CHAPTER 17 COMPOSITES

Transcription

1 CHAPTER 17 COMPOSITES EXERCISE 64, Page Calculate the stiffness matrix (Q) and derive the (A), (B) and (D) matrices fr a sheet f aluminium 1 mm thick. The fllwing material prperties can be assumed: E = 7 GPa and =.35 and fr an istrpic material: E G 21 A Qh and D Q 3 h 12 E 7 G = GPa ) E E 1 (Q) E E G i.e. 7.35(7) 1 (Q).35(7) 7 in GPa (25.93) i.e. (Q) = ingpa A Q h where h = 1. mm i.e. A Q (1.) The units f (A) are: (GPa) mm = MPa mm 1 N / mm mm (A) = inkn / mm

2 (B) = due t symmetry 3 h D Q Q 1. Q The units f (D) are: (GPa)(mm 3 ) = 1 3 MPa (mm 3 ) 1 3 N / mm 2 (mm 3 ) 1 3 Nmm (D) = inknmm Shw that the inverse f (A) in Prblem 1 is: a in mm/n Frm Prblem 1, (A) = in kn / mm hence, C A C i.e A

3 C T and A The determinate f A, A = 79.77(268) ( 724) = 1448 i.e. r a A a a a in mm/n in mm/n 3. An aluminium sheet is 1 mm thick 3 mm lng and 2 mm wide. The sheet is clamped at ne end, and a tensile lad f 6 kn is applied acrss the 2 mm width (i.e. the x directin). Using the stiffness matrix (Q) (fr questin 1 abve), determine the strains and hence the stress in the x and y directins, given the matrix (a). Fr the aluminium assume that E = 7 GPa, =.35, and fr an istrpic material: E G Als given: a in mm/n N A B M B D and 1 A B N B D M and (B) = due t symmetry Hence, Therefre, N A (B)( ) (a)(n)

4 Nw, N 6 / 2 = 3 N/mm X Hence, 3 N in N / mm 3 (a) units: (mm/n)(n/mm) in units f strain where (Q) is given by: i.e. (Q)( ) E E 1 (Q) E E G Q in GPa and x y xy x 3 y MPa xy

5 4. The aluminium sheet described in Prblem 1 abve is 1 mm thick, 3 mm lng and 2 mm wide. The sheet is clamped at ne end frming a 3 mm lng cantilever. A lad is applied which prduces a pure bending mment f 2 N mm acrss the 2 mm width f the sheet (i.e. M x ). Using the stiffness matrix (Q) (fr Prblem 1 abve), determine the strain n the tp and bttm f the sheet. Use this t determine the stress n the tp and bttm f the sheet and sketch the graph f stresses in the x and y directins, given the matrix (d) d in units N 1 mm in units f mm, in MPa, in MPa TOP BOT N A B M B D and 1 A B N a b N B D M h d M Hence, M B (D)( ) and (B) = due t symmetry and (h) = = κ h N d M Lading f 2 N mm ver 2 mm width M x = 2/2 = 1 N mm/mm (i.e. Newtns) d N 1 mm 1 hn

6 i.e mm 1 εz ε (z) κ = Tp surface z =.5 mm (z is +ve dwnwards) (ε) =.5( ) ε.3 strain (σ) = Q N/mm TOP MPa Mid-plane: z = (k) = s () MIDPLANE = Bttm surface z = +.5 mm

7 BOTTOM MPa

8 EXERCISE 65, Page A laminate, shwn belw, cnsists f 4 plies f unidirectinal high mdulus carbn fibres in an epxy resin, each.125 mm thick and arranged as (/9) s. Given that E 1 = 18 GPa, E 2 = 8 GPa, G 12 = 5 GPa and 12 =.3, calculate the reduced stiffness matrices (Q) fr the single plies alng the fibres (i.e. at ). 2 E 1 = 18 GPa, E 2 = 8 GPa, G 12 = 5 GPa, 12 =.3, v E E 8 v 21 = and.3 Q Q 9 v = E1 21E1 1 E E G 12 = (18 1.3(8) (5) Q Q ingpa 5 i.e. 2. Fr the laminate described in Prblem 5, write ut the transfrmatin matrices (T) and (T) 1

