51e64ae0e4b08833fb23efa5-shkrina-1374061788922-sfbmdiagrams.ppt

8/14/2019 51e64ae0e4b08833fb23efa5-shkrina-1374061788922-sfbmdiagrams.ppt

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Shear Force and Bending Moment

Shear Force: is the algebraic sum of thevertical forces acting to the left or right of

a cut section along the span of the beam

Bending Moment: is the algebraic sum ofthe moment of the forces to the left or to

the right of the section taken about thesection


2/17


3/17

Longitudinal strainLongitudinal stress

Location of neutral surfaceMoment-curvature equation


4/17

It is important to distinguishbetween pure bending andnon-uniform bending.

Pure bending is thedeformation of the beam under

a constant bending moment.Therefore, pure bendingoccurs only in regions of abeam where the shear force iszero, because V = dM/dx.

Non-uniform bending isdeformation in the presence ofshear forces, and bendingmoment changes along the axisof the beam.

Bending of Beams


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Causes compression on one face and tension onthe other

Causes the beam to deflect

How much deflection?

How muchcompressive stress?

How muchtensile stress?

What the Bending Moment does to the Beam


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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


7/17

Pure Bending

Pure Bending: Prismatic memberssubjected to equal and opposite couplesacting in the same longitudinal plane


8/17

Symmetric Member in Pure Bending

MdAyM

dAzM

dAF

xz

xy

xx

0

0

These requirements may be applied to the sumsof the components and moments of thestatically indeterminate elementary internalforces.

Internal forces in any cross section areequivalent to a couple. The moment of the

couple is the section bending moment.

From statics, a couple M consists of two equaland 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 andzero about any axis contained in the plane.


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Bending Deformations

Beam with a plane of symmetry in

pure bending:

bends uniformly to form a circular arc

cross-sectional plane passes through arc centerand remains planar

length of top decreases and length of bottomincreases

a neutral surfacemust exist that is parallel tothe upper and lower surfaces and for which thelength does not change

stresses and strains are negative (compressive)above the neutral plane and positive (tension)

below it

member remains symmetric


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Strain Due to Bending

Consider a beam segment of lengthL.After deformation, the length of the neutralsurface remainsL. At other sections,

maximum strainin a cross section

ex< 0 shortening compression (y>0, k 0 elongation tension (y0)

L

L

mx

m

m

x

c

y

c

c

y-yy

L

yyLL

yL

or

linearly)ries(strain va


11/17

Curvature

A small radius

of curvature, ,implies largecurvature of thebeam, , andvice versa. Inmost cases ofinterest, thecurvature issmall, and wecan approxima-te dsdx.

q q dq

q dq

q

dq


12/17


13/17

I

My

c

y

S

M

I

Mc

c

IdAy

cM

dAc

yydAyM

xmx

m

mm

A

m

A

x

ngSubstituti

2

A

dAyI 2

is the secondmoment of area

The moment of the resultant of the stresses dFabout the N.A.:

,

2

EI

M

EIdAyEM

dAEyydAyM

A

AA

x

Moment-curvature relationship

f i f d


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Deformation of a Beam UnderTransverse Loading

Relationship between bending momentand curvature for pure bending remainsvalid for general transverse loadings.

EI

xM )(1

Cantilever beam subjected to concentratedload at the free end,

EI

Px

1

At the free endA, AA

,01

At the supportB,PL

EIB

B

,01


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tension: stretched

compression

neutral plane

elastic curve

The deflection diagramof the longitudinal axisthat passes through thecentroid of each cross-sectional area of the

beam is called theelastic curve, which ischaracterized by thedeflection and slopealong the curve.

Elastic Curve


16/17

Moment-curvature relationship: Sign convention

Maximum curvature occurs where the moment magnitude is a maximum.


17/17

Deformations in a Transverse Cross Section

Deformation due to bending moment M isquantified by the curvature of the neutral surface

EI

M

I

Mc

EcEcc

mm

11

Although cross sectional planes remain planarwhen subjected to bending moments, in-planedeformations are nonzero,

yyxzxy

Expansion above the neutral surface andcontraction below it causes an in-plane curvature,

curvaturecanticlasti1

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