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Beams: Pure Bending 1

Beams: Pure Bending(4.1-4.5)MAE 314 Solid Mechanics

Xiaoning Jiang

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Beams: Pure Bending 2

Beams in Pure Bending Prismatic beams subject to equal and opposite couples acting in the

same plane are in pure bending.

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Beams: Pure Bending 3

Pure vs. Non-Uniform Bending Pure bending: Shear force (V) = 0 over the section

Non-uniform bending: V 0 over the section

Pure bending

Moment normal stresses

Non-uniform bending

Moment normal stresses Shear force shear stresses (Ch. 6)

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Beams: Pure Bending 4

Pure Bending: Assumptions Beam is symmetric about the x y plane

All loads act in the x y plane

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Beams: Pure Bending 5

Pure Bending:

Curvature Sections originally perpendicular to longitudinal

(y-z) axis remain plane and perpendicular:Plane sections remain plane.

Sign convention

Positive bending moment:beam bends towards +y direction

Negative bending moment:beam bends towards -y direction

Right angle

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Beams: Pure Bending 6

Pure Bending: Deformation Since angles do not change

(remain plane), there is no

shear stress.

The top part of the beam contracts

in the axial direction. The bottom part of the beam expands

in the axial direction.

There exists a line in the beam that

remains the same length called the

neutral line. Set y = 0 at the neutral line.

= radius of curvature

x< 0 for y > 0 and x> 0 for y < 0

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Beams: Pure Bending 7

Pure Bending: Axial Strain The length of DE is LDE=

The length of JK is LJK= (-y)

Axial strain at a distance y from

the neutral axis (x):

Maximum compressive strainoccurs on the upper surface.

Maximum tensile strainoccurs on the lower surface.

)( y

L

LL

LDE

DEJK

x

y

x

LDE

LJK

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Beams: Pure Bending 8

Pure Bending: Axial Strain

c = maximum distance between the neutral axis and the upper or

lower surface

When c is the distance to the surface in compression

When c is the distance to the surface in tension

cmax

cmax

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Beams: Pure Bending 9

Pure Bending: Transverse Strain Recall there are no transverse stresses since

the beam is free to move in the y and z directions.

However, transverse strains (in the y and z

directions) exist due to the Poissons ratioof the material.

' = radius of anticlastic curvature = /

xy

yy

xz

yz

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Beams: Pure Bending 10

Pure Bending: Normal Stress Let us now assume that the beam is made of a linear-elastic material.

The normal stress varies linearly with the distance from the neutralsurface.

EyE

xx

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Beams: Pure Bending 11

Pure Bending: Normal Stress Recollecting that the applied loading is a pure moment, we calculate

resultant loads on the cross-section.

The resultant axial force must be equal to zero.

0 AAx

A

x ydA

E

dAEdA

0 A

ydA

which is the definition of the centroid, so theneutral axis is just the centroid of the section.

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Beams: Pure Bending 12

Pure Bending: Normal Stress The resultant moment about the z-axis must be equal to the applied

moment M.

MdAyE

dAyEdAy

AA

x

A

x

2

Definition of the second moment of inertia, I

EIM

I

Myx

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13

Pure bending

Neutral line (surface)

Normal strain and stress (bending)

Moment of inertia

Reading for the next lecture: A.1-A.5, 4.6

Key Concepts