The combination of a 4-node quadrilateral mixed interpolation of tensorial components element (MITC4) and the cell-based smoothed finite element method (CSFEM) was formulated and implemented in this work for the analysis of free vibration and unidirectional buckling of shell structures. This formulation was applied to numerous numerical examples of non-woven fabrics. As CSFEM schemes do not require coordinate transformation, spurious modes and numerical instabilities are prevented using bilinear quadrilateral element subdivided into two, three and four smoothing cells. An improvement of the original CSFEM formulation was made regarding the calculation of outward unit normal vectors, which allowed to remove the integral operator in the strain smoothing operation. This procedure conducted both to the simplification of the developed formulation and the reduction of computational cost. A wide range of values for the thickness-to-length ratio and edge boundary conditions were analysed. The developed numerical model proved to overcome the shear locking phenomenon with success, revealing both reduced implementation effort and computational cost in comparison to the conventional FEM approach. The cell-based strain smoothing technique used in this work yields accurate results and generally attains higher convergence rate in energy at low computational cost.
Thin shells are of paramount importance in many engineering applications, namely in civil, mechanical, textile, aeronautical and maritime areas. Shell structures can be found in large-span roofs, liquid reservoirs and arch domes in civil engineering applications. Shells can also be used in pressure vessels, pipes and turbine disks, as examples of mechanical engineering applications. Non-woven fabrics are examples found in many textile products that can be identified in ceiling covering panels, roof linings for automotive interiors and geotextile mats [
Mathematically, a shell structure can be analysed as a plate structure (or sheet) with curved middle surface upon different boundary conditions. Plates can typically endure transverse loads resulting in bending. Shells, on the other hand, withstand loads in any direction.
The formulations currently employed to describe the mechanical behaviour of shells use: (A) curved shells elements, which are established on the basis of the general shell theory [
The MITC allows employing a full quadrature (or integration) in the components of the stiffness matrix, thus eliminating detected inconsistencies of the used interpolation functions for displacements and strains without considering the assumption of additional degrees of freedom. This is the reason why the MITC method has recently been receiving much attention, in particular for the implementation of shell elements and locking-free plates [
The cell-based smoothed finite element method (CSFEM), a typical S-FEM model proposed by Nguyen-Xuan et al. [
In the SFEM approach, the strain values in an element are modified by smoothing the compatible strain fields over smoothing domains, which originates important softening results. This technique is now known as the cell-based FEM, or simply CSFEM [
In this work a combination of the 4-node quadrilateral MITC element (MITC4) and cell-based FE method (CSFEM) was formulated and implemented (in Python) to analyse the mechanical response of a rectangular non-woven fabric structure under free vibration and unidirectional buckling. The original formulation of CSFEM was modified in the sense that the unit outward normal vectors are substituted by outward normal vectors, which allowed to remove the integral operator in the strain smoothing method. A wide range of values for the thickness-to-length ratio was analysed considering different edge boundary conditions and various domain discretization in standard numerical examples to demonstrate the robustness of the developed numerical formulation. The achieved results were compared with analytical and standard FEM approaches, with significant gains in implementation and computational costs.
Consider the linear elastic domain
Assume that the displacements induced in the domain
with
which is known as the compatibility equation.
