Cable
Shape & Strength: Telford and The Flat Form Bridge
1826 Menai Straits Bridge
by Thomas Telford
The purpose of this laboratory
is to understand important features of bridges that help to determine
their form and cost. Thomas Telford's greatest work, a cable suspension
bridge, is shown above. Telford introduced the flat form for cable
and arch iron bridges. The Menai Straits Bridge is quite flat
with a cable sag (d) of only 43 feet for a span (L) of 579 feet.
The form factor (L/d) is 13.5. Besides a flat form, two factors
affecting the appearance and cost of this suspension bridge are the
shape and thickness of the cable. This laboratory will explore
both of these aspects.
Cable Shape
The correct shape of the main
cable of a suspension bridge is parabolic. The parabolic shape
results when the total weight (deck & cable) is distributed uniformly
along the span. You will determine the correct cable shape in
the laboratory by hanging weights at regular horizontal intervals along
a cable and by recording the cable sag at the location of each weight.
Your measurements will be compared to the computed form of a parabola.
V
One of the attractions of working
with cables is that the direction of force can be determined by visual
inspection. This occurs because a cable has strength only in tension.
It does not have any strength to resist shear or bending. As a
result, all forces are axial, that is, all forces are directed along
the cable. One can therefore see how the bridge works. The deck
is supported by hangers which bring the deck load to the cable. The
hangers are supported by the cable which brings the deck load to the
towers. And the cable is held up by the towers through vertical and
horizontal reaction forces at the top of the towers. The vertical
force goes directly through the towers to the ground whereas the horizontal
force is taken by anchorage with which the cable is fixed at some distance
from the towers. All of these forces are shown below.
V
H
H
Forces in a Cable Bridge
Arch bridges are similar to
cable bridges. The arch form is the complement of a cable form.
The cable and the arch differ since the cable is in pure tension and
the arch in pure compression. Horizontal abutment forces are now
inward, in contrast to the tower forces in cable bridges that were outward.
The correct form for a uniformly distributed deck load is again a parabola,
but this time with the parabola arcing downward.
H
H
V
V
Forces in an Arch Bridge
Telford's designed many arch
bridges, the earliest which survives is the Craigallachie Bridge
built in Scotland in 1814 as shown below.
1814 Craigallachie Bridge by
Thomas Telford
Telford's arch is essentially
a correct form. Its span is 150 feet. This form is to be contrasted
with the Roman-Arch semicircular form of the first metal bridge, Iron
Bridge built by Abraham Darby III in 1779. Iron Bridge has a 100
feet span which extends over the River Severn in England as shown below.
1779 Iron Bridge by Abraham
Darby III
Because it is the wrong form,
Iron Bridge's arch experiences a bending moment and as a result it must
be stiffer, more massive, and more expensive than it would be if it
were of the correct form.
If the load of the deck were
not uniformly distributed it would be possible for the correct form
of the arch to be semicircular rather than parabolic. To understand
this concept you will in laboratory figure out how to adjust the weights
in the hanging cable such that the cable takes on a circular form.
If this same weight distribution were used in an arch supported deck,
the arch would be semicircular and in pure compression. The bridge therefore
could be made with less material at a lower cost and still preserve
the semicircular form.
The Romans knew that the semicircular
shape was wrong. They used this shape nevertheless because it
was easier to manufacture. In the Roman arch stone segments, known
as voussoirs, which were all cut at the same angle are used to construct
the bridge (see below).
By loading the arch with an
uneven distribution of rubble the Romans could cause the voussoirs to
be in pure compression. This is a clever solution to a difficult problem.
Cable Strength
The laboratory will also explore
the breaking strength of steel. The strength of material is the
determining specification that sets the minimum safe cable cross sectional
area. The minimum safe cable cross-section is a factor affecting the
cost and appearance of the cable suspension bridge.
You will test out three different
steel cables and subject them to increasing tension loading until they
break. By recording both the breaking load (in lbf) and the area
(in square inches), you will be able to calculate the breaking stress
(in lbf / sq. in. which is often abbreviated "psi") for each
of the three wires. If all goes well you will find that the breaking
stress in all three cases will be the same. The breaking stress
is a material property. The stronger the cable material, the smaller
the cable diameter that is needed. The Menai Straits Bridge was
constructed of wrought iron which has a breaking stress of about 30,000
psi. Modern high strength steel has a breaking stress of about
250,000 psi. A minimum diameter steel cable therefore could support
the deck with a more transparent appearance than wrought iron.
