Shear Force and Moment Diagrams
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1 C h a p t e r 9 Shear Force and Moment Diagrams In this chapter, you will learn the following to World Class standards: Making a Shear Force Diagram Simple Shear Force Diagram Practice Problems More Complex Shear Force Diagrams with Multiple Loads Complex Shear Force Diagram Practice Problems Drawing a Shear Force Diagram for a Simple Cantilever Truss Drawing a Bending Moment Diagram for a Simple Truss Draw Free Body, Shear Force and Bending Moment Diagrams 9-1
2 Making a Shear Force Diagram To determine the point where the supported truss is most prone to breakage, we use a shear force diagram to analyze the beam. The diagram starts at the zero point on both the left and right ends of the graph and when we see the data line cross the zero point somewhere in between the shear diagram, we see the point where the beam is weak. Designers and engineers need to examine the cross sectional area and the material at that point of the beam to determine if the truss is strong enough to bear the assessed load. For CAD specialist, the Shear Force diagram is another type of drawing that can help the engineer analyze an assembly. Before making the shear force or the moment diagram Figure 9.1 Simple Truss When we solve a simple supported truss, we will examine the problem in equilibrium. In the problem in Figure 9.1, we sum the forces in the Y direction, with the weight of 600 pounds acting in a positive direction and the two reactionary forces R1 and R2 repulsing the weight with in a negative path. With just this equation, we are unable to solve for the unknowns. We need to write a second formula, the sum of the moments where we scrutinize the forces acting around support A. Both equations are shown below: ΣFy = 600 lbs R1 R2 = 0 ΣMa = (600 lbs 10 ft ) (R2 20 ft) = 0 The sum of the moment around support A is the 600 lbs times 10 feet or 6000 ft lbs. Divide the 6000 ft lbs by 20 feet and we see the answer for R2 as 300 pounds. 9-2
3 R2 20 ft = 6000 ft lbs R2 = 6000 ft lbs 20 ft = 300 lbs We use the equation for the sum of forces in the Y direction to find R1, by plugging in the value of R2 into the formula. As shown below, we see the amount for R1 is 300 lbs. R1 = 600 lbs - R2 R1 = 600 lbs lbs = 300 lbs Below the beam in the CAD file, draw a 20-foot long line. Draw a line on the left end of the 20- foot long line that is 30 units high for the 300 pound force at R1. We are making each unit represent 10 lbs of force. Draw another line to the right 10 feet (120 inches). Draw the next line down 60 units for the 600 lb force. Draw the next line to the right 10 feet (120 inches). The last line goes up 30 units high for the 300 pound force at R2. We now observe our first shear force diagram. Figure 9.2 Simple Shear Force Diagram Whenever we make a truss diagram and solve for the unknown units, we now have the capability to draw a Shear Force Diagram directly below the first layout. By lining up the two diagrams, we can easily apply the changes in the Shear Force diagram in the correct locations. Many designers find the task easier, if they can use projection lines from the free body diagram on the top to draw the vertical changes on the Shear Force diagram. Always properly label the drawing and save the file in the project folder for future reference. Next, we have a series of practice problems to reinforce our methods. 9-3
4 Simple Shear Force Diagram Practice Problems In our CAD program, draw the truss to scale, placing the weight in the correct position. Solve for the unknown reactive forces. Draw a Shear Force diagram below each truss layout. Span (S) Length (L) Weight (W) R1 R2 1 25' 9' ' 18' '-9" 8'-2" '-10" 11'-6" '-6" 19'-2" 1050 Figure 9.3 Simple Truss Practice Problems More Complex Shear Force Diagrams with Multiple Loads As in the previous chapter, trusses and beams can have multiple loads acting upon the structure. When there is more than one load, the Shear Force diagram can become more interesting to draw and interpret. In figure 9.3, we have four loads on the beam, where one of the load is not a point load, but is spread over the length of the beam that supports the weight. 9-4
