Exaple of CL for Syetric Laate with Mechaical Loadg hese are two probles which detail the process through which the laate deforatios are predicted usg Classical Laatio theory. he first oly cludes echaical loads o a syetric laate; the secod aalyzes a asyetric laate ad corporates the effects of oisture ad teperature chages. Itroductory rearks he below is a exaple of all the calculatios ecessary for a geeral case to calculate the deforatio of a laate udergog echaical loadg, teperature chages ad oisture absorptio. Recogizg special cases ca reduce the calculatios ecessary to perfor the predictios, e.g. if there is o oisture absorptio, the etirety of those calculatios ca be eglected. As well, if the laate is syetric, the B atrix does ot eed to be calculated, it will be ad if the laate is balaced, A 16 A 26 D 16 D 26. Siilarly, if the chage teperature or oisture is zero, the fictitious forces ad oets ca be oitted. Fally, all the atrices are syetric, thus oly the upper triagle is calculated, the lower triagle is idetical ad therefore is siply reflected across the diagoal. Proble Stateet If a (/45/9/-45) S which is subjected to loadg of y M y y] 3 lb 1 lb lb 5 lb lb lb ] What are the resultat idplae stras, curvatures ad effective properties? Assue the laa aterial properties are; E 1 2 1 6 psi E 2 1.5 1 6 psi, G 12 1 1 6 psi, ν 12.34 α 1.2 1 6 F, α 2 2 1 6 F, β 1.2 1 4, β 2 2 1 4, thickess.75 Solutio: Fdg the Q ad Q atrices o solve this proble, we first eed to fd the Q atrix which is defed as: Hadout: Exaple of CL, p. 1 214
ν 12 E 2 Q 11 Q 12 Q 16 1 ν 12 ν 21 1 ν 12 ν 21 Q] Q 21 Q 22 Q 26 ] ν 12 E 2 E 2 Q 61 Q 62 Q 66 1 ν 12 ν 21 1 ν 12 ν 21 G 12 ] I order to evaluate this expressio, we first eed to fd ν 21. Fro before, the expressio is Isertg the appropriate values we have: E 1 ν 21 ν 12 E 2 E 1.34 ( 1.5 2 ).255 2 1 6.34 1.5 1 6 1.34.255 1.34.255 Q].34 1.5 1 6 1.5 1 6 1.34.255 1.34.255 1 1 6 ] he, for each uique ply, the Q atrix eeds to be foud. he expressios are: Q 11 Q 11 cos 4 θ + 2(Q 12 + 2Q 66 ) s 2 θ cos 2 θ + Q 22 s 4 θ Q 12 Q 21 (Q 11 + Q 22 4Q 66 ) s 2 θ cos 2 θ + Q 12 (s 4 θ + cos 4 θ) Q 22 Q 11 s 4 θ + 2(Q 12 + 2Q 66 ) s 2 θ cos 2 θ + Q 22 cos 4 θ 2.17 1 6.514 1 6.514 1 6 1.513 1 6 ] psi 1 1 6 Q 16 Q 61 (Q 11 Q 12 2Q 66 ) s θ cos 3 θ + (Q 12 Q 22 + 2Q 66 ) s 3 θ cos θ Q 26 Q 62 (Q 11 Q 12 2Q 66 ) s 3 θ cos θ + (Q 12 Q 22 + 2Q 66 ) s θ cos 2 θ Q 66 (Q 11 + Q 22 2Q 12 2Q 66 ) s 2 θ cos 2 θ + Q 66 (s 4 θ + cos 4 θ) Fro trigooetry, the ply, Q ] Q], ad for the 9 ply, sce we have cos 9, s 9 1, oly the Q 11 ad Q 22 expressios are reversed. Evaluatg the expressios for θ, 9, 45, 45 the four Q ] atrices are thus: 2.17 1 6.514 1 6 1.513 1 6.514 1 6 Q ].514 1 6 1.513 1 6 ], Q ] 9.514 1 6 2.17 1 6 ] 1 1 6 1 1 6 6.68 1 6 4.68 1 6 4.67 1 6 6.68 1 6 4.68 1 6 4.67 1 6 Q ] 45 4.68 1 6 6.68 1 6 4.67 1 6 ], Q ] 45 4.68 1 6 6.68 1 6 4.67 1 6 ] 4.67 1 6 4.67 1 6 5.16 1 6 4.67 1 6 4.67 1 6 5.16 1 6 Note how the Q 16 ad Q 26 ters are equal ad opposite Q ] 45 ad Q ] 45, this is why a balaced laate will ot have shear-extesio couplg; the ters will cacel out if there equal ubers of ± θ Hadout: Exaple of CL, p. 2 214
