The multiple coupling of composite laminates has a unique advantage in improving the macro mechanical properties of composite structures. A total of three hygro-thermally stable multi-coupled laminates with extension-twisting coupling were presented, which were conducive to the formation of passive adaptive structures. Then, the multi-coupled laminates were used to design the bending-twisting coupled box structure, in which the configuration of laminate and box structure could be extended to variable cross-section configuration. The optimal design of stacking sequence was realized, the optimization objectives of which were to maximize bending-twisting coupling of box structure and extension-twisting coupling of laminate, respectively. The effects of multiple coupling on hygro-thermal stability, coupling, failure strength, buckling load, robustness and other comprehensive mechanical properties of laminates and box structures were analyzed by parametric modeling method. The results show that the extension-twisting coupling of laminate and the bending-twisting coupling of box structures can be greatly improved by 450% and 260% at maximum, respectively. Meanwhile, it would have a negative impact on the failure strength and buckling load, which, however, can be minimized by a reasonable paving method. Multi-coupled laminates have good robustness, and the bending-twisting coupling helps improve robustness. Finally, the hygro-thermal stability and mechanical properties were verified by numerical simulation with finite element method.
Passive adaptive structure has become the forefront and research hotspot in the present era with its unique advantages in the field of aerospace and wind power. For instance, the bending-twisting coupled principle bearing structure can control the aeroelastic deformation of the forward-swept wing aircraft, thereby inhibiting the rapid decline of the critical velocity caused by aeroelastic divergence and improving design performance and aeroelastic properties [
The stiffness characteristics of the passive adaptive structure can be realized by the paving design of composite laminate [
The above-mentioned research is mainly aimed at the single coupling of laminate, and strives to achieve the best mechanical properties of laminates or composite structures formed by laminates. The single coupling of laminate limits the range of the paving angle to a large extent despite the fact that it is easy to design, resulting in the inability of the laminate to effectively give play to its advantages in improving mechanical properties in the form of free layup. Moreover, the single coupling of laminate is likely to cause limitations, such as bending-twisting coupling that may reduce buckling load [
The multi-coupled laminates refer to laminates that contain two or more couplings that can exist independently at the same time. There are few studies on multi-coupled laminates in existing research. Moore et~al. [
The innovation of this article lies in improving the comprehensive mechanical properties of the composite structure from the perspective of the multi-couplings of the laminate. The universal analytical conditions for hygro-thermally stable multi-coupled laminates with extension-twisting coupling were established. Then, the multi-coupled laminates were used to design hygro-thermally stable adaptive structures, in which the configuration of laminate and structure could be extended to variable cross-section configuration. The optimal design was realized for the multi-coupled laminates and adaptive structures. Meanwhile, the effect of multi-coupling on laminates and adaptive structures was explored in terms of coupling effect, failure strength, bucking load and robustness.
There are many kinds of multi-coupled laminates. Considering the extension-twisting coupling (a kind of extensively studied coupling) of laminate plays a notable role in the design of adaptive structures, the multi-coupled laminates with such coupling are studied in the following research. In order to maximize the advantages of laminates, the paving form of laminates should be no longer limited to special forms such as standard layups and symmetrical layups. The possible coupling, except extension-twisting coupling, may possess include one or more of the following couplings: extension-shearing, extension-bending, bending-twisting and shearing-twisting couplings. It is worth noting that the coexistence of extension-shearing coupling and extension-twisting coupling will not be conducive to the improvement of the adaptive ability of the structure. It mainly reflects that the paving orders of the upper and lower flanges of adaptive box structure are completely the opposite. Thus, other couplings here do not include extension-shearing coupling.
