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Telford and The Flat Form Bridge

CEE 102 Structures Concepts   03/06/02

CEE 102 Structures Lab 1B   03/06/02

CEE 102 Structures Lab 1B   03/06/02

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

    1. For most parts of the course, we use historical units, eg. Pounds, feet, etc.
    2. Use consistent units when adding, multiplying, etc.
    3. 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 

    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. 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. 
    7. 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.
    8. 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.
    9. 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.
    10. 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.
    11. 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?
    12. 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.

    1. Remove all weights from the hanging chain, and then remove the chain from the hook at (-12,0) and (12,0). 
    2. With a wet paper towel, remove all marks from the acetate.  Be sure to dry the board after it has been cleaned. 
    3. 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).
    4. Suspend the chain from the hooks at (-12,0) and (12,0) so that the sag is 12 inches.  
    5. 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.    
       
       
       
       
       
       

             
      Weight (ounces) X Coordinate Y Coordinate  
        -9    
             
        -6    
             
        -3    
             
        0    
             
        3    
             
        6    
             
        9    
             
     
     

                             Table 2: Data Table for Part II of the Cable Experiment 

  1. 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. 

  1. 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.
  2. 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.
  3. Repeat this procedure with the next largest diameter of steel wire. 
  4. Repeat again with the largest diameter of steel wire. 
  5. Using Excel, plot the tensile capacity (breaking load) of the wires against the cross-sectional area.
  6. 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)

Steel

       
Steel        
Steel        

Table 3: Data Table for Cable Strength Experiment 

 

 


 

 

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