Differential quadrature element for second strain gradient beam theory

Differential quadrature element for second strain gradient beam theory

Md Ishaquddin Corresponding author: E-mail address: ishaq.isro@gmail.com S.Gopalakrishnan E-mail address: krishnan@iisc.ac.in; Phone: +91-80-22932048
Abstract

In this paper, first we present the variational formulation for a second strain gradient Euler-Bernoulli beam theory for the first time. The governing equation and associated classical and non-classical boundary conditions are obtained. Later, we propose a novel and efficient differential quadrature element based on Lagrange interpolation to solve the eight order partial differential equation associated with the second strain gradient Euler-Bernoulli beam theory. The second strain gradient theory has displacement, slope, curvature and triple displacement derivative as degrees of freedom. A generalize scheme is proposed herein to implement these multi-degrees of freedom in a simplified and efficient way. The proposed element is based on the strong form of governing equation and has displacement as the only degree of freedom in the domain, whereas, at the boundaries it has displacement, slope, curvature and triple derivative of displacement. A novel DQ framework is presented to incorporate the classical and non-classical boundary conditions by modifying the conventional weighting coefficients. The accuracy and efficiency of the proposed element is demonstrated through numerical examples on static, free vibration and stability analysis of second strain gradient elastic beams for different boundary conditions and intrinsic length scale values.

Keywords: Differential quadrature element, second strain gradient elasticity, eighth order partial differential equation, weighting coefficients, non-classical, curvature, Lagrange interpolation.

Department of Aerospace Engineering,Indian Institute of Science Bengaluru 560012, India

1.0 Introduction

The differential quadrature method (DQM) was first introduced by Bellman et al.[1] to solve the linear and non-linear partial differential equations. In this technique, the derivative of the function at a grid point is assumed as a weighted linear sum of the function values at all other gird points in the computational domain, leading to a set of algebraic equations [2]-[3]. The computational efficiency and accuracy of differential quadrature method has been established in the literature in comparison with the other numerical methods. However, to circumvent the shortcomings related to the imposing multi-boundary conditions and its applications to the generic structural systems, many improved and efficient versions were proposed in recent years [4]-[15]. A comprehensive survey can be found in review paper by Bert et.al. [16] on various aspects of the differential quadrature methods developed by different authors. Most of the above cited research focussed on developing efficient models for classical beam and plate theories which are governed by fourth order partial differential equations [17]-[22]. Some research addressed the solution methodologies for sixth and eighth order partial differential equations by employing strong from of governing equation in conjunction with the Hermite interpolations [23]-[25]. Recently, authors have proposed two novel differential quadrature elements for the solution of sixth order partial differential equations encountered in non-classical higher order continuum theories[26]-[27]. These elements are based on strong and weak form of governing equations and employees Lagrange and Hermite interpolations respectively.

The non-classical gradient elasticity theories are well established in the literature for modelling micro-structural behaviour of materials in contrast to the classical theories[28]-[35]. These theories are generalized versions of linear elasticity theories incorporating higher-order terms to account for scale effects. In the non-classical gradient theories, the strain energy depends upon the elastic strain and its gradients[36]-[38]. The first strain gradient theory which is based on strain tensor and its first gradient generates Cauchy’s and double stress tensors. This theory renders sixth order partial differential equation and have been extensively applied to study the static and dynamic behaviour of beams and plates [39]-[43]. Further, various numerical models have been reported in the literature based on this theory to study the behaviour of beams [44]-[45]. The second strain gradient theory, which is an extension of the first gradient theory accounting for second gradient of strain tensor have been used limitedly in the literature for few applications [46]-[49]. The research on second srain gradient beams is missing in the literature as per the authors knowledge. In this paper, the variational formulation for the second strain gradient Euler-Bernoulli beam is presented for the first time and the associated classical and non-classical boundary conditions are discussed. Employing the above theoretical basis, we develop a differential quadrature element for the second strain gradient Euler-Bernoulli beam theory which is governed by eighth order partial differential equation. Here we use strong form of the governing equations with Lagrange interpolations as test functions. The present element is the extension of the earlier work by authors for sixth order partial differential equation [26]. A novel way to enforce the classical and non-classical boundary conditions in the context of differential quadrature framework is presented. The procedure to compute the higher order weighting coefficients for the proposed element are explained in detail. The efficiency of the proposed differential quadrature element is demonstrated through numerical examples on bending, free vibration and stability analysis.

