The bending and free vibration of porous functionally graded (PFG) beams resting on elastic foundations are analyzed. The material features of the PFG beam are assumed to vary continuously through the thickness according to the volume fraction of components. The foundation medium is also considered to be linear, homogeneous, and isotropic, and modeled using the Winkler-Pasternak law. The hyperbolic shear deformation theory is applied for the kinematic relations, and the equations of motion are obtained using the Hamilton’s principle. An analytical solution is presented accordingly, assuming that the PFG beam is simply supported. Comparisons with the open literature are implemented to verify the validity of such a formulation. The effects of the elastic foundations, porosity volume percentage and span-to-depth ratio are finally discussed in detail.

Nowadays functionally graded materials (FGMs) are an alternative materials widely used in aerospace, nuclear, automotive civil, biomechanical, mechanical, electronic, chemical, and shipbuilding industries. FGMs have been proposed, developed and successfully used in industrial applications since 1980’s [

Several studies have been performed to analyze the mechanical responses of FG structures. Simsek et al. [

In FGM fabrication, micro voids or porosities can occur within the materials during the process of sintering. This is because of the large difference in solidification temperatures between material constituents [

Structures resting on an elastic foundation are used in a variety of fields, including missile and rocket launchers in the military and aerospace industries, various applications in technology, civil and mechanical engineering, industry. Therefore, it is critical to include the superstructure-foundation-soil interaction in modern structural design and analysis for the applications to be adequately served.

In the years that followed, scientists experimented with different soil continuity factors to make the Winkler foundation model more realistic One of these models is the Winkler-Pasternak type foundation model with two parameters which includes shearing layer and Winkler layer has been the most widely utilized of these models [

According to the preceding literature review, while numerous researchers have been performed for mechanical behaviors of FG structures with or without elastic foundations, the number of studies on the bending and free vibration of FG beams including the effects of two-parameter elastic foundations and porosity simultaneously is still limited. Hence, the current study attempts to address the banding and free vibration of imperfect FG beams resting on two-parameter elastic foundations. The impacts of the two-parameter elastic foundation, porosity volume fraction, type of porosity models, and and span-to-depth ratio, on the bending and fundamental natural frequency of the beams, are investigated in detail.

Consider a functionally graded beam with length

Beam’s material is made from metal with a volume fraction _{m}(_{c}(

The characteristic of material made beams are written

In this study, two types of porosity are considered, some of them present an evenly distribution (called hereafter Imperfect I), whereas the other one are characterized by an unevenly distribution (Imperfect II), along the beam thickness direction (

The various expressions of the porosity distribution are presented in the following equations:

Imperfect-I:

Imperfect-II:

The material properties of FGM porous beam such as the elastic modulus

Imperfect-I:

Imperfect-II:

_{cm} = _{c} − _{m}, _{cm} = _{c} − _{m}, _{cm} = _{c} − _{m} and the Poisson ratio

The assumed displacement field is as follows:

The strains associated with the displacements in

By assuming that the material of FG beam obeys Hooke’s law, the stresses in the beam become

where

Hamilton’s principle is used herein to derive the equations of motion. The principle can be stated in analytical form as [_{1} and _{2} are the initial and end time, respectively; _{ef} the potential energy of elastic foundation; _{x}, _{xz} are the stress resultants defined as

The variation of the potential energy of elastic foundation given by_{W} and _{P} are the transverse and shear stiffness coefficients of the foundation, respectively.

The variation of work done by externally transverse load

The variation of the kinetic energy can be expressed as_{0}, _{1}, _{1}, _{2}, _{2}, _{2}) are the mass inertias defined as

Substituting the expressions for _{ef},_{0}, _{b} and _{s} to zero separately, the following equations of motion are obtained:

Introducing _{0}, _{b},_{s}) and the appropriate equations take the form

where _{11}, _{11}, etc., are the beam stiffness, defined by

Navier-type analytical solutions are obtained for the bending and free vibration analysis of functionally graded beams resting on two parameter elastic foundation. According to the Navier-type solution technique, the unknown displacement variables are expanded in a Fourier series as given below:_{m}, _{bm}, and _{sm} are arbitrary parameters to be determined,

The transverse load _{m} is the load amplitude calculated from

The coefficients _{m} are given below for some typical loads. For the case of a sinusoidally distributed load, we have

Substituting

For free vibration problem:

For static problems, we obtain the following operator equation:

and

with

This section looks at the current situation using numerical examples. The imperfect/perfect FG beams on elastic foundations supposed to be composed of Al (metal) and Al_{2}O_{3} (ceramic). (ceramic The material properties are taken to be _{m} = 70 GPa; _{m} = 2702 kg/m^{3}; _{c} = 380 GPa; _{c} = 3960 kg/m^{3} and the subsequent non-dimensional forms are used

Finally, the non-dimensional elastic foundation parameters are:

Note that _{w} = _{p} = 0, _{w} = 0.1, _{p} = 0 and _{w} = _{p} = 0.1 represents the foundationless case, Winkler foundation case and Winkler-Pasternak foundation case,

k | _{w} |
_{p} |
Theory | L/h = 5 | L/h = 20 | ||||
---|---|---|---|---|---|---|---|---|---|

