Structures ST Shainal Sutaria Student Number: 1059965 Wednesday, 14 th Jan, 2011 Abstract An experiment to find the characteristics of flow under a sluice gate with a hydraulic jump, also known as a standing wave is to be concluded. To acquire the properties of the hydraulic jump, a graph of the energy line in a rectangular channel through regions of gradually varied flow and rapidly varied flow is to be plotted. The value of the Froude number is also to be determined upstream and downstream of the hydraulic jump from the results. Thus the energy head loss can be calculated due to the jump. Then the theoretically estimated and measured values of downstream depth and energy head can be compared.
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Structures
ST
Shainal Sutaria
Student Number: 1059965
Wednesday, 14
th Jan, 2011
Abstract
An experiment to find the characteristics of flow under a sluice gate with a
hydraulic jump, also known as a standing wave is to be concluded. To acquire
the properties of the hydraulic jump, a graph of the energy line in a rectangular
channel through regions of gradually varied flow and rapidly varied flow is to
be plotted. The value of the Froude number is also to be determined upstream
and downstream of the hydraulic jump from the results. Thus the energy head
loss can be calculated due to the jump. Then the theoretically estimated and
measured values of downstream depth and energy head can be compared.
Contents
Page
Number
1. Introduction 3
2. Theory 4
3. Experimental Procedure and Results
3.1.1 Experimental Procedure – Objectives 5
3.1.2 Experimental Procedure – Apparatus and Method 5
3.2 Results
3.2.1 Test Data Tables 6
3.2.2 Example Calculations 6
3.2.3 Graph of Energy Head against Depth 7
3.2.4 Calculation of Loss of Specific Energy Head 8
3.2.5 Hydraulic Jump Classification 8
4. Discussion 9
5. Conclusions 10
6. References 10
1. Introduction
An experiment to find the buckling loads of a simple Euler strut is to be carried
out. To acquire the properties of the strut, values of the deflection of the beam
will be measured for various loads at the end of the strut. Also the deflection for
central loading of the strut for three different end loads also calculated.
Therefore we are able to calculate the objective of the experiment, the buckling
loads of the strut when loaded centrally and end loaded.
When calculating the deflection and buckling load in theory, the strut is
assumed to be initially straight. In reality there is and initial deflection and
curve to the beam. This makes the values obtained to differ from theoretical
solutions and more accurate.
2. Theory
For a loaded strut the magnitude of the critical load Pcr can be acquired
experimentally by considering a load-deflection curve, it is typically obtained
by drawing the horizontal asymptote to the curve. However this procedure has
numerous drawbacks. In reality the curve of the beam does not flatten, so the
calculated value of Pcr is generally measured on the final reading obtained and
the rest of the experimental values are not taken into account, thus an error is
incurred. A more accurate procedure for determining the critical load from the
experimental data within the elastic region was introduced by R.V. Southwell.
(Richard Vynne Southwell (born: Norwich 1888, died 1970) was an English
mathematician who specialized in applied mechanics. He studied in Cambridge
and in 1925, he returned to Cambridge University as a mathematics lecturer.
Then In 1929, he became a professor of engineering at Oxford University. In
1942, he became rector of Imperial College. He retired in 1948.) [2]
Consider an unloaded pin-ended strut. In theory before loading the beams
remains straight; however in reality this is not true. The strut initially has a
small curvature, such that at any value of the displacement, x the curvature is v0.
Fig. 1.
Assume that L
xav sin0
where a is the initial central displacement of the strut. This equation complies
with the boundary conditions that v0=0 when x=0, and x=L, and also d v0/dx=0
at x=L/2. Thus the assumed deflection is therefore practical.
Due to the initial curvature in the strut, a axial load P instantly create bending of
the beam and consequently additional displacements, v occur which is measured
from the original displaced position. The bending moment, M at any point in the
section is )( 0vvPM . (2.01)
When the strut is initially unstressed the bending moment at any section is
proportional to the change in curvature at that section from its initial
arrangement and is not its absolute value.
Thus2
2
dx
vdEIM , and hence )( 02
2
vvEI
P
dx
vd . (Note that P is not the buckling
load for the strut.)
Substituting for v0, we get.
L
xa
EI
Pv
EI
P
dx
vdsin
2
2
(2.02)
The solution of this equation is,
L
x
L
axBxAv
sinsincos
2
2
2
2
,
(2.03)
in which, EI
P2 .
From the initial boundary conditions in equation (2.01), where v=0 at x=0, we
get A=0 and v=0 x=L we get LB sin0
The strut in this case is in stable equilibrium as only bending is involved, sinμL
cannot be zero.
Therefore B=0, which gives the equation,
L
x
L
av
sin
122
2
,
(2.04)
Since A=0 and B=0
Substitute from above equations thatEI
P2 and
L
xav sin0 into equation
(2.04) we get the equation
12
2
PL
EI
vv o
Now crPL
EI
2
2, is the buckling load for a perfectly straight pin-ended strut.
