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Priit Põdra 3. Strength of Components under Shearing Load 1

STRENGTH OF MATERIALSSTRENGTH OF MATERIALS

3. Strength of Components under Shearing Load

3. Strength of Components under Shearing Load

Mehhanosüsteemide komponentide õppetool

3.1

Shearing Load in Joints

3.2

Short Bar Under Transverse Load

3.3

Internal Forces and Stress Analysis

3.4

Strength Calculations



3.1. Shearing Load in Joints3.1. Shearing Load in Joints

Priit Põdra

Force

Bar

Shearing Load

3. Strength of Components under Shearing Load 3

Shearing Load Action in GeneralShearing Load Action in General

In transverse direction to the bar’ axis

Zoom

Very short distance ---bending is irrelevant

Cut or „Sheared“ Bar

Cutting Surface

As if cut using scissors or cutter

Shear Area


Components under Shearing Action Components under Shearing Action

Pinned Joint

F

Pins

Riveted Joint

F F

Rivets

Welded Joint

F WeldmentBearing Support

F

Axle

Roller

Belt

Pinned Joint

Pin

F F



3.2. Short Bar Under Transverse Load3.2. Short Bar Under Transverse Load


Shearing DeformationShearing Deformation

Short Bar under Transverse ActionShearing Action Area

Bending deformation is negligibly small

F

Axle

Roller

BeltHousing

Shear Area

ZoomRoller

Axle

Transverse loading to the Axle

Axle material parallel internal layers

(surfaces) slide past each other


Shearing Sections of Pure ShearShearing Sections of Pure Shear

Shear Stresses on Cross-Sections

EFBC

Other sections, EC and BF, must also

have shear stresses

EFBCECBF

C

BF

E

Shear Stresses on Cross-Sections

BC EF

C

BF

E

Element in Equilibrium

BC EF

BC

EC

C

B F

E

Requirement of Equilibrium

BCEF is NOT in Equilibrium

BCEF is now in Equilibrium


Principle of Complementary Shear StressesPrinciple of Complementary Shear Stresses

A shear stress is always accompanied by a balancing shear stress across the section at right angles – they both have the same sign and the same magnitude

(they both are directed towards to or outwards from the common edge of these planes)

(This is valid for every laoding case)

Shear stresses always act IN PAIRS:

For each shear stress there is a COMPLEMENTARY

shear stress

xy is accompanied by equivalentyx

yz is accompanied by equivalent zy

xzzx

zyyz

yxxy

y

z

xzy (+)

yz (+)yx (-)

xy (-)

xz (+)zx (+)

Principle of Complementary Shear Stresses

zx is accompanied by equivalent xz


Shearing Deformations of a Short BarShearing Deformations of a Short Bar

Structural Model of the Bar and Transverse Deformation

NB! Principle of complementary Shear Stresses

Shear Strains

uvxy vuyx

yxxyyxxy Total Shear Strain

u

v

u

v

xy

yx

y

x

F

v

v

y

xxy

Shear Stress in Cross-Section

uuy

x

yx

Complementary Shear Stress

Priit Põdra 3. Strength of Components under Shearing Load

Bearing Stress in the JointBearing Stress in the Joint

Compressive stresses are acting in the bearing area

Compressive deformations

F

Axle

Roller

BeltHousing

Shear Area Bearing Action Area

y

z

F

Roller

Belt

Belt loading action is transferred to the axle through the roller-axle

bearing/contact surface

NB! There is a danger of bearing surface damage in the joint

10



3.3. Internal Forces and Stress Analysis3.3. Internal Forces and Stress Analysis


Internal Forces of a Bar under ShearInternal Forces of a Bar under Shear

F

1st Shear Area

2nd Shear Area

Mechanical system

loaded by gravity

Bar under transverse loading: Two shear areas

in this case

Load is divided equally between two shear areas

221

FFF

Acts transversly to the bar’ axis

External Force F2

F2

Internal Force

Load to the 2nd

shear area

Interaction between the particles of a loaded bar,

resisting the shearing deformations and fracture

Is a result of the load F action

Zoom


Shear ForceShear Force

Shear Force Q, [N] = resultant of the particles’ translatory and in-plane

interaction on a given section

Cross-Section

External Force = Load F

Q

Interaction between the particles

Shear Force

Transverse to the bar’ axis

Resultant of the particles’ interaction

forces due to transverse loading

In English/US literature usually denoted with V


Shear Force at the Bar’ Cross-SectionShear Force at the Bar’ Cross-Section

Shear forces (in the bar’ shear areas) are determined using the method of sections

