# Static Analysis of Stiffened Shells Using an Edge-Based Smoothed MITC3 (ES-MITC3) Method

Authors: Quoc Hoa Pham; Phu-Cuong Nguyen
Journal: Mathematical Problems in Engineering
 (2021)

---

## Abstract

A novel approach for solving the stiffened shell structures by using an edge-based smoothed MITC3 finite element method (ES-MITC3) is presented in this paper. The ES-MITC3 method is an efficient finite element method by combining the edge-based smoothed finite element method (ES-FEM) with the original MITC3 triangular element to not only significantly improve the accuracy but also overcome the shear-locking phenomenon in the Reissner–Mindlin shell analysis. In this study, the ES-MITC3 method is applied for shell structures and then reinforced by stiffeners based on the Timoshenko beam theory to achieve more durability and strength structures. The transverse displacements of the shell structures and stiffeners at the contact positions are assumed compatible. Numerical results of the ES-MITC3 element are compared with those of available other numerical results to demonstrate a good convergence and accuracy of the present method.


---

## Body

## 1. Introduction

Many engineering structures based on plate/shell models are often reinforced by stiffeners, named stiffened plates/shells, to achieve more durability and strength [1,2]. Hence, the study of stiffened shell structures has unendingly received heavy interest from many researchers over the world in recent decades.For analytical methods, Mecitoğlu [3] firstly presented a free vibration analysis of stiffened conical thin shells based on Donnell–Mushtari theory. The evaluation of free vibration of cylindrical stiffened shells with rings and stringers by using and comparing Donnell’s, Love’s, Sanders’, and Flugge’s thin shell theories was addressed by Ruotolo [4]. Pan et al. [5] developed the classical thin shell theory [6] to investigate free vibration of cylindrical stiffened shells with arbitrary boundary conditions. Nevertheless, the analytical methods are difficult to analyze for stiffened shells due to the complexity of structural shapes in practice. As a result, numerical methods are preferred for solving the stiffened plates/shells, especially the finite element (FE) methods.In analyzing the stiffened shell structures using FE methods, a favored and popular approach is separating the stiffened shell structures into two parts: a background shell element and stiffener elements while maintaining the monolithic structure state. Palani et al. [7] introduced eight- and nine-node isoparametric plate/shell elements combined with three-node isoparametric beam elements for static and vibration analyses of the stiffened plates/shells. An investigating behavior of stiffened laminated cylindrical shells is proposed by Venkatesh and Rao [8]. In this work, a four-node laminate thin shell element using 48 degrees of freedom (d.o.f) is incorporated with an anisotropic beam element of 16 d.o.f. Liao and Reddy [9] developed a three-dimensional continuum shell element combining with curved beam elements to investigate the geometric nonlinear analysis of laminated stiffened shells. In order to solve the elastic static behavior of isotropic stiffened shells, Prusty and Satsangi [10] combined a three-node beam element for the stiffeners and an eight-node quadrilateral element for the shell structures. Using a shallow shell model, Sinha et al. [11] proposed a stiffened triangular shell element with 36 d.o.f in which the stiffener elements are placed anywhere in the shell elements.As discussed above, almost FE methods proposed for analyzing stiffened shell structures used higher-order elements based on the degenerated shell or shallow thin shell theories. These kinds of elements require a large number of d.o.f, which leads to complicated analysis and high computational cost. As an alternative method, a Reissner–Mindlin plate/shell theory combined flat shell models can provide a simpler derivation and promising method for solving the stiffened shell structures. Samanta and Mukhopadhyay [2,12] incorporated the Allman and DKT plate elements for the geometric nonlinear static and vibration analysis of stiffened shells. Nguyen et al. [13,14] developed the cell-based smoothed discrete shear gap method (CS-DSG3) to investigate for static, free vibration, and buckling of stiffened plates/shells.Recently, in order to improve the accuracy of the original MITC3 triangular element, an efficient method by combining the edge-based smoothed finite element method (ES-FEM) [15–20] and the MITC3 triangular element, so-called ES-MITC3 [21–25], is recently presented. The numerical results are conducted by these studies show that the ES-MITC3 is almost found to achieve better convergence and accuracy than the original MITC3 triangular element, and existing free shear-locking shell elements such as DSG3 [26], MITC3 [27], and MITC4 [28].This paper now extends the application of ES-MITC3 to the static analysis of the stiffened shells by incorporating the shell element ES-MITC3 and a linearly isotropic two-node Timoshenko beam element, respectively. In this work, we prove that the triangular ES-MITC3 element is a strong competitor to other existing shell elements in the literature.