In addition to extension-twisting coupling, other coupling of double-coupled laminates may include one of the following couplings: extension-bending coupling, shearing-twisting coupling and bending-twisting coupling. The
The stiffness coefficients
The relationships of hygro-thermal stability for laminates are [
However,
Then, considering
Inserting
Substituting
Furthermore, the extension-bending coupling must co-exist with shearing-twisting coupling (
The shearing-twisting coupling will produce the extension-bending coupling (
The stiffness equation of
Substituting
Substituting
Furthermore, the stiffness equation of
Substituting
However,
Similarly, for composite
Substituting
Substituting
In brief, when the extension-twisting coupled laminates with three or more couplings are used to construct hygro-thermally stable composite structures, only
Type | Number of couplings | Necessary and sufficient conditions |
---|---|---|
1 | ||
2 | ||
3 | ||
4 |
There are two types of typical composite adaptive structures, namely bending-twisting coupled structure and extension-twisting coupled structure [
The geometric shape of bending-twisting coupled variable cross-section box structure is shown in
Considering the outstanding performance of extension-twisting coupling of laminate on constituting the bending-twisting coupling of box structure, the laminates with extension-twisting coupling in
The torsion deformation of the box structure can be measured by the vertical (
The wing of an executive jet transport (EJT) was used as a research object. In the light of the physical parameters of EJT, the wing size can be set as the lengths of the wing root, wing tip and wingspan are 4.57 m, 1.52 m and 8.26 m, respectively [
Elastic modulus, GPa |
161.0 | Axial compressive strength, MPa |
1761 |
Elastic modulus, GPa |
11.38 | Transverse compressive strength, MPa |
347 |
Shear modulus, GPa |
5.17 | Shear strength, MPa |
120 |
Poisson's ratio |
0.38 | Thickness of each lamina, mm |
0.14 |
Axial tensile strength, MPa |
2647 | Thermal expansion coefficient μ/°C α1 | −0.0181 |
Transverse tensile strength, MPa |
127 | Thermal expansion coefficient μ/°C α2 | 24.3 |
The upper and lower flanges of the wing should produce torsion deformation in the same direction. The bending moment of box structure can be equivalent to the force of the same magnitude and opposite direction. Therefore, the paving angle of the upper and lower flanges should be opposite to each other. Compared with the upper and lower flanges, the web laminate is very small in size, and the laminate [60/−60/−60/0/0/0/60/60/0/−60/60/60/−60/−60/−60/0/0/60]T without any couplings is taken. The external bending moment
Owing to the strong nonlinear equality constraints, the DE_CMSBHS algorithm was used for optimization combined with the penalty function [
