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Week 11 Recap
Different Forms of Loading § Normal stress due
to Axial Load § Normal stress due to
Bending § Shear stress due to
Bending § Shear stress due to
Torsion (circular cross section)
§ Shear stress due to Torsion (noncircular)
§ Normal stress due to Axial Load
§ Normal stress:
§ Deflection:
§ Deflection for constant N
Dienstag, 13. Mai 14 2 Mechanics II/ Philipp Fisch/ Group 6
σ x =NA
dδdx
=N(x)A(x)E Figure 1
δ =PiLiAiEi
∑
Different Forms of Loading § Normal stress due to
Axial Load § Normal stress due
to Bending § Shear stress due to
Bending § Shear stress due to
Torsion (circular cross section)
§ Shear stress due to Torsion (noncircular)
§ Normal stress due to Bending
§ Normal stress:
§ Deflection:
§ Combination: Normal stress due to Axial Load and Bending § Normal stress:
§ Deflection:
Dienstag, 13. Mai 14 3 Mechanics II/ Philipp Fisch/ Group 6
σ x =−Mb
Iy
dv2
dx2=Mb(x)EI
σ x =NA−Mb
Iy
dv2
dx2=Mb(x)EI
dδdx
=NAE
Figure 2
Figure 3
Different Forms of Loading § Normal stress due to
Axial Load § Normal stress due to
Bending § Shear stress due to
Bending § Shear stress due to
Torsion (circular cross section)
§ Shear stress due to Torsion (noncircular)
§ Shear stress due to Bending
§ Shear stress: § t(y) = width
§ I = moment of inertia
§ Q :
§ Maximum shear stress at y=0
Dienstag, 13. Mai 14 4 Mechanics II/ Philipp Fisch/ Group 6
τ xy =VQIt
Q = ycΔA
Q = η∫∫ dA
Figure 5
Figure 4
Different Forms of Loading § Normal stress due to
Axial Load § Normal stress due to
Bending § Shear stress due to
Bending § Shear stress due to
Torsion (circular cross section)
§ Shear stress due to Torsion (noncircular)
§ Shear stress due to Torsion (circular cross section)
§ Shear stress:
§ Maximum shear stress at r = R
§ Angle of twist:
§ Angle of twist for T(x)=T=const.:
Dienstag, 13. Mai 14 5 Mechanics II/ Philipp Fisch/ Group 6
τφx =TJr
dφdx
=T (x)GJ
φ =TiLiGJ∑
Different Forms of Loading § Normal stress due to
Axial Load § Normal stress due to
Bending § Shear stress due to
Bending § Shear stress due to
Torsion (circular cross section)
§ Shear stress due to Torsion (noncircular)
§ Shear stress due to Torsion (noncircular)
Dienstag, 13. Mai 14 6 Mechanics II/ Philipp Fisch/ Group 6
Stress concentration factors § Stress
concentration factors
§ Mohr’s circle: Plane stress
§ Mohr’s circle: Plane strain
§ Strain rosettes
§ Generally: § Axial load: § Torsion: § Bending:
Dienstag, 13. Mai 14 7 Mechanics II/ Philipp Fisch/ Group 6
σmax = KNA
K =σmax
σ avg
σmax = KTcJ
σmax = KMcI
Stress concentration factors: Axial load § Stress
concentration factors
§ Mohr’s circle: Plane stress
§ Mohr’s circle: Plane strain
§ Strain rosettes
Dienstag, 13. Mai 14 8 Mechanics II/ Philipp Fisch/ Group 6
