Luận án Phân tích dao động dầm, tấm Sandwich 2D-FGM hai và ba pha bằng phương pháp phần tử hữu hạn

Nội dung text: Luận án Phân tích dao động dầm, tấm Sandwich 2D-FGM hai và ba pha bằng phương pháp phần tử hữu hạn

1. 120 [23] A.E. Alshorbagy, M.A. Eltaher, and F.F. Mahmoud. Free vibration chatacteristics of a functionally graded beam by finite element method. Applied Mathematical Modelling, 35:412–425, 2011. [24] A. Shahba, R. Attarnejad, M. T. Marvi, and S. Hajilar. Free vibration and stability analysis of axially functionally graded tapered Euler-Bernoulli beams. Shock and Vibration, 18:683–696, 2011. [25] A. Shahba, R. Attarnejad, M. T. Marvi, and S. Hajilar. Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions. Composites: Part B, 42:801–808, 2011. [26] Y. Huang and X.-F. Li. A new approach for free vibration of axially functionally graded beams with non-uniform cross-section. Journal of Sound and Vibration, 329(11):2291–2303, 2010. [27] Y. Huang, L.-E. Yang, and Q.-Z. Luo. Free vibration of axially functionally graded Timoshenko beams with non-uniform cross-section. Composites Part B: Engineering, 45(1):1493–1498, 2013. [28] Y. Zhao, Y. Huang, and M. Guo. A novel approach for free vibration of axially functionally graded beams with non-uniform cross-section based on Chebyshev polynomials theory. Composite Structures, 168:277–284, 2017. [29] P.S. Ghatage, V.R. Kar, and P.E. Sudhagar. On the numerical modelling and analy- sis of multi-directional functionally graded composite structures: A review. Com- posite Structures, 236:111837, 2020. [30] Z. Wang, X. Wang, G. Xu, S. Cheng, and T. Zeng. Free vibration of two- directional functionally graded beams. Composite Structures, 135:191–198, 2016. [31] D. Hao and C. Wei. Dynamic characteristics analysis of bi-directional functionally graded timoshenko beams. Composite Structures, 141:253–263, 2016. [32] A. Karamanlı. Free vibration analysis of two directional functionally graded beams using a third order shear deformation theory. Composite Structures, 189:127–136, 2018. [33] T.A. Huynh, X.Q. Lieu, and J. Lee. NURBS-based modeling of bidirectional functionally graded Timoshenko beams for free vibration problem. Composite Structures, 160:1178–1190, 2017.
2. 122 [44] Y. Yang, C. C. Lam, K. P. Kou, and V. P. Iu. Free vibration analysis of the func- tionally graded sandwich beams by a meshfree boundary-domain integral equa- tion method. Composite Structures, 117:32–39, 2014. [45] T.P. Vo, H.-T. Thai, T.-K. Nguyen, F. Inam, and J. Lee. A quasi-3D theory for vibration and buckling of functionally graded sandwich beams. Composite Struc- tures, 119:1–12, 2015. [46] A.I. Osofero, T.P.Vo, T.-K. Nguyen, and J. Lee. Analytical solution for vibra- tion and buckling of functionally graded sandwich beams using various quasi-3D theories. Journal of Sandwich Structures & Materials, 18(1):3–29, 2016. [47] P. Tossapanon and N. Wattanasakulpong. Stability and free vibration of function- ally graded sandwich beams resting on two-parameter elastic foundation. Com- posite Structures, 142:215–225, 2016. [48] L.C. Trinh, T.P. Vo, A.I.Osofero, and J. Lee. Fundamental frequency analysis of functionally graded sandwich beams based on the state space approach. Compos- ite Structures, 156:263–275, 2016. [49] V. Kahya and M. Turan. Vibration and stability analysis of functionally graded sandwich beams by a multi-layer finite element. Composites Part B: Engineering, 146:198–212, 2018. [50] A.S. Sayyad and P.V. Avhad. On static bending, elastic buckling and free vibra- tion analysis of symmetric functionally graded sandwich beams. Journal of Solid Mechanics, 11(1):166–180, 2019. [51] M. S¸ims¸ek and T. Kocaturk.¨ Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load. Composite Structures, 90:465–473, 2009. [52] M. S¸ims¸ek. Vibration analysis of a functionally graded beam under a moving mass by using different beam theory. Composite Structures, 92:904–917, 2010. [53] M. S¸ims¸ek, T. Kocaturk,¨ and D.S¸. Akbas¸. Dynamic behavior of an axially func- tionally graded beam under action of a moving harmonic load. Composite Struc- tures, 94:2358–2364, 2012. [54] M. S¸ims¸ek and M. Al-shujairi. Static, free and forced vibration of functionally graded (FG) sandwich beams excited by two successive moving harmonic loads. Composites Part B, 108:18–34, 2017.