Optimization objectives | Number | Stacking sequences/° | | |
|
---|---|---|---|---|
12 | [74.78/−13.54/−27.38/65.86/58.36/−51.20/ −35.99/28.93/24.41/−64.63/19.64/−78.08]T | 1.08 × 10−5 | 4.06 × 10−5 | |
13 | [−18.25/71.39/−90.00/56.47/−22.64/−26.71/ 39.10/1.45/−64.63/−70.70/36.49/12.95/−76.89]T | 9.22 × 10−6 | 1.99 × 10−5 | |
Max | |
14 | [−90.00/−18.80/65.44/−90.00/11.60/−24.49/ −13.18/25.70/19.23/90.00/−42.88/90.00/ −84.06/13.61]T | 4.81 × 10−6 | 7.55 × 10−6 |
15 | [4.87/−74.49/25.77/ −69.77/90.00/3.46/ 46.07/−40.79/−26.98/39.52/−90.00−90.00/ −12.66/67.17/−11.15]T | 7.06 × 10−6 | 1.14 × 10−5 | |
16 | [88.19/2.64/69.09/ −29.40/−28.17/−26.79/ 49.73/42.18/32.47/ 90.00/−65.13/−67.78/ 18.39/−31.79/90.00/16.45]T | 6.00 × 10−6 | 7.27 × 10−6 | |
12 | [74.78/−13.54/−27.38/65.86/58.36/−51.20/ −35.99/28.93/24.41/−64.63/19.64/−78.08]T | 1.08 × 10−5 | 4.06 × 10−5 | |
13 | [−18.25/71.39/−90.00/56.47/−22.64/−26.71/ 39.10/1.45/−64.63/−70.70/36.49/12.95/ −76.89]T | 9.22 × 10−6 | 1.99 × 10−5 | |
Max |
14 | [−90.00/−18.80/65.44/−90.00/11.60/ −24.49/−13.18/25.70/ 19.23/90.00/−42.88/ 90.00/−84.06/13.61]T | 4.81 × 10−6 | 7.55 × 10−6 |
15 | [4.87/−74.49/25.77/−69.77/90.00/3.46/ 46.07/−40.79/−26.98/39.52/−90.00−90.00/ −12.66/67.17/−11.15]T | 7.06 × 10−6 | 1.14 × 10−5 | |
16 | [88.19/2.64/69.09/−29.40/−28.17/−26.79/ 49.73/42.18/32.47/90.00/−65.13/−67.78/ 18.39/−31.79/90.00/16.45]T | 6.00 × 10−6 | 7.27 × 10−6 |
Optimization objectives | Number | Stacking sequences/° | | |
|
---|---|---|---|---|
Max | |
12 | [74.91/−21.11/−23.97/62.59/55.32/−52.34/ −34.29/32.27/27.13/−66.93/19.92/−72.37]T | 1.16 × 10−5 | 4.33 × 10−5 |
13 | [73.41/71.62/−17.91/−19.70/−19.49/−56.42/ 44.94/35.35/27.70/−56.24/−69.05/90.00/13.37]T | 9.30 × 10−6 | 2.74 × 10−5 | |
14 | [17.67/−69.59/−67.25/−64.73/29.36/34.08/ 39.83/−31.37/ 53.55/−34.38/−27.82/69.31/ −21.49/74.83]T | 1.22 × 10−5 | 3.45 × 10−5 | |
15 | [−17.49/−18.85/81.55/−22.94/75.35/70.51/ 90.00/47.13/26.65/ 15.67/−62.05/7.72−67.86/ −69.75/2.39]T | 1.01 × 10−5 | 2.25 × 10−5 | |
16 | [−77.71/−76.55/9.65/−72.66/13.05/9.15/26.96/ 48.66/−29.99/−25.34/62.01/−90.00/−90.00/ −15.84/−14.51/79.04]T | 7.28 × 10−6 | 1.70 × 10−5 | |
12 | [74.91/−21.11/−23.97/62.59/55.32/−52.34/ −34.29/32.27/27.13/−66.93/19.92/−72.37]T | 1.16 × 10−5 | 4.33 × 10−5 | |
13 | [71.11/−15.93/67.20/−23.07/45.55/−34.95/ −42.80/90.00/17.62/−63.44/28.32/25.63/−73.87]T | 1.10 × 10−5 | 2.72 × 10−5 | |
Max |
14 | [−22.80/−25.38/69.57/ 65.67/60.73/−16.73/ 47.82/−55.29/34.68/−58.68−61.47/21.87/ −66.48/16.86]T | 1.28 × 10−5 | 3.02 × 10−5 |
15 | [−16.21/−17.95/66.15/63.23/ 59.91/−29.00/43.11/ 90.00/−46.02/−53.59/ 14.10/−66.24/−70.48/ 23.54/21.35]T | 1.20 × 10−5 | 2.15 × 10−5 | |
16 | [−77.71/−76.55/9.65/ −72.66/13.05/9.15/ 26.96/48.66/−29.99/ −25.34/62.01/−90.00/ −90.00/−15.84/ −14.51/79.04]T | 7.28 × 10−6 | 1.70 × 10−5 |
Optimization objectives | Number | Stacking sequences/° | | |