1 Second strain gradient Euler-Bernoulli beam theory

In the present study we consider the simlified second strain gradient micro-elasticity theory with two classical and two non-classical material constants [46, 49]. The two classical material coefficients correspond to Lam constants and the non-classical ones are of dimension length which are introduced to account for non-local effects. The potential energy density function for a second strain gradient theory is represented as:

(1)

The stress-strain relations for 1-D second strain gradient elastic theory are expressed as [46, 49]

(2)

where, , are Lam constants and , are the strain gradient coefficients of dimension length. is the Laplacian operator and I is the unit tensor. , and denotes Cauchy, double and triple stress respectively, and () are the classical strain and its trace which are expressed in terms of displacement vector w as:

(3)

From the above equations the constitutive relations for a second strain gradient Euler-Bernoulli can be expressed as

(4)

Based on the above constitutive relations the strain energy is written as

(5)

The potential energy of the applied load is given by

(6)

The kinetic energy is given as

(7)

where, , and are the Young’s modulus, area, moment of inertia, respectively. and are the transverse load and displacement of the beam. , , and are shear force, bending moment, double and triple moment acting on the beam.

Using the The Hamilton’s principle[50]

(8)

and performing the integration-by-parts. The governing equation of motion for a second strain gradient Euler-Bernoulli beam is obtained as

(9)

and the associated boundary conditions are:

Classical :

(10)

Non-classical :

(11)

The list of classical and non-classical boundary conditions employed in the present study for a second strain gradient Euler-Bernoulli beam are as follows

Simply supported :
classical :   ,    non-classical :   at

Clamped :
classical :   ,    non-classical :   at

Cantilever :
classical :     at ,    at     
non-classical :   at ,    at     

Propped cantilever :
classical :     at , at     
non-classical :     at ,    at     

Free-free :
classical :    at     
non-classical :  at

2 Differential quadrature element for second strain gradient Euler-Bernoulli beam

The th order derivative of the displacement at location for a N-node 1-D beam element is assumed as

(12)

are the Lagrangian interpolation functions defined as[3, 2],

(13)

where


The first order derivative of the above shape functions can be written as

(14)

The higher order conventional weighting coefficients are defined as

(15)

Where, , and are weighting coefficients for second, third, and fourth order derivatives, respectively.

Figure 1: A typical differential quadrature element for second strain gradient Euler-Bernoulli beam with N=5.

A N-noded second strain gradient Euler-Bernoulli beam element is shown in the Figure 1. Each interior node has displacement as the only degree of freedom (dof), and the boundary nodes has four degrees of freedom , , and . These extra boundary degrees of freedom related to higher gradients of displacement are introduced in to the formulation through modifying the conventional weighting coefficients. The new displacement vector now includes the slope, curvature and triple derivative of displacement as additional dofs at the element boundaries as: . The modified weighting coefficient matrices accounting for multi-degrees of freedom at the boundaries are derived as follows:

First order derivative matrix:

(16)

Second order derivative matrix:

(17)
(18)
(19)

Third order derivative matrix:

(20)
(21)
(22)

Fourth order derivative matrix:

(23)

Fifth order derivative matrix:

(24)
(25)
(26)

Sixth order derivative matrix:

(27)

Seventh order derivative matrix:

(28)
(29)
(30)
(31)

Eight order derivative matrix:

(32)

Here, , , , , , , and are first to eight order modified weighting coefficients matrices, respectively. Using the above Equations (16)-(32), the governing differential Equation (9), at inner grid points interms of the differential quadrature framework is written as

(33)

The boundary forces given by Equations (1)-(1), are expressed as

Shear force:

(34)

Bending moment:

(35)

Double moment:

(36)

Triple moment:

(37)

here and correspond to the left support and right support of the beam, respectively.

After applying the respective boundary conditions the system of equations are given as:

(38)

where the subscript and indicates the boundary and domain of the beam element. , and , are the boundary and domain forces and displacements of the beam, respectively. For static analysis the system of equations are written as

(39)

Expressing the system of equations in terms of domain dofs , we obtain

(40)

The solution of the above system of equations gives the unknown displacements at the domain nodes of the beam element. The boundary displacements are computed from Equation (39), and forces are computed from the Equations (34)-(36).

Similarly for free vibration analysis , and the system of equations are reduced to

(41)

and finally for stability analysis the system of equations are given by

(42)

The Equations (41) and (42) represents an Eigen value problem and the solution provides the frequencies and buckling load.

3 Numerical Results and Discussion

The accuracy and convergence characteristic of the proposed differential quadrature beam element is verified through numerical examples on static, free vibration and stability analysis. The results are compared with the analytical solutions obtained in the Appendix-I for two different combinations of length scale parameters, and . Single DQ element is used in the present study. The grid employed in the present analysis is unequal Gauss–Lobatto–Chebyshev points given by

(43)

where is the number of grid points and are the coordinates of the grid.

