# turbine stresses

```Axial Turbines
and Vibrations
Turbomachinery: Turbines -58
AE4803
also applies to
compressors
– centrifugal stresses (spinning rotor)
• Cycling
– thermal
– mechanical, including vibrations
• Dominant contribution in rotor and rotor disk
– centrifugal stresses
centrifugal
• limits N, h, rm,…
• AND lower stresses  longer lifetime
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1
Creep Rupture Strength
• For rotor blades, the limiting specific tensile strength
(𝜎 𝜌)𝑚𝑎𝑥 (known as allowable strength-to-weight ratio) is
based on creep
rupture strength
based on 100-hr life,
– maximum
0.1% creep
1000-hr life limits
tensile  mat’l
often used
Ti
can tolerate
alloy
w/o failure due
to creep for 
const. for given
time at given T
• Typically use 50% of
this as limit
• For compressor
rotor, low T
 different matls
can be used
Mechanics and Thermodynamics of Propulsion, Hill and Peterson
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• Increasing temperature results in decreased
time a given material can tolerate a fixed 
before hitting creep limit
• Tradeoff in temperature and time captured in
the Larson-Miller parameter
𝑇𝑎𝑏𝑠 𝐶 + log 𝑡ℎ𝑟𝑠
– C~25 for turbine disk alloys (Wilson and
Korakianitis, 1998)
– example: for disk material that hits creep
limit in 1000 hr at 1400 K
• how long if raised to 1500 K? only 13.6 hrs!!
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AE4803
2
Centrifugal Stresses
• As noted before, largest rotor stress
contribution is centrifugal
c
• Maximum stress at rotor  c  Fc Ahub
r
• Centrifugal force
r
r A
r 
c
  2  blade rdr
neglecting cooling
r
Ahub
passage volume)
 taper
• Assuming linear taper
At /Ah taper ratio
turbine rotors
r 
c
r  rh 
A 
~0.4-1
1  t  rdr
  2  1 
compressor
rotors
r
 rt  rh  Ah 
~0.7-1
t
h
t
h
t
h
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AE4803
Centrifugal Stresses
• Integrating
rt 2  rh2 
A  1  rt3 rt 2 rh   rh3 rh2 rh 
 

 1  t 
 
2
2   3
2 
 Ah  rt  rh  3
r 2  rh2 
A  1  2rt3  rh3  3rh rt 2 
 t
 1  t 


2
6

 Ah  rt  rh 
rt  rh
rt  rh 
 rt  rh 2 2rt  rh  
2


6
 rt  rh  
At  2rt  rh 


 rt  rh 
  1  

 2   Ah  6 
c
2

 3r  r  2r  r  A  2r  r 
 rt  rh  t h  t h   t  t h 
6
6  Ah  6 

 rt  2rh  At  2rt  rh 
 rt  rh 
 

 6  Ah  6 
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3
Centrifugal Stresses
 r  2rh  At  2rt  rh 
c
  2 rt  rh  t
 

 6  Ah  6 
– assuming
rt + 2rh  rh + 2rt  3rm
and (axial) Flow Area
Az  2 rm (rt rh)
• So
given material,
c
At  for
2 Az 
centrifugal
stress scales



1  A  with blade angular speed,

4

(II.59b)
h 

flow area and taper ratio
• Leads to what is known as the AN2 rule
a turbine
– design limit for maximum allowed AzN2 for
material at
max temp.
