This article presents closed form natural frequency solutions for two classes of 

mechanically coupled laminate with: extension-bending or; extension-twisting and shearingbending.  Details on the derivation of these two laminate classes are  given; all of which 

contain combinations of standard  ply angles, e.g. +45, -45, 0 and 90°.  Upper and lower 

bounds on the non-dimensional natural frequencies are shown graphically for each class of 

laminate over a range of aspect ratios.  Finally, differences are highlighted between these 

bounds and others obtained by the simplifying assumption that the two laminate classes 

contain only cross plies or angle plies, respectively

Li, D.

York, C.B.

English

Enlighten

Enlighten – Research publications by members of the University of Glasgow http://eprints.gla.ac.uk        Li, D., and York, C. (2012) Natural frequency prediction for laminated rectangular plates with extension-bending or extension-twisting and shearing-bending coupling. In: 1st International Conference on Mechanics of Nano, Micro and Macro Composite Structures, 18-20 June 2012, Torino, Italy  http://eprints.gla.ac.uk/66509/  Deposited on: 3rd July 2012   1st International Conference on Mechanics of Nano, Micro and Macro Composite Structures  N AT U R A L F R E Q U E N C I E S  F O R  L A M I N AT E D  R E C TA N G U L A R  P L AT E S  W I T H  EXTENSION -BENDING  O R  EXTENSION - T W I S T I N G  A N D  S H E A R I NG -BENDING  COUPLING    DAOKUI LI  College of Aerospace and Material Engineering, National University of Defence Technology, Changsha, Hunan, P.R. China, 410073. lidaokui@nudt.edu.cn   CHRISTOPHER B. YORK  School of Engineering, University of Glasgow, James Watt Building, Glasgow, G12 8QQ, Scotland.  christopher.york@glasow.ac.uk     Natural Frequencies for laminated rectangular plates with Extension-Twisting and Shearing-Bending coupling 2 PRESENTATION CONTENTS  • Background on some applications of mechanically coupled laminates. • Characterization of laminated composite materials. • Derivation of Extension-Twisting (and Shearing-Bending) coupled laminates. • Form of stacking sequences - standard angle-ply and cross-ply sub-sequences; combinations which are contrary to the previously assumed form for this class of laminate, i.e., anti-symmetric angle-ply laminates. • Dimensionless parameters - from which the extensional, coupling and bending stiffness terms are readily calculated for any fiber/matrix system.   • Lamination parameters (Ply orientation dependent dimensionless parameters) are shown graphically to illustrate the extent of the design space with up to 21 plies. • Hygro-Thermally Curvature Stable laminates - a special sub-group that can be manufactured flat under a standard elevated temperature curing process. • Bounds on the natural frequency are assessed using a closed form solution, applicable to all the laminate groups presented.   Daokui Li & Christopher B. York 3 BACKGROUND  Exotic mechanical coupling responses are not present in conventional materials, such as metals, and therefore represent an important and significant enabling technology.   However ….. many mechanically coupled laminate designs lead to undesirable warping distortions as a result of the high temperature curing process.  One family of coupled laminate, with immunity to thermal warping, has been used extensively for application to tilt-rotor blade design.   Extension-Twisting coupling, at the structural or blade level, is used to develop optimized twist distribution along the blade for both hover and forward flight: a change in rotor speed, and the resulting centrifugal forces, provides the required twist differential between the two flight regimes.   This behaviour can be achieved from laminate level Extension-Shearing coupling through off-axis alignment of a fully uncoupled laminate.   Natural Frequencies for laminated rectangular plates with Extension-Twisting and Shearing-Bending coupling 4 CHARACTERIZATION  Laminated composite plates are typically characterized in terms of their response to mechanical or thermal loading, which is generally associated with a description of the coupling behaviour, described by the ABD relation:  Couplings generally exist between: • in-plane and out-of-plane actions, when Bij ≠ 0, (5 distinct forms for matrix B!) • Shearing and Extension, when A16 = A26 ≠ 0, and  • Bending and Twisting, when D16 = D26 ≠ 0.  ⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧xyyxNNN = ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡662622161211ASym.AAAAA⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧xyyxγεε + ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡662622161211BSym.BBBBB⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧xyyxκκκ ⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧xyyxMMM = ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡662622161211BSym.BBBBB⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧xyyxγεε + ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡662622161211DSym.DDDDD⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧xyyxκκκ Daokui Li & Christopher B. York 5 Unrestrained thermal (contraction) response of square, initially flat, composite laminates, B = 0: Uncoupled in Extension Extension-Shearing  Uncoupled in Bending Bending-Twisting Bending-Twisting  Uncoupled in Bending  [+/−2//+2/−]T  Simple laminate [+/−/−/+]T  B-T [+/+]T  E-S;B-T [±//−//−3//−3//+]T  E-S Uncoupled (B = 0)  Off-axis alignment of uncoupled (Simple) laminates leads to significant Bending-Twisting coupling at the laminate level, and to detrimental effects on the compression buckling strength of the blade1; which represents an important static design constraint.                                                           1 York, C. B. (2012). Extension-Twisting coupled laminates for aero-elastic compliant blade design. Proc. 53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Paper No. AIAA-2012-1457, Honolulu, Hawaii. Natural Frequencies for laminated rectangular plates with Extension-Twisting and Shearing-Bending coupling 6 By contrast, laminate level Extension-Twisting coupling can also be achieved with immunity to the thermal warping distortions, but special tailoring strategies must be employed. Note: the vast majority of configurations in this class require specially curved tooling in order to possess the required shape after high temperature curing!    Unrestrained thermal (contraction) response of square, initially flat, composite laminates, B16, B26 ≠ 0: Uncoupled in Extension Extension-Shearing (AF)  Uncoupled in Bending Bending-Twisting Bending-Twisting Uncoupled in Bending  [+/−]T  B-S-T-E [+/−3/+2]T  B-S;B-T [+/+/−]T  E-S;B-S;B-T [±//−3//−//−3/±//+]T  E-S;B-S Extension-Twisting, & Shearing-Bending  Daokui Li & Christopher B. York 7 How do we obtain these special laminates?    DERIVATION OF STACKING SEQUENCE DATA  • The general rules of symmetry are relaxed!    Cross plies (0° and 90°) and angle plies (45° and -45°) are not constrained to be symmetric or anti-symmetric about the laminate mid-plane.    • Symbolic notation is adopted to allow for non-standard ply orientations.  All sequences have an angle-ply (+) on one surface (1st ply) of the laminate, but the other surface ply may have equal (+) or opposite (−) orientation or it may indeed be a cross ply ( or ) of 0 or 90° orientation.   • Non-dimensional parameters are adopted to allow for any fiber/matrix system.   Natural Frequencies for laminated rectangular plates with Extension-Twisting and Shearing-Bending coupling 8 DEVELOPMENT OF NON-DIMENSIONAL PARAMETERS The derivation of non-dimensional coupling stiffness parameters is readily demonstrated for the 9-ply laminate, with non-symmetric stacking sequence [+//−//+2/−2/]T, where the coupling stiffness terms, Bij = ∑=nk 1Qʹ′ij,θk(zk2 – zk-12)/2 Bij = {Qʹ′ij+((-7t/2)2 – (-9t/2)2) + Qʹ′ij((-5t/2)2 – (-7t/2)2) + Qʹ′ij−( (-3t/2)2 – (-5t/2)2)  + Qʹ′ij((-t/2)2 – (-3t/2)2) + Qʹ′ij+((t/2)2 – (-t/2)2) + Qʹ′ij+((3t/2)2 – (t/2)2)  + Qʹ′ij−((5t/2)2 – (3t/2)2) + Qʹ′ij−((7t/2)2 – (5t/2)2) + Qʹ′ij((9t/2)2 – (7t/2)2)}/2 where subscripts i,j = 1, 2, 6. The coupling stiffness contributions from the different ply orientations are:  Bij+ = 6t2/2 × Qʹ′ij+ = χ+t2/4 × Qʹ′ij+ χ+ = 12 Bij− = -6t2/2 × Qʹ′ij− = χ−t2/4 × Qʹ′ij− χ− = -12 Bij = 0t3/2 × Qʹ′ij = χt3/4 × Qʹ′ij χ = 0   -9t/2 7t/2 Ply: 1 2 3 4 5 6 7 8 9 Daokui Li & Christopher B. York 9 CALCULATION OF THE ABD MATRIX The calculation procedure for the [+//−//+2/−2/]T laminate ABD matrix, using these non-dimensional parameters (derived for each stacking sequence discovered), is as follows:  Aij = {n+Qʹ′ij+ + n−Qʹ′ij− + nQʹ′ij+ nQʹ′ij} × t Bij = {χ+Qʹ′ij+ + χ−Qʹ′ij− + χQʹ′ij + χQʹ′ij} × t2/4 Dij = {ζ+Qʹ′ij+ + ζ−Qʹ′ij− + ζQʹ′ij + ζQʹ′ij} × t3/12   For Extension-Twisting (and Shearing-Bending) coupling, i.e., B16, B26 ≠ 0 (with all other Bij = 0):  χ+ = -χ− and χ = χ = 0 ü  This does not imply that cross plies ( or ) are absent!  For uncoupled Extensional stiffness properties, i.e., A16 = A26 = 0: n+ = n− ü  For uncoupled Bending stiffness properties, D16 = D26 = 0,  ζ+ = ζ− ü   Natural Frequencies for laminated rectangular plates with Extension-Twisting and Shearing-Bending coupling 10 NUMBER OF SOLUTIONS Table 4 – Number of E-T-S-B coupled angle-ply laminates for each ply number (n) grouping.  