The classical and non-classical boundary conditions used in this study for different end supports are listed in the Section 1. The non-classical boundary conditions employed for simply supported gradient beam are at , the equations related to curvature and triple displacement derivative are eliminated. For the cantilever beam the non-classical boundary conditions used are at and at . The equation related to curvature and triple displacement derivative at are eliminated and the equation related to double and triple moment at are retained. Similarly, for clamped and propped cantilever beam the non-classical boundary conditions remains the same, at . The numerical data used for the analysis of beams is as follows: Length , Young’s modulus , Poission’s ratio , density and load .

3.1 Static analysis of second strain gradient Euler-Bernoulli beam

In this section, the capability of the element is demonstrated for static analysis of gradient elastic beams subjected to uniformly distributed load. Three support conditions are considered in this analysis, simply supported, clamped and cantilever. The performance of the element is verified by comparing the classical (deflection and slope) and the non-classical (curvature, triple displacement derivative, double moment and triple moment) quantities with the exact values. The results reported here for beams with udl are nondimensional as, deflection : , bending moment: , curvature :, triple derivative of displacement :, double moment : and triple moment : .

N
,
1.1242 0.1397 0.4401 3.6074 0.0063
1.1656 0.1471 0.4540 3.6884 0.0061
1.1631 0.1463 0.4537 3.4360 0.0113
1.1479 0.1432 0.4501 3.6968 0.0599
1.1559 0.1438 0.4537 4.5624 0.1118
1.1676 0.1455 0.4578 4.9474 0.1245
1.1724 0.1462 0.4593 5.0214 0.1245
1.1739 0.1465 0.4597 5.0277 0.1226
1.1742 0.1465 0.4598 5.0263 0.1220
Exact 1.1743 0.1465 0.4598 5.0252 0.1218
0.9463 0.1154 0.3749 6.7135 0.0015
0.9562 0.1164 0.3854 7.9528 0.2522
0.9614 0.1155 0.3934 11.1079 0.7041
0.9883 0.1194 0.4025 11.6260 0.7199
0.9931 0.1201 0.4039 11.6126 0.7056
0.9936 0.1202 0.4040 11.6054 0.7054
0.9936 0.1202 0.4040 11.6045 0.7052
0.9936 0.1202 0.4040 11.6045 0.7052
0.9936 0.1202 0.4040 11.6045 0.7052
Exact 0.9936 0.1202 0.4040 11.6045 0.7052
Table 1: Comparison of deflection, slope, curvature, double and triple moment for a simply supported beam under a udl.
N
0.1133 0.0994 4.2731 0.0137
0.0987 0.0900 5.1124 0.0518
0.0967 0.0865 4.5254 0.0483
0.0854 0.0800 5.1261 0.0278
0.0784 0.0761 6.4657 0.1115
0.0789 0.0764 6.8750 0.1278
0.0802 0.0771 6.8948 0.1248
0.0808 0.0774 6.8743 0.1222
0.0810 0.0776 6.8642 0.1211
Exact 0.0810 0.0776 6.8604 0.1208
0.0390 0.3749 4.6644 0.2576
0.0520 0.0520 4.9381 0.3009
0.0268 0.0320 9.6015 0.3329
0.0295 0.0340 9.6332 0.3415
0.0311 0.0352 9.4460 0.3174
0.0313 0.0354 9.4153 0.3133
0.0313 0.0354 9.4124 0.3129
0.0313 0.0354 9.4123 0.3129
0.0313 0.0354 9.4123 0.3129
Exact 0.0313 0.0354 9.4123 0.3129
Table 2: Comparison of deflection, curvature, double and triple moment for a clamped beam under a udl.
N
,
7.3448 0.3876 0.3636 2.4728 30.9229 0.0668
7.5689 0.4380 0.4491 1.7241 37.1076 0.2550
8.1486 0.5067 0.5299 1.7889 33.0394 0.2136
7.8110 0.4891 0.5399 1.9633 37.2511 0.3858
7.5423 0.4698 0.5330 1.5194 47.0093 0.9621
7.5465 0.4658 0.5279 1.8599 49.9742 1.0436
7.5837 0.4661 0.5259 1.8391 50.1746 1.0103
7.6023 0.4666 0.5253 1.8458 50.0539 0.9870
7.6088 0.4668 0.5251 1.8450 49.9893 0.9782
Exact 7.6106 0.4668 0.5251 1.8453 49.9620 0.9750
4.9134 0.2996 0.3520 1.9194 40.1363 1.1514
6.0607 0.4268 0.5255 1.5420 48.8664 0.6205
5.0586 0.3674 0.5204 1.1617 78.6422 4.0284
5.2501 0.3633 0.5048 1.3175 76.8798 3.5652
5.3320 0.3653 0.5027 1.3299 75.4694 3.3091
5.3453 0.3657 0.5025 1.3331 75.2408 3.2678
5.3460 0.3657 0.5025 1.3333 75.2120 3.2635
5.3454 0.3657 0.5024 1.3332 75.2040 3.2629
5.3596 0.3668 0.5039 1.3352 75.3480 3.2695
Exact 5.3437 0.3658 0.5022 1.3357 75.2160 3.2634
Table 3: Comparison of deflection, slope, curvature, triple displacement derivative, double and triple moment for a cantilever beam under a udl.

