2D Static Analysis

6.8. 2D Static Analysis#

The general procedure for solving 2D static analysis problems (two-dimensional particle equilibrium) is that you determine each force vector component and then solve for the equilibrium in each component direction. The process follows the same five-step method for creating a free-body diagram, followed by steps to solve your equilibrium equations.

Draw a Free-Body Diagram#

Below are the steps to draw a free-body diagram.[2]

  1. Select and isolate the particle. The “free-body” in free-body diagram means that a concurrent force particle or connection must be isolated from the supports that are physically holding it in place. This means creating a separate free-body diagram from your problem sketch.

  2. Establish a coordinate system. Draw a right-handed coordinate system to use as a reference for your equilibrium equations. Look ahead and select a coordinate system that minimizes the number of force components. This will simplify your vector algebra. The choice is technically arbitrary, but a good choice will simplify your calculations and reduce your effort.

  3. Identify all loads. Add force vectors to your free-body diagram representing each applied load pushing or pulling the body, in addition to the body’s weight, if it is non-negligible. If a force vector has a known direction, draw it. If its direction is unknown, assume one, and your later algebra will check your assumption. Every vector should have a descriptive variable name and a clear arrowhead indicating its direction.

  4. Identify all reactions. Reactions represent the resistance of the physical supports you cut away by isolating the body in step 1. All particle supports are some type of two-force members with tension or compression reaction forces. These reactions will all be concurrent with the body loads from Step 2. Label each reaction with a descriptive variable name and a clear arrowhead. Again, if a vector’s direction is unknown, just assume one.

  5. Label the diagram. Verify that every dimension, angle, force, and moment is labeled with either a value or a symbolic name if the value is unknown. In our eyes, dimensioning is optional. Having the information needed for your calculations is helpful, but don’t clutter the diagram up with unneeded details. Your final free-body diagram should be a stand-alone presentation and is the basis of your equilibrium equations.

Create and Solve Equilibrium Equations#

Below are the steps to create and solve scalar equations of equalibrium.[2]

  1. Break vectors into components. Compute each force’s \(x\) and \(y\) and components using right-triangle trigonometry.

  2. Write equilibrium equations. Now represent your free-body diagram as two equilibrium equations:

\[ \sum F_x = 0 \text{ and } \sum F_y = 0 \]

  1. Count knowns and unknowns. At this point, you should have at most two unknown values. If you have more than two, reread the problem and look for overlooked information.

  2. Solve for unknowns. Use algebra to simplify the equilibrium equations and solve for unknowns. All answers in Statics will have units - unless you have solved for a dimensionless value, like a friction coefficient. Finally, box your answers.

  3. Check your work. If you add the components of the forces, do they add to zero? Do the results seem reasonable given the situation? Have you included appropriate units?

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