being
The local equations of equilibrium for a static problem is formulated through,
While the boundary conditions in terms of stress dictate
with
The purpose of the so-called boundary-value problem is to identify the displacement value
which contemplates the work performed by the body forces
For all displacements that satisfy the boundary conditions, i.e., those that are adequate to the boundary-value problem (
which are used in the following to get a generalized solution to the differential equation through the principle of minimum total potential energy. Thus, for the solution
with
The principle of minimum potential energy formulated in
exists, then it refers to the first variation of
then
According to
which means that for fixed values of
which, as mentioned above, represents a variation of
Considering
which configures the variational equation of the structural problem. The first term of
In this equation,
with
This equation has a unique solution,
Consider
with
The matrices of membrane strain
and the displacement components
while the membrane strain is defined by
Considering
being
and, based on the of the displacement assumption (
The pre-buckling in-plane stress is expressed as,
being,
The general displacement solution
with
in which,
The discrete curvature field
with,
and
The nodal unknowns can be expressed by considering the discretized system of equations of the Mindlin/Reissner theory for shells. Therefore, using the FEM procedure for a static analysis, yields
with the stiffness at the element domain
and the corresponding load components defined,
considering
with
To analyse the buckling effect, the discretized governing equation is
with
where for the present case,
Mindlin–Reissner type plate elements are affected by an intrinsic limitation known as locking, i.e., the presence of spurious stresses in plates of small thickness, which can occur in the last term of
Shear strain components in the natural coordinate system
with
considering that
The modification of the compatible strain fields
For a sake of simplicity,
in which
being
with
It should be noted that the strains are constant within the domain of each cell
It should be noted that the integration points in
where,
The geometrical element stiffness matrix can be defined as follows,
Two examples have been analysed regarding the application of the CS-FEM and MITC4 formulations. The performed analyses comprise the problematic of free vibration and buckling of a square non-woven fabric sheet (
Mesh | CS-FEM | MITC | FEM | Analytical [ |
|||
---|---|---|---|---|---|---|---|
0.179090 | 0.179453 | 0.179634 | 0.179815 | 0.180056 | 0.180047 | 0.1754 | |
0.17701 | 0.177168 | 0.177247 | 0.177325 | 0.17743 | 0.177425 | ||
0.176296 | 0.176384 | 0.176428 | 0.176472 | 0.17653 | 0.176527 | ||
