The document discusses methods for analyzing statically determinate truss structures, including the Method of Joints and Method of Sections. It uses the example of a truss bridge in Brockport, NY to demonstrate these methods. The Method of Sections involves cutting the truss into rigid bodies and applying equilibrium equations to solve for member forces. Reactions can first be found using the Method of Joints, then sections are cut to solve for internal forces.

1. Method of Sections Erie Canal Lift Bridge – Main Street, Brockport, NY Statics ©2007 Coon All Rights Reserved

2. Supports A pinned support can support a structure in two dimensions. A roller support can support a structure in only one dimension.

3. Static Determinacy Formula

4. Statically Indeterminate A truss is considered statically indeterminate when the static equilibrium equations are not sufficient to find the reactions on that structure. There are simply too many unknowns. Try It Did you notice the two pinned connections?

5. Statically Determinate A truss is considered statically determinate when the static equilibrium equations can be used to find the reactions on that structure. Try It Is the truss statically determinate now?

6. Static Determinacy Example Each side of the main street bridge in Brockport, NY has 19 joints, 35 members and three reaction forces (pin and roller) making it a statically determinate truss. What if these numbers were different?

7. Equilibrium Equations The sum of all forces in the X- direction is zero. The sum of all forces in the Y- direction is zero. The sum of the moments about a given point is zero.

8. Moment ary Review A moment is a twisting or turning force sometime referred to as torque. A moment arm is nothing more than a lever. The wheelbarrow pictured to the right is a third class lever. Given the following information, you could calculate how much force would be needed to lift the handles of the wheelbarrow. Distance from the fulcrum (A) to the effort (C) Distance from the fulcrum (A) to the resistance (B) . Resistance Load (B) RESISTANCE ARM L r EFFORT ARM L e F e EFFORT FORCE F r RESISTANCE FORCE A D C

9. Now lets replace the wheelbarrow with a truss. Likewise, joint A would be the fulcrum, the load is applied at joint D , and the reaction at joint C is counteracting force F D . Remember the truss is in static equilibrium, therefore, all forces must sum to zero. If we sum the moments about point A , we can find the reaction force R CY at point C . Moment ary Review RESISTANCE ARM L r EFFORT ARM L e F e EFFORT FORCE F r RESISTANCE FORCE 3’ 7’

10. Finding the Reaction Forces “For every action, there is an equal and opposite reaction” - Sir Isaac Newton

11. Using Moments to Find R CY A force that would cause a clockwise moment is negative. A force that causes a counterclockwise moment is positive. F D is negative because it causes a clockwise moment. R CY is positive because it causes a counterclockwise moment. 3’ 7’

12. Sum the Y Forces to Find R AY We know two out of the three forces acting in the Y-direction. By simply summing those forces together we can find the unknown reaction at point A. Please note that F B is a shown as a negative because of its direction. See Cartesian coordinate system.

13. Sum the X Forces to Find R AX Because joint A is pinned, it could react to a force applied in the X-direction. However, Since the only load applied to this truss ( F B ) has no X-component, R AX must be zero.

14. A B C D E F Load Load If you can solve a truss using the Method of Joints, you can solve a truss using the Method of Sections.

15. A B C D E F Load Load R AY R AX R FY Calculate Reaction Forces R AX , R AY & R FY

16. A B C D E F Load Load R AY R AX R FY Let’s find the force in member CD. F CD known known known

17. A B C D E F Load Load R AY R AX R FY Cut across two or three members, but no more than three. known known known

18. A B C Load R AY R AX Treat this cut section as a RIDGID BODY. known known

19. A B C Load R AY R AX Assume the forces on cut members act as external forces on the cut F BD Assumed Compression F CD Assumed Tension F CE Assumed Tension known known

20. A B C Load R AY R AX Treat left section as a RIDGID BODY. F BD F CD F CE 3 unknowns BD , CD X & BC 1 unknown CD y can be found 1 unknown FE can be found known known

21. The sum of all forces in the X- direction is zero. The sum of all forces in the Y- direction is zero. The sum of the moments about a given point is zero. Why can we only cut three members?

22. D E F Load R FY You could use the right side of the truss as well. Start by cutting through two or three members. F BD F CD F CE Assumed Compression Assumed Tension Assumed Compression known

23. D E F Load R FY You could use the right side of the truss as well. Start by cutting through two or three members. F BD F CD F CE 3 unknowns BD , CD X & BC 1 unknown CD y can be found 1 unknown FE can be found known

24. Good Luck! Erie Canal Lift Bridge – Main Street, Brockport, NY ©2007 Coon All Rights Reserved