MECHENG 743 Page 1 of 7 THE UNIVERSITY OF AUCKLAND SEMESTER ONE 2018 Campus: City MECHANICAL ENGINEERING Composite Materials (Time Allowed: THREE hours) NOTE: Attempt any FIVE questions. All questions carry equal marks. The marks allocated to parts of questions are indicated. A list of formulae is given in the APPENDICES BOOKLET. MECHENG 743 Page 2 of 7 1. (a) (i) Define the state of plane stress. (ii) What are the conditions that allow the assumption of a plane stress state? (4 marks) (b) Cylindrical fibres in an idealised unidirectional, continuous fibre-reinforced composite material may be packed either in a square or a hexagonal pattern. Both cross-sectional arrangements are shown in Figure 1. S S d S d S Figure 1 (i) Determine the fibre volume fraction of the composite material in each case as a function of the fibre-to-fibre spacing between the centres of the fibres, S and the fibre diameter, d. (ii) If cylindrical carbon fibres of 10 µm diameter are packed in epoxy in both of the arrangements shown in Figure 1 and the fibre volume fraction of the composite is 0.5, calculate the fibre-to-fibre spacing between the centres of the fibres in each case. (8 marks) MECHENG 743 Page 3 of 7 (c) A unidirectional, continuous fibre-reinforced composite lamina is subjected to the stresses shown in Figure 2. σ11 σ22 τ12 Figure 2 (i) Derive the compliance coefficients in the material coordinate system for the lamina as functions of the engineering constants. (ii) The composite lamina shown in Figure 2 has the following compliance coefficients in the material coordinate system: S11 = 4.15x10-12 Pa-1; S12 = S21 = -1.25x10-12 Pa-1; S22 = 41.5x10-12 Pa-1; S66 = 96.7x10-12 Pa-1 Find the engineering constants of the lamina. (8 marks) 2. (a) (i) Describe briefly the meaning of the term “prepreg material”. (ii) Why are prepregs typically used in the production of high-performance composites? (4 marks) (b) Two unidirectional, continuous fibre-reinforced composite materials consist of the same matrix but different fibres; both composite materials have a fibre volume fraction of 0.5. In the first case, Ef /Em = 50 and in the second case, Ef /Em = 25. (i) In each case, find the ratio of the tensile modulus parallel to the fibres to the tensile modulus perpendicular to the fibres. (ii) Compare the calculated ratios and briefly discuss your conclusions. (8 marks) (c) (i) A short fibre-reinforced composite, consisting of completely aligned fibres is loaded in a direction parallel to the fibres. Show that the tensile strength of the composite in the direction of the fibres is given by when the length of the fibres, l, is greater than the fibre critical length, lc. (1 ) 2 v vcc fu f m m l l σ σ σ ′= − + MECHENG 743 Page 4 of 7 (ii) A carbon fibre-polycarbonate composite consists of completely aligned, short fibres of diameter 7 µm; the volume fraction of the fibres in the composite is 0.4. The tensile strength of the fibres, shear strength of the fibre- matrix interface, and the stress in the matrix at the failure strain of the fibres are 2.5 GPa, 12.5 MPa and 30 MPa, respectively. Calculate the critical fibre length. For the specific case in which the length of the fibres in the composite is ten times the critical fibre length, estimate the tensile strength of the composite in the direction of the fibres. (8 marks) 3. A composite laminate, constructed from identical continuous fibre-reinforced laminae made of E-glass fibres and epoxy, has the stacking sequence: [45º/-45º/45º/-45º/0º/0º/-45º/45º/-45º/45º]. Each lamina, of 0.125 mm thickness, has the following engineering constants: E11 = 40 GPa; E22 = 9.8 GPa; G12 = 2.8 GPa; ν12 = 0.3. (a) Find the elements of the extensional stiffness matrix. (10 marks) (b) Calculate the in-plane Young’s moduli of the laminate in the X and Y directions of the global co-ordinate system. (10 marks) 4. The stress state in the critical lamina of a component made from a composite laminate has been calculated to be: σ1 = 550 MPa, σ2 = -175 MPa, τ12 = 0. The strengths of all laminae in the laminate are equal and are as follows: �1= 1000 MPa, �1 = 600 MPa, �2= 75 MPa, �2= 200 MPa, ̂12= 73 MPa. (a) Calculate the factor of safety against failure using the maximum stress failure criterion. (2 marks) (b) Calculate the factor of safety against failure using the Tsai-Hill failure criterion. (3 marks) (c) Calculate the factor of safety against failure using the Tsai-Wu failure criterion. (5 marks) (d) Sketch the approximate failure loci in the σ1-σ2 plane using both the maximum stress and the Tsai-Hill failure criteria. Indicate where the actual state of stress for the critical lamina is on the diagram. (4 marks) MECHENG 743 Page 5 of 7 (e) Show on the σ1-σ2 failure loci (that you have drawn for part (d)) where the stress state would be if the stresses in the lamina increased by 30%, and identify the resulting failure mode. Clearly differentiate it from the original stress state on the loci. (4 marks) (f) Illustrate with a sketch four different modes of failure initiation for a unidirectional composite under uniaxial tension. (2 marks) 5. A four point bending test is to be undertaken to determine the shear strength of a polymeric foam (SAN foam). A sandwich beam 60 mm wide is constructed for this purpose with facesheets made from a bi-axial E-glass/epoxy composite. Suitable load pads have been selected to prevent localised facesheet and core failures. The beam and loading scenario are illustrated in Figure 3. Section AA is at the centre of the beam, and the loading and support points are at equal distances from this location. Section BB is mid-way between the loading and support points. The relevant material properties for the facesheets and core are provided in Table 1. Figure 3 Table 1 E-Glass/epoxy skins M130 SAN Foam Core Orthotropic Isotropic Thickness 0.4 mm /ply Thickness 30 mm E1 = E2 