khdeir and Reddy FSDT CS-FEM-DSG3 i 0 0. 01 0. 02 0.003 0. . 07 0.008 -30 -15 0 15 30 Khdeir and Reddy FSDT CS-FEM-DSG3 Time (sec) i Time (sec) i x x  x  x  x  0 0. 1 0. 02 0. 03 0.004 0.005 0.0 6 0.0 7 0.0 8 -36 -18 0 18 36 Khdeir and Reddy FSDT CS-FEM-DSG3 Step 0 0. 1 0. 02 0. 03 0.004 0.005 0.0 6 0.0 7 0.0 8 -32 -16 0 16 32 Khdeir and Reddy FSDT CS-FEM-DSG3 triangular 0 0.001 0.002 0.003 0.004 .005 .006 .007 0.008 -20 -10 0 10 20 Khdeir and Red y FSDT CS-FEM-DSG3 Sine Exponential 0 0.001 0.002 0.003 0.004 .005 .006 .007 0.008 -30 -15 0 15 30 Khdeir and Red y FSDT CS-FEM-DSG3 Time (sec) Time (s c) Time (sec) Time (s c) x x  x  x  x  (d) Fig. 3. Variation of the center deflection as a function of time for various pulses loading: a) sine; b) step; c) exponential; d) triangle a) b) c) d) Fig. 3. Variation of the center deflection as a func ion of time for various pulses loading: a) sine; b) step; c) exponen ial; d) triangle. Fig. 3 shows the central deflection as a function of time corresponding to different pulse loadings by the CS-FEM-DSG3, DSG3 (meshing 12122 triangular elements) and previous published results. It is s en that the results by the CS-FEM-DSG3 agree well with those from Khdeir and Reddy [1] using analytical solution based on the FSDT th ory, and outperform those by the DSG3. This is because the cell-based gradient smoothing technique of the CS-FEM-DSG3 helps soften the over-stiffness of the DSG3 as shown in Ref [11], and hence make the results by the CS-FEM-DSG3 more accurate than those by the DSG3. 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Exponential 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -0.4 -0.2 0 0.2 0.4 0.6 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Sine 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) Ti e (sec) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Step 0 0.001 0.0 2 0.003 4 . 5 0. 6 0. 7 . 08 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n (i n) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 triangular . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) Time (sec) Time (sec) Time (sec) 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Exponential 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -0.4 -0.2 0 0.2 0.4 0.6 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Sine 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n (i n) Ti e (sec) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Step 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n (i n) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 triangular . C en tr al d ef le ct io n (i n) . C en tr al d ef le ct io n (i n) . C en tr al d ef le ct io n (i n) Time (sec) Time (sec) Time (sec) 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Exponential 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -0.4 -0.2 0 0.2 0.4 0.6 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Sine 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) Ti e (sec) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Step 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n (i n) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 triangular . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) Time (sec) Time (sec) Time (sec) 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Exponential 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -0.4 -0.2 0 0.2 0.4 0.6 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Sine 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) Ti e (sec) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Step 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 triangular . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) Time (sec) Time (sec) Time (sec) 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -36 -18 0 18 36 Khdeir and Reddy FSDT CS-FEM-DSG3 Step 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -32 -16 0 16 32 Khdeir and Reddy FSDT CS-FEM-DSG3 triangular 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -20 -10 0 10 20 Khdeir and Reddy FSDT CS-FEM-DSG3 Sine Exponential 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -30 -15 0 15 30 Khdeir and Reddy FSDT CS-FEM-DSG3 Time (sec) Time (sec) Time (sec) Time (sec) x x  x  x  x  0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -36 -18 0 18 36 Khdeir and Reddy FSDT CS-FEM-DSG3 Step 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -32 -16 0 16 32 Khdeir and Reddy FSDT CS-FEM-DSG3 triangular 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -20 -10 0 10 20 Khdeir and Reddy FSDT CS-FEM-DSG3 Sine Exponential 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -30 -15 0 15 30 Khdeir and Reddy FSDT CS-FEM-DSG3 Time (sec) Time (sec) Time (sec) Time (sec) x x  x  x  x  (a) a) b) c) d) Fig. 3. Variation of the center deflection as a function of time for various pulses loading: a) sine; b) step; c) exponential; d) triangle. Fig. 3 shows the central deflection as a function of time corresponding to different pulse loadings by the CS-FEM-DSG3, DSG3 (meshing 12122 triangular l ments) and previous published results. It is seen that the results by the CS-FEM-DSG3 agr e well with those from Khdeir and Reddy [1] using a alytical solution based on the FSDT theory, and outperform those by the DSG3. This is because the cell-based gradient smoothing technique of the CS-FEM-DSG3 helps soften the over-stiffnes of the DSG3 as shown in Ref [11], and henc make the results by the CS-FEM-DSG3 more accurate than those by the DSG3. 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Exponential 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -0.4 -0.2 0 0.2 0.4 0.6 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Sine 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n (i n) Ti e (sec) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Step 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n (i n) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 triangular . C en tr al d ef le ct io n (i n) . C en tr al d ef le ct io n (i n) . C en tr al d ef le ct io n (i n) Time (sec) Time (sec) Time (sec) 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n (i n) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Exponential 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -0.4 -0.2 0 0.2 0.4 0.6 Time (sec) C en tr al d ef le ct io n (i n) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Sine 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n (i n) . C en tr al d ef le ct io n (i n) Ti e (s c) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Step 0 0.001 0. 02 0.0 3 0.004 . .006 0. 07 .008 -1 -0.5 0 0.5 1 Tim (sec) C en tr al d ef le ct io n (i n) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 triangular . C en tr al d ef le ct io n (i n) . C en tr al d ef le ct io n (i n) . C en tr al d ef le ct io n (i n) Time (sec) i e ( c) Time (sec) 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Exponential 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -0.4 -0.2 0 0.2 0.4 0.6 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Sine 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n (i n) Ti e (sec) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Step 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n (i n) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 triangular . C en tr al d ef le ct io n (i n) . C en tr al d ef le ct io n (i n) . C en tr al d ef le ct io n (i n) Time (sec) Time (sec) Time (sec) 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Exponential 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -0.4 -0.2 0 0.2 0.4 0.6 Time (sec) C en tr al d ef le ct io n ( in ) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Sine 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) Ti e (s c) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 Step 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 Time (sec) C en tr al d ef le ct io n (i n) Khdeir and Reddy FSDT CS-FEM-DSG3 DSG3 triangular . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) . C en tr al d ef le ct io n ( in ) Time (sec) Time (sec) Time (sec) 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -36 -18 18 36 Khdeir and Reddy FSDT CS-FEM-DSG3 Step 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -32 -16 0 16 32 Khdeir and Reddy FSDT CS-FEM-DSG3 triangular 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -20 -10 0 10 20 Khdeir and Reddy FSDT CS-FEM-DSG3 Sine Exponential 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -30 -15 0 15 30 Khdeir and Reddy FSDT CS-FEM-DSG3 Time (sec) Time (sec) Time (sec) Time (sec) x x  x  x  x  0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -36 -18 0 18 36 Khdeir and Reddy FSDT CS-FEM-DSG3 Step 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -32 -16 0 16 32 Khdeir and Reddy FSDT CS-FEM-DSG3 triangular 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -20 -10 0 10 20 Khdeir and Reddy FSDT CS-FEM-DSG3 Sine Exponential 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -30 -15 0 15 30 Khdeir and Reddy FSDT CS-FEM-DSG3 Time (sec) Time (sec) Time (sec) Time (sec) x x  x  x  x  (b) a) b) c) d) Fig. 4. Variation of the normal stress x x  (a/2, a/2, h/2)/q0 as a function of time for various pulse loadings: a) sine; b) step; c) exponential; d) triangle. Similarly, Fig. 4 presents the normal stress as a function of time for the various pulse loadings, obtained by