|
---|---|---|---|---|
12 | [71.36/−16.33/−34.02/62.50/56.47/−46.65/ −37.57/30.03/24.71/−66.66/19.17/−77.19]T | 1.11 × 10−5 | 4.10 × 10−5 | |
13 | [−73.67/13.76/36.23/−65.89/−52.23/ 16.18/47.87/−90.00/ −20.22/−18.75/64.00/ −17.00/80.16]T | 9.95 × 10−6 | 2.31 × 10−5 | |
Max | |
14 | [−71.30/15.90.00/90.00/18.72/−52.51/26.31/ 35.39/−40.23/−38.81/65.58/−30.46/72.86/ 75.06/−8.06]T | 9.30 × 10−6 | 2.33 × 10−5 |
15 | [−75.12/17.79/18.48/−41.74/90.00/90.00/ 22.47/24.12/−43.03/−36.73/62.31/−15.06/ 77.59/78.87/−9.00]T | 6.08 × 10−6 | 1.33 × 10−5 | |
16 | [75.34/−19.53/−21.16/67.32/62.73/−29.47/ 43.34/−39.86/−46.41/38.53/36.27/−63.29/ −66.98/21.62/17.25/−74.04]T | 1.06 × 10−5 | 2.64 × 10−5 | |
Max |
12 | [71.36/−16.33/−34.02/62.50/56.47/−46.65/ −37.57/30.03/24.71/−66.66/19.17/−77.19]T | 1.11 × 10−5 | 4.10 × 10−5 |
13 | [−73.67/13.76/36.23/−65.89/−52.23/16.18/ 47.87/−90.00/−20.22/−18.75/64.00/ −17.00/80.16]T | 9.95 × 10−6 | 2.31 × 10−5 | |
14 | [−71.30/15.90.00/90.00/ 18.72/−52.51/26.31/ 35.39/−40.23/−38.81/ 65.58/−30.46/72.86/ 75.06/−8.06]T | 9.30 × 10−6 | 2.33 × 10−5 | |
15 | [−40.92/−3.40/64.01/ 57.11/53.79/−32.33/ −90.00/−2.88/−55.46/ 14.71/90.00/−64.23/ 10.47/90.00/9.58]T | 9.63 × 10−6 | 7.94 × 10−6 | |
16 | [75.34/−19.53/−21.16/ 67.32/62.73/−29.47/ 43.34/−39.86/−46.41/ 38.53/36.27/−63.29/ −66.98/21.62/ 17.25/−74.04]T | 1.06 × 10−5 | 2.64 × 10−5 |
Optimization objectives | Number | Stacking sequences/° | | |
|
---|---|---|---|---|
12 | [−67.12/19.42/23.16/−56.58/−66.98/42.02/ 49.85/−33.40/62.49/−24.56/−21.22/73.12]T | 1.24 × 10−5 | 4.48 × 10−5 | |
13 | [16.77/18.31/−79.55/−75.93/90.00/ −65.50/44.20/−25.10/−11.89/−8.03/ −5.35/73.07/74.62]T | 9.88 × 10−6 | 2.91 × 10−5 | |
Max | |
14 | [−20.34/71.64/−25.21/64.96/60.08/−23.23/ 46.97/−51.29/−56.48/28.41/24.24/−68.34/ −71.10/16.02]T | 1.26 × 10−5 | 3.40 × 10−5 |
15 | [−14.73/−90.00/−18.03/72.01/65.72/−26.38/ 44.87/53.17/−45.90.00/−55.65/15.00/18.36/ −71.60/15.06/−75.33]T | 9.73 × 10−6 | 2.44 × 10−5 | |
16 | [−11.05/76.65/−13.55/72.25/−90.00/−13.80/ 53.66/43.10/−36.04/−52.97/−57.78/ 21.68/19.19/−69.64/90.00/14.10]T | 8.49 × 10−6 | 1.68 × 10−5 | |
12 | [−67.12/19.42/23.16/ −56.58/−66.98/42.02/ 49.85/−33.40/62.49/ −24.56/−21.22/73.12]T | 1.24 × 10−5 | 4.48 × 10−5 | |
13 | [−15.23/66.65/62.93/ −24.18/−29.72/90.00/ 51.33/−56.02/15.53/20.06/−72.59/−75.50/ 12.54]T | 1.11 × 10−5 | 2.90 × 10−5 | |
Max |
14 | [−20.34/71.64/−25.21/ 64.96/60.08/−23.23/ 46.97/−51.29/−56.48/ 28.41/24.24/−68.34/ −71.10/16.02]T | 1.26 × 10−5 | 3.40 × 10−5 |
15 | [74.11/−10.84/70.42/ −15.46/64.11/−26.23/ −40.11/−53.99/34.97/ 27.99/90.00/−68.82/ 16.45/14.41/−76.61]T | 1.02 × 10−5 | 2.43 × 10−5 | |
16 | [−11.05/76.65/−13.55/ 72.25/−90.00/−13.80/ 53.66/43.10/−36.04/ −52.97/−57.78/21.68/ 19.19/−69.64/90.00/14.10]T | 8.49 × 10−6 | 1.68 × 10−5 |