Numbers in parentheses indicate non-symmetric stacking sequences. n 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21  1 - 2 - 4 - 7 - 16 - 35(4) - 84(20) - 194(70) - 512(256) - 1,352(850) -  Table 5 – Number of E-T-S-B coupled laminates for each ply number (n) grouping containing combinations of standard ply orientations: 0°, 90° and/or ±45°.   n 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21  1 2 4 8 16 30 63 128 256 536 1,085 2,412 4,774 11,292 21,994 58,502 111,908 329,412 632,996 2,095,268  Daokui Li & Christopher B. York 11 LAMINATION PARAMETERS Lamination parameters can be used to reveal the extent of the 2D and 3D design space; these introduce ply angle dependency to the previously derived non-dimensional parameters:  ξ1 = ξ1A = {n+cos(2θ+) + n−cos(2θ−) + ncos(2θ) + ncos(2θ)}/n  ξ2 = ξ2A = {n+cos(4θ+) + n−cos(4θ−) + ncos(4θ) + ncos(4θ)}/n  ξ5 = ξ1B = { χ+cos(2θ+) + χ−cos(2θ−) + χcos(2θ) + χcos(2θ)}/n2  ξ6 = ξ2B = { χ+cos(4θ+) + χ−cos(4θ−) + χcos(4θ) + χcos(4θ)}/n2  ξ7 = ξ3B = { χ+sin(2θ+) + χ−sin(2θ−) + χsin(2θ) + χsin(2θ)}/n2 Note that ξ8 = ξ4B = 0 if the angle ply orientations are restricted to ±45, 0 and 90°.    ξ9 = ξ1D = {ζ+cos(2θ+) + ζ−cos(2θ−) + ζcos(2θ) + ζcos(2θ)}/n3  ξ10 = ξ2D = {ζ+cos(4θ+) + ζ−cos(4θ−) + ζcos(4θ) + ζcos(4θ)}/n3 Natural Frequencies for laminated rectangular plates with Extension-Twisting and Shearing-Bending coupling 12 LAMINATION PARAMETERS  A11= U1+ξ1U2+ξ2U3 H B11= ξ5U2+ξ6U3 H2/4 D11= U1+ξ9U2+ξ10U3 H3/12 A12=A21= -ξ2U3+U4 H B12=B21= -ξ6U3 H2/4 D12=D21= U4-ξ10U3 H3/12 A16=A61= ξ3U2 2 +ξ4U3 H B16=B61= ξ7U2 2 +ξ8U3 H2/4 D16=D61= ξ11U2 2 +ξ12U3 H3/12 A22= U1-ξ1U2+ξ2U3 H B22= -ξ5U2+ξ6U3 H2/4 D22= U1-ξ9U2+ξ10U3 H3/12 A26=A62= ξ3U2 2 -ξ4U3 H B26=B62= ξ7U2 2 -ξ8U3 H2/4 D26=D62= ξ11U2 2 -ξ12U3 H3/12 A66= -ξ2U3+U5 H B66= -ξ6U3 H2/4 D66= -ξ10U3+U5 H3/12  Laminate thickness (H) = number of plies (n) × ply thickness (t);  Ui are laminate invariant (material) properties. Daokui Li & Christopher B. York 13 Lamination parameters design spaces:   (a)  (b) Figure 2 – Lamination parameter design space representing the 42,601 E-T-S-B laminates with up to (n =) 16 plies for standard combinations of ply orientations, i.e., ±45, 0 and 90°, representing (a) extensional stiffness and (b) bending stiffness. -1.0-0.50.00.51.0-1.0 -0.5 0.0 0.5 1.0ξ 2ξ1-1.0-0.50.00.51.0-1.0 -0.5 0.0 0.5 1.0ξ 10ξ9Natural Frequencies for laminated rectangular plates with Extension-Twisting and Shearing-Bending coupling 14   Figure 3 – Three-view illustration of the coupling lamination parameter design space for E-T-S-B laminates representing all stacking sequences with up to 21 plies, with standard combinations of ply orientations, i.e., ±45, 0 and 90°.     ξ5 ξ7 ξ6 Daokui Li & Christopher B. York 15 HYGRO-THERMALLY CURVATURE-STABLE (HTCS) LAMINATES Here, attention is focused on E-B-S-T;B-T coupled extensionally isotropic laminates, which give rise to the E-T-S-B coupling behaviour of interest; following off-axis alignment.    Note:    Extensional isotropy implies thermal isotropy (but not vice versa!);  Bending stiffness properties have no influence on HTCS behaviour; Non-dimensional parameters are unrevealing in the search for HTCS laminates, hence; Ply-angle-dependent lamination parameters or equivalent Aij and Bij must be interrogated.   Natural Frequencies for laminated rectangular plates with Extension-Twisting and Shearing-Bending coupling 16 CONDITIONS FOR HTCS BEHAVIOUR These are dependent on off-axis alignment, β, of the E-B-S-T parent laminate (described in first column), which for coupled laminates with extensional isotropy are: Lamination parameters and stiffness relationships with respect to material axis alignment, β. β = mπ/2  β = π/8 + mπ/2 β ≠ mπ/2, π/8 + mπ/2  (m = 0, 1, 2, 3)  ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡1111111111B-000BB-0B-B ξ5 = ξ7 = ξ8 = 0 E-B-S-T ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡0B-BB-00B0016161616 ξ5 = ξ6 = ξ7 = 0 E-T-S-B ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡111616161111161111B-B-BB-BB-BB-B ξ5 = ξ7 = 0 E-B-E-T-S-B-S-T The design space for HTCS laminates, achieved by off-axis alignment, β = π/8, of the E-B-S-T parent laminate contains the following number of solutions: n 8 12 16 20 E-T-S-B  (E-T-S-B;B-T) 6 (6) 20 (280) 252  (23,652) 3,076 (2,379,722) Daokui Li & Christopher B. York 17 NATURAL FREQUENCY ASSESSMENT Bounds on the natural frequencies are compared for:    angle-ply laminates (with symmetric and non-symmetric configurations);    new laminates containing standard ply orientations and;   HTCS laminates.   