A steel cable would be about or about 1/3rd of the diameter
of an iron cable supporting the same load.
Important
Equations
Variables:
L = span (ft, m)
d = sag (ft, m)
Q = Total load (kip, N)
q = distributed load (lbf/in,
N/m) = Q/L
qL
= applied live distributed load
qD
= distributed dead load of structure
As = cross sectional
area of a cable (in2, cm2)
f = stress in cable at midspan
(kip/in2, N/cm2)
S.F. = safety factor
Equations
For a load, q, distributed
uniformly along the entire span:
= horizontal reaction (kip, N), constant along bridge
=
vertical reaction (kip, N), varies linearly along bridge
Q = qL
As =
/ 4 * (diameter)2
f = H / As
S.F. =
breaking stress
allowable stress
Useful Conversions
1 kip = 1000 lbf
1 lbf = 4.448 N
1 inch = 2.54 cm
1 ft = 0.3048 m
100 cm = 1 m
16 ozf = 1 lbf
Notes
- For most parts of
the course, we use historical units, eg. Pounds, feet, etc.
- Use consistent units
when adding, multiplying, etc.
- Percent error calculation
= theoretical – experimental * 100%
theoretical
APPARATUS
This experiment
is in three parts. The first two parts, as seen in Figure 1, involve
measurements using a board with hooks along the top. A chain supported
by these hooks will be loaded in various ways using weights. The
third part, as seen in Figure 2, involves a pail and a container that
unloads steel beads. The pail hangs from a sample of the material
to be tested, thus putting the wire in tension. A white safety rope
supports the pail after the sample has failed. The pail filled
with steel beads is then weighed on a large scale.
Figure
1: Cable Shape Apparatus
Figure 2:
Cable Breaking Apparatus
PROCEDURE AND CALCULATIONS
Part I:
The shape of a cable under uniform loading
The first part
of the lab uses the cable shape apparatus to derive the shape of a cable
under uniform horizontal loading. Uniform loading is characteristic
of the conventional cable suspension bridge as shown in Figure 3.
The cable shape apparatus board is covered with graph paper incremented
in tenths of an inch.
Figure
3: Shape of Cable in a Suspension Bridge
The ��cable�� in this experiment
is a length of chain. You will measure four chain configurations.
Chain configurations correspond to 1, 3, 5, and 7 equal loads hung at
equal intervals in the X direction as shown in figure 4 below.
In each configuration the chain will be adjusted in length such that
the sag at mid-span is kept constant. X and Y values of the chain
at the supports and at the location of each load will be recorded with
the help of a video camera and Excel.
N=7
N=5
N=3
N=1
Figure 4:
Four Loading Conditions for Cable Shape Experiment
- The graph origin
is (0,0). Positive x values lie to the right of this point,
negative x to the left and negative y below. The
x-direction measurements relate to the span of a bridge and the
y-direction measurements relate to the sag of the chain. Any
red marks on the acetate from previous measurements should be removed.
Do this with a damp paper towel.
- Suspend the chain
from the hooks at (-12,0) and (12,0) so that the sag of the chain at
mid-span is about 11 inches and several links hang over each end.
The shape of the chain under only the load of its own weight is a curve
known as a catenary.
- To approximate a
uniform load, you will hang clusters of n loads (4-ounces
each), as seen in Figure 4. Start with the entire span as
one segment, N = 1. The center of this segment is the origin,
x = 0. Hang a 4-ounce load near the midpoint of the chain
using one of the long suspender wires between the chain and weight.
The suspender wires allow the weights to hang freely below the board.
The shape of the chain will change from a catenary to a triangle. Adjust
the length of the chain and the position of the load so that the lowest
point on the chain is as close as possible to coordinate (0, -12). This
corresponds to a sag of 12 inches. The 12 inch sag must be maintained
throughout the experiment.