Considering that equipment and human errors are often unavoidable when laminates are laid by existing processing technologies, it is meaningful to test the deformation deviation caused by the paving angle deviation for free-layer laminate. Taking the 14-ply laminates with maximum
Assume that the angle deviation of the
The hygro-thermal performance of laminates and box structures was tested by using the FEM, as shown in
The free shrinkage deformation results under cooling of laminates and the bending-twisting coupled structure were solved by the software MSC.Nastran, as shown in
In addition, laminates with different paving angles produce the same thermal linear strains, which suggests that the hygro-thermal linear strain of laminate is independent of the stacking sequence. In view of the fact that the effect of humidity changes on laminates is similar to that of temperature changes, only the thermal coefficient needs to be replaced with the humidity coefficient. Thus, the reasons for this phenomenon were analyzed only from the perspective of temperature changes. The thermal linear strain of the laminate with extension-twisting coupling is expressed as
The material thermal constant
Substituting
Substituting
Inserting the common analytical conditions
Then Substituting
From
In order to verify the stiffness performance of the laminates, the finite element model of a rectangular laminate with the size of 0.18 by 0.1 m was established. The purpose of using rectangular laminates for verification was to facilitate the extraction of the distortion of laminates. The external load was the axial uniform tension (400 N/m). 800 shell elements were divided and the geometric center of laminate was fixed. The size of the box structure was the same as in the optimal design process, and the bending moment of the free end of the structure
The torsion angle
Type of laminate | Number of layers | Analytical |
Numerical |
Percentage difference∗ |
---|---|---|---|---|
12 | 1.62 × 10−2 | 1.61 × 10−2 | 0.62% | |
13 | 0.80 × 10−2 | 0.80 × 10−2 | 0 | |
14 | 0.30 × 10−2 | 0.30 × 10−2 | 0 | |
15 | 0.46 × 10−2 | 0.46 × 10−2 | 0 | |
16 | 0.29 × 10−2 | 0.29 × 10−2 | 0 | |
12 | 1.73 × 10−2 | 1.72 × 10−2 | 0.58% | |
13 | 1.10 × 10−2 | 1.10 × 10−2 | 0 | |
14 | 1.38 × 10−2 | 1.37 × 10−2 | 0.72% | |
15 | 0.90 × 10−2 | 0.90 × 10−2 | 0 | |
16 | 0.68 × 10−2 | 0.68 × 10−2 | 0 | |
12 | 1.64 × 10−2 | 1.63 × 10−2 | 0.61% | |
13 | 0.92 × 10−2 | 0.92 × 10−2 | 0 | |
14 | 0.93 × 10−2 | 0.93 × 10−2 | 0 | |
15 | 0.53 × 10−2 | 0.53 × 10−2 | 0 | |
16 | 1.06 × 10−2 | 1.06 × 10−2 | 0 | |
12 | 1.79 × 10−2 | 1.78 × 10−2 | 0.56% | |
13 | 1.16 × 10−2 | 1.16 × 10−2 | 0 | |
14 | 1.36 × 10−2 | 1.35 × 10−2 | 0.74% | |
15 | 0.98 × 10−2 | 0.98 × 10−2 | 0 | |
16 | 0.67 × 10−2 | 0.67 × 10−2 | 0 |
Note: ∗: The percentage difference between numerical solution and analytical solution.
