FiberMatrix
5.6 Numerical results and discussions
5.6.2 Effective elastic properties of graded composite lamina
5.6.3.2 Bending responses of GLCPs
Figure 5.15 illustrates the transverse deflection (W) vs. load (Q) curves for symmetric and asymmetric CLCPs/GLCPs. For any of the plates (symmetric/asymmetric), Fig. 5.15 shows lesser rigidity of GLCP as compared to that of corresponding CLCP. Moreover, the rigidity of GLCP decreases as its graded plies are made of higher power-law exponent (n). For liner and nonlinear bending deformations of symmetric CLCP/GLCP, two different mechanical loads (p = 10 N/m2, 5 kN/m2) are considered from Fig.
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5.15(a). Corresponding to these mechanical loads, the distributions of dimensionless stresses (x,y) across the thickness of the symmetric GLCP/CLCP are demonstrated in Figs. 5.16(a)-(d). It may be observed from Figs. 5.16(a)-(d) that there is a significant stress-mismatch at the interface of 0o and 90o plies of CLCP. But, this stress-mismatch could significantly be reduced by the use of present graded composite plies. Also, this stress-mismatch could almost be eliminated using the graded plies of higher power-law exponent (n). Similar stress-distributions for asymmetric GLCP/CLCP are also presented in Fig. 5.17. In this case, the values of mechanical load (Q) corresponding to linear and nonlinear deformations of asymmetric GLCP/CLCP are chosen from Fig.
5.15(b). The results in Fig. 5.17 exhibit similar observations as those are obtained for symmetric plates (Fig. 5.16).
Fig. 5.15 Variations of dimensionless transverse deflection (W) at the middle point of (a) symmetric
(
00/
900/ )
00 and (b) asymmetric(
900/
00/
900/ )
00 cross-ply laminated composite plates with the dimensionless transverse mechanical load (Q).Chapter 5:Design of laminated composite plates .... composite plies
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Fig. 5.16 Distributions of the dimensionless stresses
(
x,
y)
across the thickness of the graded/conventional symmetric cross-ply (00/900/00) laminated composite plate under the (a)-(b) linear (p = 10 N m/
2) and (c)-(d) nonlinear (p = 5 kN m/
2) bending deformations.Chapter 5:Design of laminated composite plates .... composite plies
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5.6.4 Conventional-graded laminated composite plates
The continuous stresses in Figs. 5.16-5.17 indicate similar nature (continuous) of variations of material properties across the thickness of laminate. But, the discrepancies like the increase of maximum value of every stress (Figs. 5.16-5.17) and the decrease of laminate-rigidity (Figs. 5.15) arise in forming a GLCP through direct substitution of conventional plies of a CLCP. In order to eliminate these demerits retaining the merit of continuous variations of stresses/material properties, a new lamination scheme is proposed in the present study by utilizing graded as well as conventional composite Fig. 5.17 Distributions of the dimensionless stresses
(
x,
y)
across the thickness of the graded /conventional asymmetric cross-ply (900/00/900/00) laminated composite plate under the (a)-(b) linear (p= 10 N m/
2) and (c)-(d) nonlinear (p= 5 kN m/
2) bending deformations.Chapter 5:Design of laminated composite plates .... composite plies
147
plies within the same laminate. These two (graded and conventional) different types of plies are utilized within a laminate in such a manner that the graded plies work mainly for achieving continuous variations of material properties while the conventional plies strengthen the laminate. Since this type of laminated plate is composed of graded as well as conventional laminas, it (laminated plate) is presently designated as conventional- graded laminated composite plate (CGLCP). However, in this strategy of lamination, a graded ply is considered to have maximum stiffness at one of its top and bottom surfaces while the other surface of the same has minimum stiffness. So, the graded ply illustrated in Fig. 5.11 is utilized.
0 Layero 90 Layero
0 Layero
h h/3
h/3
h/3 o
0 Layer 0 Layero
90 Layero
0 THBS Layero 90 BHTS Layero
90 THBS Layero 0 BHTS Layero
(a)
h/4
h h/4
h/4 90 Layero
0 Layero
90 Layero 0 Layero
90 Layero 90 BHTS Layero
0 THBS Layero
0 Layero
(b)
h/5 h/10 h/10 h/5
h/10 h/10
h/5
h/3
h/6 h/6
h/3 h/4
Fig. 5.18 Modified stacking sequences for conventional-graded composite plates (a) symmetric and (b) asymmetric laminated.
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Fig. 5.19 Variations of elastic constants (Cij) across the thickness of the symmetric conventional-graded laminated composite plate (CGLCP, n 2).
It should be noted that this graded composite ply (Fig. 5.11) can be utilized in its two forms. The first one is for its stiffer top-surface and softer bottom-surface. The next one is just reverse of the first one. Corresponding to these two forms of the graded ply (Fig.