- You will next record
the shape using Microsoft Excel. Open the file Cable_Shape_Experiment.xls
from the desktop. When asked, enable macros. Once this file is open,
save the file WITH A NEW FILE NAME in your lab group��s folder located
within the CEE 102 or CEE 262 – Lab Groups Data folder. Press
��ctrl-t�� and the macro will open up a UserForm Window and the ��VidCap��
Window. Do not click within the image window of the UserForm as this
will load arbitrary ordered pairs into the spread sheet. Select
the UserForm Window and initialize it by clicking on the ��load image��
button. Now press the button labeled ��1 weight��. The UserForm window
should go blank with the Windows Media logo in the center. Now switch
to the VidCap Window. The ��VidCap�� program enables the video
camera. ��VidCap should be in the ��Preview�� mode. To
check this, wave in front of the camera and see if you can see it on
the computer screen. If not, pull down the ��Options�� menu
in ��VidCap�� and select ��Preview��. To capture an
image select the ��Capture�� – ��Frames�� from the pull down menu.
Then press ��Capture��. The frames count should advance from
0 to 1. Press the ��Close�� button. VidCap should remain
open during all of your measurements. Move the VidCap Window to
the side until later or ��Minimize�� it.
- To transfer the
image that you have just captured to the UserForm, click on the UserForm
button labeled ��Load Image��. The image that you captured in
VidCap should now appear in the box that was previously black.
Move the cursor to any point on the chain and note that the X and Y
positions of the cursor appears in the white textboxes below the image.
Place the cursor pointer finger on the upper left point of support of
the chain (location (-12,0) on the graph paper) and click the left mouse
button. The X and Y positions of the cursor will be transferred
automatically to the Excel Worksheet 1. Continue with ��cursor
place and mouse click�� at the bottom of the chain (location (0,-12)),
and then at the upper right point of support (location (12,0)).
Look at Worksheet 1 to see if three table entries have been made.
You have now completed the first measurement of the cable shape with
one weight. BE VERY CAREFUL NOT TO MOVE THE COMPUTER OR THE CAMERA FOR
THE REMAINING MEASUREMENTS. IF THE CAMERA GETS BUMPED, ALL MEASUREMENTS
WILL HAVE TO BE REPEATED.
- Next hang two more
4-ounce weights on the chain with suspender wires (N = 3).
Place them so that the three loads freely hang as close as possible
to x-coordinates of –6, 0, 6 respectively. Also, adjust
the chain length by adding a link or two at the support hooks so that
sag at mid-span is as close to twelve inches as possible, with the lowest
point on the chain in the same position, as with N=1 (see the horizontal
dashed line in Fig 4). Reposition the loads on the chain as required
so that they freely hang as close as possible to their respective
x-coordinates of –6, 0, 6. Adjust chain length as required to
maintain the 12 inch sag. This may take some trial and error.
To record the positions of the three load points on the chain and the
two points of support, press the ��3 weight�� button on the UserForm.
The image should change to black. THIS STEP IS VERY IMPORTANT – IF
YOU DO NOT HAVE A BLACK IMAGE, THE VIDCAP PROGRAM WILL GENERATE AN ERROR
DURING CAPTURE. Switch to the VidCap Window and capture another image
as you did in the previous step. After ��Capturing�� one frame
and ��Closing��, return to the UserForm Window and load the image.
As you ��Point and Click�� the data will be recorded in the ��D��
and ��E�� columns of the Worksheet.
- Repeat the above
step with N = 5. The five weights should freely hang at x-coordinates
of -8, -4, 0, 4, 8. You will need to adjust the chain again so
that the sag at mid-span is maintained at 12 inches. Press the
��5 weight�� button. Record the positions as before.
- Repeat once again
with seven weights. The seven weights should freely hang at
x-coordinates of -9, -6, -3, 0, 3, 6, 9. Don��t forget to press
the ��7 weights�� button before running VidCap.
- Close the UserForm
and VidCap Windows. Graph your results by first highlighting every row
and column within of X and Y positions of the seven-weight case.
- Then left click
on the chart wizard icon, the chart type window should appear.
Select XY (Scatter) as your chart type by left clicking on
XY (Scatter) and then left click on Next >. The
graph should appear in the following window (Chart Source Data).
Left click on Next > for the Chart Options window to appear.
Give the graph a title and be sure to label the x and y axes.
Left click on Next > to move to the final chart wizard window.
In this window, left click on the bubble beside As new sheet,
set the new sheet to Chart2 and then left click Finish. Your
graph should appear on the screen.
