(Luận văn thạc sĩ) nghiên cứu dao động và ổn định dầm rỗng phân lớp chức năng dùng lý thuyết bậc cao hai biến

BỘ GIÁO DỤC VÀ ĐÀO TẠO TRƯỜNG ĐẠI HỌC SƯ PHẠM KỸ THUẬT THÀNH PHỐ HỒ CHÍ MINH LUẬN VĂN THẠC SĨ PHAN TUẤN BÌNH NGHIÊN CỨU DAO ĐỘNG VÀ ỔN ĐỊNH DẦM RỖNG PHÂN LỚP CHỨC NĂNG DÙNG LÝ THUYẾT BẬC CAO HAI BIẾN NGÀNH: KỸ THUẬT XÂY DỰNG - 8580201 SKC007998 Tp Hồ Chí Minh, tháng 3/2023 BỘ GIÁO DỤC VÀ ĐÀO TẠO TRƯỜNG ĐẠI HỌC SƯ PHẠM KỸ THUẬT THÀNH PHỐ HỒ CHÍ MINH LUẬN VĂN THẠC SĨ PHAN TUẤN BÌNH NGHIÊN CỨU DAO ĐỘNG VÀ ỔN ĐỊNH DẦM RỖNG PHÂN LỚP CHỨC NĂNG DÙNG LÝ THUYẾT BẬC CAO HAI BIẾN NGÀNH: KỸ THUẬT XÂY DỰNG – 8580201 Hướng dẫn khoa học: TS NGUYỄN NGỌC DƯƠNG Tp Hồ Chí Minh, tháng 03/2023 LÝ LỊCH KHOA HỌC I LÝ LỊCH SƠ LƯỢC Họ tên: Phan Tuấn Bình Giới tính: Nam Ngày, tháng, năm sinh: 10/11/1994 Nơi sinh: Tiền Giang Quê quán: Xã Tân Tây, Gò Công Đông, Tiền Giang Dân tộc: Kinh Chỗ riêng địa liên lạc: 677, Tổ 1, Ấp 2, Tân Tây, Gị Cơng Đơng, Tiền Giang Điện thoại quan: 028.3925.1005 Điện thoại cá nhân: 0393514013 Fax: Email: binhtp94@gmail.com II QUÁ TRÌNH ĐÀO TẠO Đại học Hệ đào tạo: Chính quy Thời gian đào tạo từ 09/2012 đến 06/2016 Nơi học (trường, thành phố): Trường Đại Học Sư Phạm Kỹ Thuật TP Hồ Chí Minh Ngành học: Cơng Nghệ Kỹ Thuật Cơng Trình Xây Dựng Tên đồ án, luận án môn thi tốt nghiệp: Chung cư cao cấp D’EVELYN TOWER Ngày & nơi bảo vệ đồ án, luận án thi tốt nghiệp: -Ngày bảo vệ: 23/06/2016, nơi bảo vệ: Trường Đại Học Sư Phạm Kỹ Thuật TP HCM Người hướng dẫn: TS Nguyễn Ngọc Dương III Q TRÌNH CƠNG TÁC CHUN MƠN KỂ TỪ KHI TỐT NGHIỆP ĐẠI HỌC Thời gian Nơi công tác Công việc đảm nhiệm 07/201604/2017 Công ty TNHH Tư Vấn Xây Dựng AVDH Thiết kế, thẩm tra dự án nhà cao tầng, nhà thấp tầng 04/2017- Công ty TNHH Tư Vấn Xây Dựng Span Việt Nam Thiết kế, thẩm tra dự án nhà cao tầng, nhà thấp tầng đến i Buckling and Free Vibration Analysis 447 elements based on EBBT The Euler-Bernoulli beam theory disregards shear deformation; so EBBT is only appropriate for beams had a length-to-height ratio large The shear deformation plays an important role with medium and thick beams, therefore, FOBT is proposed Chen et al [3, 4] presented the static bending, elastic buckling, and vibration of shear deformable functionally graded porous beams based on FOBT The authors investigated the non-uniform porosity distribution of FGP beams, applied the Lagrange equation method combined with Ritz trial functions, and the Newmark-β method used to express the relation of the equation of motion in the time Wu et al [5, 6] analysed line elastic and dynamic behaviours of FGP structures through the framework of the finite element method (FEM) based on FOBT Lei et al [7] investigated the dynamic behaviours of single-span and multi-span FGP beams with flexible boundary constraints by considering the displacement of two-dimensional and rotational Through above the studies, the FOBT is used to analyse static, dynamic behaviours of FGP beams, however, a shear correction coefficient is required on the model, and it is very difficult to determine To make good FOBT’s shortcomings, investigators proposed higher-order beam theory (HOBT) Wattanasakulpong et al [8] used the third-order beam theory to analyse the free vibration of FGP beams by the Chebyshev collocation approach The uniform and non-uniform porosity distributions of FGP beams are considered, and the displacement of the fields is due to a translational and a rotational Zghal et al [9] analysed the static bending of FGP beams using a refined mixed finite element beam model based on Reddy’s theory Fazzolari [10] investigated the free vibration and elastic stability behaviour of FG sandwich beams on Winkler-Pasternak elastic foundations by developing the refined exponential, polynomial, and trigonometric HOBT The governing equations were developed from