## 2. A Brief on the ES-MITC3 Formulation for the Stiffened Shell

A flat shell structural element with the global coordinateOxyz and local coordinate O^x^y^z^ systems is shown in Figure 1. Let u^,β^x,v^,β^y and w^,β^z are the rotations and displacements at the middle surface of the shell element in the x^, y^, and z^ directions, respectively. The unknown nodal vector of displacement field with six independent variables can be given as follows:(1)u^T=u^,v^,w^,β^x,β^y,β^z.Figure 1
A shell element model in the local coordinateO^x^y^z^ and global system Oxyz.In the well-known Reissner–Mindlin shell formulation, the membrane strainε^m, the bending strain κ^, and the shear strains γ^ in the local coordinate system O^x^y^z^ are defined, respectively, by(2)ε^m=∂u^∂x^∂v^∂y^∂u^∂y^+∂v^∂x^=Lmu^,withLm=∂∂x^000000∂∂y^0000∂∂y^∂∂x^0000,κ^=∂β^x∂x^∂β^y∂y^∂β^x∂y^+∂β^y∂x^=Lbu^,withLb=000∂∂x^000000∂∂y^0000∂∂y^∂∂x^0,γ^=∂w^∂x^+β^x^∂w^∂y^+β^y^=Lsu^,withLs=00∂∂x^10000∂∂y^010.The shell structures are reinforced by stiffeners to achieve more durability and strength with a minimal weight increase. In general, the stiffeners are represented by the beam structures placed parallel to the edges of shell structures. It should be noted that the phenomenon of warping with beams is not considered in our study [14]. Figure 2 shows a stiffener placed askew an angle ϕ to the x^ -axis of the flat shell element, wherein the plane O^r^s^ in the local coordinate O^r^s^z^ of the stiffener and the plane O^x^y^ in the local coordinate O^x^y^z^ of the flat shell element are assumed to be coincident [29]. As presented in [14], the elastic strain field of a stiffener in local coordinate O^x^y^z^ based on the Timoshenko beam theory is given as(3)ε^st=n⋅∇nxny0enxeny000nxny001nxn⋅∇nyn⋅∇000−nynx︸Lstu^stv^stw^stβ^st⋅xβ^st⋅y︸u^st=Lstu^st,where n=nx,ny is the direction cosine vector, e is the eccentricity between the middle surface of the flat shell element and the centroid of the stiffener as plotted in Figure 2, and u^st=u^st,v^st,w^st,β^st⋅x,β^st⋅yT is the displacement fields of the stiffener in the local coordinate O^x^y^z^. It should be noted that the transverse displacements of the shell structures and stiffeners at the contact positions are assumed compatible in this study [14,29]. Then, the displacement fields of the shell structure and stiffeners u^,u^st in the local coordinate O^x^y^z^ are transformed to u,ust in the global coordinate Oxyz by the transformation matrix Λ presented in detail in [21,23].Figure 2
Model of an eccentric stiffener element.By using Hamilton’s principle, the weak form for the static analysis of the Reissner–Mindlin shell reinforced by a stiffener with a problem domainΩ⊂ℝ3 can be established as [14](4)∬ΩδuTLmTDmLm+LbTDbLb+LsTDsLsudΩ+∫lδustTLstDstLstustdl=∬ΩδuTbdΩ,where b is the body force of the shell and Dm, Db, Ds, and Dst are the material matrices given, respectively, as follows:(5)Dm=Eh1−υ21υ0υ10001−υ2,Db=Eh3121−υ21υ0υ10001−υ2,Ds=khG1001,(6)Dst=diagEAst,EIs,kGAst,GJ,in which G,k=5/6,h,E, and υ are the shear modulus, shear correction factor, the thickness of the shell, Young’s modulus, and Poisson’s ratio, respectively. In equation (6), Ast, Is, and J are the cross area, second moment of the cross-sectional area, and the torsional modulus of the stiffener, respectively, as expressed in [14].Next, the finite element (FE) approximation for weak form (4) can be expressed as follows [14]:(7)Kd=F,where K,F, and d are the global stiffness matrix, load vector, and the displacement vector in global coordinate Oxyz, respectively. The global stiffness matrix K in equation (7) is computed based on a combination between the global stiffness matrix of shell structure Ksh and stiffeners Kst as(8)K=Ksh+∑i=1nstTiTKst⋅iTi,where nste is the number of elements of the stiffener and Kst⋅i and Ti are the global stiffness and transformation matrices of the ith stiffener as presented in [14]. In equation (8), the stiffness matrix of the shell structure Ksh is computed based on the following membrane, bending, and shearing components [21,23]:(9)Ksh=Km+Kb+Ks=∬ΩBmTDmBmdΩ+∬ΩBbTDbBbdΩ+∬ΩBsTDsBsdΩ,where Bm,Bb, and Bs are the FE membrane, bending, and shearing strain gradient matrices [21,23].In order to avoid the shear-locking phenomenon in the Reissner–Mindlin shell theory, the MITC3 triangular element is applied for the shear strain field in this study. As a result, the shear strain gradient matrixBs in equation (9) is replaced by the MITC3 shear strain gradient matrix B˘s as presented in [21,23]. The detailed expressions of the MITC3 triangular element can be found in [21,23,24,27].Recently, an efficient finite element method for shell structure analysis based on a combination of the MITC3 triangular shell element and the edge-based smoothed method (ES-FEM), named as ES-MITC3, is proposed by Pham et al. [21]. Then, the ES-MITC3 is successfully applied to solve the geometrically nonlinear analysis of functionally graded shell structures and transient response of composite shells [22,23,30]. In the present paper, the ES-MITC3 method is further extended for the static analysis of stiffened shell structures to enhance the accuracy of the original MITC3 shell elements. By applying the ES-MITC3 method, the stiffness matrix of the shell structure Ksh in equations (8) and (9) is replaced by a smoothed stiffness matrix K˜ES−MITC3 [21]. For more detailed information, readers can find in [21].Finally, to deal with the drill rotation locking issue as discussed in [31], the null values of the stiffness corresponding to the drilling degree of freedom are then replaced by approximate values. This approximate value is taken to be equal to 10−3 times the maximum diagonal value in the element stiffness matrix [32].