In Table 1, convergence of nondimensional deflection, slope, curvature, double and triple moment are given for two different combinations of length scale values , and , . The results are compared with exact solutions obtained in Appendix-I. The deflection and curvature are computed at the center of the beam , the slope, double moment and triple moment at . The convergence is seen faster for both classical and non-classical quantities obtained using the present element. Similar convergence trend is noticed in Tables 2-3 for clamped and cantilever beam. A good agreement with the exact solution is seen with 15 grid points for , values and converged solutions are obtained using 21 grid points. For , converged solution are obtained using 15 grid points.

present Exact present Exact present Exact present Exact
0.0000 0.0000 0.0000 0.1202 0.1507 0.0000 0.0000 0.0000 0.0000
0.0125 0.0377 0.0377 0.1202 0.1502 -0.0021 -0.0021 -0.3187 -0.3191
0.0495 0.1487 0.1487 0.1198 0.1435 -0.0266 -0.0266 -0.9373 -0.9375
0.1091 0.3249 0.3249 0.1162 0.1188 -0.0961 -0.0961 -1.3049 -1.3050
0.1882 0.5447 0.5447 0.1045 0.0687 -0.1997 -0.1997 -1.2507 -1.2506
0.2830 0.7657 0.7657 0.0804 0.0000 -0.3036 -0.3036 -0.9197 -0.9196
0.3887 0.9317 0.9317 0.0440 -0.0687 -0.3775 -0.3775 -0.4758 -0.4758
0.5000 0.9936 0.9936 0.000 -0.1188 -0.4040 -0.4040 0.0000 0.0000
0.6113 0.9317 0.9317 -0.0440 -0.1435 -0.3776 -0.3777 0.4758 0.4758
0.7169 0.7657 0.7657 -0.0804 -0.1502 -0.3036 -0.3036 0.9197 0.9196
0.8117 0.5447 0.5447 -0.1045 -0.1507 -0.1997 -0.1997 1.2507 1.2506
0.8909 0.3249 0.3249 -0.1162 -0.1507 -0.0961 -0.0961 1.3049 1.3049
0.9505 0.1487 0.1487 -0.1198 -0.1507 -0.0266 -0.0266 0.9373 0.9375
0.9875 0.0377 0.0377 -0.1202 -0.1507 -0.0021 -0.0021 0.3187 0.3191
1.0000 0.0000 0.0000 -0.1202 -0.1507 0.0000 0.0000 0.0000 0.0000
Table 4: Comparison of deflection, slope, curvature and triple displacement derivative for a simply supported beam along the length.

In Table 4, comparison is made for classical and non-classical quantities computed along the length of a simply supported beam subjected to udl. The results are obtained using 15 grid points for , values. Excellent match with the exact solutions is exhibited for all the classical and non-classical quantities along the length of the beam.

From the above tabulated results it can be concluded that the solutions obtained using the proposed element with 15 grid points are in excellent agreement with the exact solutions for all the boundary conditions and values considered. Hence, a single element with fewer nodes can be efficiently applied to study the static behaviour of a second strain gradient Euler-Bernoulli beam for any choice of intrinsic length and boundary condition.

3.2 Free vibration analysis of gradient elastic beams

The applicability of the proposed beam element for free vibration analysis of second strain gradient beam will be verified in this section. The first six elastic frequencies obtained for different boundary conditions are compared with the analytical solutions computed in the Appendix-I for , and , . Four different boundary conditions are considered in this analysis, simply supported, clamped, cantilever and free-free. In Table 5, convergence behaviour of the first six frequencies for a simply supported gradient beam are shown. The frequencies obtained using 15 grid point are in close agreement with the analytical solutions for both combinations of values. Similar convergence trend and accuracy is noticed in the Tables 6-8, for clamped, cantilever and free-free beams, respectively.

