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# FEM: Nonlinear Beam Deflection Model (with Temperature)

Derivation and solution of the nonlinear finite element model of a beam under thermal loading.

#WikiCourses
https://wikicourses.wikispaces.com/TopicX+Nonlinear+Solid+Mechanics
https://eau-esa.wikispaces.com/Topic+Nonlinear+Solid+Mechanics

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### FEM: Nonlinear Beam Deflection Model (with Temperature)

1. 1. FEM: Nonlinear Beam Deflection Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Introduction to the Finite Element Method Nonlinear Beam Deflection
2. 2. FEM: Nonlinear Beam Deflection Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Nonlinear Strain Displacement Relation
3. 3. FEM: Nonlinear Beam Deflection Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Nonlinear Strain Displacement Relation • The von Karman large deflection strain- displacement relation for the deflections u, and w can be written as follows 2 22 2 1 x w z x w x u x                        zm 
4. 4. FEM: Nonlinear Beam Deflection Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Using the relations …      }{ xx1 32 aH axw w       }{ 1 bH bxu u         bwbbw wNwTHw  1        mummu wNwTHu  1
5. 5. FEM: Nonlinear Beam Deflection Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com For the derivatives …       bbb w wBwT dx dH dx dw        1       bbbb w wBwT dx Hd dx wd        1 2 2 2 2       mmmm u wBwT dx dH dx du        1
6. 6. FEM: Nonlinear Beam Deflection Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Giving the strain relation … • Where: • Is a scalar function of x                bbbmm m wzww z BBB       2 1 }{   x w   
7. 7. FEM: Nonlinear Beam Deflection Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Inplane Forces and Bending Moments
8. 8. FEM: Nonlinear Beam Deflection Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Inplane Forces and Bending Moments • For homogeneous symmetric cross sections:                           T T M N D A M N   0 0 EAQhA  EIQ h D  12 3    TEAdzzyxTQN h h T    2/ 2/ ),,(    0),,( 2/ 2/   h h T zdzzyxTQM 
9. 9. FEM: Nonlinear Beam Deflection Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Expanding equation                       Tm Tbmm Tm NNN NwAwA NAN            BB 2 1 ][     bb wDDM B][}]{[  
10. 10. FEM: Nonlinear Beam Deflection Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com The Potential Energy        Volume height TT ijij dzMNdVU }{}{                                  dz wDw NwAwA ww U height bb T b T b Tbmm TTT b T m T m                                     BB BB BB ][ 2 1 *      
11. 11. FEM: Nonlinear Beam Deflection Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Skipping the details … • We get the element equation as:                                                                m b mb bmnm TN m b W W n n nn k T k k 00 02 3 1 01 11 2 1 00 0 0 0                  Tm b p p 0 0
12. 12. FEM: Nonlinear Beam Deflection Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Where … • The linear stiffness matreces    dzDk height b T bb  BB ][][     dzAk height m T mm  BB][
13. 13. FEM: Nonlinear Beam Deflection Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Where … • The nonlinear stiffness matreces      dzAnn height T m T bmmb   BB]1[]1[       dzBNn height m T nm  B1         dzAn height TT    BB 2 3 ]2[
14. 14. FEM: Nonlinear Beam Deflection Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Where … • The thermal effect terms     dzBNk height T T TN  B][      dzNp height T T mTm  B
15. 15. FEM: Nonlinear Beam Deflection Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com In a compact form …              TTN PPWNNKK         2 3 1 1 2 1
16. 16. FEM: Nonlinear Beam Deflection Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Note • For details for the derivation, follow the link on the pages: • https://wikicourses.wikispaces.com/Topic+Nonlinear+Solid+Mechani cs • https://eau-esa.wikispaces.com/Topic+Nonlinear+Solid+Mechanics
17. 17. FEM: Nonlinear Beam Deflection Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Solving the Equations for Post-Buckling Deflection
18. 18. FEM: Nonlinear Beam Deflection Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com When no external loading exists • We may write the equations in the form: • Introducing a new function:            TTN PWNNKK         2 3 1 1 2 1                02 3 1 1 2 1         TTN PWNNKKW
19. 19. FEM: Nonlinear Beam Deflection Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Using truncated Taylor expansion • Linear approximation: • Where:          W dW Wd WWW                 tan21 KNNKK dW Wd TN   
20. 20. FEM: Nonlinear Beam Deflection Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Using Newton Algorithm: Repeat, until:               TiiiTNi PWNNKKW         2 3 1 1 2 1      iii WWK 1tan        ii WKW i    1 tan1       11   iii WWW    toliW  1max
21. 21. FEM: Nonlinear Beam Deflection Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Maximum deflection vs. Temp.
• #### AKIB-KHAN

Aug. 18, 2020
• #### PranithaPalampalli

Jul. 19, 2020
• #### ssuserbb3d39

Mar. 31, 2016

Derivation and solution of the nonlinear finite element model of a beam under thermal loading. #WikiCourses https://wikicourses.wikispaces.com/TopicX+Nonlinear+Solid+Mechanics https://eau-esa.wikispaces.com/Topic+Nonlinear+Solid+Mechanics

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