## 3. Numerical Results

In this section, four numerical examples of the stiffened shell are considered to show the high accuracy of the proposed element. For comparison, free shear-locking shell elements such as DSG3 [26], MITC3 [27], and MITC4 [28] are used.


### 3.1. A Square Plate Stiffened by a Single Beam

We consider a simply supported square plate stiffened by one stiffener as in Figure3. The plate is subjected to a uniformly distributed load of q=6.89476×10−3 N/mm2. The material parameters of the plate and stiffened beam are given by Young’s modulus E=1.1721×105 N/mm2 and Poisson’s ratio ν=0.3. Four uniform meshes of the plate corresponding to 2×2, 4×4, 6×6, and 8×8 elements are used.Figure 3
A simply supported square plate stiffened by a single stiffener.The convergence of central displacement for both eccentric and concentric plates using the present ES-MITC3 element is shown in Figures4 and 5. In these figures, the results based on commercial software NASTRAN [33] and SAP2000 [13] are used as the reference solutions. It can be seen that the results of the ES-MITC3 element present better convergence than those of the MITC3 and DSG3 elements and are virtually equal with those of the superconvergence quadrilateral MITC4 element.Figure 4
The convergence of central displacements of the eccentric stiffened square plate by different methods.Figure 5
The convergence of central displacements of the concentric stiffened square plate by different methods.


