311 ch13
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- 1.
- IV .) Shear and Bending Moment in Beams
- A.) Reaction Forces (Statics Review)
- 1.) Replace Supports with unknown
- reaction forces (free body diagram)
2.
- a.) Roller - produces a reaction forceperpendicular to the support plane.
R Y 3.
- b.) Pin (or Hinge) - produces avertical and horizontal reaction.
R y R x 4.
- c.) Fixed - produces a reaction force in any direction and Moment.
R y R x M 5.
- 2.) Apply laws of equilibrium to findR AX,R AY,R BY
- F x= 0 F y= 0 M z= 0
R AX R AY R BY 6.
- B.) Internal Shear
- 1.) Shear - find by cutting a section at the point of interest and F y= 0 on the FBD.
R y R x F.B.D. V 7.
- B.) Internal Bending Moment
- 2.) Moment - find by cutting a section at the point of interest and M= 0 on the FBD.
R y R x F.B.D. V M 8.
- If you were to find the internal shear and moment at several locations along the length of a beam, you could plot a graph shear vs. length and a graph of moment vs. length and find where the maximum shear and moment occur.
9. 4 4 4 8 8 2k/ft 3k 4k A B V(k) 10. 4 4 4 8 8 2k/ft 3k 4k A B M (k-ft) 11.
- C.) Shear Diagram - Simpler Way to Draw
- 1.)Sketch the beam with loads andsupports shown (this is the LOADDIAGRAM).
- 2.)Compute the reactions at the supportsand show them on the sketch.
12.
- 3.)Draw Shear Diagram baseline
- (shear = zero) below the load diagram
- a horizontal line.
- 4.)Draw vertical lines down from the load
- diagram to the shear diagram at:
- a.)supports
- b.)point loads
- c.)each end of distributed loads
13.
- 5.)Working fromlefttoright , calculate
- the shear on each side of each
- support and point load and at each
- end of distributed loads:
- a.)For portions of a beam that have
- no loading, the shear diagram is
- a horizontal line.
14.
- b.) Point loads (and reactions) cause
- a vertical jump in the shear
- diagram.
- - The magnitude of the
- jump is equal to the magnitude of the load (or reaction).
- - Downward loads cause a negativechange in shear.
15.
- c.)For portions of a beam under
- distributedloading:
- i.) the slope of the shear diagram isequal to the intensity (magnitude) ofthe uniformly distributed load (w).
- ii.) the change in shear between two
- points is equal to the area underthe load diagram between thosetwo points.
16.
- V = wL (uniformly distributed)
- V = (wL)/2 (triangular distribution)
- Note:If the distributed load is acting
- downward w is negative.
- 6.)Locate points of zero shear using a
- known shear value at a known location
- and the slope of the shear diagram(w)
17. 4 4 4 8 8 2k/ft 3k 4k A B 18.
- D.) Moment Diagram - Simpler Method
- 1.)Moment = 0 at ends of simplysupported beams.
- 2.)Peak Moments occur where the shear
- diagram crosses through zero.There
- can be more than one peak moment
- on the diagram.
19.
- 3.)Extend the vertical lines below the
- shear diagram and draw the Moment
- Diagram baseline (moment = 0), a
- horizontal line.Also, extend vertical
- lines down from points of zero shear.
- 4.)Working left to right, calculate the
- moment at each point the shear was
- calculated and at points of zero shear:
20.
- a.)the change in moments between
- two points is equal to the area under
- the shear diagram between those
- points.
- b.)determine the slope of the moment
- diagrams as follows:
21.
- i.)if the shear is positive and constant,
- the slope of the moment diagram is
- positive and constant.
Negative, constant shear (-) (+) Positive, constant shear (+) 0 (-) V 22. Positive, constant slope Negative, constant slope (+) 0 (-) M 23.
- ii.)if the shear is positive and increasing,
- the slope of the moment diagram is
- positive and increasing.
- Positive, decreasing shear
Negative, decreasing shear (+) 0 (-) V 24. Positive, decreasing slope Negative,decreasing slope (+) 0 (-) M 25. 4 4 4 8 8 2k/ft 3k 4k A B 26.