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
APPLICATIONS
These diagrams plot the
internal forces with respect to
x along the beam.
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
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Find Shear and Bending
Moment diagram for the beam
Support A is thrust bearing (Ax, Ay)
Support C is journal bearing (Cy)
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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
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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