All are E-T-S-B coupled laminates; hence the following closed form solution2 is applicable:  ω2 = (π4/ρ){T33 + (2T12T23T13 – T22T132 – T11T232)/(T11T22 – T122)} where  T11 = A11(mπ/a)2 + A66(nπ/b)2 T12 = (A12 + A66)(mπ/a)(nπ/b) T13 = –(3B16(mπ/a)2 + B26(nπ/b)2)(nπ/b) T22 = A22(nπ/b)2 + A66(mπ/a)2 T23 = –(B16(mπ/a)2 + 3B26(nπ/b)2)(mπ/a) T33 = D11(mπ/a)4 + 2(D12 + 2D66)(mπ/a)2(nπ/b)2 + D22(nπ/b)4  Results are expressed in terms of the non-dimensional frequency: ! =! b2! 2"DIso                                                         2 Jones R M. (1999). Mechanics of composite materials, Taylor & Francis, Philadelphia. Natural Frequencies for laminated rectangular plates with Extension-Twisting and Shearing-Bending coupling 18 Other non-dimensional expressions3 are also used: !" =! a2H!E2; instead of  Fundamental natural frequency for the laminate [45°/-45°/45°/-45°]T, a/b Ωʹ′ Ωʹ′ Ω ΩUpper bound ΩLower bound 1 23.53 23.46 2.0801 2.0801 1.2941 2 53.74 53.65 1.1878 1.1878 0.7477 3 98.87 98.78 0.9713 0.9713 0.6192 2nd natural frequency for the laminate [45°/-45°/45°/-45°]T, a/b Ωʹ′ Ωʹ′ Ω ΩUpper bound ΩLower bound 1 53.74 53.60 4.7514 4.7514 2.9910 2 94.11 93.89 2.0801 2.0801 1.2941 3 147.65 147.38 1.4504 1.4504 0.9062 3rd natural frequency for the laminate [45°/-45°/45°/-45°]T, a/b Ωʹ′ Ωʹ′ Ω ΩUpper bound ΩLower bound 1 94.11 53.60 4.7514 4.7514 2.9910 2 147.65 147.31 3.2634 3.2634 2.0390 3 211.75 211.31 2.0801 2.0801 1.2941 Results have also been validated against NASTRAN FEM analysis.                                                        3 Khdeir, A. A., (1988). Free vibration of antisymmetric angle-ply laminated plates including various boundary conditions, Journal of Sound and Vibration, 122, 377-388. ! =!b2! 2"DIsoDaokui Li & Christopher B. York 19  (a)  (b)  (c) Bounds on Fundamental Natural Frequency curves for simply supported E-T-S-B coupled plates composed of 12-ply laminates with: (a) angle-ply configurations, with ply orientations +45 and -45°, (b) standard configurations with ply orientations +45, -45, 0 and 90° and (c) hygro-thermally curvature-stable configurations with ply orientations +45, -45, 0 and 90° and π/8 off-axis alignment.  FIL 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 0.5 1 1.5 2 2.5 3 3.5 4 Non-dimensional frequency, ! Plate aspect ratio (a/b) Natural Frequencies for laminated rectangular plates with Extension-Twisting and Shearing-Bending coupling 20  (a)  (b)  (c) Bounds on 2nd Natural Frequency curves for simply supported E-T-S-B coupled plates composed of 12-ply laminates with: (a) angle-ply configurations, with ply orientations +45 and -45°, (b) standard configurations with ply orientations +45, -45, 0 and 90° and (c) hygro-thermally curvature-stable configurations with ply orientations +45, -45, 0 and 90° and π/8 off-axis alignment.  Daokui Li & Christopher B. York 21  (a)  (b)  (c) Bounds on 3rd Natural Frequency curves for simply supported E-T-S-B coupled plates composed of 12-ply laminates with: (a) angle-ply configurations, with ply orientations +45 and -45°, (b) standard configurations with ply orientations +45, -45, 0 and 90° and (c) hygro-thermally curvature-stable configurations with ply orientations +45, -45, 0 and 90° and π/8 off-axis alignment.  Natural Frequencies for laminated rectangular plates with Extension-Twisting and Shearing-Bending coupling 22 CONCLUSIONS New Extension-Twisting and Shearing-Bending coupled laminates (E-T-S-B), with up to 21 plies, have been derived from combinations of standard ply orientations, i.e., +45, -45, 0 and 90°, and have been shown to exist for all ply number groupings.    The design space for laminates with standard ply orientations has been shown to be vast in comparison to the previously assumed anti-symmetric angle-ply configurations; non-symmetric angle-ply configurations were also identified.  The design space is increased yet further with the introduction of HTCS laminates.    Natural frequency results for angle-ply and standard-ply laminates, using a closed form solution, reveal an increase in the bounds for standard ply configurations in comparison to angle-ply laminates; with both anti-symmetric and non-symmetric form.    HTCS laminates , which can be manufactured flat after high temperature curing, are constrained within tighter bounds, about the equivalent isotropic laminate, than for general E-T-S-B laminates.   By contrast, HTCS laminates fall on or below the buckling strength of the equivalent isotropic laminate.  Increasing the mechanical Extension-Twisting coupling behaviour has a negative effect on the compression buckling performance, which is not reflected in the natural frequency response. 