0.175968 | 0.176024 | 0.176052 | 0.176080 | 0.176117 | 0.176115 | ||
0.175790 | 0.175829 | 0.175848 | 0.175868 | 0.175893 | 0.175891 |
Mesh | CS-FEM | MITC | FEM | Analytical [ |
|||
---|---|---|---|---|---|---|---|
0.097077 | 0.097179 | 0.097230 | 0.097282 | 0.097350 | 0.097350 | 0.0963 | |
0.096635 | 0.096680 | 0.096702 | 0.096725 | 0.096755 | 0.096755 | ||
0.096480 | 0.096506 | 0.096518 | 0.096531 | 0.096548 | 0.096548 | ||
0.096409 | 0.096425 | 0.096433 | 0.096441 | 0.096452 | 0.096452 | ||
0.096370 | 0.096382 | 0.096387 | 0.096393 | 0.096400 | 0.096400 |
Mesh | CS-FEM | MITC | FEM | Analytical [ |
|||
---|---|---|---|---|---|---|---|
1.621835 | 1.624356 | 1.625613 | 1.626868 | 1.628537 | 1.625867 | 1.5940 | |
1.604610 | 1.605709 | 1.606258 | 1.606806 | 1.607536 | 1.606331 | ||
1.598654 | 1.599268 | 1.599575 | 1.599882 | 1.600290 | 1.599605 | ||
1.595908 | 1.596300 | 1.596496 | 1.596692 | 1.596952 | 1.596512 | ||
1.594419 | 1.594691 | 1.594827 | 1.594962 | 1.595143 | 1.594836 |
Mesh | CS-FEM | MITC | FEM | Analytical [ |
|||
---|---|---|---|---|---|---|---|
0.937637 | 0.938577 | 0.939046 | 0.939516 | 0.940141 | 0.939935 | 0.9300 | |
0.933542 | 0.933954 | 0.934161 | 0.934367 | 0.934641 | 0.934551 | ||
0.932112 | 0.932343 | 0.932458 | 0.932574 | 0.932727 | 0.932677 | ||
0.931450 | 0.931598 | 0.931671 | 0.931745 | 0.931843 | 0.931812 | ||
0.931091 | 0.931193 | 0.931245 | 0.931296 | 0.931364 | 0.931342 |
Mode | |
CS-FEM | MITC | FEM | Mindlin [ |
|||
---|---|---|---|---|---|---|---|---|
1 | 1–1 | 0.931091 | 0.931193 | 0.931245 | 0.931296 | 0.931364 | 0.931342 | 0.930 |
2 | 2–1 | 2.227249 | 2.227508 | 2.227744 | 2.227979 | 2.228222 | 2.228032 | 2.219 |
3 | 1–2 | 2.227249 | 2.227720 | 2.227849 | 2.227979 | 2.228222 | 2.228032 | 2.219 |
4 | 2–2 | 3.416492 | 3.417814 | 3.418475 | 3.419136 | 3.420016 | 3.418949 | 3.406 |
5 | 3–1 | 4.185884 | 4.186313 | 4.186797 | 4.187281 | 4.187746 | 4.187053 | 4.149 |
6 | 1–3 | 4.185884 | 4.186852 | 4.187067 | 4.187281 | 4.187746 | 4.187053 | 4.149 |
7 | 3–2 | 5.238504 | 5.240597 | 5.242123 | 5.243647 | 5.245359 | 5.242132 | 5.206 |
8 | 2–3 | 5.238504 | 5.241556 | 5.242602 | 5.243647 | 5.245359 | 5.242132 | 5.206 |
9 | 4–1 | 6.009895 | 6.011010 | 6.011933 | 6.012854 | 6.013836 | 6.013836 | 6.520 |
10 | 1–4 | 6.009895 | 6.011743 | 6.012299 | 6.012854 | 6.013836 | 6.013836 | 6.520 |
11 | 3–3 | 6.620909 | 6.621499 | 6.622231 | 6.622964 | 6.623648 | 6.622002 | 6.834 |
12 | 4–2 | 6.620909 | 6.622374 | 6.622669 | 6.622964 | 6.623648 | 6.622002 | 7.446 |
13 | 2–4 | 6.878402 | 6.883519 | 6.886075 | 6.888628 | 6.892027 | 6.883471 | 7.446 |
Mode | |
CS-FEM | MITC | FEM | Mindlin [ |
|||
---|---|---|---|---|---|---|---|---|
1 | 1–1 | 0.096370 | 0.096382 | 0.096387 | 0.096393 | 0.096400 | 0.096400 | 0.0963 |