2
10
2
2
AzN |max= 0.510 10 in RPM
= 0.3  6 107 m2 RPM 2
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Maximum Hub Speed Limits
• Maximum allowed rotational speed leads to
• For example at turbine rotor hub
U h ,max  rh N max  30
– typical (conventional) values Uh,max
• HPT: 300 – 500 m/s (1000 – 1500 ft/s)
• LPT: 150 – 300 m/s (500 – 1000 ft/s)
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4
Bending Stresses
• Simple model for cantilevered blade in a stage
 bend
pavg
c ho1,3 1  rtip 


 z
U tip c pTo1 2  t max 
2
(II.60)
• Bending stresses increase with
– higher flow coefficient (at tip)
– lower solidity
Turbomachinery: Turbines -66
*after Kerrebrock, Aircraft Engines and Gas Turbines, 2 nd ed.,
MIT Press (1992); and Farokhi, Aircraft Propulsion
AE4803
Example: Turbine Rotor Stresses
• Given:
rm= 0.45 m Um=526 m/s
– turbine stage design from
MW=28.7 N=11,160 rpm
previous example(s)
=1.33
c1=cz=447 m/s
taper ratio of 1/2,
tmax/h =1/6, protor=7.5bar To1= 1800 K
m = 0.85
• Find:
 = 1.15
1. Required creep rupture
T2=1550 K
strength-to-weight ratio
Nozzle
Rotor
2. Rotor bending stress
r = 1.6
• Assume: same as previous (cz
const, tpg/cpg,…)
Turbomachinery: Turbines -67
h=5.0cm
disk
AE4803
5
Example: Turbine Rotor Stresses
• Required rupture strength-to-weight ratio
– needed to match centrifugal stress 2

A 
 r h A 
1  t    N  m 1  t 
 Ah   30  2  Ah 
2
2


 0.450.050 m
1  0.5
 11160 s 1 
30 
2

A = 0.14 m2
• from II.59b  c   2 Az
4
z
 2.3110 4
2
2
m kg m
kPa
 23.1
2
s kg m
kg m 3
 max Trotor &lt; 1200-1350 K depending
on mat’l. used
 T2=1550 K, so will need cooling/TBC
1
also AzN2 ~ 2107; within typical range
AE4803
Turbomachinery: Turbines -68
Example: Turbine Rotor Stresses
• Rotor bending stress
 bend
c ho1,3 1
• from II.60
 z
pavg U tip c pTo1 2
1
 rtip 


 t max 
2
1
 h 2
U
r
cz
0.025 
  0.851 
 m m  m m  m 1 
  0.805
U tip
U tip
rtip
rm 
0.45 


ho1,3
U m2
 1.15526 / s 
2

 0.151

c pTo1    1 RTo1 4289.7 J kgK 1800 K
rtip rm  h 2 rm h  1 2 0.45 0.050  1 2



 57
t max
t max
t max h
16
 bend  7.5bar 0.8050.151
1
57 2  92MPa
3 .2
comparing to centrifugal stress, using steel alloys ~ 8200 kg/m3
c ~ 23.1 (8200) kPa ~ 190 MPa  c ~ 2 bend at our pressure
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6
Thermal Stresses
• Based on thermal strains in material caused
by a temperature difference T
E = modulus of elasticity
 t  E t  ET
 = thermal strain
t
• For turbine disk, simple
 = coeff. of linear
model is constant thickness, thermal expansion
no central hole, linear temperature profile in r
T
ET  r 
1   T +T
 t ,r 
– tangential stress
stress at rmin (disk center)
 r 
h 

ET 
r
1  2 

3 
rh 
0
3
 t ,
T0
0
r
rh
(II.61)
AE4803
Turbomachinery: Turbines -70
Example: Disk Thermal Stresses
• Nickel-alloy disk
–~
10.210-6
 t ,r 
ET 
3
r
1  
 rh 
 t , 
ET 
r
1  2 
3 
rh 
in/in F at 1400F (1033K)
– E ~ 20.5106 psi at 1400F
– T
• 200 F (100 K)
– @ r  0, t,r  t,  14 ksi  2 Mpa
– @ r  rh, t,r  0, t,   2 MPa
acceptable, only 3% of
yield strength
(0.2% offset) at 900 K
to reduce thermal stresses, need lower T
 disk material should have high thermal conductivity,
e.g., nickel alloys
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7
• Blades (nozzle or rotor) essentially cantilevered beams
Thamizh, et al. (2016) DOI 10.1088/1757-899X/152/1/012008
• Exhibit resonant
(vibrational)
mode shapes,
each with its own
natural frequency
– bending (1st, 2nd,..)
– torsional (1st,…)
increasing
frequency
– coupled bending/torsion
• Disks and shafts also have
vibrational resonances
• Need to avoid running turbine at rotational
speeds that “match” these resonances
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Campbell Diagram (Comp. Rotor)
2nd bending
mode
1st torsional
mode
1st bending
mode
adapted from Allied-Signal, now part of Honeywell
•
•
8/Rev
7/Rev
6/Rev
multiples
of shaft
freq
(rev/sec)
properties (e.g., stiffness)
can change with rpm
Avoid design/operation at shaft speed that has integer multiples (up
to # stators, # rotor, # struts) that match any structural resonances
Same considerations true for turbine
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8
• Cooling of at least 1st stage HPT nozzle and
rotor typically required to maintain sufficient
material strength
– high temp., turbines, sometimes 2nd stage too
• Maximum stresses typically at rotor hub
• Most difficult regions to cool are blade tip and
trailing edge