According to the optimization results in
The results show that: (1) Compared with the single-coupled
Type of laminate | Number of layers | | |
Improvement∗ | Improvement∁ | |
---|---|---|---|---|---|
12 | 4.06 × 10−5 | — | 1.08 × 10−5 | — | |
13 | 1.99 × 10−5 | — | 9.22 × 10−6 | — | |
14 | 7.55 × 10−6 | — | 4.81 × 10−6 | — | |
15 | 1.14 × 10−5 | — | 7.06 × 10−6 | — | |
16 | 7.27 × 10−6 | — | 6.00 × 10−6 | — | |
12 | 4.33 × 10−5 | 6.65% | 1.16 × 10−5 | 7.41% | |
13 | 2.74 × 10−5 | 37.69% | 9.30 × 10−6 | 0.87% | |
14 | 3.45 × 10−5 | 356.95% | 1.22 × 10−5 | 153.64% | |
15 | 2.25 × 10−5 | 97.37% | 1.01 × 10−5 | 43.06% | |
16 | 1.70 × 10−5 | 133.84% | 7.28 × 10−6 | 21.33% | |
12 | 4.10 × 10−5 | 0.99% | 1.11 × 10−5 | 2.78% | |
13 | 2.31 × 10−5 | 16.08% | 9.95 × 10−6 | 7.92% | |
14 | 2.33 × 10−5 | 208.61% | 9.30 × 10−6 | 93.35% | |
15 | 1.33 × 10−5 | 16.67% | 6.08 × 10−6 | −13.88% | |
16 | 2.64 × 10−5 | 263.14% | 1.06 × 10−5 | 76.67% | |
12 | 4.48 × 10−5 | 10.34% | 1.24 × 10−5 | 14.81% | |
13 | 2.91 × 10−5 | 46.23% | 9.88 × 10−6 | 7.16% | |
14 | 3.40 × 10−5 | 350.33% | 1.26 × 10−5 | 161.95% | |
15 | 2.44 × 10−5 | 114.04% | 9.73 × 10−6 | 37.82% | |
16 | 1.68 × 10−5 | 131.09% | 8.49 × 10−6 | 41.50% |
Notes: ∗: Improvement of
∁: Improvement of
When the axial external load of laminate increases gradually, the destruction will occur first on one lamina. This will affect the rigidity of the entire laminate, and may cause further damage to other laminae. Thus, the failure load should be considered during the design process. The tension and compression external loads of first ply failure
Type of laminate | Number of layers | | |
Improvement∗ | Improvement∁ | |
---|---|---|---|---|---|
12 | 4.06 × 10−5 | — | 1.08 × 10−5 | — | |
13 | 1.99 × 10−5 | — | 9.22 × 10−6 | — | |
14 | 7.55 × 10−6 | — | 4.81 × 10−6 | — | |
15 | 1.14 × 10−5 | — | 7.06 × 10−6 | — | |
16 | 7.27 × 10−6 | — | 6.00 × 10−6 | — | |
12 | 4.33 × 10−5 | 6.65% | 1.16 × 10−5 | 7.41% | |
13 | 2.72 × 10−5 | 36.68% | 1.10 × 10−5 | 19.31% | |
14 | 3.02 × 10–5 | 300.00% | 1.28 × 10–5 | 166.11% | |
15 | 2.15 × 10–5 | 88.60% | 1.20 × 10–5 | 69.97% | |
16 | 1.70 × 10−5 | 133.84% | 7.28 × 10−6 | 21.33% | |
12 | 4.10 × 10−5 | 0.99% | 1.11 × 10−5 | 2.78% | |
13 | 2.31 × 10−5 | 16.08% | 9.95 × 10−6 | 7.92% | |
14 | 2.33 × 10−5 | 208.61% | 9.30 × 10−6 | 93.35% | |
15 | 7.94 × 10−6 | −30.35% | 9.63 × 10−6 | 36.40% | |
16 | 2.64 × 10−5 | 263.14% | 1.06 × 10−5 | 76.67% | |
12 | 4.48 × 10−5 | 10.34% | 1.24 × 10−5 | 14.81% | |
13 | 2.90 × 10−5 | 45.73% | 1.11 × 10−5 | 20.39% | |
14 | 3.40 × 10−5 | 350.33% | 1.26 × 10−5 | 161.95% | |
15 | 2.43 × 10−5 | 113.16% | 1.02 × 10−5 | 44.48% | |
16 | 1.68 × 10−5 | 131.09% | 8.49 × 10−6 | 41.50% |
Notes: ∗: Improvement of
∁: Improvement of
Then, the strain of each lamina is expressed as
