Free Vibration of Functionally Graded Sandwich Shallow Shells on Winkler and Pasternak Foundations with General Boundary Restraints

<div>Variation of volume fraction<i> V</i><sub><i>c</i></sub> through the thickness in various types: (a) Type <span class="nowrap"><svg height="8.39864pt" id="M36" style="vertical-align:-0.04981995pt" version="1.1" viewbox="-0.0498162 -8.34882 17.2718 8.39864" width="17.2718pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"></path></g><g transform="matrix(.013,0,0,-0.013,6.24,0)"><path d="M300 253H71L57 198H286L300 253Z"></path></g><g transform="matrix(.013,0,0,-0.013,10.885,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"></path></g></svg>;</span> (b) Type <span class="nowrap"><svg height="8.39864pt" id="M37" style="vertical-align:-0.04981995pt" version="1.1" viewbox="-0.0498162 -8.34882 17.2718 8.39864" width="17.2718pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"></path></g><g transform="matrix(.013,0,0,-0.013,6.24,0)"><path d="M300 253H71L57 198H286L300 253Z"></path></g><g transform="matrix(.013,0,0,-0.013,10.885,0)"><path d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"></path></g></svg>;</span> (c) Type <span class="nowrap"><svg height="8.39864pt" id="M38" style="vertical-align:-0.04981995pt" version="1.1" viewbox="-0.0498162 -8.34882 17.2718 8.39864" width="17.2718pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"></path></g><g transform="matrix(.013,0,0,-0.013,6.24,0)"><path d="M300 253H71L57 198H286L300 253Z"></path></g><g transform="matrix(.013,0,0,-0.013,10.885,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"></path></g></svg>;</span> (d) Type <span class="nowrap"><svg height="8.39864pt" id="M39" style="vertical-align:-0.04981995pt" version="1.1" viewbox="-0.0498162 -8.34882 17.2718 8.39864" width="17.2718pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"></path></g><g transform="matrix(.013,0,0,-0.013,6.24,0)"><path d="M300 253H71L57 198H286L300 253Z"></path></g><g transform="matrix(.013,0,0,-0.013,10.885,0)"><path d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"></path></g></svg>.</span></div>

Mathematical Problems in Engineering

fig4

Figure 4

Figure 4: Free Vibration of Functionally Graded Sandwich Shallow Shells on Winkler and Pasternak Foundations with General Boundary Restraints 