Shear Force and Bending Moment -...

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Transcript of Shear Force and Bending Moment -...

Page 1: Shear Force and Bending Moment - libvolume6.xyzlibvolume6.xyz/.../bendingmomentandshearforceinbeamspresentatio… · between pure bending and non-uniform bending. Pure bending is

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

Page 2: Shear Force and Bending Moment - libvolume6.xyzlibvolume6.xyz/.../bendingmomentandshearforceinbeamspresentatio… · between pure bending and non-uniform bending. Pure bending is

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

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Longitudinal strain Longitudinal stress

Location of neutral surface Moment-curvature equation

Page 4: Shear Force and Bending Moment - libvolume6.xyzlibvolume6.xyz/.../bendingmomentandshearforceinbeamspresentatio… · between pure bending and non-uniform bending. Pure bending is

� 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

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� 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

Page 6: Shear Force and Bending Moment - libvolume6.xyzlibvolume6.xyz/.../bendingmomentandshearforceinbeamspresentatio… · between pure bending and non-uniform bending. Pure bending is

� 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

Page 7: Shear Force and Bending Moment - libvolume6.xyzlibvolume6.xyz/.../bendingmomentandshearforceinbeamspresentatio… · between pure bending and non-uniform bending. Pure bending is

Pure Bending

Pure Bending: Prismatic members

subjected to equal and opposite couples

acting in the same longitudinal plane

Page 8: Shear Force and Bending Moment - libvolume6.xyzlibvolume6.xyz/.../bendingmomentandshearforceinbeamspresentatio… · between pure bending and non-uniform bending. Pure bending is

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.

Page 9: Shear Force and Bending Moment - libvolume6.xyzlibvolume6.xyz/.../bendingmomentandshearforceinbeamspresentatio… · between pure bending and non-uniform bending. Pure bending is

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

Page 10: Shear Force and Bending Moment - libvolume6.xyzlibvolume6.xyz/.../bendingmomentandshearforceinbeamspresentatio… · between pure bending and non-uniform bending. Pure bending is

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

c

y-yy

L

yyLL

yL

εε

ερε

κρρθ

θδε

θρθθρδθρ

−=

==

=−=−==

−=−−=−′=

−=′

or

linearly) ries(strain va

Page 11: Shear Force and Bending Moment - libvolume6.xyzlibvolume6.xyz/.../bendingmomentandshearforceinbeamspresentatio… · between pure bending and non-uniform bending. Pure bending is

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

Page 12: Shear Force and Bending Moment - libvolume6.xyzlibvolume6.xyz/.../bendingmomentandshearforceinbeamspresentatio… · between pure bending and non-uniform bending. Pure bending is

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

Page 13: Shear Force and Bending Moment - libvolume6.xyzlibvolume6.xyz/.../bendingmomentandshearforceinbeamspresentatio… · between pure bending and non-uniform bending. Pure bending is

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

Page 14: Shear Force and Bending Moment - libvolume6.xyzlibvolume6.xyz/.../bendingmomentandshearforceinbeamspresentatio… · between pure bending and non-uniform bending. Pure bending is

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

Page 15: Shear Force and Bending Moment - libvolume6.xyzlibvolume6.xyz/.../bendingmomentandshearforceinbeamspresentatio… · between pure bending and non-uniform bending. Pure bending is

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

Page 16: Shear Force and Bending Moment - libvolume6.xyzlibvolume6.xyz/.../bendingmomentandshearforceinbeamspresentatio… · between pure bending and non-uniform bending. Pure bending is

Moment-curvature relationship: Sign convention

Maximum curvature occurs where the moment magnitude is a maximum.

Page 17: Shear Force and Bending Moment - libvolume6.xyzlibvolume6.xyz/.../bendingmomentandshearforceinbeamspresentatio… · between pure bending and non-uniform bending. Pure bending is

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