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Samenvatting ALLE formules MMVC opgelijst per hoofdstuk
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HF 2 Vectoren & tensorenTensor product transformatie !𝑎# ⨂ 𝑏#' ∙ 𝑐̅ = 𝑎# ∙ !𝑏# ∙ 𝑐̅'
Ontbinding tensor in symm/antisymm 1 1
𝑇!" = !𝑇!" + 𝑇"! ' + !𝑇!" − 𝑇"! '
2 2
projectietensor (𝑒̅ ⨂ 𝑒̅ )
HF 3a Kracht vh evenwicht
Evenwicht star lichaam 2D 4 𝐹# = 0, 4 𝐹$ = 0 , 4 𝑀% = 0
Isostatisch stelsel #onbekenden = #evenwcihtsvgl
Hypostatisch stelsel #onbekenden < #evenwichtsvgl
Hyperstatisch stelsel #onbekenden > #evenwichtsvgl
HF 3b Concept spanning
Spanningsprincipe van Cauchy ∆𝑭
lim = 𝑻* = 𝑠𝑝𝑎𝑛𝑛𝑖𝑛𝑔𝑠𝑣𝑒𝑐𝑡𝑜𝑟
(kracht) ∆'→) ∆𝐴
Spanningsprincipe van Cauchy ∆𝑀
lim =0
(moment) ∆'→) ∆𝐴
Spanning na ontbinden F ∆𝐹+
𝜎++ = lim
(z component) ∆'→) ∆𝐴
Spanning na ontbinden F ∆𝐹# ∆𝐹$
𝜏+# = lim , 𝜏+$ = lim
(x en y component) ∆'→) ∆𝐴 ∆'→) ∆𝐴
Index notatie uitleg 1ste index = richting normaal op oppervlak
2de index = richting kracht
Stelling Tetraëder formule + uitleg (-)
𝑇! = 𝜎"! 𝑛"
componenten T = spanningsvector op arbitrair schuin vlak met normaal n, 𝜎!"
zijn spanningscomponenten geassocieerd aan de OG vlakken
Basisformule trekspanning 𝐹/
𝜎=
𝐴%
Basisformule schuifspanning 𝐹0
𝜏=
𝐴%
Gelijkheid van schuifspanningen 𝜏#$ = 𝜏$#
Spanning op schuin vlak 𝜎#1 = 𝜎# cos 2 𝜃
𝜏#1$1 = −𝜎# sin 𝜃 cos 𝜃
Normaalspanning op schuin vlak uit gegeven 1 1
𝜎!" = %𝜎! + 𝜎# ' + %𝜎! − 𝜎# ' cos 2𝜃 + 𝜏!# sin 2𝜃
spanningen van 2 ⊥ vlakken (tetraëder 2D) 2 2
1
, Schuifspanning op schuin vlak uit gegeven 1
𝜏#1$1 = − !𝜎# − 𝜎$ ' sin 2𝜃 + 𝜏#$ cos 2𝜃
spanningen van 2 ⊥ vlakken (tetraëder 2D) 2
Hoofdspanningen 𝜎$ + 𝜎% 𝜎$ − 𝜎% &
𝜎","" = ± () &
+ + 𝜏$%
2 2
HF 4 Vervorming & rek
Lagrange rek ∆𝐿
𝜀=
𝐿!
Euler rek ∆𝐿
𝜀=
𝐿3
Gemiddelde normale rek ∆𝑆 1 − ∆𝑆
𝜀'45 =
∆𝑆
Lengte na vervorming indien axiale rek ∆𝑆 1 ≈ (1 + 𝜀)∆𝑆
gekend is
Afschuifhoek 𝜋
𝛾= − 𝜙 = 𝜃6 + 𝜃2
2
Verband rek en afschuifhoek 1
𝜀#$ = 𝛾#$
2
Relatieve verplaatsing in 2D probleem 𝑑𝑢# 𝜀## 𝜀#$ 𝑑𝑥 0 −𝜔 𝑑𝑥
X𝑑𝑢 [ = \𝜀 𝜀$$ ] X𝑑𝑦[+\𝜔 ]X [
$ $# 0 𝑑𝑦
HF 5 Materiaaleigenschappen
Engineering stress & strain 𝑃 ∆𝑙
𝜎7 = , 𝜀7 =
𝐴) 𝑙)
True stress 𝑃
𝜎8 =
𝐴
Verband true/engineering stress 𝜎8 = 𝜎7 (1 + 𝜀7 )
True strain 𝑙3
𝜀8 = ln = ln(1 + 𝜀7 )
𝑙)
Uitspraak true vs engineering strain “True strains are additive, engineering
strains are not.”
Wet van Hooke 𝜎 = 𝐸𝜀
E = Elasticiteitsmodulus
Poisson ratio (dwarscontractiecoë\iciënt) 𝜀9:/
𝜈=−
𝜀9%-;
Wet van Hooke voor afschuiving 𝜏 = 𝐺𝛾
G = glijdingsmodulus
Verband materiaalconstanten 𝐸
𝐺=
2(1 + 𝜈)
2
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