Shear Areas

Bar

Section I Section II

F

Bar under Shearing Action

Bar

F

F1 F2

Active Load

Support Reactions

Free Body Diagram

21 FFF

Shear Forces in Sections

QI = F1

F1Section I

QII = F2

F2Section II

m = 2

In the cases of symmetrical loading pattern, the shear forces at sections are all equal

m

FQQQQ m ...III

Shear Force at a Section

Number of Equally Loaded Sections


Bar’ Load State — ShearBar’ Load State — Shear

F

Shear Areas

Bar

F

Bar under Shearing Action Material „layers“ slide

past each other in the shear areas

As if in cutting process

Shear = (simple) load state of a bar, where in the shear areas:• only the shear force is considered in the cross-section (no bending),• sections slide in-plane and perpendicular to the bar’ axis relative to

each other,• sections remain flat and parallel.


Section’ Shear Stress DistributionSection’ Shear Stress Distribution

F Active Load

Bar’ Structural Model

SectionShear Force

Section’ Internal Force

F

Q

ACross-Section

Area

A

Q

STRESS = Intensity of Internal Force

Shear stress distribution on sections is considered uniform

Diagrams of Shear

Stress

A

A

Q

AA

Q

This is a simplification and valid only, if the material state is very

close to the limiting one


Contact ForcesContact Forces

1st Bearing Area

Axle

F

Bar under Transverse Loading

Housing

Rod2nd Bearing

Area

3rd Bearing Area

FC

Zoom

External Force

Interaction between the particles of a loaded bodies, resisting the

deformations and fracture

„Internal“ Force

Axle

Housing

Loading of the 2nd

Bearing Area

Symmetrical loading pattern

Loading of the 1st

Bearing Area is identical

Area distributed load is acting in

the contact between bodies

There is a danger of contact

deformations

This danger increases with the decrease of bearing area size


Real and Projected Bearing AreaReal and Projected Bearing Area

Real Bearing Area

Axle

FC Projected Bearing Area

FC

AC

b

dAxle

Real bearing area is replaced by the

projected one

bdA C

Projected Bearing Area, AC = projection of the real bearing/contact are on the axial section

Real Bearing Area Projected Bearing Area


Contact’ Bearing Stress DistributionContact’ Bearing Stress Distribution

This is a simplification -- contact stress analysis is not a subject of

Strength of Materials

b

d

F

Axle

Real Bearing Area

Loading of the Bearing Area

bdA C

F Real Bearing Area

Projected Bearing Area

STRESS = Intensity of Internal Force

Bearing stress distribution on the contact bearing area is considered uniform C

C A

F

Bearing Area Load

Projected Bearing Area



3.4. Strength Calculations3.4. Strength Calculations


Strength ConditionsStrength Conditions

Strength Analysis for Shear when Shear Force Q is the most dangerous internal force

(Other internal forces are either absent or their action is negligible)

Strength Condition for Shear: The real values of the shear stress in the bar must not exceed the value of the design

stress (for shear)

Strength Analysis for Bearing when the load acts through a contact area(There is a danger of surfaces’ damage)

Strength Condition for Bearing: The real values of the bearing stress on any surface of any component must not

exceed the respective values design stress for bearing CC

Design Stress for Shear

Design Stress for Bearing

Priit Põdra

Shear Sections

F F

Shear Sections of Pin

m = 4

3. Strength of Components under Shearing Load 22

Pin Joint: Pin Calculation for ShearPin Joint: Pin Calculation for Shear

FF

d

bR

bL

FF

Pin Joint

Pin Pin’ Strength Condition for Shear

2

4

dm

F

mA

F

A

Q

or

m

Fd

4

Four sheared sections

All sheared sections are

equally loaded

Shear stress distribution on each sheared

section is uniform

Required diameter of the pin

m

FQ

4

2dA

Pin’ cross-section area


Pin Joint: Joint Calculation for BearingPin Joint: Joint Calculation for Bearing

Left Side of Pin

F/2

Deformed Pin

F/2

F/3

F/3

F/3

bR

bL

mR = 2mL = 3

RightC,RR

RightC,

LeftC,LL

LeftC,

dbm

Fdbm

F

RightC,RR

LeftC,LeftLeft

bm

Fbm

F

d

Strength Conditions

Right Side of Pin

Required diameter of the pin

Three bearing areas on the left side of the pin

Two bearing areas on the right side of the pin

Joint’ Design Stress for Bearing = lower value of design stresses of the two materials in contact

