PDF 2. Analyzing stress: Definitions and Concepts - Seoul National University

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2. Analyzing stress:

Definitions and Concepts

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2.1 Introduction

• Stress and strain

- The fundamental and paramount subject in mechanics of materials - Chapter 2: Stress

- Chapter 3: Strain

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2.2 Normal stress under axial loading

Force Stress =

Area

- (Magnitude of) stress = force intensity:

- Normal stress: stress normal to a plane (σ) - Axial loading: tension or compression load

acting along the long axis of a straight structural member

0

, lim

avg A

F F

A A

 

 

  



- Sign convention of normal stress of the textbook:

tensile - positive, compressive - negative

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2.3 Shearing stress in connections

- Shear stress: stress tangential to a plane (σ)

- Punching shear: ex) an action of a punch in forming rivet holes in a metal plate

0

, lim

avg A

V V

A A

 

 

  



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2.4 Bearing stress

- Bearing stress: compressive normal stress occurring on the surface of contact between two interacting members

ex) stress on the surfaces of contact between a bolt head and a top plate

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2.5 Units of stress

- In SI units (newtons and meters): N/m² = Pa (pascal), kPa, MPa - In US units: lb/in.² = psi, ksi (= 1,000 psi = kip per square inch)

ex) 1 lb ≈ 4.45 N, 1 inch = 2.54 cm = 1/12 feet, 1 psi ≈ 6.89 kPa

• Example Problem 2-1

- Normal stress (ksi) on mid-points between A-B, B-C, and C-D A cross-sectional area is 3.0 in.²

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2.5 Units of stress

- Diameter required for each of the sections when the allowable axial stresses are 125 MPa in the steel, 70MPa in the brass, and 85MPa in the aluminum.

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2.5 Units of stress

- Rigid collar: The minimum collar diameter when the bearing stress of collar must not exceed 10 ksi and force on the bearing plate is 50 kip.

- Flexible collar: Calculate and plot σ_max vs. collar diameter (2.5 in. ≤ d_c≤ 5.0 in.)

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2.6 Stresses on an inclined plane in an axially loaded member

- Equilibrium of force:

 

2

,

cos , sin

/ cos

/ cos 1 cos 2

2

/ sin cos sin 2

2

n

n n

P F F N V

N P V P

A A

P P

N A A A

P P

V A

A A

 



  

   

  



   

    

 Stress depends on force and area:

stress is not a vector quantity and therefore, the law of vector addition cannot be applied to stresses on

different planes.

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2.6 Stresses on an inclined plane in an axially loaded member

max min

0 when ,

 

     

- Maximum shear stress is a half of the maximum normal stress:

- Shear stress is zero on the planes of max. or min. normal stress:

max max 2

 

- Maximum shear stress is 45 apart from max. or min. normal stress.

- Magnitude of shear stress at θ is the same as that at 90+ θ.

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2.6 Stresses on an inclined plane in an axially loaded member

   

Moment equilibrium ( Mz 0):

yx xy

dx dz dy dy dz dx

 

 

 



- Equality of shearing stress on orthogonal planes:

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- Normal and shear stresses on section a-a (diameter = 1.25 in.).

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When A = 200 100-mm, normal stress on section a-a = 12 MPa (C),

- Load P

- Shear stress on section a-a

- Maximum normal and shear stresses in the block

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2.7 Stress at a general point in an arbitrarily loaded member

- Stress (force) at a point in an arbitrary plane can be resolved into a normal and shear components.

- When an x – axis is set to be normal to the plane the shear component can be resolved into two shear stresses coinciding with y- or z-axis.

- It is customary to show the stresses on positive and negative surfaces through a point using a small element.

(tensile/compressive is positive/negative)

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2.8 Two-dimensional or plane stress

- A state of plane stress occurs where forces can be resolved into only two components: within thin plates where z-dimension of the body is small and

the z-components of force are zero.

ex) z  _zx( _xz)  _zy( _yz)  0.

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2.9 The stress transformation equations for plane stress

- Normal and shear stresses on a plane can be calculated by using the force equilibrium of a free-body diagram.

0 : cos 2 sin 2

2 2

0 : sin 2 cos 2

2

x y x y

n n xy

x y

t nt xy

F

   

   

 

   

 

   

    



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2.9 The stress transformation equations for plane stress

- Normal and shear stresses on section a-b.

- Normal and shear stresses on section c-d.

- Stresses on section a-b and c-d using a small element.

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2.9 The stress transformation equations for plane stress

- Normal and shear stresses on section b-b.

