# Rotation-Free Based Numerical Model for Nonlinear Analysis of Thin Shells

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Proposed Numerical Formulation

#### 2.1. Discretisation

#### 2.2. Calculation of Strain at Gaussian Points

_{i}, all nodes have their current coordinates. For the purposes of simplicity and computational efficiency, in addition to the global coordinate system, a local coordinate system is introduced, which is fixed to node 0 of the observed triangle as it is presented in Figure 2. Coordinates (x, y, z) refer to the global coordinate system, coordinates ($\stackrel{-}{x}$, $\stackrel{-}{y}$, $\stackrel{-}{z}$) refer to the initial local coordinate system, and coordinates ($\stackrel{~}{x}$, $\stackrel{~}{y}$, $\stackrel{~}{z}$) refer to the current local coordinate system.

**F**is then obtained as [25]:

**E**

_{c}in the middle surface can be obtained from the deformation gradient

**F**, as follows [25]:

**I**is the second-order identity tensor.

_{i}is the $\stackrel{~}{z}$ coordinate of the i-th Gaussian point:

#### 2.3. Material Model

**Stress–strain relation in a linear-elastic regime**. Neglecting the stress perpendicular to the shell surface, Cauchy stress tensor

**T**and the Green St. Venant strain tensor

**E**at the Gaussian points are presented as follows [31]:

**T**is decomposed into the deviatoric stress tensor

**T**

_{d}, which causes the change of shape, and the hydrostatic stress tensor

**T**

_{h}, which causes the change of volume as follows [30]:

**I**is the third-order identity tensor. In the proposed numerical model, the hydrostatic and deviatoric stress tensors are presented in an incremental form given by:

**T**

_{h}and Δ

**T**

_{d}represent the increment of the hydrostatic and deviatoric stress tensors between time t and time t−Δt, respectively.

**I**represents the third-order identity tensor and G and K represent the shear modulus and bulk modulus, respectively, which are associated with the Poisson’s ratio ν and Young’s modulus of elasticity E as follows:

_{v}is the change in the volumetric deformation between two time steps presented by:

**E**

_{e}presents the elastic increment of the Green St. Venant strain tensor given as:

**E**

_{p}being the plastic increment of the Green St. Venant strain tensor. Taking into account the assumption that the material is incompressible in the plastic range, the deviatoric elastic increment of the Green St. Venant strain tensor dev(Δ

**E**

_{e}) is presented by:

**I**is the third-order identity tensor. As long as the material is not subject to plastic deformation, the plastic part of the increment of the Green St. Venant strain tensor Δ

**E**

_{p}equals zero and Δ

**E**

_{e}= Δ

**E**.

**Plasticity model**. In the proposed numerical model, yield criterion is defined with respect to equivalent stress σ

_{eff}in the form of [30]:

_{y}represents yield stress, which depends on the equivalent plastic strain ε

_{p,eff}, given by:

**E**

_{p}may be expressed as [30]:

_{p,eff}within the time increment Δt is given by:

**E**within the time increment Δt is fully elastic. Following that, deviatoric

**T**

_{d,tr,t}and hydrostatic

**T**

_{h,t}Cauchy stress tensors at time t are presented as follows:

**T**

_{d,tr,t}has to be corrected pursuant to the flow rule.

_{y,t−Δt}/dε

_{p,eff,t−Δt}being the slope of the hardening law at the current point, presumed to be constant during the time increment Δt. Said method may lead to many instabilities when the nonlinear terms of the constitutive flow law become significant [32]. In the proposed method, the iterative Newton–Raphson and bisection procedure is used to solve the Equation (31), which is an approach similar to the one proposed by Ming and Pantalé [32].

_{zz}must be recalculated in each iteration considering the plastic deformation pursuant to the following equation:

_{$\stackrel{~}{y}\stackrel{~}{y}$,p}and Δε

_{$\stackrel{~}{x}\stackrel{~}{x}$,p}being the increment of plastic strain in $\stackrel{~}{y}$ and $\stackrel{~}{x}$ direction, given as:

_{y}(ε

_{p,eff}) defined by the equation [30]:

#### 2.4. Stress Presentation over Nodal Forces

_{i}represents the weight coefficients, whereas n represents the number of Gaussian integration points. The last term in the Equation (36) represents the contribution of damping where μ

_{a}is the normal damping coefficient, whereas $\stackrel{.}{\epsilon}$

_{ij}(i,j = $\stackrel{~}{x}$,$\stackrel{~}{y}$) is the deformation rate. Critical normal damping μ

_{a,cr}for the highest normal frequency in the system is presented by [25]:

_{$\stackrel{~}{x}$}and t

_{$\stackrel{~}{y}$}are the components of the normal on the edge of the deformed configuration of the triangle. Traction forces acting on the observed triangular edge are distributed in the corresponding nodes in the form of equivalent nodal forces, as displayed in Figure 4b.