Stress concentration factors: Torsion § Stress
concentration factors
§ Mohr’s circle: Plane stress
§ Mohr’s circle: Plane strain
§ Strain rosettes
Dienstag, 13. Mai 14 9 Mechanics II/ Philipp Fisch/ Group 6
Stress concentration factors: Bending § Stress
concentration factors
§ Mohr’s circle: Plane stress
§ Mohr’s circle: Plane strain
§ Strain rosettes
Dienstag, 13. Mai 14 10 Mechanics II/ Philipp Fisch/ Group 6
Mohr’s circle: Plane stress § Stress concentration
factors § Mohr’s circle: Plane
stress § Mohr’s circle: Plane
strain § Strain rosettes
Dienstag, 13. Mai 14 11 Mechanics II/ Philipp Fisch/ Group 6
Mohr’s circle: Plane stress § Stress concentration
factors § Mohr’s circle: Plane
stress § Mohr’s circle: Plane
strain § Strain rosettes
§ Important formulas
Dienstag, 13. Mai 14 12 Mechanics II/ Philipp Fisch/ Group 6
σ1,2 =σ x +σ y
2±
σ x −σ y
2"
#$
%
&'
2
+τ xy2
σ avg =σ x +σ y
2
τmax =σ x −σ y
2"
#$
%
&'
2
+τ xy2
tan(2θ p ) =τ xy
(σ x −σ y ) / 2
σ x ' =σ x +σ y
2+σ x −σ y
2cos(2θ )+τ xy sin(2θ )
σ y ' =σ x +σ y
2−σ x −σ y
2cos(2θ )−τ xy sin(2θ )
τ x ' y ' = −σ x −σ y
2sin(2θ )+τ xy cos(2θ )
Mohr’s circle: Plane strain § Stress concentration
factors § Mohr’s circle: Plane
stress § Mohr’s circle: Plane
strain § Strain rosettes
Dienstag, 13. Mai 14 13 Mechanics II/ Philipp Fisch/ Group 6
Mohr’s circle: Plane strain § Stress concentration
factors § Mohr’s circle: Plane
stress § Mohr’s circle: Plane
strain § Strain rosettes
§ Important formulas
Dienstag, 13. Mai 14 14 Mechanics II/ Philipp Fisch/ Group 6
ε1,2 =εx +εy2
±εx −εy2
"
#$
%
&'
2
+γ xy2
"
#$
%
&'
2
εavg =εx +εy2
γmax2
=εx −εy2
"
#$
%
&'
2
+γ xy2
"
#$
%
&'
2
tan(2θ p ) =γ xy
εx −εy
εx ' =εx +εy2
+εx −εy2
cos(2θ )+γ xy2sin(2θ )
εy ' =εx +εy2
−εx −εy2
cos(2θ )−γ xy2sin(2θ )
γ x ' y '2
= −εx −εy2
sin(2θ )+γ xy2cos(2θ )
Strain rosettes § Stress concentration
factors § Mohr’s circle: Plane
stress § Mohr’s circle: Plane
strain § Strain rosettes
§ Generally
Dienstag, 13. Mai 14 15 Mechanics II/ Philipp Fisch/ Group 6
εa = εx cos2(θa )+εy sin
2(θa )+γ xy cos(θa )sin(θa )
εb = εx cos2(θb )+εy sin
2(θb )+γ xy cos(θb )sin(θb )
εc = εx cos2(θc )+εy sin
2(θc )+γ xy cos(θc )sin(θc )
Strain rosettes § Stress concentration
factors § Mohr’s circle: Plane
stress § Mohr’s circle: Plane
strain § Strain rosettes
§ 45°
§ 60°
Dienstag, 13. Mai 14 16 Mechanics II/ Philipp Fisch/ Group 6
εx = εaεy = εcγ xy = 2εb − (εa +εc )
εx = εa
εy =13(2εb + 2εc −εa )
γ xy =23(εb −εc )
References § Figure 1: http://me-lrt.de/1-dimensionierung-trager-statische-beanspruchung § Figure 2: http://me-lrt.de/1-dimensionierung-trager-statische-beanspruchung § Figure 3: http://me-lrt.de/1-dimensionierung-trager-statische-beanspruchung
Dienstag, 13. Mai 14 17 Departement/Institut/Gruppe
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