N
,
10.2230 74.7250 367.1055
10.3637 45.6637 102.3530 525.1573 680.9263
10.4383 47.6502 127.1765 267.7521 357.8939 2311.9832
10.515 48.3026 127.6144 266.3728 474.5031 1364.4916
10.4838 48.1502 128.1376 277.3628 526.9411 776.2329
10.4340 47.8289 127.7450 272.6296 505.0123 920.0973
10.4134 47.6803 127.4444 271.7701 506.5422 854.8488
10.4073 47.6300 127.3290 271.3294 505.6394 862.4963
10.4058 47.6156 127.2946 271.2002 505.4833 860.6482
Analyt. 10.4058 47.6156 127.2946 271.1597 505.4257 860.6195
10.7250 126.6505 138.7740
11.4542 78.7448 160.1790 438.9373 769.3881
11.5201 63.8629 183.8644 643.4299 1473.2662 2918.2181
11.3653 62.2388 197.0683 424.9276 736.9888 3072.7803
11.3372 61.6128 193.7879 489.2711 1172.3459 1448.7316
11.3342 61.5469 193.8910 472.0980 970.9535 2234.6097
11.3340 61.5405 193.8719 473.9725 991.7657 1801.7299
11.3340 61.5401 193.8715 473.8101 989.1708 1860.3001
11.3340 61.5401 193.8714 473.8181 989.3820 1850.6519
Analyt. 11.3340 61.5401 193.8714 473.8175 989.3680 1851.5906
Table 5: Comparison of first six frequencies for a simply supported gradient beam.
N
,
49.9249 49.9249 75.8825
36.1724 149.5939 256.7750 256.7750 417.2657
37.1702 117.8567 231.6243 1054.0463 1120.8910 1120.8910
39.4632 125.4928 262.9499 759.4087 876.5466 2153.0935
41.1352 131.1305 288.4611 511.1651 745.5959 2792.7604
40.9844 127.7012 283.4289 565.7662 1184.2651 1184.2651
40.6729 125.6902 280.7206 535.0132 904.316 1799.6178
40.5314 124.9045 279.3736 532.1407 914.4509 143.1897
40.4857 124.6561 278.9482 530.5742 910.6078 1462.6078
Analyt. 40.4857 124.6561 278.9482 530.1349 910.2262 1455.7444
53.6137 53.6137 88.3409
51.6586 150.3018 377.3499 377.3499 807.8832
69.1149 242.0681 898.0838 898.0838 1055.0828 3200.8126
67.3022 231.4358 482.3860 1251.1029 2401.0403 8404.3042
65.6599 226.28010 575.4737 958.2332 1370.1105 5498.4399
65.4442 221.6120 542.5355 1223.8458 2498.9469 2493.0270
65.4249 221.4972 545.1122 1117.9998 2006.7471 4623.3352
65.4237 221.4775 544.9018 1130.6769 2106.0718 3404.5865
65.4236 221.4765 544.9096 1129.3365 2090.5403 3609.1100
Analyt. 65.4235 221.4764 544.9086 1129.4247 2091.9224 3573.1262
Table 6: Comparison of first six frequencies for a clamped gradient beam.
N
,
4.3401 25.5081 65.3349
4.4726 27.9495 81.6317 272.0331 272.0331
4.3794 28.4598 89.6828 212.6240 212.6240 1286.9187
4.4826 30.0222 91.9593 205.0360 559.2756 559.2756
4.5560 30.4765 94.4327 214.8785 397.4738 627.8038
4.5506 30.2841 94.1460 212.1549 408.0297 729.5127
4.5381 30.1294 93.7373 210.7456 403.3498 697.6757
4.5321 30.0660 93.5528 210.1586 402.3776 698.0781
4.5301 30.0459 93.4918 209.9751 401.9982 696.9199
Analyt. 4.5320 30.0400 93.4723 209.9177 401.8808 696.6921
5.1344 48.7656 48.7656
5.1339 33.8646 91.9452 318.4791 602.5529
5.6158 41.1355 149.6005 325.1466 667.0974 864.7130
5.4907 38.9018 132.2349 321.3672 855.4734 855.4734
5.4463 38.2281 130.4734 331.3741 693.3906 1115.4421
5.4394 38.1358 130.0263 326.6949 701.9729 1393.5781
5.4391 38.1273 129.9964 326.8027 698.8957 1330.1310
5.4393 38.1268 129.9941 326.7796 699.1914 1341.5299
5.4323 38.1255 129.9935 326.7793 699.1701 1340.1697
Analyt. 5.4324 38.0950 129.9835 326.7750 699.1683 1340.2756
Table 7: Comparison of first six frequencies for a cantilever gradient beam.
N
,
18.0353
22.5762 59.5169 104.9781
23.1959 71.3456 177.7259 240.5436 271.3764
23.3713 71.5657 159.0272 342.6494 584.0780 633.2793
23.4183 71.9297 161.3527 304.6324 507.0495 1243.2544
23.4335 71.9950 161.3431 309.6856 543.5879 835.1189
23.4382 72.0158 161.3771 309.2322 538.0707 885.8146
23.4393 72.0217 161.3850 309.2702 538.5042 875.6494
23.4398 72.0231 161.3869 309.2707 538.4716 876.7202
Analyt. 23.4398 72.0235 161.3873 309.2713 538.4729 876.6273
18.3813
23.7026 64.6455 123.6199
24.2394 82.0112 233.2016 315.7059 384.1724
24.3203 81.4475 201.4857 497.4937 1035.1812
24.3256 81.6035 203.8481 433.2935 810.0301 2392.5067
24.3258 81.5935 203.6459 440.1209 866.9113 1482.3796
24.3259 81.5927 203.6495 439.3965 858.9223 1570.1713
24.3256 81.5927 203.6493 439.4347 859.5209 1554.4556
24.3239 81.5928 203.6493 439.4332 859.4868 1556.0376
Analyt. 24.3230 81.5930 203.6489 439.4334 859.4885 1555.9236
Table 8: Comparison of first six elastic frequencies for a free-free gradient beam.

