HSE
Health & Safety
Executive
The effects of dynamic loading on structural
integrity assessments
Prepared by UMIST and TWI Ltd for the
Health and Safety Executive 2004
RESEARCH REPORT 208
HSE
Health & Safety
Executive
The effects of dynamic loading on structural
integrity assessments
Professor F M Burdekin, Dr W Zhao and Dr Y Tkach
Manchester Centre for Civil and Construction
Engineering
UMIST
P.O.Box 88
Manchester
M60 1QD
Dr C S Wiesner and Dr W Xu
TWI Ltd
Granta Park
Great Abington
Cambridge
CB1 6AL
Although there are well-established procedures for assessing the significance of defects in welded
structures in a number of countries, there are no clear guidelines for such assessments under dynamic
loading. In principle standard procedures can be applied for any rate of loading, but there is little or no
experience as to how to allow for the effects of dynamic loading on load magnitude or rate of loading in
such assessments. In this project, work has been carried out jointly by UMIST and TWI to investigate
effects of dynamic loading on structural integrity assessments for fracture. The UMIST work has
concentrated on the effects of dynamic loading on the crack tip severity or driving force aspect of
fracture mechanics analyses in structural integrity assessments. This has included finite element
analyses of large scale structures under dynamic loading due to waves or earthquakes, and the
determination of local crack tip driving forces for assumed defects under such conditions. The UMIST
work has also included a brief investigation of crack arrest behaviour using a finite element cohesive
zone model applied to the behaviour of local brittle zones in weldments. The TWI work has included a
review of effects of dynamic loading on tensile and fracture toughness properties of steels and
weldments and some finite element analyses of fracture and plastic collapse behaviour of fracture
toughness specimens and wide plates under dynamic loading.
This report and the work it describes were funded by the Health and Safety Executive (HSE). Its
contents, including any opinions and/or conclusions expressed, are those of the authors alone and do
not necessarily reflect HSE policy.
HSE BOOKS
© Crown copyright 2004
First published 2004
ISBN 0 7176 2831 0
All rights reserved. No part of this publication may be
reproduced, stored in a retrieval system, or transmitted in
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written permission of the copyright owner.
Applications for reproduction should be made in writing to: Licensing Division, Her Majesty's Stationery Office, St Clements House, 2-16 Colegate, Norwich NR3 1BQ or by e-mail to hmsolicensing@cabinet-office.x.gsi.gov.uk
ii
ACKNOWLEDGEMENTS
The work presented here is part of a joint program between UMIST and TWI under a
grant award from EPSRC with additional support from Corus, HSE, BNFL Magnox
Electric plc, EQE Ltd. and Ove Arup and Partners.
iii
iv
CONTENTS 1
INTRODUCTION ............................................................................................................................1
2
AIMS AND OBJECTIVES ..............................................................................................................2
3
UMIST WORK ON OFFSHORE STRUCTURES UNDER WAVE LOADING AND BRACE LOADING...................................................................................................................................................3
3.1
3.1.1
3.1.2
3.1.3
3.1.4
3.2
3.3
3.3.1
3.3.2
3.4
3.4.1
3.4.2
3.4.3
3.5
3.6
3.7
GENERAL ....................................................................................................................................3
Design codes of practice.......................................................................................................3
Wave types: Stokes 5th order and gridded wave data ...........................................................3
Method for study fracture response of dynamic loaded structures.......................................4
FE model and sub-models ....................................................................................................4
CRACKED PLATE UNDER SINUSOIDAL LOADING ..........................................................................4
CRACKED TUBES UNDER SINUSOIDAL LOADING .........................................................................6
Axial dynamic response ........................................................................................................6
Flexural dynamic response...................................................................................................7
OFFSHORE JACKET STRUCTURE UNDER WAVE LOADING .............................................................8
Submodels of cracked brace members and multiplanar joint...............................................8
Results for cracked brace members ......................................................................................9
Results for cracked multi-planar joint:..............................................................................10
CRACKED WELDMENT K-JOINT .................................................................................................11