Calculatg the A, B, D atrix After we have the Q atrix for each ply, we ca asseble the A, B ad D atrices which are defed as A] Q ] k (z k z ) Q ] k t k, B] 1 2 Q ] k (z 2 2 k z ), D] 1 3 Q ] k (z 3 3 k z ) I order to calculate these atrices, we eed the appropriate values of z k. o fd the z positios we eed to kow the total thickess of the laate ad the thickess of each laa. he total thickess is the su of all the laa thickesses. I this case, all of those are the sae, so h 4(laa) 2(Syetric).75 ( ).6 ches laa Sce z is defed to be alog the idplae, ad z is egative, we ca the defe z ad z z h 2.3 2.3 ches, z h.3 ches 2 he, the rest of the z values ca be filled. he desired z values are the top ad botto of each laate, so the ext value sequece is siply the additio of the previous z value ad the thickess of the laa questio; hus we have z k+1 z k + t k z.3, z 1.3 +.75.225, z 2.15, z 3.75, z 4 z 5.75, z 5.15, z 6.225, z 7.3 Havg the values of z k, we ca for exaple fd the first etry of the A atrix as; A 11 Q 11 t + Q 11 45 t + Q 11 9 t + Q 11 45 t + Q 11 45 t + Q 11 9 t + Q 11 45 t + Q 11 t For coveiet siplificatio of coplicated laates are syetric, you ca calculate the first half of the A atrix etries ad the siply ultiply by two his eds up beg; A 11 (Q 11 t + Q 11 45 t + Q 11 9 t + Q 11 45 t ) 2 A 11 (2.17 1 6 (.75) + 6.68 1 6 (.75) + 1.513 1 6 (.75) + 6.68 1 6 (.75)) 2 A 11 3.13 1 5 lb Note that the superscript is defg which ply Q ] atrix the value is cog fro while the subscript defes the appropriate etry. As well, although the superscript is the agle of the ply used to calculate the Q ] atrix, ot the explicit value of k which this case would vary fro to 4 or to 8. Hadout: Exaple of CL, p. 3 214
Calculatg the rest of the values usg the sae ethodology; ABD] 5.26 1 5 1.56 1 5 1.56 1 5 5.26 1 5 1.85 1 6 2.5 1 3 3.27 1 1 2.36 1 1 3.27 1 1 9.3 1 1 2.36 1 1 2.36 1 1 2.36 1 1 4.14 1 1 ] Fdg the effective theral ad oisture forces ad oets Because we have o chage teperature or oisture cotet, we ca eglect the calculatios ecessary to fd the theral ad oisture forces/oets, all of these calculatios will be zero. Calculatio of Midplae Stras ad Curvatures Now we ca asseble the etire syste, the total applied loadg is the su of the echaical, theral ad oisture loads; N M ]otal N M ]Mechaical + N M ]heral + N M ]Moisture 3 1 N M ]otal + + 5. ] ] ] Usg a uerical tool (or basic lear algebra) to vert the 6x6 ABD atrix, we obta the abd atrix which ca be used to detere the idplae stras ad curvatures by ultiplyg; abd] N M ]otal ε κ ] ε x ε y γ xy κ x κ y κ xy] 5.64 1 4 2.3 1 5 2.15 1 2 5.197 1 3 9.34 1 3 ] Fdg the total stra at ay pot the thickess of the laate 3 1 5. ] hese ca the be utilized to fd the stra at ay pot through the expressio ε ε + z κ For exaple, to fd the stra alog the upper surface, we have z.1875 thus the stras are ε op Surface x 5.64 1 4 2.15 1 2 8.2 ε y ] 2.3 1 5 ] + (.3) 5.19 1 3 ] 17.9] 1 5 γ xy 9.34 1 3 27.9 Hadout: Exaple of CL, p. 4 214