Natural frequency prediction for laminated rectangular plates with extension-bending or extension-twisting and shearing-bending coupling

Enlighten: Publications

Enlighten – Research publications by members of the University of Glasgow 
http://eprints.gla.ac.uk 
 
 
 
 
 
 
 
Li, D., and York, C. (2012) Natural frequency prediction for laminated 
rectangular plates with extension-bending or extension-twisting and 
shearing-bending coupling. In: 1st International Conference on 
Mechanics of Nano, Micro and Macro Composite Structures, 18-20 June 
2012, Torino, Italy 
 
http://eprints.gla.ac.uk/66509/ 
 
Deposited on: 3rd July 2012 
 
 
1st International Conference on Mechanics of Nano, Micro and Macro Composite Structures 
 
N AT U R A L F R E Q U E N C I E S  F O R  L A M I N AT E D  
R E C TA N G U L A R  P L AT E S  W I T H  EXTENSION -BENDING  
O R  EXTENSION - T W I S T I N G  A N D  S H E A R I NG -BENDING  
COUPLING  
 
 
DAOKUI LI  
College of Aerospace and Material Engineering, 
National University of Defence Technology, 
Changsha, Hunan, P.R. China, 410073. 
lidaokui@nudt.edu.cn  
 
CHRISTOPHER B. YORK  
School of Engineering, 
University of Glasgow, 
James Watt Building, 
Glasgow, G12 8QQ, Scotland. 
 
christopher.york@glasow.ac.uk  
 
 
 
Natural Frequencies for laminated rectangular plates with Extension-Twisting and Shearing-Bending coupling 
2 
PRESENTATION CONTENTS 
 
• Background on some applications of mechanically coupled laminates. 
• Characterization of laminated composite materials. 
• Derivation of Extension-Twisting (and Shearing-Bending) coupled laminates. 
• Form of stacking sequences - standard angle-ply and cross-ply sub-sequences; combinations which 
are contrary to the previously assumed form for this class of laminate, i.e., anti-symmetric angle-ply 
laminates. 
• Dimensionless parameters - from which the extensional, coupling and bending stiffness terms are 
readily calculated for any fiber/matrix system.   
• Lamination parameters (Ply orientation dependent dimensionless parameters) are shown 
graphically to illustrate the extent of the design space with up to 21 plies. 
• Hygro-Thermally Curvature Stable laminates - a special sub-group that can be manufactured flat 
under a standard elevated temperature curing process. 
• Bounds on the natural frequency are assessed using a closed form solution, applicable to all the 
laminate groups presented.   
Daokui Li & Christopher B. York 
3 
BACKGROUND 
 
Exotic mechanical coupling responses are not present in conventional materials, such as metals, and 
therefore represent an important and significant enabling technology.   
However ….. many mechanically coupled laminate designs lead to undesirable warping distortions as a 
result of the high temperature curing process. 
 
One family of coupled laminate, with immunity to thermal warping, has been used extensively for 
application to tilt-rotor blade design.   
Extension-Twisting coupling, at the structural or blade level, is used to develop optimized twist 
distribution along the blade for both hover and forward flight: a change in rotor speed, and the resulting 
centrifugal forces, provides the required twist differential between the two flight regimes.   
This behaviour can be achieved from laminate level Extension-Shearing coupling through off-axis 
alignment of a fully uncoupled laminate.   
Natural Frequencies for laminated rectangular plates with Extension-Twisting and Shearing-Bending coupling 
4 
CHARACTERIZATION 
 
Laminated composite plates are typically characterized in terms of their response to mechanical or 
thermal loading, which is generally associated with a description of the coupling behaviour, described by the 
ABD relation: 
 
Couplings generally exist between: 
• in-plane and out-of-plane actions, when Bij ≠ 0, (5 distinct forms for matrix B!) 
• Shearing and Extension, when A16 = A26 ≠ 0, and  
• Bending and Twisting, when D16 = D26 ≠ 0.  
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⎪
⎨
⎧
xy
y
x
γ
ε
ε
 + 
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
66
2622
161211
BSym.
BB
BBB
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
xy
y
x
κ
κ
κ
 
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
xy
y
x
M
M
M
 = 
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
66
2622
161211
BSym.
BB
BBB
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
xy
y
x
γ
ε
ε
 + 
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
66
2622
161211
DSym.
DD
DDD
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
xy
y
x
κ
κ
κ
 
Daokui Li & Christopher B. York 
5 
Unrestrained thermal (contraction) response of square, initially flat, composite laminates, B = 0: 
Uncoupled in Extension Extension-Shearing  
Uncoupled in 
Bending Bending-Twisting Bending-Twisting  Uncoupled in Bending 
 
[+/−2//+2/−]T 
 
Simple laminate 
[+/−/−/+]T 
 
B-T 
[+/+]T 
 
E-S;B-T 
[±//−//−3//−3//+]T 
 
E-S 
U
ncoupled 
(B
 = 0) 
 
Off-axis alignment of uncoupled (Simple) laminates leads to significant Bending-Twisting coupling at the 
laminate level, and to detrimental effects on the compression buckling strength of the blade1; which 
represents an important static design constraint.   
  
                                                      
1 York, C. B. (2012). Extension-Twisting coupled laminates for aero-elastic compliant blade design. Proc. 53rd AIAA/ASME/ASCE/AHS/ASC 
Structures, Structural Dynamics and Materials Conference, Paper No. AIAA-2012-1457, Honolulu, Hawaii. 
Natural Frequencies for laminated rectangular plates with Extension-Twisting and Shearing-Bending coupling 
6 
By contrast, laminate level Extension-Twisting coupling can also be achieved with immunity to the 
thermal warping distortions, but special tailoring strategies must be employed. 
Note: the vast majority of configurations in this class require specially curved tooling in order to possess 
the required shape after high temperature curing!   
 
Unrestrained thermal (contraction) response of square, initially flat, composite laminates, B16, B26 ≠ 0: 
Uncoupled in Extension Extension-Shearing (AF) 
 
Uncoupled in 
Bending Bending-Twisting Bending-Twisting Uncoupled in Bending 
 
[+/−]T 
 
B-S-T-E 
[+/−3/+2]T 
 
B-S;B-T 
[+/+/−]T 
 
E-S;B-S;B-T 
[±//−3//−//−3/±//+]T 
 
E-S;B-S 
E
xtension-
T
w
isting, &
 
Shearing-B
ending 
 
Daokui Li & Christopher B. York 
7 
How do we obtain these special laminates?   
 
DERIVATION OF STACKING SEQUENCE DATA 
 
• The general rules of symmetry are relaxed!   
 
