Mechanics of Materials Laboratory
Pressure Vessel Stresses
A. Introduction and Objective
The objectives of this experiment are to compute hoop and axial stresses for a cylindrical pressure vessel and to transform the stress state to various directions; compute strains; and compare the strain values to measurements performed on a pressure vessel using instrumented with strain gages.
B. Theory
1.0 Cylindrical Pressure Vessel Stress State
The state of stress on a thinwalled cylindrical pressure vessel can be found by equilibrium equations and the assumption that the stresses in the wall thickness have a uniform distribution. The stresses in the hoop and axial directions are derived in the lecture.
The stresses in the hoop and axial directions are

o = p r t

o = p r 2t
(1)
(2)
where p is the internal pressure, r is the inside radius, and t is the wall thickness of the pressure vessel.
When plotted on Mohr’s Circle, these points are on the horizontal diameter of the 2 D circle, since there is no shearing stress in the axialhoop coordinate system—i.e. these stress are inplane principal stresses.
2D Circle
( 0 , 0 )
( sA , 0 )
s
( sH , 0 )
+t
2.0 Stress Transformations
Stresses can be transformed from the axial and hoop coordinate system through the use of stress transformation equations or Mohr’s circle.
sy’
tx’y’
sH

x sA
y’ x’
q
sx’
ave 2 4 t
α = 28
On the surface of the pressure vessel where o_{z} = 0 , the strains in the x’ – y’ coordinate system can be found through 3D Hooke’s Law:
s = osu — u oyu
su E E
s = oyu — u osu
yu
ysuyu
E E
= vsuyu
G
C. Lab Procedure
Turn on the FL 152 strain indicator. Ensure that it is set up for the FL 130 pressure vessel experiment. Check that you and anyone else in the area are wearing safety glasses.
IMPORTANT: Turn all adjustment knobs very gently to avoid damaging the pressure vessel.
Check that the piston handwheel is fully out for a closedended pressure vessel, and the relief knob is adjusted so that there is no pressure in the system. A diagram is attached at the end of this report. Take your initial zero reading for the gages and write them down in the table.
IMPORTANT: DO NOT EXCEED A MAXIMUM PRESSURE OF 30 BAR UNDER ANY CIRCUMSTANCES.
Close the pressure valve and increase the pressure in the pressure vessel and note the strain gage readings according to the attached table. It will only take a fraction of a pump to increase the pressure. Note any discrepancies between the target pressure reading and the actual pressure reading.
Raw Measurements Strain Gage Readings (micro‐strain)  
θ°  30  45  60  90  
bar (target)  bar (actual)  1  2  3  4  5 
5 

10 

15 

20 

25 

30 

IMPORTANT: DO NOT EXCEED A PRESSURE OF 30 BAR UNDER ANY CIRCUMSTANCES 
Release the pressure in the pressure vessel and note the reading of the strain gages.
Determine the net strain readings for each gage.
Net Adjusted Strain Gage Readings (micro‐strain) (Raw ‐ Average Zero)  
θ°  30  45  60  90  
MPa (target)  MPa (actual)  1  2  3  4  5 
0.0 

0.5 

1.0 

1.5 

2.0 

2.5 

3.0 

0.0 
Note: The material of the pressure vessel is aluminum: E = 70 GPa, ν = 0.33, G = (E/2)/(1+ν).
Note: wall thickness, t = 2.5 mm, outside diameter = 75 mm.

Lab Report
Check to see is the strain readings are linear with internal pressure: plot actual pressure on the yaxis and each gage strain reading on the xaxis. All of the gages will plotted on this same plot. Fit the data for each gage with a straight line (linear regression trend line). Note any deviations from linearity in your discussion.
Calculate the strain in the x’ direction for each gage direction and complete the table below for the calculated results using the highest actual pressure. Include a sample calculation for 30º and 60º.
Calculated Strain Gage Readings (micro‐strain) (from theory) at highest pressure  
θ°  30  45  60  90  
MPa (actual)  1  2  3  4  5  
Compare your calculated and experimental strain values at 3.0 MPa (or the highest actual value of pressure) and determine the percent error; complete the table below.
Comparison of Theory and Experiment‐‐Strain readings at highest MPa (microstrain)  
θ°  30  45  60  90  
MPa (actual)  1  2  3  4  5  
Theory 

