# Experimental and Numerical Evaluation of Residual Displacement and Ductility in Ratcheting and Shakedown of an Aluminum Beam

^{*}

## Abstract

**:**

## Featured Application

**The presented method allows applying the limit analysis of structures and reformulates the method in terms of displacements and permanent strain. In this perspective, it results that the method is stated directly considering the strain rather than the stress. Hence when one calculates the limit load factor, i.e., collapse load limit or shakedown load limit, one obtains the Melan residual strain immediately. The main disadvantage of limit analysis of structure is the uncertainty about the dissipated energy. As a consequence, the limit analysis does not occupy, within the structural analysis practice, the position it deserves. The proposed method, which can be extended to the general Finite Element Method, allows us to calculate the dissipation and to evaluate the permanent displacement compatible with it. It is, in our opinion, a useful tool to assess the structural safety in the masonry building, seismic analysis of frames to support push over and step-by-step approaches. The analogous application concerns the mechanical forecast of structural integrity under severe loads when plastic deformation is expected to occur, and one wants to allow the structure to dissipate extra-energy to prevent hazards.**

## Abstract

## 1. Introduction

## 2. Fundamentals of Limit Analysis

_{i}have denoted the i

^{th}component of a vector in the reference frame, two subscripts have referred to the two components of a second-order tensor and so on; moreover, subscripts following the comma indicate partial derivatives with respect to the coordinate directions.

## 3. Lower Bound Limit Analysis

**x**and calculate, at any point

**y**, the resulting stress ${\zeta}_{ij}$. The linearity of the elastic problem allows writing the ${\zeta}_{ij}$ in terms of the linear operator ${Z}_{ijhk}$. In numerical application a discrete approximation of the operator ${Z}_{ijhk}$ can be obtained using the boundary integral equation method [20]

_{c}, is summarized in

**x**, of the structure. In Equation (19), the objective function can be any linear combination of the permanent strain; hence, any displacement can be calculated using (19) even if it is not a variable of the inequalities provided linear relationship holds between displacements and dislocations.

## 4. Numerical Example

_{1}and F

_{2}(see Figure 2), have varied accordingly with the load program whose diagram has been drawn in Figure 3. In Figure 4, a flow-chart explains the numerical procedure used in the paper.

#### 4.1. Numerical Results

**Z**matrix. It is pursued calculating the bending moment on the structure due to applied concentrated rotation at any point of it.

**Z**. Since the formulation in terms of generalized vectors of rotations and bending moments, the operator assumes the form of a square matrix here called the influence matrix. The influence matrix can be calculated directly in the case of one-dimensional structures applying the definition given in Equation (11).

**Z**:

_{ij}of

**Z**is of the bending moment at ${P}_{i}$ due to the rotation $\Delta {\phi}_{j}$ at ${P}_{j}$.

**Z**, obtained using the assigned data, was calculated and is

Z = | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

0 | −4.18 × 10^{5} | −4.18 × 10^{5} | −8.36 × 10^{5} | −8.36 × 10^{5} | −4.18 × 10^{5} | −4.18 × 10^{5} | 0 | |

0 | −4.18 × 10^{5} | −4.18 × 10^{5} | −8.36 × 10^{5} | −8.36 × 10^{5} | −4.18 × 10^{5} | −4.18 × 10^{5} | 0 | |

0 | −8.36 × 10^{5} | −8.36 × 10^{5} | −1.67 × 10^{6} | −1.67 × 10^{6} | −8.36 × 10^{5} | −8.36 × 10^{5} | 0 | |

0 | −8.36 × 10^{5} | −8.36 × 10^{5} | −1.67 × 10^{6} | −1.67 × 10^{6} | −8.36 × 10^{5} | −8.36 × 10^{5} | 0 | |

0 | −4.18 × 10^{5} | −4.18 × 10^{5} | −8.36 × 10^{5} | −8.36 × 10^{5} | −4.18 × 10^{5} | −4.18 × 10^{5} | 0 | |

0 | −4.18 × 10^{5} | −4.18 × 10^{5} | −8.36 × 10^{5} | −8.36 × 10^{5} | −4.18 × 10^{5} | −4.18 × 10^{5} | 0 | |

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

_{3}. The correlation between permanent rotation and displacement is obtained by solving the structure under applied dislocation in P

_{3}:

_{3}, rotation, and dissipated energy reported in Table 7:

#### 4.2. Experimental Results

_{3}have been reported. The measures have been recorded when, after some cycles of loading, the displacement did cease increasing, and the load was removed, ${\widehat{\mathit{v}}}_{{\mathit{P}}_{\mathbf{3}}}^{\mathbf{0}}$. The measure has been repeated when the displacement, ${\widehat{\mathit{v}}}_{{\mathit{P}}_{\mathbf{3}}}^{\mathbf{\infty}}$, reduced, completing its a recovery. Finally, under the loads indicated by the (*), the displacement did not stabilize. In this case, the experiment finished after many cycles, and the residual displacement has been recorded. Even in this case, a new measure has been recorded after time to keep the memory of the recovering effect. Table 8 contains the experimental results, as reported in [15].

