Luận án Nghiên cứu ứng xử tĩnh, ổn định và dao động dầm composite với tiết diện khác nhau

beams ( L/ h 5 ) with all BCs. 
 60 
 h
 /
 z
 a. 00/900 b. 00/900/00 
 Figure 4.1. Distribution of nondimensional transverse displacement through the 
thickness of (00/900) and (00/900/00) composite beams with S-S boundary condition 
 (MAT II.4). 
 0.5 0.5
 L/h=5 L/h=5
 L/h=10 L/h=10
 h
 0 L/h=50 / 0 L/h=50
 z
 -0.5 -0.5
 11 12 13 14 15 16 2 3 4 5 6 7
 Nondimensional transverse displacement Nondimensional transverse displacement 
 a. 00/900 b. 00/900/00 
 Figure 4.2. Distribution of nondimensional transverse displacement through the 
thickness of (00/900) and (00/900/00) composite beams with C-F boundary condition 
 (MAT II.4). 
 61 
 h
 /
 z
 a. 00/900 b. 00/900/00 
 Figure 4.3. Distribution of nondimensional transverse displacement through the 
thickness of (00/900) and (00/900/00) composite beams with C-C boundary condition 
 (MAT II.4). 
4.3.2. Angle-ply beams 
 This example is extended from previous one. The ( 00 / / 0 0 ) and ( 00 /  ) 
beams are considered. Tables 4.8, 4.9, 4.10 and 4.11 present variation of 
nondimensional fundamental frequencies, critical buckling loads, mid-span 
displacements ( x L/ 2, z 0 ) and stresses of beams used Quasi-3D theory respect 
to the angle-ply of beams. It can be seen that the present results in Table 4.10 and 
4.11 are close with those of Vo et al. [19]. 
 62 
Table 4.8. Nondimensional fundamental frequencies of (00/ /00) and (00/ ) 
composite beams (MAT I.4, E1/E2 = 40). 
 Lay-up BC L/h Angle-ply ( ) 
 00 300 600 900 
 00/ /00 S-S 5 9.5498 9.4487 9.2831 9.2083 
 10 13.9976 13.8130 13.6729 13.6099 
 50 17.7844 17.4788 17.4558 17.4493 
 C-F 5 4.3628 4.3047 4.2484 4.2231 
 10 5.6259 5.5403 5.5059 5.4909 
 50 6.3803 6.2697 6.2635 6.2622 
 C-C 5 12.0240 11.8341 11.6020 11.4992 
 10 20.4355 20.1923 19.8335 19.6723 
 50 38.4410 37.8172 37.6863 37.6333 
 00/ S-S 5 9.5498 6.8336 6.2215 6.1400 
 10 13.9976 7.9772 7.0561 6.9475 
 50 17.7844 8.5069 7.4191 7.2971 
 C-F 5 4.3628 2.7077 2.4173 2.3819 
 10 5.6259 2.9428 2.5837 2.5428 
 50 6.3803 3.0359 2.6474 2.6043 
 C-C 5 12.0240 10.4823 10.0347 9.9435 
 10 20.4355 15.0934 13.8389 13.6637 
 50 38.4410 19.0914 16.6995 16.4323 
 63 
Table 4.9. Nondimensional critical buckling loads of (00/ /00) and (00/ ) composite 
beams (MAT I.4, E1/E2 = 40). 
 Lay-up BC L/h Angle-ply ( ) 
 00 300 600 900 
 00/ /00 S-S 5 9.2665 9.0709 8.7542 8.6131 
 10 19.9125 19.3911 18.9974 18.8217 
 50 32.0563 30.9641 30.8826 30.8596 
 C-F 5 4.9708 4.8413 4.7432 4.6994 
 10 7.0644 6.8417 6.7856 6.7622 
 50 8.1715 7.8897 7.8762 7.8739 
 C-C 5 12.7118 12.3736 11.8718 11.6517 
 10 37.0660 36.2838 35.0169 34.4524 
 50 119.0990 115.2235 114.5322 114.2601 
 00/ S-S 5 9.2665 4.8298 4.0242 3.9211 
 10 19.9125 6.5116 5.1007 4.9455 
 50 32.0563 7.3361 5.5800 5.3981 
 C-F 5 4.9708 1.6246 1.2717 1.2333 
 10 7.0644 1.7845 1.3672 1.3236 
 50 8.1715 1.8418 1.3998 1.3543 
 C-C 5 12.7118 9.6517 8.7780 8.6154 
 10 37.0660 19.3221 16.1028 15.6927 
 50 119.0990 28.8970 22.0782 21.3712 
 64 
Table 4.10. Nondimensional mid-span displacements of (00/ /00) and (00/ ) 
composite beams under a uniformly distributed load (MAT II.4). 