5.11), it is designated either as top-hard-bottom-soft (THBS) ply or as bottom-hard-top- soft (BHTS) ply. Utilizing these two forms of the graded ply, two conventional laminated composite plates of symmetric (0o/90o/0o) and asymmetric (90o/0o/90o/0o) stacking sequences are modified as illustrated in Fig. 5.18. The total thickness of the laminate is not altered for this modification of stacking sequence and the thickness of THBS layer or BHTS layer is considered as half of that for the conventional composite- ply within the same laminate. It may be observed from Fig. 5.18 that a pair of THBS and
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BHTS layers is used between two conventional layers in order to reduce the mismatch of material properties at their inter-surface. For symmetric laminate (Fig. 5.18(a)), a pair of THBS and BHTS layers is inserted at every inter-surface of 0o and 90o layers. If the same procedure is followed in case of asymmetric laminate (Fig. 5.18(b)), then three pairs of THBS and BHTS layers are required. But, as the stiffness of a THBS/BHTS layer is lesser than that of a conventional layer and a constant laminate-thickness is considered in this conversion, too many THBS/BHTS layers within a laminate may cause considerable reduction of its (laminate) rigidity. Thus, the stacking sequence of asymmetric conventional laminate is modified considering only one pair of THBS and BHTS layers as illustrated in Fig. 5.18(b).
For a particular value of power-law exponent (n2), the variations of different elastic coefficients (Cij) across the thickness of laminates (CGLCPs) are demonstrated in Fig. 5.19. It may be observed from Fig. 5.19 that the elastic coefficients continuously vary across the thickness of the laminate (CGLCP). Also, the symmetric nature of variation of elastic properties of CLCP does not alter as it (CLCP) is converted to CGLCP. For the conversion of asymmetric (90o/0o/90o/0o) CLCP into a CGLCP, the corresponding variations of elastic coefficients (Cij) across the thickness of CGLCP are illustrated in Fig.
5.20. This figure shows that the asymmetric CLCP remains as asymmetric laminate after its conversion into CGLCP.
Figure 5.21 illustrates the deflection (W) vs. load (Q) curves for symmetric and asymmetric CGLCPs (Figs. 5.18). Similar responses of corresponding symmetric and asymmetric CLCPs are also illustrated in the same figure. In comparison to the previous results (Fig. 5.15), it may be observed from Fig. 5.21 that the rigidity of the symmetric/asymmetric CGLCP is significantly increased towards the same of corresponding symmetric/asymmetric CLCP. This occurs because of the use of conventional plies. It may also be observed that there is no significant change in the rigidity of a CGLCP due to the change of the value of power-law exponent (n) because the stiffness of every graded ply in CGLCP is not a major issue in its (CGCLP) design.
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Fig. 5.20 Variations of elastic constants (Cij) across the thickness of the asymmetric conventional-graded laminated composite plate (CGLCP, n 2 ).
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151
For two different loads (p = 10 N/m2, 5 kN/m2) corresponding to linear and nonlinear bending deformations of symmetric CGLCP, the distributions of stresses (x,y) across the thickness of the laminate are illustrated in Fig. 5.22. It may be observed from Fig. 5.22 that the stresses are almost continuously distributed across the thickness of the plate. It is also interesting to observe in comparison to Fig. 5.16 that the maximum value of every stress component does not alter for the conversion of a symmetric CLCP into a symmetric CGLCP. Similar to the previous results (Figs. 5.16), a higher value of power-law exponent (n) causes lesser stress-discontinuity at the interlaminar-surfaces in CGLCP. In the case of conversion of an asymmetric CLCP into an asymmetric CGLCP, the stress distributions are demonstrated in Fig. 5.23 that exhibits similar observations as those are obtained from the symmetric case (Fig. 5.22).
The foregoing results (Figs. 5.19-5.23), show that the proposed lamination scheme provides continuous variations of material properties and stresses across the thickness of laminated plate. This advantage could be achieved with ample changes of laminate-
Fig. 5.21 Variations of dimensionless transverse deflection (W) at the middle point of (a) symmetric and (b) asymmetric cross-ply laminated composite plates with the dimensionless transverse mechanical load (Q).
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rigidity and maximum values of stresses in the laminate. In this conversion there is no appearance of stress-concentration.
Fig. 5.22 Distributions of the dimensionless stresses
(
x,
y)
across the thickness of the conventional-graded (Fig. 5.18(a))/conventional symmetric (00/900/00) laminated composite plate under its (a)-(b) linear (p= 10 N m/
2) and (c)-(d) nonlinear (p= 5 kN m/
2) bending deformations.Chapter 5:Design of laminated composite plates .... composite plies
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Fig. 5.23 Distributions of the dimensionless stresses
(
x,
y)
across the thickness of the conventional-graded (Fig.5.18(b))/ conventional asymmetric cross-ply (900/00/900/00)laminated composite plate under its (a)-(b) linear (p= 10 N m