ME 760-FEM HW Probs _1-24-11_

8/7/2019 ME 760-FEM HW Probs _1-24-11_

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ME 760 FEM Homework 1

ME 760 FEM Homework

Problem FEM 1

10"

Thickness = 0.1 inch

A

B

P = 50,000 lbs

A

B

Loading 1

2.5"

Geometry Loading 2

P = 5,000 lbs/in

A

B

P = 50,000 lbs

(applied using aparabolic profile)

A

B

Loading 3

Material: A - Steel E = 30x106 psiµ = 0.3

1. Compute (by hand) the maximum stresses xx and xy at points A and B.

2. Perform a finite element analysis using plane-stress, isoparametric elements. Use a uniformgrid with and an element size that is no larger than 0.5x2 inches.

4. Generate the stress contour lines for the structure for each loading condition. Note that for3 you will probably have to compute equivalent nodal point loads by integrating segments of the parabolic profile.

5. Determine the maximum stresses ( xx and xy) and their locations using FEM for eachloading condition.

6. Compare your hand and FEM calculations, and discuss (briefly) the accuracy of both resultsfor each loading condition.

7. For loading case 1, redo the problem with a variable length and determine how long the beamhas to be before your hand and FEM stresses agree at a point 10” from the left end. If you doit iteratively, start with a length that is at least 20 inches long.

8/7/2019 ME 760-FEM HW Probs _1-24-11_



Problem FEM 2

Assume that the wrench is used to loosen the fixed nut as shown. The wrench is drawn full scaleand is 1/4-inch thick. The yield stress for the wrench steel is 80,000 psi. Determine the factor of safety in the wrench design using a hand analysis and finite elements.

Assume that the bold head is 13/16-inch, and treat the bolt as fixed. Use coupled nodes or gapelements between the wrench and the steel bolt.

F = 200 lbs

13/16"

For your write-up, do the following.

1. Explain your hand analysis and indicate your assumptions and results.

2. Draw the Von-Mises stress contours for the FEM analysis.

3. Determine the maximum stresses and their locations.

4. Compare your hand anlaysis with your FEM analysis.

Note: The objective in this assignment is to illustrate the analysis of contact using either gapelements or coupled nodes. If small modifications in the geometry will help you in the analysis,it is acceptable to make the changes. However, maintain the general size of the bolt and wrench.

8/7/2019 ME 760-FEM HW Probs _1-24-11_



Problem FEM 3

X

Y

2"

4"

16" 1"

2"A

B

A

100 lb/in

Thickness = 0.05 in

Very Rigid

Materials: A - Steel E = 30x106 psiµ = 0.3

B - Plastic E = 5x105 psiµ = 0.45

1. Compute (by hand) the strain in each material

2. Compute (by hand) the stress in each material

3. Perform a finite element analysis using plane-stress, isoparametric elements. Use elementswhich are 2x4 inches (except at the end) and use symmetry. At the end, you may use aregion of very stiff elements or apply a unit displacement to all of the nodes and scale theresults.

4. Generate the contour for the x stresses using both nodal point averaging and gauss point

stresses (if possible). Comment on the differences between the two sets of contour plots, and

indicate which is correct.

5. Discuss the accuracy of your results; i.e., if your stress contours are wrong, determine whatthe stress contours should look like versus what you got.

6. Determine the maximum stresses and their locations.

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Problem FEM 4

5"

15"

1" diameter holecentrally located

200 lbs/in 200 lbs/in

Case 1

5"

15"

1" diameter holecentrally located

1000 lbs1000 lbs

Case 2

Materials: Steel E = 30x106 psi

µ = 0.3

Thickness 0.2 in

1. Compute (by hand) the maximum stress using Case 7 on Page 785 of Young and Budynas

2. Perform a finite element analysis for both cases

3. Discuss the accuracy of your results.

4. Determine the maximum stresses and their locations in the FEM analysis.

5. Plot the maximum principal stress on contour plots

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Problem FEM 5

F = 10,000 lbs

F = 10,000 lbs

1.0"

3.0"

1"

1/2"

3.0"

3.0"

3.0"

Thickness = 1/2"

1"

Material: Steel E = 30x106 psiµ = 0.3

1. Perform a finite element analysis using plane-stress, isoparametric elements.

2. Generate the stress contour lines for the structure.

3. Determine the maximum stress and its location using FEM.

8/7/2019 ME 760-FEM HW Probs _1-24-11_



Problem FEM 6

5"

10"

F = 4000 lb/in

2 " dia. holecentrally located

The plate thickness is 0.25 in.

