Basic Stress Equations - Fairfield University

Dr. D. B. Wallace

Basic Stress Equations

Internal Reactions:

Centroid of Cross Section

6 Maximum

y

(3 Force Components

& 3 Moment Components)

"Cut Surface"

Centroid of Cross Section

Shear Forces ( )

y

x

"Cut Surface"

Bending Moments ()

x

Vx Vy

P

z

Normal Force ()

Force Components

Mx My

T

z

Torsional Moment or Torque ( )

Moment Components

Normal Force:

Centroid y "Cut Surface"

x

Pz Axial Force

Axial Stress

= P A

l Uniform over the entire cross section. l Axial force must go through centroid.

Shear Forces:

y "y" Shear Force

x

Vy

z

y "x" Shear Force

x

Vx

z

Cross Section V

Point of interest LINE perpendicular to V through point of interest

b = Length of LINE on the cross section

y

Aa

Aa = Area on one side of the LINE Centroid of entire cross section Centroid of area on one side of the LINE

I = Area moment of inertia of entire cross section about an axis pependicular to V.

b g = V Aa y Ib

y = distance between the two centroids

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:

flange web max = V A web

Thin-walled tube:

max = 2 V A

Basic Stress Equations

Dr. D. B. Wallace

Torque or Torsional Moment:

Solid Circular or Tubular Cross Section:

y

"Cut Surface"

x

Tz Torque

= Tr J

max

=

16 T D3

for solid circular shafts

e j max

=

16 T Do Do4 - Di4

for hollow shafts

r = Distance from shaft axis to point of interest

R = Shaft Radius

D = Shaft Diameter

J = D4 = R4

e j 32

2

J = Do4 - Di4

32

for solid circular shafts for hollow shafts

Rectangular Cross Section:

y Centroid

"Cut Surface"

x

Tz Torque

2 1

Torsional Stress

Cross Section: b

Note:

a

a > b

Method 1:

b g e j max = 1 = T 3a +1.8 b a2 b2

ONLY applies to the center of the longest side

Method 2:

1,2

=

1,2

T a b2

a/b

1

2

1.0

.208

.208

1.5

.231

.269

2.0

.246

.309

Use the appropriate from the table

3.0

.267

.355

on the right to get the shear stress at

4.0

.282

.378

either position 1 or 2.

6.0

.299

.402

8.0

.307

.414

10.0 .313

.421

.333

----

Other Cross Sections: Treated in advanced courses.

2

Basic Stress Equations

Dr. D. B. Wallace

Bending Moment

"x" Bending Moment

y x

Mx

z

= Mx y and = My x

Ix

Iy

"y" Bending Moment

y

x

My

z

Moments of Inertia:

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

b

c h

I = b h3 h is perpendicular to axis 12

Z = I = b h2 c6

D

c

R

I = D4 = R4

64

4

Z = I = D3 = R3

c 32

4

Parallel Axis Theorem:

new axis

d

Area, A I = I + A d2

centroid

I = Moment of inertia about new axis

I = Moment of inertia about the centroidal axis

A = Area of the region d = perpendicular distance between the two axes.

Maximum Bending Stress Equations:

max =

Mc I

=

M Z

b g max

=

32 M D3

Solid Circular

a f max

=

6M b h2

Rectangular

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:

= Ey

E = Modulus of elasticity of the beam material

M

M

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

Basic Stress Equations

Dr. D. B. Wallace

Bending Moment in Curved Beam:

Geometry:

nonlinear stress

distribution

M

o centroidal

axis

e

co

ci

i

rn

neutral axis

ri

centroid

y

ro

r

zrn =

A dA

area

e = r - rn

Stresses:

Any Position:

b g = -M y e A rn + y

Inside (maximum magnitude):

i

=

M ci e A ri

A = cross sectional area

rn = radius to neutral axis r = radius to centroidal axis

e = eccentricity

Outside:

o

=

-M co e A ro

Area Properties for Various Cross Sections:

Cross Section

Rectangle

r

t

ri

h

ro

Trapezoid

r

ti

to

ri

h

ro

Hollow Circle

r

a

r

ri

+

h 2

bb g g ri

+

h ti +2to 3 ti + to

For triangle: set ti or to to 0

r

z dA

area

t

lnFHG

ro ri

IKJ

FHG IKJ to

-

ti

+

ro

ti

- h

ri

to

ln

ro ri

LNM OQP 2 r2 - b2 - r2 - a2

b

A ht h ti + to

2

e j a2 - b2

4

Basic Stress Equations

Dr. D. B. Wallace

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.

Centroidal Axis

c

M

r

Inside Fiber:

i =

Ki

M I

c

M

Outside Fiber:

o =

Ko

M I

c

4.0

3.5

3.0

Curvature Factor

2.5

2.0 Ki

1.5

B

b/8

Values of Ki for inside fiber as at A

B U or T

Round or Elliptical

Trapezoidal

b/6

B I or hollow rectangular

AB

b/4

A

B

b/2

B

b/3

A

B

A

b

c

A

c

b

A

c

b

A

c r

1.0

Ko

0.5

U or T

I or hollow rectangular

Round, Elliptical or Trapezoidal Values of Ko for outside fiber as atB

0

1 2 3 4 5 6 7 8 9 10 11

Amount of curvature, r/c

5

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