Lecture 5-6Beam

Mechanics of Materials Laboratory Sec. 3-4

Jiangyu LiUniversity of Washington

Mechanics of Materials Lab

Inclined Load

Notice the sign convention: positive Mz compress upper part, negative stress; positive My extend front part, positive stress!

Inclined Load

z

z

y

yx I

yMI

zM

Stress

Neutral axis

0z

z

y

yx I

yMI

zM

yz

zy

IM

IM

zy tan

Asymmetrical Beam

The origin of y and z axes must be placed at centroid C; orientation isarbitrary.

0ydA

0zdA

Sign Convention for Curvature

yy

Ey

Similar equation apply toBending toward z axis

Note difference with sign convention in bending moment

Asymmetric Beam

zy

yxz

EI

dAyEydAM

2

yzy

yxy

EI

yzdAEzdAM

If z is a principal axis, My=0, bending in x-y plane, analogous to a symmetric beam

z

yz

z

y

I

I

M

M

When z axis is the neutral axis;

Asymmetric Beam

yzz

zxz

EI

yzdAEydAM

yz

zxy

EI

dAzEzdAM

2

y

yz

y

z

I

I

MM

If y is a principal axis, Mz=0, bending in x-z plane, analogous to a symmetric beam

When y axis is the neutral axis;

Asymmetric Beam

• When an asymmetric beam is in a pure bending, the plane in which the bending moments acts is perpendicular to the neutral surface only if the y and z axes are principle centroidal axes and the bending moment acts in one of the two principle plane. In such case, the principle plane in which bending moment acts becomes the plane of bending and the usual bending theory is valid

Analysis of Asymmetric Beam

• Locating the centroid, and constructing a set of principal axes

• Resolving bending moment into My and Mz

• Superposition

z

z

y

yx I

yMI

zM

tantany

z

yz

zy

II

IM

IM

zy

Principle Axes

dAyIx2 dAxI y

2

xydAIxy

2sin2cos221 xyyxyx

x IIIII

I

2cos2sin211 xyyx

yx III

I

yx

xyp II

I

2

2tan

Analysis of Asymmetric Beam

A channel section C 10x15.3

c=0.634

Iy=2.28 in4, Iz=67.4 in4

yA=5.00 in, zA=-2.6+0.634=-1.966 in

Calculating bending stress

Locating neutral axis

Analysis of Asymmetric Beam

ink605.2sin MM y

ink77.14cos MM z

psi3340z

z

y

yx I

yMI

zM

o

y

z

II

zy

1.79

212.5tantan

Normal Stress in Beam

yy

Ey

Curved Beams

Neutral axis is no longer the centroidal axis

Positive M

i

ii AerMc

o

oo Aer

Mc

)(

1/

yrAeMy

dAr

Ar

n

n

Curved Beam

in641.3

1/

i

o

n

r

rIn

h

drrbbh

dAr

Ar

)( yrAeMy

AF

n

cFrM rry n

Curved Beams

rr

IMs

ArI

e

yrAeMy

c

c

n

)(

rrs c

Curvature is large, e is small, rn is cloase to rc

Recover to straight beam

Curved Beam

sbsrb

rc

srsb

sbsr

s

e

c

c

Pay attention to the sign of s

Curved Beam

srsb

sbsr

s

e

c

c

222 sRb

Pay attention to the sign of s

• Read Mechanics of Materials Lab Sec. 4

• 4.26(e), 4.72 posted online

Assignment