STATICS AND STRENGTH OF MATERIALS REVIEW: MODULE 2
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STATICS AND STRENGTH OF MATERIALS REVIEW:
MODULE 2
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TYPES OF BEAMS
• If a beam is freely supported at its ends with
either pins or rollers, it is called a simply
supported beam or simple beam.
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TYPES OF BEAMS
• The beam that is fixed at one end and free
at the other is called a cantilever beam.
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TYPES OF BEAMS
• Supports for beams
with overhanging
ends are either pins
or rollers.
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TYPES OF BEAMS
• These beams, are all determinate because the
three unknown reactions for each beam can be
determined by the equations of static equilibrium.
THEY YIELD 3 UNKNOWNS
3 EQUATIONS ARE AVAILABLE
ΣM
ΣFX
ΣFY
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TYPES OF BEAMS
• Examples of statically indeterminate beams.
A fixed beam in which
both ends are fixed.
A propped beam in which
one end is fixed and the
other end supported by
a roller.
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SHEAR FORCE SIGN CONVENTION
• Shown here is the sign
convention for positive internal
shear force.
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SHEAR FORCE SIGN CONVENTION
• Shown here is the sign convention
for negative internal shear force.
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BENDING MOMENTS
• Shown here is the direction
of positive internal bending
moments.
– compression on the top of the beam and tension on the
bottom of the beam.
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BENDING MOMENTS
• The direction of negative
bending moments is
reversed.– Negative bending moments tend to
cause tension on the top of the
beam and compression on the
bottom of the beam.
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SHEAR AND BENDING-MOMENT DIAGRAMS
• Concentrated loads on
the beam:
– Shear diagram consists of
straight horizontal lines
broken only at new load.
– The moment diagram
consists of straight sloping
lines broken at new loads.
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SHEAR AND BENDING-MOMENT DIAGRAMS
• Uniform loads on the
beam:
– the shear diagram consists
of straight sloping
(diagonal) lines.
– The moment diagram
consists of curved lines
(second-degree curves).
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• The maximum/minimum
moments occur at points
where the value of shear is
zero.
RELATIONSHIP BETWEEN SHEAR AND BENDING MOMENT DIAGRAMS
FOR DETAILED PROCEDURES ON DRAWING SHEAR AND BENDING
MOMENT DIAGRAMS: SEE WEBSITE FOR DOWNLOADABLE DOCUMENT
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BEAMS IN BENDING
Bending stress at any distance y is:
Maximum bending stress occurs at
extreme fibers and max. bending stress:
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HORIZONTAL SHEAR STRESS DUE TO BENDING
Looking at the “stacked” planks in (a) it can be seen
that if they are individually place and NOT attached to
each other, slippage will occur between them.
Each plank is bending individually;
C in top fibers
T in bottom fibers
If an adhesive is applied that bonds the planks
together, then the planks will bend as one beam;
Cmax in top fibers of top plank
Tmax in bottom fibers of bottom blank
The adhesive will then be resisting the “shearing
stress” that occurs in those planes due to vertical
loading.
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• “General Shear Formula” Ss = VQ
I b
Where; V is computed vertical shear force at cross section being considered
Q is the statical moment about the neutral axis of the area outside
the horizontal plane being evaluated
I is moment of inertiab is the width of the cross section in the horizontal plane where the
shear stress is being calculated
HORIZONTAL SHEAR STRESS DUE TO BENDING
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VERTICAL SHEAR STRESS
***For equilibrium to exist: Vertical shear is
equal to horizontal at any given point.
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CALCULATING BEAM DEFLECTIONS
• Deflection, moment and shear equations for
various beams are shown in the Beam Deflection
Tables on the class website.
• Many textbooks will also provide deflection
equations in the Appendices.
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DEFLECTION CALCULATION:SUPERPOSITION METHOD
Consider the cantilever beam shown.
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DEFLECTION CALCULATION:SUPERPOSITION METHOD
Deflection, moment, shear can be calculated by the
method of superposition:
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