3. 124 [65] M.E. Fares, M.K. Elmarghany, and D. Atta. An efficient and simple refined theory for bending and vibration of functionally graded plates. Composite Structures, 91(3):296–305, 2009. [66] H.-T. Thai and D.-H. Choi. A refined shear deformation theory for free vibration of functionally graded plates on elastic foundation. Composites Part B: Engineer- ing, 43(5):2335–2347, 2012. [67] H.-T. Thai, T. Park, and D.-H. Choi. An efficient shear deformation theory for vibration of functionally graded plates. Archive of Applied Mechanics, 83(1):137– 149, 2013. [68] H.-T. Thai and T.P. Vo. A new sinusoidal shear deformation theory for bend- ing, buckling, and vibration of functionally graded plates. Applied Mathematical Modelling, 37(5):3269–3281, 2013. [69] F.Z. Zaoui, D. Ouinas, and A. Tounsi. New 2D and quasi-3D shear deformation theories for free vibration of functionally graded plates on elastic foundations. Composites Part B: Engineering, 159:231–247, 2019. [70] H. Matsunaga. Free vibration and stability of functionally graded plates according to a 2-D higher-order deformation theory. Composite structures, 82(4):499–512, 2008. [71] X. Zhao, Y.Y. Lee, and K.M. Liew. Free vibration analysis of functionally graded plates using the element-free kp-ritz method. Journal of Sound and Vibration, 319(3-5):918–939, 2009. [72] S. Hosseini-Hashemi, H.R.D. Taher, H. Akhavan, and M. Omidi. Free vibration of functionally graded rectangular plates using first-order shear deformation plate theory. Applied Mathematical Modelling, 34(5):1276–1291, 2010. [73] S.S. Akavci. An efficient shear deformation theory for free vibration of function- ally graded thick rectangular plates on elastic foundation. Composite Structures, 108:667–676, 2014. [74] S.S. Vel and R.C. Batra. Three-dimensional exact solution for the vibration of functionally graded rectangular plates. Journal of Sound and Vibration, 272(3- 5):703–730, 2004. [75] S. Chakraverty and K.K. Pradhan. Free vibration of functionally graded thin rect- angular plates resting on Winkler elastic foundation with general boundary condi-
4. 126 [86] H.-T. Thai, T.-K. Nguyen, T.P. Vo, and J. Lee. Analysis of functionally graded sandwich plates using a new first-order shear deformation theory. European Jour- nal of Mechanics-A/Solids, 45:211–225, 2014. [87] A. Tounsi, M.S.A. Houari, S. Benyoucef, and A.A. El Bedia. A refined trigono- metric shear deformation theory for thermoelastic bending of functionally graded sandwich plates. Aerospace Science and Technology, 24(1):209–220, 2013. [88] R. Ye, N. Zhao, D. Yang, J. Cui, O. Gaidai, and P. Ren. Bending and free vibration analysis of sandwich plates with functionally graded soft core, using the new re- fined higher-order analysis model. Journal of Sandwich Structures & Materials, 23(2):680–710, 2021. [89] Z. Belabed, A. A. Bousahla, M. S. A. Houari, A. Tounsi, and S. R. Mahmoud. A new 3-unknown hyperbolic shear deformation theory for vibration of functionally graded sandwich plate. Earthquakes and Structures, 14(2):103–115, 2018. [90] A.A. Daikh and A.M. Zenkour. Effect of porosity on the bending analysis of various functionally graded sandwich plates. Materials