Bridge Force Analysis: Method of Joints

Transcription

1 Bridge Force Analysis: Method of Joints A bridge truss is made up of single bars, which are either in compression, tension or no-load. In order to design a bridge that is stable and can meet the load requirements, it is necessary to calculate the force on the member. The method of joints is one method. It is based on Newton s First Law and uses moment of forces. Newton s First Law: Static system F x,y = 0 or F x, = 0 and F,y = 0 The bridge is not in motion (or static), therefore the forces in both the x and y direction are balanced. Newton s Third Law For every action, there is an equal and opposite reaction. The statement means that in every interaction, there is a pair of forces acting on the two interacting objects. The size of the forces on the first object equals the size of the force on the second object. In the case of bridge, the structure exerts a force on the supports. The supports exert an equal and opposite force, called the reaction or support force, on the structure. Moment of Force: (often just moment) is the tendency of a force to twist or rotate an object. A moment is calculated as the product of the force and the moment arm. m = 0 Again, since the bridge is not in motion, there is no rotation so the moments at any node is 0. Reaction Forces Background for Moments Example: Imagine two people pushing on a door at the doorknob from opposite sides. If both of them are pushing with an equal force then there is a state of equilibrium. If one of them would suddenly jump back from the door, the push of the other person would no longer have any opposition and the door would swing away. The person who was still pushing on the door created a moment.

2 Elements of a Moment The Moment of a force is a measure of its tendency to cause a body to rotate about a specific point or axis. In the diagram on the left, the two forces, represented by the arrows are on the same line. The magnitude of forces are equal. There is no motion nor a tendency to twist. In the diagram on the right, the magnitude of the forces are again equal, but not on the same line. There is a tendency for the bar to rotate or twist around the lower point. Moment = Force x Distance or M = (F)(d) The magnitude of the moment of a force acting about a point or axis is directly proportional to the distance of the force from the point or axis. It is defined as the product of the force (F) and the moment arm (d). The moment arm or lever arm is the perpendicular distance between the line of action of the force and the center of moments. A moment is expressed in units of foot-pounds or newton-meters. The Center of Moments may be the actual point about which the force causes rotation. You can select the most convenient point to calculate the moment. Convention for Signs A moment also has a sense of direction. In doing problems, it is important to be consistent. A counter clockwise rotation about the center of moments will be considered a positive moment; A clockwise rotation about the center of moments will be considered negative. Counter Clockwise (Different reference sources may have different conventions.) Example from the Playground The dogs are balanced on the see-saw. Consider the pivot point. Since there is no rotation: m pivot = 0. Find the weight of the dog on the right side of the see-saw.

3 Example from Physics: Predict the forces acting on a bridge structure. The apparatus below is identical to a beam bridge with an off-center load of weight W. The bar is supported by spring weights at Positions 1 and 2. Problem is to determine the forces R1 and R2 The Forces are represented by R1 and R2 (R is used for reaction forces for bridges, F is used for forces in Physics) R 1 R 2 x 1 x 2 W Suspend a 3 N weight from the metre rule as shown. Set x1 = 0.2 m and x2 = 0.6 m. Determine the forces R1 and R2. Ignore the t the weight of the ruler The bar is not in motion so that the equilibrium conditions are met. There are no forces in the x direction. (The sign convention is that upward is positive. In order for equilibrium R1 and R2 are in the opposite direction than the W. Conditions for equilibrium: Fy, = 0 0 = R1 + R2 -W or R1 + R2 = W = 3N (Eqn 1) There is also no rotation so m = 0 Select a point to calculate the rotation. In this case point 1. The moment arm for R1 =0 since that is the point of rotation. The moment arm for W is x1 and The moment arm for R2 is x1 + x2 m1 = 0 = R1 (0 ) + x1 W - R2(x1 + x2) 0 = 0.2m (3N) R2(0.2m +0.6m) R2 = 0.2m (3N)/ (0.2m +0.6m) = 0.75N Since from Eqn 1 R1 + R2 = W = 3N; R1 = 3.0N 0.75 N = 2.25N R2 = 0.75 N and R1= 2.25 N Repeat the problem using the point at which the weight is suspended as the center of moment: mw = 0 and get the same answer.

