S9-1NAS105, Section 9, May 2005 SECTION 9 COMPOSITES

S9-2NAS105, Section 9, May 2005

S9-3NAS105, Section 9, May 2005 TABLE OF CONTENTS SectionPage OVERVIEW………………………………………………………………………………………..9-5 CLASSICAL LAMINATION THEORY (CLT)……………………………………………………9-6 COMPOSITE MATERIALS………………………………………………………………………9-7 CLASSIFICATION OF COMPOSITE MATERIALS……………………………………………9-8 UNIDIRECTIONAL FILAMENTARY LAMINA………………………………………………….9-9 LAMINATE CONSTRUCTION………………………………………………………………… LAMINA ARRANGEMENT IN A 0/90/0 LAMINATE………………………………………… COMPOSITES…………………………………………………………………………………… ROTATION TO MATERIAL COORDINATE SYSTEM……………………………………….9-13 CALCULATION OF COMPOSITE ELEMENT PROPERTIES……………………………….9-14 SYMMETRY IN COMPOSITES………………………………………………………………….9-19 FINITE ELEMENT ANALYSIS OF COMPOSITE MATERIAL STRUCTURES…………….9-20 MSC.NASTRAN INPUT FOR COMPOSITE ANALYSIS…………………………………… TYPICAL ELEMENT – PCOMP – MAT RELATIONSHIP…………………………………….9-22 SAMPLE LAMINATE DEFINITION…………………………………………………………… SPECIFICATION OF REFERENCE DIRECTION…………………………………………… PROPERTY AND MATERIAL INPUT…………………………………………………………..9-25

S9-4NAS105, Section 9, May 2005 TABLE OF CONTENTS SectionPage PLATE ELEMENTS FOR COMPOSITE MATERIAL ANALYSIS……………………………9-30 ELEMENT INPUT…………………………………………………………………………………9-31 OUTPUT……………………………………………………………………………………………9-35 PLY STRESS AND STRAIN OUTPUT…………………………………………………………9-36 ELEMENT FORCE AND STRAIN OUTPUT………………………………………………… COMPOSITE FAILURE INDICES………………………………………………………………9-38 ALLOWABLE STRESSES……………………………………………………………………….9-39 FAILURE THEORIES FOR COMPOSITE MATERIALS…………………………………… HILLS THEORY………………………………………………………………………………… HOFFMANS THEORY………………………………………………………………………… HOFFMAN FAILURE THEORY………………………………………………………………….9-43 TENSOR POLYNOMIAL THEORY (TSAI-WU THEORY)……………………………………9-44 TSAI-WU THEORY……………………………………………………………………………….9-45 INTERLAMINAR SHEAR FAILURE INDEX……………………………………………………9-46 CONCLUSION…………………………………………………………………………………….9-47 REFERENCES ……………………………………………………………………………………9-48

S9-5NAS105, Section 9, May 2005 OVERVIEW n Classical lamination theory is used. n Family of plate elements, QUAD4, QUAD8, TRIA3, and TRIA6 available for modeling composites. n User input is simple. n Stress output for user-requested plies is available. n Can be used in optimization (SOL 200) n Failure indices for elements can be requested.

S9-6NAS105, Section 9, May 2005 CLASSICAL LAMINATION THEORY (CLT) n By this theory, equations for laminate are derived from those of laminas. n Each individual lamina is in plane stress. n The laminate is presumed to consist of perfectly bonded lamina. allowing no relative slip between layers. n A distinct feature of MSC.NASTRAN plate elements is the provision for including transverse shear stiffness: n The effective transverse shear stiffness matrix (G3) for composite plate elements is evaluated on the assumption of the applicability of elementary beam theory type equations for plates. This introduces an approximation that the effects of twisting moments are negligible. In the vast majority of cases such an approximation is satisfactory.

S9-7NAS105, Section 9, May 2005 COMPOSITE MATERIALS n Composite material is defined as one where two or more materials are combined on a macroscopic scale. n This is done to obtain the best qualities of the constituent materials (and in some cases, additional qualities that the constituents do not have). n The following properties are improved and are of major interest: Strength Stiffness Lower weight Tailored properties

S9-8NAS105, Section 9, May 2005 CLASSIFICATION OF COMPOSITE MATERIALS n Lamina is a group of unidirectional fibers (or sometimes woven fibers) arranged to form a flat or curved load resisting member by the use of a matrix. n The principal material axes are parallel and perpendicular to the fiber directions. n The principal directions are also referred to as: u fiber direction, longitudinal direction or 1-direction u matrix direction, transverse direction or 2-direction

S9-9NAS105, Section 9, May 2005 UNIDIRECTIONAL FILAMENTARY LAMINA

S9-10NAS105, Section 9, May 2005 LAMINATE CONSTRUCTION n A laminate is a stack of lamina arranged with the principal directions of each lamina at different orientations so as to obtain the desired strength and stiffness properties. n The various layers of a laminate are bonded together by the same matrix material that is used in the lamina. n Curing bonds the lamina together, usually in the presence of heat and pressure.