Hamilton’s principle and were calculated by using the Ritz functions After that, the convergence and the precision of beams are analysed by testing 86 quasi-3D beam theories Amir et al [11] analysed size-dependent free vibration of FGP beams by combining higher-order beam theory with the modified couple stress theory The motion equations were developed from Hamilton’s principle and were calculated by Navier’s method By using a shear function in the beam modal, HOBTs don’t need a shear correction coefficient for the beam The investigators show that the precision of HOBT depends on shear functions A large numbers of shear functions are developed, and includes: polynomial [12, 13], trigonometric [14, 15], hyperbolic [16, 17], and exponential [18, 19] Recently, Nguyen et al [20] proposed a new twovariable shear deformation theory for bending, free vibration, and buckling analysis of functionally graded porous beams This paper presents a new shear function for higher-order beam theory to analyse the buckling and vibration of FGP beams Three porosity distributions are investigated Ritz method is applied to solve problems The numerical results are carried out to verify the precision and efficiency of the proposed theory The influences of length-to-height ratio, boundary condition, porosity parameter, and porous distribution rules on beams’ frequency and critical buckling load are investigated 448 T.-B Phan and N.-D Nguyen Theoretical Formulation 2.1 Characterization of FGP Beams A functionally graded porous beam described with unit width b, thickness h, and length L with porosity distribution along the thickness direction h is shown in Fig Related to porosity distribution, considering three typical patterns of FGP beams as shown in Fig Fig 3D-model of a FGP beam Fig The cross-section of three typical patterns of FGP beam Young Modulus E(z) and mass density ρ(z) with three typical porosity distribution patterns of FGP beams as followings: Pattern 1: Uniform porosity distribution (D1) E(z) = π ρ(z) = π − r0 − + × E1 (1a) − r0 − + × ρ1 (1b) Pattern 2: Porosity distribution is more at the neutral axis of beams (D2) πz h πz ρ(z) = − rm cos h E(z) = − r0 cos × E1 (2a) × ρ1 (2b) Pattern 3: Porosity distribution is less at the neutral axis of beams (D3) E(z) = − r0 cos πz π − h × E1 (3a) Buckling and Free Vibration Analysis ρ(z) = − rm cos πz π − h × ρ1 449 (3b) where: E1 is the maximum values√of Young’s modulus; ρ1 is mass density; r0 is the porosity parameter; and rm = − − r0 is the porosity parameter for mass density 2.2 Stress-Strain Relation The elastic behaviour equations of FGP beams are written as followings: σx σxz ∗ Q11 ∗ Q55 = ∗ where: Q11 = εx γxz (4) E(z) E(z) ∗ ; Q = 55 − ν2 2(1 + ν) (5) 2.3 Kinematic Energy Based on a consistent beam theory of Krishna Murty [21], the displacement fields are shown as followings: u(x, z, t) = −z ∂w1 (x, t) + ∂x h sin e 2π 2π z h ∂w2 (x, t) = −zw1,x + f (z)w2,x ∂x w(x, z, t) = w1 (x, t) + w2 (x, t) (6a) (6b) where: the deflection at a point on the neutral axis due to bending component is w1 (x, t) sin 2π z h h and transverse shearing component is w2 (x, t) And f (z) = 2π e is proposed shear function It can be seen that the beam model has shear stress at the top and bottom surfaces of beams to vanish The strain-displacement field equations of beams are formed as: εx = ∂u = −zw1,xx + f (z)w2,xx ∂x (7a) γxz = ∂u ∂w + = (1 + g(z))w2,x ∂z ∂x (7b) where: g(z) = f,z 2.4 Energy Formulation The strain energy U1 of the FGP beam can be described as: U1 = V (σx εx + σxz γxz )dV 450 T.-B Phan and N.