A FIRST COURSE IN THE FINITE ELEMENT METHOD

Transcription

1 INSTRUCTOR'S SOLUTIONS MANUAL TO ACCOMANY A IRST COURS IN TH INIT LMNT MTHOD ITH DITION DARYL L. LOGAN

2 Contents Chapter 1 1 Chapter 3 Chapter 3 3 Chapter 17 Chapter Chapter 6 81 Chapter Chapter Chapter Chapter Chapter Chapter 1 1 Chapter 13 3 Chapter 1 73 Chapter 15 9 Chapter Appendix A 55 Appendix B 555 Appendix D 561

3 Chapter A finite element is a small body or nit interconnected to other nits to model a larger strctre or system. 1.. Discretization means dividing the body (system) into an eqivalent system of finite elements with associated nodes and elements The modern development of the finite element method began in 191 with the wor of Hrennioff in the field of strctral engineering. 1.. The direct stiffness method was introdced in 191 by Hrennioff. However, it was not commonly nown as the direct stiffness method ntil A matrix is a rectanglar array of qantities arranged in rows and colmns that is often sed to aid in expressing and solving a system of algebraic eqations As compter developed it made possible to solve thosands of eqations in a matter of mintes The following are the general steps of the finite element method. Step 1 Divide the body into an eqivalent system of finite elements with associated nodes and choose the most appropriate element type. Step Choose a displacement fnction within each element. Step 3 Relate the stresses to the strains throgh the stress/strain law generally called the constittive law. Step Derive the element stiffness matrix and eqations. Use the direct eqilibrim method, a wor or energy method, or a method of weighted residals to relate the nodal forces to nodal displacements. Step 5 Assemble the element eqations to obtain the global or total eqations and introdce bondary conditions. Step 6 Solve for the nnown degrees of freedom (or generalized displacements). Step 7 Solve for the element strains and stresses. Step 8 Interpret and analyze the reslts for se in the design/analysis process The displacement method assmes displacements of the nodes as the nnowns of the problem. The problem is formlated sch that a set of simltaneos eqations is solved for nodal displacements or common types of elements are: simple line elements, simple two-dimensional elements, simple three-dimensional elements, and simple axisymmetric elements. 1.1 Three common methods sed to derive the element stiffness matrix and eqations are (1) direct eqilibrim method () wor or energy methods (3) methods of weighted residals The term degrees of freedom refers to rotations and displacements that are associated with each node. 1 1 Cengage Learning. All Rights Reserved. May not be scanned, copied or dplicated, or posted to a pblicly accessible website, in whole or in part.

4 1.1. ive typical areas where the finite element is applied are as follows. (1) Strctral/stress analysis () Heat transfer analysis (3) lid flow analysis () lectric or magnetic potential distribtion analysis (5) Biomechanical engineering ive advantages of the finite element method are the ability to (1) Model irreglarly shaped bodies qite easily () Handle general load conditions withot difficlty (3) Model bodies composed of several different materials becase element eqations are evalated individally () Handle nlimited nmbers and inds of bondary conditions (5) Vary the size of the elements to mae it possible to se small elements where necessary 1 Cengage Learning. All Rights Reserved. May not be scanned, copied or dplicated, or posted to a pblicly accessible website, in whole or in part.

5 Chapter.1 (a) 1 1 [ (1) ] 1 1 [ () ] [ 3 (3) ] [K] [ (1) ] + [ () ] + [ (3) ] [K] (b) Nodes 1 and are fixed so 1 and and [K] becomes Î Ð [K] 1 {} [K] {d} 3x x Î Ð 3 1 Î 3 3 Ð 1 Î 3 3 Ð {} [K] {d} [K 1 ] {} [K 1 ] [K] {d} [K 1 ] {} {d} Using the adjoint method to find [K 1 ] C C 1 ( 1) 3 ( ) C 1 ( 1) 1 + ( ) C Cengage Learning. All Rights Reserved. May not be scanned, copied or dplicated, or posted to a pblicly accessible website, in whole or in part.

