Formula Sheet
Hook’s law
𝜎!! = 2𝜇𝜀!! + 𝜆(𝜀!! + 𝜀"" + 𝜀## )
Hook’s law (plane stress)
𝜎"" = 2𝜇𝜀"" + 𝜆(𝜀!! + 𝜀"" + 𝜀## )
$%
$%
$
𝜎"" = '*%! 𝜀!! + '*%! 𝜀""
𝜎## = 2𝜇𝜀## + 𝜆(𝜀!! + 𝜀"" + 𝜀## )
$
𝜏"# = +('(%) 𝛾"#
𝜏"# = 𝜇𝛾"# , 𝜏!# = 𝜇𝛾!# and 𝜏!" = 𝜇𝛾!"
$%
$
𝜎!! = '*%! 𝜀!! + '*%! 𝜀""
$
Elastic instability (buckling axis is Z )
where 𝜆 = ('(%)('*+%), 𝜇 = 𝐺 = +('(%)
.! $
Theories of failure
𝜎' ≤ 𝜎"1 , 𝜎2 ≥ 𝜎",
𝑃,- = (/0)""
!
𝜎3 = 𝜎' − 𝜎2 ≤ 𝜎"
𝑈 = ∫7 +$ 𝑑𝑥
𝜎4 = 3
Strain energy of elastic beam
0 6!
%
(5# *5! )! ((5! *5$ )! ((5$ *5# )!
+
where principal stresses:
𝑈=
≤ 𝜎"
0 3!
𝑈 = ∫7 +9
𝜎' > 𝜎+ > 𝜎2
'
:'
'
{𝐸7 , 𝐸# , 𝐸" } = ∬< 𝐸 {1, 𝑌, 𝑍 }𝑑𝐴
:'
+
{𝐸## , 𝐸"" , 𝐸"# } = ∬< 𝐸 {𝑌 , 𝑍 , 𝑌𝑍}𝑑𝐴
'
:'
𝜀 = 𝑢? − 𝑣 ?? 𝑌 − 𝑤 ?? 𝑍
6
$
8 ($
8
)) ""
)"
$
+
5
= 1 − : ;( ;
*
Torsion of elastic beam (twisting axis is X )
𝐺!! = ∬< 𝐺𝑅+ 𝑑𝐴
𝛾 = 𝑅𝜑 ?
𝜏 = 𝐺𝛾
??
𝑇 = ∬< 𝜏𝑅 𝑑𝐴 = 𝐺!! 𝜑 ?
𝑄? + 𝑔! = 0
𝑇 ? + 𝑡! = 0
𝑀"?? + 𝑔# = 0 and 𝑉" = −𝑀#?
and
2D Truss element
𝑉# = 𝑀"?
+
𝑙 = 3X𝑋C − 𝑋A Z + X𝑌C − 𝑌A Z
Transverse shear stress
*@&
$)" 4" *$)) 4)
𝜎′ = 𝐸 $ − 𝐸 $
%
!
)) $"" *$)"
$"" 4" *$)" 4)
𝑌+𝐸 $
!
)) $"" *$)"
𝑞A = ∬< 𝜎 ? 𝑑𝐴
+
1+
*
)"
𝑀# = − ∬< 𝜎𝑌𝑑𝐴 = 𝐸## 𝑣 ?? + 𝐸"# 𝑤 ??
and
=1−;
)) ""
𝑀" = ∬< 𝜎𝑍𝑑𝐴 = −𝐸"# 𝑣 − 𝐸"" 𝑤
𝜏A =
)
5(
8
??
B+
5
= 1 − ;(
8 ($
𝑄 = ∬< 𝜎𝑑𝐴 = 𝐸7 𝑢?
𝑀#?? − 𝑔" = 0
𝑑𝑥
𝜎< = 𝐵𝑁=*>
)) "
)" "
𝜀 = $ − $)" $) *$
𝑌 + $"" $) *$
𝑍
!
!
%
&&
Fatigue
Axial and bending of elastic beam
+
0 8!
∫7 +$ 𝑑𝑥
""
𝐹 = ∬< 𝜏 𝑑𝐴
Thermal strain and stress (bending axis is Z )
𝑄1G = − ∬< 𝐸𝛼Δ𝑇 𝑑𝐴
𝑀1G = ∬< 𝐸𝛼Δ𝑇𝑌 𝑑𝐴
𝑍
𝑐=
!, *!+
D
and 𝑠 =
𝒖3𝒆 = [𝑢A
𝑣A
", *"+
D
𝑢C
𝑣C ]
𝑐+
<$
𝑲𝒆 = D a 𝑐𝑠+
−𝑐
−𝑐𝑠
𝑐𝑠
𝑠+
−𝑐𝑠
−𝑠 +
−𝑐 +
−𝑐𝑠
𝑐+
𝑐𝑠
$
𝑐
𝑠]𝒖𝒆
𝜎F = D [−𝑐
−𝑠
+
−𝑐𝑠
−𝑠 + b
𝑐𝑠
𝑠+