Material: Steel E = 30x106 psiµ = 0.3

1. Compute (by hand) the maximum stress in the part.

2. Perform a finite element analysis using plane-stress, isoparametric elements.

3. Generate the maximum principal stress contour lines for the structure.

4. Compare your hand and FEM calculations, and discuss (briefly) the accuracy of both results.

5. Determine the maximum stress and its location using FEM.

6. Repeat the analysis with a 20,000 lb load applied at the center of the left and right ends.

7. Compare the results for the two cases (use a table).

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Problem FEM 7

One of the ideas presented for reducing stress concentrations involves stress flow guides. Testthis concept by doing two finite element analyses on the following geometries, and compare themaximum stresses in each case. Assume the material in each case is steel with E = 30x106 psi

and µ = 0.3. The plate is 0.5 inch thick.

Case 1:

10"

20"

σ = 50 ksi

2 " Dia. Holecentrally located

Case 2: Same as Case 1 except for the two 1-inch holes as shown.

σ = 50 ksi

1" Dia.

3"

1" Dia.

3"

1. Perform a finite element analysis using plane-stress isoparametric elements on Case 1.

2. Repeat the analysis for Case 2.

3. Based on the results from Cases 1 and 2, comment on the effect of tstress-flow guides.

4. Plot the maximum principal stress on contour plots.

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Problem FEM 8

One of the ideas presented for reducing stress concentrations involves stress flow guides. Testthis concept by doing three finite element analyses on the following geometry, and compare the

maximum stresses in each case. Assume the material in each case is steel with E = 30x106

psiand µ = 0.3. The plate is 0.5 inch thick.

5"

15"

100 lbs/in 100 lbs/in

x x1"

0.5"

0.5" 0.5

"

1"

0.02"r (typ.)

Case 1:

Preform the analysis without the two 0.5" notches present.

Case 2:

Perform the analysis with x = 0.

Case 3:

Perform the analysis with x = 0.5"

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Problem FEM 9

The symmetric truss shown is made entirely of round steel bars of diameter 0.25 inch. The loadP is 1000 lbs. Use beam finite elements, and find the stress in each member.

Comment on the anti-redundancy principle and the direct-path principle.

P

10 in

10 in

8/7/2019 ME 760-FEM HW Probs _1-24-11_



Problem FEM10

The isosceles truss shown is made entirely ofsquare steel bars of diameter 0.25 inch. The load Pis 1000 lbs. Use beam finite elements, and find the stress in each member.

P

10 in

10 in

1/4" square section

8/7/2019 ME 760-FEM HW Probs _1-24-11_



Problem FEM 11

The truss (a) and beam (b) are both 0.25” thick and both are made of steel. The load P is 1000lbs. Use symmetry for the truss in (a) and analyze both members using the same FEM mesh.

Compare the stresses at the base of each member. Comment on the supplementary shapeprinciple.

P

2 in 2 in

10 in10 in

60˚ 60˚

P/2

2 in

10 in

60˚

(a) (b)

8/7/2019 ME 760-FEM HW Probs _1-24-11_



Problem FEM 12

Assume that the beam is 0.15” thick and is inserted into a rigid wall. The beam is made of steeland the load P is 100 lbs. Compute the stresses and determine the force or pressure distributionat the top and bottom surfaces of the beam for the following conditions:

a) When x = 1.5 in

b) When x = 10 in

Comment on stress pattern shift.

8/7/2019 ME 760-FEM HW Probs _1-24-11_



Problem FEM 13

A link similar to that used in the Silver Bridge is shown schematically in the following. Assumethat the link is made out of 1040 steel, and it is two inches thick. Using FEM, compute the ratioof the maximum stress in the eye of the link to that in the long body. Assume that the loading is

provided through a pin at each eye section. Represent this by assuming that half the eye is incontact with the pin, and fix the nodes on the inside circumference of the eye in the radialdirection.

Check your results with a hand analysis.

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Problem FEM 14

One of the ideas presented for reducing stress concentrations involves stress flow guides. Testthis concept by doing three finite element analyses on the following geometry, and compare the

maximum stresses in each case. Assume the material in each case is steel with E = 30x106

psiand µ = 0.3. The plate is 0.5 inch thick, and the notches are placed symmetrically.

Case 1:

Preform the analysis without the two 0.5" notches present.

Case 2:

Perform the analysis with x = 0.