_{b}is the damping coefficient due to bending, while κ

_{ij}(i,j = $\stackrel{~}{x}$,$\stackrel{~}{y}$) is the rate of change of curvature. The critical bending damping μ

_{b,cr}for the highest normal frequency in the system is presented by [25]:

_{n}on the respective side of the examined triangular element, as displayed in Figure 5a, is presented by:

_{n}is then transformed into nodal forces presented by:

#### 2.5. Time Integration of Transient Dynamics

_{i}, a

_{i}and f

_{i}are the mass, acceleration, and total force which correspond to the i-th degree of freedom, respectively.

_{i}and x

_{i}are velocity and coordinates of i-th degree of freedom, respectively.

## 3. Numerical Verification

#### 3.1. Uniaxial Tensile Test

^{−8}, 1 × 10

^{−7}, and 1 × 10

^{−6}. All numerical results were compared with the analytical solution. The material properties of the triangle are adopted from Ming and Pantalé [32] and are summarised in Table 1.

^{−7}, the Newton–Raphson method provides the solution with an accuracy of 0.28% and an average of 1.996 iterations per time step, while the bisection method provides the solution with an accuracy of 0.42% and an average of 16.00 iterations per time step. The direct solution method provides the solution with an accuracy of 0.78%. Adopting a 0.5% error as acceptable, it can be concluded that, in terms of computational efficiency, it is recommended the Newton–Raphson iterative procedure be used for the evaluation of the plastic correction parameter Γ with a tolerance error t

_{NR}= 1 × 10

^{−7}.

#### 3.2. Four-Element Bending Test

#### 3.3. Square Plate Subjected to Gravity Load

^{3}. Gravity constant was g = 10.0 m/s

^{2}.

#### 3.4. Cantilever Beam Exposed to End Moment

_{0}= EI/L. The maximum end moment m

_{max}is 2πm

_{0}at which the cantilever will bend into a circle. The analyses are performed with the lengths L = 10 m and L = 12 m. Figure 13b presents the FE mesh used in the performed analyses, which consists of 16 × 4 FEs, while Figure 13c shows the deformed configurations of the cantilever with the length L = 10 m at different stages. The central node in the presented finite element mesh is added because the spatial second order polynomial defined by relations (4) and (5), which approximates the cantilever geometry near the observed triangle, cannot be performed with only two nodes in one direction.

#### 3.5. Slit Annular Plate Exposed to Lifting Force

_{max}= 0.8 N/m at one end of the slit, while the other end of the slit is fully clamped. This issue has been considered in many references [35]. The thickness of the plate is 0.03 m, while the Poisson’s ratio and Young’s modulus equal v = 0.0 and E = 21.0 MPa, respectively.

_{max}.

#### 3.6. Cantilever Beam Exposed to End Shear Force

_{max}= 4 N. In the analysis with nonlinear material, whose material properties are presented in Table 1, the adopted thickness t was 0.1 m, while the maximal end shear force was equal to p

_{max}= 2.0 MN. The FE mesh used in the numerical analyses and displayed in Figure 17b consisted of 10 × 4 finite elements, while Figure 17c shows the deformed FE mesh for the linear material behaviour under the maximum force.

_{tip}) and the vertical (w

_{tip}) displacement of the cantilever tip relative to the vertical force for linear-elastic material behaviour in comparison with the reference solutions, i.e., three-noded triangular shell elements MITC3+ presented by Jeon et al. [37] and quadrilateral flat shell finite elements presented by Tang et al. [38]. Reference solution [35] was obtained with the programme package ABAQUS by using 16 × 1 S4R shell FEs with 6 DOFs per node, which yields 192 DOFs. It can be observed that the results of the presented shell element are in excellent agreement with the other elements. It should be noted that, in the same example, Jeon et al. [37] used 16 × 2 MITC3+ finite elements with 5 DOFs per node and 2 additional rotational degrees of freedom in the centre of the finite element, which yields 224 DOFs, while Tang et al. [38] used 10 × 1 quadrilateral four-noded FEs with 6 DOFs per node, which yields 120 DOFs. In this paper, we used 40 triangular three-noded FEs with 3 DOFs per node, which yields only 90 DOFs.

_{tip}) and the vertical (w

_{tip}) displacement of the cantilever tip relative to the vertical force for nonlinear material compared to reference solutions. The reference solution [35] in this case was obtained from the programme package ABAQUS by using 20 × 1 S4R FEs. It shows a very good agreement of the results with the reference solution.