The choice of strength theory depends on the actual laminate composite material. Studies have shown that for glass fiber/epoxy composite materials and composite materials with the same tensile strength and compressive strength, the Hill-Tsai strength theory is close to the experimental value in both qualitative and quantitative aspects, while the Tsai-Wu criterion is suitable for composite materials with different tensile and compressive strengths [
Substituting the paving angle of each lamina into the stiffness equation, we can obtain the flexibility matrix coefficient of laminates. Then, combining
Taking the laminates with maximum
Type of laminate | Number of layers | Improvement∗ | Improvement∁ | ||
---|---|---|---|---|---|
12 | 173.35 | — | 228.30 | — | |
13 | 294.20 | — | 327.60 | — | |
14 | 516.29 | — | 640.78 | — | |
15 | 380.27 | — | 468.67 | — | |
16 | 378.35 | — | 491.95 | — | |
12 | 156.90 | −9.49% | 194.34 | −14.88% | |
13 | 207.50 | −29.47% | 281.43 | −14.09% | |
14 | 155.44 | −69.89% | 164.55 | −74.32% | |
15 | 249.71 | −34.33% | 256.70 | −45.23% | |
16 | 253.41 | −33.02% | 362.18 | −73.62% | |
12 | 162.37 | −21.74% | 198.73 | −12.95% | |
13 | 240.37 | −18.29% | 312.94 | −4.47% | |
14 | 228.18 | −55.80% | 280.24 | −56.27% | |
15 | 311.30 | −18.14% | 394.90 | −15.74% | |
16 | 145.80 | −61.46% | 177.89 | −36.16% | |
12 | 163.00 | −5.97% | 180.22 | −21.06% | |
13 | 269.24 | −8.48% | 268.69 | −17.98% | |
14 | 162.05 | −68.61% | 163.85 | −74.43% | |
15 | 199.00 | −47.67% | 248.50 | −46.98% | |
16 | 256.76 | −32.14% | 305.75 | −62.15% |
Notes: ∗: Improvement of
∁: Improvement of
However, the negative impact of strength performance can be minimized through reasonable paving methods. Taking 14-ply
Stacking sequences | | |
||||
---|---|---|---|---|---|
309.77 | 384.49 | [−19.76/−40.92/56.26/54.89/57.83/ −37.84/41.29/−50.62/49.59/−46.93/ −60.40/29.86/−55.51/24.63]T | 2.05 × 10−5 | 348.05 | 465.72 |
361.40 | 448.55 | [−17.61/58.99/−35.89/48.90/47.57/ 64.50/−50.53/−39.78/−45.21/−60.07/ 43.68/20.41/−62.82/30.20]T | 1.91 × 10−5 | 393.81 | 534.43 |
413.03 | 512.62 | [−23.87/−25.05/54.68/−60.11/57.56/ 60.10/46.13/−38.16/46.25/−53.59/ −63.10/24.44/18.50/−52.98]T | 1.79 × 10−5 | 429.74 | 620.84 |
464.66 | 576.70 | [−72.77/−50.79/28.23/43.20/26.01/ −54.73/56.22/33.09/−44.57/−19.40/ −40.92/−32.33/66.24/62.83]T | 1.68 × 10−5 | 466.60 | 586.88 |
516.29 | 640.78 | [−80.68/23.43/29.91/−42.01/−57.52/ 30.03/−46.35/68.39/46.65/−44.59/ −28.24/47.93/−16.46/76.40]T | 1.65 × 10−5 | 533.13 | 733.45 |
With the increase of the in-plane compression load, the multi-coupled laminate entered an unstable equilibrium. Once the load continued to increase, buckling occurred, which further affected the mechanical properties. Multi-couplings may increase the out-of-plane deformation under the in-plane load, which will lead to earlier buckling. Thus, it is very important to determine the influence of multiple coupling on the buckling load.
