LECTURE #13 :3.11 MECHANICS OF MATERIALS F03

INSTRUCTOR : Professor Christine OrtizOFFICE : 13-4022 PHONE : 452-3084WWW : http://web.mit.edu/cortiz/www

• Review : Torsion and Beam Bending 1&2• Beam Bending 3 : Stresses in Beams

1. Torsion : loading of a bar by an external torque or twisting moment that tends to produce rotation about the longitudinal (z) axis of the bar2. Pure Torsion :ASSUMPTIONS :•every cross-section of the bar is identical and subjected to same internal torque•all cross-sections remain plane, the same shape, and the radii remain straight as they rotate around the longitudinal axis•if the angle of rotation is small, the change in length and radius can be neglected •only twisting and pure shear stresses, no bending or warping (assume bar is weightless)•linear elastic, isotropicParameters :L=bar lengthφ=angle of twist at end of barγφz=shear strainδ =shear deformationr=radial coordinatez=axial coordinateR=cross-sectional area radiusT=torqueτφz=shear stress

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τφz (r)= τmax(r/R)

T cross-sectionalarea

τmax

z

z=0z=L

φ

rr=Rr=0

γ

SOLID BAR

=double arrow notation

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Review Lecture 10 : Torsion

δ

rigid support

z

z z

1. Geometrical Statement : d

2. Shear Strain Displacment Relation : =rdz

d 3. Constitutive Law : = =Gr

dz4. Equations of Static Equilibrium (Rotational) :

TLconstant T : =

JG[ radians

rd

G

φ

φ φ

δ φφ

γ

φτ γ

φ

π

=

o o

4

4

4 4o i

2

=180 , 1 rad=57.3 ] J(m ) = polar moment of inertia,

rsolid bar, circular cross-sectional area : J=

2(r -r )

hollow bar : J=2

JGJG(Nm )="torsional rigidity", k(Nm)= ="torsiona

L

π

π

( )z

fz f

l stiffness"

Tr5. Shear Stresses : =

JT r

= =modulus of rupture in torsionJ

φ

φ

τ

τ

DERIVATION:

Review Lecture 10 : Torsion

©2001 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning ™ is a trademark used herein under license.©2001 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning ™ is a trademark used herein under license.©2001 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning ™ is a trademark used herein under license.

I. Basics : BEAM : structural member subjected to lateral loads, i.e. forces or moments having their vectors perpendicular to the axis of the bar

nomenclature :L=length or span, b=width, h=heightNA=neutral axis passes through centroid (x,y=0) of cross-sectionI=moment of inertia of cross section about NA

Review : Beam Theory

x

yz

Lb

h

NAweb

flange

flange

1. simply supported

2. overhanging

P

A ByA=0 yB=0

P

A ByA=0 yB=0

PA

ByA=0, θA=0

I-beamcross-section

rectangularcross-section

II. Types of Supports and Boundary Conditions

NA

web

flange T-beamcross-section

3. cantilever

3 4

rectangular circularbh rI = , I = 12 4

EI=flexural modulus

π

Review : Beam Theory (Cont'd)

III. Types of Loading

0 •a

0 •a

0 •a

Po

0 •a

slope=k

0 •a

L

L

L

L

L

Mo

Po

concentrated moment or

couple

concentrated load

distributed, uniform load

linearly varying load

parabolically varying load

IV. Successive Integration Method / Shear and Bending Moment Diagrams :

V. Sign Conventions :

Review : Beam Theory (Cont’d)

(+)

V

V

(-)

V

V

(+)

(-)M MM M

compression

tensioncompression

tension

1

1 2

21

2 3

dVq(x) q=- q= loading function

dx

dMV(x) =- q(x)dx+C V= V=shear force

dx

dM(x) = V(x)dx+C x +C M= M=bending moment

C x1(x) = M(x)dx+ +C x+C

EI 5

dxθ

θ

∫

∫

∫3 2

1 23 4

dV =EI =curvature=slope of y-displacement curve

dx

C x C xy(x) = q(x)dx + + +C x +C y=vertical displacement

6 5

θ θ

∫

Beam Theory 3 : Normal Stresses and Strains

y

σx (y)x

σy =0

NAMo

(+ moment)

σx (max)c

σx (max)T

x

yz

?

N.A. compression

tension

m p

n q

dθρ

dxe fy

Mo

Mo

Flexure formula :

where : normal stress in x-direction

internal bending momenty = vertical distance from NA axis(see Gere Chapter 12, Appendix D, p 321)I = moment of inertia of cross-sect

ox

x

o

M yI

M

σ

σ

= −

=

=

( ) [ ]

( ) [ ]

maxmaxmax

max

maxmaxmax

ional area

( )

for rectangular beams : 2

( ),

"flexural modulus"

ox

oox x

M x y

Ih

y

M x yM yEI EI

EI

σ

ε ε

= −

=

= − = −

=

Beam Theory (Cont'd) : Shear Stresses and Strains

y

z

NA

22

Derivation in Gere Section 5.8 :

(rectangular cross section)2 4

where : shear stress

shear forceh = height of cross sectional areay= distance from NA

(rectangular cross sect

xy

xy

xy

V h yI

V

τ

τ

τ

= − −

=

=

max

3ion)

2cross sectional area

VA

A

= −

=

τxy(y)x

Beam Theory 3 : Stresses and Strains

x

y

z

y

z

Beam Theory 3 : Stresses and Strains

x

y

z

y

z

?

Beam Theory 3 : Stresses and Strains

x

yz

? O

N.A. compression

tension

m p

n q

dθρ

dxe fy

Mo

Mo

Beam Theory 3 : Relate Stress to Moment

y

σx (y)x

σy =0

NAMo

(+ moment)

σx (max)c

σx (max)T