fea_assignment_3_solution

Assignmnet No: 3

Subject : Finite Element Analysis

Submitted By : Syed Farasat Ali Shah

Submitted To: Dr. Shahzad Rahman Dated:

10/03/20210

Department Of Civil Engineering

NWFP University Of Engineering & Technology, Peshawar.

Problem: A thick-walled cylinder shown above is subjected to internal pressure pi.

Determine the distribution of radial stress σr and tangential stress σθ through the thickness of the cylinder due to applied internal pressure. Use appropriate 2-D elements from SAP Element Library to model and solve the structure in the most efficient manner (your grade will depend upon the efficiency of modeling approach. Draw a neat sketch of your model with boundary conditions and applied loadings. a) Make a suitable mesh to capture the stress gradients and utilize the symmetry of the structure to solve the problem. b) Refine the mesh so that numerical results are within 5% of the exact solution. c) Plot a comparison of the analytical radial stress and tangential stress as compared to numerically obtained stresses.

Exact solution is:

Solution: Given Values: Modulus of Elasticity, E = 29,000,000 Psi Poison’s Ratio, υ = 0.2 Internal Pressure, Pi = 1000 Psi Internal Diameter, a = 12 in.

External Diameter, b = 16 in.

Thickness, t = b-a =16-12 = 4in Height = 12in. (Assumed) Required Values: Radial Stresses across the thickness, σr =?

Tangential Stresses across the thickness, σt =?

Model of Cylinder as drawn in SAP alongwith applied loading is shown in figure below.

Coarse Mesh ( 1”)

Fine Mesh (0.5”)

Stress Gradient for Tangential Stresses as obtained in SAP is shown in figure below.

Stresses σ33 for 1” element mesh

Stresses σ33 for 0.5” element mesh

Values of Tangential Stresses across the thickness of Cylinder are shown in table below.

Tangential Stresses (σ33) Radius (R) (inches)

(Psi)

Exact Solution

Numerical Solution

Numerical Solution

Coarse Mesh (1”)

Finer Mesh (0.5”)

Percentage Error

Percentage Error

(coarse Mesh)

(Fine Mesh)

12

3572

3590

3590

-0.5

-0.5

13

3233

3260

3240

-0.8

-0.2

14

2965

2980

2970

-0.5

-0.2

15

2749

2760

2760

-0.4

-0.4

16

2572

2540

2560

1.24

0.5

Stress Gradient for Radial Stresses as obtained in SAP is shown in figure below

Stresses σ11 for 1” element mesh

Stresses σ11 for 0.5” element mesh

Values of Radial Stresses across the thickness of Cylinder are shown in table below.

Radial Stresses (σ11) (Psi) Radius (inches) Exact Solution

Numerical Solution

Numerical Solution

Coarse Mesh (1”)

Finer Mesh Mesh (0.5”)

Percentage Error

Percentage Error

(coarse Mesh)

(Fine Mesh)

12

-1000

-800

-950

20

5

13

-662

-506

-630

24

4.8

14

-394

-269

-380

32

3.6

15

-177

-76

-176

57

0.56

16

0

-76

-33

Graph comparision of numerically and analytically obtained values of Tangential and Radial Stresses is Shown Below.