CONCLUSIONS ON OFFSHORE STRUCTURE RESPONSE................................................................12
RECOMMENDATIONS FOR OFFSHORE APPLICATIONS.................................................................13
4
UMIST WORK ON EFFECTS OF EARTHQUAKE LOADING ON FRACTURE AT
WELDED JOINTS...................................................................................................................................14
4.1
4.2
4.3
4.4
4.4.1
4.4.2
4.5
5
GENERAL .................................................................................................................................14
DESCRIPTION OF FINITE ELEMENT MODELS .............................................................................15
MATERIAL PROPERTIES ............................................................................................................16
RESULTS ..................................................................................................................................16
Applied CTOD Distributions from Dynamic Sub Model Analyses .....................................16
Applied CTOD Distributions from UBC Equivalent Static Loading Sub Model Analyses .17
CONCLUSIONS ON DYNAMIC LOADING DUE TO EARTHQUAKES ................................................17
TWI WORK ON MATERIAL RESPONSE TO DYNAMIC LOADING .................................23
5.1
TENSILE PROPERTIES UNDER DYNAMIC LOADING .....................................................................23
5.1.1
General ...............................................................................................................................23
5.1.2
Empirical yield strength-temperature-strain rate equation................................................25
5.2
FRACTURE TOUGHNESS UNDER DYNAMIC LOADING .................................................................27
5.2.1
Definition of strain rate in fracture specimens ...................................................................27
5.2.2
General trends of strain rate effects on fracture toughness transition Curve ....................27
5.2.3
Empirical methods for fracture toughness transition temperature shift .............................28
5.2.4
An empirical equation for fracture toughness under dynamic loading ..............................30
5.3
CONCLUSIONS ON MATERIAL PROPERTY ASPECTS ....................................................................32
6
TWI WORK ON LOCAL APPROACH IN FRACTURE TOUGHNESS SPECIMENS ........34
6.1
6.2
6.3
6.4
6.5
7
GENERAL .................................................................................................................................34
TENSILE PROPERTIES AND FRACTURE TOUGHNESS OF STEEL 450EMZ.....................................34
FINITE ELEMENT MODEL .........................................................................................................37
PREDICTION OF FRACTURE TOUGHNESS AT INTERMEDIATE LOADING SPEED ............................39
DISCUSSION .............................................................................................................................42
TWI WORK ON LIMIT LOAD SOLUTIONS ...........................................................................43
7.1
7.2
7.3
GENERAL .................................................................................................................................43
FINITE ELEMENT MODELS AND ANALYSES ................................................................................43
RESULTS ..................................................................................................................................43
v
8
TWI WORK ON FAILURE ASSESSMENT DIAGRAMS ........................................................45
9
UMIST WORK ON CRACK ARREST........................................................................................47
9.1
9.2
9.3
9.4
10
GUIDANCE ON DYNAMIC STRUCTURAL INTEGRITY ASSESSMENT..........................53
10.1
10.2
10.3
10.4
11
OBJECTIVES .............................................................................................................................47
COMPUTATIONAL METHODOLOGY AND FINITE ELEMENT MODELS .........................................47
ANALYSIS OF CRACK GROWTH AND ARREST ...........................................................................48
CONCLUSIONS OF CRACK ARREST ...........................................................................................52
GENERAL ISSUES ......................................................................................................................53
SINGLE LOAD APPLICATIONS – IMPACT OR BLAST CONDITIONS ................................................54
REPEATED SINUSOIDAL LOADING ............................................................................................54
CRACK ARREST ANALYSIS .......................................................................................................55
REFERENCES................................................................................................................................56
vi
EXECUTIVE SUMMARY
INTRODUCTION
Although there are well-established procedures for assessing the significance of defects in
welded structures in a number of countries, there are no clear guidelines for such assessments
under dynamic loading.