Excellent fit with the analytical solutions is noticed in the fundamental frequencies obtained using the proposed element with fewer number of grid points. This consistency is maintained for all the boundary conditions and length scale parameters. Hence, based on the above findings it can be stated that the present element can be efficiently applied for free vibration analysis of second strain gradient Euler-Bernoulli beam for any choice of boundary conditions and values.

3.3 Stability analysis of gradient elastic beams

In the earlier sections the efficiency of the proposed beam element was verified for static and free vibration analysis of gradient elastic beams. Here, we validate the applicability of the element for stability analysis of second strain gradient Euler-Bernoulli beam under different support conditions. The DQ results are compared with the analytical values obtained in Appendix-I for different support conditions. The convergence of critical buckling load for a simply supported beam obtained for , and , are shown in Table 9. It can be noticed that the convergence of buckling load is rapid and approaches to analytical values with 15 grid points for all the values. Similar convergence behaviour is noticed in Table 10, for a clamped, cantilever and propped cantilever beam. Hence, these observations validate the effectiveness of the proposed beam element for buckling analysis of second strain gradient elastic prismatic beams.

N
10.6471 11.7216
11.0387 13.2749
11.0386 13.4545
11.2035 13.0837
11.1359 13.0163
11.0302 13.0091
10.9867 13.0085
10.9737 13.0085
10.9705 13.0085
Analytical 10.9704 13.0084
Table 9: Comparison of normalized buckling load for a simply supported gradient beam.
N Clamped Cantilever Propped cantilever
93.8695 157.5518 3.0490 3.8694 38.1265 84.9298
78.4093 172.4970 3.3739 3.7910 29.0109 66.7547
87.8053 221.4184 3.1775 4.0903 31.3016 63.4955
94.4424 240.1186 3.1755 4.0818 33.1909 56.1652
98.8326 230.7690 3.2643 4.0608 33.3466 54.2149
98.4481 230.0158 3.2759 4.0571 32.7474 53.9577
97.6336 229.4688 3.2707 4.0567 32.4233 53.9347
97.2638 229.9457 3.2674 4.0571 32.3046 53.9334
97.1446 229.9458 3.2663 4.0566 32.2688 53.9334
Analytical 97.1445 229.9456 3.2661 4.0565 32.2686 53.9331
Table 10: Comparison of normalized buckling load for a clamped, cantilever and propped-cantilever gradient beams.

4 Conclusion

Variational formulation for a second strain gradient Euler-Bernoulli beam theory was presented for the first time, and the governing equation and associated classical and non-classical boundary conditions were obtained. A novel differential quadrature beam element was proposed to solve a eight order partial differential equation which governs the second strain gradient Euler-Bernoulli beam theory. The element was formulated using the strong form of the governing equation in conjunction with the Lagrange interpolation functions. A new way to account for the non-classical boundary conditions associated with the gradient elastic beam was introduced. The efficiency and accuracy of the proposed element was established through application to static, free vibration and stability analysis of gradient elastic beams for different support conditions and length scale parameters.

