• 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]
(1)STRENGTH OF MATERIALS
TRAN MINH TU -University of Civil Engineering, Giai Phong Str 55, Hai Ba Trung Dist Hanoi, Vietnam
(2)4
(3)Contents
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)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 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 components,
, , , , , , ) normal stresses shearing stresses (Note:
x y z xy yz zx
xy yx yz zy zx xz
s s s 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: of principal stresses equal to zeros
- Plane state of stress: of principal stresses equal to zeros
(6)• 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
x s t s t t 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
txy
sy
y
tyx x
y
sx
txy
sy
O
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Sign Convention：
• Normal Stress: positive: tension; negative: compression
• 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：
(8)0
u
F
2
2
cos cos sin sin sin cos
u x xy
y yx
A A A
A A
s s t
s t
su >0 – pull out
t uv - clockwise
2
uv xy x
τ A - τ Acos α - σ Acosαsinα
v
F 0
x y sx txy sy O u
styyx sxv
u
A
A sin
su tuv
txy
4.2 Plane Stress
4.2.2 Stresses on Inclined Planes：
Sign Convention：
(9)1/10/2013 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 cos xy sin s s s s
s 2 t 2
2 2
x y
uv sin 2 xy cos 2
2 s s
(10)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 x y 2 d
0 => tg2 =-d
t
s
s s
2
2 1,2(3)
2 2 xy
x y x y
max, min
s s s s
s s t