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Theory of elasticity and plasticity (Equations sheet part 01) Att 8676
1. Theory of elasticity and plasticity
Equations sheet, Part 1
Theory of stress
Cauchy stress formula
σT
· n = t, or
σxx σyx σzx
σxy σyy σzy
σxz σyz σzz
nx
ny
nz
=
tx
ty
tz
where tx, ty and tz are the traction components; nx = cos(n, x), ny = cos(n, y) and nz = cos(n, z) are
the direction cosines of the outward normal vector.
Normal and shear components of the stress (traction) vector
tn = t · ˆn = tini, ts = |t|2 − t2
n
Hint: Unite normal to the plane
ˆn =
φ
| φ|
Principal stresses
(σ − λI)n = 0
Characteristic equation
λ3
− I1λ2
+ I2λ − I3 = 0
Stress invariants
I1 = σxx + σyy + σzz, I2 =
σxx σxy
σyx σyy
+
σxx σxz
σzx σzz
+
σyy σyz
σzy σzz
, I3 =
σxx σxy σxz
σyx σyy σyz
σzx σzy σzz
Principal directions
(σ − λiI)ni = 0
1
4. Classical beam and bar theories
TPE functionals for bars and beams
ΠT P E(u) =
0
EA
2
du
dx
2
dx −
0
qxudx
ΠT P E(w) =
0
EI
2
d2
w
dx2
2
dx −
0
qwdx
BCs for beams
• Fixed end- w = 0 and dw
dx = 0
• Free end- M = −EI d2
w
dx2 = M0 and Q = −EI d3
w
dx3 = Q0
• Pinned end- w = 0 and M = −EI d2
w
dx2 = 0
Ritz method
Displacement approximation functions
u = u0 +
n
i=1
aiui, v = v0 +
n
i=1
bivi, w = w0 +
n
i=1
ciwi
where the terms of u0, v0 and w0 are chosen to satisfy any non-homogeneous displacement BCs and ui,
vi and wi satisfy the corresponding homogeneous displacement BCs.
The TPE functional
ΠT P E = ΠT P E(ai, bi, ci)
Minimum conditions
∂ΠT P E
∂ai
= 0,
∂ΠT P E
∂bi
= 0,
∂ΠT P E
∂ci
= 0
4