Composite structure and failure theories
DE=16; FG=22; HI=46
Microsoft PowerPoint - SEM302_CLASS_W10_Composite_Analysis_2.pptx SEM302: Advanced Stress Analysis Dr Mathew W Joosten Week 10: Analysis of Composite Structures 2 1 Weekly Objectives Analysis of Composite Structures 2 Understand how we can determine lamina properties when the lamina is orientated at an angle Understand why we must use classical laminate theory to determine the stresses in a composite laminate 2 Analysis of Composite Structures 2 Composite structures can have unexpected behaviour if the designer/analyst does not understand the mechanical properties of composite laminates….. 3 This panel was consolidated using a flat tool. What happened?!? Overview of Composite Analysis (Elastic) 4 Analysis of Composite Structures 2 Fibres Matrix LaminatePly Covered in Week 9 This week Composite Lamina Properties at an Angle • Since unidirectional plies can be orientated at any angle we must transform the stresses from the global coordinate system to the ‘local’ (or ply) coordinate system • How do we do this? Answer: We use stress transformations 5 Analysis of Composite Structures 2 Note: We can only apply stress transformations when all plies are orientated in the same direction. y x θº Analysis of Composite Structures 2 6 Transformation Equations 7 Analysis of Composite Structures 2 2 2 2 2 2 2 2 2 2 2 2 σx τxy τxy σy σy σx +veθ Covered in Week 4 Composite Lamina Properties at an Angle • We can write the transformation equations in an alternate form (to make computation easier) 8 Analysis of Composite Structures 2 2 1 cos 2 2 1 cos 2 2 2 1 cos 2 2 1 cos 2 2 sin 2 2 Using these trigonometric identities Composite Lamina Properties at an Angle • We can write the transformation equations in an alternate form (to make computation easier) 9 Analysis of Composite Structures 2 2 2 Composite Lamina Properties at an Angle • We can write the transformation equations in matrix form 10 Analysis of Composite Structures 2 2 ? ? ? ? ? ? cos sin 2 Composite Lamina Properties at an Angle • We can write the transformation equations in matrix form 11 Analysis of Composite Structures 2 2 2 cos sin We can write the above expression in contracted form: Composite Lamina Properties at an Angle Exercise: You are conducting a tensile test on a four ply composite plate [60]4. The orientation of all plies is 60º from the horizontal x axis. The thickness of each ply is 0.25mm. The width of the sample is 25mm and the applied load is 5kN. 1. Calculate the axial stress in the global coordinate system 2. Determine the stress in the local (or ply) coordinate system 12 Analysis of Composite Structures 2 2 2 60º Analysis of Composite Structures 2 Composite Lamina Properties at an Angle • We can also relate the strain transformation using the transformation matrix, A: 2 2 • The factor of ½ on the shear strain is inelegant • The factor of ½ is due to the definition of shear strain which is twice the tensorial strain (Covered in Week 5) • We can overcome this by introducing Reuters Matrix, [R] 13 Analysis of Composite Structures 2 Composite Lamina Properties at an Angle • Reuters Matrix, [R], is given by: 1 0 0 0 1 0 0 0 2 and 1 0 0 0 1 0 0 0 0.5 • We can therefore relate our local and global strains using: or 14 Analysis of Composite Structures 2 Composite Lamina Properties at an Angle • The compliance matrix for a composite lamina relates the stresses to strain by: = • We can rotate the compliance matrix from the local (ply) to the global (x,y) coordinate system using: ̅ = 15 Ply compliance matrix (covered last) week 1,2,3 coordinate system Ply compliance matrix in the x,y,z coordinate system (C bar) The stiffness matrix in the inverse of the compliance matrix = ̅ Rotating strain from the 1,2,3 to the ply x,y,z coordinate system Composite Lamina Properties at an Angle • We can perform some matrix multiplication and simplification to produce a more convenient form for the stiffness matrix: = 2 4 4 2 4 2 2 2 16 Analysis of Composite Structures 2 Composite Lamina Properties at an Angle Exercise: Calculate the reduced stiffness matrices for the following two lamina: • Single ply orientated at 0º, and (we covered this in last week’s seminar) • Single ply orientated at 90° 17 Analysis of Composite Structures 2 Composite Lamina Properties at an Angle Exercise: Calculate the reduced stiffness matrices for the following two lamina: • Single ply orientated at 0º, and • Single ply orientated at 90° 18 Analysis of Composite Structures 2 Result for the 0º ply (units in MPa) Result for the 90º ply (units in MPa) What is important about this result? 