- Use the trend-line
function in Excel to find the formula of the final chain shape.
Right-click on a data-point and choose ��Add trend-line��, select
polynomial and order 2, then ��Display equation on chart�� from Options.
What is the shape of the final chain? What does the inverted
shape of this cable resemble? What cable shape would you expect if you
were supporting a bridge deck by a very large number of suspenders?
- Finally look at
Chart 1 which has been set up to display all of the data for the 4 configurations.
Note that all data points should lie on a universal curve – i.e.,
a parabola that goes through points (-12,0), (0,-12), and (12,0).
Part II:
The Semicircular Form
The second part of
the lab will again use the board apparatus; however, now you will distribute
the weights on the chain
so that the shape of the chain is semicircular. This experiment will
illustrate that a cable can only take on a circular form when
loaded with a non-uniformly distributed load.
- Remove all weights
from the hanging chain, and then remove the chain from the hook at (-12,0)
and (12,0).
- With a wet paper
towel, remove all marks from the acetate. Be sure to dry the board
after it has been cleaned.
- Next you will draw
a semicircle on the board with the help of a flat metal compass from
the hook at (0,0). Place the tip of the pen in the compass hole
and sweep out a semicircle. The semicircle should extend from (-12,0)
to (12,0) and pass through (0,-12).
- Suspend the chain
from the hooks at (-12,0) and (12,0) so that the sag is 12 inches.
- Using any combination
of 4, 2, or 1-ounce weights, distribute the weights so that the shape
of the chain mirrors the drawn semicircle. Place the weights at
the X-locations –9,-6,-3,0,3,6, and 9. To achieve close to a
semicircular shape the distributions of weights will no longer be uniform.
6. The
chain should now have a circular form. Record the position and weights
of each hanging weight.
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Weight (ounces) |
X Coordinate |
Y Coordinate |
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-9 |
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-6 |
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-3 |
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0 |
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3 |
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6 |
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9 |
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Table 2: Data Table for Part II of the Cable Experiment
- Where must the majority
of the total load be placed on a suspension bridge in order for its
cable to take on a semicircular shape? Are suspension bridges ever
subjected to this loading scenario?
Part III:
The strength of a cable
The third part
of the lab uses the pail apparatus to determine the tensile load necessary
to cause failure of a cable. From this we can determine the breaking
strength of the material. Three different diameters of steel cables
will be tested.
- Measure the diameter
of all three samples using calipers. Ask the instructor to verify the
diameter. Determine the cross-sectional area A of each sample, and record
both sets of values in Table 3.
- Wear safety glasses
while using the pail apparatus.
There are two thumb screws on the apparatus. The top thumb screw has
a gray knob, and the bottom thumb screw is attached to the bucket handle.
Snap the bucket��s carabiner to the apparatus with the head of the
thumb screw pointing to the right. Turn both thumb screws counterclockwise
until they stop. Starting with the smallest diameter wire, cut a piece
of wire 14�� long. Stick one end of the wire into the hole in the side
of the bottom thumb screw and turn the screw clockwise until it stops
while keeping light tension on the wire. Do the same with the other
end of the wire and the top thumb screw. Pull sharply upwards on the
black knob, and immediately move your hands away from the apparatus
when the steel beads start pouring into the bucket. Keep hands away
until the wire breaks. Notice that as the bucket weight increases
the wire begins to stretch, and as it elongates it also gets visually
thinner. Use caution because the wire snaps catastrophically. When the
wire breaks, the safety rope should catch the pail and the steel beads
should stop pouring out. Weigh the bucket and record this value T in
Table 3. Pour the steel beads back into the top of the apparatus.
- Repeat this procedure
with the next largest diameter of steel wire.
- Repeat again with
the largest diameter of steel wire.
- Using Excel, plot
the tensile capacity (breaking load) of the wires against the cross-sectional
area.
- Calculate the stress
of each cable using the formula f=T/A. Should these values be
the same for all three samples? How different are they?
Material |
Diameter
(in)
|
Cross-Sectional
Area A (in2) |
Breaking tension,
T (kip) |
Stress, f
(k/in2)
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Steel |
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Steel |
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Steel |
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Table 3:
Data Table for Cable Strength Experiment