Case 3:

Perform the analysis with x = 0.5"

Case 4:

Conduct a hand analysis of Case 1 by computing a nominal stress and correcting it with a stressconcentration factor. Compare your hand results with the FEM analysis.

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Problem FEM 15

Assume that the round bar is loaded as shown. Use an axisymmetric model and assume that theload is distributed as a uniform tensile load on each end. Use symmetry when you do the model.Determine the stress concentration factor at the notch for the following cases:

1) L = 1”2) L = 1/4”3) L = 1/16”

8/7/2019 ME 760-FEM HW Probs _1-24-11_



Problem FEM 16

Assume that the rectangular bar is loaded in torsion as shown. The material is steel, and theshear strength is 40,000 psi. Do the following:

1)

Find the maximum torque that the bar can carry.2) For the maximum torque value computed, compute the maximum shear stress at thewall

3) Discuss the best way(s) to apply the loading.4) Conduct a FEM analysis, and compare the results with the hand calculations.

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Problem FEM 17

The main figure in the drawing is symmetric left and right and top and bottom. All dimensionsare in inches, and the part is made of steel. The thickness is 0.5 inches. Assume that the load is20,000 lbs. The boundary is rigid, and the part has a slip fit with the boundary. Conduct finite

element analyses for each of the following cases:

a) When L = 5 inchb) When L = 0.5 inch.

Compute the stress concentration factor for each case and compare results. Note that the contactforces between the boundary and the part can be compressive only. Does the peninsula principleapply to this situation?

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Problem FEM 18

The main figure in the drawing is symmetric left and right and top and bottom. All dimensionsare in inches, and the part is made of steel. The thickness is 0.5 inches. Assume that the load is20,000 lbs. Conduct finite element analyses for each of the following cases:

a) When L = 5 inchb) When L = 0.5 inch.

Compute the stress concentration factor for each case and compare results. Note that the contactforces between the boundary and the part can be compressive only. Comment on the peninsulaprinciple.

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Problem FEM 19

The bar shown is made from 0.2” flat steel stock. Assume that the material is 1020 steel, andanalyze the problem using finite elements.

Compute the stress concentration factor for the geometry given and compare it with the typicalvalue given in the literature.

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Problem FEM 20

Based on simple bending theory, the maximum stress in a beam is equal to Mc/I. The three crosssections below, will have the same moment of inertia about the horizontal neutral axis. To check this assumption, perform a finite element analysis for the two loading conditions shown.

Compute the maximum stress on the top surface 1 inch from the fixed support. Also determine thedeformations. If the results are not substantially the same for the three models for each loadingcondition, explain why. You may apply the loading however you want as long as it is equivalent tothe two cases shown; but, explain how you actually applied the loading in each case.

Assume that the material is steel with E = 30x106 psi and = 0.3.

Problem FEM 21

8/7/2019 ME 760-FEM HW Probs _1-24-11_



The tension member below has notches that are symmetrically located both horizontally. Holes aredrilled through the member to reduce the stress concentration in the second model.

Find the theoretical stress concentration factor for both models. You may pick any material you

want and any convenient loading for this.

Analyze the model for both plane stress and plane strain conditions using FEM. For this, assumethat the thickness is 0.1 inch for the plain stress case and 4.0 inch for the plane strain case.Compute the stress concentration factor for both plane stress and plane strain for the notch with andwithout the holes. For the stress concentration factors, use the net area through the notch whencomputing the nominal stress.

8/7/2019 ME 760-FEM HW Probs _1-24-11_



Problem FEM 22

The tension member below has notches that are symmetrically located horizontally. All dimensionsare in inches.

Find the theoretical stress concentration factor for the model from Peterson. You may pick anymaterial you want and any convenient loading value for this.

Analyze the model for both plane stress and plane strain conditions using FEM. For this, assumethat the thickness is 0.1 inch for the plain stress case and 4.0 inch for the plane strain case.Compute the stress concentration factor for both plane stress and plane. For the stressconcentration factors, use the net area through the notch when computing the nominal stress. Donot use the FEM values for computing the nominal stress.

8/7/2019 ME 760-FEM HW Probs _1-24-11_



Problem FEM 23

The symmetric trusses shown are made entirely of round steel bars of diameter 0.25 inch. Theload P is 1000 lbs. Use beam finite elements, and find the maximum stress in each member foreach case.

Comment on the anti-redundancy principle and the direct-path principle.

ME 760-FEM HW Probs _1-24-11_

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