#### 3.7. Hinged Cylindrical Roof under Monotonic Increasing Central Force

#### 3.8. Free Vibration of Hinged Cylindrical Roof under Constant Central Force

#### 3.9. Buckling of Axially Loaded Rectangular Plate

^{−1}s

^{−1}, 2.5 × 10

^{−3}s

^{−1}, and 2.5 × 10

^{−5}s

^{−1}. All analyses were performed with the nonlinear and linear-elastic material behaviour, with the material characteristic presented in Table 1.

_{0}equal to w

_{0}/L = 1.0 × 10

^{−9}. Equation (46) ensures that the initial imperfection at the edges of the plate is zero while at the rest of the plate it equals the cosine function with the highest value of w

_{0}at the centre of the plate.

_{x,cr}= 74.31 MPa.

^{−1}s

^{−1}and the linear-elastic material behaviour, the buckling axial stress is approximately 18 times higher as compared to the buckling axial stress at the strain rate of 2.5 × 10

^{−5}s

^{−1}. It can also be observed that for higher strain rates, in the specific case of 2.5 × 10

^{−1}s

^{−1}, the buckling can appear after reaching the yielding stress, resulting in significantly lower buckling stress, considering the nonlinear material behaviour as compared to the buckling stress in materials with linear-elastic behaviour. This example further emphasises the need to consider material nonlinearities in the stability analysis of plate or shell structures due to the dynamic loading condition. It derives from the presented results that by decreasing the strain rate, the numerically obtained stress at which the plate loses stability converges according to the analytical solution obtained for the static activity. In this particular case, for the strain rate of 2.5 × 10

^{−5}s

^{−1}, the numerically obtained buckling axial stress is 2.5% higher as compared to the analytical solution; however, more accurate results could be obtained with smaller strain rates when the dynamic effects would be even less pronounced. Figure 24, Figure 25 and Figure 26 show the shape of the plate in different time steps after the buckling for strain rates 2.5 × 10

^{−1}s

^{−1}, 2.5 × 10

^{−3}s

^{−1}, and 2.5 × 10

^{−5}s

^{−1}, respectively. It derives from the presented results that, at lower strain rates, the buckling occurs in the first eigenmode, while at higher strain rates, the buckling occurs at higher eigenmodes.

## 4. Conclusions

- The presented numerical model is appropriate for the dynamic, quasi-static, and stability analysis of thin shell structures.
- In the tested examples, the performance of the proposed formulations is similar to or better than the one with standard triangular or quadrilateral shell elements measured in terms of the number of DOFs.
- Finally, it cannot be concluded that the presented numerical model is better or worse than other numerical models of a similar type or that it can do something that could not be done with other numerical models. However, one of the main advantages of the presented numerical model may be sought in the robustness of its formulation. Considering that the time integration transient dynamics is solved explicitly for each degree of freedom, there is no need for introducing stiffness and mass matrix at the system level or to solve the large-scale system of equations. The determination of stress and strain at the Gaussian integration points is based on the geometry of the control domain determined by the observed triangle and three neighbouring triangles, which implies that no information related to other triangular FEs in the system is required. In view of the above, the presented formulation seems appropriate for the parallel programming to be executed at the level of the control domain.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Forces on the triangular finite element due to membrane carrying mechanism: (

**a**) normal forces; (

**b**) equivalent nodal forces.

**Figure 5.**Forces resulting from bending carrying mechanism: (

**a**) bending moment on the observed edge; (

**b**) equivalent nodal forces.

**Figure 7.**Stress–strain relation in the triangle obtained by: (

**a**) Newton–Raphson; and (

**b**) bisection iterative procedure for evaluation of plastic correction parameter Γ.

**Figure 8.**Triangle plate: (

**a**) geometry and loading condition; and (

**b**) relation between bending moment at the edge of the centre finite element and the corresponding curvature.

**Figure 10.**Structured finite element meshes: (

**a**) mesh S1 with 36 FE; (

**b**) mesh S2 with 144 FE; (

**c**) mesh S3 with 576 FE; (

**d**) mesh S4 with 2304 FE.

**Figure 11.**Unstructured finite element meshes: (

**a**) mesh U1 with 36 FE; (

**b**) mesh U2 with 160 FE; (

**c**) mesh U3 with 608 FE; (

**d**) mesh U4 with 2268 FE.

**Figure 13.**Cantilever exposed to end moment: (

**a**) geometry; (

**b**) discretisation; (

**c**) deformed finite element mesh under different loading stages.

**Figure 14.**Load–displacement curves for the cantilever plate exposed to end moment for the lengths (

**a**) L = 10 m and (

**b**) L = 12 m.

**Figure 16.**(

**a**) Load–deflection curves for the slit annular plate lifted by line force p; (

**b**) the deformed mesh at p = p

_{max}.

