DBA
Design by Analysis
Analysis Type:
Analysis Details
Page
7.2(S)
Example 1.1
GPD-Check and PD-Check
Direct Route using Elastic Compensation
Member:
Strathclyde
FE-Software:
ANSYS 5.4
Element Types:
8 – node, 2-D structural axisymmetric solid.
Boundary Conditions:
No vertical displacement in the undisturbed end of the shell remote
from the flat end.
Axisymmetry from elements
Model and Mesh:
Number of elements – 339
Height of model – 500 mm
Results:
Maximum internal pressure according to the GPD-check: PSmax GPD = 60.1 MPa
Check against PD: Shakedown limit pressure PSmax GPD = 102.1 MPa
DBA
Design by Analysis
Analysis Details
Example 1.1
Page
7.3(S)
1. Finite Element Mesh
Element divisions for the finite element mesh were defined parametrically to allow modification of
the element density. Analysis was carried out for a coarse mesh density and a much finer mesh
density to allow any alterations on the results to be noted. Finite element models were created using
linear 4-node 2-D axisymmetric solids, the analysis was repeated with higher order 8-node
structural 2-D axisymmetric solids, any alterations this may have on the results could also be noted.
The allowable pressure according to GPD and the shakedown pressure according to PD were
calculated from the model giving the greatest lower bounds. Here, the fine density mesh with 8node higher order elements gave the highest lower bounds.
Boundary conditions applied to the model reflect axisymmetry, applied via a key option when
defining the element type in the FE-software (axisymmetry around the vertical axis Y). The nodes
at the undisturbed end of the cylindrical shell have their vertical degree of freedom constrained to
zero to ensure that plane sections remain plane.
2. Material properties
Material strength parameter RM = 255 MPa , modulus of elasticity E = 212 GPa .
3. Determination of the maximum admissible pressure according to the GPD-Check
Using the application rule in prEN 13445-3 Annex B.9.2.2 to check against GPD, the principle is
fulfilled when for any load case the combination of the design actions do not exceed the design
resistance. This may be shown by calculating the limit load. The limit load has to be determined
using Tresca‘s yield condition and associated flow rule. As elastic compensation is based on a
series of elastic equilibrium stress fields, it is a relatively simple procedure to derive a lower bound
limit load directly from the Tresca yield model.
From prEN 13445-3 Annex B, Table B.9-3 the partial safety factor, γR on the resistance is 1.25.
Therefore, the design material strength parameter is given by RM/γR = 204 MPa. The analysis was
carried out using the elastic
compensation method conforming to
the direct route rules for GPD in
Annex B: linear elastic - ideal plastic
material
law,
Tresca’s
yield
condition and associated flow rule
and first order theory.
In each elastic compensation
iteration an equilibrium stress field is
produced where the elastic modulus
of each subsequent iteration is
defined by the previous elastic
solution. In this way some regions
in
the
FE-model
may
be
systematically stiffened or weakened
depending
upon
the
stress
magnitudes in the previous solution.
The result is, that every equilibrium
stress field is a lower bound of the
Figure 7.2.1-1: Limit Stress Field (Tresca)
Analysis Details
DBA
Design by Analysis
Example 1.1
Page
7.4(S)
limit load, the stress field giving the greatest limit load is taken as the closest to the actual limit state
and defines the limit load in the analysis. Convergence of the equilibrium stress field will occur
typically between 8 and 15 iterations, i.e. no further benefit in the limit load will be noted with
subsequent iterations. The total computing time to run the analysis on a 300 MHz Pentium two
processor with 128 Mb RAM Windows NT workstation was 70 seconds. The stress field was
shown to converge after eight iterations giving a lower bound on the pressure limit of 72.1 MPa.
