PROBLEMS ON PRESSURE VESSEL

Question 1

A sectional impression of a pressure vessel used in a radioactive material processing plant is shown in Figure Q1. The pressure vessel needs to be installed in a rigid concrete (assume Grade 40, density 2400 kg/m³) cover as shown and closed with a hemispherical cover dome. A flexible leak protection seal will be placed around the concrete dome. Assume there is no clearance between the vertical walls of the vessel and the concrete enclosure walls. The vessel is resting on the bottom concrete slab as shown.

The initial conditions of the vessel estimated as ambient pressure, 1 bar and ambient temperature, 22 ⁰C. Major design dimensions are shown and the thickness T needs to be determined by a FE Analysis. You need to refine the meshes appropriately. The vessel needs to withstand 0.45 MPa internal pressure and temperature increased up to 90⁰C (at steady operational state). The vessel needs to be designed for 25% pressure surges and maximum stress in the vessel walls needs to be minimized as possible. You need to check your answers with appropriate manual calculations. (Assume the concrete cover adjacent to vessel’s outer vertical walls and the interface at concrete dome and the vessels top dome reach 50⁰C at the steady state. Determine a suitable trial value for thickness “T “.Use the trial value for FEA and refine it until you meet the optimal design criteria. You need to (list all your assumptions):

1. Assume a trial thickness and perform simple manual calculations for a pressure vessel i.e. hoop and axial stresses at ambient conditions.

Question 2 (100/300 Marks)

Figure Q2 shows an engineer’s layout diagram of a proposed vertical stainless-steel shaft which will drive a mixer of a chemical plant. The larger end of the shaft will be coupled to a 4.5kW, 4000 rpm electrical motor through a flexible coupling and the smaller end will be coupled to the shaft of the mixer. The length of bearing surfaces can be assumed as indicated in the drawing. Bearing A and the connection at mixer do not allow axial movements. The mixer being used for a thick slurry which has varying densities and large material pieces, a 30 mm thick steel fly-wheel is attached to the shaft for smoother operation as indicated in the figure. The surface of the shaft is fully insulated by safety covers (to cover hot surface and not shown here) as indicated in the figure. The portion of largest diameter of the shaft and flywheel surfaces will be exposed to ambient air (convection coefficient of air can be taken as 50 W/m²/⁰C) at 30C

.

Your task is to find the stresses/strain/temperature distribution of the shaft under the operational condition:

1. Create an appropriate FEA model on Creo Simulate 6.0.

Figure 1 VERTICAL STAINLESS-STEEL SHAFT

2. Assume and justify any missing details you required. Include the weight of the shaft and flywheel in the analysis appropriately. You need to list assumptions with your justification in the report. You need to refine the meshes appropriately.

Heat load

convection coefficient of air = 50 W/m²/⁰C

Outside Temp. (To) = 30 ⁰C

Body Temp (Tb) = 90⁰C

Area of rod = 92191.61 mm2 = 0.0921 m2

Q = h A (Tb-To) = 50*0.0921* (90-30) = 276.3W

3. Perform (i) static, (ii) thermal and (iii) static +thermal analyses. You need to select appropriate stress/strain, temperature contour plots and include in the report. If the yield strength of stainless steel is 290MPa, determine the overall safety factor of the shaft. (Note: Since the shaft is rotating at 4000rpm, you need to add centrifugal force in to your FEA model). If the stress levels under operational conditions are beyond yield strength of stainless steel, list your recommendations to reduce the stresses below yield strength.

As given shaft is rotating at 4000 RPM

1 RPM = 0.10472 rad/s

4,000 RPM = 418.87902 rad/s

Figure 2 STATIC ANALYSIS

Figure 3 THERMAL ANALYSIS

Figure 4 THERMAL ANALYSIS PLOT

Figure 5 STATIC ANALYSIS PLOT

Yield strength of stainless steel = 290MPa

Max stress = 5.345E+07 Pa

Overall safety factor = 290MPa / 53.45 MPA = 5.425

4. Perform a vibration analysis for the shaft without support (fre-free) and with the supports. Show results of first 6 natural frequencies and mode shapes of the shaft.

Figure 6 VALUES OF FIRST 6 FREQUENCIES

Figure 7 RESULTS OF FIRST 6 FREQUENCIES

5. Determine the highest safe rotational speed of the motor considering the vibration characteristic of the shaft and the possible yielding of the pulley.

6. Manually check the 19 mm shaft for possible buckling under operational loads using Euler buckling criteria.

F = π²xExI/(kxL)²

k=(1/n)^1/2

n = 0.25 (Since the shaft is free at one end)

k = 2

E = 200000000 pa

I = π D⁴ / 64

I = 6393.8740625 mm4 = 6.39×10-9 m4

F = 4108.58 N

MEC3302-CMD S1, 2021

Question 3 (80/300 Marks)

Q3.1. Write a short note (not more than half A4 page) on your understanding on the St. Venant’s

Principal. (10 Marks)

Saint–Venant’s principle, named after Adhémar Jean Claude Barré de Saint–Venant, a French elasticity theorist, may be expressed as follows: … the difference between the effects of two different but statically equivalent loads becomes very small at sufficiently large distances from load.

Venant’s Principle state that if the forces acting on a small portion of the surface of an elastic body are replaced by another statically equivalent system of forces acting on the same portion of the surface, this redistribution of loading produces substantial changes in the stresses locally but has a negligible effect on the stresses at distances which are large in comparison with the linear dimensions of the surface on which the forces are changed. The difference between the effects of two different but statically equivalent loads becomes very small at sufficiently large distances from load.