Research Express, 6(6):065703, 2019. [91] S. S. Akavci. Mechanical behavior of functionally graded sandwich plates on elastic foundation. Composites Part B: Engineering, 96:136–152, 2016. [92] L. Iurlaro, M. Gherlone, and M. DiSciuva. Bending and free vibration analysis of functionally graded sandwich plates using the refined zigzag theory. Journal of Sandwich Structures & Materials, 16(6):669–699, 2014. [93] S. Pandey and S. Pradyumna. Analysis of functionally graded sandwich plates using a higher-order layerwise theory. Composites Part B: Engineering, 153:325– 336, 2018. [94] C.F. Lu,¨ C.W. Lim, and W.Q. Chen. Semi-analytical analysis for multi-directional functionally graded plates: 3-D elasticity solutions. International Journal for Nu- merical Methods in Engineering, 79(1):25–44, 2009. [95] V. Tahouneh and M.H. Naei. A novel 2-D six-parameter power-law distribu- tion for three-dimensional dynamic analysis of thick multi-directional functionally graded rectangular plates resting on a two-parameter elastic foundation. Mecca- nica, 49(1):91–109, 2014.
5. 128 [107] Trần Thị Thơm. Mô hình phần tử hữu hạn trong phân tích dao động của dầm có cơ tính biến đổi theo hai chiều. Luận án Tiến sĩ, Học viện Khoa học và Công nghệ, VAST, Hà Nội, 2019. [108] Nguyễn Ngọc Huyên. Phân tích dao động và chẩn đoán vết nứt dầm FGM. Luận án Tiến sĩ, Học viện Khoa học và Công nghệ, VAST, Hà Nội, 2017. [109] Ngô Trọng Đức. Phân tích dầm Timoshenko có nhiều vết nứt bằng vật liệu cơ tính biến thiên (FGM) và ứng dụng vào nhận dạng tham số. Luận án Tiến sĩ, Trường Đại học Xây dựng, Hà Nội, 2018. [110] Nguyễn Bá Duy. Analysis of functionally graded sandwich beams under hygro- thermo-mechanical loads. Luận án Tiến sĩ, Đại học Sư phạm Kỹ thuật, Thành phố Hồ Chí Minh, 2019. [111] Lê Thị Ngọc Ánh. Mô hình phần tử hữu hạn trong phân tích kết cấu dầm sandwich FGM. Luận án Tiến sĩ, Học viện Khoa học và Công nghệ, VAST, Hà Nội, 2022. [112] T.-K. Nguyen, T.P. Vo, and H.-T. Thai. Static and free vibration of axially loaded functionally graded beams based on the first-order shear deformation theory. Composites Part B: Engineering, 55:147–157, 2013. [113] T.-K. Nguyen, T.-P. Nguyen, T.P. Vo, and H.-T. Thai. Vibration and buckling analysis of functionally graded sandwich beams by a new higher-order shear de- formation theory. Composites Part B, 76:273–285, 2015. [114] T.-K. Nguyen, T.P. Vo, B.-D. Nguyen, and J. Lee. An analytical solution for buckling and vibration analysis of functionally graded sandwich beams using a quasi-3D shear deformation theory. Composite Structures, 156:238–252, 2016. [115] T.-K. Nguyen and B.-D. Nguyen. A new higher-order shear deformation theory for static, buckling and free vibration analysis of functionally graded sandwich beams. Journal of Sandwich Structures & Materials, 17(6):613–631, 2015. [116] T.-K. Nguyen, B.-D. Nguyen, T.P. Vo, and H.-T. Thai. Hygro-thermal effects on vibration and thermal buckling behaviours of functionally graded beams. Com- posite Structures, 176:1050–1060, 2017. [117] N.T. Khiem and N.N. Huyen. A method for crack identification in functionally graded Timoshenko beam. Nondestructive Testing and Evaluation, 32(3):319– 341, 2017.
