Lecture Note C01 - Chapter 3 5 - Load and Stress Analysis Failures Resulting From Static Loading

7/23/2019 Lecture Note C01 - Chapter 3 5 - Load and Stress Analysis Failures Resulting From Static Loading

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Mechanical Design

Chapter 3 Load & StressAnalysis

&Chapter 5 Failures

Resulting from Static

Loading

En. Wan Anuar in Wan !assan



Chapter 3 Load & StressAnalysis



"ormal Stress

F

(Eq. 3-22, page 88)



#ending Stress

b

h r

c = h/2 c = r

(Eq. 3-26a, page 90)



$rans%erse Shear Stress

(Table 3.2, page 97)



$orsional Stress

(Eq. 3-37, page 101)

r

c

b

or

Circular Cross Section Rectangular Cross

Section



Force F causes t'isting & ending of a () mm diameter shaft*+ed to a support at the origin of the reference system. ,nactuality the support may e an inertia that 'e 'ish to rotateut for the purposes of a stress analysis 'e can consider this a

static prolem.

a- Calculate the %alue of all forces moments and torues that acton the shaft A# and the arm #C.

- Calculate the ma+imum torsional stress and ending stress inthe arm #C and indicate 'here these act.

c- Calculate the torsional stress and ending stress at the topsurface of the shaft at A.



a- Shaft A#/

at # F 0 1.3 2"

0 1.3 2" + ).)34 m 0 6. ".m

$ 0 1.3 2" + ).1 m 0 13) ".mat A F 0 1.3 2"

0 1.3 2" + ).173 m 0 (11.6 ".m

$ 0 1.3 2" + ).1 m 0 13) ".m

Arm #C/

at C F 0 1.3 2"

0 1.3 2" + ) m 0 ) ".m

$ 0 1.3 2" + ).)34 m 0 6. ".mat # F 0 1.3 2"

0 1.3 2" + ).1 m 0 13) ".m

$ 0 1.3 2" + ).)34 m 0 6. ".m



- ,n arm #C oment at C 0 ) ".m 'hile oment at # 013) ".m. $herefore ma+imum ending stress occurs at #

in memer #C.

0 13) ".m 8).)15 m-

).))7 m 8).)3) m-3

1(

0 1. 9a

$he ma+imum torsional stress occurs in the middle of the3) mm side. 8page 1)( second paragraph last

sentence.-

0 6. ".m + 8 3 : 1.4 -

).)3) m 8).))7 m-( ).)3) m ; ).))7 m

0 153.76 9a



c- ,n shaft A# at A oment 0 (11.6 ".m 'hile $orue 013) ".m. a+imum ending stress at the top at A

0 (11.6 ".m 8).)1) m-

π 8).)1) m-

0 (76.4) 9a

a+imum torsional stress at the top A.

0 13) ".m 8).)1) m-

π 8).)1) m-

(

0 4(.<7 9a



σ x / y / z = Bending Stress + Normal Stress

= Mc + F

I Aτ xy / yz / zx = Torsional Stress + Transverse Stress

= Tr + Transverse Stress

J Refer to Table 3.2, page 97

9rinciple Stress

(Eq. 3-13, page 81)



$he ar is loaded y the forces F 0 ).55 2" P

0 4.) 2" and T 0 3) ".m. Calculate thestresses 8σ x and τ xy) at point A and B.



=i%en F 0 ).55 2" therefore M 0 ).552" + ).1 m 0 55".m.

P 0 4))) " V 0 55) " T 0 3) ".m and M 0 55 ".m.

For stresses at A

( )( ) ( )

MPa 49.95020.00004

020.055!2

4!2

2!

2!

=×+=⇒

+=+=+=

π π

σ

π π

σ σ σ

x

nor b xd

P

d

M

A

P

I

MC

( )

MPa "0."9020.0

!0"#"#

!! =×

===

π π

τ

d

T

J

Tr xy



For stresses at #

( )MPa 4#.25

020.0

0004

4

2

2

=×

=⇒

===

π

σ

π

σ σ

x

nor xd

P

A

P

( ) ( )MPa 4!.2"

020.04

550

!

4

020.0

!0"#

!

4"#

2!! =

+×

=+= π π π

τ

A

V

d

T xy



Chapter 5 Failures

Resulting from StaticLoading



Ductile Materials (yield criteria)

a+imum Shear Stress 8SS-

>istortion Energy 8>E-

>uctile Coulom? ohr 8>C-

Brittle Materials (fracture criteria)

a+imum "ormal Stress 8"S-#rittle Coulom?ohr 8#C-

odi*ed ohr 8-



Failures Theories

Structural metal eha%iour is typicallyductile or rittle.

>uctile aterial/

εf ≥ ).)5@

,denti*ale ield Strength 8Syt0Syc0Sy-.

#rittle aterial/

εf B ).)5

>o not e+hiit an ,denti*ale ield Strength@ $ypically classi*ed y ultimate tensilestrength Sut and ultimate compressi%e

strength Suc 8in :%e %alue-.



Selection of FailureCriteria

14



>uctile aterials



a+imum Shear Stress 8SS- $heory

$he principal stresses gi%en y E. 83?13- are

assumed as $A and $B. T%en t%e& are ordered 'it% t%e(ero prinipal stress aording to t%e onvention $"*$2*$!.

Assme t%at $A*$B.

,or design prposes-

(Eq. 5-3, page 220)



>istortion Energy 8>E- $heory

on ises Stress $.

(Eq. 5-12, page 223)

(Eq. 5-13, page 223)

(Eq. 5-14, page 223)

(Eq. 5-15, page 223)

(Eq. 5-19, page 223)



>uctile Coulom?ohr 8>C- $heory"ot all materials ha%e compressi%e strengths

eual to their tensile strengths.

S t = S yt and S c = S yc

(Eq. 5-26, page 229)



#rittle aterials



a+imum "ormal Stress 8SS- $heory

(Eq. 5-30, page 235)



#rittle Coulom?ohr 8#C- $heory

(Eq. 5-31, page 236)



odi*ed ohr 8- $heory

(Eq. 5-32, page 236)

I t d ti t F t



Introduction to FractureMechanicsCrac2 odes & Stress ,ntensity Factor

$hree distinct modes of crac2propagation e+ist.

◦ ode ,/ the opening crac2

propagation mode

◦ ode ,,/ the sliding mode

◦ ode ,,,/ the tearing mode.



Stress ,ntensity Factor K I

'here β is the Stress ,ntensity odi*cation

FactorFrom Figures 5?(5 to 5?3).

Critical Stress ,ntensity Factor K Ic

also called as Fracture $oughness of thematerial.



Figure 5.(5/ D?center crac2 in a plate inlongitudinal tension.



Figure 5.(7/ 9late loaded in longitudinaltension 'ith a crac2 at the edge.



$he strength?to?stress ratio K Ic ; K , can e

used as a factor of safety as@

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Lecture Note C01 - Chapter 3 5 - Load and Stress Analysis Failures Resulting From Static Loading