### 3.2. A Cantilever Cylinder Shell Stiffened by Concentric Beams

In this example, the static behavior of a cantilever cylindrical shell stiffened by three concentric stiffeners is investigated. The dimensions, material parameters, and loaded point are shown in Figure6. Four uniform meshes of the plate corresponding to 2 × 2, 4 × 4, 6 × 6, and 8 × 8 elements are also examined. The numerical results of the ES-MITC3 are compared with those of the DSG3, MITC3, and MITC4 elements. In this problem, the results performing by the four-node shell element given by SAP2000 are also used as references, wherein the mesh 12 × 12 is used. Figures 7 and 8 show the convergence of the radial and tangential deflections at the loaded point. These figures indicate that the results of the ES-MITC3 are close to the reference solution and converge much better than those of the DSG3, MITC3, and MITC4 elements.Figure 6
Geometry, material parameters, and loaded point of the cantilever cylinder shell stiffened by concentric stiffeners.Figure 7
The convergence of radial displacements at the loaded point of the concentric stiffened cantilever shell by different methods.Figure 8
The convergence of tangential displacements at the loaded point of the concentric stiffened cantilever shell by different methods.


### 3.3. A Cantilever Cylinder Shell Stiffened by Eccentric and Concentric Beams

We continue to study the stiffened cantilever cylindrical shell subjected to concentrated load. Unlike the previous example when the stiffened cantilever cylindrical used all three concentric stiffeners, in this example, there are two eccentric stiffeners at two the curved edges and only a concentric stiffener at the straight edge of the cylindrical shell, as shown in Figure9. Figures 10 and 11 show the convergence of the radial and tangential displacements at the loaded point given by the proposed ES-MITC3 element and other elements to the reference solution also given by SAP2000. It can be seen that the results of the proposed method present an excellent performance compared to the MITC3, DSG3, and MITC4 elements.Figure 9
Geometry, material parameters, and loaded point of the cantilever cylinder shell stiffened by eccentric and concentric stiffeners.Figure 10
The convergence of radial displacements at the loaded point of the eccentrically stiffened cantilever shell using structured meshes by different methods.Figure 11
The convergence of tangential displacements at the loaded point of the eccentrically stiffened cantilever shell using structured meshes by different methods.


### 3.4. Spherical Shell with Cross Stiffeners

Finally, a simply supported spherical shell stiffened by two cross concentric stiffeners is investigated. The dimensions and the material properties of the spherical stiffened shell are shown in Figure12. Four uniform meshes of shell corresponding to 2 × 2, 4 × 4, 6 × 6, and 8 × 8 triangular and quadrilateral elements are used. In this problem, the vertical displacement at the center load is compared, and a reference solution of w=−0.0434m presented by Prusty and Satsangi [10] with 12 × 12 eight-noded isoparametric quadrilateral elements is used. Figure 13 compares the convergence of the vertical deflections at the central load by different methods. It can be again seen that, with the same d.o.f, the triangular shell element ES-MITC3 outperforms all other elements, even the MITC4 element, in terms of accuracy and convergence speed.Figure 12
The geometry of the stiffened spherical shell panel with simply supported on the boundaries.Figure 13
The convergence of vertical displacement at the center of stiffened spherical shell panel.


## 3.1. A Square Plate Stiffened by a Single Beam

We consider a simply supported square plate stiffened by one stiffener as in Figure3. The plate is subjected to a uniformly distributed load of q=6.89476×10−3 N/mm2. The material parameters of the plate and stiffened beam are given by Young’s modulus E=1.1721×105 N/mm2 and Poisson’s ratio ν=0.3. Four uniform meshes of the plate corresponding to 2×2, 4×4, 6×6, and 8×8 elements are used.Figure 3
A simply supported square plate stiffened by a single stiffener.The convergence of central displacement for both eccentric and concentric plates using the present ES-MITC3 element is shown in Figures4 and 5. In these figures, the results based on commercial software NASTRAN [33] and SAP2000 [13] are used as the reference solutions. It can be seen that the results of the ES-MITC3 element present better convergence than those of the MITC3 and DSG3 elements and are virtually equal with those of the superconvergence quadrilateral MITC4 element.Figure 4
The convergence of central displacements of the eccentric stiffened square plate by different methods.Figure 5
The convergence of central displacements of the concentric stiffened square plate by different methods.