6. 130 [128] D.K. Nguyen, Q.H. Nguyen, T.T. Tran, and V.T. Bui. Vibration of bi-dimensional functionally graded Timoshenko beams excited by a moving load. Acta Mechan- ica, 228(1):141–155, 2017. [129] D.K. Nguyen and T.T. Tran. Free vibration of tapered BFGM beams using an efficient shear deformable finite element model. Steel and Composite Structures, 29(3):363–377, 2018. [130] D.K. Nguyen and T.T. Tran. A corotational formulation for large displacement analysis of functionally graded sandwich beam and frame structures. Mathemati- cal Problems in Engineering, 2016, 2016. [131] C.I. Le, N.A.T. Le, and D.K. Nguyen. Free vibration and buckling of bidi- rectional functionally graded sandwich beams using an enriched third-order shear deformation beam element. Composite Structures, 261261:113309, 2020. [132] D.K. Nguyen, T.T. Tran, V.N. Pham, and N.A. Le. Dynamic analysis of an in- clined sandwich beam with bidirectional functionally graded face sheets under a moving mass. European Journal of Mechanics-A/Solids, 88:104276, 2021. [133] A.N.T. Vu, , N.A. Le, and D.K. Nguyen. Dynamic behaviour of bidirectional functionally graded sandwich beams under a moving mass with partial foundation supporting effect. Acta Mechanica, 232(4), 2021. 021-02948-z. [134] M. Nemat-Alla, K.I.E. Ahmed, and I. Hassab-Allah. Elastic–plastic analysis of two-dimensional functionally graded materials under thermal loading. Interna- tional Journal of solids and Structures, 46(14-15):2774–2786, 2009. [135] A. Karamanlı. Bending behaviour of two directional functionally graded sand- wich beams by using a quasi-3d shear deformation theory. Composite Structures, 174:70–86, 2017. [136] W.B. Bickford. A consistent higher order beam theory. Development of Theoreti- cal and Applied Mechanics, 144:341–356, 1982. [137] J.N. Reddy. A simple higher-order theory for laminated composite plates. Journal of Applied Mechanics, 51:745–752, 1984. [138] R.P. Shimpi and H.G. Patel. Free vibrations of plate using two variable refined plate theory. Journal of Sound and Vibration, 296(4-5):979–999, 2006.
7. 132 [151] F. Ebrahimi, M. Nouraei, and A. Dabbagh. Thermal vibration analysis of embed- ded graphene oxide powder-reinforced nanocomposite plates. Engineering with Computers, 36(3):879–895, 2020. [152] A.N.T. Vu, N.A.T. Le, and D.K. Nguyen. Dynamic behaviour of bidirectional functionally graded sandwich beams under a moving mass with partial foundation supporting effect. Acta Mechanica, pages 1–23, 2021. [153] P. Solín. Partial differential equations and the finite element method. John Wiley & Sons Inc., Hoboken, 2006. [154] Y.S. Hsu. Enriched finite element methods for timoshenko beam free vibration analysis. Applied Mathematical Modelling, 40(15-16):7012–7033, 2016. [155] G.N. Praveen and J.N. Reddy. Nonlinear transient thermoelastic analysis of func- tionally graded ceramic-metal plates. International Journal of Solids Strutures, 33:4457 4476, 1998. [156] Q. Song, J. Shi, and Z. Liu. Vibration analysis of functionally graded plate with a moving mass. Applied Mathematical Modelling, 46:141–160, 2017. [157] M. Nemat-Alla. Reduction of thermal stresses by developing two-dimensional functionally graded materials. International Journal of Solids and Structures, 40:7339–7356, 2003. [158] M. Nemat-Alla, K.I.E. Ahmed, and I. Hassab-Allah. Elastic–plastic analysis of two-dimensional functionally graded materials under thermal loading. Interna- tional Journal of Solids and Structures, 46(14-15):2774–2786, 2009. [159] R.D. Mindlin. Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. ASME Journal of Applied Mechanics, 18:31–38, 1951. [160] O.C. Zienkiewicz, Z. Xu, L.F. Zeng, A. Samuelsson, and N.-E. Wiberg. Linked interpolation for reissner-mindlin plate elements: Part i—a simple quadrilateral. International Journal for Numerical Methods in Engineering, 36(18):3043–3056, 1993. [161] D. Ribaric´ and G. Jelenic.´ Higher-order linked interpolation in quadrilateral thick plate finite elements. Finite elements in analysis and design, 51:67–80, 2012.