Saint-Venant’s Principle simply states that the stress measured at any point on an axially loaded cross section is uniform given that the measured location is far enough away from the point of load application or any discontinuity in the member’s cross section. In other words, when we calculate stress by conventional methods, i.e.,

σ = P / A

we have assumed that we are reasonably far away from the point of application or any discontinuity such that the normal stress is uniform.

In reality, when a point load is applied to a surface, the stress is concentrated at the point of application and eventually evens out as the distance from the point is increased. This increase in stress, also known as a stress riser, also occurs during abrupt changes in the material’s cross section.

Q3.2. Figure Q3, shows a steel bracket of an earth moving equipment. Perform FE analysis as indicated below on Creo 6.0 Simulate to observe stress strain distribution in the bracket due to estimated operational loads. You need to simplify the problem using the symmetry. It is advisable to create a coarse mesh first and then refine the mesh appropriately. Plate is 9 mm thick and P= 60 kN. E=200 GPa and Poisson ratio 0.3 for steel. You need to provide appropriate stress and strain contour plots for your answers. You need to refine the meshes appropriately.

Figure 8 STRAIN CONTOUR PLOT

Figure 9 STRESS CONTOUR PLOT

Perform a 3D solid FEA analysis for the bracket:

^{Q3.2 (a) Estimate the “Stress Concentration Factor” K}t ^{(Hoop Stress at A/Far field uniform} stress) at stress raisers areas identified by letter A (or C) and B.

Diameter of hole (D) = 10mm

Width of plate (W) = 200 mm

The value of the stress concentration factor is calculated by:

Stress Concentration Factor (K_t): 2.85

Q3.2 (b) Determine the maximum minimum principal stress and the locations.

Figure 10 MAX PRINCIPAL STRESS

Figure 11 MIN. PRINCIPAL STRESS

Q3.2 (c) Determine the local stress and strain in X and Y directions (XX, YY, XY and XY @45 deg) at points A (or C) and B

LOCAL STRESS

POINTS	XX	YY	XY	ZZ
A	9.58E+07	1.181E+07	1.212E+07	1.436E+06
B	1.492E+08	4.753E+07	4.069E+07	7.753E+06
C	9.58E+07	1.181E+07	2.640E+07	1.436E+06

Figure 12 LOCAL STRESS

LOCAL STRAIN

POINTS	XX	YY	XY	ZZ
A	1.913E-04	2.183E-11	2.00E-04	9.104E-05
B	9.546E-04	1.864E-04	9.003E-04	5.260E-05
C	1.913E-04	2.183E-11	2.00E-04	9.104E-05

Q3.2 (d) What is the safety factor? Provide your reasons.

In engineering, a factor of safety (FoS), also known as (and used interchangeably with) safety factor (SF), expresses how much stronger a system is than it needs to be for an intended load. Safety factors are often calculated using detailed analysis because comprehensive testing is impractical on many projects, such as bridges and buildings, but the structure’s ability to carry a load must be determined to a reasonable accuracy.

Perform 2D Plane Stress Analysis for the bracket:

Q3.2 (e) Perform 2D Plane stress analysis for the bracket.

Figure 13 2D PLANE STRESS ANALYSIS FOR THE BRACKET

^{Tabulate stress (XX, YY, & XY) & strain (XX, YY & XY) at points A (or C) & B}

	STRESS			STRAIN
POINTS	XX	YY	XY	XX	YY	XY
A	9.598E-07	1.561E+07	1.592E+07	3.618E-04	-2.785E-05	1.844E-04
B	1.885E+08	4.310E+07	5.458E+07	8.993E-04	1.735E-04	7.713E-04
C	8.814E+07	1.193E+07	1.209E+07	4.528E-4	-2.113E-05	1.531E-04

^{Estimate the “Stress Concentration Factor” K}t (Hoop Stress at the point/Far field uniform stress) at areas identified by letter A (or C) and B.

^{Stress Concentration Factor}

^{At Point A:-}

^{= 1.971e+08 / 8.829e+07}

^{= 2.23}

^{At Point B:-}

^{= 1.971e+08 / 1.856e+08}

^{= 1.06}

^{At Point C:-}

^{= 1.971e+08 / 8.468e+07}

^{= 2.32}

Q3.2 (f) Determine the maximum & minimum principal and Von Mises stresses at point A (or

C) and B.

Figure 14 Max. Principal stress for 2D stress analysis

Figure 15 Min. principal stress for 2D stress analysis

Perform 2D Plane Strain Analysis for the bracket:

Q3.2 (g) Perform 2D Plane strain analysis for the brackets.

Figure 16 2D strain analysis for the bracket

Stress (XX, YY, & XY) & Strain (XX, YY & XY) at points A (or C) & B.

	STRESS			STRAIN
POINTS	XX	YY	XY	XX	YY	XY
A	7.068E+08	1.769E+08	1.700E+08	3.020E-03	-2.238E-04	2.006E-03
B	1.590E+09	4.124E+08	4.572E08	7.959E-03	1.190E-03	6.479E-03
C	7.975E+08	8.587E+07	1.041E+08	3.372E-03	-7.542E-04	1.715E-03

Force P acting on this surface

Y

100 mm

Force P

50 mm

200 mm

X

C

Ø10.00 mm

Figure Q3 Bracket