Shear Force and Bending Moment
� Shear Force: is the algebraic sum of thevertical forces acting to the left or right ofa cut section along the span of the beam
� Bending Moment: is the algebraic sum ofthe moment of the forces to the left or tothe right of the section taken about thesection
V = +P
Mm ax= -PL
L
P P
L
V = +P/2
V = -P/2
Mm ax= PL/4
L
P = wL
Vmax = +P/2
Vmax = -P/2
Mm ax= PL/8
= wL2/8
L
P = wL
Vmax = +P
Mm ax= -PL/2
= -wL2/2
SFD & BMD Simply Supported Beams
Longitudinal strain Longitudinal stress
Location of neutral surface Moment-curvature equation
� It is important to distinguish between pure bending and non-uniform bending.
� Pure bending is the deformation of the beam under a constant bending moment. Therefore, pure bending occurs only in regions of a beam where the shear force is zero, because V = dM/dx.
� Non-uniform bending is deformation in the presence of shear forces, and bending moment changes along the axis of the beam.
Bending of Beams
� Causes compression on one face and tension on the other
� Causes the beam to deflect
How much deflection?
How much compressive stress?
How much tensile stress?
What the Bending Moment does to the Beam
� It depends on the beam cross-section
how big & what shape?
is the section we are using as a beam
�We need some particular properties of the section
How to Calculate the Bending Stress
Pure Bending
Pure Bending: Prismatic members
subjected to equal and opposite couples
acting in the same longitudinal plane
Symmetric Member in Pure Bending
∫∫∫
=−=
==
==
MdAyM
dAzM
dAF
xz
xy
xx
σ
σ
σ
0
0
� These requirements may be applied to the sums
of the components and moments of the
statically indeterminate elementary internalforces.
� Internal forces in any cross section are
equivalent to a couple. The moment of the
couple is the section bendingmoment.
� From statics, a couple M consists of two equal
and opposite forces.
� The sum of the components of the forces in any
direction is zero.
� The moment is the same about any axis
perpendicular to the plane of the couple and
zero about any axis contained in the plane.
Bending Deformations
Beam with a plane of symmetry inpure bending:
� bends uniformly to form a circular arc
� cross-sectional plane passes through arc center
and remains planar
� length of top decreases and length of bottom
increases
� a neutral surface must exist that is parallel to
the upper and lower surfaces and for which the
length does not change
� stresses and strains are negative (compressive)
above the neutral plane and positive (tension)
below it
�member remains symmetric
Strain Due to Bending
Consider a beam segment of length L.
After deformation, the length of the neutral surface remains L. At other sections,
maximum strain
in a cross section
ex < 0 ⇒ shortening ⇒ compression (y>0, k <0)ex > 0 ⇒ elongation ⇒ tension (y<0, k >0)
L
L’
( )( )
mx
m
m
x
c
y
cρ
c
y-yy
L
yyLL
yL
εε
ερε
κρρθ
θδε
θρθθρδθρ
−=
==
=−=−==
−=−−=−′=
−=′
or
linearly) ries(strain va
Curvature
A small radius
of curvature, ρρρρ,implies largecurvature of thebeam, κκκκ, andvice versa. In
most cases ofinterest, thecurvature issmall, and wecan approxima-
te ds ≅dx.
qqqq q+q+q+q+dqqqq
q+q+q+q+dqqqq
qqqq
dqqqq
Stress Due to Bending
0=⇒ ∫A
dAy
�For a linearly elastic material,
linearly) varies(stressm
mxx
c
y
Ec
yE
yE
σ
ερ
εσ
−=
−=−==
�For static equilibrium,
⇒−=
−=−===
∫
∫∫∫
A
AAA
xx
dAyE
dAyEdAy
EdAF
ρ
κρ
σ
0
)()(0
First moment with respect toneutral plane (z-axis) is zero.Therefore, the neutral surfacemust pass through the sectioncentroid.
maximum stress
in a cross section
I
My
c
y
S
M
I
Mc
c
IdAy
cM
dAc
yydAyM
xmx
m
mm
A
m
A
x
−=⇒−=
==
==
−−=−=
∫
∫∫
σσσ
σ
σσ
σσ
ngSubstituti
2
∫=A
dAyI 2
is the ‘second moment of area’
� The moment of the resultant of the stresses dF about the N.A.:
( )
,
2
EI
M
EIdAyEM
dAEyydAyM
A
AA
x
=
==
−−=−=
∫
∫∫
κ
κκ
κσ
Moment-curvature relationship
Deformation of a Beam Under Transverse Loading
� Relationship between bending moment
and curvature for pure bending remains
valid for general transverse loadings.
EI
xM )(1==
ρκ
� Cantilever beam subjected to concentrated
load at the free end,
EI
Px−=
ρ1
� At the free end A, ∞== A
A
ρρ
,01
� At the support B,PL
EIB
B
=≠ ρρ
,01
tension: stretched
compression
neutral “plane”
elastic curve
The deflection diagramof the longitudinal axisthat passes through thecentroid of each cross-sectional area of thebeam is called theelastic curve, which ischaracterized by thedeflection and slopealong the curve.
Elastic Curve
Moment-curvature relationship: Sign convention
Maximum curvature occurs where the moment magnitude is a maximum.
Deformations in a Transverse Cross Section
�Deformation due to bending moment M is
quantified by the curvature of the neutral surface
EI
M
I
Mc
EcEccmm
=
===11 σε
ρ
� Although cross sectional planes remain planar
when subjected to bending moments, in-plane
deformations are nonzero,
ρν
νεερ
ννεε
yyxzxy =−==−=
� Expansion above the neutral surface and
contraction below it causes an in-plane curvature,
curvaturec anticlasti 1
==′ ρ
νρ
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