4 Example for Bridges: Reaction Forces Find the Reaction forces for the bridge shown below. Bridge Diagram Note that the moment arm from A to D is the perpendicular distance between the line of action of the force and the center of moment A and is 1m Force Diagram No Motion R x, = 0, R,y = 0 M = 0 Point A: Fixed Node: Reaction forces in both x and y direction. Point B Rolling Node: Reaction Force only in y direction. Select A as the point of rotation: Moment arm at A: 0 m Moment arm at D: 1m Moment arm at B: 2m M = 0 = -10N*1m =2m*R by so R by = 5N R x, = 0 = R ax (no forces in x direction) R,y = 0 = R ay + R by - 10N so R ay = 10N - 5N = 5N

5 Procedure for Analyzing Truss Method of Joints The previous example showed the first steps in the Method of Joints used to calculate the forces on each member. If the truss is in equilibrium, each of its joints is in equilibrium, Write the equations of equilibrium for each of the joint. R x = 0, R,y = 0, m = 0. Method of Joints 1. Determine the support reaction forces if possible. 2. Draw the free body diagram for each joint. 3. Write the equations of equilibrium for each of the joint. F x = 0, F,y = 0, m = If possible, begin solving the truss at the joint where at least two or more unknown forces are acting, this helps out in solving the truss, otherwise it is not a necessary step. 5. Always assume the unknown forces acting on members to be in tension. 6. Solve the joint equations of equilibrium simultaneously, typically using advanced calculator or a computer. First, Determine the Support forces on each truss Ay and Cy

6 The truss has a fixed node at A and a rolling node at C. The problem is similar to the previous example, except that now the applied force is in the x direction. The point of rotation will be selected as Point A. The forces at A and C are then calculated as shown below.

7 Compare to JHU Simulator This result agrees with the JHU simulator result. Below, a load of 50 lb. at Point A and Point B was used, so the forces on the members are reduced by a factor of 10. Point B Point A Point C FAB = 49 lb f FAC = 50 lb f FBC = 70 lb f Since both the calculated result and the JHU simulator used the same equations, what is the reason for the small difference?

8 Problems 1. Determine the reaction forces at H and I. Note that H is on a fixed node and I is on a rolling node. Review carefully, the definition of the moment above and determine the line of action of the force. 3 lb f 3 lb f 3lb f 3 lb f (MIT) 2. Determine the reaction forces at Points A and B: Rax, Ray, Rbx, Rby Show all steps, equations, calculations and signs. (Note that A is on a fixed node and B on a rolling node.) 3. Determine the reaction. forces at points A (fixed node) and D (rolling node). mathlin

9 Bridge: Stability/Calculation by Method of Joints Stability The resistance offered by a structure to undesirable movement like sliding, collapsing and over turning etc. is called stability. Stability depends upon the supports conditions and arrangements of members. Stability does not depend upon loading. A structure is said to be stable if it can resist the applied load without moving or A structure is said to be stable if it has sufficient number of reactions to resist the load without moving. A structure which has not sufficient number of reactions to resists the load without moving is called unstable structures. For complete stability the should be both internally and externally stable External Stability: A truss or structure is externally unstable: Reaction forces are all parallel Reaction forces are all concurrent passing through the same point. Internal stability of a truss can often be checked by careful inspection of the arrangement of its members (by kinematical analysis). If each joint is fixed so that it cannot move in a rigid body sense with respect to the other joints, then the truss will be internally stable. Example: The four member structure below will collapse unless a diagonal member is added for support, such as bar Therefore, a simple truss is geometrically stable when is constructed by having a basic triangular element such as and adding members to form an additional element. (2011 S. Parvanova, University of Architecture, Civil Engineering and Geodesy - Sofia 9 saved as method of joints, stability