2 | 2–1 | 0.241523 | 0.241554 | 0.241583 | 0.241612 | 0.241642 | 0.241642 | 0.2406 |
3 | 1–2 | 0.241523 | 0.241580 | 0.241596 | 0.241612 | 0.241642 | 0.241642 | 0.2406 |
4 | 2–2 | 0.386116 | 0.386295 | 0.386385 | 0.386475 | 0.386594 | 0.386593 | 0.3847 |
5 | 3–1 | 0.485757 | 0.485819 | 0.485889 | 0.485959 | 0.486026 | 0.486025 | 0.4807 |
6 | 1–3 | 0.485757 | 0.485897 | 0.485928 | 0.485959 | 0.486026 | 0.486025 | 0.4807 |
7 | 3–2 | 0.629412 | 0.629742 | 0.629982 | 0.630223 | 0.630493 | 0.630487 | 0.6246 |
8 | 2–3 | 0.629412 | 0.629893 | 0.630058 | 0.630223 | 0.630493 | 0.630487 | 0.6246 |
9 | 4–1 | 0.832602 | 0.832706 | 0.832835 | 0.832964 | 0.833084 | 0.833081 | 0.8156 |
10 | 1–4 | 0.832602 | 0.832860 | 0.832912 | 0.832964 | 0.833084 | 0.833081 | 0.8156 |
11 | 3–3 | 0.871134 | 0.872050 | 0.872508 | 0.872966 | 0.873576 | 0.873557 | 0.8640 |
12 | 4–2 | 0.974930 | 0.975444 | 0.975913 | 0.976381 | 0.976864 | 0.976848 | 0.9592 |
13 | 2–4 | 0.974930 | 0.975868 | 0.976125 | 0.976381 | 0.976864 | 0.976848 | 0.9592 |
Mode | |
CS-FEM | MITC | FEM | R-R [ |
[ |
|||
---|---|---|---|---|---|---|---|---|---|
1 | 1–1 | 1.594419 | 1.594691 | 1.594827 | 1.594962 | 1.595143 | 1.594836 | 1.5940 | 1.5582 |
2 | 2–1 | 3.055547 | 3.056102 | 3.056459 | 3.056817 | 3.057240 | 3.056336 | 3.0390 | 3.0182 |
3 | 1–2 | 3.055547 | 3.056263 | 3.056540 | 3.056817 | 3.057240 | 3.056336 | 3.0390 | 3.0182 |
4 | 2–2 | 4.282932 | 4.284875 | 4.285845 | 4.286815 | 4.288107 | 4.284832 | 4.2650 | 4.1711 |
5 | 3–1 | 5.079738 | 5.080694 | 5.081174 | 5.081652 | 5.082289 | 5.080391 | 5.0350 | 5.1218 |
6 | 1–3 | 5.128075 | 5.128999 | 5.129459 | 5.129920 | 5.130534 | 5.128918 | 5.0780 | 5.1594 |
7 | 3–2 | 6.009895 | 6.011010 | 6.011933 | 6.012854 | 6.013836 | 6.013836 | 6.0178 | |
8 | 2–3 | 6.009895 | 6.011743 | 6.012299 | 6.012854 | 6.013836 | 6.013836 | 6.0178 | |
9 | 4–1 | 6.128879 | 6.131764 | 6.133569 | 6.135371 | 6.137529 | 6.130685 | 7.5169 | |
10 | 1–4 | 6.128879 | 6.132492 | 6.133933 | 6.135371 | 6.137529 | 6.130685 | 7.5169 | |
11 | 3–3 | 7.162518 | 7.166630 | 7.168686 | 7.170740 | 7.173477 | 7.173477 | 7.7288 | |
12 | 4–2 | 7.541235 | 7.542099 | 7.542860 | 7.543621 | 7.544414 | 7.541495 | 8.3985 | |
13 | 2–4 | 7.541235 | 7.542758 | 7.543190 | 7.543621 | 7.544414 | 7.541495 | 8.3985 |
Mode | |
CS-FEM | MITC | FEM | R-R [ |
[ |
|||
---|---|---|---|---|---|---|---|---|---|
1 | 1–1 | 0.175790 | 0.175829 | 0.175848 | 0.175868 | 0.175893 | 0.175891 | 0.1754 | 0.1743 |
2 | 2–1 | 0.359804 | 0.359896 | 0.359957 | 0.360017 | 0.360089 | 0.360082 | 0.3576 | 0.3576 |
3 | 1–2 | 0.359804 | 0.359926 | 0.359972 | 0.360017 | 0.360089 | 0.360082 | 0.3576 | 0.3576 |
4 | 2–2 | 0.529658 | 0.530026 | 0.530210 | 0.530394 | 0.530639 | 0.530616 | 0.5274 | 0.5240 |