The analytical solution of buckling load
Taking the laminates with maximum
Type of laminate | Number of layers | First-order buckling factor | Improvement∗ | |
---|---|---|---|---|
12 | 5.32 | 2.13 × 103 | — | |
13 | 6.71 | 2.69 × 103 | — | |
14 | 10.94 | 4.37 × 103 | — | |
15 | 9.09 | 3.64 × 103 | — | |
16 | 12.53 | 5.01 × 103 | — | |
12 | 4.97 | 1.99 × 103 | −6.58% | |
13 | 7.25 | 2.90 × 103 | 8.05% | |
14 | 7.53 | 3.01 × 103 | −31.17% | |
15 | 6.59 | 2.63 × 103 | −27.50% | |
16 | 15.28 | 6.11 × 103 | 21.95% | |
12 | 5.12 | 2.05 × 103 | −3.76% | |
13 | 6.56 | 2.63 × 103 | −2.24% | |
14 | 8.59 | 3.44 × 103 | −21.48% | |
15 | 10.10 | 4.04 × 103 | 11.11% | |
16 | 11.13 | 4.45 × 103 | −11.17% | |
12 | 4.15 | 1.66 × 103 | −21.99% | |
13 | 5.88 | 2.35 × 103 | −12.37% | |
14 | 5.80 | 2.32 × 103 | −46.98% | |
15 | 9.47 | 3.79 × 103 | 4.18% | |
16 | 11.44 | 4.58 × 103 | −8.70% |
Note: ∗: Improvement of
However, the negative impact of buckling can be minimized through reasonable paving design [
Stacking sequences | | |
||
---|---|---|---|
2.62 × 103 | [5.99/−76.30/−74.03/−69.43/12.76/14.95/18.67/ −28.22/60.23/67.35/73.07/−90.00/−22.68/−21.46]T | 2.43 × 10−5 | 2.68 × 103 |
3.06 × 103 | [−89.97/3.03/16.91/−89.57/−76.69/13.37/−52.96/ 17.19/−37.62/−37.89/51.15/60.66/70.71/−25.16]T | 1.98 × 10−5 | 3.07 × 103 |
3.50 × 103 | [21.78/−77.86/−75.25/90.00/26.51/26.56/5.13/−41.55/ −30.86/−23.32/−90.00/−14.78/62.89/66.97]T | 1.80 × 10−5 | 3.56 × 103 |
3.93 × 103 | [73.45/89.83/−18.36/−16.56/41.68/−36.28/54.47/ −29.66/10.30/32.65/−70.13/−71.22/−85.62/18.72]T | 1.74 × 10−5 | 4.05 × 103 |
4.37 × 103 | [−89.80/−11.27/78.62/61.63/−37.59/−24.30/−5.78/ 44.58/30.56/−0.88/−75.50/−69.99/20.75/−81.41]T | 1.67 × 10−5 | 4.39 × 103 |
There are four types of hygro-thermally stable laminates with extension-twisting coupling, including three multi-coupled laminates. They have the same thermal strain in the two main axis directions, which is only related to the material constant, the thermal material constant and the temperature difference. Multi-couplings significantly improve the extension-twisting coupling of laminates (up to 450%). At the same time, multi-couplings generally reduce the buckling load and failure strength. However, the negative impact can be minimized by reasonable paving methods. Furthermore, multi-coupled laminates have good robustness, and the bending-twisting coupling helps improve robustness.
The design optimization of the variable cross-section bending-twisting coupled box structure was achieved based on the multi-coupled laminates. The parametric modeling method is combined with finite element method to achieve high precision and high efficiency of optimal design. The multi-couplings of laminates can obviously improve the bending-twisting coupling of box structure (up to be more than 260%). The design method is applicable to other types of laminates. The optimal design method can also be extended to the static deformation of composite structures with arbitrary complex configurations.