Lecture Strength of Materials I: Chapter 4 - PhD. Tran Minh Tu

• Shear Stress: positive: the direction associated with its subscripts are plus-plus or minus-minus; negative: the directions are plus-minus or minus-plus. 4.2[r]

STRENGTH OF MATERIALS

TRAN MINH TU -University of Civil Engineering, Giai Phong Str 55, Hai Ba Trung Dist Hanoi, Vietnam

CHAPTER

4.1 State of stress at a point 4.2 Plane Stress

4.3 Mohr’s Circle 4.4 Special cases of plane stress 4.5 Stress – Strain relations

4.6 Strength Hypotheses

4.1 State of stress at a point

K

x

y z

n





• External loads applied to the body =>

The body is deformed =>The stress is

occurred

• At a point K on the arbitrary section, there

are 2 types of stress: normal stress s and

shearing stress t

• The state of stress at a point K is a set of

all stresses components acting on all

sections, which go through this point

• The most general state of stress at a point

may be represented by 6 components,

, ,

, ,

, , )

normal stresses shearing stresses (Note:

x y z

xy yz zx

s s s

t t t

t t t t t t

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• Principal planes: no shear stress acts on

4.1 State of stress at a point

• Principal directions: the direction of the principal planes

• Principal stresses: the normal stress act on the principal plane

• There are three principal planes , which are perpendicular to each other and go through a point

• Three principal stresses: s1, s2, s3 with: s1 ≥ s2 ≥ s3

• Types of state of stress:

- Simple state of stress: 2 of 3 principal

stresses equal to zeros

- Plane state of stress: 1 of 3 principal

stresses equal to zeros

- General state of stress: all 3 principal

stresses differ from zeros

• Plane Stress – the state of stress in which two

faces of the cubic element are free of stress For the illustrated example, the state of stress

is defined by

0 ,

, y xy and z  zx  zy 

s

• State of plane stress occurs in a thin plate subjected to the forces acting in the mid-plane

of the plate

4.2 Plane Stress

sx

t xy

sy

y

y

sx

t xy

sy

O

tyx

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Sign Convention：

• Shear Stress: positive: the direction associated with its subscripts are plus-plus or minus-minus; negative: the directions are plus-minus or minus-plus

4.2 Plane Stress

y

4.2.1 Complementary shear stresses：

• The shear stresses with the same subscripts

in two orthogonal planes (e.g txy and tyx)

are equal

u



2

2

cos cos sin sin sin cos 0

 su >0 – pull out

 t uv - clockwise

2

τ A - τ Acos α - σ Acosαsinα

v

F   0



x y

sx

t xy

sy

O

u

s ty yx

sxv

u



A

A sin

su

tuv



t xy

4.2 Plane Stress

4.2.2 Stresses on Inclined Planes：

Sign Convention：

  >0 - counterclockwise；

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4.2 Plane Stress

tyx

sy

su

tuv



sy

sx

txy

4.2.2 Stresses on Inclined Planes：

x

y

v u

-  > 0: counterclockwise from the x axis to u axis

x y x y

u s s s s cos xy sin

s     2  t  2 

uv sin 2 xy cos 2

2

s s

t    t  

4.2 Plane Stress

4.2.3 Principal stresses are maximum and minimum stresses ：

By taking the derivative of su to  and setting it equal to zero:

xy u

p

2

d

0 => tg2 =-d

t

2

2 1,2(3)

max, min

s  s          t

0

1, 2

90

p

p p

p









   