5 | 3–1 | 0.649644 | 0.649843 | 0.649943 | 0.650043 | 0.650176 | 0.650162 | 0.6402 | 0.6465 |
6 | 1–3 | 0.652801 | 0.652986 | 0.653078 | 0.653171 | 0.653294 | 0.653282 | 0.6432 | 0.6505 |
7 | 3–2 | 0.810772 | 0.811409 | 0.811815 | 0.812220 | 0.812702 | 0.812651 | 0.8015 | |
8 | 2–3 | 0.810772 | 0.811583 | 0.811902 | 0.812220 | 0.812702 | 0.812651 | 0.8015 | |
9 | 4–1 | 1.049627 | 1.049839 | 1.050032 | 1.050225 | 1.050425 | 1.050402 | 1.0426 | |
10 | 1–4 | 1.049627 | 1.050013 | 1.050119 | 1.050225 | 1.050425 | 1.050402 | 1.0426 | |
11 | 3–3 | 1.080217 | 1.081735 | 1.082493 | 1.083251 | 1.084260 | 1.084141 | 1.0628 | |
12 | 4–2 | 1.199590 | 1.200769 | 1.201372 | 1.201969 | 1.202761 | 1.202662 | 1.1823 | |
13 | 2–4 | 1.204826 | 1.205979 | 1.206542 | 1.207110 | 1.207870 | 1.207780 | 1.1823 |
Mode | |
CS-FEM | MITC | FEM | Mindlin | |||
---|---|---|---|---|---|---|---|---|
1 | 1–1 | 1.302448 | 1.302584 | 1.302706 | 1.302828 | 1.302955 | 1.302788 | 1.302 |
2 | 2–1 | 2.401693 | 2.402257 | 2.402536 | 2.402815 | 2.403189 | 2.402591 | 2.398 |
3 | 1–2 | 2.901311 | 2.901601 | 2.901906 | 2.902211 | 2.902511 | 2.901959 | 2.888 |
4 | 2–2 | 3.854944 | 3.856364 | 3.857266 | 3.858167 | 3.859240 | 3.856982 | 3.852 |
5 | 3–1 | 4.266729 | 4.267812 | 4.268188 | 4.268563 | 4.269174 | 4.267844 | 4.237 |
6 | 1–3 | 4.990535 | 4.990964 | 4.991465 | 4.991966 | 4.992443 | 4.991209 | 4.936 |
7 | 3–2 | 5.491102 | 5.494350 | 5.495749 | 5.497146 | 5.499156 | 5.493867 | |
8 | 2–3 | 5.836633 | 5.838743 | 5.840387 | 5.842028 | 5.843823 | 5.838722 | |
9 | 4–1 | 6.009895 | 6.011010 | 6.011933 | 6.012854 | 6.013836 | 6.013836 | |
10 | 1–4 | 6.009895 | 6.011743 | 6.012299 | 6.012854 | 6.013836 | 6.013836 | |
11 | 3–3 | 6.653747 | 6.655322 | 6.655766 | 6.656210 | 6.657029 | 6.654570 | |
12 | 4–2 | 7.162518 | 7.166630 | 7.168686 | 7.170740 | 7.173477 | 7.173477 | |
13 | 2–4 | 7.271811 | 7.277049 | 7.279860 | 7.282666 | 7.286271 | 7.274263 |
Mode | |
CS-FEM | MITC | FEM | Mindlin | |||
---|---|---|---|---|---|---|---|---|
1 | 1–1 | 0.141422 | 0.141439 | 0.141454 | 0.141470 | 0.141486 | 0.141486 | 0.1411 |
2 | 2–1 | 0.267700 | 0.267779 | 0.267821 | 0.267863 | 0.267917 | 0.267913 | 0.2668 |
3 | 1–2 | 0.340137 | 0.340180 | 0.340228 | 0.340276 | 0.340323 | 0.340320 | 0.3377 |
4 | 2–2 | 0.462838 | 0.463060 | 0.463219 | 0.463378 | 0.463558 | 0.463546 | 0.4608 |
5 | 3–1 | 0.502601 | 0.502778 | 0.502846 | 0.502913 | 0.503017 | 0.503011 | 0.4979 |
6 | 1–3 | 0.637848 | 0.637927 | 0.638025 | 0.638123 | 0.638215 | 0.638208 | 0.6279 |
7 | 3–2 | 0.687135 | 0.687708 | 0.687996 | 0.688284 | 0.688667 | 0.688639 | |
8 | 2–3 | 0.761328 | 0.761721 | 0.762071 | 0.762422 | 0.762786 | 0.762758 | |
9 | 4–1 | 0.844625 | 0.844933 | 0.845031 | 0.845128 | 0.845296 | 0.845283 | |