References

  • [1] Bellman RE, Casti J., Differential quadrature and long-term integration. Journal of Mathematical Analysis and Applications 1971; 34:235–238.
  • [2] C. Shu, Differential Quadrature and Its Application in Engineering,. Springer-Verlag, London, 2000.
  • [3] Xinwei Wang, Differential Quadrature and Differential Quadrature Based Element Methods Theory and Applications,.Elsevier, USA, 2015
  • [4] Bert, C. W., and Malik, M., 1996,“Differential Quadrature Method in Compu-tational Mechanics: A Review,”. ASME Appl. Mech. Rev., 49(1), pp. 1–28.
  • [5] Bert, C. W., Malik, M., 1996,“The differential quadrature method for irregular domains and application to plate vibration.”. International Journal of Mechanical Sciences 1996; 38:589–606.
  • [6] H. Du, M.K. Lim, N.R. Lin, Application of generalized differential quadrature method to structural problems,. Int. J. Num. Meth.Engrg. 37 (1994) 1881–1896.
  • [7] O. Civalek, O.M. Ulker., Harmonic differential quadrature (HDQ) for axisymmetric bending analysis of thin isotropic circular plates,. Struct. Eng. Mech.17 (1) (2004) 1–14.
  • [8] X. Wang, H.Z. Gu, Static analysis of frame structures by the differential quadrature element method,. Int. J. Numer. Methods Eng. 40 (1997) 759–772.
  • [9] Wang Y, Wang X, Zhou Y., Static and free vibration analyses of rectangular plates by the new version of differential quadrature element method, International Journal for Numerical Methods in Engineering 2004; 59:1207–1226.
  • [10] Y. Xing, B. Liu, High-accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain, Int. J. Numer. Methods Eng. 80 (2009) 1718–1742.
  • [11] Karami G, Malekzadeh P., A new differential quadrature methodology for beam analysis and the associated differential quadrature element method. Computer Methods in Applied Mechanics and Engineering 2002; 191:3509–3526.
  • [12] A.G. Striz, W.L. Chen, C.W. Bert, Static analysis of structures by the quadrature element method (QEM),. Int. J. Solids Struct. 31 (1994) 2807–2818.
  • [13] W.L. Chen, A.G. Striz, C.W. Bert, High-accuracy plane stress and plate elements in the quadrature element method, Int. J. Solids Struct. 37 (2000) 627–647.
  • [14] Wu TY, Liu GR, The generalized differential quadrature rule for fourth-order differential equations, International Journal for Numerical Methods in Engineering 2001; 50:1907–1929
  • [15] H.Z. Zhong, Z.G. Yue, Analysis of thin plates by the weak form quadrature element method, Sci. China Phys. Mech. 55 (5) (2012) 861–871.
  • [16] C.W. Bert, M. Malik, Differential quadrature method in computational mechanics:. A review, Appl. Mech. Rev. 49 (1996) 1–28
  • [17] Malik M., Differential quadrature element method in computational mechanics: new developments and applications. Ph.D. Dissertation, University of Oklahoma, 1994.
  • [18] X. Wang, Y. Wang, Free vibration analysis of multiple-stepped beams by the differential quadrature element method, Appl. Math. Comput. 219 (11) (2013) 5802–5810.
  • [19] O. Civalek, Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns. Eng. Struct. 26 (2) (2004) 171–186.
  • [20] H. Du, M.K. Lim, N.R. Lin, Application of generalized differential quadrature to vibration analysis,. J. Sound Vib. 181 (1995) 279–293.
  • [21] Karami G, Malekzadeh P., Application of a new differential quadrature methodology for free vibration analysis of plates. Int. J. Numer. Methods Eng. 2003; 56:847–868.
  • [22] Chunhua Jin, Xinwei Wang, Luyao Ge, Novel weak form quadrature element method with expanded Chebyshev nodes, Applied Mathematics Letters 34 (2014) 51–59.
  • [23] T.Y. Wu, G.R. Liu, Application of the generalized differential quadrature rule to sixth-order differential equations, Comm. Numer. Methods Eng. 16 (2000) 777–784.
  • [24] Y. Wang, Y.B. Zhao, G.W. Wei, A note on the numerical solution of high-order differential equations, J. Comput. Appl. Math. 159 (2003) 387–398.
  • [25] G.R. Liu a , T.Y. Wu b, Differential quadrature solutions of eighth-order boundary-value differential equations, Journal of Computational and Applied Mathematics 145 (2002) 223–235.
  • [26] Md. Ishaquddin, S. Gopalakrishnan, Novel differential quadrature element method for higher order strain gradient elasticity theories, http://arxiv.org/abs/1802.08115.