Cross plies (0° and 90°) and angle plies (45° and -45°) are not constrained to be symmetric or anti-
symmetric about the laminate mid-plane.   
 
• Symbolic notation is adopted to allow for non-standard ply orientations. 
 
All sequences have an angle-ply (+) on one surface (1st ply) of the laminate, but the other surface ply 
may have equal (+) or opposite (−) orientation or it may indeed be a cross ply ( or ) of 0 or 90° 
orientation.  
 
• Non-dimensional parameters are adopted to allow for any fiber/matrix system. 
  
Natural Frequencies for laminated rectangular plates with Extension-Twisting and Shearing-Bending coupling 
8 
DEVELOPMENT OF NON-DIMENSIONAL PARAMETERS 
The derivation of non-dimensional coupling stiffness parameters is readily demonstrated for the 9-ply 
laminate, with non-symmetric stacking sequence [+//−//+2/−2/]T, where the coupling stiffness terms, 
Bij = ∑
=
n
k 1
Qʹ′ij,θk(zk
2 – zk-12)/2 
Bij = {Qʹ′ij+((-7t/2)
2 – (-9t/2)2) + Qʹ′ij((-5t/2)
2 – (-7t/2)2) + Qʹ′ij−( (-3t/2)
2 – (-5t/2)2)  
+ Qʹ′ij((-t/2)
2 – (-3t/2)2) + Qʹ′ij+((t/2)
2 – (-t/2)2) + Qʹ′ij+((3t/2)
2 – (t/2)2)  
+ Qʹ′ij−((5t/2)
2 – (3t/2)2) + Qʹ′ij−((7t/2)
2 – (5t/2)2) + Qʹ′ij((9t/2)
2 – (7t/2)2)}/2 
where subscripts i,j = 1, 2, 6. 
The coupling stiffness contributions from the different ply orientations are: 
 
Bij+ = 6t
2/2 × Qʹ′ij+ = χ+t
2/4 × Qʹ′ij+ χ+ = 12 
Bij− = -6t
2/2 × Qʹ′ij− = χ−t
2/4 × Qʹ′ij− χ− = -12 
Bij = 0t
3/2 × Qʹ′ij = χt
3/4 × Qʹ′ij χ = 0 
  
-9t/2 
7t/2 
Ply: 1 
2 
3 
4 
5 
6 
7 
8 
9 
Daokui Li & Christopher B. York 
9 
CALCULATION OF THE ABD MATRIX 
The calculation procedure for the [+//−//+2/−2/]T laminate ABD matrix, using these non-
dimensional parameters (derived for each stacking sequence discovered), is as follows: 
 
Aij = {n+Qʹ′ij+ + n−Qʹ′ij− + nQʹ′ij+ nQʹ′ij} × t 
Bij = {χ+Qʹ′ij+ + χ−Qʹ′ij− + χQʹ′ij + χQʹ′ij} × t
2/4 
Dij = {ζ+Qʹ′ij+ + ζ−Qʹ′ij− + ζQʹ′ij + ζQʹ′ij} × t
3/12 
 
 
For Extension-Twisting (and Shearing-Bending) coupling, i.e., B16, B26 ≠ 0 (with all other Bij = 0):  
χ+ = -χ− and χ = χ = 0 ü 
 
This does not imply that cross plies ( or ) are absent! 
 
For uncoupled Extensional stiffness properties, i.e., A16 = A26 = 0: 
n+ = n− ü 
 
For uncoupled Bending stiffness properties, D16 = D26 = 0,  
ζ+ = ζ− ü 
  
Natural Frequencies for laminated rectangular plates with Extension-Twisting and Shearing-Bending coupling 
10 
NUMBER OF SOLUTIONS 
Table 4 – Number of E-T-S-B coupled angle-ply laminates for each ply number (n) grouping.  Numbers in 
parentheses indicate non-symmetric stacking sequences. 
n 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 
 1 - 2 - 4 - 7 - 16
 
- 
35
(4
) 
- 
84
(2
0)
 
- 
19
4(
70
) 
- 
51
2(
25
6)
 
- 
1,
35
2(
85
0)
 
- 
 
Table 5 – Number of E-T-S-B coupled laminates for each ply number (n) grouping containing combinations 
of standard ply orientations: 0°, 90° and/or ±45°.   
n 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 
 1 2 4 8 16
 
30
 
63
 
12
8 
25
6 
53
6 
1,
08
5 
2,
41
2 
4,
77
4 
11
,2
92
 
21
,9
94
 
58
,5
02
 
11
1,
90
8 
32
9,
41
2 
63
2,
99
6 
2,
09
5,
26
8 
 
Daokui Li & Christopher B. York 
11 
LAMINATION PARAMETERS 
Lamination parameters can be used to reveal the extent of the 2D and 3D design space; these introduce 
ply angle dependency to the previously derived non-dimensional parameters: 
 ξ1 = ξ1A = {n+cos(2θ+) + n−cos(2θ−) + ncos(2θ) + ncos(2θ)}/n 
 ξ2 = ξ2A = {n+cos(4θ+) + n−cos(4θ−) + ncos(4θ) + ncos(4θ)}/n 
 ξ5 = ξ1B = { χ+cos(2θ+) + χ−cos(2θ−) + χcos(2θ) + χcos(2θ)}/n
2 
 ξ6 = ξ2B = { χ+cos(4θ+) + χ−cos(4θ−) + χcos(4θ) + χcos(4θ)}/n
2 
 ξ7 = ξ3B = { χ+sin(2θ+) + χ−sin(2θ−) + χsin(2θ) + χsin(2θ)}/n
2 
Note that ξ8 = ξ4B = 0 if the angle ply orientations are restricted to ±45, 0 and 90°.   
 ξ9 = ξ1D = {ζ+cos(2θ+) + ζ−cos(2θ−) + ζcos(2θ) + ζcos(2θ)}/n
3 
 ξ10 = ξ2D = {ζ+cos(4θ+) + ζ−cos(4θ−) + ζcos(4θ) + ζcos(4θ)}/n
3 
Natural Frequencies for laminated rectangular plates with Extension-Twisting and Shearing-Bending coupling 
12 
LAMINATION PARAMETERS 
 
