Chapter 3 - Islamic University of...

Chapter 3

Load and

Stress Analysis

Shear Force and Bending

Moments in Beams

Internal shear force V & bending moment M

must ensure equilibrium

Sunday, February 24, 2019

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

2

Sign Conventions for Bending

and Shear


Distributed Load on Beam

Distributed load q(x) = load intensity

Units of force per unit length



4

Relationships between Load, Shear,

and Bending



5

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

moment relations for the beam shown.



6

Example 3-2



7

Singularity Functions


Mohammad Suliman Abuhaiba,

Ph.D., P.E.

8

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.



9

Example 3-3



10

Assignment #3-1

3, 6Program Shear and

moment diagram



11

Stress elementChoosing coordinates which result in zero

shear stress will produce principal stresses



12

Cartesian Stress Components

Shear stress is resolved into components:

1st subscript = direction of surface normal

2nd subscript = direction of shear stress



13

Plane stress occurs = stresses on one surface

are zero


Cartesian Stress Components


14

Plane-Stress Transformation Equations



15

Principal Stresses for Plane Stress

principal directions

principal stresses

Zero shear stresses at principal surfaces

Third principal stress = zero for plane stress



16

Extreme-value Shear Stresses for

Plane Stress

Max shear stresses: on surfaces that are

±45º from principal directions

Two extreme-value shear stresses:



17

Maximum Shear Stress

If principal stresses are ordered so that

s1 > s2 > s3

tmax = t1/3



18

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]


2

2

2

x y

xyRs s

t


19

Mohr’s

Circle



20

Example 3-4

A stress element has σx =

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

principal stresses & directions,

and show on a stress element

correctly aligned wrt xycoordinates.

b. Draw another stress element

to show t1 & t2, find

corresponding normal

stresses, and label drawing.

c. Repeat part a using

transformation equations only.



21

General 3-D Stress

Principal stresses are

found from the roots of

the cubic equation



22

Principal stresses are ordered such that

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


General 3-D Stress


23

Assignment #3-2

15 (a, d), 20

Program Mohr Circle 2D and

3D using SW, Mathematica,

or Matlab



24

Elastic Strain Hooke’s law

For axial stress in x direction,

Table A-5: values for common materials



25

For a stress element undergoing sx, sy, and

sz, simultaneously,


Elastic Strain


26

Hooke’s law for shear:

Shear strain g = change in a right angle ofa stress element when subjected to pure

shear stress

G = shear modulus of elasticity

For a linear, isotropic, homogeneous

material,


Elastic Strain


27

For tension and compression,

For direct shear (no bending present),

Uniformly Distributed Stresses



28

• x axis = neutral axis

Normal Stresses for Straight Beams in

Bending



29

• 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



30


Normal Stresses for Beams in

Bending


31

Example 3-5


A beam having a T section

is subjected to a bending

moment of 1600 N·m,

about the negative z axis,

that causes tension at the

top surface.

1. Locate the neutral axis

2. Find max tensile &

compressive bending

stresses.

Assignment #3-3

25, 29, 34.C, 44Due Tuesday 14/2/2018

Solve and program using mathematical software



33

Bending in both xy & xz planes

Cross sections with one or two planes of

symmetry only

Max bending stress For solid circular

section,

Two-Plane Bending



34

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 max

tensile & 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 max bending stress.



35

Example 3-6Sunday, February 24, 2019


36

Shear Stresses for Beams in Bending



37

Transverse Shear Stress (TSS)

TSS is always accompanied

with bending stress



38

Transverse Shear Stress in a

Rectangular Beam



39

Maximum Values of TSS


Table 3−2


40

Maximum Values of TSS


Table 3−2


41

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),



42

Significance of TSS Compared

to Bending Figure 3–19



43

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


Significance of TSS Compared to Bending


44

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.



45

Example 3-7



46

Torsion

Angle of twist for a solid round bar



47

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



48

Torsional Shear in Rectangular

Section

Shear stress does not vary linearly with

radial distance

Shear stress is zero at corners

Max shear stress is at middle of longest

side



49

Torsional Shear in Rectangular

Section

For rectangular b×c bar, b is longest

side



50

Power, Speed, and Torque

A convenient conversion with speed in rpm

H = power, W

n = angular velocity, rpm



51

Power, Speed, and Torque

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



52

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

directions of coordinate axes on these diagrams.

b. Compute max 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 upon this

element.

d. Determine max normal & shear stresses at A.