  • [27] Md. Ishaquddin, S. Gopalakrishnan, Novel weak form quadrature elements for non-classical higher order beam and plate theories, http://arxiv.org/abs/1802.05541.
  • [28] Mindlin, R.D., 1965.1964.Micro-structure in linear elasticity. Arch. Rat. Mech. Anal. 16, 52–78.
  • [29] Fleck, N.A., Hutchinson, J.W., A phenomenological theory for strain gradient effects in plasticity. 1993. J. Mech. Phys. Solids 41 (12), 1825–1857.
  • [30] Mindlin, R., Eshel, N., 1968.On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 4, 109–124.
  • [31] Mindlin, R.D., 1965.1964.Micro-structure in linear elasticity. Arch. Rat. Mech. Anal. 16, 52–78.
  • [32] Koiter, W.T., 1964.Couple-stresses in the theory of elasticity, I & II. Proc. K. Ned.Akad. Wet. (B) 67, 17–44.
  • [33] F. Yang, A.C.M. Chong, D.C.C. Lam,P. Tong, Experiments and theory in strain gradient elasticity Journal of the Mechanics and Physics of Solids 51 (2003) 1477–1508.
  • [34] D.C.C. Lam, F. Yang, A.C.M. Chong, J. Wang, P. Tong, Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures 39 (2002) 2731–2743.
  • [35] J.N. Reddy, , Nonlocal theories for bending, buckling, and vibration of beams, Int. J. Eng. Sci. 45 (2007) 288–307.
  • [36] Harm Askes, Elias C. Aifantis, Gradient elasticity in statics and dynamics: An overview of formulations,length scale identification procedures, finite element implementations and new results Int. J. Solids Struct. 48 (2011) 1962–1990
  • [37] Aifantis, E.C., Update on a class of gradient theories. 2003.Mech. Mater. 35,259e280.
  • [38] Altan, B.S., Aifantis, E.C., On some aspects in the special theory of gradient elasticity. 1997. J. Mech. Behav. Mater. 8 (3), 231e282.
  • [39] Papargyri-Beskou, S., Tsepoura, K.G., Polyzos, D., Beskos, D.E., Bending and stability analysis of gradient elastic beams. 2003. Int. J. Solids Struct. 40, 385e400.
  • [40] S. Papargyri-Beskou, D. Polyzos, D. E. Beskos, Dynamic analysis of gradient elastic flexural beams. Structural Engineering and Mechanics, Vol. 15, No. 6 (2003) 705–716.
  • [41] A.K. Lazopoulos, Dynamic response of thin strain gradient elastic beams, International Journal of Mechanical Sciences 58 (2012) 27–33.
  • [42] K.A. Lazopoulos,A.K. Lazopoulos, Bending and buckling of thin strain gradient elastic beams, European Journal of Mechanics A/Solids 29 (2010) 837e843.
  • [43] Vardoulakis, I., Sulem, J., Bifurcation Analysis in Geomechanics. 1995. Blackie/Chapman and Hall, London.
  • [44] I. P. Pegios · S. Papargyri-Beskou · D. E. Beskos, Finite element static and stability analysis of gradient elastic beam structures, Acta Mech 226, 745–768 (2015), DOI 10.1007/s00707–014–1216–z.
  • [45] Tsinopoulos, S.V., Polyzos, D., Beskos, D.E, Static and dynamic BEM analysis of strain gradient elastic solids and structures, Comput. Model. Eng. Sci. (CMES) 86, 113–144 (2012).
  • [46] Mindlin, R.D, 1965. Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1, 417–438.
  • [47] Chien, H.Wu, 1965.Cohesive elasticity and surface phenomena.Quarterly of applied mathematics, Vol L, Nmber 1, March 1992, pp 73–103 .
  • [48] Castrenze Polizzotto, Gradient elasticity and nonstandard boundary conditions. Int. J. Solids Struct. 2003, 40,7399–7423.
  • [49] Markus Lazar, Gerard A. Maugin , Elias C. Aifantis, Dislocations in second strain gradient elasticity. Int. J. Solids Struct. 2006, 43,1787–1817.
  • [50] J.N. Reddy, Energy Principles and Variational Methods in Applied Mechanics, Second Edition, John Wiley, NY, 2002.
  • [51] S.P. Timoshenko, D.H. Young, Vibration Problem in Engineering, Van Nostrand Co., Inc., Princeton, N.J., 1956.
  • [52] S.P. Timoshenko, J.M. Gere, Theory of Elastic Stability, Second Edition, McGraw-Hill, 1985.

Appendix

Analytical solutions for second strain gradient Euler-Bernoulli beam

In this section we obtain the analytical solutions for bending, free vibration and stability analysis of second strain gradient Euler-Bernoulli beam for different support conditions and length scale parameters.

Bending analysis

Let us consider a beam of length L subjected to a uniformly distributed load q. To obtain the static deflections of the second gradient elastic Euler-Bernoulli beam which is governed by Equation (9), we assume a solution of the form