A11= U1+ξ1U2+ξ2U3 H B11= ξ5U2+ξ6U3 H2/4 D11= U1+ξ9U2+ξ10U3 H3/12 
A12=A21= -ξ2U3+U4 H B12=B21= -ξ6U3 H2/4 D12=D21= U4-ξ10U3 H3/12 
A16=A61= ξ3U2 2 +ξ4U3 H B16=B61= ξ7U2 2 +ξ8U3 H2/4 D16=D61= ξ11U2 2 +ξ12U3 H3/12 
A22= U1-ξ1U2+ξ2U3 H B22= -ξ5U2+ξ6U3 H2/4 D22= U1-ξ9U2+ξ10U3 H3/12 
A26=A62= ξ3U2 2 -ξ4U3 H B26=B62= ξ7U2 2 -ξ8U3 H2/4 D26=D62= ξ11U2 2 -ξ12U3 H3/12 
A66= -ξ2U3+U5 H B66= -ξ6U3 H2/4 D66= -ξ10U3+U5 H3/12 
 
Laminate thickness (H) = number of plies (n) × ply thickness (t);  
Ui are laminate invariant (material) properties. 
Daokui Li & Christopher B. York 
13 
Lamination parameters design spaces: 
 
 
(a) 
 
(b) 
Figure 2 – Lamination parameter design space representing the 42,601 E-T-S-B laminates with up to (n =) 16 
plies for standard combinations of ply orientations, i.e., ±45, 0 and 90°, representing (a) extensional stiffness 
and (b) bending stiffness. 
-1.0
-0.5
0.0
0.5
1.0
-1.0 -0.5 0.0 0.5 1.0
ξ 2
ξ1
-1.0
-0.5
0.0
0.5
1.0
-1.0 -0.5 0.0 0.5 1.0
ξ 1
0
ξ9
Natural Frequencies for laminated rectangular plates with Extension-Twisting and Shearing-Bending coupling 
14 
 
 
Figure 3 – Three-view 
illustration of the coupling 
lamination parameter design 
space for E-T-S-B laminates 
representing all stacking 
sequences with up to 21 
plies, with standard 
combinations of ply 
orientations, i.e., ±45, 0 and 
90°.   
  
ξ5 
ξ7 
ξ6 
Daokui Li & Christopher B. York 
15 
HYGRO-THERMALLY CURVATURE-STABLE (HTCS) LAMINATES 
Here, attention is focused on E-B-S-T;B-T coupled extensionally isotropic laminates, which give rise to the 
E-T-S-B coupling behaviour of interest; following off-axis alignment.   
 
Note:   
 
Extensional isotropy implies thermal isotropy (but not vice versa!);  
Bending stiffness properties have no influence on HTCS behaviour; 
Non-dimensional parameters are unrevealing in the search for HTCS laminates, hence; 
Ply-angle-dependent lamination parameters or equivalent Aij and Bij must be interrogated.  
 
Natural Frequencies for laminated rectangular plates with Extension-Twisting and Shearing-Bending coupling 
16 
CONDITIONS FOR HTCS BEHAVIOUR 
These are dependent on off-axis alignment, β, of the E-B-S-T parent laminate (described in first column), 
which for coupled laminates with extensional isotropy are: 
Lamination parameters and stiffness relationships with respect to material axis alignment, β. 
β = mπ/2  β = π/8 + mπ/2 β ≠ mπ/2, π/8 + mπ/2 
 (m = 0, 1, 2, 3)  
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
11
1111
1111
B-00
0BB-
0B-B
 
ξ5 = ξ7 = ξ8 = 0 
E-B-S-T 
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
0B-B
B-00
B00
1616
16
16
 
ξ5 = ξ6 = ξ7 = 0 
E-T-S-B
 
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
111616
161111
161111
B-B-B
B-BB-
BB-B
 
ξ5 = ξ7 = 0 
E-B-E-T-S-B-S-T
 
The design space for HTCS laminates, achieved by off-axis alignment, β = π/8, of the E-B-S-T parent 
laminate contains the following number of solutions: 
n 8 12 16 20 
E-T-S-B  
(E-T-S-B;B-T) 
6 
(6) 
20 
(280) 
252  
(23,652) 
3,076 
(2,379,722) 
Daokui Li & Christopher B. York 
17 
NATURAL FREQUENCY ASSESSMENT 
Bounds on the natural frequencies are compared for:  
  angle-ply laminates (with symmetric and non-symmetric configurations);  
  new laminates containing standard ply orientations and; 
  HTCS laminates.   
All are E-T-S-B coupled laminates; hence the following closed form solution2 is applicable: 
 ω2 = (π4/ρ){T33 + (2T12T23T13 – T22T132 – T11T232)/(T11T22 – T122)} 
where  
T11 = A11(mπ/a)2 + A66(nπ/b)2 
T12 = (A12 + A66)(mπ/a)(nπ/b) 
T13 = –(3B16(mπ/a)2 + B26(nπ/b)2)(nπ/b) 
T22 = A22(nπ/b)2 + A66(mπ/a)2 
T23 = –(B16(mπ/a)2 + 3B26(nπ/b)2)(mπ/a) 
T33 = D11(mπ/a)4 + 2(D12 + 2D66)(mπ/a)2(nπ/b)2 + D22(nπ/b)4 
 