8. 134 for i=1:nG r=PT2(i); x=le*(1+r)/2; % in x direction x1=(ne-1)*le+x; % x measured from beam left end for j=1:nG t=PT2(j); z1=(h1+h0)/2 + (h1-h0)*t/2; % in [h0,h1] z2=(h2+h1)/2 + (h2-h1)*t/2; % in [h1,h2] z3=(h3+h2)/2 + (h3-h2)*t/2; %in [h2,h3] % V1, V2 of layers 1,2,3 hs=(1-x1/(2*L))^nx; V21=((z1-h1)/(h0-h1))^nz*hs; V11=1-V21; V22=0; V12=1-V22; V23=((z3-h2)/(h3-h2))^nz*hs; V13=1-V23; % switch Model case 1 %Voigt model % EL1,EL2,EL3: E of layers 1,2,3 EL1=E1*V11+E2*V21; EL2=E1*V12+E2*V22; EL3=E1*V13+E2*V23; % nuL1,nuL2,nuL3: G layers 1,2,3 nuL1=nu1*V11+nu2*V21; nuL2=nu1*V12+nu2*V22; nuL3=nu1*V13+nu2*V23; % roL1,roL2,roL3: ro layers 1,2,3 roL1=ro1*V11+ro2*V21; roL2=ro1*V12+ro2*V22; roL3=ro1*V13+ro2*V23; case 2 % Mori-Tanaka scheme [EL1,nuL1]=Enu_MoriT(V11,VL); [EL2,nuL2]=Enu_MoriT(V12,VL); [EL3,nuL3]=Enu_MoriT(V13,VL); % roL1,roL2,roL3: layers 1,2,3 roL1=ro1*V11+ro2*V21; roL2=ro1*V12+ro2*V22; roL3=ro1*V13+ro2*V23;
9. 136 le=HH(1);h0=HH(2);h1=HH(3); h2=HH(4);h3=HH(5);b=HH(6); L=HH(9);h=HH(10); E1=VL(1);E2=VL(2);E3=VL(3); ro1=VL(7);ro2=VL(8);ro3=VL(9); [PT2,WT2] = GaussRule(nG); Ke1=zeros(22,22); Ke2=zeros(22,22); Ke3=zeros(22,22); Me1=zeros(22,22); Me2=zeros(22,22); Me3=zeros(22,22); % Determinant of Jacobian matrix det1=0.25*le*(h1-h0); det2=0.25*le*(h2-h1); det3=0.25*le*(h3-h2); for i=1:nG r=PT2(i); x=le*(1+r)/2; % Phep doi can theo phuong x x1=(ne-1)*le+x; %x trong V tinh tu dau trai dam for j=1:nG t=PT2(j); z1=(h1+h0)/2 + (h1-h0)*t/2; %Doi can in [h0,h1] z2=(h2+h1)/2 + (h2-h1)*t/2; %Doi can in [z1,z2] z3=(h3+h2)/2 + (h3-h2)*t/2; %Doi can in [z2,z3] %EL1,EL2,EL3: E tren lop 1,2,3 [EL1,E12,E13]=pro(z1,x1,E1,E2,E3,h0,h1,h2,h3,nz,nx,L); [E21,EL2,E23]=pro(z2,x1,E1,E2,E3,h0,h1,h2,h3,nz,nx,L); [E31,E32,EL3]=pro(z3,x1,E1,E2,E3,h0,h1,h2,h3,nz,nx,L); % below matrix k [hsE1,hsT1]=hesoKeTe_Voigt(x,z1,VL,HH); [hsE2,hsT2]=hesoKeTe_Voigt(x,z2,VL,HH); [hsE3,hsT3]=hesoKeTe_Voigt(x,z3,VL,HH); Ke1=Ke1 + WT2(i)*WT2(j)*EL1*hsE1*det1;