S9-11NAS105, Section 9, May 2005 LAMINA ARRANGEMENT IN A 0/90/0 LAMINATE

S9-12NAS105, Section 9, May 2005 COMPOSITES n Composites generally specify lamina properties as a 2-D orthotropic material n The stress-strain relations in principal lamina material directions are n There are four independent constants in the relationship, E 1, E 2, or, G 12 n In many references, this is also written as

S9-13NAS105, Section 9, May 2005 ROTATION TO MATERIAL COORDINATE SYSTEM n To form laminate properties, first the lamina properties are rotated to the element material coordinate system [x m Y m ] Using the equation: where = lamina properties rotated to material coordinate system. [u] = the stress-transformation matrix for transforming stresses from the 1-2 system to the x-y system that is given by

S9-14NAS105, Section 9, May 2005 CALCULATION OF COMPOSITE ELEMENT PROPERTIES n The for the stacked lamina are then integrated through the thickness, to relate the curvatures and mid-surface strains with Forces and Moments: where

S9-15NAS105, Section 9, May 2005 CALCULATION OF COMPOSITE ELEMENT PROPERTIES (Cont.) and where u A ij – Extensional Stiffness u B ij – Coupling Stiffness u D ij – Bending Stiffness

S9-16NAS105, Section 9, May 2005 CALCULATION OF COMPOSITE ELEMENT PROPERTIES (Cont.) n MSC.NASTRAN uses the G1, G2, G3, and G4 matrices to define element properties. n The relation between forces and strains used for MSC.NASTRAN plate elements is where membrane forces per unit length bending moments per unit length transverse shear force per unit length membrane strains in mean plane curvatures; transverse shear strains

S9-17NAS105, Section 9, May 2005 CALCULATION OF COMPOSITE ELEMENT PROPERTIES (Cont.) n Note that where = Laminate thickness = I = Bending inertia

S9-18NAS105, Section 9, May 2005 CALCULATION OF COMPOSITE ELEMENT PROPERTIES (Cont.) n Note that MSC.NASTRANS M x,M y, and M xy terms are reversed from classical lamination theory. NASTRAN Forces In Plate Elements Classical Lamination Theory Forces In Plate Elements

S9-19NAS105, Section 9, May 2005 SYMMETRY IN COMPOSITES n [B] is zero for symmetric laminator symmetry that occurs if for each lamina above the midplane, there is an identical ply (in properties and orientation) located at the same distance below the midplane. n In MSC.NASTRAN, [G 4 ] is similar to [B]. n Examples of symmetric laminates u +45/0/ +45 u +45/ +45/ -45/ -45/ -45/ -45/ +45/ +45/ u +45/ -45/ +45/ -45/ -45/ +45/ -45/ +45/

S9-20NAS105, Section 9, May 2005 FINITE ELEMENT ANALYSIS OF COMPOSITE MATERIAL STRUCTURES n 2-D analysis using lamination theory is found to give good results where the laminate is thin relative to its length. n Otherwise, a full 3-D anisotropic material analysis is desirable. n 3-D analysis is also needed near free edges. n 3-D analysis uses HEXA elements to represent either single lamina or sets of lamina

S9-21NAS105, Section 9, May 2005 MSC.NASTRAN INPUT FOR COMPOSITE ANALYSIS n Executive Control – No changes required. n Case Control – No changes required. n Bulk Data u Plate elements QUAD4, QUAD8, TRIA3, TRIA6 referring to PCOMP property entry. u PCOMP may refer to MAT1, MAT2, or MAT8 material property entries. However, currently QUAD8 and TRIA6 only works with isotropic materials (MAT1) only.

S9-22NAS105, Section 9, May 2005 TYPICAL ELEMENT –PCOMP – MAT RELATIONSHIP Output If ECHO = SORT CQUAD PCOMP MAT1MAT2MAT8 EQUIV PSHELL MID1 MAT2 G1G1 MID2 MAT2 G2G2 MID3 MAT2 G3G3 MID4 MAT2 G4G4

S9-23NAS105, Section 9, May 2005 SAMPLE LAMINATE DEFINITION 2 ply material CQUAD4, 101, 1, A, B, C, D, 5 PCOMP, 1,,,5000., STRN,,,,+ +, 2,.003, 0.,YES, 1,.005, 45.,YES,+ +, 1,.005, 0.,YES, 1,.005,-45.,YES,+ +, 3,.007,90.,YES MAT8, 1, 1.+7, 1.+7,.05, 1.+6, 1.+6, 1.+6,,+ +,,,,.007,.006,.007,.006,.001 MAT8, 2, 2.+7, 2.+6,.35, 1.+6, 1.+6, 1.+6,,+ +,,,,.007,.006,.007,.006,.001 MAT8, 3, 8.+6, 8.+6,.05, 7.+5, 7.+5, 1.+6,,+ +,,,,.006,.005,.006,.005,.001 CORD2R, 5,,0., 0., 0., 0., -1., 0.,+CORD +CORD, 0., 0., 1.