-D Nguyen = L B1 (w1,xx )2 − 2B2 w1,xx w2,xx + B3 (w2,xx )2 + B4 (w2,x )2 dx (8) where: h/2 (B1 , B2 , B3 ) = −h/2 h/2 ∗ Q11 (z , zf (z), f (z))dz; B4 = −h/2 ∗ Q55 (1 + g(z))2 dz (9) The work U of external axial compression force F is written as followings: U2 = L (w1,x + w2,x )2 × F0 dx (10) The kinetic energy U of the FGP beams can be determined as followings: U3 = = (˙u2 + w˙ ) × ρ(z)dV V L (I0 (w˙ )2 + I1 w˙ 1,x 2 + 2I0 w˙ w˙ − 2I2 w˙ 1,x w˙ 2,x + I0 (w˙ )2 + I3 w˙ 2,x )dx (11) where: the dot superscript convention represents the respective velocity components; I0 , I1 , I2 , I3 are the inertia coefficients are described as followings: h/2 (I0 , I1 , I2 , I3 ) = −h/2 (1, z , zf (z), f (z)) × ρ(z)dz (12) The total energy U of the FGP beam is described as followings: U =U1 + U2 − U3 U= + L B1 w1,xx − 2B2 w1,xx w2,xx + B3 w2,xx + B4 w2,x + F0 (w1,x + w2,x )2 dx L I0 (w˙ )2 + I1 (w˙ 1,x )2 + 2I0 w˙ w˙ + I0 (w˙ )2 + I3 (w˙ 2,x )2 − 2I2 w1,x w2,x dx (13) Buckling and Free Vibration Analysis 451 2.5 Using the Solution by Ritz Method Apply the Ritz’s approximation functions, the trial functions of the displacement field are shown as followings:   m [w1 (x, t), w2 (x, t)] =  m ξj (x)w1j e iωt , j=1 j=1 ξj (x)w2j eiωt  (14) where: w1j and w2j are unknown Ritz values and need to be found; ξj (x) is Ritz’s approximation function; ω is frequency; i2 = −1 is the imaginary unit In this paper, the typical types of boundary conditions (BCs) are considered as: pinned-pinned (PP), clamped-pinned (C-P), clamped-free (C-F), clamped-clamped (C-C) The approximation functions for different boundary conditions (BCs) are shown as followings in Table Table Approximation functions for each boundary condition of beams BCs ξj (x)/e−jx/L P-P − Lx Lx C-F x L C-P − Lx C-C x − Lx L x=0 x=L w1 = w2 = 0; - w1 = w2 = 0; w1,x = w2,x = w1 = w2 = w1 = w2 = 0; w1,x = w2,x = w1 = w2 = 0; w1,x = w2,x = w1 = w2 = w1 = w2 = w1,x = w2,x = x L Combine Eq (13) with Eq (14) and using the Lagrange equation method: ∂U d ∂U − ∂qj dt ∂ q˙ j =0 (15) Which qj representing the values of (w1j , w2j ), so the buckling and free vibration behaviours of FGP beams are written as followings: K 11 K 12 T K 12 K 22 − ω M 11 M 12 T M 12 M 22 w1 w2 =0 (16) where the mass matrix M, and stiffness matrix K have the components are written as followings: L L Kij11 =B1 ξi,xx ξj,xx dx + F0 ξi,x ξj,x dx; o 452 T.-B Phan and N.-D Nguyen L Kij12 = − B2 L ξi,xx ξj,xx dx + F0 L Kij22 =B3 Mij11 =I0 L ξi,xx ξj,xx dx + B4 =I0 ξi,x ξj,x dx; ξi,x ξj,x dx; ξi,x ξj,x dx; L Mij22 =I0 ξi,x ξj,x dx + F0 L ξi ξj dx − I2 0 L L L Mij12 L ξi ξj dx + I1 ξi,x ξj,x dx; L ξi ξj dx + I3 ξi,x ξj,x dx (17) Numerical Results In this section, the precision of the proposed theory will be assessed through numerical examples The influences of length-to-height ratio, BCs, porosity parameter, and porous distribution rules on beam’s frequency and critical buckling load are investigated Specifications of FGP beam are hypothesized to be E = 2x10 11 N/m2 , ν = 1/3, ρ = 7850 kg/m3 , b = h = 0.1 m Non-dimensional parameters are determined: ρ1 − ν ω = ωL E1 (18) − ν F0 = E1 h (19) F cr 3.1 Convergence Test For purpose of testing convergence, considering FGP beams (D2, L/h = 20, r0 = 0.5) None-dimensional critical load and frequency of beams with various BCs are shown in Table with respect to m It is seen that the results converge at m = for all BCs Therefore, m = is used in the following numerical examples Buckling and Free Vibration Analysis 453 Table Convergence assessment for D2-beams (L/h = 20, r0 = 0.5) BCs m 10 12 Non-dimensional critical buckling load (x10–4 ) P-P 34.415 33.639 33.404 33.400 33.400 33.400 C-C 131.997 130.587 130.338 130.338 130.338 130.338 C-P 74.483 67.859 67.735 67.735 67.735 67.735 C-F 8.881 8.403 8.403 8.403 8.403 8.403 Non-dimensional fundamental frequency P-P 0.1446 0.1427 0.1418 0.1418 0.1418 0.1418 C-C 0.3191 0.3175 0.3171 0.3171 0.3171 0.3171 C-P 0.2284 0.2210 0.2209 0.2209 0.2209 0.2209 C-F 0.0515 0.0508 0.0508 0.0508 0.0508 0.0508 3.2 Buckling Analysis Verification