In principle standard procedures can be applied for any rate of loading, but there is little or no
experience as to how to allow for the effects of dynamic loading on load magnitude or rate of
loading in such assessments. In this project, work has been carried out jointly by UMIST and
TWI to investigate effects of dynamic loading on structural integrity assessments for fracture.
The UMIST work has concentrated on the effects of dynamic loading on the crack tip severity
or driving force aspect of fracture mechanics analyses in structural integrity assessments. This
has included finite element analyses of large scale structures under dynamic loading due to
waves or earthquakes, and the determination of local crack tip driving forces for assumed
defects under such conditions. The UMIST work has also included a brief investigation of crack
arrest behaviour using a finite element cohesive zone model applied to the behaviour of local
brittle zones in weldments. The TWI work has included a review of effects of dynamic loading
on tensile and fracture toughness properties of steels and weldments and some finite element
analyses of fracture and plastic collapse behaviour of fracture toughness specimens and wide
plates under dynamic loading.
OBJECTIVES
The objectives of the project were as follows:(i)
To provide guidance on how and when to treat dynamic loading in structural
integrity assessments with respect to severity of loading and the effect of strain
rate on the response of the material, with particular reference to high strength
steels and welded joints.
(ii)
To provide guidance on use of dynamic toughness measurements for
assessment of crack arrest at short crack lengths by tough weld deposits,
including performance of high strength steels.
WORK CARRIED OUT
Offshore structures under wave loading and brace loading
This part of research project presents the numerical investigation on dynamic integrity
assessment of an offshore structure under normal and wave loads. To understand the principles,
initial work was carried out on a cracked plate under sinusoidal loading at different frequencies.
This was then extended to the behaviour of cracked tubes under sinusoidal loading at different
frequencies. Wave loading was then applied to a full jacket structure by either gridded wave or
Stokes 5th order wave methods with gravity loads also present. Global structural dynamic
analyses of a whole offshore jacket have been carried out in the first step, then, a set of sub­
models of joints selected from the worst locations. These were chosen as where the maximum
stresses occurred or to place a crack at the weldment between chord and brace. Through
thickness cracks were analysed under the loading extracted from the global analysis. Some
analyses were carried out under force control for investigations of local dynamic responses. A
vii
data analysing process of the sub-model dynamic results was created and carried out to obtain
the fracture response in terms of crack tip opening displacement (CTOD) variation. Related
values of K, J and their rates of application were also calculated to investigate load rate effect.
The analysis methodology was proved to be applicable to this kind of study. Some analysis
results were compared to the available analytical solution and reasonably good agreement was
found between them. These results are used to give guidance on methods for design and
assessment of structural integrity under dynamic loading. In a final stage the behaviour of
cracked tubular members under sinusoidal loading was investigated.
Effects of earthquake loading on fracture at welded joints
In this investigation, finite element analysis has been used to study the local behaviour of
connections with defects within a complete building frame under the dynamic loading of an
earthquake. The purpose of the FEM analyses was to identify the effects of crack length,
connection design, and material properties on the local behaviour of sub model connections
located in the full steel frame building under such loading. The stress distribution in the region
of the column and beam flange connection was also considered.
Global model analyses were carried out on two building frames, one of four storeys height and
one of seventeen storeys height, representative of those involved in the Northridge earthquake.
The results in this report are for the four-storey building. These models were two-dimensional
representing one frame of the complete building. The earthquake loading applied was the time
history record of the ground movement recorded at the Canoga station during the Northridge
earthquake. In addition loadings were applied representing 95% of the energy content of the
loading spectrum, and representing the equivalent static design load, which would be used in
design with the UBC code
Five different sub model analyses were carried out consisted of five alternative designs of beam
to column connections, each series being defined by the configuration of the connection as
follows:
·
·
·
·
·
Basic connection as used at Northridge with no additional strengthening.