Hence
1
0
P
P
vv
cr
(2.05)
The effect of the load P will increase the initial deflection by a factor of
1
1
P
Pcr
As P approaches Pcr, v tends to infinity. This is not possible in practice as the
strut would breakdown before reaching Pcr. Due to this we consider the
displacement at the mid-point of the strut
1P
Pav cr
c
(2.06)
This can be rearranged to give
aP
vPv c
crc
(2.07)
This is in the form of a straight line equation cmxy , which represents a linear
relationship between vc and vc/P. A graph of vc against vc/P will produce a
straight line as the critical condition is approached. The gradient of the straight
line represents Pcr and the intercept on the vc axis is equal to a (initial central
displacement). This graph is acknowledged as a Southwell plot.
When a strut is loaded centrally and without an end load, the strut becomes a
simply supported beam. Thus the mean deflection can be calculated as,
EI
WLdeflection
48
3
, where W is the weight of the central load, L is the length of the
beam, E the Young’s Modulus and I the second moment of area. Due to the
graph of load against deflection being a straight line, EI
WL
48tan
3
. Therefore
tanθ=deflection. (where θ=angle between deflection angle and the line of the
graph).
Fig. 2.
Part 2:Unsymmetrical Bending
2.2 Unsymmetrical Bending
If the plane containing the applied bending moment is not parallel to the
principal axis of the section, the simple bending formula cannot be applied to
find the bending stress. For unsymmetrical sections the direction of the principal
axes has first to be determined. In this experiment a beam made up of angle
section is used and it is required to find the principal axes.
Let,
OX and OY – Perpendicular axis through the centroid.
OU and OV – Principal axis.
δA is a elemental area at distance u and v from the OU and OV axis
respectively.
The rectangular second moment of area IUV is given by (13)
Where,
u = x cos θ + y sin θ
and v = y cos θ – x sin θ
Thus giving (14)
The condition for a principle axis state that IUV = 0, Thus
(15)
Tan2θ =
The second moment of area for the X and Y axis has to be determined. This can
be done by the use of the following equation: (16)
Ix=
Iy=
Ixy=bdxy
Where x and y – in this case is the distance from the surface of the beam to the
neutral axis.
b – The breadth of the strut
d – The Depth of the strut
Ix and Iy – Second moment of area.
2.3 Shear centre
When a beam or section does not have a vertical axis of symmetry, then the
horizontal shear stressed on the beam are not in complete equilibrium, they will
create a couple causing the section to twist. The twist leads to the development
of torsional stresses in the member. The shear centre is the point where the
torsional stresses are zero, and only bending occurs. In a channel section, there
is no vertical axis of symmetry, and so, the shear centre needs to be determined
For a channel section under a shearing force, F, at a distance d from the centre
of the web, the shearing stress at any point is given by:
=
(3)
This can help derive the equation for d, the distance from the centre of the web,
which will give the position of the shear centre [at d, there will be no torque,
just pure bending]:
d
3. Experimental Procedure and Results
3.1.1 Experimental Procedure – Apparatus and Method
(i) For the first experiment a strut of aluminium extruded section was set up,
with one end pinned and the other end free for loading weights.
(ii) The length of the strut was measured three times and the average
recorded. The width and thickness were taken in three different positions
using a micrometer, and then the mean calculated for each.
(iii) A dial gauge was placed at the centre of the strut to measure the central
deflection of the beam (the dial gauge had an accuracy of 0.01mm).
(iv) The strut used had a Young’s Modulus = 70 GN/m2 (10 x 106 lb/sq.in).
(v) The dial gauge was set to measure the central deflection of the strut,
taking readings of the deflection for loads of 0-380N, in increments of
100N from 0-200N, 5N from 200-300N, 20N from 300-360N and 10N
from 360-380N. Then the deflection was also recorded on unloading in
the same increments. From this the mean deflection for each load could
be found.
(vi) Now a graph of load P against central deflection can be plotted, and
hence determine the buckling load.
(vii) From the calculated results of P and vc, plot a graph of vc/P against vc.
Hence determine the buckling load and the initial deflection.
(viii) For the second experiment the strut was now loaded centrally and an end
load P applied, the strut.
(ix) For the first central loading no end load was applied, therefore P=0kg.
Loads were then applied centrally for loads of 0-12N in increments of 1N
each time, and the deflection noted on loading and then on unloading
also. From this the mean deflection for each load could be found.
(x) Now the procedure will be repeated a further two times for end loads of
10kg and 20kg.
(xi) A graph of W against deflection can now be plotted for all three loads
(0,10,20kg) on the same graph
3.2 Results
3.2.1 Test Data Tables
Length of strut, L (mm)
1 2 3 Mean
990 990 990 990
Width of strut, b (mm)
1 2 3 Mean
25 25 25 25
Thickness of strut, d (mm)
1 2 3 Mean
6.40 6.37 6.38 6.38
Table 3.2.1.1
Second moment of Area of Strut, I (mm4) = 541.03
E = 70x109 N/m2
381.37
N
Table 3.2.1.2
End Load (P) Central Deflection (mm) Deflection, Vc