Basic stress equations

 Dr. D. B. Wallace

Basic Stress Equations Centroid of  Cross Section

Internal Reactions:

Centroid of  Cross Section

Shear Forces (τ )

y

6 Maximum (3 Force Components  & 3 Moment Components)

y

x

"Cut Surface"

Vx

x

"Cut Surface"

Mx

z

P

Vy

Bending Moments ( σ)

My

 Normal Force (σ)

T

z

Torsional Moment or Torque ( τ )

F orce Components  Components 

M oment oment Components  Components 

Normal Force: Centroid y

σ

x

"Cut Surface"

z P Axial Force

P

σ=

l

A

l

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

Axial Stress

Shear Forces: Cross Section y

"y" Shear Force

Point of interest LINE perpendicular to V through point of interest

V

x

b = Length of LINE on the cross section z

Vy

Aa = Area on one side of the LINE

τ y

y

Centroid of entire cross section

"x" Shear Force

Vx

Centroid of area on one side of the LINE

Aa

x

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

τ z

y = distance between the two centroids

τ=

b

V ⋅ Aa ⋅ y

g

I⋅b

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

Cross Section:

Cross Section:

τ max = 3

Rectangular:

I-Beam or   H-Beam:

flange

web τ max =

2⋅V A

V A web

Solid Circular:

Thin-walled tube:

τ max = 4

τ max =

3⋅ V A

2⋅V A

 Basic Stress Equations

Dr. D. B. Wallace

Torque or Torsional Moment: Solid Circular or Tubular Cross Section: y x

"Cut Surface"

τ=

T ⋅ r  J

z

T Torque

τ max = τ max =

π⋅D

3

π⋅D

32 π⋅

for solid circular shafts

16 ⋅ T ⋅ Do π⋅

J=

τ

16 ⋅ T

r = Distance from shaft axis to point of interest R = Shaft Radius D = Shaft Diameter 

J=

4

4

=

π ⋅ R 

2

e Do4 − Di 4 j 32

for solid circular shafts

for hollow shafts

for hollow shafts

e Do 4 − D i 4 j

Rectangular Cross Section: Cross Section: b

τ2

y Centroid x

"Cut Surface"

τ1

z T Torque

Note: a>b

a

Torsional Stress

Method 1: τ max = τ1 = T ⋅

b3 ⋅ a + 1.8 ⋅ bg ea 2 ⋅ b 2 j

ONLY applies to the center of the longest side

Method 2: τ1 ,2 =

T α1 ,2 ⋅ a ⋅ b

2

Use the appropriate α from the table  on the right to get the shear stress at  either position 1 or 2.

a/b 1.0 1.5 2.0 3.0 4.0 6.0 8.0 10.0 ∞

Other Cross Sections: Treated in advanced courses.

2

α

.208 .231 .246 .267 .282 .299 .307 .313 .333

α

.208 .269 .309 .355 .378 .402 .414 .421 ----

 Basic Stress Equations

Dr. D. B. Wallace

Bending Moment "x" Bending Moment

y

x

Mx

σ=

σ

z

Mx ⋅ y Ix

and

My ⋅ x

σ=

Iy

"y" Bending Moment y

where: Mx and My are moments about indicated axes y and x are perpendicular from indicated axes

x z

My

Ix and Iy are moments of inertia about indicated axes σ

Moments of Inertia:  b

I=

c

 b ⋅ h3 12

h

I

Z=

c

=

D

h is perpendicular to axis  b ⋅ h

c

I =

R 

2

Z =

6

π⋅ D

64 I c

=

4

4

=

π ⋅ R 

π⋅D

32

4 3

3

=

π ⋅ R 

4

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

new axis Area, A

d

I  = Moment of inertia about the centroidal axis

I = I + A ⋅d2

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

centroid

Maximum Bending Stress Equations: σ max =

M⋅c I

=

M

σ max =

Z

32 ⋅ M π⋅D

3

bSolid Circular g

σ max =

6⋅ M  b ⋅ h 2

a Rectangular f

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

σ= After:

E⋅

y ρ

E = Modulus of elasticity of the beam material M

ρ

y = Perpendicular distance from the centroidal axis to the  point of interest (same y as with bending of a straight beam with Mx).

M

ρ =

3

radius of curvature to centroid of cross section

 Basic Stress Equations

Dr. D. B. Wallace

Bending Moment in Curved Beam: Geometry: σo

nonlinear  stress distribution

centroid

centroidal axis

y

co

e

ci σi

rn

neutral axis

r n =

z 

ro

A = cross sectional area

r n = radius to neutral axis

area ρ

ρ r

ri

A dA

r  = radius to centroidal axis e = eccentricity

e = r − r n

M

Stresses: Any Position: σ=

Inside (maximum magnitude):

−M ⋅ y

b

e ⋅ A ⋅ rn + y

σi =

g

Outside:

M ⋅ ci

σo =

e ⋅ A ⋅ r i

− M ⋅ co

e ⋅ A ⋅ r o

Area Properties for Various Cross Sections:

Cross Section Rectangle r  ρ

r i +

t r i

dA

area

h

t ⋅ ln

2

A

ρ

F r o I  GH r i J K 

h⋅t

h r o

Trapezoid r  ρ

z 

r 

r i +

ti

r i

to

b

h ⋅ ti + 2 ⋅ to

b

3⋅ ti + t o

g

g to − ti +

ro ⋅ t i − ri ⋅ t o h

For triangle: 

h

⋅ ln

F r o I  GH r i J K 

h⋅

ti + t o 2

set ti or to to 0

r o

Hollow Circle r  a ρ

r 

2⋅π

 b

4

L MN

r 2 − b2 −

r 2 − a2

O PQ

π⋅

ea 2 − b2 j

 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

σi =

K i ⋅

r

Outside F iber: 

σo =

K o ⋅

M

I M⋅c I

B 

A

 b/8

4.0

B 

A  b

 b/4

c

Values of Ki for inside fiber as at A

3.5

B

B 

A

A

c

U or T  b/2

Curvature Factor 

Round or Elliptical

 b

A

B 

c

2.5 Trapezoidal

 b/3

 b/6

2.0

B

 b

B

A

A

c

I or hollow rectangular 

K i

M

M⋅c

I nside Fiber: 

3.0

c

r

1.5 1.0 K o

I or hollow rectangular  U or T

0.5

Round, Elliptical or Trapezoidal

Values of Ko for outside fiber as at B 

0 1

2

3

4

5

6

7

Amount of curvature, r/c

5

8

9

10

11