ME 309 ANSYS Material Plasticity in Pure Bending

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ME309 Homework 5

Problem 2: A Study of Material Plasticity in Pure Bending

Instructor: Prof. Sheri Sheppard

Student: Cheng-Chieh Chao

[email protected]

05361779

6.11.2007
mailto:[email protected]:[email protected]


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Table of Contents

I. Problem Statement................................................................................................ 3

II . Objective of the analysis....................................................................................... 3III. Physical description of the part to be analyzed.................................................. 4

IV. Finite element program and computer system used for the analysis............... 4

V. Finite element model............................................................................................. 5

VI . Result plots (Stress contour plots) ....................................................................... 8

VII. Stresses and displacements for critical sections of the model ......................... 10

VIII. Theoretical solutions........................................................................................... 12

IX. Conclusions and recommendations................................................................... 13

Appendix: Log file for the solution of both models...................................................... 14

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I. Problem StatementConsider a rectangular beam loaded in pure bending between the two forces, as shown

(this is referred to as the four point bend specimen). For elastic-perfectly plastic

stress-strain behavior, show using finite element analysis, that the beam remains elastic at

Myp=ypbh2/6 and is completely plastic at Mult=1.5 Myp.

II . Objective of the analysisThe objective of this problem is to understand plastic-elastic deflection behavior of the

specimen under load. The finite element analysis simulation result will be compared to

the theoretical computation.

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III. Physical description of the part to be analyzed

The beam specimen will be loaded with the configuration of four-point bending testing

with adjustable forces F and supports at both ends. It is usually used in a test to

determine the yield strength of the material. The cross section of the specimen is a

rectangle, dimensions are shown in figure. The coordinate axes are shown in the figure,

where x axis passes through the neutral axis of the beam with the origin at the left end. y

and z axes are oriented as shown.

Dimensions:

b=1, h=2

Material properties:

Youngs Modulus E=30e6 psi

Poisson Ratio =0.3

Yield Strength yp=36000 psi

Loading conditions:

Two external forces, F, of equal magnitude and direction, are applied from the top of the

beam. Supports in the y direction are at the two ends of the specimen.

IV. Finite element program and computer system used for the analysisANSYS v7 is used for the FEM analysis in the elaine cluster at Terman.

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V. Finite element model(a) Analysis using PLANE42 element

Element used: PLANE42

Real constraint: thickness 1

Mesh density: 0.5x0.5 per element.

Boundary conditions:

UX=UY=0 at node 1 (bottom left corner)

UY=0 at node 2 (bottom right corner).

Applied loads:

The load F required for applying Myp=ypbh2/6=24000 lb-in is F=6000 lb, and the load

required for Mult=1.5 Myp=36000 lb-in is F=9000 lb.

The load F is applied incrementally as 5999, 6001, 6500, 7000, 7500, 8000, 8500, and

9000 lbs to avoid the sudden loading that may lead to the divergence of solution in

ANSYS. The F is applied at nodes 38 and 46 as in the figure above.

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Assumptions: It is assumed the material behavior of this steel beam is bilinear with zero

tangent modulus after yield point. Bilinear kinematic hardening plasticity behavior is

used for solving this problem in ANSYS.

Bilinear material behavior with zero tangent modulus, yield strength y=36000 psi

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(b) Analysis using SOLID45 element

Element used: SOLID45

Mesh density: 0.5x0.5x0.5 per element

Boundary conditions:

UX=UY=0 for all nodes at x=y=0 (bottom left corner)

UY=0 for all nodes at x=12, y=0 (bottom right corner)

Applied loads:

The load F required for applying Myp=ypbh2/6=24000 lb-in is F=6000 lb, and the load

required for Mult=1.5 Myp=36000 lb-in is F=9000 lb.

The load F is applied incrementally as 5999, 6001, 6500, 7000, 7500, 8000, 8500, and

9000 lbs to avoid the sudden loading that may lead to the divergence of solution in

ANSYS. One F is uniformly distributed along all the 3 nodes at x=4, y=2 with the

magnitude of F/3. The other F is also uniformly distributed along all the 3 nodes at x=8,

y=2 with the magnitude of F/3, as in the figure above.

Assumptions: It is assumed the material behavior of this steel beam is bilinear with zero

tangent modulus after yield point. Bilinear kinematic hardening plasticity behavior is

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used for solving this problem in ANSYS.

VI . Result plots(a) Analysis using PLANE42 element

xx stress contour plots using PLANE42 element at F=8500 lbs

Discussion: The figure shows the xx stresses when the beam is under the load of F=8500

lbs (1.42 Myp). The region between x=4 and x=8 is under the stresses over 30000 psi,

indicating the structure behavior is nonlinear. At this time, the beam is in the

elastic-plastic region.

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xx stress contour plots using SOLID45 element at F=8500 lbs

Discussion: The figure shows the xx stresses when the beam is under the load of F=8500

lbs (1.42 Myp). The region between x=4 and x=8 is under the stresses over 30000 psi,

indicating the structure behavior is nonlinear. At this time, the beam is in the

elastic-plastic region.