 Lay-up BC L/h Reference Angle-ply ( ) 
 00 300 600 900 
 00/ /00 S-S 5 Present 1.7930 2.0140 2.3030 2.4049 
 Vo et al. [19] 1.7930 2.0140 2.3030 2.4049 
 10 Present 0.9222 0.9946 1.0700 1.0965 
 Vo et al. [19] 0.9222 0.9946 1.0700 1.0965 
 50 Present 0.6370 0.6608 0.6650 0.6661 
 Vo et al. [19] 0.6370 0.6608 0.6650 0.6661 
 C-F 5 Present 5.2683 5.8705 6.5930 6.8442 
 Vo et al. [19] 5.2774 5.8804 6.6029 6.8541 
 10 Present 2.9647 3.1810 3.3871 3.4511 
 Vo et al. [19] 2.9663 3.1828 3.3889 3.4605 
 50 Present 2.1599 2.2402 2.2529 2.2562 
 Vo et al. [19] 2.1602 2.2405 2.2531 2.2565 
 C-C 5 Present 1.0866 1.2616 1.4711 1.5431 
 Vo et al. [19] 1.0998 1.2670 1.4766 1.5487 
 10 Present 0.3958 0.4459 0.5098 0.5323 
 Vo et al. [19] 0.3968 0.4469 0.5108 0.5332 
 50 Present 0.1367 0.1431 0.1462 0.1473 
 Vo et al. [19] 0.1367 0.1431 0.1462 0.1472 
 00/ S-S 5 Present 1.7930 3.6681 4.6312 4.7645 
 Vo et al. [19] 1.7930 3.6634 4.6135 4.7346 
 10 Present 0.9222 2.7463 3.6070 3.6942 
 Vo et al. [19] 0.9222 2.7403 3.5871 3.6626 
 50 Present 0.6370 2.4454 3.2725 3.3446 
 Vo et al. [19] 0.6370 2.4406 3.2540 3.3147 
 C-F 5 Present 5.2683 11.6981 14.8708 15.2595 
 65 
 Lay-up BC L/h Reference Angle-ply ( ) 
 00 300 600 900 
 Vo et al. [19] 5.2774 11.6830 14.8020 15.1540 
 10 Present 2.9647 9.1667 12.0630 12.3387 
 Vo et al. [19] 2.9663 9.1499 12.0020 12.2440 
 50 Present 2.1599 8.3044 11.1059 11.3428 
 Vo et al. [19] 2.1602 8.2916 11.0540 11.2580 
 C-C 5 Present 1.0866 1.5673 1.8524 1.9164 
 Vo et al. [19] 1.0998 1.5755 1.8575 1.9193 
 10 Present 0.3958 0.7797 0.9771 1.0050 
 Vo et al. [19] 0.3968 0.7783 0.9726 0.9983 
 50 Present 0.1367 0.4987 0.6646 0.6790 
 Vo et al. [19] 0.1367 0.4974 0.6608 0.6733 
Table 4.11. Nondimensional stresses of (00/ /00) and (00/ ) composite beams with 
S-S boundary condition under a uniformly distributed load (MAT II.4). 
