Basic Stresses

8/2/2019 Basic Stresses

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Dr. D. B. Wallace

Basic Stress Equations

Internal Reactions:

6 Maximum(3 Force Components

& 3 Moment Components)

Normal Force

)(

)(

Shear Forces

z

x

y

PVy

Vx

Torsional Moment)(

)(Bending Moments

z

x

y

TMy

Mx

or TorqueForce Components Moment Components

"Cut Surface" "Cut Surface"

Centroid ofCross Section

Centroid ofCross Section

Normal Force:

Axial Force

z

x

y

P

Centroid

Axial Stress

"Cut Surface" =P

Al Uniform over the entire cross section.

l Axial force must go through centroid.

Shear Forces:Cross Section

y

Aa

Point of interest

LINE perpendicular to V through point of interest

= Length of LINE on the cross section

= Area on one side of the LINE

Centroid of entire cross section

Centroid of area on one side of the LINE

= distance between the two centroids

= Area moment of inertia of entire cross sectionabout an axis pependicular to V.

V

b

Aa

y

I

"y" Shear Force

z

x

y

Vy

"x" Shear Force

z

x

y

Vx

=

V A y

I b

ab g

Note: The maximum shear stress for common cross sections are:

Cross Section: Cross Section:

Rectangular:

max = 3 2 V A Solid Circular: max = 4 3 V A

I-Beam or

H-Beam: webflange max = V Aweb

Thin-walled

tube: max = 2 V A


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Basic Stress Equations Dr. D. B. Wallace

Torque or Torsional Moment:

Solid Circular or Tubular Cross Section:

r = Distance from shaft axis to point of interest

R = Shaft RadiusD = Shaft Diameter

JD R

JD D

for solid circular shafts

for hollow shaftso i

=

=

=

4 4

4 4

32 2

32

e jTorque

z

x

y

T

"Cut Surface"

=T r

J

max

max

=

=

16

16

3

4 4

T

D

T D

D D

for solid circular shafts

for hollow shaftso

o ie j

Rectangular Cross Section:

Torque

z

x

y

T

Centroid

Torsional Stress

"Cut Surface"

1

2

a

b

Note:a > b

Cross Section:

Method 1:

max .= = + 12 23 1 8T a b a bb g e j ONLY applies to the center of the longest side

Method 2:

1 21 2

2,

,

=

T

a b

a/b

1.0 .208 .208

1.5 .231 .269

2.0 .246 .309

3.0 .267 .355

4.0 .282 .378

6.0 .299 .402

8.0 .307 .414

10.0 .313 .421

.333 ----

Use the appropriate from the table

on the right to get the shear stress at

either position 1 or 2.

Other Cross Sections:

Treated in advanced courses.

2


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Bending Moment

"x" Bending Moment

z

x

y

z

x

y

Mx

My

"y" Bending Moment

=

=

M y

I and

M x

I

x

x

y

y

where: Mx and My are moments about indicated axes

y and x are perpendicular from indicated axes

Ix and Iy are moments of inertia about indicated axes

Moments of Inertia:

h

c

b D

c RIb h

h

Z Ic

b h

is perpendicular to axis=

= =

3

2

12

6

ID R

Z Ic

D R

=

=

= =

=

4 4

3 3

64 4

32 4

Parallel Axis Theorem:I = Moment of inertia about new axis

I I A d= + 2

centroid

d

new axis

Area, A I = Moment of inertia about the centroidal axisA = Area of the region

d = perpendicular distance between the two axes.

Maximum Bending Stress Equations:

max =

323

M

D

Solid Circularb g max =

62

M

b h

Rectangulara fmax =

=M c

I

M

Z

The section modulus, Z, can be found in many tables of properties of common cross sections (i.e., I-beams,

channels, angle iron, etc.).

Bending Stress Equation Based on Known Radius of Curvature of Bend, .

The beam is assumed to be initially straight. The applied moment, M, causes the beam to assume a radius of

curvature, .

Before:

After:

M M

= Ey

E = Modulus of elasticity of the beam material

y = Perpendicular distance from the centroidal axis to the

point of interest (same y as with bending of a

straight beam with Mx).

= radius of curvature to centroid of cross section

3


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Bending Moment in Curved Beam:

Geometry:

r

r

e

rrn

o

i

centroidcentroidal

neutral axis

axis

o

i

ynonlinear

stressdistribution

M

c

ci

o

rA

dA

e r r

n

area

n

=

=

z

A = cross sectional area

rn = radius to neutral axis

r = radius to centroidal axis

e = eccentricity

Stresses:

Any Position: Inside (maximum magnitude): Outside:

=

+

M y

e A r ynb gi

i

i

M c

e A r=

o

o

o

M c

e A r=

Area Properties for Various Cross Sections:

Cross Section rdA

area z A

t

hriro

r

Rectangle

rh

i +2

tr

r

o

i

FHG

IKJ

ln h t

t

hri

ro

r

Trapezoid

ot i

rh t t

t ti

i o

i o

+ +

+

2

3

b gb g

For triangle:

set ti

or to

to 0

t tr t r t

h

r

ro i

o i i o o

i

+

FHG

IKJ

ln ht ti o

+

2

r

Hollow Circle

a

b

r 22 2 2 2

LNM

OQP r b r a

a b2 2e j

4


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Bending Moment in Curved Beam (Inside/Outside Stresses):

Stresses for the inside and outside fibers of a curved beam in pure bending can be

approximated from the straight beam equation as modified by an appropriate

curvature factor as determined from the graph below [i refers to the inside, and o

refers to the outside]. The curvature factor magnitude depends on the amount of

curvature (determined by the ratio r/c) and the cross section shape. r is the radius

of curvature of the beam centroidal axis, and c is the distance from the centroidal

axis to the inside fiber.

M

M

CentroidalAxis

r

c

Inside Fiber: i iKM c

I=

Outside Fiber: o oKM c

I=

0

1.0

0.5

1.5

2.0

2.5

3.0

3.5

4.0

1 2 3 4 5 6 7 8 9 10 11

K

i

o

Curvature

Factor

Amount of curvature, r/c

K r

b

c

b

c

b

c

c

b/2

b/3

b/4b/8

b/6

B A B A

B A

BA

B A

B A B A

Values of K for inside fiber as ati A

Values of K for outside fiber as at

oB

U or T

Round or Elliptical

Trapezoidal

I or hollow rectangular

Round, Elliptical or Trapezoidal

U or TI or hollow rectangular

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Basic Stresses

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