Chapter 3site.iugaza.edu.ps/mhaiba/files/2012/01/Ch-3-Load-and-Stress-Analys… · a. Using...

Chapter 3

Load and

Stress Analysis

Chapter Outline

Wednesday, January 31,

2018

Equilibrium & FBDs

Shear Force & Bending

Moments in Beams

Stress

Cartesian Stress

Components

Mohr’s Circle for Plane

Stress

General 3-D Stress

Elastic Strain

Uniformly Distributed

Stresses

Normal Stresses for Beams

in Bending

Shear Stresses for Beams in

Bending

Torsion

Stress Concentration

Stresses in Pressurized

Cylinders

Stresses in Rotating Rings

Press and Shrink Fits

Temperature Effects

Curved Beams in Bending

Contact Stresses

Mohammad Suliman Abuhaiba, Ph.D., P.E.

2

Shear Force and Bending

Moments in Beams

Internal shear force V & bending moment M

must ensure equilibrium


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3

Sign Conventions for Bending

and Shear


Distributed Load on Beam

Distributed load q(x) = load intensity

Units of force per unit length


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5

Relationships between Load, Shear,

and Bending


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6

Example 3-2Derive the loading, shear-force, and bending-

moment relations for the beam shown.


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Example 3-2


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Example 3-3

The Figure shows the loading diagram for a

beam cantilevered at A with a uniform load of 20

lbf/in acting on the portion 3 in ≤ x ≤ 7 in, and a

concentrated ccw moment of 240 lbf.in at x = 10

in. Derive the shear-force and bending moment

relations, and the support reactions M1 and R1.


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Example 3-3


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Assignment #3-1

3, 6Due Saturday 3/2/2018

Program Shear and

moment diagram


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11

Stress elementChoosing coordinates which result in zero

shear stress will produce principal stresses


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12

Cartesian Stress Components

Shear stress is resolved into components:

1st subscript = direction of surface normal

2nd subscript = direction of shear stress


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13

Plane stress occurs = stresses on one surface

are zero


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Cartesian Stress Components


14

Plane-Stress Transformation Equations


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Principal Stresses for Plane Stress

principal directions

principal stresses

Zero shear stresses at principal surfaces

Third principal stress = zero for plane stress


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Extreme-value Shear Stresses for

Plane Stress

Max shear stresses: on surfaces that are

±45º from principal directions

Two extreme-value shear stresses:


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Maximum Shear Stress

If principal stresses are ordered so that

s1 > s2 > s3

tmax = t1/3


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Mohr’s Circle Diagram

Relation between x-y stresses and principal

stresses is a circle with center at:

C = (s, t) = [(sx+ sy)/2, 0]


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2

2

2

x y

xyRs s

t


19

Mohr’s

Circle

Diagram


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20

Example 3-4

A stress element has σx =

80 MPa & τxy = 50 MPa cw.a. Using Mohr’s circle, find the

principal stresses & directions,

and show these on a stress

element correctly aligned wrt

xy coordinates. Draw another

stress element to show t1 &

t2, find corresponding normal

stresses, and label the

drawing completely.

b. Repeat part a using

transformation equations only.


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General 3-D Stress

Principal stresses are

found from the roots of

the cubic equation


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Principal stresses are ordered such that

s1 > s2 > s3, in which case tmax = t1/3


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General 3-D Stress


23

Assignment #3-2

15 (a, d), 20Due Sunday 11/2/2017

Program Mohr Circle 2D and

3D using SW, Mathematica,

or Matlab


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Elastic StrainHooke’s law

For axial stress in x direction,

Table A-5: values for common materials


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25

For a stress element undergoing sx, sy, and

sz, simultaneously,


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Elastic Strain


26

Hooke’s law for shear:

