18 Equilibrium equation of a laminated plate (a laminate)

Transcription

1 z ν = 1 ν = ν = 3 ν = N 1 ν = N z 0 z 1 z N 1 z N z α 1 < 0 α > 0 ν = 1 ν = ν = 3 Such laminates are often described b an orientation code [α 1 /α /α 3 /α 4 ] For eample [0/-45/90/45/0/0/45/90/-45/0] Short hand [0/-45/90/45/0] s Other eamples of short hand [0/90] 4 or [0/±45/90], [0/45 /30] etc. 18 Equilibrium equation of a laminated plate (a laminate) h ν = 1 ν = ν = 3 z 0 z 1 ν = 1 ν = ν = 3 z 0 z 1 z d ν = N 1 ν = N z N 1 z N z d ν = N 1 ν = N z N 1 z N z α 1 < 0 d d 9 Cut out the element d d and appl the internal forces and moments as resultants of applied stresses

2 h σ σ z σ σ σ z σ ( + d, z, ) z As h is finite the stresses are unknown functions of z. On the other hand the dimensions d and d are infinitesimall small and we ma approimate the functions according to Talor series σ ab ( + d,, z) = σ ab + σ ab, d and σ ab (, + d, z) = σ ab + σ ab, d where b σ ab we understand σ ab = σ ab (,, z) Now, to write equilibrium equations we need forces and moments acting upon the element. The acting generalized forces are the resultants of the stresses N = N = N = N = Q z = Q z = M = M = h h h h h h h h σ dz σ dz σ dz σ dz σ z dz σ z dz zσ dz zσ dz 10

3 M = M = h h zσ dz zσ dz From the definition of these quantities we see that the are not forces or moments but in fact linear densities of these forces and moments. To get a real forces we need to multipl them b the width of the appropriate area of the element. 11

4 The forces acting in the -direction and the equilibrium equation N, + N, = 0 () N N N +N, d N + N, d z The forces acting in the -direction and the equilibrium equation N, + N, = 0 () N N + N, d N N + N, d z 1

5 The forces acting in the z-direction and the equilibrium equation p = p(, ) p + Q z, + Q z, = 0 (z) Q z Q z Q z + Q z, d Q z + Q z, d z The moments acting in the -direction and the equilibrium equation +resultant moment of the couple Q z -Q z M, + M, Q z = 0 (m) M M M +M, d M + M, d z 13

6 The moments acting in the -direction and the equilibrium equation +resultant moment of the couple Q z -Q z M, + M, Q z = 0 () M M + M, d M M + M, d z Puting these equilibrium equations together we get M ab,ab = p and N ab,a = 0 where a, b =, There are three equations for si unknown. We need a compatibilit equation. The most common one is Kirchhoff hpothesis resulting in Classical lamination theor. 19 Classical lamination theor In Classical lamination theor we assume Kirchhoff hpothesis that sas that points on a normal to an undeformed middle plane sta on a normal to the deformed middle plane. Following the Kirchhoff hpothesis shown on the figure below u o = u o (, ) v o = v o (, ) w = w o = w(, ) u = u o zw, v = v o zw, 14

7 z A o B o z A o B o z w(, ) w, u o A u B Kirchhoff hpothesis z w, z w(, ) w, v o A v B Kirchhoff hpothesis z w, From Cauch s strain tensor formula ε ab = 1 (u a,b + u b,a ) we have ε = u, = u o, zw, ε = v, = v o, zw, ε = 1 (u o, + v o, ) zw, ε z = 0 ε z = 0 ε zz = 0 The last epression is in contrariet with the assumption of plane stress... Now, we are to epress the stresses using the Hooke s law for plane stress state. Wh plane stress when the Kirchhoff hpothesis leads to plane strain we will discuss later. According to (..) we have where and ε= ε ε ε = 1 u o, v o, σ= σ= E ε σ σ σ (u o, + v o, Note the change in the ± sign due to the definition of the curvature vector κ. For the generalized forces N = or, as ε o and κ do not depend on z, where A = h E dz = h N ν=1 z σ dz = zν w, w, w, h = ε o + zκ E (ε o + zκ)dz N = Aε o + Bκ z ν 1 T σ ν(α ν ) ν E T ε ν(α ν )dz 15

8 and B = z E dz = h i.e. B = Similarl for the moments M = or, as ε o and κ do not depend on z, where D = h z E dz = i.e. D = N zν zt σ ν(α ν ) E ν T ε ν(α ν )dz ν=1 z ν 1 N zν zν 1 T σ ν(α ν ) E ν T ε ν(α ν ) ν=1 h N ν=1 N ν=1 z σ dz = zν h z E (ε o + zκ)dz M = Bε o + Dκ z ν 1 z T σ ν(α ν ) ν E T ε ν(α ν )dz zν 3 zν 1 3 T σ 3 ν(α ν ) E ν T ε ν(α ν ) 0 Smmetric laminate Smmetric laminate is a laminate for which for ever ν there is a µ such that α ν = α µ and z ν = z µ 1 Then B = 0 and N = Aε o M = Dκ Using (here) and (there) we get D abcd w abcd = p and A abcd u o c,ad = 0 Add comment on coupling... 1 Balanced laminate Solved problems not onl on B = 0 case 3 Buckling analsis of laminated plates Let us consider smmetric laminate B = 0. For this case we have from above D abcd w,abcd = 0 N ab,a = 0 These equations of equilibrium have been derived under the undeformed geometr configuration. As in the case of column buckling we need to look at the case of deformed shape. 16