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