Figure7.2.1-1 shows the limit stress (intensity) field according to the Tresca criterion. The limit
pressure is given by scaling the limit stress field so that the stress anywhere in the model does not
exceed the design materialstrength, 204 MPa, i.e. the applied pressure is scaled by the factor
(204/28.304) 7.207. According to prEN 13445-3 Annex B, Table B.9-2 for pressure loads (without
natural limit) the partial safety factor γp is 1.2. Thus, the internal pressure limit according to failure
by GPD is
PS max GPD =
72.1
= 60.1 MPa
1.2
It is also possible to determine a limit pressure from the check against PD. In this case, the elastic
compensation is based on Mises‘ yield condition. The partial safety factor on the resistance γR is
not applied for the PD-check. However, as the analysis is wholly elastic it is possible to scale the
stress fields at any time (similarly as was done above). The maximum ratio of Mises' equivalent
stress to Tresca's equivalent stress for the same load is 2/√3. Therefore, applying a factor of √3/2 to
the yield stress in the Mises analysis (or to the limit load, as the analysis is elastic) will always lead
to a conservative result. From the Mises analysis the limit load was found to be 102.1 MPa, and
with the partial safety factors γR = 1.25 and γp = 1.2, the internal pressure limit according to GPD
can be found as
PS max GPD =
102.1
3
⋅
= 58.95MPa
γ p ⋅γ R 2
4. Check against PD
In this check the principle in prEN13445-3 B.9.3.1 is fulfilled if the
structure can be shown to shake
down. When a structure has been
shown to shake down, the failure
modes of progressive plastic
deformation
and
alternating
plasticity can not occur.
In elastic compensation the load at
which the structure will shake down
is simple to calculate. Based on
Melan’s shakedown theorem, the
self-equilibrating residual stress field
that would result after a loading
cycle can be calculated by
subtracting the linear-elastic stress
field at the limit pressure from the
limit stress field. The residual stress
Figure 7.2.1-2: Residual Stress Field (Mises)
DBA
Design by Analysis
Analysis Details
Example 1.1
Page
7.5(S)
field is in effect the resulting stress from an elastic unloading from the limit state back to zero
pressure. If no stress in the residual field violates the yield condition, i.e. if there is no equivalent
stress above the material (yield) parameter, then the shakedown load is equal to the limit load.
Where the residual stress field of a structure does exceed the yield condition, the shakedown limit
can be calculated easily from the stress plots because of the linearity.
The residual stress field of the
compensation analysis using Mises’
criterion is shown in Figure 7.2.1-2.
Because the applied load is arbitrary
and the resulting stress fields are scaled
to the yield condition, the maximum
residual stress is then compared to the
maximum stress in the limit stress field,
Figure 7.2.1-3. The maximum residual
equivalent stress of 21.246 MPa is
smaller than the 24.971 MPa for the
limit stress field. The shakedown limit
is therefore the same as the calculated
limit load from the Mises condition,
given by scaling up the load by a factor
of material yield parameter to
maximum stress in the limit field
(255/24.971) = 10.21. With an applied
load of 10 MPa the shakedown limit is
102.1 MPa.
Figure 7.2.1-3: Limit Stress Field (Mises)
5. Check against GPD Using Non-linear Analysis
A check against GPD was also
performed for the same FE - model
using
conventional
non-linear
analysis.
In this way, a direct
comparison may be made between the
two limit approaches. The FE geometry, mesh and boundary
conditions are the same as those used
in the elastic compensation analysis.
Material non-linearities were applied
corresponding to the material strength
parameter, 204 MPa and perfect
plasticity. A ramped load is applied
and the analysis runs until the applied
load is such that convergence can no
longer occur due to unrestrained
displacement
–
Gross
Plastic
Deformation. It is assumed that the
last converged solution is the limit
Figure 7.2.1-4: Limit Stress Field (non-linear analysis)
Analysis Details
DBA
Design by Analysis
Example 1.1
Page
7.6(S)
load. Here, the last converged solution was at a load of 87 MPa, using the same method as above
the allowable pressure according to the GPD-Check using Mises' criterion is
87 3
⋅
= 62.8MPa
γp 2
The result offers a small benefit to the allowable pressure calculated using elastic compensation,
however the analysis is more difficult. Figure 7.2.1-4 shows the Mises equivalent stress at the limit
load. Analysis time to calculate limit load using non-linear FE - analysis was 290 seconds.
PS max GPD =
6. Additional Comments
Additional analysis was completed to ascertain any effect on the results for different mesh density
and for lower order elements (4-node). Limit loads and shakedown were calculated using Mises'
condition, the results are summarised in Table 7.2.1-1.