 Lay-up L/h Reference Angle-ply ( ) 
 00 300 600 900 
 a. Normal axial stress 
 00/ /00 5 Present 0.9556 1.0062 1.0556 1.0732 
 Vo et al. [19] 0.9498 1.0010 1.0500 1.0670 
 10 Present 0.7998 0.8325 0.8459 0.8504 
 Vo et al. [19] 0.8002 0.8326 0.8459 0.8502 
 50 Present 0.7520 0.7785 0.7803 0.7806 
 Vo et al. [19] 0.7523 0.7788 0.7806 0.7809 
 00/ 5 Present 0.9556 0.3736 0.2476 0.2380 
 Vo et al. [19] 0.9498 0.3746 0.2510 0.2428 
 10 Present 0.7998 0.3655 0.2445 0.2346 
 66 
 Lay-up L/h Reference Angle-ply ( ) 
 00 300 600 900 
 Vo et al. [19] 0.8002 0.3661 0.2464 0.2375 
 50 Present 0.7520 0.3631 0.2436 0.2336 
 Vo et al. [19] 0.7523 0.3633 0.2449 0.2358 
 b. Shear stress 
 00/ /00 5 Present 0.6668 0.5721 0.4456 0.4013 
 Vo et al. [19] 0.6679 0.5729 0.4462 0.4017 
 10 Present 0.7078 0.6070 0.4751 0.4286 
 Vo et al. [19] 0.7100 0.6088 0.4762 0.4295 
 50 Present 0.7439 0.6377 0.5006 0.4521 
 Vo et al. [19] 0.7434 0.6373 0.5003 0.4518 
 00/ 5 Present 0.6668 0.7545 0.8646 0.9052 
 Vo et al. [19] 0.6679 0.7598 0.8703 0.9117 
 10 Present 0.7078 0.7902 0.9046 0.9476 
 Vo et al. [19] 0.7100 0.7913 0.9039 0.9474 
 50 Present 0.7439 0.8234 0.9418 0.9869 
 Vo et al. [19] 0.7434 0.7434 0.8085 0.8481 
 Figs. 4.4-4.6 show the displacements of the ( 00 / / 0 0 ) and ( 00 /  ) thick beams 
 L/ h 3 increase with the increase of angle-ply . There are significant differences 
between the results of HOBT and quasi-3D solutions. 
 67 
 8 
 HOBT, 0o/o/0o
 o o o
 7.5 Quasi-3D, 0 / /0 , z=0
 Quasi-3D, 0o/o/0o, z=-h/2
 o o
 7 HOBT, 0 /
 Quasi-3D,0o/o, z=0
 o o
 6.5 Quasi-3D, 0 / , z=-h/2
 6
 5.5
 5
 4.5
 Nondimensional transverse displacement transverse Nondimensional
 4
 3.5 
 0 10 20 30 40 50 60 70 80 90
 o 
 Figure 4.4. The nondimensional mid-span transverse displacement with respect to 
the fiber angle change of composite beams with S-S boundary condition ( L/ h 3, 
 MAT II.4). 
 68 
 22 
 20
 HOBT, 0o/o/0o
 Quasi-3D, 0o/o/0o, z=0
 18 Quasi-3D, 0o/o/0o, z=-h/2
 HOBT, 0o/o
 Quasi-3D,0o/o, z=0
 16
 Quasi-3D, 0o/o, z=-h/2
 14
 12
 Nondimensional transverse displacement transverse Nondimensional
 10
 8 
 0 10 20 30 40 50 60 70 80 90
 o 
Figure 4.5. The nondimensional mid-span transverse displacement with respect to the 
fiber angle change of composite beams with C-F boundary condition ( L/ h 3, MAT 
 II.4). 
 69 
 4.2 
 4
 3.8
 3.6
 3.4
 3.2
 3 HOBT, 0o/o/0o
 Quasi-3D, 0o/o/0o, z=0
 2.8
 Quasi-3D, 0o/o/0o, z=-h/2
 HOBT, 0o/o
 Nondimensional transverse displacement transverse Nondimensional 2.6
 Quasi-3D,0o/o, z=0
 o o
 2.4 Quasi-3D, 0 / , z=-h/2
 2.2 
 0 10 20 30 40 50 60 70 80 90
 o 
Figure 4.6. The nondimensional mid-span transverse displacement with respect to the 
 fiber angle change of composite beams with C-C boundary condition ( L/ h 3, 
 MAT II.4). 
4.3.3. Arbitrary-ply beams 
 The example aims to analyse behaviours of composite beams with arbitrary-ply. 
