4.2 Strains and stresses

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The deformation of a plate is described by the displacement vector

u(x, y) =

_

u(x, y)

v(x, y)

_

horizontal displacement

vertical displacement (4.19)

εxx = ∂u

∂x

εyy = ∂v

∂y

γxy = ∂v

∂x

+ ∂u

∂y

εxy =

1

2 γxy . (4.20)

⎡

⎣

σxx

σyy

τxy

⎤

⎦

_ _ _

σ

= E

1 − ν2

⎡

⎣

1 ν 0

ν 1 0

0 0 (1− ν)/2

⎤

⎦

_ _ _

E

⎡

⎣

εxx

εyy

γxy

⎤

⎦

_ _ _

ε

(4.21)

and in a state of plane strain,

⎡

⎣

σxx

σyy

τxy

⎤

⎦ = E

(1 + ν)(1 − 2 ν)

⎡

⎣

(1 − ν) ν 0

ν (1 − ν) 0

0 0 (1− 2 ν)/2

⎤

⎦

⎡

⎣

εxx

εyy

γxy

⎤

⎦. (4.22)

To recover the strains from the stresses, the formula

⎡

⎣

εxx

εyy

γxy

⎤

⎦ =

⎡

⎣

1/E −ν/E 0

−ν/E 1/E 0

0 0 1/G

⎤

⎦

⎡

⎣

σxx

σyy

τxy

⎤

⎦ (4.23)

of the individual points. The stresses (see Fig. 4.5) are not proportional to

the magnitude of the displacements, but to the change in the displacements

per unit length, that is the gradient (strains) of the displacement field

In a state of plane stress (see Fig. 4.6), where σzz = τyz = τxz = 0,

4.2 Strains and stresses 335

Fig. 4.6. Stress distribution in a wall. The distance between the stress resultants

is proportional to the magnitude of the internal bending moment

Fig. 4.7. Principal stresses in a plate

Fig. 4.8. At free edges the principal stresses always run parallel to the edge. This

provides a visual check on the FE results

336 4 Plane problems

is used, where G = 0.5E/(1 + ν) is the shear modulus of the material.

In rubber-like materials where Poisson’s ratio is close to 0.5, the stresses

become infinite in a state of plane strain. Special efforts are necessary to

deliver useful results with an FE program close to this point; see Sect. 4.17,

p. 393.

Table 4.2. Critical angles for a plate; stresses become infinite if the angle of the

boundary point exceeds these values.

Boundary conditions angle

fixed–fixed 180

◦

fixed–roller 90

◦

fixed–tangential 90

◦

fixed–free 61.7

◦

ν = 0.29 plane stress state

roller–roller 90

◦

roller–tangential 45

◦

roller–free 90

◦

tangential–tangential 90

◦

tangential–free 128.73

◦

free–free 180

◦

The angle

tan 2 ϕ =

2 τxy

σxx − σyy

(4.24)

defines the orientation of the principal planes where the principal stresses

σI,II = σxx + σyy

2

±

._

σxx − σyy

2

_2

+ τ 2

xy (4.25)

are acting. The shear stresses are zero in these planes. They attain their

maximum values if the planes are rotated by 45◦. The stress trajectories (see

Fig. 4.7 and Fig. 4.8) provide a graphic description of the stress state.

If un and us denote the edge displacements in the normal and tangential

direction and tn and ts the tractions in these directions, four combinations of

support conditions are possible

un = us = 0 fixed edge

un = 0, ts = 0 roller support

us = 0, tn = 0 tangential support

tn = ts = 0 free edge

The stress singularities at corner points depend on these boundary conditions

and on the angle of the corner points; see Table 4.2 [206], [252].

4.3 Shape functions 337

Fig. 4.9. CST element, E = 2.1 ・ 109 kN/m2, ν = 0.2, t = 0.1 m. Displayed are the

edge loads (kN/m) necessary to push the lower left node to the right while all other

nodes are kept fixed

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