The softer material deforms

Strength conditions must be fulfilled at both sides of the pin


Quality of a Riveted JointQuality of a Riveted Joint

Quality of riveted joint = the joint is equally strong for shear, bearing and tear

Simplification = friction in the contacts between components due to the rivet tension is neglected

F F

FF

Shear of Rivet

F F

FF

Bearing of Components

F F

FF

Tear of Plate


Riveted Joint: Calculation for Shear and BearingRiveted Joint: Calculation for Shear and Bearing

Load is divided equally between all bearing areas (at left and right sides

respectively)

FF

bR

bL

d1

FF

Riveted Joint

Number of shear sections: m = 4Number of bearing areas at the left side of rivet: mL = 3Number of bearing areas at the right side of rivet: mR = 2

Per ONE rivet

Strength Conditions

21

4

dnm

F

nmA

F

RightC,1RR

RightC,

LeftC,1LL

LeftC,

dbnm

Fdbnm

F

RightC,1RR

LeftC,1LL

21

4

dbm

Fdbm

Fdm

F

n

nm

FQ

L/RL/RC, nm

FF

All rivets are equally loaded

Load is divided equally between

all shearing areas

Shear stress distribution is

uniform A

Q

Bearing stress distribution is

uniform C

CC A

F

Rivet Hole Diameter


Riveted Joint: Calculation for TearRiveted Joint: Calculation for Tear

Section I Section II

The plate has been weakened by rivet holes

AGross

Section I

Cross-Section Gross Area

ANet

Section II

Cross-Section Net Area

b

d1

The impacy of holes to the plate strength must be checked

Rivet Hole Diameter

11GrossNet bdnAA

Number of Holes in

One (given)

Section

11GrossNet bdnA

N

A

N

N

Adn

b Gross11

1

Strength Conditions


Welded Joint: Internal Forces AnalysisWelded Joint: Internal Forces Analysis

Weldments on both sides of the plate

F

z 0B

LW1

LW2 LW2

Q1

Q2h

Weldment are under shearing action

hQFzM

QQFF

20B

21

20

20

h

zFQ 0

1 12

h

zFQ 0

2 2

Shear forces Q and shear

stresses act at the sections of weldments

Load F acts along the axis of L-beams

L-beams are subjected to tension

Equilibrium conditions at the shearing section are used in order to calculate the internal forces


Shearing Section of a WeldShearing Section of a Weld

F

Fracture of a Welded Joint Under Shear

Shear Section

WminW,W LhA Nominal Shear Section Area

hW

45

WWminW, 7.045cos hhh

Geometry of Weld = Right Triangle

Outer surface of weld

Minimum Thickness

of Weld

The weld fractures under shear at the section of minimum thickness, this is 45 inclined relative to the fillet weld side


Welded Joint: Calculation for ShearWelded Joint: Calculation for Shear

F

Welded Joint Under Shear Action

Q1

WeldsShear Force

Weld Length

WminW,W LhA Nominal Shear Section Area

Weld Nominal Length

WWW 7.0 Lh

Q

A

Q

Shear Stress at the Shear SectionWeld Shear

Force

Strength Condition for Shear

WW7.0 Lh

Q

W1W

1

7.0 Lh

Q

W2W

2

7.0 Lh

Q

WW

11W 7.0

hh

QL

WW

22W 7.0

hh

QL

Weld Lengths

Strength condition must be fulfilled for

each weld separately

Load

Weld’ real length is made a bit longer



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STRENGTH OF MATERIALS - data.vk.edu.eedata.vk.edu.ee/RDER/RDER32/Tugevusopetus I/MHE0011 Strength...

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