5 Figure 9.3 Simple Supported Truss with More Than One Load In the first formula, we add the three loads which are 950 pounds, 450 pounds and 500 pounds and subtract reactive forces at the supports R1 and R2. In the second equation, we have clockwise moments of 950 lbs 6 ft, 450 lbs ft, and 500 lbs feet. We measured to the center of gravity of the 500 pound weight to get feet. The last moment is in a counterclockwise direction and is R2 20 ft. ΣFy = 950 lbs lbs lbs R1 R2 = 0 ΣMa = (950 lbs 6 ft ) + (450 lbs ft ) + (500 lbs ft ) (R2 20 ft) = 0 First, we solve the sum of the moment equation, by finding the foot pounds quantity for each load and then divide by 20 feet to force R2. The value for R2 is lbs as shown below. ΣMa = (950 lbs 6 ft ) + (450 lbs ft ) + (500 lbs ft ) (R2 20 ft) = 0 R2 = 5700 ft. lbs ft. lbs ft. lbs 20 ft. R2 = ft. lbs 20 ft. = lbs. After solving for force R2, we utilize the sum of forces in the Y direction to find R1, by plugging in the value of R2 into the formula. As shown below, we see the amount for R1 is lbs. ΣFy = 950 lbs lbs lbs R1 R2 = 0 ΣFy = 950 lbs lbs lbs R = 0 R1 = 1900 lbs lbs = lbs 9-5
6 As you can see in figure 9.4, we draw both the free body diagram and the Shear Force diagram to line up with each other. Whenever there is a change in force on the top layout, we can observe the alteration in the graph below. Just up to the six-foot mark on the left side of the Shear Force graph, the reading is lbs, but at the 6-foot mark, the graph drops 950 lbs to lbs with the addition of the first load on the beam. A Shear Force diagram changes abruptly with point loads, however when loads are uniform as the 500 lbs on the right side of the truss, the slope of the graph is at an angle, changing incrementally along the area of contact. Figure 9.4 Shear Force Diagram for Truss with More Than One Load Always take the time to label the Shear Force diagram, because we might decide to make a CAD drawing unit equal 10 lbs or even 100 lbs. By indicating the points of change, the designer or engineer who reads your drawing file later will be able to interpret your data correctly. 9-6
7 Complex Shear Force Diagrams Practice Problems Figure 9.5 has five problems to compute truss and Shear Force diagrams when each case has three loads. Solve for the unknown reactive forces. Draw a Shear Force diagram below each truss layout. Span (S) Length (L1) Weight (W1) Length (L2) Weight (W2) Length (L3) Weight (W3) 1 24'-0" 3'-0" 700 4'-0" 375 6'-6" '-0" 3'-0" 900 5'-6" 150 3'-0" '-6" 2'-6" 350 2'-4" 225 1'-9" '-6" 2'-3" 640 5'-10" 410 2'-3" '-8" 8'-0" 720 3'-6" 605 4'-6" 375 R1 R2 Figure 9.5 Practice Problem for a Supported Truss with More Than One Load Drawing a Shear Force Diagram for a Simple Cantilever Truss When drawing a Shear Force diagram for a cantilever truss, we follow the same steps as before, doing the free body diagram by computing the sum of the forces in the Y direction and examining the torque in the sum of the moments. 9-7
8 Figure 9.6 Cantilever Beam with One Load In the first formula, we add the load, which is 375 pounds and subtract reactive force at the support R1. In the second equation, we have clockwise moments of 375 lbs feet. ΣFy = 375 lbs R1 = 0 ΣMa = (375 lbs ft ) = 0 First, we solve the sum of the moment equation, by finding the foot pounds quantity for the load. However, we see that the moment arm is ft lbs clockwise. Since the beam is static, back at point A, we will draw a moment with a negative ft lbs torque to counter the torque and keep the beam from twisting. ΣMa = (375 lbs ft ) = 0 ΣMa = ft -lbs After solving for sum of the moments, we utilize the sum of forces in the Y direction to find R1. As shown below, we see the amount for R1 is 375 lbs. ΣFy = 375 lbs R1 = 0 R1 = 375 lbs 9-8
9 Again, we return to our CAD layout and draw the missing moment arm at point A. For the sum of the moments to equal zero, the torque at the supported end of the beam will be opposite from the rotation of the load. In this case, we draw the ft-lbs counterclockwise as shown in Figure 9.7. Figure 9.7 Cantilever Beam with Moment Arm Drawing a Bending Moment Diagram for a Simple Truss To create a Bending Moment diagram, we begin with analyzing the truss with the free body diagram as we have done throughout this chapter and the last. Designers and Engineers use this graph to quickly analyze the design for bending moments. That is the force times the distance to the loads at any point of the beam. 9-9