Number of
Elements
119
339
119
339
Element Type
4 node
4 node
8 node
8 node
Number of
Iterations
8
8
8
8
Lower Bound
Limit Load
56.9
58.8
57
58.95
Shakedown
Load
98.5
101.9
98.75
102.1
Processor
Time
40.7
107
70.2
190.3
Table 7.2.1-1. Limit and shakedown analysis summary
Both the lower bound limit load and the shakedown load were calculated using the same method as
described above. As can be seen from Table 7.2.1-1, essentially no difference is noted between the
results for the different element types. For this geometry, higher order elements offer no benefit
over the lower order elements. The geometry is simple and at the smaller mesh density the 4-node
elements fit the curvature well, therefore little change in the results would be expected. A
difference in the results can be noted between the two mesh densities. A slightly larger shakedown
and limit load result is obtained from the analysis using the higher mesh density, although small,
approximately 3%. In general, the results for both the limit load and shakedown calculations show
very little sensitivity to the element type and element density for this geometry.
As the geometry in this problem is outside the scope of DBF, the DBA calculations are a quick and
simple alternative for this simple problem.
The two elastic compensation methods used to calculate the lower bound limit load (direct from
Tresca's criterion or via a correction of Mises') show good correlation; the Mises corrected value is
slightly conservative as would be expected. However, carrying out the Mises elastic compensation
will give both the limit load and shakedown load in one analysis.
Good agreement was shown between the elastic compensation results and the non-linear analysis
results. Processing time is longer for the non-linear analysis however, due to the simplicity of the
model this time was also short.
DBA
Design by Analysis
Analysis Type:
Analysis Details
Page
7.7(A)
Example 1.1
GPD - Check and PD - Check
Member:
Direct Route using elasto-plastic FE calculations
A&AB
FE-Software:
ANSYS® 5.4
Element Types:
4 – node, 2 – D axisymmetric solid PLANE42
Boundary Conditions:
Zero vertical displacement in the nodes at the undisturbed end of
the shell.
Symmetry boundary conditions in the nodes in the centre of the flat
end.
Model and Mesh:
Whole height of model:
751.2 mm
Number of elements:
1294
Results: Maximun allowable pressure according to the GRD-Check: PSmaxGPD =60.87 Mpa
Shakesown limit pressure:
PSmaxGPD =101.45 Mpa
DBA
Design by Analysis
Analysis Details
Example 1.1
Page
7.8(A)
1. Elements, mesh fineness, boundary conditions
The model of the structure is shown on the preceeding page, a total number of 1294 4-node
axisymmetric solid elements, PLANE42 in ANSYS® 5.4, was used. The linear shape function of
these elements is sufficient, since
• the number of elements in the (linear-elastic) high stressed region is large,
• the computation time in an analysis using nonlinear material properties is much larger
for elements with midside nodes and quadratic shape functions, although, close to the
limit load, the results are almost identical compared to those using elements with a linear
shape function,
• there is no need to compute linear-elastic peak stresses very exactly, because the check
against PD can be carried out using the stress-concentration-free structure (according to
prEN 13445-3 Annex B.9.3.2) and the structure’s geometry is modelled exactly in the
example considered.
The boundary conditions applied to the model are symmetry ones in the nodes in the centre of the
plate (where the horizontal direction is perpendicular to the plane of symmetry) and constraining the
vertical degree of freedom in the nodes at the undisturbed end of the cylindrical shell to zero.
2. Determination of the maximum admissible pressure according to the GPD-Check
The partial safety factor γ R according to prEN-13445-3 Annex B, Table B.9-3 is 1.25 . Therefore,
the analysis using Tresca’s yield condition (delivered by an ANSYS® distributor) was carried out
with a linear-elastic ideal-plastic material law, a design material strength parameter of 204 MPa
(corresponding to a material strength parameter of RM = 255 MPa according to EN 10222-2) for
the shell and the plate, associated flow rule, and first order theory.
The elastic modulus used in all calculations is E = 212 GPa .
The analysis was carried out using the arc-length method, which showed faster convergence for the
considered structure than the Newton - Raphson method; since at the limit load the structure is fully
plastified in the shell and in the adjacent part of the plate, a maximum horizontal displacement of 10
mm at the upper face of the shell was used as termination criterion.
To restrict the computation time in a reasonable manner, the analysis was terminated after 17 hours
on a Compaq® Professional Workstation 5000 with two Pentium Pro® processors and 256 MB
RAM. The last convergent solution showed an internal pressure of 73.05 MPa – this pressure was
used as limit pressure.