120,773 2,415 0 2,415 7,548 0 0 0 3,900 7,548 2,415 0 2,415 120,773 0 0 0 3,900 Composite Laminate Under Generalised Loading Conditions • Let’s consider a ‘N’ ply generic laminate… 19 Analysis of Composite Structures 2 What conditions must our solution be capable of analysing? Ply 1 Ply 2 Ply N Ply kPlies are ‘counted’ from bottom to top z x y Composite Laminate Under Generalised Loading Conditions • Let’s consider a 6 ply (N=6) generic laminate… 20 Analysis of Composite Structures 2 Ply 1 Ply 2 Ply N Ply k Laminate mid-plane z y h0 h1 h2 hk hN Note: Plies are ‘counted’ from bottom to top Composite Lamina Properties • You have seen this behaviour before. • If you have two materials of differing stiffness, the materials carry different loads. • An example is a steel tube filled with concrete, loaded with an axial load. • We assume displacement equivalence between the two materials • This is why we must use the global (x,y) stiffness to determine composite laminate strains (and stresses) • How can we do this? 21 Analysis of Composite Structures 2 Similar to the Chapter 4 recommended problem (Week 0) Classical Laminate Theory • We can use classical laminate theory to determine the behaviour of composite laminates 22 Analysis of Composite Structures 2 Classical Laminate Theory Assumptions – Each lamina is quasi-homogeneous (we consider the stresses in the plies, not the fibres and the matrix) – The total thickness of the laminate is thin compared to its dimensions • Plane stress!!! – Displacements are continuous though the laminate – Linear-elastic material response – Strain-displacement relationship is linear – Kirchoff assumpations (more on this in FE next year!) • Plane sections remain plane • Bending only results in out of plane displacement • Through thickness shear strains are zero 23 Analysis of Composite Structures 2 Classical Laminate Theory • The sign convention for loads and moments is shown below 24 Analysis of Composite Structures 2 Analysis of Composite Structures 2 Classical Laminate Theory • Let’s consider a simple case, a 2 ply [90,0] laminate loaded in-plane of cross-sectional area, A. 25 N N 0°ply 90°ply How can we calculate the axial stress due to the applied load? Classical Laminate Theory We can derive an expression relating the forces and moments to deformation of the laminate…… 26 Analysis of Composite Structures 2 Analysis of Composite Structures 2 Classical Laminate Theory (ABD Matrix) 27 = κ κ κ 2 3 i, j x, y, s Ply 1 Ply 2 Ply N Ply k z x y Analysis of Composite Structures 2 Classical Laminate Theory (ABD Matrix) Exercise • Let’s analyse a 2 ply [90,0] laminate loaded in-plane using classical laminate theory (use the material properties we calculated last week in the seminar) • Laminate thickness, t =.5mm • We can calculate the first term in the A matrix, Axx 28 i, j x, y, s Classical Laminate Theory (ABD Matrix) Exercise • Let’s analyse a 2 ply [90,0] laminate loaded in-plane using classical laminate theory (thickness, t =.5mm) • We can calculate the first term in the A matrix, Axx 29 Analysis of Composite Structures 2 i, j x, y, s 0 2 2 0 7,548 0.25 120,773 0.25 1,887 30,193 32,080 Ply 1 Ply 2 z y h0=-t/2 h1=0 h2=t/2 Units? Composite Laminate Stacking Sequence • Let’s analyse a 2 ply [90,0] laminate loaded in-plane using classical laminate theory • It is convenient to use MATLAB to perform matrix operations – You can use any software package that you are familiar with (python, maple, mathmatica ….) 30 Analysis of Composite Structures 2 MATLAB 32,080 Our Calculation We have the same result! Composite Laminate Stacking Sequence How can we design the laminate to eliminate unwanted coupling between in-plane loading and laminate curvature? 31 Analysis of Composite Structures 2 Answer: We can design the laminate to ensure that the coupling matrix (B matrix) is zero Composite Laminate Stacking Sequence We will introduce some definitions related to the composition of the laminate • Balanced • Symmetric • Quasi-Isotropic 32 Analysis of Composite Structures 2 Composite Laminate Stacking Sequence We will introduce some definitions related to the composition of the laminate • Balanced – Any laminate that contains one ply of minus theta orientation, for every identical ply with a plus theta orientation (e.g. a laminate with a principal axis of 0º combined with an equal number of plies that have -45º and +45º orientations). • Symmetric – Any laminate where the stacking sequence is symmetrical about the mid-plane of the laminate (eg. [0,30,45,30,0,30,45,30,0]) • Quasi-Isotropic – A balanced and symmetric laminate. The resulting mechanical properties allow the laminate to have quasi-isotropic material behaviour. A common quasi-isotropic laminate is [45,90,-45,0]S. Where the ‘S’ represents symmetry about the mid plane. This laminate would have 8 plies [45,90,-45,0,0,-45,90,45]. 33 Analysis of Composite Structures 2 Analysis of Composite Structures 2 • Example for each laminate type are shown below: • Balanced – [454,-454] • Symmetric – [0,30,45,30,0,30,45,30,0]) • Balanced and Symmetric – [35,-35]2S • Quasi-isotropic – [45,90,-45,0]S • The ABD Matrices for each of these configurations are shown on the following slides 34 Analysis of Composite Structures 2 Laminate 1 – Balanced: [454,-454] 35 Analysis of Composite Structures 2 Laminate 2 – Symmetric: [0,30,45,30,0,30,45,30,0] 36 Analysis of Composite Structures 2 Laminate 3 – Balanced and Symmetric: [35,-35]2S 37 Analysis of Composite Structures 2 Laminate 4 – Quasi-isotropic: [45,90,-45,0]S 38 SEM422: Advanced Stress Analysis Summary 39 Analysis of Composite Structures 2 Now you understand how we can determine lamina properties when the lamina is orientated at an angle You now understand why we must use classical laminate theory to determine the stresses in a composite laminate Homework 40 No homework this week (yay!) Use the time to reflect on your learning experiences in this unit and identify any areas that require further – Eg. If you have difficulty with buckling, please review the lecture slides, seminar notes and textbook and revise the