Basic stress equations

(1)

Basic Stress Equations

Internal Reactions: Internal Reactions: 6 Maximum 6 Maximum (3 Force Components (3 Force Components & 3 Moment Components) & 3 Moment Components)

Normal Force Normal Force )) τ τ τ τ (( )) σ σ σ σ (( Shear Forces Shear Forces zz x x y y P P V Vyy V V_x_x Torsional Moment Torsional Moment )) τ τ τ τ (( )) σ σ σ σ (( Bending Moments Bending Moments zz x x y y T T M M_y_y M M_x_x or Torque or Torque F

F orce Comporce Components onents MM omeoment Component Components nts "Cut Surface"

"Cut Surface" "Cut Surface""Cut Surface" Centroid of

Centroid of Cross Section

Cross Section Centroid of Centroid of Cross SectionCross Section

Normal Force:

Axial Force Axial Force zz x x y y P P Centroid Centroid σ σ σ σ Axial Stress Axial Stress "Cut Surface" "Cut Surface" _σ_{σ =}₌

P

A

l

l Uniform over tUniform over the entire cross he entire cross section.section.

l

l Axial force Axial force must go through must go through centroid.centroid.

Shear Forces:

Cross Section Cross Section y y A Aaa Point of interest Point of interest

LINE perpendicular to V through point of LINE perpendicular to V through point of interestinterest

= Length of LINE on the cross section = Length of LINE on the cross section

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

Centroid of area on one side of the LINE Centroid of area on one side of the LINE

= distance between the two centroids = distance between the two centroids = Area moment of inertia of

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

V V b b A Aaa y y II "y" Shear Force

"y" Shear Force

zz x x y y V Vyy "x" Shear Force "x" Shear Force zz x x y y V V_x_x τ τ τ τ τ τ τ τ τ τ == ⋅ ⋅ ⋅⋅ ⋅⋅

V

V A

A

y

II b

b

aa

b b

gg

Note

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

Crroosss s SSeeccttiioonn:: CCrroosss s SSeeccttiioonn::

Rectangular:

Rectangular: ττ_max_max = =

3 3 2

2 V

⋅⋅

V A

A

Solid Circular:Solid Circular: ττ_max_max = =

4 3

4 3 V

⋅⋅

V A

A

I-Beam or I-Beam or H-Beam:

H-Beam: _flange_flange _web_web ττ_max_max ==

V

V A

A

_web_web

Thin-walled Thin-walled

tube:

(2)

Torque or Torsional Moment:

Solid Circular or Tubular Cross Section:

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

D = Shaft Diameter

J

D

R

J

D

for solid circular shafts

for hollow shafts

o i = ⋅ = ⋅ = ⋅ − π π π 4 4 4 4

32

2

32 e

j

Torque z x y T "Cut Surface" τ τ τ =

T r

⋅

J

τ π τ π max max = ⋅ ⋅ = ⋅ ⋅ ⋅ −

16

3 4 4

T

D

T D

D

for solid circular shafts

for hollow shafts o

o i

e

j

Rectangular Cross Section:

Torque z x y T Centroid τ τ Torsional Stress "Cut Surface" τ τ₁ 2 a b Note: a > b Cross Section: Method 1:

τ _max= τ₁ = ⋅ ⋅ + ⋅

T

b

3 a

1 8

.

b

g

e j

a

2⋅

b

2 ONLY applies to the center of the longest side Method 2: τ α 1 2 1 2 2 , , = ⋅ ⋅

T

a b

a/b α α αα 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 αα from the table on the right to get the shear stress at either position 1 or 2.

Other Cross Sections:

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Bending Moment

"x" Bending Moment z x y z x y M_x σ σ σ σ M_y

"y" Bending Moment

σ = ⋅ σ = ⋅

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 R

I

b h

h

Z

I

c

b h

is perpendicular to axis = ⋅ = = ⋅ 3 2

12

6 I

D

R

Z

I

c

D

R

= ⋅ = ⋅ = = ⋅ = ⋅ π π π π 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 axis} A = Area of the region

d = perpendicular distance between the two axes.

Maximum Bending Stress Equations:

σ π max = ⋅ ⋅

32

3

M

D

b

Solid Circular

g

σmax =

⋅ ⋅

6

2

M

b h

a

Rectangular

f

σ_max =

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,

ρρ

.

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

E

⋅

y

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). ρ

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Bending Moment in Curved Beam:

Geometry:

r

e

r

_n σ σ σ σo i centroid centroidal neutral axis axis o i

y

nonlinear stress distribution

M

c

_i o ρ ρ

r

A

dA

e

r

n area n = = −

z

ρ

A = cross sectional area

r

_{n = radius to neutral axis}

r

= radius to centroidal axis e = eccentricity

Stresses:

Any Position: Inside(maximum magnitude): Outside:

σ = − ⋅ ⋅ ⋅ +

M y

e A r

b

_n

y

g

σi i i

M c

e A r

= ⋅ ⋅ ⋅ σo o o

M c

e A r

= − ⋅ ⋅ ⋅

Area Properties for Various Cross Sections:

Cross Section

r

_area

dA

ρ

z

A

t

h

r

_i

r

_o

r

Rectangle

ρ

_r

h

i +

2 t

r

o i ⋅

F

H G

I

K J

ln

h t

⋅

t

h

r

_i

r

_o

r

Trapezoid

ρ o

t

_i

r

h t

t

i i o i o + ⋅ + ⋅ ⋅ +

2

3 b

g

b

g

For triangle: set ti or to to

0 t

t

r

t

r t

h

r

o i o i i o o i − + ⋅ − ⋅ ⋅

F

H G

I

K J

ln

h

⋅

t

i +

t

o

2 r

Hollow Circle

ρ

a

b

r

2

⋅π

L

NM

r

2 −

b

2 −

r

2 −

a

2

QP

O

π ⋅

a

−

b

2 2

e

j

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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.