0 | 0 | 0 | Present | 2.5019 | 3.0913 | 0.4755 | 2.2839 | 12.171 | 0.4760 |

Sayyad et al. [ |
2.5019 | 3.0922 | 0.4800 | 2.2839 | 12.171 | 0.4806 | |||

Reddy [ |
2.5020 | 3.0916 | 0.4769 | 2.2838 | 12.171 | 0.4774 | |||

Timoshenko [ |
2.0523 | 3.0396 | 0.2653 | 2.2839 | 12.158 | 0.2653 | |||

0.1 | 0 | Present | 2.3547 | 2.9093 | 0.4475 | 1.1935 | 6.3607 | 0.2488 | |

Sayyad et al. [ |
2.3547 | 2.9102 | 0.4517 | 1.1935 | 6.3608 | 0.2511 | |||

Reddy [ |
2.3547 | 2.9096 | 0.4488 | 1.1935 | 6.3606 | 0.2495 | |||

Timoshenko [ |
2.3205 | 2.8607 | 0.2499 | 1.1929 | 6.3539 | 0.1387 | |||

0.1 | 0.1 | Present | 1.4894 | 1.8402 | 0.2830 | 0.2089 | 1.1136 | 0.0436 | |

Sayyad et al. [ |
1.4894 | 1.8407 | 0.2857 | 0.2090 | 1.1136 | 0.0440 | |||

Reddy [ |
1.4894 | 1.8403 | 0.2839 | 0.2090 | 1.1136 | 0.0437 | |||

Timoshenko [ |
1.4756 | 1.8093 | 0.1589 | 0.2089 | 1.1124 | 0.0243 | |||

1 | 0 | 0 | Present | 4.9458 | 4.7851 | 0.4755 | 4.5774 | 18.814 | 0.4760 |

Sayyad et al. [ |
4.9441 | 4.7867 | 0.5248 | 4.5774 | 18.814 | 0.5245 | |||

Reddy [ |
4.9458 | 4.7857 | 0.5243 | 4.5773 | 18.813 | 0.5249 | |||

Timoshenko [ |
4.8807 | 4.6979 | 0.5376 | 4.5734 | 18.792 | 0.5376 | |||

0.1 | 0 | Present | 4.4015 | 4.2586 | 0.4232 | 1.6169 | 6.6456 | 0.1681 | |

Sayyad et al. [ |
4.4015 | 4.2600 | 0.4657 | 1.6169 | 6.6458 | 0.1851 | |||

Reddy [ |
4.4015 | 4.2591 | 0.4666 | 1.6169 | 6.6456 | 0.1854 | |||

Timoshenko [ |
4.3499 | 4.1871 | 0.4791 | 1.6164 | 6.6418 | 0.1900 | |||

0.1 | 0.1 | Present | 2.1100 | 2.0415 | 0.2029 | 0.2189 | 0.9001 | 0.0228 | |

Sayyad et al. [ |
2.1100 | 2.0422 | 0.2232 | 0.2190 | 0.9001 | 0.0251 | |||

Reddy [ |
2.1100 | 2.0417 | 0.2237 | 0.2190 | 0.9001 | 0.0251 | |||

Timoshenko [ |
2.0981 | 2.0195 | 0.2311 | 0.2190 | 0.8998 | 0.0257 | |||

5 | 0 | 0 | Present | 7.7715 | 6.6047 | 0.3840 | 6.9539 | 25.794 | 0.3847 |

Sayyad et al. [ |
7.7739 | 6.6079 | 0.5274 | 6.9541 | 25.795 | 0.5313 | |||

Reddy [ |
7.7723 | 6.6057 | 0.5314 | 6.9540 | 25.794 | 0.5323 | |||

Timoshenko [ |
7.5056 | 6.4382 | 0.9942 | 6.9373 | 25.752 | 0.9942 | |||

0.1 | 0 | Present | 6.5072 | 5.5302 | 0.3216 | 1.8389 | 6.8211 | 0.1017 | |

Sayyad et al. [ |
6.5089 | 5.5327 | 0.4416 | 1.8389 | 6.8212 | 0.1397 | |||

Reddy [ |
6.5078 | 5.5310 | 0.4450 | 1.8389 | 6.8211 | 0.1408 | |||

Timoshenko [ |
6.3198 | 5.4210 | 0.8371 | 1.8377 | 6.8221 | 0.2634 | |||

0.1 | 0.1 | Present | 2.4974 | 2.1224 | 0.1234 | 0.2226 | 0.8258 | 0.0123 | |

Sayyad et al. [ |
2.4976 | 2.1231 | 0.1694 | 0.2226 | 0.8258 | 0.0170 | |||

Reddy [ |
2.4975 | 2.1226 | 0.1708 | 0.2226 | 0.8258 | 0.0170 | |||

Timoshenko [ |
2.4693 | 2.1181 | 0.3271 | 0.2226 | 0.8264 | 0.0319 | |||

10 | 0 | 0 | Present | 8.6526 | 7.9069 | 0.4208 | 7.6421 | 30.923 | 0.4215 |

Sayyad et al. [ |
8.6539 | 7.9102 | 0.4237 | 7.6422 | 30.923 | 0.4263 | |||

Reddy [ |
8.6530 | 7.9080 | 0.4226 | 7.6421 | 30.999 | 0.4233 | |||

Timoshenko [ |
8.3259 | 7.7189 | 1.2320 | 7.6215 | 30.875 | 1.2320 | |||

0.1 | 0 | Present | 7.1138 | 6.5008 | 0.3459 | 1.8838 | 7.6224 | 0.1039 | |

Sayyad et al. [ |
7.1147 | 6.5033 | 0.3484 | 1.8838 | 7.6225 | 0.1051 | |||

Reddy [ |
7.1141 | 6.5016 | 0.3474 | 1.8838 | 7.5606 | 0.1043 | |||

Timoshenko [ |
