# Non-Linear Analysis of Structures Utilizing Load-Discretization of Stiffness Matrix Method with Coordinate Update

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Review the Literature of the Studies

#### 2.1. Dynamic Relaxation Method (DRM)

#### 2.2. Non-Linear Approach by Kwan

## 3. Establishing the Proposed Approach

#### 3.1. Formulation of the Technique

#### 3.2. Stiffness Matrix Establishment

## 4. An Example for Illustrating the Technique

## 5. Results and Discussion

#### 5.1. Structure 1

#### 5.2. Structure 2

#### 5.3. Structure 3

#### 5.4. Structure 4

^{2}, shown in Figure 10, is examined to validate the accuracy of the current technique for analyzing geometrically non-linear flexural members. The results show that the technique is roughly in agreement with the previous non-linear technique by Lewis et al. (1984), as shown in Table 5. It can be seen that the linear stiffness method has zero horizontal displacements, while the current approach and non-linear is clearly giving some results that are concise with the reality.

## 6. Conclusions

- ➢
- The results of the proposed technique are in good agreement with the non-linear techniques’ ones, with a slight discrepancy, while the dissimilarity between of the nodal displacements of SM and the quoted non-linear methods is 228%;
- ➢
- It can be concluded that the proposed technique is accurate and applicable for geometrically non-linear structures;
- ➢
- The accuracy of the technique is enhanced by increasing the number of iterations; however, the number of iterations could be assumed to be 15 to get reasonable results.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Geometry and loading of Structure 1, the y-coordinate of joint 2 is 10 mm below the other joints.

**Figure 7.**Geometry and loading of a three-dimensional Structure, z-coordinate of joint 2 is 10 mm below the other joints.

**Figure 9.**Geometry and loading of a three-dimensional Structure, z-coordinate of joint 2 is 10 mm below the other joints.

**Table 1.**Step-by-step loading and corresponding displacements of Structure 1 in Figure 1.

Iterations | y-Coordinates of J2 mm | Load N | Vertical Displacement of J2 mm |
---|---|---|---|

1 | −10 | 533.33 | −2.4318 |

2 | −12.4318 | 533.33 | −1.5741 |

3 | −14.0059 | 533.33 | −1.2406 |

4 | −15.2465 | 533.33 | −1.0472 |

5 | −16.2936 | 533.33 | −0.9171 |

6 | −17.2107 | 533.33 | −0.8222 |

7 | −18.0329 | 533.33 | −0.7491 |

8 | −18.7820 | 533.33 | −0.6906 |

9 | −19.4726 | 533.33 | −0.6427 |

10 | −20.1153 | 533.33 | −0.6024 |

11 | −20.7176 | 533.33 | −0.5679 |

12 | −21.2856 | 533.33 | −0.5381 |

13 | −21.8237 | 533.33 | −0.5120 |

14 | −22.3357 | 533.33 | −0.4889 |

15 | −22.8246 | 533.33 | −0.4682 |

Total | 8000 | −13.2929 |

**Table 2.**The displacement of y-coordinate of joint 2 of Structure 1 using the current technique and the quoted methods.

Methods | Joint | SM (mm) | Current Study (mm) | Kwan (1998) (mm) | Lewis et al., (1984) (mm) |
---|---|---|---|---|---|

Downward displacement (mm) | 2 | −36.4770 | −12.9887 | −11.1120 | −11.1168 |

**Table 3.**The displacement of y-coordinate of joint 2 of Structure 2 using the current technique and the quoted methods.

Displacement of Joint (2) | SM (mm) | Current Study (mm) | Kwan (1998) (mm) | Lewis et al., (1984) (mm) |
---|---|---|---|---|

x | −0.03 | −0.03 | −0.03 | −0.03 |

Y | −0.03 | −0.03 | −0.03 | −0.03 |

Z | −36.47 | −12.87 | −11.11 | −11.11 |

**Table 4.**Vertical displacements in mm of Joints 5 and 6 of Structure 3, using the proposed approach and the quoted methods.

Joint | Direction | Methods | |||
---|---|---|---|---|---|

SM (mm) | Curren Study (mm) | Kwan (1998) (mm) | Lewis et al., (1984) (mm) | ||

5 | x | 3.75 | 1.989 | 1.99 | 1.988 |

y | −27.857 | −28.144 | −27.61 | −28.221 | |

6 | x | −6 | −7.601 | −7.54 | −7.812 |

y | −25.607 | −25.487 | −24.95 | −25.510 |

Methods | ||||
---|---|---|---|---|

Joint | Direction | SM (mm) | Current Study (mm) | Lewis et al., (1984) (mm) |

2 | _{X} | 0 | −6.05 | −6.01 |

_{Y} | −135 | −134.64 | −134.05 |

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**MDPI and ACS Style**

Saeed, N.; Manguri, A.; Szczepanski, M.; Jankowski, R.
Non-Linear Analysis of Structures Utilizing Load-Discretization of Stiffness Matrix Method with Coordinate Update. *Appl. Sci.* **2022**, *12*, 2394.
https://doi.org/10.3390/app12052394

**AMA Style**

Saeed N, Manguri A, Szczepanski M, Jankowski R.
Non-Linear Analysis of Structures Utilizing Load-Discretization of Stiffness Matrix Method with Coordinate Update. *Applied Sciences*. 2022; 12(5):2394.
https://doi.org/10.3390/app12052394

**Chicago/Turabian Style**

Saeed, Najmadeen, Ahmed Manguri, Marcin Szczepanski, and Robert Jankowski.
2022. "Non-Linear Analysis of Structures Utilizing Load-Discretization of Stiffness Matrix Method with Coordinate Update" *Applied Sciences* 12, no. 5: 2394.
https://doi.org/10.3390/app12052394