Figure 7.2.2-1 shows the horizontal displacement in the upper end of the shell versus the internal
pressure.
Analysis Details
DBA
Design by Analysis
Example 1.1
Page
7.9(A)
Figure 7.2.2-1
Figure 7.2.2-2 shows the distribution
of Tresca's equivalent stress at this
limit pressure. Because of the almost
full plastification in the shell there is a
small region where, due to numerical
effects, the equivalent stress exceeds
204 MPa , but this has no effect on
further analyses.
The maximum absolute value of the
principal strains in the structure at this
limit pressure is 1.4 %, smaller than
5%, as required in the standard.
According prEN 13445-3 Annex B,
Table B.9-2, the partial safety factor
for pressure (without natural limit), γ P ,
is 1.2.
Figure 7.2.2-2
Therefore, the (internal) allowable pressure according to GPD is given by
PS max GPD =
73.05
= 60.87 MPa.
1.2
A less time-consuming method to determine a limit pressure according to GPD is given by usage of
the limit pressure result from the check against PD (see chapter 2 of section 3 - Procedures). With
the partial safety factors γ R = 1.25 and γ P = 1.2 , the internal limit pressure according to GPD is, in
this approach,
PS max GPD =
102.89 3
102.89
3
⋅
=
⋅
= 59.4 MPa .
γ P ⋅ γ R 2 1.2 ⋅ 1.25 2
DBA
Design by Analysis
Analysis Details
Example 1.1
Page
7.10(A)
The maximum absolute value of the principal strains in the structure, at the limit pressure used here,
and for the design material strength parameter of the check against PD-check, is 4 %, smaller than
5%, as required in the standard.
3. Check against PD
The elasto-plastic FE analysis was carried out as stated in prEN 13445-3 Annex B, Sec. B.9.3.1,
using Mises’ yield condition and associated flow rule, a linear-elastic ideal-plastic constitutive law
with a material strength parameter of 255 MPa for shell and plate, and first order theory.
By defining and using load cases in
ANSYS®, the superposition of stress
fields can be done easily. Therefore the
first load step of the analysis was
defined at a very low load level (5
MPa), so that there was linear-elastic
response of the structure. All other
linear-elastic stress fields can then be
determined easily by multiplication
with a suitable scale-up factor.
Again, the analysis was carried out
using the arc-length method; since at
the limit load the structure is fully
plastified in the shell and in the
adjacent part of the plate, a maximum
horizontal displacement of 10 mm at
the upper face of the shell was used as
termination criterion.
Figure 7.2.2-3
The computation time of the limit load
was 1 hour and 15 minutes on the
Compaq® Professional Workstation
5000 with two Pentium Pro®
processors and 256 MB RAM.
The termination criterion was fulfilled
for an internal pressure of 102.89 MPa
– this pressure was used as limit
pressure. Figure 7.2.2-3 shows the
horizontal displacement in the
undisturbed shell versus the internal
pressure; according to this figure the
limit state is reached.
Figure 7.2.2-4 shows the elasto-plastic
Mises equivalent stress distribution at
the limit pressure of 102.89 MPa .
Figure 7.2.2-4
DBA
Design by Analysis
Analysis Details
Example 1.1
Figure 7.2.2-5 shows the linearelastic Mises equivalent stress
distribution for the limit pressure
– the stress maximum is located
in the fillet.
Figure 7.2.2-5
Figure 7.2.2-6 shows the Mises
equivalent stress distribution of
the corrected residual stress field,
the scaling factor β is given by
0.96 (see subsection 3.3.2.5 of
section 3 – Procedures). The site
of the stress maximum is now
located at the outer surface of the
cylinder.
Figure 7.2.2-6
Page
7.11(A)
Analysis Details
DBA
Design by Analysis
Example 1.1
Figure 7.2.2-7 shows the Mises
equivalent stress distribution at
the lower bound shakedown
limit. The scaling factor α is
given by 0.986 (see subsection
3.3.2.5 of section 3 – Procedures).
Figure 7.2.2-7
Thus, the shakedown limit pressure is given by
PS max SD = 0.986 ⋅ 102.89 = 101.45 MPa .