The critical buckling load of D1, and D2-beams (r0 = 0.5) with different BCs and the L/h ratio are investigated, and compared with those of Nguyen et al [20] and Ansys software It can be found that the present results are approximately close to the results of Ansys software, Ref [20] as displayed in Table Table Non-dimensional critical buckling load of FGP beams (r0 = 0.5) Distribution pattern D1 BCs P-P C-C C-F D2 P-P C-C L/h ratio References Present Ansys Nguyen et al [20] 10 0.010627 0.010670 0.010574 50 0.000435 0.000435 0.000435 10 0.039749 0.039524 0.038942 50 0.001735 0.001739 0.001734 10 0.002705 0.002705 0.002702 50 0.000109 0.000109 0.000109 10 0.013034 – 0.012953 50 0.000538 – 0.000538 10 0.047643 – 0.046433 50 0.002144 – 0.002142 (continued) 454 T.-B Phan and N.-D Nguyen Table (continued) Distribution pattern BCs C-F L/h ratio References Present Ansys Nguyen et al [20] 10 0.003340 – 0.003335 50 0.000135 – 0.000135 Parameter study The critical buckling loads of FGP beams are considered to clarify the influences of porosity parameters, distribution patterns, and BCs The results of the three patterns of beams are shown in Table and Fig From Table 4, and Fig 3, it can be observed that the critical buckling loads decrease as the porosity parameters increase for all FGP beams, L/h ratio and BCs The reason for this change is that the stiffness of beams decreases as the porosity parameter increases Figure described the influence of the L/h ratio on the critical buckling load of FGP beams The higher the L/h ratio with different BCs, and porosity distribution patterns, the lower the critical buckling load of FGP beams In addition, the D2-beams have the highest buckling load while the D3-beams have the lowest ones Table Non-dimensional critical buckling load (x10–4 ) of FGP beams with various BCs, porosity parameters and L/h ratio L/h ratio Distribution pattern BCs r0 0.2 D1 D2 D3 20 D1 0.4 0.6 0.8 P-P 522.572 440.592 352.415 252.256 C-F 139.709 117.792 94.217 67.440 C-C 1696.470 1430.332 1144.067 818.922 P-P 552.269 502.386 449.323 388.063 C-F 148.694 136.477 124.100 110.978 C-C 1764.988 1569.189 1353.077 1092.972 P-P 498.690 395.619 290.994 184.242 C-F 132.503 104.348 76.075 47.646 C-C 1644.663 1330.442 1003.087 655.874 P-P 35.562 29.998 23.982 17.167 C-F 8.932 7.530 6.023 4.311 (continued) Buckling and Free Vibration Analysis 455 Table (continued) L/h ratio Distribution pattern BCs r0 0.2 D2 D3 0.4 0.6 0.8 C-C 139.709 117.792 94.218 67.441 P-P 37.891 34.901 31.893 28.834 C-F 9.521 8.776 8.033 7.281 C-C 148.585 136.477 124.100 110.977 P-P 33.669 26.460 19.244 12.017 C-F 8.452 6.639 4.825 3.011 C-C 132.503 104.348 76.075 47.647 Fig Non-dimensional critical buckling load of FGP beams with various BCs and porosity distribution patterns with respect to the influence of porosity parameter (L/h = 20) 3.3 Vibration Analysis Verification The fundamental frequencies of D1, and D2-beams (r0 = 0.5) with different BCs, and L/h ratio are investigated The present results are approximately close to the results of Ansys software, and references as shown in Table Parameter study The fundamental frequencies of FGP beams are considered to clarify the influences of porosity parameters, distribution patterns, and BCs The present results are shown in Table and Fig It can be said that the fundamental frequencies of FGP beams change due to the stiffness of the beam changing as the porosity parameter increases However, the fundamental frequencies decrease for D1, and D3-beams, and increase for D2-beams due to the moment of inertia of D1, D3-beams decrease faster than that of D2-beams 456 T.-B Phan and N.