Shear and Moment Diagrams
Today’s Objective
:
Students will be able to:
1.
Derive shear and bending moment
diagrams for a loaded beam using
a) piecewise analysis
b) differential/integral relations
These diagrams plot the
internal forces with respect to
x along the beam.
APPLICATIONS
They help engineers analyze
where the weak points will
be in a member
General Technique
•
Because the shear and
bending moment are
discontinuous near a
concentrated load, they
need to be analyzed in
segments between
discontinuities
Detailed Technique
•
1) Determine all reaction forces
•
2) Label x starting at left edge
•
3) Section the beam at points of
discontinuity of load
•
4) FBD each section showing V and M in
their positive sense
•
5) Find V(x), M(x)
•
6) Plot the two curves
SIGN CONVENTION FOR SHEAR, BENDING MOMENT
Sign convention for:
Shear:
+ rotates section clockwise
Moment:
+ imparts a U shape on section
Normal:
+ creates tension on section
(we won't be diagraming nrmal)
Example
•
Find Shear and Bending
•
Moment diagram for the beam
•
Support A is thrust bearing (Ax, Ay)
•
Support C is journal bearing (Cy)
•
PLAN
•
1) Find reactions at A and C
•
2) FBD a left section ending at x where (0<x<2)
•
3) Derive V(x), M(x)
•
4) FBD a left section ending at x where (2<x<4)
•
5) Derive V(x), M(x) in this region
•
6) Plot
Example, (cont)
•
1) Reactions on beam
•
2)
FBD of left section in AB
–
note sign convention
•
3) Solve: V = 2.5 kN
M = 2.5x kN

m
•
4) FBD of left section ending in BC:
•
5) Solve: V =

2.5 kN

2.5x+5(x

2)+M = 0
M = 10

2.5x
Example, continued
•
Now, plot the curves in
their valid regions:
•
Note disconinuities due
to mathematical ideals
Example2
•
Find Shear and Bending
•
Moment diagram for the beam
•
PLAN
•
1) Find reactions
•
2) FBD a left section ending at x, where (0<x<9)
•
3) Derive V(x), M(x)
•
4) FBD a left section ending somewhere in BC (2<x<4)
•
5) Derive V(x), M(x)
•
6) Plot
Example2, (cont)
•
1) Reactions on beam
•
2)
FBD of left section
–
note sign convention
•
3) Solve:
Example 2, continued
•
Plot the curves:
•
Notice Max M occurs
•
when V = 0?
•
could V be the slope of M?
A calculus based approach
•
Study the curves in the previous slide
•
Note that
•
1) V(x) is the area under the loading
curve plus any concentrated forces
•
2) M(x) is the area under V(x)
•
This relationship is proven in your text
•
when loads get complicated, calculus
gets you the diagrams quicker
derivation assumes
positive distrib load
Examine a diff beam
section
Example3
•
Reactions at B
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