The first, the symmetric single-layered C-F beams of 150 and 300 ply (MAT III.4) are 
considered. Their first four natural frequencies are displayed in Table 4.12 and 
compared with those from Chen et al. [84] and experiment results of Abarcar and 
Cunniff [128]. It is seen that there is consistency between present results and those 
from [84] and [128], especially the first mode of vibration. 
 70 
 Table 4.12. Fundamental frequencies (Hz) of single-layer composite beam with C-F 
 boundary condition (MAT III.4). 
Lay Theory Reference Mode 
-up 
 1 2 3 4 
150 HOBT Present 82.19 512.86 1426.29 - 
 Quasi-3D Present 82.22 513.09 1427.12 - 
 Chen et al. [84] 82.55 515.68 1437.02 - 
 Experiment Abarcar and Cunniff [128] 82.50 511.30 1423.40 1526.90* 
300 HOBT Present 52.63 329.13 918.51 1791.22 
 Quasi-3D Present 52.67 329.43 919.48 1793.62 
 Chen et al. [84] 52.73 330.04 922.45 1803.01 
 Experiment Abarcar and Cunniff [128] 52.70 331.80 924.70 1766.90 
 Note: ‘*’ denotes: the results are the torsional mode 
 Table 4.13. Nondimensional fundamental frequencies of arbitrary-ply laminated 
 composite beams (MAT IV.4). 
 Lay-up Theory Reference BC 
 S-S C-F C-C 
 450/-450/450/-450 HOBT Present 0.7961 0.2849 1.7592 
 Chandrashekhara 0.8278 0.2962 1.9807 
 and Bangera [27] 
 Quasi-3D Present 0.7962 0.2852 1.7629 
 Chen et al. [84] 0.7998 0.2969 1.8446 
 300/-500/500/-300 HOBT Present 0.9726 0.3486 2.1255 
 Quasi-3D Present 0.9728 0.3489 2.1281 
 Chen et al. [84] 0.9790 0.3572 2.2640 
 Next, the un-symmetric (450/-450/450/-450) and (300/-500/500/-300) beams (MAT 
 IV.4) with various BCs are considered, and their responses on fundamental 
 frequencies are reported in Table 4.13. Good agreements of the present theory and 
 71 
 / 
 previous studies are again found. Finally, the symmetric s composite beams 
 (MAT IV.4) are considered. 
 The effects of angle-ply variation on the frequency, buckling and displacement 
 are again illustrated in Table 4.14. In addition, the nondimensional fundamental 
 frequencies are also shown in Fig. 4.7. It can be seen that the present frequencies are 
 closer to those of [47, 84] and smaller than to those of [78, 130] which neglected the 
 Poisson’s effect, especially for beams with arbitrary-ply. This phenomenon can be 
 explained as follows. In present study, Poisson’s effect is incorporated in the 
 constitutive equations by assumingy  xy  yz 0 . It means that the strains ( y , yz
 0
 , xy ) are nonzero. For beams with arbitrary-ply (30 ), when the Poisson’s effect is 
 considered, the beam’s stiffness constants are much smaller than when the Poisson’s 
 effect is neglected. This causes beams more flexible [85]. It leads to the conclusion 
 that the Poisson’s effect is quite significant to the arbitrary-ply laminated beams, and 
 that the neglect of this effect is only suitable for the cross-ply laminated beams. It 
 should be also noted that there is deviation between the present critical buckling load 
 and those from Wang et al. [33]. This situation occurred because Wang et al. [33] 
 mentioned the rotation of the normal to the mid-plane in y -direction in displacement 
 field. 
 Table 4.14. Nondimensional fundamental frequencies, critical buckling loads and 
 / 
 mid-span displacements of s composite beams (MAT IV.4). 