Fdg the effective properties Additioally, abd] ca be eployed to fd the effective laate properties which have bee defed as; E x 1 h a 11 7.99 1 6 psi, E y 1 h a 22 7.99 1 6 psi, G xy 1 h a 66 3.8 1 6 psi, ν 12 a 21 a 11.296 Also, as stackg sequece strogly effects the bedg stiffess of the laate, the effective flexural stiffess is also calculated. E x f 12 h 3 d 11 12.9 1 6 psi, E y f 12 h 3 d 22 4.33 1 6 psi, G f xy 12 h 3 1.59 1 5 psi, ν f d 12 d 21.242 66 d 11 Note that this is a quasi-isotropic laate, but this oly applies to the plae stiffess values, the flexural stiffess the two directios is ot idetical. As discussed previously, the flexural stiffess is depedet o the stackg sequece, which results differet D 11 ad D 22 values. Hadout: Exaple of CL, p. 5 214
Exaple of CL 2 Itroductory rearks he below is a exaple of all the calculatios ecessary for a geeral case to calculate the deforatio of a laate udergog echaical loadg, teperature chages ad oisture absorptio. Recogizg special cases ca reduce the calculatios ecessary to perfor the predictios, e.g. if there is o oisture absorptio, the etirety of those calculatios ca be eglected. As well, if the laate is syetric, the B atrix does ot eed to be calculated, it will be ad if the laate is balaced, A 16 A 26 D 16 D 26. Siilarly, if the chage teperature or oisture is zero, the fictitious forces ad oets ca be oitted. Fally, all the atrices are syetric, thus oly the upper triagle is calculated, the lower triagle is idetical ad therefore is siply reflected across the diagoal. Proble Stateet If a (/45/9/45/-45) which is cured at 35 F ad is ow service at 8 F ad has absorbed 1% oisture, what will the resultat stras ad curvatures, assug a loadg state of; Assue the laa aterial properties are; y M y y] 3 lb 1 lb 2 lb 5 lb 3 lb 1 lb ] E 1 2 1 6 psi E 2 1.5 1 6 psi, G 12 1 1 6 psi, ν 12.34 α 1.2 1 6 F, α 2 2 1 6 F, β 1.2 1 4, β 2 2 1 4, thickess.75 Solutio: Fdg the Q ad Q atrices o solve this proble, we first eed to fd the Q atrix which is defed as: ν 12 E 2 Q 11 Q 12 Q 16 1 ν 12 ν 21 1 ν 12 ν 21 Q] Q 21 Q 22 Q 26 ] ν 12 E 2 E 2 Q 61 Q 62 Q 66 1 ν 12 ν 21 1 ν 12 ν 21 G 12 ] E 1 Hadout: Exaple of CL, p. 6 214
I order to evaluate this expressio, we first eed to fd ν 21. Fro before, the expressio is Isertg the appropriate values: ν 21 ν 12 E 2 E 1.34 ( 1.5 2 ).255 2 1 6.34 1.5 1 6 1.34.255 1.34.255 Q].34 1.5 1 6 1.5 1 6 1.34.255 1.34.255 1 1 6 ] he, for each uique ply, the Q atrix eeds to be foud. he expressios are: Q 11 Q 11 cos 4 θ + 2(Q 12 + 2Q 66 ) s 2 θ cos 2 θ + Q 22 s 4 θ Q 12 Q 21 (Q 11 + Q 22 4Q 66 ) s 2 θ cos 2 θ + Q 12 (s 4 θ + cos 4 θ) Q 22 Q 11 s 4 θ + 2(Q 12 + 2Q 66 ) s 2 θ cos 2 θ + Q 22 cos 4 θ 2.17 1 6.514 1 6.514 1 6 1.513 1 6 ] psi 1 1 6 Q 16 Q 61 (Q 11 Q 12 2Q 66 ) s θ cos 3 θ + (Q 12 Q 22 + 2Q 66 ) s 3 θ cos θ Q 26 Q 62 (Q 11 Q 12 2Q 66 ) s 3 θ cos θ + (Q 12 Q 22 + 2Q 66 ) s θ cos 2 θ Q 66 (Q 11 + Q 22 2Q 12 2Q 66 ) s 2 θ cos 2 θ + Q 66 (s 4 θ + cos 4 θ) Fro trigooetry, for the ply, Q ] Q], ad for the 9 ply, sce we have cos 9, s 9 1, oly the Q 11 ad Q 22 expressios are reversed. Evaluatg the expressios for θ, 9, 45, 45 the four Q ] atrices are thus: 2.17 1 6.514 1 6 1.513 1 6.514 1 6 Q ].514 1 6 1.513 1 6 ], Q ] 9.514 1 6 2.17 1 6 ] 1 1 6 1 1 6 6.68 1 6 4.68 1 6 4.67 1 6 6.68 1 6 4.68 1 6 4.67 1 6 Q ] 45 4.68 1 6 6.68 1 6 4.67 1 6 ], Q ] 45 4.68 1 6 6.68 1 6 4.67 1 6 ] 4.67 1 6 4.67 1 6 5.16 1 6 4.67 1 6 4.67 1 6 5.16 1 6 Note how the Q 16 ad Q 26 ters are equal ad opposite Q ] 45 ad Q ] 45, this is why a balaced laate will ot have shear-extesio couplg; the ters will cacel out if there equal ubers of ± θ Calculatg the A, B, D atrix After we have the Q atrix for each ply, we ca asseble the A, B ad D atrices which are defed as A] Q ] k (z k z ), B] 1 2 Q ] k (z 2 2 k z ), D] 1 3 Q ] k (z 3 3 k z ) Hadout: Exaple of CL, p. 7 214
I order to calculate these atrices, we eed the appropriate values of z k. o fd the z positios we eed to kow the total thickess of the laate ad the thickess of each laa. he total thickess is the su of all the laa thickesses. I this case, all of those are the sae, so h 5(laa).75 ( ).375 ches laa Sce z is defed to be alog the idplae, ad z is egative, we ca the defe z ad z z h 2.375 2.1875 ches, z h.1875 ches 2 he, the rest of the z values ca be filled. he desired z values are the top ad botto of each laate, so the ext value sequece is siply the additio of the previous z value ad the thickess of the laa questio; hus we have z k+1 z k + t k z.1875, z 1.1875 +.75.1125, z 2.375 z 3.375, z 4.1125, z 5.1875 Note that oe of the values are z k. his is because there is a odd uber of plies, the ceterle of the 9 ply lies alog the ceterle ad sce we oly care about the z positios o the edges of each ply, we igore this locatio. Havg the values of z k, we ca for exaple fd the first etry of the A atrix as; A 11 Q 11 (z 1 z ) + Q 11 45 (z 2 z 1 ) + Q 11 9 (z 3 z 2 ) + Q 11 45 (z 4 z 3 ) + Q 11 45 (z 5 z 4 ) A 11 2.17 1 6 (.1125 (.1875)) + 6.68 1 6 (.375 (.1125)) + 1.513 1 6 (.375 (.375)) + 6.68 1 6 (.1125.375) + 6.68 1 6 (.1875.1125) A 11 2.17 1 6.75 + 6.68 1 6.75 + 1.513 1 6.75 + 6.68 1 6.75 + 6.68 1 6.75 Note that the superscript is defg which Q ] atrix the value is cog fro while the subscript defes the appropriate etry. As well, although the superscript is the agle of the ply used to calculate the Q ] atrix, ot the explicit value of k which this case would vary fro to 5. As you ay have oted, z k z is equal to the thickess of each laa. For cosistecy with the B ad D atrix expressios the differece expressio is retaed, although for efficiecy, siply ultiplyg the appropriate Q ] ter by the thickess of the ply questio will suffice. his shortcut caot be used for B]ad D]. Calculatg the value; A 11 3.13 1 5 lb Calculatg the rest of the values usg the sae ethodology; Hadout: Exaple of CL, p. 8 214