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Koiter, W.T. General theorems for elastic-plastic solids. Prog. Solid Mech.
**1960**, 1, 165–221. [Google Scholar] - Lubliner, J. Plasticity Theory; Macmillan: New York, NY, USA, 1990. [Google Scholar]
- Hassan, T.; Kyriakides, S. Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior. Int. J. Plast.
**1994**, 10, 149–184. [Google Scholar] [CrossRef] - Hassan, T.; Kyriakides, S. Ratcheting of cyclically hardening and softening materials: II. Multiaxial behavior. Int. J. Plast.
**1994**, 10, 185–212. [Google Scholar] [CrossRef] - Mandal, N.K. On the low cycle fatigue failure of insulated rail joints (IRJs). Eng. Fail. Anal.
**2014**, 40, 58–74. [Google Scholar] [CrossRef] - Melan, E. Der Spannungenszustand eines Henky-Misesschen Kontinuums bei Veranderlicher Beelastung. Sitzungsberichte der Accademic Wissenschaften
**1938**, 147–173. [Google Scholar] - Casciaro, R.; Garcea, G. An iterative method for shakedown analysis. Comput. Methods Appl. Mech. Eng.
**2002**, 191, 5761–5792. [Google Scholar] [CrossRef] - Garcea, G.; Armentano, G.; Petrolo, S.; Casciaro, R. Finite element shakedown analysis of two-dimensional structures. Int. J. Numer. Methods Eng.
**2005**, 63, 1174–1202. [Google Scholar] [CrossRef] - Yu, H.S.; Salgado, R.; Sloan, S.; Kim, J.M. Limit Analysis versus Limit Equilibrium for Slope Stability. J. Geotech. Geoenviron. Eng.
**1998**, 124, 1–11. [Google Scholar] [CrossRef] - Ponter, A.R.; Engelhardt, M. Shakedown limits for a general yield condition: Implementation and application for a Von Mises yield condition. Eur. J. Mech. A/Solids
**2000**, 3, 19–423. [Google Scholar] - Björkman, G.; Klarbring, A. Shakedown and residual stresses in frictional systems. In Proceedings of the 2nd International Symposium on Contact Mechanics and Wear of Rail/Wheel Systems II, University of Rhode Island, Kingston, RI, USA, 7–10 July 1986; pp. 27–39. [Google Scholar]
- Seshadri, R.; Mangalaramanan, S. Lower bound limit loads using variational concepts: The mα-method. Int. J. Press. Vessel. Pip.
**1997**, 71, 93–106. [Google Scholar] [CrossRef] - Ponter, A.R.S. An Upper Bound on the Small Displacements of Elastic, Perfectly Plastic Structures. J. Appl. Mech.
**1972**, 39, 959–963. [Google Scholar] [CrossRef] - Ponter, A.R.S.; Williams, J.J. Work Bounds and Associated Deformation of Cyclically Loaded Creeping Structures. J. Appl. Mech.
**1973**, 40, 921–927. [Google Scholar] [CrossRef] - Guarracino, F.; Frunzio, G.; Minutolo, V.; Pan, L.G.; Nunziante, L. Residual effects in shakedown: A theoretical approach and experimental results. In Proceedings of the AEPA, Beijing, China, 29 June–2 July 1994; Xu, B., Yang, W., Eds.; pp. 579–586. [Google Scholar]
- Koning, J. Shakedown of Elastic-Plastic Structures; Elsevier: Amsterdam, The Netherlands; Oxford, UK; New York, NY, USA; Tokyo, Japan, 1987. [Google Scholar]
- Ponter, A.; Martin, J. Some extremal properties and energy theorems for inelastic materials and their relationship to the deformation theory of plasticity. J. Mech. Phys. Solids
**1972**, 20, 281–300. [Google Scholar] [CrossRef] - Capurso, M. Some Upper Bound Principles to Plastic Strains in Dynamic Shakedown of Elastoplastic Structures. J. Struct. Mech.
**1979**, 7, 1–20. [Google Scholar] [CrossRef] - Mura, T.; Barnett, D.M. Micromechanics of Defects in Solids. J. Appl. Mech.
**1983**, 50, 477. [Google Scholar] [CrossRef] - Ruocco, E.; Letizia, F.; Minutolo, V. Shakedown and residual displacements by BIEM. In Advances in Boundary Element Techniques V; Leitão, V.M.A., Aliabadi, M.H., Eds.; EC Ltd: Lisbon, Portugal, 2004; pp. 343–348. [Google Scholar]
- Maier, G. Complementary plastic work theorems in piecewise-linear elastoplasticity. Int. J. Solids Struct.
**1969**, 5, 261–270. [Google Scholar] [CrossRef] - Minutolo, V.; Ruocco, E.; Migliore, M.R. On the influence of in-plane displacements on the stability of plate assemblies. J. Strain Anal. Eng. Des.
**2004**, 39, 213–223. [Google Scholar] [CrossRef] - Ruocco, E.; Mallardo, V.; Minutolo, V.; Di Giacinto, D. Analytical solution for buckling of Mindlin plates subjected to arbitrary boundary conditions. Appl. Math. Model.
**2017**, 50, 497–508. [Google Scholar] [CrossRef] - Ilyushin, A.A. On a postulate of plasticity. Prikl. Math. Mekh
**1961**, 18, 503–507. [Google Scholar] - Lubliner, J. A simple theory of plasticity. Int. J. Solids Struct.
**1974**, 10, 313–319. [Google Scholar] [CrossRef] - Owen, D.R. Thermodynamics of materials with elastic range. Arch. Ration. Mech. Anal.
**1968**, 31, 91–112. [Google Scholar] [CrossRef] - Reyes, J.C.; Chopra, A.K. Three-Dimensional Modal Pushover Analysis of Unsymmetric-Plan Buildings Subjected to Two Components of Ground Motion. Seism. Behav. Des. Irregul. Complex Civ. Struct. III
**2012**, 24, 203–217. [Google Scholar]

