# Mechanics of Solids I

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18 Ιουλ 2012 (πριν από 5 χρόνια και 10 μήνες)

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Mechanics of Solids I Mechanics of Solids I
Analysis and Design of Beams for Bending
Introduction
Introduction
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perpendicular to their longitudinal axis are called
beams
Introduction
Introduction
perpendicular to their longitudinal axis are called
beams
Introduction
Introduction
o Applied loads result in internal forces
consisting of a shear force (from the shear
stress distribution) and a bending couple
(from the normal stress distribution).
o Normal stress is often the critical design criteria
x m
M
c M
My
I I S
σ σ= − = =
o Requires determination of the location and
magnitude of largest bending moment.
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Shear and Bending Moment Diagrams
o Determination of maximum normal and
shearing stresses requires identification of
maximum internal shear force and bending
couple.
o passing a section through the beam and
applying an equilibrium analysis.
o Shear and bendin
g
-moment functions must
be determined for each region of the beam
Shear and Bending Moment Diagrams
o Si
g
n conventions:
g
4
Example Example 55..11
shown, draw the shear and bending
moment dia
g
rams and determine the
maximum normal stress due to bending.
Relations among Load, Shear, and Bending Relations among Load, Shear, and Bending
MomentMoment
o Relationship between load and shear:
(
)
〺 0
y
F V V V w x
V w x
=
− + Δ − Δ =
Δ = − Δ

dV
w
dx
= −
slope of SFD = - w
D
C
x
D C
x
V V w dx− = −

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Relations among Load, Shear, and Bending Relations among Load, Shear, and Bending
MomentMoment
o
Relationship between shear and bending moment:
( )
( )
2
1
2
0:0
2
C
x
M M M M V x w x
M V x w x

Δ
=
+ Δ − − Δ + Δ =
Δ = Δ − Δ

dM
V
=
o
Relationship between shear and bending moment:
slo
p
e of BMD = shear at the
p
oint
D
C
x
D C
x
V
dx
M
M Vdx− =

p p
area under SFD = moment difference
Relations among Load, Shear, and Bending
Relations among Load, Shear, and Bending
MomentMoment
o
Example: Draw SFD and BMD for simply
supported beams as shown
o
Example: Draw SFD and BMD for simply
-
supported beams as shown
L
w
P
a
2a
C
a
2a
C
M
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Example Example 55..22
Draw the shear and bending-moment
.
Example
Example 55..33
The structure shown is constructed of a W250 ×
167 rolled-steel beam. (a) Draw the shear and
bending-moment diagrams for the beam and
b
) Determine normal stress
.
(
b
) Determine normal stress
in sections just to the right and left of point D.
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6.2 GRAPHICAL METHOD FOR CONSTRUCTING SHEAR
AND MOMENT DIAGRAMS
Design of Beams for BendingDesign of Beams for Bending
o The largest normal stress is found at the surface where the maximum
bending moment occurs.
max max
m
M c M
I S
σ = =
o A safe design requires that the maximum normal stress be less than
the allowable stress for the material used.
max
m all
M
S
σ σ≤
o Among beam section choices which have an acceptable section
modulus, the one with the smallest weight per unit length or cross-
sectional area will be the least expensive and the best choice.
max
min
all
S
σ
=
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Example Example 55..44
A simply supported steel beam is to carry
shown. Knowing that the allowable normal
stress for the grade of steel to be used is 160
MPa, select the wide-flange shape that
should be used.
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Principle Stresses in a BeamPrinciple Stresses in a Beam
• Prismatic beam subjected to transverse
σ σ
τ τ
= − =
= − =
x m
xy m
M
y Mc
I I
VQ VQ
It It
• Determine if the maximum normal stress
within the cross-section is larger than
σ =
m
M
c
I
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Principle Stresses in a BeamPrinciple Stresses in a Beam
• Cross-section shape results in large values of
near the s rface here
is also large
τ
xy
near the s
u
rface
w
here
σ
x
is also large
.
• σ
max
may be greater than σ
m
.
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Example Example 88..11
A 160-kN force is applied at the end
of a W200 × 52 rolled-steel beam.
Neglecting the effects of fillets and of
stress concentrations, determine
whether the normal stresses satisfy a
design specification that they be
equal to or less than 150 MPa at
section A−A’.
Example
Example 88..22
The overhanging beam supports a
uniforml
y