6.8914 | 6.3891 | 1.0197 | 1.8825 | 7.6262 | 0.3043 | |||

0.1 | 0.1 | Present | 2.5819 | 2.3594 | 0.1256 | 0.2233 | 0.9035 | 0.0123 | |

Sayyad et al. [ |
2.5820 | 2.3601 | 0.1264 | 0.2233 | 0.9035 | 0.0125 | |||

Reddy [ |
2.5819 | 2.3596 | 0.1261 | 0.2233 | 0.8934 | 0.0124 | |||

Timoshenko [ |
2.5520 | 2.3660 | 0.3776 | 0.2233 | 0.9045 | 0.0361 |

k | |||||||||
---|---|---|---|---|---|---|---|---|---|

L/h | _{w} |
_{p} |
Theory | 0 | 1 | 2 | 5 | 10 | ∞ |

5 | 0 | 0 | Present | 5.1527 | 3.9904 | 3.6265 | 3.4014 | 3.2817 | 2.6773 |

Sayyad et al. [ |
5.1453 | 3.9826 | 3.6184 | 3.3917 | 3.2727 | 2.6734 | |||

0.1 | 0 | Present | 5.3114 | 4.2299 | 3.9047 | 3.7170 | 3.6192 | 3.0987 | |

Sayyad et al. [ |
5.3038 | 4.2216 | 3.8961 | 3.7066 | 3.6094 | 3.0942 | |||

0.1 | 0.1 | Present | 6.6783 | 6.1088 | 5.9935 | 5.9983 | 6.0059 | 5.7979 | |

Sayyad et al. [ |
6.6689 | 6.0973 | 5.9810 | 5.9830 | 5.9909 | 5.7903 | |||

20 | 0 | 0 | Present | 5.4603 | 4.2051 | 3.8361 | 3.6485 | 3.5389 | 2.8371 |

Sayyad et al. [ |
5.4603 | 4.2050 | 3.8361 | 3.6484 | 3.5389 | 2.8371 | |||

0.1 | 0 | Present | 7.5533 | 7.0752 | 7.0185 | 7.0949 | 7.1280 | 6.9259 | |

Sayyad et al. [ |
7.5533 | 7.0751 | 7.0184 | 7.0948 | 7.1279 | 6.9259 | |||

0.1 | 0.1 | Present | 18.052 | 19.224 | 19.753 | 20.390 | 20.704 | 21.022 | |

Sayyad et al. [ |
18.052 | 19.224 | 19.752 | 20.390 | 20.703 | 21.022 |

In the present work, bending and free vibration of imperfect FG beams resting on two-parameter elastic foundations was investigated. To accomplish this, the material properties of the beam are assumed to change continuously along the thickness direction based on the volume fraction of constituents defined by the modified rule of the mixture. In addition, to describe the elastic foundation’s response on the imperfect FG beam, the foundation medium is assumed to be linear, homogenous, and isotropic, and it has been modeled using the Winkler-Pasternak model with two parameters. Moreover, in the kinematic relationship of the imperfect FG beam resting on a two-parameter elastic foundation, hyperbolic shear deformation theory is used, and the equations of motion are derived using Hamilton’s principle. For the bending and free vibration analysis of imperfect FG beams resting on a two-parameter elastic foundation with simply supported edges, an analytical solution is obtained. The impacts of the two-parameter elastic foundation, porosity volume fraction, type of porosity models, and aspect ratio, on the bending and fundamental natural frequency of the beams, are investigated in detail. Finally, it is concluded that the types of adopted, two-parameter elastic foundation porosity model, porosity volume fraction, aspect ratio, plays significant role on the bending and free vibration of the FG beams. The negative effects of porosity may be reduced by adopting suitable values for said parameters, considerably.

Volume fraction index

Porosity volume fraction

_{W}

Transverse coefficient of the foundation

_{P}

Shear stiffness coefficient of the foundation

Non-dimensional transverse displacement

Non-dimensional axial stress

Non-dimensional transverse shear stress

Non-dimensional natural frequencies

_{2}–NiCr functionally graded material by powder metallurgy