-D Nguyen Fig Non-dimensional critical buckling load of FGP beams with various BCs, and porosity distribution patterns with respect to the influence of L/h ratio (r0 = 0.5) Table Non-dimensional fundamental frequencies of FGP beams (r0 = 0.5) Distribution pattern BCs D1 P-P C-C C-F D2 P-P C-C C-F L/h ratio References Present Ansys Nguyen et al [20] Chen et al [4] Wattanasakulpong et al [1] 10 0.2529 0.2524 0.2523 – 0.2523 50 0.0514 0.0513 0.0514 – – 10 0.5505 0.5456 0.5428 – 0.5349 50 0.1162 0.1163 0.1162 – – 10 0.0909 0.0910 0.0908 – – 50 0.0183 0.0183 0.0183 – – 10 0.2799 0.2799 0.2791 0.2798 0.2791 50 0.0571 0.569 0.0571 0.0571 – 10 0.6016 0.6050 0.5908 0.5944 0.5787 50 0.1292 0.1310 0.1291 0.1291 – 10 0.1009 0.1009 0.1007 0.1008 – 50 0.0204 0.0206 0.0204 0.0204 - Figure described the influence of the L/h ratio on the non-dimensional fundamental frequencies of FGP beams Following that, the higher the L/aseh ratio with different BCs, and porosity distribution patterns, the lower the fundamental frequencies of FGP beams Besides, the C-C beams have the highest, and the C-F beams have the lowest fundamental frequencies when the boundary conditions are considered Buckling and Free Vibration Analysis 457 Table Non-dimensional fundamental frequencies of FGP beams with various BCs, porosity parameters and L/h ratio L/h ratio Distribution pattern D1 D2 D3 20 D1 D2 D3 BCs r0 0.2 0.4 0.6 0.8 P-P 0.518 0.497 0.470 0.432 C-F 0.191 0.183 0.173 0.159 C-C 1.049 1.005 0.950 0.874 P-P 0.533 0.531 0.530 0.536 C-F 0.196 0.196 0.197 0.202 C-C 1.069 1.052 1.032 1.009 P-P 0.507 0.471 0.427 0.370 C-F 0.186 0.172 0.156 0.134 C-C 1.033 0.970 0.891 0.785 P-P 0.137 0.131 0.124 0.114 C-F 0.049 0.047 0.044 0.041 C-C 0.307 0.294 0.278 0.256 P-P 0.141 0.142 0.143 0.148 C-F 0.050 0.051 0.051 0.053 C-C 0.317 0.317 0.319 0.328 P-P 0.133 0.123 0.111 0.096 C-F 0.048 0.044 0.040 0.034 C-C 0.299 0.277 0.250 0.215 Fig The non-dimensional fundamental frequency of FGP beams with various BCs, and porosity distribution patterns with respect to the influence of the porosity parameter (L/h = 20) 458 T.-B Phan and N.-D Nguyen Fig The non-dimensional fundamental frequency of FGP beams with various BCs, and porosity distribution patterns with respect to the influence of the L/h ratio (r0 = 0.5) For more data to compare with numerical results, this paper uses the results of Ansys 2020 R1 The Ansys is used to obtain the finite element results by dividing the beam model into several layers of the same thickness The model Ansys with a total number of 10-layers are used for three typical porosity distribution patterns of FGP beams, the different distribution of material parameters along the thickness direction, as shown in Fig 7a–c The best precision will be achieved with the number of enough mesh elements for different beam lengths The 20 nodes with 186 elements of Solid are used in this paper Fig a 10-layer model for D1-beams b 10-layer model for D2-beams c 10-layer model for D3-beams Buckling and Free Vibration Analysis 459 Conclusions This paper presents a new higher-order beam theory to analyse the vibration and buckling of functionally graded porous beams Three porosity distributions are investigated Ritz method is applied to solve problems The numerical results are carried out to verify the precision and efficiency of the proposed theory The influences of length-to-height ratio, boundary condition, porosity parameter, and porous distribution rules on beams’ frequency and critical buckling load are investigated The results indicated that the present theory is efficient to analyse buckling and free vibration of porous beams References Wattanasakulpong, N., Ungbhakorn, V.: Linear and nonlinear vibration analysis of elastically restrained ends FGM beams with porosities Aerosp Sci Technol 32(1), 111–120 (2014) Eltaher, M.A., Fouda, N., El-midany, T., Sadoun, A.M.: Modified porosity model in analysis of functionally graded 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