10. 138 % below shape functions for Wb H0=1-3*x^2/le^2+2*x^3/le^3; H1=x-2*x^2/le+x^3/le^2; H2=3*x^2/le^2-2*x^3/le^3; H3=-x^2/le+x^3/le^2; H4=sqrt(5/128)*(1-xi^2)^2; H5=sqrt(7/128)*(1-xi^2)^2*xi; H6=1/6*sqrt(9/128)*(1-xi^2)^2*(-7*xi^2+1); H7=1/2*sqrt(11/128)*(1-xi^2)^2*(3*xi^2-1)*xi; Nwb = [0 H0 H1 zeros(1,6) H4 H5 H6 H7 zeros(1,5) H2 H3 zeros(1,2)]; % 22 d.o.f Nws = [zeros(1,3) H0 H1 zeros(1,8) H4 H5 H6 H7 zeros(1,3) H2 H3]; % 22 d.o.f % xi1=2/le; % Derivative of Nu respect to x N0x=-1/le; N1x=1/le; N2x=1/2*sqrt(3/2)*2*xi1*xi; N3x=1/2*sqrt(5/2)*xi1*(3*xi^2-1); N4x=1/8*sqrt(7/2)*xi1*(20*xi^3-12*xi); N5x=1/8*sqrt(9/2)*xi1*(35*xi^4-30*xi^2+3); Nux=[N0x zeros(1,4) N2x N3x N4x N5x zeros(1,8) N1x zeros(1,4)]; % 22 d.o.f % Dao ham bac nhat ham dang Nwb theo x H0x=-6*x/le^2+6*x^2/le^3; H1x=1-4*x/le+3*x^2/le^2; H2x=6*x/le^2-6*x^2/le^3; H3x=-2*x/le+3*x^2/le^2; H4x=-4*sqrt(5/128)*xi1*xi*(1-xi^2); H5x=sqrt(7/128)*xi1*(5*xi^4-6*xi^2+1); H6x=1/6*sqrt(9/128)*xi1*(-42*xi^5+60*xi^3-18*xi);
11. 140 A2.2. Function tính ma trận k và m tính theo mô hình Maxwell function [k,m]=KeMesandwichbeam_MoriT(HH,VL,nG,nx,nz,ne) % Stiffness and mass matrices using enrichment % Sinusoidal theory with Maxwell m odel % nG - number of Gauss points % HH=[le h0 h1 h2 h3 b A I LT h nel]; % VL=[E1 E2 E3 G1 G2 G3 ro1 ro2 ro3 psi nu1 nu2 nu3]; % ne- element number ne le=HH(1);h0=HH(2);h1=HH(3); h2=HH(4);h3=HH(5);b=HH(6); L=HH(9);h=HH(10); E1=VL(1);E2=VL(2);E3=VL(3); ro1=VL(7);ro2=VL(8);ro3=VL(9); [PT2,WT2] = GaussRule(nG); Ke1=zeros(22,22); Ke2=zeros(22,22); Ke3=zeros(22,22); Me1=zeros(22,22); Me2=zeros(22,22); Me3=zeros(22,22); % Determinant of Jacobian matrix det1=0.25*le*(h1-h0);det2=0.25*le*(h2-h1);det3=0.25*le*(h3-h2); for i=1:nG r=PT2(i); x=le*(1+r)/2; % change integral in x direction x1=(ne-1)*le+x; % x in potential V for j=1:nG t=PT2(j); z1=(h1+h0)/2 + (h1-h0)*t/2; % change integrals limits z2=(h2+h1)/2 + (h2-h1)*t/2; % Change integrals limits z3=(h3+h2)/2 + (h3-h2)*t/2; % Change integrals limits % EL1,EL2,EL3; nuL1,nuL2,nuL3: E and nu of layers 1,2,3 [V11,V21,V31]=V123_layer1_MoriT(z1,x1,h0,h1,h2,h3,nz,nx,L);