Shear strain g = change in a right angle of astress element when subjected to pure

shear stress

G = shear modulus of elasticity

For a linear, isotropic, homogeneous

material,


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Elastic Strain


27

For tension and compression,

For direct shear (no bending present),

Uniformly Distributed Stresses


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• x axis = neutral axis

Normal Stresses for Straight Beams in

Bending


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• Neutral axis coincides with

centroidal axis of x-section

• xz plane = neutral

plane

Assumptions for Normal Bending Stress

Pure bending

Material is isotropic & homogeneous

Material obeys Hooke’s law

Beam is initially straight with constant x-section

Beam has axis of symmetry in plane of

bending

Failure is by bending rather than crushing,

wrinkling, or sidewise buckling

Plane cross sections remain plane during

bending


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2018

Normal Stresses for Beams in

Bending


31

Max bending stress is where y is greatest

c = magnitude of greatest y

Z = I/c = section modulus


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Normal Stresses for Beams in

Bending


32

Example 3-5


A beam having a T section

with the dimensions shown

is subjected to a bending

moment of 1600 N·m,

about the negative z axis,

that causes tension at the

top surface. Locate the

neutral axis and find

maximum tensile &

compressive bending

stresses.

Assignment #3-3

25, 29, 34.C, 44Due Tuesday 15/2/2017

Solve and program using mathematical software


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34

Consider bending in both xy & xz planes

Cross sections with one or two planes of

symmetry only

Maximum bending stress For solid circular

cross section,

Two-Plane Bending


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Example 3-6

Beam OC is loaded in the xy plane by a uniform

load of 50 lbf/in, and in the xz plane by a

concentrated force of 100 lbf at end C. The

beam is 8 in long.

a. For the cross section shown determine the

max tensile and compressive bending

stresses and where they act.

b. If the cross section was a solid circular rod of

diameter, d = 1.25 in, determine the

magnitude of the maximum bending stress.


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Example 3-6Wednesday, January 31,

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37

Shear Stresses for Beams in Bending


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Transverse Shear Stress (TSS)

TSS is always accompanied

with bending stress


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Transverse Shear Stress in a

Rectangular Beam


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Maximum Values of TSS


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Table 3−2


41

Maximum Values of TSS


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Table 3−2


42

Significance of TSS Compared

to Bending

Figure 3–19: Plot of max shear stress for a

cantilever beam, combining effects of

bending & TSS

Max shear stress, including bending stress

(My/I) and transverse shear stress (VQ/Ib),


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Significance of TSS Compared

to Bending Figure 3–19


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44

Critical stress element (largest tmax) willalways be either due to:

bending, on outer surface (y/c=1), TSS = 0

TSS at neutral axis (y/c=0), bending is zero

Transition at some critical value of L/h


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Significance of TSS Compared to Bending


45

Example 3-7A beam 12” long is to support a load of 488 lbf

acting 3” from the left support. The beam is an I

beam with cross-sectional dimensions shown.

Points of interest are labeled a, b, c, and d. At

the critical axial location along the beam, find

the following information:

a. profile of distribution of TSS, obtaining values

at each of the points of interest.

b. bending stresses at points of interest.

c. max shear stresses at points of interest, and

compare them.


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46

Example 3-7


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47

Torsion

Angle of twist for a solid round bar


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48

Assumptions for Torsion Equations

Pure torque

Remote from any discontinuities or point of

application of torque, Material obeys Hooke’s

law

Adjacent cross sections originally plane &

parallel remain plane & parallel

Radial lines remain straight: Depends on axi-

symmetry, so does not hold true for noncircular

cross sections

Only applicable for round cross sections


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49

Torsional Shear in Rectangular

Section

Shear stress does not vary linearly

with radial distance

Shear stress is zero at the corners

Max shear stress is at the middle of

the longest side


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50

Torsional Shear in Rectangular

Section

For rectangular b×c bar, where b is

longest side


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51

Power, Speed, and Torque

A convenient conversion with speed in rpm

H = power, W

n = angular velocity, rpm


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Power, Speed, and Torque

U.S. Customary units (built in unit conversion)


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Example 3-8The Figure shows a crank loaded by a force F = 300

lbf that causes twisting and bending of a

0.75”diameter shaft fixed to a support at the origin of

the reference system.

a. Draw FBDs of shaft AB & arm BC, and compute values

of all forces, moments, and torques that act. Label

the directions of coordinate axes on these diagrams.

b. Compute maxima of torsional stress and bending

stress in arm BC and indicate where these act.

c. Locate a stress element on top surface of shaft at A,

and calculate all stress components that act upon

this element.

d. Determine max normal & shear stresses at A.