53

Example 3-8Sunday, February 24, 2019


54

Example 3-9

1.5”- diameter solid steel shaft is simply

supported at ends. Two pulleys are keyed to

shaft where pulley B is of diameter 4.0 in &

pulley C is of diameter 8.0 in. Considering

bending & torsional stresses only, determine

locations & magnitudes of greatest tensile,

compressive, and shear stresses in shaft.



55

Example 3-9



56

Example 3-9



57

Example 3-9



58

Example 3-9



59

Example 3-9



60

Closed Thin-Walled Tubes

t << r

t × t = constant

t is inversely proportional to t



61

Total torque T is

Am = area enclosed by section median line

Solving for shear stress




62

Angular twist (radians) per unit length

Lm = length of the section median line




63

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 allowable torque T

b. Estimate angle of twist due to the torque



64

Example 3-10



65

Example 3-11

Compare the shear stress on a circular

cylindrical tube with an outside diameter of

1” and an inside diameter of 0.9”,

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

by Eq. (3–45).



66

Open Thin-Walled Sections When 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



67

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



68

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)



69

Assignment #3-4

49, 51, 52, 57, 64Due Wednesday

14/2/2018



70

Stress Concentration

Localized increase

of stress near

discontinuities

Kt = Theoretical

(Geometric) Stress

Concentration

Factor



71

Theoretical Stress

Concentration Factor

A-15 and A-16

Peterson’s Stress-Concentration

Factors



72

Theoretical Stress




73

Theoretical Stress




74

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



75

Techniques to Reduce Stress

Concentration

Increase radius

Reduce disruption

Allow “dead zones” to shape flow lines

more gradually



76


Concentration



77


Concentration



78

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?



79

Example 3-13



80

Example 3-13

Fig. A−15 −1



81

Fig. A−15−5

Example 3-13



82

Assignment #3-5

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



83

Stresses in Pressurized

Cylinders

Tangential and radial stresses



84

Special case of zero outside pressure, po = 0


Stresses in Pressurized Cylinders


85

If ends are closed, then longitudinal stresses

also exist


Stresses in Pressurized Cylinders


86

Thin-Walled Vessels

Cylindrical pressure vessel with wall

thickness ≤ 1/10 the radius

Radial stress is small compared to

tangential stress

Average tangential stress



87

Thin-Walled Vessels

Maximum tangential stress

Longitudinal stress (if ends are closed)



88

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

permissible tangential stress is 12 kpsi &

theory for thin-walled vessels is assumed

to apply?

b. On the basis of pressure found in part

(a), compute stress components using

theory for thick-walled cylinders.



89

Stresses in Rotating Rings

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



90

Stresses in Rotating Rings



91

Press & Shrink Fits

Two cylindrical parts are assembled with

radial interference d

Pressure at interface



92

Press and Shrink Fits

If both cylinders are of the same material



93

Eq. (3-49) for pressure cylinders applies




94

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

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




95

Shrink Fit Bonus

Make a shrink assembly

Calculate the torque capacity

Validate the torque capacity

experimentally.

Due Wednesday 21/2/2018

Optional

Points: 3



96

Assignment #3-6

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



97

Temperature Effects

Normal strain due to expansion from

temperature change

a = coefficient of thermal expansion



98

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



99

Coefficients of Thermal Expansion

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



100

Curved Beams in Bending

In thick curved beams:

Neutral axis & centroidal axis are not

coincident

Bending stress does not vary linearly with

distance from neutral axis



101




102

Location of neutral axis

Stress distribution




103

Stress at inner and outer surfaces




104

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.



105

Example 3-15



106

Formulas for Sections of

Curved Beams (Table 3-4)



107

Formulas for Sections of

Curved Beams (Table 3-4)



108

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



109

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



110

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.



111

Contact (Hertzian) Stresses

Two bodies with curved surfaces pressed

together

Point or line of contact changes to area

contact

Stresses developed are 3-D

Common examples

Wheel rolling on rail

Mating gear teeth

Rolling bearings



112

Spherical Contact Stress

Two solid spheres of diameters d1 & d2

are pressed together with force F

Circular area of contact of radius a



113


Pressure distribution is

hemispherical

Max pressure at center

of contact area



114

Max stresses on z axis

Principal stresses




115


From Mohr’s circle, max shear stress is

For poisson ratio of 0.30,

tmax = 0.3 pmax at depth of z = 0.48a



116




117

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



118


Half-width b

Max pressure



119

Max stresses on z axis




120




121

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



122

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