**Figure 17.**Cantilever exposed to line force at the free end: (

**a**) geometry; (

**b**) discretisation; (

**c**) deformed finite element mesh for the linear material behaviour under the maximum force.

**Figure 18.**Tip deflection in relation to the end force: (

**a**) linear-elastic material; and (

**b**) nonlinear material.

**Figure 19.**Hinged cylindrical roof exposed to the central pinching force: (

**a**) problem description; and (

**b**) meshes used for the numerical analysis.

**Figure 20.**Vertical deflection in relation to the point force of the cylindrical roof: (

**a**) linear analysis; (

**b**) non-linear analysis.

**Figure 21.**Load–deflection curves of the hinged cylindrical roof for: (

**a**) linear-elastic material; (

**b**) nonlinear material.

**Figure 22.**Rectangular plate exposed to axial loading: (

**a**) geometry and loading condition; (

**b**) finite element mesh.

**Figure 23.**Relation between average axial stress and shortening of the plate for strain rates: (

**a**) 2.5 × 10

^{−1}s

^{−1}; (

**b**) 2.5 × 10

^{−3}s

^{−1}; and (

**c**) 2.5 × 10

^{−5}s

^{−1}.

**Figure 24.**Z coordinates of nodes for strain rate 2.5 × 10

^{−5}s

^{−1}and horizontal shortening of the plate of: (

**a**) 0.75 mm; (

**b**) 1.125 mm; and (

**c**) 1.5 mm.

**Figure 25.**Z coordinates of nodes for strain rate 2.5 × 10

^{−3}s

^{−1}and horizontal shortening of the plate of: (

**a**) 1.5 mm; (

**b**) 3.2 mm; and (

**c**) 5.0 mm.

**Figure 26.**Z coordinates of nodes for strain rate 2.5 × 10

^{−1}s

^{−1}and horizontal shortening of the plate of: (

**a**) 10.0 mm; (

**b**) 14.2 mm; and (

**c**) 28.0 mm.

E, GPa | ν | A, MPa | B, MPa | n |
---|---|---|---|---|

206.9 | 0.29 | 806 | 614 | 0.168 |

Rel. Error (%) | Iterations | |
---|---|---|

t_{bis} = 1 × 10^{−8} | 0.24 | 19.982 |

t_{bis} = 1 × 10^{−7} | 0.42 | 16.000 |

t_{bis} = 1 × 10^{−6} | 1.23 | 13.000 |

t_{NR} = 1 × 10^{−8} | 0.24 | 2.993 |

t_{NR} = 1 × 10^{−7} | 0.28 | 1.996 |

t_{NR} = 1 × 10^{−6} | 0.74 | 1.000 |

Direct | 0.78 | 1.000 |

Moment, MNm | Rel. Error (%) | |
---|---|---|

3 Gaussian points | 6.2967 | 11.29 |

5 Gaussian points | 6.8607 | 3.34 |

7 Gaussian points | 7.0275 | 0.99 |

9 Gaussian points | 7.0981 | - |

Average Length of the FE Sides | NDOF | Deflection d, mm | Relative Error, % | |||
---|---|---|---|---|---|---|

FE Mesh Pattern | FE Mesh Pattern | FE Mesh Pattern | ||||

Structured | Unstructured | Structured | Unstructured | Structured | Unstructured | |

h = L/4 | 75 | 81 | 8.87636 | 8.33342 | 2.19 | 8.85 |

h = L/8 | 255 | 294 | 9.06101 | 8.93905 | 0.10 | 1.47 |

h = L/16 | 969 | 1014 | 9.07437 | 9.03185 | 0.04 | 0.43 |

h = L/32 | 3603 | 3597 | 9.06939 | 9.06250 | 0.01 | 0.09 |

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**MDPI and ACS Style**

Smoljanović, H.; Balić, I.; Munjiza, A.; Hristovski, V.
Rotation-Free Based Numerical Model for Nonlinear Analysis of Thin Shells. *Buildings* **2021**, *11*, 657.
https://doi.org/10.3390/buildings11120657

**AMA Style**

Smoljanović H, Balić I, Munjiza A, Hristovski V.
Rotation-Free Based Numerical Model for Nonlinear Analysis of Thin Shells. *Buildings*. 2021; 11(12):657.
https://doi.org/10.3390/buildings11120657

**Chicago/Turabian Style**

Smoljanović, Hrvoje, Ivan Balić, Ante Munjiza, and Viktor Hristovski.
2021. "Rotation-Free Based Numerical Model for Nonlinear Analysis of Thin Shells" *Buildings* 11, no. 12: 657.
https://doi.org/10.3390/buildings11120657