10 | 1–4 | 0.980502 | 0.981555 | 0.982236 | 0.982917 | 0.983720 | 0.983653 | |
11 | 3–3 | 1.019250 | 1.020339 | 1.020764 | 1.021189 | 1.021835 | 1.021780 | |
12 | 4–2 | 1.039509 | 1.039637 | 1.039802 | 1.039968 | 1.040120 | 1.040107 | |
13 | 2–4 | 1.163856 | 1.164455 | 1.165072 | 1.165688 | 1.166298 | 1.166243 |
Mode | |
CS-FEM | MITC | FEM | Mindlin | |||
---|---|---|---|---|---|---|---|---|
1 | 1–1 | 1.083162 | 1.083349 | 1.083424 | 1.083492 | 1.083582 | 1.083199 | 1.089 |
2 | 2–1 | 1.746471 | 1.746999 | 1.747200 | 1.747396 | 1.747694 | 1.746189 | 1.758 |
3 | 1–2 | 2.674374 | 2.674770 | 2.674936 | 2.675088 | 2.675288 | 2.674425 | 2.673 |
4 | 2–2 | 3.211195 | 3.211917 | 3.212335 | 3.212749 | 3.213262 | 3.210839 | 3.216 |
5 | 3–1 | 3.304154 | 3.305615 | 3.306088 | 3.306550 | 3.307325 | 3.304131 | 3.318 |
6 | 1–3 | 3.837385 | 3.839567 | 3.840268 | 3.840947 | 3.841998 | 3.841998 | 4.615 |
7 | 3–2 | 4.576690 | 4.579256 | 4.580417 | 4.581571 | 4.583178 | 4.575977 | |
8 | 2–3 | 4.786956 | 4.787602 | 4.787887 | 4.788151 | 4.788496 | 4.786879 | |
9 | 4–1 | 5.295560 | 5.296426 | 5.297076 | 5.297724 | 5.298441 | 5.295077 | |
10 | 1–4 | 5.348007 | 5.349635 | 5.350219 | 5.350772 | 5.351524 | 5.351524 | |
11 | 3–3 | 5.360165 | 5.362655 | 5.363387 | 5.364104 | 5.365381 | 5.360180 | |
12 | 4–2 | 5.768408 | 5.773567 | 5.775097 | 5.776610 | 5.779196 | 5.779196 | |
13 | 2–4 | 6.439317 | 6.442665 | 6.444737 | 6.446802 | 6.449275 | 6.437471 |
Mode | |
CS-FEM | MITC | FEM | Mindlin | |||
---|---|---|---|---|---|---|---|---|
1 | 1–1 | 0.116925 | 0.116946 | 0.116953 | 0.116959 | 0.116969 | 0.116919 | 0.1171 |
2 | 2–1 | 0.195190 | 0.195269 | 0.195296 | 0.195322 | 0.195365 | 0.195190 | 0.1951 |
3 | 1–2 | 0.310577 | 0.310632 | 0.310649 | 0.310666 | 0.310692 | 0.310580 | 0.3093 |
4 | 2–2 | 0.375546 | 0.375677 | 0.375744 | 0.375810 | 0.375897 | 0.375632 | 0.3740 |
5 | 3–1 | 0.394394 | 0.394645 | 0.394718 | 0.394790 | 0.394918 | 0.394641 | 0.3931 |
6 | 1–3 | 0.570349 | 0.570868 | 0.571073 | 0.571278 | 0.571585 | 0.570997 | 0.5695 |
7 | 3–2 | 0.604879 | 0.604989 | 0.605025 | 0.605059 | 0.605113 | 0.604901 | |
8 | 2–3 | 0.662529 | 0.662714 | 0.662838 | 0.662961 | 0.663104 | 0.662798 | |
9 | 4–1 | 0.691010 | 0.691521 | 0.691657 | 0.691790 | 0.692045 | 0.691682 | |
10 | 1–4 | 0.848069 | 0.848863 | 0.849284 | 0.849704 | 0.850245 | 0.849370 | |
11 | 3–3 | 0.867421 | 0.868582 | 0.868964 | 0.869343 | 0.869978 | 0.869211 | |
12 | 4–2 | 1.003934 | 1.004128 | 1.004195 | 1.004258 | 1.004355 | 1.003997 | |
13 | 2–4 | 1.057172 | 1.057424 | 1.057622 | 1.057820 | 1.058035 | 1.057703 |
Mesh | CS-FEM | MITC | FEM | Analytical [ |
|||
---|---|---|---|---|---|---|---|
4.066699 | 4.075307 | 4.079611 | 4.083914 | 4.066699 | 4.066699 | 4.000 | |
4.029397 | 4.033152 | 4.035030 | 4.036908 | 4.029397 | 4.029396 | ||