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Example 3-8


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Example 3-9

The 1.5”-diameter solid steel shaft is simply

supported at the ends. Two pulleys are

keyed to the shaft where pulley B is of

diameter 4.0 in and pulley C is of diameter

8.0 in. Considering bending and torsional

stresses only, determine the locations and

magnitudes of the greatest tensile,

compressive, and shear stresses in the shaft.


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Example 3-9


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Example 3-9


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Example 3-9


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Example 3-9


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Example 3-9


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61

Closed Thin-Walled Tubes

t << r

t × t = constant

t is inversely proportional to t


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Total torque T is

Am = area enclosed by section median line

Solving for shear stress


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63

Angular twist (radians) per unit length

Lm = length of the section median line


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64

Example 3-10

A welded steel tube is 40 in long, has a 1/8 in

wall thickness, and a 2.5-in by 3.6-in

rectangular x-section. Assume an allowable

shear stress of 11.5 kpsi and a shear modulus

of 11.5 Mpsi.

a. Estimate the allowable torque T

b. Estimate angle of twist due to the torque


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65

Example 3-10


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Example 3-11

Compare the shear stress on a circular

cylindrical tube with an outside diameter of

1 in and an inside diameter of 0.9 in,

predicted by Eq. (3–37), to that estimated

by Eq. (3–45).


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Open Thin-Walled SectionsWhen the median wall line is not closed,

the section is said to be an open section

Torsional shear stress

T = Torque, L = length of median line, c =

wall thickness, G = shear modulus, and q1 =angle of twist per unit length


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Open Thin-Walled Sections

For small wall thickness, stress and

twist can become quite large

Example:

Compare thin round tube with and

without slit

Ratio of wall thickness to outside

diameter of 0.1

Stress with slit is 12.3 times greater

Twist with slit is 61.5 times greater


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Example 3-12

A 12” long strip of steel is 1/8”

thick and 1” wide. If the

allowable shear stress is 11500

psi and the shear modulus is 11.5

Mpsi, find the torque

corresponding to the allowable

shear stress and the angle of

twist, in degrees,

a. using Eq. (3–47)

b. using Eqs. (3–40) and (3–41)


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Assignment #3-4

49, 51, 52, 57, 64Due Sunday 21/2/2016


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71

Stress Concentration

Localized increase

of stress near

discontinuities

Kt = Theoretical

(Geometric) Stress

Concentration

Factor


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Theoretical Stress

Concentration Factor

A-15 and A-16

Peterson’s Stress-Concentration

Factors


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Theoretical Stress



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Theoretical Stress



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75

Stress Concentration for Static

and Ductile Conditions

With static loads and ductile materials

Highest stressed fibers yield (cold work)

Load is shared with next fibers

Cold working is localized

Overall part does not see damage unless

ultimate strength is exceeded

Stress concentration effect is commonly

ignored for static loads on ductile

materials


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Techniques to Reduce Stress

Concentration

Increase radius

Reduce disruption

Allow “dead zones” to shape flow lines

more gradually


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Concentration


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Concentration


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Example 3-13The 2-mm-thick bar shown is loaded axially

with a constant force of 10 kN. The bar

material has been heat treated and

quenched to raise its strength, but as a

consequence it has lost most of its ductility. It

is desired to drill a hole through the center of

the 40-mm face of the plate to allow a cable

to pass through it. A 4-mm hole is sufficient for

the cable to fit, but an 8-mm drill is readily

available. Will a crack be more likely to

initiate at the larger hole, the smaller hole, or

at the fillet?