10 Figure 9.8 Simple Truss Diagram When we solve a simple supported truss, we will examine the problem in equilibrium. In the problem in Figure 9.8, we sum the forces in the Y direction, with the weight of 415 pounds acting in a positive direction and the two reactionary forces R1 and R2 repulsing the weight with in a negative path. With just this equation, we are unable to solve for the unknowns. We need to write a second formula, the sum of the moments where we scrutinize the forces acting around support A. Both equations are shown below: ΣFy = 415 lbs R1 R2 = 0 ΣMa = (415 lbs 7 ft ) (R2 14 ft) = 0 The sum of the moment around support A is the 415 lbs times 7 feet or 2905 ft lbs. Divide the 2905 ft lbs by 14 feet and we see the answer for R2 as pounds. R2 14 ft = 2905 ft lbs R2 = 2905 ft lbs 14 ft = lbs We use the equation for the sum of forces in the Y direction to find R1, by plugging in the value of R2 into the formula. As shown below, we see the amount for R1 is lbs. R1 = 415 lbs - R2 R1 = 415 lbs lbs = lbs 9-10
11 Now, a bending moment diagram shows the total force times the distance or moment arm at any point along the length of the truss. To compute the quantities for this diagram, we can work from left to right as we did in the Shear Force diagram. For our beam, we will start at point A. At the initial end, the distance to R1 is zero, so the moment arm is equal to zero. Next, we will examine the beam at a location one foot to the right of point A. In our work across the beam, we will examine the forces and distances to the left as we move to the right. At the one foot mark we calculate lbs times 1 foot or ft-lbs. We continue to move to the right one foot at a time to compute the moment arm at each incremental position. Figure 9.8 Bending Moment Diagram for a Simple Truss Bending Moment diagrams are somewhat more difficult to draw than Shear Force graphs. A Bending Moment illustration is more likely to have sloping angles and curves, so we want to examine the data at smaller increments to get an accurate picture of what stresses are occurring in the truss. A good method to follow is to make a chart showing each calculation. There are textbooks that give solutions to a series of common applications and we can buy computer software that will do the analysis for even more difficult problems. We will make a chart to continue the assessment of the beam. We choose one foot increases across the truss, however if a force vector or uniform load starts at a point not on an even position, we have to examine that spot along the way. 9-11
12 Position Moment Arm Answer Point A lbs 0 ft 0 ft-lbs 1 foot lbs 1 ft ft-lbs 2 foot lbs 2 ft 415 ft-lbs 3 foot lbs 3 ft ft-lbs 4 foot lbs 4 ft 830 ft-lbs 5 foot lbs 5 ft ft-lbs 6 foot lbs 6 ft 1245 ft-lbs 7 foot lbs 7 ft 415 lbs 0 ft ft-lbs 8 foot lbs 8 ft 415 lbs 1 ft 1245 ft-lbs 9 foot lbs 9 ft 415 lbs 2 ft ft-lbs 10 foot lbs 10 ft 415 lbs 3 ft 830 ft-lbs 11 foot lbs 11 ft 415 lbs 4 ft ft-lbs 12 foot lbs 12 ft 415 lbs 5 ft 415 ft-lbs 13 foot lbs 13 ft 415 lbs 6 ft ft-lbs Point B lbs 14 ft 415 lbs 7 ft 0 ft-lbs Figure 9.9 Computing Bending Moments for a Simple Truss Draw Free Body, Shear Force and Bending Moment Diagrams Now that we know the methods for producing Free Body diagrams and we have the expertise to layout the Shear Force and Bending Moment graphs, we can create a drawing file and do all three together. Analyze the data given in figure 9.10 to find the points of interest in this truss. Figure 9.10 Draw All Three Diagrams; Free Body, Shear Force and Bending Moment 9-12
13 * World Class CAD Challenge * - Draw a Free Body, Shear Force and Bending Moment Diagram of the truss in Figure 9.10 and indicate the points of interest in 30 minutes. Save the file as Beam Analysis.dwg. Continue this drill four times using some beams problems you have determined, each time completing the drawing file in under 30 minutes to maintain your World Class ranking. 9-13
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