BC Theory Reference Angle-ply ( ) 
 00 300 600 900 
a. Fundamental frequency 
S-S HOBT Present 2.649 0.999 0.731 0.729 
 Aydogdu [47] 2.651 1.141 0.736 0.729 
 Nguyen et al. [130] 2.656 2.103 1.012 0.732 
 FOBT Chandrashekhara et al. [78] 2.656 2.103 1.012 0.732 
 Quasi-3D Present 2.650 0.999 0.731 0.730 
 72 
BC Theory Reference Angle-ply ( ) 
 00 300 600 900 
C-F HOBT Present 0.980 0.358 0.261 0.261 
 Aydogdu [47] 0.981 0.414 0.262 0.260 
 Nguyen et al. [130] 0.983 0.768 0.363 0.262 
 FOBT Chandrashekhara et al. [78] 0.982 0.768 0.363 0.262 
 Quasi-3D Present 0.980 0.358 0.262 0.262 
C-C HOBT Present 4.897 2.180 1.620 1.615 
 Aydogdu [47] 4.973 2.195 1.669 1.619 
 Nguyen et al. [130] 4.912 4.131 2.202 1.621 
 FOBT Chandrashekhara et al. [78] 4.849 4.098 2.198 1.620 
 Quasi-3D Present 4.901 2.183 1.626 1.625 
 Chen et al. [84] 4.858 2.345 1.671 1.623 
b. Critical buckling load 
S-S HOBT Present 10.709 1.522 0.816 0.813 
 Quasi-3D 10.713 1.523 0.816 0.813 
C-F HOBT Present 2.973 0.386 0.206 0.205 
 Quasi-3D 2.974 0.387 0.206 0.206 
 FOBT Wang et al. [33] 2.971 0.712 0.208 0.205 
C-C HOBT Present 30.689 5.747 3.154 3.136 
 Quasi-3D 30.726 5.758 3.168 3.160 
 FOBT Wang et al. [33] 30.592 10.008 3.187 3.136 
c. Mid-span displacement 
S-S HOBT Present 1.196 8.437 15.745 15.811 
 Quasi-3D 1.195 8.432 15.733 15.796 
C-F HOBT 3.987 28.611 53.452 53.675 
 Quasi-3D 3.983 28.570 53.170 53.208 
C-C HOBT 0.355 1.815 3.289 3.308 
 Quasi-3D 0.355 1.812 3.272 3.276 
 73 
 y
 c
 n
 e
 u
 q
 e
 r
 f
 y
 r
 a
 t
 n
 e
 m
 a
 d
 n
 u
 f
 l
 a
 n
 o
 i
 s
 n
 e
 m
 i
 d
 n
 o
 N
 Figure 4.7. Effects of the fibre angle change on the nondimensional fundamental 
 / 
 frequency of s composite beams (MAT IV.4). 
4.4. Conclusions 
 The new approximation functions which combined polynomial and exponential 
functions are presented to study the free vibration, buckling and static behaviours of 
laminated composite beams. The displacement field is based on a quasi-3D theory 
accounting for a higher-order variation of both axial and transverse displacements. 
Poisson’s effect is incorporated in beam model. Numerical results for different BCs 
are obtained to compare with previous studies and investigate effects of material 
anisotropy, Poisson’s ratio and angle-ply on the natural frequencies, buckling loads, 
displacements and stresses of composite beams. The obtained results show that: 
 - The transverse normal strain effects are significant for un-symmetric and thick 
 beams. 
 74 
- The Poisson’s effect is quite significant to the laminated beams with arbitrary 
 lay-up, and the neglect of this effect is only suitable for the cross-ply laminated 
 beams. 
- The present model is found to be appropriate for vibration, buckling and 
 bending analysis of cross-ply and arbitrary-ply composite beams. 
 75 
 Chapter 5. SIZE DEPENDENT BEHAVIOURS OF MICRO GENERAL 
LAMINATED COMPOSITE BEAMS BASED ON MODIFIED COUPLE 
STRESS THEORY 6 
5.1. Introduction 
 In Chapters Two, Three and Four, the macro composite beams are analysed using 
classical continuum mechanics theories. However, the use of composite materials 
with microstructure in micro-electro-mechanical systems such as microswitches and 
microrobots has recently motivated many researchers [131-133] to study the 
behaviour materials in order of micron and sub-micron. The results obtained by these 
studies show that the classical continuum mechanics theories can not describe the 
behaviour of such micro-structures due to their size dependencies. 