th CS-FEM-DSG3 and by Khdeir and Reddy [1]. The two results match perfectly. This hence illustrates again the accuracy and robustness of the present method. a) b) Fig. 5. Central deflection and normal stress of the plate by the CS-FEM-DSG3 (as a function of time for exponential pulse loading) subjected to various number of layers and fiber orientations; (a) Central deflection (in) and b) Normal stress x x  (a/2, a/2, h/2)/q0. Fig. 5 illustrates the central deflection and normal stress of the plate by the CS-FEM- DSG3 (as a function of time for exponential pulse loading) subjected to various numbers of layers and fiber orientations. It can be seen that the plate stiffness is directly proportional to the number of layers, in which the cross-ply contributes less stiffness than the angle-ply. The example demonstrates the dependence of the plate stiffness to the number of layers and fiber orientations, and hence an optimization problem for determining the optimal number of layers and fiber orientations should be applied to optimize the stiffness of the composite laminated plate. 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -36 -18 0 18 36 Khdeir and Reddy FSDT CS-FEM-DSG3 Step 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -32 -16 0 16 32 Khdeir and Reddy FSDT CS-FEM-DSG3 triangular 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -20 -10 0 10 20 Khdeir and Reddy FSD CS-FEM-DS 3 Sine Exponential 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -30 -15 0 15 30 Khdeir and Reddy FSDT CS-FEM-DSG3 Time (sec) Time (sec) Time (sec) Time (sec) x x  x  x  x  0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -36 -18 0 18 36 Khdeir n Reddy FSDT CS-FEM-DSG3 Step 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 - - 32 ir and Reddy FSDT - - S 3 triangular 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -20 -10 0 10 20 Khdeir and Reddy FSDT CS-FEM-DSG3 Sine Expo ential 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -30 -15 0 15 30 Khdeir and Reddy FSDT CS-FEM-DSG3 i e (sec) Time (sec) i e (sec) Time (sec) x x  x  x  x  0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 00/900/00 00/900/900/00 450/-450/450 450/-450/-450/450 . C en tr al d ef le ct io n (i n) Time (sec) 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -30 -15 0 15 30 Time (sec) 00/900/00 00/900/900/00 450/-450/450 450/-450/-450/450 x  0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 00/900/00 00/900/900/00 450/- 0/450 450/-450/-450/450 . C en tr al d ef le ct io n (i n) Time (sec) 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -30 -15 0 15 30 Time (sec) 00/900/00 00/90 /90 /00 450/-450/450 450/-450/-450/450 x  ( ) a) b) c) d) Fig. 4. Variation of the normal stress x x  (a/2, a/2, h/2)/q0 s a function of time for various pulse loadings: a) sine; b) step; c) exponential; d) triangle. Similarly, Fig. 4 pres nts he normal stress a function f time for the various p lse loadings, obtained by the CS-FEM-DSG3 and by Khdeir and Reddy [1]. The two results match perfectly. This hence illustrates ag in the accuracy and robustness of he pr sent method. a) b) Fig. 5. Central deflection and normal stress of the plate by the CS-FEM-DSG3 (as a function of time for expone tial pulse loading) subject d to various number of layers and fiber orientations; (a) Central defl ction (in) and b) Normal stress x x  (a/2, a/2, h/2)/q0. Fig. 5 illustrates the central defl ction a d ormal stre s of the plate by the CS-FEM- DSG3 (as a function of time for exponential pulse loading) subjected to various n mbers of layers and fiber orientations. It can be s en tha the plate s iffness i d rectly proportional to the number of layers, in which the cross-ply contributes less tiffness than the angle-ply. The example demonstrates the dep ndence of the plate stiffness to the number of layers and fiber orienta ions, and hence an optim zation problem for determining the optimal number of layers and fiber orientations hould be applied to optimize the stiffness of the composite laminated plate. 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -36 -18 0 18 36 Khdeir and Reddy FSDT CS-FEM-DSG3 Step 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -32 -16 0 16 32 Khdeir and Reddy FSDT CS-FEM-DSG3 triangular 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -20 -10 0 10 20 Khdeir and Reddy FSDT CS-FEM-DSG3 Sine Exponential 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -30 -15 0 15 30 Khdeir and Reddy FSDT CS-FEM-DSG3 Time (sec) Time (sec) Time (sec) Time (sec) x x  x  x  x  0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -36 -18 0 18 36 Khdeir and Reddy FSDT CS-FEM-DSG3 Step 0 .001 0.002 0.003 0.004 .005 0.006 .007 .008 -32 -16 0 16 32 Khdeir and Reddy FSDT CS-FEM-DSG3 triangular 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -20 -10 0 10 20 Khdeir and Reddy FSDT CS-FEM-DSG3 Sine Exponential 0 0.001 .002 .003 .004 .005 .006 .007 .008 -30 -15 0 15 30 Khdeir and Reddy FSDT CS-FEM-DSG3 Time (sec) Time (sec) Time (sec) Time (sec) x x  x  x  x  0 .0 1 .0 2 .003 . 