Page
7.12(A)
DBA
Design by Analysis
Analysis Details
Page
7.13(T)
Example 1.1
Analysis Type:
Member:
Stress Categorization Route
FE-Software:
BOSOR
Element Types:
Axisymmetric shell elements.
TKS
Model and Mesh:
As shown in diagram – Shell numbers in calculation model
Remark:
In this example the structure is very thick walled. As BOSOR operates with thin-walled elements
the pressure acting on element 3 and 4 has been reduced with the factor Ri / Rm = 250.4 / 301.2
Results:
Maximum admissible action according to the Stress Categorization Route:
With reduction factor (Calc NO 11E) - Internal pressure PSmax SC = 69.7 MPa
Analysis Details
DBA
Design by Analysis
Page
7.14(T)
Example 1.1
-5.22E+01
1.25E+03
MIN:
MAX:
JOB
NO11E
99-08-25
10.54.26
0
4.26E+02
MIN:
MAX:
WINDOW (X,Y):
UNDEFORM.
DEFORM.
ALL SHELLS
*GEOMETRY PLOT
POSTBOSOR 1.04
The following figure shows the deformed model, the figures thereafter the distribution of stresses
– membrane and membrane and membrane plus bending - in the surfaces of the various parts of
the model.
With the designation list in subsection 3.6 the various plots are self-explaining.
The plots are for a pressure of 50 MPa. The limiting part is the cylindrical shell, the general
membrane stress criterion is the governing one – see the membrane stress distribution in shell 4
on the last page of this contribution.
2
.20
.40
.40
.60
.60
.80
.80
1.00
1.00
1.20
1.20
1.40
1.40
10
10
2
2
JOB
NO11E
99- 08-18
14.09.44
-8.03E+01
1.32E+02
10
.40
.50
.60
.70
.80
.90
1.00
1.10
1.20
1.30
-.75
-.50
2
.20
.20
.40
.40
.60
.60
.80
.80
1.00
1.00
1.20
1.20
1.40
1.40
10
10
2
2
-8.03E
+01
1.32E+02
3.54E+01
1.32E+02
JOB
NO11E
99-08-18
14.11.10
MIN:
MAX:
FUNC.VALUES:
"OUTSIDE"
"INSIDE"
SHELL 1
TRE SCA
*STRESSES*
POSTBOSOR 1.04
JOB
NO1 1E
99-08-18
14.09.05
MIN:
MAX:
FUNC.VALUES:
"OUTSIDE"
"INSIDE"
SHELL 1
GEN.DIR
*STRESSES*
POSTBOSOR 1.04
Example 1.1
MIN:
MAX:
FUNC.VALUES:
"OUTSIDE"
"INSIDE"
SHELL 1
CIRCUMF
*STRESSES*
POSTBOSOR 1.04
JOB
NO11E
99-08-18
14.08.15
2.61E+01
2.61E+01
2
Analysis Details
-.75
-.50
-.25
.00
.25
.50
.75
1.00
1.25
10
.20
MIN:
MAX:
-.25
.00
.25
.50
.75
1.00
1.25
10
Shell No. 1
FUNC.VALUES:
"MEMBR"
SHELL 1
COMP.-ST
*STRESSES*
POST BOSOR 1.04
DBA
Design by Analysis
Page
7.15(T)
Stresses
1.50
1
1.50
1.75
1.75
2.00
2.00
2.25
2.25
2.50
2.50
2.75
2.75
3.00
3 .00
3.25
3.25
3.50
3.50
10
10
2
2
1.12E+01
2.61E+01
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
10
2
JOB
NO11E
99-08-18
14.14.30
-3.08E+01
8.29E+01
.20
.40
.60
.80
1.00
1.20