## 3.2. A Cantilever Cylinder Shell Stiffened by Concentric Beams

In this example, the static behavior of a cantilever cylindrical shell stiffened by three concentric stiffeners is investigated. The dimensions, material parameters, and loaded point are shown in Figure6. Four uniform meshes of the plate corresponding to 2 × 2, 4 × 4, 6 × 6, and 8 × 8 elements are also examined. The numerical results of the ES-MITC3 are compared with those of the DSG3, MITC3, and MITC4 elements. In this problem, the results performing by the four-node shell element given by SAP2000 are also used as references, wherein the mesh 12 × 12 is used. Figures 7 and 8 show the convergence of the radial and tangential deflections at the loaded point. These figures indicate that the results of the ES-MITC3 are close to the reference solution and converge much better than those of the DSG3, MITC3, and MITC4 elements.Figure 6
Geometry, material parameters, and loaded point of the cantilever cylinder shell stiffened by concentric stiffeners.Figure 7
The convergence of radial displacements at the loaded point of the concentric stiffened cantilever shell by different methods.Figure 8
The convergence of tangential displacements at the loaded point of the concentric stiffened cantilever shell by different methods.


## 3.3. A Cantilever Cylinder Shell Stiffened by Eccentric and Concentric Beams

We continue to study the stiffened cantilever cylindrical shell subjected to concentrated load. Unlike the previous example when the stiffened cantilever cylindrical used all three concentric stiffeners, in this example, there are two eccentric stiffeners at two the curved edges and only a concentric stiffener at the straight edge of the cylindrical shell, as shown in Figure9. Figures 10 and 11 show the convergence of the radial and tangential displacements at the loaded point given by the proposed ES-MITC3 element and other elements to the reference solution also given by SAP2000. It can be seen that the results of the proposed method present an excellent performance compared to the MITC3, DSG3, and MITC4 elements.Figure 9
Geometry, material parameters, and loaded point of the cantilever cylinder shell stiffened by eccentric and concentric stiffeners.Figure 10
The convergence of radial displacements at the loaded point of the eccentrically stiffened cantilever shell using structured meshes by different methods.Figure 11
The convergence of tangential displacements at the loaded point of the eccentrically stiffened cantilever shell using structured meshes by different methods.


## 3.4. Spherical Shell with Cross Stiffeners

Finally, a simply supported spherical shell stiffened by two cross concentric stiffeners is investigated. The dimensions and the material properties of the spherical stiffened shell are shown in Figure12. Four uniform meshes of shell corresponding to 2 × 2, 4 × 4, 6 × 6, and 8 × 8 triangular and quadrilateral elements are used. In this problem, the vertical displacement at the center load is compared, and a reference solution of w=−0.0434m presented by Prusty and Satsangi [10] with 12 × 12 eight-noded isoparametric quadrilateral elements is used. Figure 13 compares the convergence of the vertical deflections at the central load by different methods. It can be again seen that, with the same d.o.f, the triangular shell element ES-MITC3 outperforms all other elements, even the MITC4 element, in terms of accuracy and convergence speed.Figure 12
The geometry of the stiffened spherical shell panel with simply supported on the boundaries.Figure 13
The convergence of vertical displacement at the center of stiffened spherical shell panel.


## 4. Conclusion

This paper presents an extension of the ES-MITC3 for the static analysis of the stiffened shells by combining the original element MITC3 [27] with a linearly isotropic two-node Timoshenko beam element. The compatibility of displacement field of stiffeners and shell is applied at the contact positions. Through the theoretical formulation and numerical results, we can withdraw some following interesting points:(i)
The present ES-MITC3 only uses three-node triangular elements that are much easily generated automatically, even for complicated geometries.(ii)
Owing to the edge-based gradient smoothing operation, the overstiff behavior in the MITC3 [24] has been resolved; hence, the proposed ES-MITC3 significantly improves the accuracy and convergence of the numerical results.(iii)
The ES-MITC3, which only uses the triangular elements, has a clear advantage compared to the four-node quadrilateral elements of MITC4 using the same set of nodes [28].(iv)
For static analysis, the results of the ES-MITC3 agree well with reference results and those by SAP2000. The ES-MITC3 is much more accurate than the DSG3 [26], MITC3 [27], and a good competitor to the MITC4 [28].



---

*Source: 3996485-2021-09-22.xml*