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VII. Stresses and displacements for critical sections of the model(a) Analysis using PLANE42 element

PLANE42

-50000

-40000

-30000

-20000

-10000

0

10000

20000

30000

40000

50000

-1 -0.5 0 0.5 1

Distance from neutral axis

S

tress

(psi)

F=5999

F=6001

F=6500

F=7000

F=7500

F=8000

F=8500

Material behavior using PLANE42 elements

node 5999 6001 6500 7000 7500 8000 8500 9000

(7, -1) -35771 -35783 -38758 -36015 -36016 -36006 -35982 N/A

(7, -0.5) -18008 -18014 -19511 -24639 -28782 -33035 -39120 N/A

(7, 0) -189.7 -189.76 -207.87 -26.236 -25.609 89.043 215.35 N/A

(7, 0.5) 17957 17963 19456 24664 28807 32948 38896 N/A

(7, 1) 36254 36266 39284 36017 36019 36003 35998 N/A

xx at nodes on x=7 vs. distance from the neutral axis in y direction under different F

Discussion:

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The stresses in the table are xx on the 5 nodes on x=7. As we can see in the figure

above, the material is elastic at F=5999 and 6001 lbs (~Myp), and xx=My/I. When the

load is increasing from 6500 lbs, the material behavior becomes nonlinear. The

solutions still converge for F=6500, 7000, 7500, 8000, 8500 lbs, indicating the material is

in the elastic-plastic region. When the load F is 9000 lbs, ANSYS fails to converge in 5


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steps, indicating that the beam is fully plastic.


SOLID45

-50000

-40000

-30000

-20000

-10000

0

10000

20000

30000

40000

50000

-1 -0.5 0 0.5 1

Distance from neutral axis

Stress

(psi)

F=5999

F=6001

F=6500

F=7000

F=7500

F=8000

F=8500

Material behavior using SOLID45 elements

node 5999 6001 6500 7000 7500 8000 8500 9000

(7, -1, 1) -35360 -35372 -38274 -36115 -36306 -36326 -36295 N/A

(7, -0.5, 1) -18856 -18862 -20452 -25071 -29238 -33715 -39524 N/A

(7, 0, 1) -82.572 -82.599 -100.61 -63.159 -102.32 -125.35 35.954 N/A

(7, 0.5, 1) 17914 17920 19415 24795 28966 33332 39163 N/A

(7, 1, 1) 36425 36437 39487 36065 36203 36348 36278 N/A

xx at nodes on x=7, z=1 vs. distance from the neutral axis in y direction under different F

Discussion:

The stresses in the table are xx on the 5 nodes on x=7. As we can see in the figure

above, the material is elastic at F=5999 and 6001 lbs (~Myp), and xx=My/I. When the

load is increasing from 6500 lbs, the material behavior becomes nonlinear. Thesolutions still converge for F=6500, 7000, 7500, 8000, 8500 lbs, indicating the material is

in the elastic-plastic region. When the load F is 9000 lbs, ANSYS fails to converge in 5

steps, indicating that the beam is fully plastic. The result agrees with PLANE42 elements.

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VIII. Theoretical solutionsReference: Case, J.; Chilver, L.; Ross, C.T.F. (1999). Strength of Materials and Structures

(4th Edition). (pp. 350-366). Elsevier

Myp=ypbh2/6

Mult=1.5 Myp

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M < Myp M > MultMyp < M < Mult


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IX. Conclusions and recommendationsThe analysis presented using PLANE42 and SOLID45 matches the theoretical material

behavior. In summary, the result is listed in the following table:

F 6000 6500 7000 7500 8000 8500 9000

Mult / Myp 1 1.083 1.167 1.25 1.333 1.417 1.5

Theory ElasticElastic-

plastic

Elastic-

plastic

Elastic-

plastic

Elastic-

plastic

Elastic-

plastic

Fully

plastic

Converge Converge Converge Converge Converge ConvergeNot

converge

ANSYS

PLANE42

&SOLID45

Elastic Elastic-plastic

Elastic-plastic

Elastic-plastic

Elastic-plastic

Elastic-plastic

Fullyplastic

Therefore, it can be concluded that ANSYS is suitable for nonlinear material plasticity

simulation. The result of finite element simulation and theoretical values matches well.

It is confirmed that a rectangular beam is pure elastic at Myp=ypbh2/6 and completely

plastic at Mult = 1.5 Myp. With proper models and settings, ANSYS can be used to

predict the material behavior for elastic and elastic-plastic.

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Appendix: Log file for the solution of both models

(a) Analysis using PLANE42 element

/SOL

solcontrol,0

neqit,5 ! MAXIMUM 5 EQUILIBRIUM ITERATIONS PER STEP

ncnv,0 ! DO NOT TERMINATE THE ANALYSIS IF THE

SOLUTION FAILS TO CONVERGE

cnvtol,u ! CONVERGENCE CRITERION BASED UPON

DISPLACEMENTS

f,46,fy,-5999

f,38,fy,-5999

solve

f,46,fy,-6001

f,38,fy,-6001

solve

*do,I,1,8

f,46,fy,-6000-(I*500)

f,38,fy,-6000-(I*500)

solve

*enddo

Finish

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/SOL

solcontrol,0neqit,5 ! MAXIMUM 5 EQUILIBRIUM ITERATIONS PER STEP

ncnv,0 ! DO NOT TERMINATE THE ANALYSIS IF THE

SOLUTION FAILS TO CONVERGE

cnvtol,u ! CONVERGENCE CRITERION BASED UPON

DISPLACEMENTS

f,163,fy,-1999.66

f,293,fy,-1999.66

f,14,fy,-1999.66

f,171,fy,-1999.66

f,285,fy,-1999.66

f,22,fy,-1999.66

solve

f,163,fy,-2000.33

f,293,fy,-2000.33

f,14,fy,-2000.33

f,171,fy,-2000.33

f,285,fy,-2000.33

f,22,fy,-2000.33

solve

*do,I,1,6

f,163,fy,-2000-(I*166.667)

f,293,fy,-2000-(I*166.667)

f,14,fy,-2000-(I*166.667)

f,171,fy,-2000-(I*166.667)

f,285,fy,-2000-(I*166.667)

f,22,fy,-2000-(I*166.667)

solve

*enddo

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FINISH

Documents

ME 309 ANSYS Material Plasticity in Pure Bending