10 Examples Internal stability of truss depends upon the arrangements of members and joints Where m = number of members, J = number of joints, R = number of unknown reactions. In the case of the JMU simulator, where there is a fixed node and a rolling node, there are 3 reaction forces General JMU Simulator If 2j =m + r internally stable 2j = m+3 (OK) If 2j > m + r internally unstable 2j > m+3 (not enough members for support) If 2j> m + r indeterminate 2j < m+3 (more unknowns than equations; external stability depends reaction forces)

11 Stability Criteria Calculation

12 Stability Problems

13 Optional Project Opportunities to Demonstrate Individual Progress Credit will be based on the quality of the work compared to earlier assessments. The credit earned depends both upon an objective evaluation of the project and the earlier baseline assessments. Each person can make their own assessment as to the upside potential. Your report should include full narrative discussion of your analysis, screen shots, and conclusions. All work is individual, no consultations, and is due by January 12. (I will not be contactable from 12/23 until near the end of the break). 1. The purpose of the ENGINEERING ENCOUNTERS Bridge Design Contest ( is to provide students with a realistic, engaging introduction to engineering. In this project, you design a bridge to cross a river. There are many variables (span width, types of building materials, truss design, additional piers). After the design is completed, there is an animated load test. The test identifies failed members and design can then be revised. All of the design options carry an associated cost. The challenge is to design a bridge that can hold the load at a minimum cost. Use the West Point Bridge Designer (a free program) to construct one or two bridges at the minimum cost. The first bridge follows a template and gets you into the program. The second bridge may use all of the features of the full program. Link to Download: To get started: File: New Design Project. This will introduce the design problem. The wizard will take you through the set-up conditions. Use all of the default conditions for the first bridge: (span length 24 m, standard abutments, no piers, no cables, medium strength concrete, standard 2 lane road, and through truss Pratt.) The wizard takes you to Step 7 and the screen below. Good luck. If you do a second bridge, you can use any settings. I m not a gamer, but it did not take long to figure it out. There is an annotated screen shot below to help get you started. Tools (same as JMU) Test (Animation) Material of Construction Drawing Board Cost Failed Members identified after load test Undo/Previous versions

14 Optional Credit -Evaluation of Other Bridge Design Simulators We used the Johns Hopkins Bridge Designer for the initial evaluations of trusses. There are at least 2 others that might be appropriate. All three of these simulators work on the same principle as demonstrated above. However, there approaches are different and the information from the model comes in a different form. The documentation of both leaves something to be desired. Evaluate one of both of the Simulators: Learn how to use it. (The notation is different, but with a little patience and the ability to leverage your knowledge, you can see that the concepts are the same.) Design or evaluate at least two truss bridges. Show screen shots or printout of the data in your report. Write instructions for a student to construct or evaluate a truss bridge. Compare/Contrast the new Simulator with the JHU designer. Comment on which of the simulators you have evaluated you would recommend to a future class. State the reasons for your recommendation. Credit will be given for resourcefulness in mastering the programs, application to truss bridges, analysis of the results, clarity of the instructions and overall quality of the work. There is no minimum or maximum length, but you can probably get the key points covered in a couple of focused pages. A lengthy, verbose report is not a plus. Harvard: Bridge Simulation (Search Harvard bridge simulation) Comments: This model looks more complicated than it is. Use only secure pins (to match your model). Run Java as indicated at the tab at the top of the page, use the Open button to view sample bridges. Once you do this, the concept becomes clear. Cornell: Find Force Reactions and Bar Forces in a Truss Bridge (Search: Find reactions and bar forces for truss) Comments: 4 different types of truss bridges can be modeled. Bar Force refers to the force on a member (compression or tension). Note that the forces can be calculated for different numbers of members. More detailed instructions at: How to Calculate Truss Design ehow.com

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