6 . [C] 3 1 and C T det [K] [K] ( 1 + ) ( + 3 ) ( ) ( ) [K] ( 1 + ) ( + 3 ) T [K 1 [ C ] ] det K [K 1 ] ( )( ) Î Ð Î 1 Ð ( 1 ) (c) In order to find the reaction forces we go bac to the global matrix [K] {d} Î Ð 1x x 3x x 1 1 Î Ð 1x x x 3 3 x 3 ( 1 ) ( 1 ) lb [ (1) ] (1) () () (3) (1) ; [ () () ] () (3) By the method of sperposition the global stiffness matrix is constrcted. 1 Cengage Learning. All Rights Reserved. May not be scanned, copied or dplicated, or posted to a pblicly accessible website, in whole or in part.

7 [K] (1) () (3) (1) () [K] (3) Node 1 is fixed 1 and 3 δ Î 1x x Ð 3x {} [K] {d} Î Ð3 x Internal forces lement (1) (1) Îf 1x () Ðf x Î 1 Ð3 Î Î À Ð Ð 1 3x (.5 ) + (1 ) 3x.5 3x ( 1 lb lb ) (.5 ) + (1 ) (1 ) 3x 5 lbs Î 1 Ð.5 (1) f 1x ( 1 lb ) (.5 ) (1) f 1x 5 lb (1) f x (1 lb ) (.5 ) (1) f x 5 lb lement () Îf Ðf () x () 3x.5 Î 3 1 Ð () x f () 3x f 5 lb 5 lb.3 (a) [ (1) ] [ () ] [ (3) ] [ () ] By the method of sperposition we constrct the global [K] and nowing {} [K] {d} we have Î 1x x 3x x Ð 5x Î 1 3 Ð Cengage Learning. All Rights Reserved. May not be scanned, copied or dplicated, or posted to a pblicly accessible website, in whole or in part.

8 (b) Î Î 3 À Ð Ð 3 3 ; Sbstitting in the eqation in the middle É 3 È Ø Ê ÙÚ È Ø ÉÊ ÙÚ 3. ; (c) In order to find the reactions at the fixed nodes 1 and 5 we go bac to the global eqation {} [K] {d} 1x 1x 5x 5x Chec Σ x 1x + 5x + + È É Ê Ø Ù Ú + (a) [ (1) ] [ () ] [ (3) ] [ () ] By the method of sperposition the global [K] is constrcted. Also {} [K] {d} and 1 and 5 δ Î 1x x 3 x x Ð 5x Î 1 3 Ð Cengage Learning. All Rights Reserved. May not be scanned, copied or dplicated, or posted to a pblicly accessible website, in whole or in part.

9 (b) 3 (1) + 3 () 3 + δ (3) rom () 3 rom (3) Sbstitting in qation () ( ) + ( ) È Ê x + Ø Ú (c) Going bac to the global eqation {} [K] {d} 1x 1x È É 5x + δ 3 Ê Ø ÙÚ + δ 5x ip ip 3 ip ip ip 3 x [ (1) ] [ (3) ] [ (5) ] d 1 d d d 1 1 ; [ () ] 1 1 d d d d 3 3 ; [ () ] 3 3 d d Cengage Learning. All Rights Reserved. May not be scanned, copied or dplicated, or posted to a pblicly accessible website, in whole or in part.

10 Assembling global [K] sing direct stiffness method [K] Simplifying [K] ip Now apply + ip at node in spring assemblage of.5..7 [K] from.5 x ip [K]{d} {} Î 1 3 Ð Î1 3 Ð where 1, 3 as nodes 1 and 3 are fixed. Using qations (1) and (3) of (A) 1 9 Î Î 9 1 Ð Ð Solving.75,.35 (A) conv. f 1x C, f x C f δ ( 1 ) f 1x ( 1 ) f x ( ) ( 1 ) f Î 1x Ðfx Î Î Ð Ð 1 [K] Î same as for Ð tensile element 8 1 Cengage Learning. All Rights Reserved. May not be scanned, copied or dplicated, or posted to a pblicly accessible website, in whole or in part.