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
10
2
"INSIDE"
2.05E+00
1.75E+02
JOB
NO11E
99-08-25
10.56.02
MIN:
MAX:
FUNC.VALUES:
"OUTSIDE"
Example 1.1
MIN:
MAX:
FUNC.VALUES:
"OUTSIDE"
"INSIDE"
SHELL 2
*STRESSES*
POSTBOSOR 1.04
SHELL 2
1.40
2
-1.23E+02
1.75E+02
JOB
NO11E
99-08-18
14.13.40
MIN:
MAX:
FUNC.VALUES:
TRESCA
1.60
10
-1.00
CIRCUMF
*STRESSES*
POST BOSOR 1.04
JOB
NO11E
99-08-18
14.13.02
MIN:
MAX:
FUNC.VALUES:
-.50
.00
.50
"INSIDE"
"OUTSIDE"
Analysis Details
- 2.00
.00
2.00
4.00
6.00
8.00
10
12.00
14.00
16.00
"MEMBR"
SHELL 2
1.00
SHELL 2
*STRESSES*
POST BOSOR 1.04
GEN.DIR
1.50
2
COMP.-ST
*STRESSES*
10
Shell No. 2
18.00
20.00
22.00
24.00
26.00
POSTBOSOR 1.04
DBA
Design by Analysis
Page
7.16(T)
Stresses
4.60
4.60
10
10
2
2
0
4.00
4.20
4.20
4.40
4.40
4.60
4.60
10
10
2
2
*STRESSES*
POSTBOSOR
"INSIDE"
60.00
JOB
NO11E
99-08-18
14.17.57
1.98E+01
8.18E+01
20.00
"INSIDE"
1.98E+01
8.18E+01
JOB
NO11E
99-08-18
14.18
.32
MIN:
MAX:
FUNC.VALUES:
"OUTSIDE"
Example 1.1
MIN:
MAX:
FUNC.VALUES:
"OUTSIDE"
1.04
JOB
NO11E
99-08-18
14.17.05
SHELL 3
3.8
4.00
SHELL 3
3.60
3.80
-1.36E+01
6.94E+01
Analysis Details
20.00
3.60
MIN:
MAX:
FUNC.VALUES:
"OUTSIDE"
"INSIDE"
SHELL 3
GEN.DIR
*STRESSES*
POST BOSOR 1.04
TRESCA
70.00
80.00
1
CIR CUMF
*STRESSES*
POSTBOSOR 1.04
JOB
NO11E
99-08-18
14.15.21
30.00
4.40
4.40
30.00
4.20
4.20
40.00
4.00
4.00
-1.00
40.00
3.80
3.80
3.72E+01
7.06E+01
.00
1.00
2.00
3.00
4.00
5.00
50.00
3.60
3.60
MIN:
MAX:
FU NC.VALUES:
"MEMBR"
SHELL 3
6.00
50.00
60.00
70.00
80.00
40.00
45.00
50.00
*STRESSES*
COMP.-ST
10
Shell No. 3
55.00
60.00
65.00
70.00
POSTBOSOR 1.04
DBA
Design by Analysis
Page
7.17(T)
Stresses
2
.60
.60
.80
.80
1.00
1.00
1.20
1.20
1.40
1.40
(σ eq )Pm ≤ f
Mem. 122MPa
10
3
10
3
6.99E+01
1.27E+02
JOB
NO11E
99-08-18
14.21.11
6.66E+01
1.29E+
02
.70
.80
.90
1.00
1.10
1.20
10
10.00
20.00
30.00
40.00
50.00
60.00
2
.60
.60
.80
.80
1.00
1.00
1.20
1.20
1.40
1.40
10
10
3
3
9.08E+00
6.52E+01
6.66E+01
1.29E+
02
JOB
NO11E
99-08-18
14.21.41
MIN:
MAX:
FUNC.VALUES:
"OUTSIDE"
"INSIDE"
SHELL 4
TRESCA
*STRESSES*
POSTBOSOR 1.04
JOB
NO11E
99-08-18
14.20.30
MIN:
MAX:
FUNC.VALUES:
"OUTSIDE"
"INSIDE"
SHELL 4
GEN .DIR
*STRESSES*
POSTBOSOR 1.04
Example 1.1
MIN:
MAX:
FUNC.VALUES:
"OUTSIDE"
"INSIDE"
SHELL 4
CIRCUMF
*STRESSES*
POSTBOSOR 1.04
JOB
NO11E
99-08-18
14.19.47
MIN:
MAX:
FUNC.VALUES:
"MEMBR"
SHELL 4
COMP.-ST
*STRESSES*
POSTBOSOR 1.04
Analysis Details
.70
.80
.90
1.00
1.10
1.20
10
.70
.80
.90
2
Shell No. 4
1.00
1.10
1.20
10
DBA
Design by Analysis
Page
7.18(T)
Stresses