12. 142 Matlab function tính các hệ số "hsE" và "hsT" trong function tính k và m theo mô hình Maxwell function [hsE,hsT]=hesoKeTe_MoriT(x,z,nu,HH) % HH=[le h0 h1 h2 h3 b A I L h] le=HH(1); h=HH(10); xi=2*x/le-1; % change variable % below shape functions for u N0=(le-x)/le; N1=x/le; N2=1/2*sqrt(3/2)*(xi^2-1); N3=1/2*sqrt(5/2)*(xi^2-1)*xi; N4=1/8*sqrt(7/2)*(xi^2-1)*(5*xi^2-1); N5=1/8*sqrt(9/2)*(xi^2-1)*(7*xi^2-3)*xi; Nu=[N0 zeros(1,4) N2 N3 N4 N5 zeros(1,8) N1 zeros(1,4)]; % below shape functions for w H0=1-3*x^2/le^2+2*x^3/le^3; H1=x-2*x^2/le+x^3/le^2; H2=3*x^2/le^2-2*x^3/le^3; H3=-x^2/le+x^3/le^2; H4=sqrt(5/128)*(1-xi^2)^2; H5=sqrt(7/128)*(1-xi^2)^2*xi; H6=1/6*sqrt(9/128)*(1-xi^2)^2*(-7*xi^2+1); H7=1/2*sqrt(11/128)*(1-xi^2)^2*(3*xi^2-1)*xi; Nwb = [0 H0 H1 zeros(1,6) H4 H5 H6 H7 zeros(1,5) H2 H3 zeros(1,2)]; % 22 d.o.f Nws = [zeros(1,3) H0 H1 zeros(1,8) H4 H5 H6 H7 zeros(1,3) H2 H3]; % 22 d.o.f % xi1=2/le; % Derivative of Nu respect to x
13. 144 H6xx=1/6*sqrt(9/128)*xi1^2*(-210*xi^4+180*xi^2-18); H7xx=1/2*sqrt(11/128)*xi1^2*(126*xi^5-140*xi^3+30*xi); Nwbxx=[0 H0xx H1xx zeros(1,6) H4xx H5xx H6xx H7xx zeros(1,5) H2xx H3xx zeros(1,2)]; % 22 d.o.f % Second order derivative of Nws respect to x Nwsxx=[zeros(1,3) H0xx H1xx zeros(1,8) H4xx H5xx H6xx H7xx zeros(1,3) H2xx H3xx]; % 22 d.o.f % fz=-z+(h/pi)*sin(pi*z/h); gz=-1+cos(pi*z/h); % Sinusoidal Theory hsE=Nux’*Nux+z^2*Nwbxx’*Nwbxx+fz^2*Nwsxx’*Nwsxx -z*Nux’*Nwbxx-z*Nwbxx’*Nux+fz*Nux’*Nwsxx+fz*Nwsxx’*Nux -z*fz*Nwbxx’*Nwsxx-z*fz*Nwsxx’*Nwbxx +(1+gz)^2/(2*(1+nu))*Nwsx’*Nwsx; hsT=Nu’*Nu+z^2*Nwbx’*Nwbx+fz^2*Nwsx’*Nwsx -z*Nu’*Nwbx-z*Nwbx’*Nu+fz*Nu’*Nwsx+fz*Nwsx’*Nu -z*fz*Nwbx’*Nwsx-z*fz*Nwsx’*Nwbx+Nwb’*Nwb+Nws’*Nws +Nwb’*Nws+Nws’*Nwb;