20 000 MPa E 176 MPa G12 4 000 MPa G 59 MPa ν12 0.30 ν 0.33 �1 300 MPa ̂ 1.98 MPa �1 250 MPa (a) Determine the minimum number of plies required in the skin to prevent a facesheet failure prior to core failure, if P=10 kN, L1 = 150 mm and L2= 450 mm. Assume thin face, weak core assumptions to be true. Hint: solve for d first. (10 marks) MECHENG 743 Page 6 of 7 (b) Sketch both the true bending stress distribution and the distribution assumed for your analysis in part (a) at section AA. (2 marks) (c) Determine the difference between the true and assumed maximum transverse shear stress in the core at section BB, for the loading case of part (a). (3 marks) (d) With the aid of sketches, describe five possible failure modes for a sandwich structure, excluding shear failure of the core, and face yielding or fracture. (5 marks) 6. (a) A uniaxial tensile load of 4.5 tons is to be applied to the edge of a composite panel. The load is to be transferred into the panel through a strap made of carbon tape, the properties of the tape being described in Table 2. Determine the required length of the strap and the minimum number of plies required to sufficiently transfer the load into the panel without exceeding either the design strain or interlaminar shear limits given. Determine an appropriate ply drop-off for the plies in the strap. Sketch the resulting strapping, labelling key dimensions. (8 marks) Table 2 Unidirectional CFRP Tape Thickness 0.3 mm/ply Width 25 mm E1 100 000 MPa 1̂ 0.55% ̂13 15 MPa (b) Sketch the cross-section of a stiffener constructed from a planar sandwich laminate with unidirectional capping applied to its free edge. Label the key features. (3 marks) (c) Sketch the cross-section of a 90° corner of a composite structure where the laminate transitions from sandwich to monolithic across the corner and back to sandwich. Label the key features. (3 marks) (d) Briefly discuss two key considerations when determining the specification of the laminate and coving of a taped joint between two pieces of composite structure. (2 marks) MECHENG 743 Page 7 of 7 (e) Select the correct answer(s) from the options provided and write them in your answer booklet. (4 marks) (i) Failure in which loading direction is more likely to be matrix dominated in a unidirectional long fibre composite? A. Tension B. Compression (ii) How is the failure strain typically affected with an increase in the modulus of a carbon fibre? A. Increases B. Decreases C. Stays the same (iii) Which of the following materials is most suitable for use as a core material in an application where large dynamic loads are expected? A. SAN Foam B. PET Foam C. Balsa wood C. Aramid Honeycomb (iv) Which of the following fibreglass/polyester laminates will provide the best performance in a highly shear (in-plane) loaded region of a structure? Assume all laminae are of equal material composition and thickness. A. [[0/90/0]2]s B. [[-45/45]2]s C. [[90/0/90]4]s D. [0/-45/45/90/-45/45/90/0]s APPENDIX MECHENG 743 Page 1 of 4 THE UNIVERSITY OF AUCKLAND SEMESTER ONE 2018 Campus: City MECHANICAL ENGINEERING Composite Materials APPENDICES BOOKLET APPENDIX MECHENG 743 Page 2 of 4 The following formulae may be of assistance. All symbols have their usual meaning.                                                12 22 11 12 2211 12 22 21 11 12 22 11 1 00 0 1 0 1          G EE EE S lll                                        12 22 11 211212 222212 112111 2112 12 22 11 )1(00 0 0 1 1            G EE EE Q lll   xx xx yy yy xy xy M M D M                           g gS     )cos(sincossin)422(2 cossin)22(cossin)22( cossin)22(cossin)22( cossincos)2(sin )cos(sinsincos)( sinsincos)2(cos 44 66 22 6612221166 3 661222 3 66121126 3 661222 3 66121116 4 22 22 6612 4 1122 44 12 22 66221112 4 22 22 6612 4 1111             SSSSSS SSSSSSS SSSSSSS SSSSS SSSSS SSSSS    g gQ     APPENDIX MECHENG 743 Page 3 of 4 )cos(sincossin)22( cossin)2(cossin)2( cossin)2(cossin)2( cossincos)2(2sin )cos(sinsincos)4( sinsincos)2(2cos 44 66 22 6612221166 3 662212 3 66121126 3 662212 3 66121116 4 22 22 6612 4 1122 44 12 22 66221112 4 22 22 6612 4 1111             QQQQQQ QQQQQQQ QQQQQQQ QQQQQ QQQQQ QQQQQ a b c d ad bc d b c a                 1 1 2 2 2 2 2 2 cos sin 2sin cos sin cos 2sin cos sin cos sin cos cos sin lT                               0g g gz    N N N M M M A A A B B B A A A B B B A A A B B B B B B D D D B B B D D D B B B D D D x y xy x y xy x y xy x y xy                                                        11 12 16 11 12 16 12 22 26 12 22 26 16 26 66 16 26 66 11 12 16 11 12 16 12 22 26 12 22 26 16 26 66 16 26 66 0 0 0                          3 1 32 1 2 1 1 3 1 , 2 1 , kkkkkk n k k zzzzzzQDB,A, 1 ˆˆˆˆ 2 ˆˆˆˆˆ 2 2 2 2 1 1 1 1 21122 12 2 12 22 2 2 11 2 1  ctctctct F                         1 2 1 2 1 2 1 2 2 2 2 2 12 2 12 2 1                i ffu c d l   2 APPENDIX MECHENG 743 Page 4 of 4          2 2dtE D ff        c c t d GS 2     D zzExM x x  d Tx xz  35.0 ccf wr f GEE






















































































































































































































































































































































































































































































































































































































































































































































































































































































































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