Experiment 

% Error 
E. Questions
Why do we need to use 3D Hooke’s Law to calculate strains for this experiment?
F. Discussion
3 Description of the device
3.1 Layout of the device
The device is designed as a handy, compact benchtop device.
All components are mounted on a base plate, so a high degree of mobility is provided. The inter face circuit for the strain gauge with auxilary resis tors is placed inside the base plate.
11 10 9 8 7 6 5 4
12 13
Item  Name  Item  Name 
1  Relieve knob  8  Closure piston 
2  Hydraulic cylinder  9  Supporting collar 
3  Pump lever  10  Threaded spindle 
4  Manometer  11  Handwheel for piston adjustment 
5  Fixed lid  12  Base plate 
6  Strain gauge application  13  Connecting socket 
7  Cylinder 
Fig. 3.1 Device view
FL 130 STRESS AND STRAIN ANALYSIS ON A THINWALLED CYLINDER
Hydraulic pump
Pump lever
Oil filling screw
Safety valve
The cylinder is filled with hydraulic oil. The desired internal pressure is generated by way of a hand operated hydraulic pump and displayed on a manometer (4).
NOTICE
Do not exceed nominal pressure 30 bar.
The membrane can be overstretched and lasting deformed. The hydraulic system may develop leaks.
Fig. 3.3 Hydraulic pump
The safety valve is actuated at approx. 35bar. Hydraulic oil can be refilled via the oil filling screw.
Uniaxial and biaxial stress state
In the outer position of the piston (8) the pres sure on the front face is supported by way of the piston and a collar (9) bolted onto the cylinder. The biaxial stress state of the closed vessel applies.
Piston outside: closed
Piston inside: open
In the inner position of the piston the pressure on the front face is supported by way of the base frame. No load is placed on the cylinder in longi tudinal direction. The uniaxial stress state of the open pipe applies.
Fig. 3.4 Piston outer and inner positions
Cylinder with strain gauge
Fig. 3.5 Strain gauge application
Channel Angle Colour
Red
On the surface of the cylinder strain gauges (7) are arranged around the circumference. Two dia metrically opposing strain gauges form a diagonal halfbridge.
This prevents disturbance from overlaid bending stresses. A total of 5 halfbridges, with the varying angular positions of 0°, 30°, 45°, 60° and 90° rel ative to the cylinder axis, are specified.
By way of the connecting socket (13) the device is connected to the G.U.N.T. multichannel measur ing amplifier FL 152. In the multichannel measur ing amplifier channels 1 – 5 are occupied.
The measuring channels are assigned to the strain gauges as shown in Fig. 3.6. The angle 0° corresponds to the axial direction, and the angle 90° to the tangential direction.
Green
Blue
White
Yellow
Fig. 3.6 Arrangement of
measuring points
Mechanics of Materials Laboratory
Stress State from Combined Loading
A. Introduction and Objective
The objectives of this experiment are to compute a stress state for combined loading and the principal stresses that result from this stress state; measure strains experimentally with a strain gage rosette; calculate stresses from the measured strains and compute the principal stresses from the experimental data.
B. Theory
1.0 Stress State from Combined Loading
We have studied states of stress from varies types of loads as they are applied independently on a structure. The formulas for the stresses are summarized below:
o = ^{F}
Æ
o = –My
I
oh = p r , oa = p r
(axial normal stress) (1)
(longitudinal bending stress) (2)
(pressure vessel stresses) (3)
tw
v = VQ
It
v = Tq
Ip
2tw
(transverse shearing stress) (4)
(torsional shearing stress). (5)
For a state of combined loading, the stress states from these individual loads can be superimposed to determine the combined loading stress state. The principle of superposition can be applied as long as the system that you are analyzing remains linear—for our purposes in this class, that means that the material cannot have yielded—i.e. when load is applied and removed, the strains return to nearly the same initial value.
The following steps should be taken to determine the stress state at the point of interest on your structure:
 Make a freebodydiagram to determine the reactions (loads and moments) that act at the point of interest in the
 Determine how each reaction contributes to the stress state at the point of
Before these two steps, you can also use statics to simplify the structure by removing various loading arms and replacing them with a statically equivalent forcemoment system. Note that forces and moments must be resolved to the centroid of the cross section of the bar on which you are determining the stress state.
Figure 1: Structure subjected to combined loading.
After the stresses are calculated, principal stress can be calculated by using Mohr’s Circle.
2.0 Strain Gage Rosette Analysis
A strain gage rosette is a pattern of two or three strain gages. A strain gage is mounted on the upper surface of the combined loading structure as shown in Figure
 (The orientation of your gage may be different than shown in the figure.) The 0º 45º90º strain gage rosette can be used to determine a complete state of surface strain (including shear strain) by measuring extensional strains in three (noncolinear) directions. The strains are determined in the gage coordinate system.
Figure 2: Strain gage rosette.
Gage 1, 2, 3 are labelled in Figure 3. Gage 1 is the 0º gage aligned with the xaxis, gage 3 is the 90º gage aligned with the yaxis, and gage 2 is the 45º gage. The gage numbers are manufactured onto the gage but are partially obscured by the wires. The rosette xy coordinate system is also manufactured onto the gage and are shown highlighted on the figure. Note that the strain gage coordinate system will usually not line up with the axis of the tube.
Figure 3: Strain gage rosette and coordinate system.
The strain components in the strain gage coordinate system are given by the following equations:
s_{s} = s_{1} (6)
s_{y} = s_{3} (7)
y_{sy} = 2 s_{2} — s_{1} — s_{3} (8)
and the stresses in the rosette coordinate system are found by inverting generalized Hooke’s Law specialized for zero stress in the zdirection:
s = ^{o}x — u oy
(9)
s E E
s = oy — u ^{o}x
(10)
y
ysy
E cxy
= G
E
(11)
Finally, calculate principal stresses from the measured experimental strains are found by Mohr’s Circle.
C. Lab Procedure
IMPORTANT: The strain gage and wires are very delicate and are time consuming to install. Please handle the test structure carefully to avoid damage.
Attached the combined loading structure to the benchtop using bolts and nuts. Check that you and anyone else in the area are wearing safety glasses and closedtoe shoes. The main possible hazard during this lab is a dropped weight.
Use a small cable tie to loop through one of the holes on the loading arm if it is not already attached. Use a weight pan, and place a foam pad under the weight pan in case the weight pan fails.
Measure the distances l1 (from the load line to the centroid of the tube) and l2 (from the centroid of the tube to the center of the strain gage rosette) as shown in Figure 1. Note that the material is 6061T6 aluminum tube with an outside diameter of 0.625 inches and a wall thickness of 0.035 inches.
Attach the strain gage wires to the P3 strain indicator as a threewire quarter bridge as shown in Figure 4 below. Wire 1 should be attached to channel 1, wire 2 should be attached to channel 2, wire 3 should be attached to channel 3. Look for markings on the wires to indicate the wire number. Set the gage factor for each channel to 2.02 for test specimens 171, 172, 173, 174, 175. Verify the settings with the Lab Assistant.
Figure 4: P3 strain indicator.
Balance the strain indicator with the weight pan in place, but with no weights attached. Record the initial strain readings, and take strain readings for the weights listed in the table below. Remove the weights and take a final zero load reading.
Raw Measurements Strain Gage Readings (micro‐strain)  
weight (lbs)  1  2  3 
2 

