Bending of Beams

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Bending of Beams (Introduction)
Beam a structural member long in one
direction relative to the other two, and loads
are in the transverse direction (( loads ? to the
long direction Beam axis )).
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Beams are usually drawn as lines only (not as
above) as shown
Beams are usually straight.
Loads
Concentrated
Distributed
Moment (of a couple)
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Supports
? one reaction
roller
? two reactions (in general)
hinge
? three reactions (in general)
fixed
Names (Types)
simply-supported cantilever fixed-fixed overhan
g continuous
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Internal forces
N normal force (already covered ignore now)
M bending moment (are covered now)
V shear force (will be covered later)
Shear Force diagram (SFD)
Bending Moment Diagram (BMD)
You have already studied one method for drawing
the SFD BMD in Statics. The FBDs and equations
were used. REVIEW!
In this course, a new method called summation
(or integration or area) method will be
developed. No need for FBDs and equations. It
is a very quick and convenient method.
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SFD BMD by Summation Method
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In the FBD, the distributed load is replaced by
an equivalent concentrated load (Resultant)
R w (x ? ?x) ?x 0 ? ? ? 1
? ? Fy 0 ?
V(x) w (x ? ?x) ?x ? V(x ?x) 0
Divide by ?x, rearrange, and let ?x ? 0

OR
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? ? Mo 0 ?
? M(x) M (x ?x) ? V(x) ?x ? w (x ??x)
(?x) (1??) (?x) 0
Divide by ?x, rearrange, and let ?x ? 0

?
OR
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Interpret the equations above geometrically
(slope area)
By integrating the two eqs. above on the interval
i and i 1
?V Vi1 Vi area under the load diagram
between xi and xi1
?M Mi1 Mi area under the shear diagram
between xi and xi1
Note that the load must be continuous.

• If the load W is of order Xn, then
• V is of order Xn1 and
• M is of order Xn2

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Concentrated Loads

? ? Fy 0 ?
P V (x ?) V (x ?) 0
? V (x ) V (x ?) ?V P
? ? Mo 0 ?
? M (x ? ?) M (x ?) ? V (x ? ?) (2 ?) ? P? 0
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? M (x ) ? M (x ?) ?M 0
Concentrated Couple
We do as we did above ?
? Fy 0 ?M 0 ?
V (x ) ? V (x ?) ?V 0
M (x ) ? M (x ?) ?M C ? Note () C (CW)
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