 A review of non-classical continuum mechanics models for size-dependent 
analysis of small-scale structures can be found in [134]. These models for size-
dependent analysis can be divided into three groups: nonlocal elasticity theory, micro 
continuum theory and strain gradient family. Nonlocal elasticity theory was proposed 
by Eringen [3, 135], Eringen and Edelen [136], and its recent applications can be 
found in [137-140]. In this theory, the stress at a reference point is considered as a 
function of strain field at all points of the body, and thus the size effect is captured by 
means of constitutive equations using a nonlocal parameter. Micro continuum theory 
in which each particle can rotate and deform independently regardless of the motion 
of the centroid of the particle was developed by Eringen [141-143]. The strain 
gradient family is composed of the strain gradient theory [133, 144], the modified 
strain gradient theory [131], the couple stress theories [4-6] and the modified couple 
stress theory (MCST) [145]. In the strain gradient family, both strains and gradient 
of strains are considered in the strain energy. The size effect is accounted for using 
material length scale parameters (MLSP). The MCST introduced an equilibrium 
condition of moments of couples to enforce the couple stress tensor to be symmetric. 
Consequently, MCST needs only one MLSP instead of two as the couple stress 
6 A slightly different version of this chapter has been published in Composite Structures in 2018 
 76 
theories, or three as the modified strain gradient theory. This feature makes the MCST 
easier to use and more preferable to capture the size effect because the determination 
of MLSP is a challenging task. 
 Chen et al. [12, 146] developed Timoshenko and Reddy beam models to analyse 
the static behaviours of cross-ply simply supported microbeams. Chen and Si [147] 
suggested an anisotropic constitutive relation for the MCST and used global-local 
theory to analyse Reddy beams using Navier solutions. By using a meshless method, 
Roque et al. [10] analysed the static bending response of micro laminated 
Timoshenko beams. A size-dependent zigzag model was also proposed by Yang et 
al. [7] for the bending analysis of cross-ply microbeams. Abadi and Daneshmehr 
[148] analysed the buckling of micro composite beams using Euler-Bernoulli and 
Timoshenko models. Mohammadabadi et al. [37] also predicted the thermal effect on 
size-dependent buckling behaviour of micro composite beams. The generalized 
differential quadrature method was used to solve with different boundary conditions 
(BCs). Chen and Li [149] predicted dynamic behaviours of micro laminated 
Timoshenko beams. Mohammad-Abadi and Daneshmehr [8] used the MCST to 
analyse free vibration of cross-ply microbeams by using Euler-Bernoulli, 
Timoshenko and Reddy beam models. Ghadiri et al. [150] analysed the thermal effect 
on dynamics of thin and thick microbeams with different BCs. Most of the above-
mentioned studies mainly focused on cross-ply microbeams. Therefore, the study of 
micro general laminated composite beams (MGLCB) with arbitrary lay-ups is 
necessary. 
 Despite the fact that numerical approaches are used increasingly [10, 21, 50, 137, 
138, 151], Ritz method is still efficient to analyse structural behaviours of beams [44, 
45, 48, 130, 152-155]. In Ritz method, the accuracy and efficiency of solution strictly 
depends on the choice of approximation functions. An inappropriate choice of the 
approximation functions may cause slow convergence rates and numerical 
instabilities [48]. The approximation functions should satisfy the specified essential 
BCs [1]. If this requirement is not satisfied, the Lagrangian multipliers and penalty 
 77 
method can be used to handle arbitrary BCs [44, 126, 156]. However, this approach 
leads to an increase in the dimension of the stiffness and mass matrices and causing 
computational costs. Therefore, the objective of this Chapter is to propose 
approximation functions for Ritz type solutions that give fast convergence rate, 
numerical stability and satisfy the specified BCs. 
 In this Chapter, new exponential approximation functions are proposed for the 
size-dependent analysis of MGLCB based on the MCST using a refined shear 
deformation theory. Lagrange’s equations are used to obtain the governing equations 
of motion. The accuracy of the present model is demonstrated by verification studies. 
Numerical results are presented to investigate the effects of MLSP, length-to-height 
ratio and fibre angle on the deflections, stresses, natural frequencies and critical 
buckling loads of micro composite beams with arbitrary lay-ups. 
5.2. Theoretical formulation 
 A MGLCB with rectangular cr