04 0. 05 0. 06 0. 07 0.008 -1 -0.5 0 0.5 1 00/900/ 0 00/900/900/00 450/-450/ 450/-450/-450/ . C en tr al d ef le ct io n (i n) Time (sec) . . . . 04 . 05 0. 06 0. 7 0.0 8 - - i (sec) / 0/ 0/ 450/-450/450 450/-450/-450/450 x  0 0.0 1 0.0 2 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 00/900/ 00/900/900/ 0 450/- 0/45 450/-450/-450/ . C en tr al d ef le ct io n ( in ) Time (sec) 0 0.001 0.002 .003 .004 .005 0. 06 0. 07 0. 08 -30 -15 0 15 30 Time (sec) 00/900/00 00/900/900/00 450/-450/450 450/-450/-450/450 x  (d) Fig. 4. Variation of the normal stress σ¯x = σx(a/2, a/2, h/2)/q0 as a function of time for various pulse loadings: a) sine; b) step; c) exponent l; ) triangle 88 Dang Trung Hau, Nguyen Thoi My Hanh, Nguyen Thoi Trung Similarly, Fig. 4 presents the normal stress as a function of time for the various pulse loadings, obtained by the CS-FEM-DSG3 and by Khdeir and Reddy [1]. The two results match perfectly. This hence illustrates again the accuracy and robustness of the present method. a) b) c) d) Fig. 4. Variation of the normal stress x x  (a/2, a/2, h/2)/q0 as a function of time for various pulse loadings: a) sine; b) step; c) exponential; d) triangle. Similarly, Fig. 4 presents the normal stress as a function of time for the various pulse loadings, obtained by the CS-FEM-DSG3 and by Khdeir and Reddy [1]. The two results match perfectly. This hence illustrates again the accuracy and robustness of the present method. a) b) Fig. 5. Central deflection and normal stress of the plate by the CS-FEM-DSG3 (as a function of time for exponential pulse loading) subjected to various number of layers and fiber orientations; (a) Central deflection (in) and b) Normal stress x x  (a/2, a/2, h/2)/q0. Fig. 5 illustrates the central deflection and normal stress of the plate by the CS-FEM- DSG3 (as a function of time for exponential pulse loading) subjected to various numbers of layers and fiber orientations. It can be seen that the plate stiffness is directly proportional to the number of layers, in which the cross-ply contributes less stiffness than the angle-ply. The example demonstrates the dependence of the plate stiffness to the number of layers and fiber orientations, and hence an optimization problem for determining the optimal number of layers and fiber orientations should be applied to optimize the stiffness of the composite laminated plate. 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -36 -18 0 18 36 Khdeir and Reddy FSDT CS-FEM-DSG3 Step 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -32 -16 0 16 32 Khdeir and Reddy FSDT CS-FEM-DSG3 triangular 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -20 -10 0 10 20 Khdeir and Reddy FSDT CS-FEM-DSG3 Sine Exponential 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -30 -15 0 15 30 Khdeir and Reddy FSDT CS-FEM-DSG3 Time (sec) Time (sec) Time (sec) Time (sec) x x  x  x  x  0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -36 -18 0 18 36 Khdeir and Reddy FSDT CS-FEM-DSG3 Step 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -32 -16 0 16 32 Khdeir and Reddy FSDT CS-FEM-DSG3 triangular 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -20 -10 0 10 20 Khdeir and Reddy FSDT CS-FEM-DSG3 Sine Exponential 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -30 -15 0 15 30 Khdeir and Reddy FSDT CS-FEM-DSG3 Time (sec) Time (sec) Time (sec) Time (sec) x x  x  x  x  0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 00/900/00 00/900/900/00 450/-450/450 450/-450/-450/450 . C en tr al d ef le ct io n ( in ) Time (sec) 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -30 -15 0 15 30 Time (sec) 00/900/00 00/900/900/00 450/-450/450 450/-450/-450/450 x  0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 00/900/00 00/900/900/00 450/-450/450 450/-450/-450/450 . C e n tr a l d e fl e c ti o n ( in ) Time (sec) 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -30 -15 0 15 30 Time (sec) 00/900/00 00/90 /90 /00 450/-450/450 450/-450/-450/450 x  (a) a) b) c) d) Fig. 4. Variation of the normal stress x x  (a/2, a/2, h/2)/q0 as a function of ti f pulse loadings: a) sine; b) step; c) exponential; d) triangle. Similarly, Fig. 4 presents the normal stres as a function of ti e for t ri loadings, obtained by the CS-FEM-DSG3 and by Khdeir and Reddy [1]. The t r l perfectly. This hence illustrates