Shear and Bending Moment Diagrams
(Credit for many illustrations is given to McGraw Hill publishers and an array ofinternet search results)
Parallel Reading
Chapter 5 Section 5.1 Section 5.2 Section 5.3 Section 5.4 Section 5.5 (Do Chapter 5 Reading Assignment Problems)
Beams are One of the Most Important Structural Members
And the Load Distribution on them can be Complex
This can mean that we have to find our maximumstress loads in order to design beams to handletheir task
Shear and Bending Moment Diagrams are One of the Tools we
Use to Find Stress Maximums
O HaGot You!
My homeworkis due When!
Interesting Fact About Forces on Beam Cross Sections
If we slice a beam at any cross section point
We will find shear and bending forces there
I bet that is why welike to do shear andbending momentdiagrams.
End Conditions Help us Figure Out Where to Start
Simple Support can push and holdin place but cannot apply a moment.
Fixed end can supply xand y forces as well asmoment
Over-hang configuration
Lets Try to Get a Diagram
Here is a nice timber beam to work with
We can get our reactions at B and D using statics
Two Methods to Get the Diagram
We can do statics up and down cross sectionsOf the beam till we puke.
Or There is a Graphical Method
Starting with the shear diagramwe note that the change in shearis equal to the additional weightadded to the next increment.
If we go over a 1 foot increment, how muchshear will be pick up?
Of Course the Easiest Case is the Jump that Occurs at a Point Load
Lets give it a try moving left to right
V
20 kN
A B
That Last One Wasn’t BadLets Try This One!
A evenly distributed loadresults in a linearslope on the sheardiagram.
Oh Ya – Like Figuring the Area of a Triangle will Scare Us
We draw a curve downto the appropriate value.
We know the curve is 2nd
order but we don’t have tobe big on detail because ourcritical load point is not goingto be on a down sloping curve.
Knowing that We Can Get the Total Change in Shear Load Over an Interval by Integrating Under the Load Curve lets Us Solve for Any Crazy Load
Distribution.
Integrate this areato get the magnitudeof
This
Semi - English TranslationIntegrate the Area of the Shear Diagram to get
total Moment Change for the Interval6 ft 8 ft 10 ft
8 ft
Note that we integrated 18 kips over 6 feet(ok you don’t have to integrate to know the area of arectangle is base X height)
To find the area of 108 positive
We now use this for the magnitude of theMoment change over that 6 ft.
When the Slope of the Shear Diagram is Zero the Slope of the Moment Diagram is a
Straight Line
The area of the shear diagram from B to C is-2 kips X 8 ft = -16 kip*ft
Working Out the Next Part of OurMoment Diagram
92 Kip*ft
We take the 16 kip*ft away from our old valueOf 108 kip*ft and get our new value of
Note also that since the shear lineHas no slope the drop in momentIs linear.
We Next Drop to -14 kips shear over 10 ft.
Integrating that(ok just getting the area of the rectangle on the shear diagram)
We get -14 kips * 10 ft = -140 kip*ft
Now We Will Put that -140 kip*ft into the Moment Diagram
+ 92 kip*ft – 140 kip*ft = -48 kip*ft
-48Again note theSteady sloped line
Our Shear Load Jumps at Point D and then tapers to zero
Our triangle has an area of12 * 8 / 2 = 48 kip*ft
Obviously a plus 48 kip*ftWill bring a minus48 kip*ft to zero
Rules for Shear and Moment Diagrams
Concentrated loads produce a jump in the shear diagram and a change inThe linear slope of the moment diagram.
Shear and Moment Diagrams
An evenly distributed load produces and linear slope in the shear diagramand the value of the shear is the slope of the line in the moment diagram.
Shear and Moment Diagram Rules
For an unevenly distributed load the value of the load is the slope of theLine in the shear diagram (and the value shear is the slope of the momentLine)
For Your SanityYou can go nuts trying to get theslope and curve right between twopoints but it is the value inflextionpoints on the moment curve thatcontrols design.
Get the values at the section endpoints right and don’t sweat tomuch about exact values in between.
Critical points
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