3.13 1 5 1.13 1 5 3.5 1 4 1.52 1 3 4.69 1 2 5.25 1 2 1.13 1 5 3.13 1 5 3.5 1 5 4.69 1 2 5.81 1 2 5.25 1 2 ABD] 3.5 1 4 3.5 1 4 1.31 1 5 5.25 1 2 5.25 1 2 4.69 1 2 1.52 1 2 4.69 1 2 5.25 1 2 52.4 13.2 3.77 4.69 1 2 5.81 1 2 5.25 1 2 13.2 2.93 3.77 5.25 1 2 5.25 1 2 4.69 1 2 3.77.377 15.4 ] Fdg the effective theral ad oisture forces ad oet Now to fd the theral forces/oets, to apply to the laate we use the expressios which have bee previously defed as; y ] Q ] k ] y k Δ(z k z ), For exaple, to fd the theral forces, we have; y ] Δ Q ] ] y + Q ] 45 ] y M y y (z 1 z ) + Q ] 45 ] y 45 ] 1 2 Q ] k ] y 45 (z 4 z 3 ) + Q ] 45 ] y (z 2 z 1 ) + Q ] 9 ] y 45 k (z 5 z 4 )] Δ(z 2 2 k z ) 9 (z 3 z 2 ) Note that the x-y trasfored coefficiets of theral expasio are ecessary for these calculatios, which have bee previously defed as Utilizg θ,45, 45,9, we have ] y ] y 45 α 1 cos 2 θ + α 2 s 2 θ ] α 1 s 2 θ + α 2 cos 2 θ ] y 2 cos θ s θ (α 1 α 2 ).2 1 6 2 1 6 ], ] F y 45 1.1 1 5 1.1 1 5 ],, ] F 1.98 1 5 y 1.1 1 5 1.1 1 5 ],, F 1.98 1 5 9 2 1 6.2 1 6 ], F he, carryg out the atrix ultiplicatio to fd the theral forces ad oets we have y M y M xy ] 226.2 226.2 16.2.24.24.24 ] Hadout: Exaple of CL, p. 9 214
Followg the sae procedure to fd the oisture forces/oets defed as β x ] Q ] k β y ] β xy y We obta k Δ(z k z ), M y y ] 1 β x 2 Q ] k β y ] β xy k Δ(z 2 2 k z ) y M y M xy ] Calculatio of Midplae Stras ad Curvatures 83.8 83.8 6..9.9.9 ] Now we ca asseble the etire syste, the total applied loadg is the su of the echaical, theral ad oisture loads; N M ]otal N M ]Mechaical + N M ]heral + N M ]Moisture 3 1 N 2 M ]otal + 5. 3. 1. ] 226.2 226.2 16.2.24.24.24 ] + 83.8 83.8 6..9.9.9 ] 157.6 42.4 21.2 4.85 3.15.85 ] Usg a uerical tool (or basic lear algebra) to vert the 6x6 ABD atrix, we obta the abd atrix which ca be used to detere the idplae stras ad curvatures by ultiplyg; abd] N M ]otal ε κ ] ε x ε y γ xy κ x κ y κ xy] 1.1 1 3 1.23 1 3 2.61 1 3 1.26 1 1 1.53 1 1 3.95 1 2 ] Fdg the total stra at ay pot the thickess of the laate hese ca the be utilized to fd the stra at ay pot through the expressio ε ε + z κ For exaple, to fd the stra alog the upper surface, we have z.1875 thus the stras are Hadout: Exaple of CL, p. 1 214
ε op Surface x ε y ] γ xy Fdg the effective properties 1.1 1 3 1.26 1 1 1.25 1 3 1.23 1 3 ] + (.1875) 1.53 1 1 ] 4.11 1 3 ] 2.61 1 3 3.95 1 2 1.87 1 3 Additioally, abd] ca be eployed to fd the effective laate properties which have bee defed as; E x 1 h a 11 4.57 1 6 psi, E y 1 h a 22 6.15 1 6 psi, G xy 1 h a 66 2.39 1 6 psi, ν 12 a 21 a 11.244 Also, as stackg sequece strogly effects the bedg stiffess of the laate, the effective flexural stiffess is also calculated. E x f 12 h 3 d 11 6.91 1 6 psi, E y f 12 h 3 d 22 2.95 1 6 psi, G f xy 12 h 3 2.57 1 6 psi, ν f d 12 d 21.716 66 d 11 Of particular ote is that the effective properties are derived assug a syetric, balaced laate, which this is ot. his assuptio is ecessary because the effective properties assue that there is o couplg betwee extesio, shear ad bedg. hus, these effective properties ca be used to give a ipressio of how stiff the laate is but does ot accout for the couplg effects. Hadout: Exaple of CL, p. 11 214