Mechanics of Materials Goal:Load Deformation Factors that affect deformation of a structure P PPP...

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Mechanics of Materials Goal: Load Deformation Factors that affect deformation of a structure P P P P ress: intensity of internal force

Transcript of Mechanics of Materials Goal:Load Deformation Factors that affect deformation of a structure P PPP...

Page 1: Mechanics of Materials Goal:Load Deformation Factors that affect deformation of a structure P PPP Stress: intensity of internal force.

Mechanics of Materials

Goal: Load Deformation

Factors that affect deformation of a structure

P PPP

Stress: intensity of internal force

Page 2: Mechanics of Materials Goal:Load Deformation Factors that affect deformation of a structure P PPP Stress: intensity of internal force.

Normal Stress ()

P

A

Definition: stresses that act in a direction perpendicular to the cut surface

P

n

P

n

tensile stresses (+) compressive stresses (-)

Uniformly distributed stresses:

P: normal force acting at the cut surfaceA: cross sectional area

Non-uniformly distributed stresses:0

limA

P

A

Page 3: Mechanics of Materials Goal:Load Deformation Factors that affect deformation of a structure P PPP Stress: intensity of internal force.

Normal Stress – Example 1Find stresses at cross sections AA and BB. The cross sectional areas of AA and BB are SAA and SBB respectively.

P1 P3

P2

A

A

B

B

Page 4: Mechanics of Materials Goal:Load Deformation Factors that affect deformation of a structure P PPP Stress: intensity of internal force.

Deformation and Normal Strain

oLL

Deformation:

P P

L0

L

oLL

Normal Strain:

Strain at one point:

change in size

change in shape

oL

0limL L

Page 5: Mechanics of Materials Goal:Load Deformation Factors that affect deformation of a structure P PPP Stress: intensity of internal force.

Normal Strain – Example 2Find the total deformation of the structure shown below if the values in the strain gauges are and 6

26

1 10200,10100 mm 100 ,mm 100 21 LL

P1

P2

L1 L2

P3

strain gauge 1strain gauge 2

Page 6: Mechanics of Materials Goal:Load Deformation Factors that affect deformation of a structure P PPP Stress: intensity of internal force.

Stress-Strain Relationship

Review:

load

stress strain

deformation

Page 7: Mechanics of Materials Goal:Load Deformation Factors that affect deformation of a structure P PPP Stress: intensity of internal force.

Constitutive Law - Stress and Strain Relationship

Tensile Test:

- Apply load - Measure strain- Plot stress vs. strain curve

Page 8: Mechanics of Materials Goal:Load Deformation Factors that affect deformation of a structure P PPP Stress: intensity of internal force.

Stress - Stain Diagram - Ductile Material (structural steel)

True Diagram Partially Enlarge Diagram

• Proportional limit

• Yield stress

• Ultimate stress

Page 9: Mechanics of Materials Goal:Load Deformation Factors that affect deformation of a structure P PPP Stress: intensity of internal force.

Stress - Stain Diagram - Ductile Material Proportional Limit: the largest value of stress for which Hooke’slaw may be applied for a given material.

Yield Point ): a critical point, after the yield point, the specimen undergoes a large deformation with a relatively small increase in the applied load.

Plastic Deformation: deformation that remains after the load is applied.

Ultimate Stress (): the maximum stress developed in a materialbefore rupture.

Breaking Stress (): stress at rupture.

Page 10: Mechanics of Materials Goal:Load Deformation Factors that affect deformation of a structure P PPP Stress: intensity of internal force.

Stress - Strain Diagram - Aluminum Alloy

- no noticeable yield point- offset method - yield occurs at offset. %2.0

Page 11: Mechanics of Materials Goal:Load Deformation Factors that affect deformation of a structure P PPP Stress: intensity of internal force.

Stress - Strain Diagram - Brittle Material

- rupture occurs without noticeable any prior change in the rate of elongation.- no difference between and .

Page 12: Mechanics of Materials Goal:Load Deformation Factors that affect deformation of a structure P PPP Stress: intensity of internal force.

Linear Elasticity

lateral strain

axial strain

Hooke’s Law:

Poisson’s ratio: the ratio of the lateral or perpendicular strain to

the longitudinal or axial strain.

E

Page 13: Mechanics of Materials Goal:Load Deformation Factors that affect deformation of a structure P PPP Stress: intensity of internal force.

Stress-Strain Relationship – Example 3

Find the total deformation of the structure shown below. Express the answer in terms of P’s, S’s, L’s and E.

P1

P2

L1 L2

P3

Page 14: Mechanics of Materials Goal:Load Deformation Factors that affect deformation of a structure P PPP Stress: intensity of internal force.

Shear Stress

Shear force: force that acts tangential to the surface.

Average shear stress:

V V

V

A

Page 15: Mechanics of Materials Goal:Load Deformation Factors that affect deformation of a structure P PPP Stress: intensity of internal force.

©2001 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.©2001 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.

Shear Stress - double shear

22

2, is the diameter of the bolt

12

4

V P Pd

A dd

Shear stress:

Bearing stress:

F.B.D. of bolt

, is the thickness of the clevis2

bb

b

F Ph

A d h

Page 16: Mechanics of Materials Goal:Load Deformation Factors that affect deformation of a structure P PPP Stress: intensity of internal force.

©2001 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.©2001 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.

Shear Stress - single shear

F.B.D. of bolt

22

4, is the diameter of the bolt

14

V P Pd

A dd

Shear stress:

Bearing stress:

, is the thickness of the bar or flangebb

b

F Ph

A d h

Page 17: Mechanics of Materials Goal:Load Deformation Factors that affect deformation of a structure P PPP Stress: intensity of internal force.

Shear Strain

2

Shear strain: changes in shape (angle).

V d

L

If deformation is small, i.e., is small, tand

L

Page 18: Mechanics of Materials Goal:Load Deformation Factors that affect deformation of a structure P PPP Stress: intensity of internal force.

Sign Conventions for Shear Stress

Positive faces: outward normal direction is in the positivedirection of a coordinate axis. Negative faces: the opposite faces

Positive shear stress:

on a positive face, acts in the positive direction of one of the coordinate axes.

on a negative face, acts in the negative direction of oneof the coordinate axes.

Negative shear stress:

on a positive face, acts in the negative direction of one of the coordinate axes.

on a negative face, acts in the positive direction of oneof the coordinate axes.

x

y

z

Page 19: Mechanics of Materials Goal:Load Deformation Factors that affect deformation of a structure P PPP Stress: intensity of internal force.

Sign Conventions for Shear Strain

Positive shear strain:

Negative shear stress:

If the angle between two positive faces (or two negativefaces) are reduced.

If the angle between two positive faces (or two negativefaces) are increased.

Page 20: Mechanics of Materials Goal:Load Deformation Factors that affect deformation of a structure P PPP Stress: intensity of internal force.

Shear Stress vs. Strain - Hooke’s Law in Shear

G

G: shear modulus or modulus of rigidity

Page 21: Mechanics of Materials Goal:Load Deformation Factors that affect deformation of a structure P PPP Stress: intensity of internal force.

Shear Stress and Strain – Example 4

A punch for making holes in steel plates is shown in the following figure. Assume that a punch having diameter d = 20 mm. is used to punch a hole in a 8 mm. plate, as shown in the cross-sectional view. If a force P = 110kNis required to create the hole, what is the average shear stress in the plate and the average compressive stress in the punch?