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2.10 Principal stresses and maximum shearing stress – plane stress

- Transformation equations for plane stress:

cos 2 sin 2 (2-12b)

2 2

sin 2 cos 2

2

x y x y

n xy

x y

nt xy

   

   

 

   

 

  

   

- Max. and min. values of σ_n:

 

= when sin 2 2 cos 2 0

tan 2 2 , 0 (2-14)

n

n p x y xy

xy

p nt

x y

       



  

 

     



  



2 2

1, 2 from (2-12b) and (2-14)

2 2

x y x y

p p xy

   

  ^  ^ ^ ^ 

 

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2.10 Principal stresses and maximum shearing stress – plane stress

- Max. value of τ_nt:

 

2 2

= when cos 2 2 sin 2 0

tan 2 , (2-17)

2 2

nt

nt p x y xy

x y x y

p xy

xy



       



   

  



     



   

       

 

1 2

max min

max

from (2-15) and (2-17) 2

in 3D cases 2

p p

p

 



 



 

  

- The direction of the maximum shear stress must oppose the larger of the two principal stresses.

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2.10 Principal stresses and maximum shearing stress – plane stress

- Invariant of stress

1 2

p p x y

   

   

2 2

1, 2

2 2

x y x y

p p xy

   

  ^  ^ ^ ^ 

 

- In engineering problems, “maximum” will always refer to the largest absolute value (largest magnitude).

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2.10 Principal stresses and maximum shearing stress – plane stress

- Principal stresses and the maximum shear stress

- Locate the planes on which these stresses act and show the stresses

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2.11 Mohr’s circle for plane stress

- Equation of Mohr’s circle

2 2

cos 2 sin 2

2 2

sin 2 cos 2 2

2 2

2

x y x y

n xy

x y

nt xy

x y x y

n nt xy

x y

R xy

   

   

 

   

   

  

 



 

  

   

 

   

        

   

  

   

 

- Horizontal coordinates: normal stress (V’,F’..) - Vertical coordinates: shear stress (F’F, V’V..)

(CW - above; CCW – below)

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2.11 Mohr’s circle for plane stress

- Line CV, CH… : plane

- Angle on the circle: twice the angle for an actual body (CW - above; CCW – below)

- Procedure for drawing Mohr’s circle:

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2.11 Mohr’s circle for plane stress

- Principal stresses and the maximum shear stress - Normal and shear stress on plane a-a

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2.11 Mohr’s circle for plane stress

- Principal stresses and the maximum shear stress

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2.12 General state of stress at a point

- Traction components on the oblique face

x x x yx zx

y y xy y zy

z z xz yz z

F S dA dA l dA m dA n F S dA dA l dA m dA n F S dA dA l dA m dA n

  

  

  

      

- Force equilibrium on an oblique face

x y z

x x xy xz

y yx y yz

z zx zy z

S S S S

S l m n

  

  

  

  

- Normal/shear stress on the oblique face

   

2 2 2

2 2

, ˆ , , , ,

2 2 2

, n

n nt n x y z

n x y z xy yz zx

nt n nt

S S n S S S l m n

l m n lm mn nl

S S

  

      

   

   

      

   

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2.12 General state of stress at a point

- Principal stress when the oblique face is a principal plane

 

, , ,

0 0 0

0

p x p y p z p

x p x p xy xz

y p yx y p yz

z p zx zy z p

x p xy xz

yx y p yz

zx zy z p

S S l S m S n

S l l m n

S m l m n

S n l m n

   

    

    

    

   

   

   

   

     

     

     



 



   

 

3 2 2 2 2

2 2 2

2 0

p x y z p x y y z z x xy yz zx p

x y z x yz y zx z xy xy yz zx

              

           

         

     

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- Maximum shear stress when σ_x, σ_y, and σ_z are principal stresses

 

2 2 2 2 2 2 2

2 2 2 2 2

2 2 2 2 2 2 2 2 2 2

, ,

x x y y z z

x y z

n x y z

nt x y z x y z

S l S m S n

S l m n

l m n

l m n l m n

  

   

      

  

  

     

2

1 2

2 2 max min

max 1 2 1 2 max

1 1 1 1

2 2 2 2 2 2

p p

p p p p

   

     ^    ^  ^    ^

 

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14 ksi 12 ksi 10 ksi

8 ksi 10 ksi 6

x y z

xy yz zx

  

  

  

   

- Normal/shear stresses on a plane whose outward normal is oriented at angles of 61.3˚, 53.1˚, and 50.2 ˚ with x-, y-, and z-axes, respectively.

- Principal stresses and the maximum shearing stress at the point

• Home work problems:

2-2, 7, 36, 55, 75, 94

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※Mohr’s circle for 3D stress