9 T c s 2sc s c -2sc = -sc sc c s T T 9 1 sc -sc c s c s 2sc s c 2sc c s 2sc 1 s c -2sc 1 -sc sc c s 1 c s 2sc T 9 s c 2sc sc -sc c s Fr the laminate described in Prblem 5, calculate the reduced stiffness matrices (Q*) and (Q*) 9 fr the single plies with respect t the glbal x-axis (i.e. = ). In this case: (T) = (I) the identity matrix (see Questin 6) (T) 9 = (T) 9 1 frm inspectin f (T) and (T) 1 Q* T QRTR 9 * Q Q Since θ = and T Q in GPa 5.

10 TR RTR Q R T R Q* T QRTR Q* i.e Q* in GPa 4. A lad f 1 N/mm is applied in the x-directin (N x ) and a mment f 1 N mm/mm in the y- directin (M y ), t the laminate described in Prblem 1, Exercise 65. Using the ABD and the abhd matrices fr the laminate described in Prblem 1 given belw: (i) Calculate the direct and the bending strains in the and the 9 layers f the laminate. (ii) Sketch the graph f the direct and the bending strains in the x and y directins thrugh each layer f the laminate. (iii) Calculate the direct and the bending stresses in the and the 9 layers f the laminate. (iv) Sketch the graph f the direct and the bending stresses in the x and y directins thrugh each layer f the laminate.

11 A in N / mm 2.5 ABD 1 B in N D in N mm a in mm / N 4 abd 1 b in1/ N d in1/ (N mm) (i) N A B M B D and 1 A B N B D M and (B) = due t symmetry Hence, N A B Nw, N x = 1 N/mm and M y = 1 N mm/mm 1 Therefre, N in N/mm and M 1 in N The direct and bending strains in the and the 9 layers must be cmpatible. z z Mid-plane (direct) strain: Therefre, 9 an bm where (b) =

12 1 a in (mm / N)(N / mm) i.e. strain strain 4 i.e inunitsf Bending strain: z BENDING and hn dm where (h) = Therefre, k (N) i.e. mm N mm 6 1 i.e. Layer 1 Tp: z =.25 mm -6 BENDING z 3245 x 1 mm/mm i.e. strain BENDING (. 25) Layer 4 Bttm: z =.25 mm

13 BENDING (. 25) (ii) (iii) Mid-plane (direct) stress: DIRECT Q* layer dependent Layer kpa DIR Layer kpa DIR Due t symmetry, layers 4 and 3 will be as fr 1 and 2 respectively.

14 Bending stress: σ zq* κ BENDING σ zq* κ 9 BENDING 9 This describes a linear variatin f stress thrugh the crss-sectin, with a different cnstant f prprtinality fr each layer (e.g. and 9) BENDING z BENDING z 2595 kpa 5 σ 9 BENDING z BENDING z 5863 kpa Tp f layer 1: z =.25 mm ( = ) Tp f layer 2: BEND (. 25) kpa

15 z =.125 mm ( = 9) (iv) 9 BEND (. 125) kpa 5. Shw, using the ABD matrix given in Prblem 4, that the abdh matrix fr the laminate is as given in Prblem 4. Nte that (B) =, hence: (b) = (h) = (A) = (a) 1 (d) = (D) A in kn/mm 2.5 A a C T 1 A A A A A C A C

16 C A T A 47.19(118.) (3.13) a A a in mm/kn D N mm 52.8 D a C T 1 D D D D Due t symmetry D C C T D D a D = +1658(167) + (25.1)( 137) + 6 D d D

17 d in mm N