**Figure 7.**Comparison between experimental and calculated residual displacements (redline: experimental displacements ${\mathit{v}}_{{\mathit{P}}_{\mathbf{3}}}^{\mathit{exp}}$, yellow points: calculated displacements through step-by-step analysis, green points: calculated upper bound displacements).

Cross-Section [mm] | Mechanical Constant | Range of Values |
---|---|---|

Young modulus E | $6250-6400\phantom{\rule{4.pt}{0ex}}\mathrm{MPa}$ | |

Limit bending moment M_{y} | $0.3276-0.3324\phantom{\rule{4.pt}{0ex}}\mathrm{kNm}$ | |

Yield stress ${\sigma}_{Y}$ | $193.16-196.00\phantom{\rule{4.pt}{0ex}}\mathrm{MPa}$ |

Load Condition | Applied Forces [N] |
---|---|

1 | F_{1} = 1000, F_{2} = 0 |

2 | F_{1} = 0, F_{2} = 1000 |

3 | F_{1} = 1000, F_{2} = 1000 |

Point | M^{max} [Nm] | M^{min} [Nm] |
---|---|---|

P_{1} | 0.000 | 0.000 |

P_{2} | 162.500 | 37.500 |

P_{3} | 162.500 | 37.500 |

P_{4} | −75.000 | 150.000 |

P_{5} | −75.000 | 150.000 |

P_{6} | 162.500 | 37.500 |

P_{7} | 162.500 | 37.500 |

P_{8} | 0.000 | 0.000 |

Load Condition | s |
---|---|

1 | 2.493 |

2 | 2.493 |

3 | 2.493 |

Shakedown | 2.099 |

P_{i} | $\mathbf{\Delta}\mathit{\phi}$ |
---|---|

P_{1} | 0 |

P_{2} | 0 |

P_{3} | 0.02093 |

P_{4} | 0 |

P_{5} | 0 |

P_{6} | 0 |

P_{7} | 0 |

P_{8} | 0 |

m | ${\mathit{v}}_{{\mathit{P}}_{3}}$ Upper Bound |
---|---|

1.01 | 4.09 |

1.03 | 4.52 |

**Table 7.**Step by step results: Displacement ${v}_{{P}_{3}}$, rotation $\Delta {\phi}_{3}$ at P

_{3}and dissipated energy ${\mathcal{E}}_{\mathrm{D}}$.