again the ac uracy and robustnes of the present t . a) b) Fig. 5. Central deflection and normal stress of the plate by the CS-FE - S 3 (as a f cti f time for exponential pulse loading) subjected to various number of layers and fiber orientations; (a) Central deflection (in) and b) Normal stress x x  (a/2, a/2, h/2)/q0. Fig. 5 illustrates the central deflection and normal stress of the plate by the S-FE - DSG3 (as a function of time for exponential pulse loading) subjected to various nu bers of layers and fiber orientations. It can be seen that the plate stiffness is directly proportional to the number of layers, in which the cross-ply contributes less stiffness than the angle-ply. The example demonstrates the dependence of the plate stiffness to the number of layers and fiber orientations, and hence an optimization problem for determining the optimal number of layers and fiber orientations should be applied to optimize the stiffness of the composite laminated plate. 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -36 -18 0 18 36 Khdeir and Reddy FSDT CS-FEM-DSG3 Step 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -32 -16 0 16 32 Khdeir and Reddy FSDT CS-FEM-DSG3 triangular 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -20 -10 0 10 20 Khdeir and Reddy FSDT CS-FEM-DSG3 Sine Exponential 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -30 -15 0 15 30 Khdeir and Reddy FSDT CS-FEM-DSG3 Time (sec) Time (sec) Time (sec) Time (sec) x x  x  x  x  0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -36 -18 0 18 36 Khdeir and Reddy FSDT CS-FEM-DSG3 Step 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -32 -16 0 16 32 Khdeir and Reddy FSDT CS-FEM-DSG3 triangular 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -20 -10 0 10 20 Khdeir and Reddy FSDT CS-FEM-DSG3 Sine xponential 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -30 -15 0 15 30 Khdeir and Reddy FSDT CS-FEM-DSG3 Time (sec) Time (sec) Time (sec) i e (sec) x x  x  x  x  0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -1 -0.5 0 0.5 1 00/9 0/ 0 00/9 0/9 0/ 0 450/-450/450 450/-450/-450/450 . C en tr al d ef le ct io n ( in ) Time (sec) 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 -30 -15 0 15 30 Ti e (sec) 00/9 0/00 00/900/900/00 450/-450/ 0 450/-450/- 0/ 0 x  0 0. 01 2 3 . 4 . 05 . 6 0.007 0.008 -1 -0.5 0 0.5 1 00/900/ 0 00/9 0/9 0/ 0 450/-450/450 450/-450/-450/450 . C en tr al d ef le ct io n ( in ) Time (sec) 0 0.001 0. 02 0. 03 0. 04 0. 05 0. 06 . 07 . 08 -30 -15 0 15 30 Time (sec) 00/900/00 00/90 /90 /00 450/-450/450 450/-450/-450/ 0 x  (b) Fig. 5. Central deflectio and normal stress of the plate by the CS-FEM-DSG3 (as a function of time for exponential pulse loading) subjected to various number of layers and fiber orientations; (a) Central deflection (in) and b) Normal stress σ¯x = σx(a/2, a/2, h/2)/q0 Fig. 5 illustrates the central deflection and normal stress of the plate by the CS- FEM-DSG3 (as a function of time for exponential pulse loading) subjected to various numbers of layers and fiber orientations. It can be seen that the plate stiffness is directly proportional to the number of layers, in which the cross-ply contributes less stiffness than the angle-ply. The example demonstrates the dependence of the plate stiffness to the number of layers and fiber orientations, and hence an optimization problem for deter- mining the optimal number of layers and fiber orientations should be applied to optimize the stiffness of the composite laminated plate. 5. CONCLUSION In this paper, the cell-based smoothed discrete shear gap method (CS-FEM-DSG3) is extended to investigate the dynamic response of laminated composite plate under the effect of blast loading modeled by some trigonometric time functions. Numerical results demonstrate that the proposed method can achieve accurate results by using only a rel- ative coarse mesh. Moreover, numerical examples also demonstrate the direct effect of the number of layers and the fiber orientation to the stiffness of plate, so an optimization algorithm should be applied to determine the optimal layer number and fiber orienta- tions in the composite laminated problem. In addition, the new results of the numerical example which are used to demonstrate the effect of the layer’s number and the fiber orientation to the dynamic response of the plate under the effect of blast loading can be served as reliable benchmark examples for later studies. The present CS-FEM-DSG3 is promising to extend to the problems with more complicated geometry domains and boundary conditions without existing available analytical solutions. 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