Results are expressed in terms of the non-dimensional frequency: ! =! b
2
! 2
"
DIso   
                                                      
2 Jones R M. (1999). Mechanics of composite materials, Taylor & Francis, Philadelphia. 
Natural Frequencies for laminated rectangular plates with Extension-Twisting and Shearing-Bending coupling 
18 
Other non-dimensional expressions3 are also used: !" =! a
2
H
!
E2
; instead of  
Fundamental natural frequency for the laminate [45°/-45°/45°/-45°]T, 
a/b Ωʹ′ Ωʹ′ Ω ΩUpper bound ΩLower bound 
1 23.53 23.46 2.0801 2.0801 1.2941 
2 53.74 53.65 1.1878 1.1878 0.7477 
3 98.87 98.78 0.9713 0.9713 0.6192 
2nd natural frequency for the laminate [45°/-45°/45°/-45°]T, 
a/b Ωʹ′ Ωʹ′ Ω ΩUpper bound ΩLower bound 
1 53.74 53.60 4.7514 4.7514 2.9910 
2 94.11 93.89 2.0801 2.0801 1.2941 
3 147.65 147.38 1.4504 1.4504 0.9062 
3rd natural frequency for the laminate [45°/-45°/45°/-45°]T, 
a/b Ωʹ′ Ωʹ′ Ω ΩUpper bound ΩLower bound 
1 94.11 53.60 4.7514 4.7514 2.9910 
2 147.65 147.31 3.2634 3.2634 2.0390 
3 211.75 211.31 2.0801 2.0801 1.2941 
Results have also been validated against NASTRAN FEM analysis.  
                                                      
3 Khdeir, A. A., (1988). Free vibration of antisymmetric angle-ply laminated plates including various boundary conditions, Journal of Sound and 
Vibration, 122, 377-388. 
! =!
b2
! 2
"
DIso
Daokui Li & Christopher B. York 
19 
 
(a) 
 
(b) 
 
(c) 
Bounds on Fundamental Natural Frequency curves for simply supported E-T-S-B coupled plates composed 
of 12-ply laminates with: (a) angle-ply configurations, with ply orientations +45 and -45°, (b) standard 
configurations with ply orientations +45, -45, 0 and 90° and (c) hygro-thermally curvature-stable 
configurations with ply orientations +45, -45, 0 and 90° and π/8 off-axis alignment.  
FIL 
0 
0.5 
1 
1.5 
2 
2.5 
3 
3.5 
4 
4.5 
5 
5.5 
6 
0.5 1 1.5 2 2.5 3 3.5 4 
N
on
-d
im
en
si
on
al
 fr
eq
ue
nc
y,
 !
 
Plate aspect ratio (a/b) 
Natural Frequencies for laminated rectangular plates with Extension-Twisting and Shearing-Bending coupling 
20 
 
(a) 
 
(b) 
 
(c) 
Bounds on 2nd Natural Frequency curves for simply supported E-T-S-B coupled plates composed of 12-ply 
laminates with: (a) angle-ply configurations, with ply orientations +45 and -45°, (b) standard configurations 
with ply orientations +45, -45, 0 and 90° and (c) hygro-thermally curvature-stable configurations with ply 
orientations +45, -45, 0 and 90° and π/8 off-axis alignment.  
Daokui Li & Christopher B. York 
21 
 
(a) 
 
(b) 
 
(c) 
Bounds on 3rd Natural Frequency curves for simply supported E-T-S-B coupled plates composed of 12-ply 
laminates with: (a) angle-ply configurations, with ply orientations +45 and -45°, (b) standard configurations 
with ply orientations +45, -45, 0 and 90° and (c) hygro-thermally curvature-stable configurations with ply 
orientations +45, -45, 0 and 90° and π/8 off-axis alignment.  
Natural Frequencies for laminated rectangular plates with Extension-Twisting and Shearing-Bending coupling 
22 
CONCLUSIONS 
New Extension-Twisting and Shearing-Bending coupled laminates (E-T-S-B), with up to 21 plies, have been 
derived from combinations of standard ply orientations, i.e., +45, -45, 0 and 90°, and have been shown to 
exist for all ply number groupings.   
 
The design space for laminates with standard ply orientations has been shown to be vast in comparison to the 
previously assumed anti-symmetric angle-ply configurations; non-symmetric angle-ply configurations were 
also identified.  The design space is increased yet further with the introduction of HTCS laminates.   
 
Natural frequency results for angle-ply and standard-ply laminates, using a closed form solution, reveal an 
increase in the bounds for standard ply configurations in comparison to angle-ply laminates; with both anti-
symmetric and non-symmetric form.   
 
HTCS laminates , which can be manufactured flat after high temperature curing, are constrained within 
tighter bounds, about the equivalent isotropic laminate, than for general E-T-S-B laminates.  
 
By contrast, HTCS laminates fall on or below the buckling strength of the equivalent isotropic laminate.  
Increasing the mechanical Extension-Twisting coupling behaviour has a negative effect on the compression 
buckling performance, which is not reflected in the natural frequency response. 


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Natural frequency prediction for laminated rectangular plates with extension-bending or extension-twisting and shearing-bending coupling

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