4.016478 | 4.018577 | 4.019626 | 4.020676 | 4.016478 | 4.016478 | ||
4.010522 | 4.011861 | 4.012531 | 4.013201 | 4.010522 | 4.010522 | ||
4.007293 | 4.008222 | 4.008686 | 4.009150 | 4.007293 | 4.007293 |
Mesh | CS-FEM | MITC | FEM | Analytical [ |
|||
---|---|---|---|---|---|---|---|
3.993495 | 4.001830 | 4.005997 | 4.010164 | 3.993495 | 3.993031 | 3.944 | |
3.957261 | 3.960899 | 3.962717 | 3.964536 | 3.957261 | 3.957059 | ||
3.944711 | 3.946745 | 3.947761 | 3.948778 | 3.944711 | 3.944599 | ||
3.938926 | 3.940223 | 3.940872 | 3.941520 | 3.938926 | 3.938854 | ||
3.935789 | 3.936688 | 3.937138 | 3.937588 | 3.935789 | 3.935739 |
Mesh | CS-FEM | MITC | FEM | Analytical [ |
|||
---|---|---|---|---|---|---|---|
3.791114 | 3.798718 | 3.802520 | 3.806321 | 3.791114 | 3.789459 | 3.786 | |
3.757742 | 3.761062 | 3.762722 | 3.764382 | 3.757742 | 3.757022 | ||
3.746181 | 3.748037 | 3.748965 | 3.749893 | 3.746181 | 3.745779 | ||
3.740851 | 3.742035 | 3.742628 | 3.743220 | 3.740851 | 3.740595 | ||
3.737961 | 3.738782 | 3.739193 | 3.739603 | 3.737961 | 3.737783 |
Mesh | CS-FEM | MITC | FEM | Analytical [ |
|||
---|---|---|---|---|---|---|---|
3.170980 | 3.176550 | 3.179335 | 3.182118 | 3.170980 | 3.166487 | 3.264 | |
3.145574 | 3.148009 | 3.149227 | 3.150444 | 3.145574 | 3.143616 | ||
3.136768 | 3.138131 | 3.138812 | 3.139493 | 3.136768 | 3.135675 | ||
3.132708 | 3.133577 | 3.134012 | 3.134447 | 3.132708 | 3.132010 | ||
3.130506 | 3.131109 | 3.131411 | 3.131712 | 3.130506 | 3.130022 |
A combination of the 4-node quadrilateral mixed interpolation of tensorial components (MITC) element (MITC4) and the cell-based smoothed finite element method (CSFEM) was formulated and implemented to endure free vibration and unidirectional buckling analyses in a rectangular non-woven fabric. A wide range of values for the thickness-to-length ratio were considered, together with edge boundary conditions that are characteristic of many fabric applications. It has been proved that the developed numerical model is able to overcome the shear locking phenomenon with success, revealing both reduced implementation effort and computational cost in comparison to the conventional FEM approach. The numerical results were compared with the ones obtained from a standard finite element computation and analytical solutions for the sake of validation regarding sub-cell integration schemes.
The first, third and fourth authors acknowledges FCT for the conceded financial support through Project UID/CTM/00264/2019 of 2C2T—Centro de Ciência e Tecnologia Têxtil, hold by National Founds of FCT/MCTES. The second author is grateful to FCT for the attributed financial funding through the reference project UID/EEA/04436/2013, COMPETE 2020 with the code POCI-01-0145-FEDER-006941.