4 

6 

8 

10 

IMPORTANT: DO NOT EXCEED 10 lbs WEIGHT 
Determine the net strain readings for each gage.
Net Adjusted Strain Gage Readings (micro‐strain) (Raw ‐ Average Zero)  
weight (lbs)  1  2  3 
2 

4 

6 

8 

10 

D. Lab Report
Check to see if the strain readings are linear with load: plot load on the yaxis and each gage strain reading on the xaxis. All of the gages will be plotted on this same plot. Fit the data for each gage with a straight line (linear regression trend line). Note any deviations from linearity in your discussion.
At 10 pounds weight, calculate the strain components in the gage coordinate system using equations (6), (7), (8). Calculate stresses through the use of equations (9), (10), (11).
Calculate the inplane principal stresses by the use of Mohr’s circle. Include these calculations in your lab report.
Calculate the stress state at the gage location using the theory—equations (1)(5) —as appropriate. Include a free body diagram and your calculations in your lab report. Use Mohr’s circle to determine the principal stresses.
Compare your calculated and experimental principal stress values at 10 pounds load and determine the percent error; complete the table below.
Comparison of Theory and Experiment  
10 lbs  Sigma_P1  Sigma_P2 
Theory 

Experiment 

% Error 
E. Questions
Why do we need to use 3D Hooke’s Law to calculate strains for this experiment?
Why don’t we use equation v =
VQ
It to calculate shear stress at the gage location?
F. Discussion
Last Updated on February 11, 2019 by EssayPro