S9-24NAS105, Section 9, May 2005 SPECIFICATION OF REFERENCE DIRECTION n Specify angle theta in element connection entries n Provision to specify coordinate system ID in theta field u The X axis is projected onto the element to define the direction of the X axis of the element material coordinate system. u The Z axis of the material coordinate system is defined by the element coordinate system Z axis (in other words, by the grid order in the element).

S9-25NAS105, Section 9, May 2005 PROPERTY AND MATERIAL INPUT Input Data Entry PCOMP: Layered Composite Element Property Description: Defines the properties of an n-ply composite material laminate. $ $ $ $ $ $ $ $ $ $ PCOMP PID Z0 NSM SB FT TREF GE LAM +AM1 +AM1 MID1 T1 THETA1 SOUT1 MID2 T2 THETA2 SOUT2 +AM2 +AM2 MID3 T3 THETA3 SOUT3..ETC 1/2 line per ply definition PIDProperty ID Z0Offset from grid points to first ply (default -1/2 element thickness) NSMNon-Structural Mass SBAllowable inter-laminar shear FTFailure Theory, Hill, Hoffman, Tsai-Wu, max strain TREFReference temperature. (Overrules MATi TREF) GEElement damping coefficient LAMSYM indicates only 1/2 plies defined and symmetry is used MIDi, Ti, THETAi Material, thickness and angle of ply SOUTiYES gives stress output at this ply

S9-26NAS105, Section 9, May 2005 PROPERTY AND MATERIAL INPUT(Cont.) Example:

S9-27NAS105, Section 9, May 2005 PROPERTY AND MATERIAL INPUT (Cont.) Input Data Entry MAT8 Material Property Definition, Form 8 Description: Defines the material property for a 2-D orthotropic material Format and Example:

S9-28NAS105, Section 9, May 2005 PROPERTY AND MATERIAL INPUT (Cont.) Input Data Entry PSHELL Shell Element Property Description: Defines the membrane, bending, transverse shear, and coupling properties of thin shell elements Format and Example:

S9-29NAS105, Section 9, May 2005 PROPERTY AND MATERIAL INPUT (Cont.) Input Data Entry MAT2 Material Property Definition, Form 2 Description: Defines the material property for a 2-D orthotropic material Format and Example:

S9-30NAS105, Section 9, May 2005 PLATE ELEMENTS FOR COMPOSITE MATERIAL ANALYSIS n QUAD4 n TRIA3 n QUAD8 LINEAR ANALYSIS with isotropic materials only n TRIA6 LINEAR AND GEOMETRIC NONLINEAR ANALYSIS

S9-31NAS105, Section 9, May 2005 ELEMENT INPUT Input Data Entry CQUAD4 Quadrilateral Element Connection Description: Defines a quadrilateral plate element (QUAD4) of the structural model. This is an isoparametric membrane-bending element. Format and Example:

S9-32NAS105, Section 9, May 2005 ELEMENT INPUT (Cont.) Input Data Entry CTRIA3 Triangular Element Connection Description: Defines a triangular plate element (TRIA3) of the structural model. This is an isoparametric membrane-bending element. Format and Example:

S9-33NAS105, Section 9, May 2005 ELEMENT INPUT (Cont.) Input Data Entry CQUAD8 Quadrilateral Element Connection Description: Defines a curved quadrilateral shell element (QUAD8) with 8 grid points. Currently CQUAD8 works with isotropic materials only. Format and Example:

S9-34NAS105, Section 9, May 2005 ELEMENT INPUT (Cont.) Input Data Entry CTRIA6 Triangular Element Connection Description: Defines a curved triangular shell element (TRIA6) with 6 grid points. Currently CTRIA6 works with isotropic materials only. Format and Example:

S9-35NAS105, Section 9, May 2005 OUTPUT n Stresses in individual lamina including approximate interlaminar shear stresses n Failure Index table n Element strains n Element forces

S9-36NAS105, Section 9, May 2005 PLY STRESS AND STRAIN OUTPUT n To obtain ply stresses and strains, use the following Case Control Commands respectively: u STRESS = u STRAIN = n To obtain Failure Index Table, allowables (X t, X c, Y t, Y c, S) must be supplied in MAT8 Bulk Data, and S b can be supplied in PCOMP Bulk Data entry. n Interlaminar shear stresses are output between the lamina. n Individual lamina stresses can be sorted (use NUMOUT1 and BIGER1 parameters). n Failure Index table can be sorted (use NUMOUT2 and BIGER2 parameters). n Available in SOL101 and 103 without alters.