$\mathit{F}\phantom{\rule{4.pt}{0ex}}\left[\mathbf{kN}\right]$ | ${\mathit{v}}_{{\mathit{P}}_{3}}\phantom{\rule{4.pt}{0ex}}\left[\mathit{m}\right]$ | $\mathbf{\Delta}{\mathit{\phi}}_{3}$ | ${\mathcal{E}}_{\mathbf{D}}$ |
---|---|---|---|

2.03 | $6.52\ast {10}^{-4}$ | $5.21\ast {10}^{-3}$ | $1.70\ast {10}^{-3}$ |

2.06 | $2.074\ast {10}^{-3}$ | $0.0166$ | $5.44\ast {10}^{-3}$ |

2.065 | $2.34\ast {10}^{-3}$ | $0.0187$ | $6.14\ast {10}^{-3}$ |

2.07 | $2.578\ast {10}^{-3}$ | $0.0206$ | $6.76\ast {10}^{-3}$ |

2.08 | $3.052\ast {10}^{-3}$ | $0.0244$ | $8.00\ast {10}^{-3}$ |

2.09 | $3.52\ast {10}^{-3}$ | $0.0282$ | $9.24\ast {10}^{-3}$ |

**Table 8.**Experimental residual displacements, before recovery ${\widehat{\mathit{v}}}_{{\mathit{P}}_{\mathbf{3}}}^{\mathbf{0}}$, complete recovery ${\widehat{\mathit{v}}}_{{\mathit{P}}_{\mathbf{3}}}^{\mathbf{\infty}}$; (*) indicates ratcheting load.

$\mathit{F}\phantom{\rule{4.pt}{0ex}}\left[\mathbf{kN}\right]$ | ${\widehat{\mathit{v}}}_{{\mathit{P}}_{3}}^{0}\phantom{\rule{4.pt}{0ex}}\left[\mathbf{mm}\right]$ | ${\widehat{\mathit{v}}}_{{\mathit{P}}_{3}}^{\mathit{\infty}}\phantom{\rule{4.pt}{0ex}}\left[\mathbf{mm}\right]$ |
---|---|---|

2.032 | 4.07 | 3.61 |

2.069 | 4.67 | 4.32 |

2.099 | 5.48 | 5.25 |

2.139 * | 7.06 | 6.72 |

2.180 * | 11.42 | 11.31 |

**Table 9.**Displacement of P

_{3}from experimental measures, ${v}_{{P}_{3}}^{exp}$; calculated residual by step-by-step approach, ${v}_{{P}_{3}}^{sbs}$, and upper bound through optimization programs, ${v}_{{P}_{3}}^{op}$.

m | $\mathit{F}\phantom{\rule{4.pt}{0ex}}\left[\mathbf{kN}\right]$ | ${\mathit{v}}_{{\mathit{P}}_{3}}^{\mathit{e}\mathit{x}\mathit{p}}\phantom{\rule{4.pt}{0ex}}\left[\mathbf{mm}\right]$ | ${\mathit{v}}_{{\mathit{P}}_{3}}^{\mathit{s}\mathit{b}\mathit{s}}\phantom{\rule{4.pt}{0ex}}\left[\mathbf{mm}\right]$ | ${\mathit{v}}_{{\mathit{P}}_{3}}^{\mathit{o}\mathit{p}}\phantom{\rule{4.pt}{0ex}}\left[\mathbf{mm}\right]$ |
---|---|---|---|---|

1.033 | 2.032 | 3.61 | 0.652 | 4.09 |

1.014 | 2.069 | 4.32 | 2.578 | 4.52 |

1.000 | 2.099 | 5.25 | 3.52 | / |

2.139 | 6.72 | 6.52 | / | |

2.180 | 11.31 | / |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Palladino, S.; Esposito, L.; Ferla, P.; Totaro, E.; Zona, R.; Minutolo, V.
Experimental and Numerical Evaluation of Residual Displacement and Ductility in Ratcheting and Shakedown of an Aluminum Beam. *Appl. Sci.* **2020**, *10*, 3610.
https://doi.org/10.3390/app10103610

**AMA Style**

Palladino S, Esposito L, Ferla P, Totaro E, Zona R, Minutolo V.
Experimental and Numerical Evaluation of Residual Displacement and Ductility in Ratcheting and Shakedown of an Aluminum Beam. *Applied Sciences*. 2020; 10(10):3610.
https://doi.org/10.3390/app10103610

**Chicago/Turabian Style**

Palladino, Simone, Luca Esposito, Paolo Ferla, Elena Totaro, Renato Zona, and Vincenzo Minutolo.
2020. "Experimental and Numerical Evaluation of Residual Displacement and Ductility in Ratcheting and Shakedown of an Aluminum Beam" *Applied Sciences* 10, no. 10: 3610.
https://doi.org/10.3390/app10103610