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Example 3-13


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Example 3-13

Fig. A−15 −1


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Fig. A−15−5

Example 3-13


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Assignment #3-5

68, 72, 84Due Monday 27/2/2017


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84

Stresses in Pressurized

Cylinders

Tangential and radial stresses


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85

Special case of zero outside pressure, po = 0


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Stresses in Pressurized Cylinders


86

If ends are closed, then longitudinal stresses

also exist


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Stresses in Pressurized Cylinders


87

Thin-Walled Vessels

Cylindrical pressure vessel with wall

thickness ≤ 1/10 the radius

Radial stress is small compared to

tangential stress

Average tangential stress


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Thin-Walled Vessels

Maximum tangential stress

Longitudinal stress (if ends are closed)


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Example 3-14An aluminum-alloy pressure vessel is made

of tubing having an outside diameter of 8 in

and a wall thickness of ¼ in.

a. What pressure can the cylinder carry if

the permissible tangential stress is 12 kpsi

and the theory for thin-walled vessels is

assumed to apply?

b. On the basis of the pressure found in part

(a), compute the stress components

using the theory for thick-walled

cylinders.


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90


Rotating rings: flywheels, blowers, disks, etc.

Tangential and radial stresses are similar to

thick-walled pressure cylinders, except

caused by inertial forces

Conditions:

Outside radius is large compared with

thickness (>10:1)

Thickness is constant

Stresses are constant over the thickness


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92


Two cylindrical parts are assembled with

radial interference d

Pressure at interface


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93


If both cylinders are of the same material


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94

Eq. (3-49) for pressure cylinders applies


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95

For the inner member, po = p and pi = 0

For the outer member, po = 0 and pi = p


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96

Shrink Fit Bonus

Make a shrink assembly

Calculate the torque capacity

Validate the torque capacity

experimentally.

Due Wednesday 1/3/2017

Optional

Points: 3


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Assignment #3-6

94, 104, 110Due Saturday 4/3/2017


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98

Temperature Effects

Normal strain due to expansion from

temperature change

a = coefficient of thermal expansion


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99

Temperature Effects

Thermal stresses occur when members are

constrained to prevent strain during

temperature change

For a straight bar constrained at ends,

temperature increase will create a

compressive stress

Flat plate constrained at edges


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100

Coefficients of Thermal Expansion

Table 3–3: Coefficients of Linear Thermal Expansion (0 –100°C)


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101


In thick curved beams:

Neutral axis & centroidal axis are not

coincident

Bending stress does not vary linearly with

distance from neutral axis


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103

Location of neutral axis

Stress distribution


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104

Stress at inner and outer surfaces


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Example 3-15

Plot the distribution of stresses across

section A–A of the crane hook shown. The

cross section is rectangular, with b = 0.75 in

and h = 4 in, and the load is F = 5000 lbf.


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Example 3-15


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Formulas for Sections of

Curved Beams (Table 3-4)


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Formulas for Sections of

Curved Beams (Table 3-4)


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Formulas for Sections of Curved Beams (Table 3-4)


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Alternative Calculations for e

Approximation for e, valid for large

curvature where e is small

Substituting Eq. (3-66) into Eq. (3-64), with rn– y = r, gives


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Example 3-16

Consider the circular section in Table 3–4

with rc = 3 in and R = 1 in. Determine e by

using the formula from the table and

approximately by using Eq. (3–66).

Compare the results of the two solutions.


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Contact Stresses

Two bodies with curved surfaces pressed

together

Point or line of contact changes to area

contact

Stresses developed are 3-D

Contact stresses or Hertzian stresses

Common examples

Wheel rolling on rail

Mating gear teeth

Rolling bearings


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Spherical Contact Stress

Two solid spheres of diameters d1 and d2

are pressed together with force F

Circular area of contact of radius a


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Pressure distribution is

hemispherical

Max pressure at the

center of contact area


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Max stresses on z axis

Principal stresses



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116


From Mohr’s circle, max shear stress is

For poisson ratio of 0.30,

tmax = 0.3 pmax at depth of z = 0.48a


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Cylindrical Contact Stress

Two right circular

cylinders with length l

and diameters d1 & d2

Area of contact is a

narrow rectangle of

width 2b and length l

Pressure distribution is

elliptical


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Half-width b

Maximum pressure


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Maximum stresses on z axis


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122

For poisson ratio of 0.30,

tmax = 0.3 pmax at depth of z = 0.786b

Assignment #3-7

129, 133, 138Due Monday 6/3/2017


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