S9-37NAS105, Section 9, May 2005 ELEMENT FORCE AND STRAIN OUTPUT n Element force output and strain output available. n Calculation of the element property data (e.g A, B, D matrices) from user supplied data for lamina is available as printed or punched output.

S9-38NAS105, Section 9, May 2005 COMPOSITE FAILURE INDICES n Composite failures are checked at the lamina level. n Failure index of a lamina checks whether the state of stress can cause a failure. n If the failure index of the element is less than or equal to 1.0, stresses in all laminas are within or on the respective failure envelopes. n If the failure index is greater than 1.0 in at least one lamina, then the element is assumed to fail.

S9-39NAS105, Section 9, May 2005 ALLOWABLE STRESSES X t = Allowable stress or strain in tension in longitudinal direction (or 1-direction or fibre direction) X c = Allowable stress or strain in compression in longitudinal direction (positive sign will be used for X c ) Y t = Allowable stress or strain in tension in transverse direction (or 2-direction or matrix direction) Y c = Allowable stress or strain in compression in transverse direction (positive sign will be used for Y c ) S = Allowable stress in shear (positive or negative shear has the same allowable) S b = Allowable shear stress of bonding material (allowable inter- laminar shear stress) X t, X c, Y t, Y c, and S are supplied in MAT8 Bulk Data entry. S b can be supplied in PCOMP Bulk Data entry.

S9-40NAS105, Section 9, May 2005 FAILURE THEORIES FOR COMPOSITE MATERIALS n HILLS THEORY n HOFFMANS THEORY n TSAI-WU THEORY n INTERLAMINAR SHEAR

S9-41NAS105, Section 9, May 2005 HILLS THEORY where X = X t if is tensile X = X c if is compressive Y = Y t if is tensile Y = Y c if is compressive For the product term, X = X t if and are of the same sign; X = X c otherwise n Basically, the equation represents a failure envelope in the stress space. If the state of stress in the orthotropic lamina (,, ) is such that the stress point is within or on the envelope, the lamina is said to be safe. If the point is outside, the lamina is said to have failed.

S9-42NAS105, Section 9, May 2005 HOFFMANS THEORY n Hills theory does not take into account the differing tensile and compressive strengths in the fiber and matrix directions. n This equation can be thought of as having been derived from Hills theory by adding linear terms to account for differing strengths in tension and compression.

S9-43NAS105, Section 9, May 2005 HOFFMANS FAILURE THEORY Is an ellipsoid in,, space:

S9-44NAS105, Section 9, May 2005 TENSOR POLYNOMIAL THEORY (TSAI-WU THEORY) n The theory of strength for anisotropic materials, proposed by Tsai and Wu, specialized to the case of an orthotropic lamina in a general state of plane stress is where and F 12 is to be determined experimentally.

S9-45NAS105, Section 9, May 2005 TSAI-WU THEORY n The magnitude of F 12 is constrained by the following inequality that is called the stability criterion associated with the theory n The need to satisfy the stability criterion together with the requirement that F 12 be determined experimentally from a combined stress-state poses difficulties. n In the absence of experimental value, Tsai recommends using: n Geometrically, this condition ensures that the strength envelope is closed. That is, the shape of the envelope must be ellipsoidal rather than parabolic or hyperbolic ensuring that the material has finite strength in all directions.

S9-46NAS105, Section 9, May 2005 INTERLAMINAR SHEAR FAILURE INDEX where = Shear stress between the i lamina and the i+1 lamina in the X direction of the element material coordinate system. = Shear stress between the i lamina and the i+1 lamina in the Y direction of the element material and coordinate system. = Allowable interlaminar shear stress that is input on the PCOMP entry.,

S9-47NAS105, Section 9, May 2005 CONCLUSION n MSC.NASTRAN layered composite analysis capability u Is user friendly u Is easy to use u Has simple input u Allows stresses in individual plies to be sorted and output u Provides failure index for individual plies

S9-48NAS105, Section 9, May 2005 REFERENCES n MSC.NASTRAN Reference Manual, Sections 6.5 and n Mechanics of Composite Materials, R.M. Jones; Scripta Book Co., Washington D.C., n Mechanics of Composite Materials, R.M. Christensen; John Wiley & Sons, New York, n Primer on Composite Materials Analysis, J.E. Ashton, J.C. Halph, P.H. Petit; Technomic Publishing Co., Inc., Stamford, Connecticut, 1969.