Connection with continuity plates added between the flanges of the column. Connection with beam flange width reduced by a "dog bone" shape away from weld. Connection with additional haunch section added underneath the beam section. Connection with cover plates added to the beam flanges and welded to the column flange. The sub models were three dimensional to allow effects across the widths of the welds to be
captured. Each sub model series was subdivided into three different cases of analysis. Case one
consisted of the sub model connection without initial defects; cases two and three consisted of
the connection with 2.5 mm and 5 mm height crack like defects in the sub model across the
underneath of the width of the bottom flange of the beam against the column face. Each series
of sub model connections was attached to the global frame model. A sensitivity study
concerning material properties was also investigated with effects of four different stress strain
curves examined. The results of the dynamic analyses for each series are also compared with
pseudo static analyses, pushover analyses, using the Uniform Building Code (UBC)-1994
method.
Material response to dynamic loading
Fundamentals of material behaviour under high loading rates have been reviewed. A significant
amount of material data under dynamic loading conditions have been collated from the open
viii
literature and generated from tests. The data included dynamic yield and tensile strength and
dynamic fracture toughness of many ferritic steels for applications in the power generation, the
construction and the offshore industries.
Material yield and tensile strength data have been analysed using a rate-temperature parameter.
An equation expressing material strength as a function of the rate parameter has been developed
and validated using the data collated.
Dynamic fracture toughness data have been analysed using methods based on a transition
temperature shift and on the mater curve approach. An equation expressing dynamic fracture
toughness as a function of the rate parameter and the static transition temperature has been
developed and validated using the data collated.
Finite element analyses have been carried out for fracture toughness test specimens and wide
plates. The effects of high loading rate on fracture toughness have been investigated using the
Beremin local approach model for a modern offshore steel. The effects of high load rates on
plastic collapse loads and on the failure assessment diagram have also been studied.
Crack arrest
Computational simulation based on the cohesive zone model has been carried out to predict
crack initiation, growth and crack arrest behaviour under static and dynamic loading conditions.
The influence of the interfacial properties on the crack growth and subsequent crack arrest has
been investigated. Two cracked geometries were considered. Firstly, crack inition, growth and
arrest were simulated in a compact tension specimen and secondly, a simulation of a double
tension wide plate crack arrest test was carried out.
CONCLUSIONS
Offshore structures under wave loading and brace loading
·
Dynamic analyses of cracked plates and tubes as individual members suggest that the
fracture mechanics based structural integrity assessment of dynamic loaded members is
significantly affected by the ratio of applied frequency to natural frequency of the member
or component. The natural frequencies of these individual components are high. Dynamic
amplification factors for crack tip driving force parameters depend also on the degree of
damping applied and material properties assumed. Typical values of peak DAF at
resonance with damping of 5% were up to about 6.5 were obtained for cracked tubes in
bending.
·
Dynamic amplification factors for offshore applications are much lower than those
recommended for onshore structure applications because of the inherent damping effects of
immersion in water.
·
Rate-dependent material properties reduce the dynamic amplification by 30 percent when
compared to the result that does not have the factor for the material selected for the analysis.
·
Global and local dynamic analysis results for an offshore structure suggest the fracture
mechanics based structural integrity assessment of cracked structural joints is not
significantly affected by the 100 year-return gridded wave. The dynamic amplification
factors obtained were less than 1.37 for the joint and 2.02 for the members. CTOD, K and J
values and their rates were also calculated for substantial through thickness cracks subjected
to this 100 year wave and low values found which were not substantially different from
ix
static values. This result is because the frequency of this wave was well removed from the
natural frequency of the structure.
·
Rates of crack tip loading are directly related to the rate (or frequency) of applied loading.
For offshore wave loading the dominant rates of loading are low and there is little effect on
material properties.
·
For design guidance purposes, under dynamic loading all the fundamental frequencies of the
global and local member, axial flexural and rotation etc. need to be considered, as they may
cause a large contribution on fracture response. The effect on offshore design is limited
however, provided the natural frequency of the structure is well removed from the
frequency of applied wave loading.
Effects of earthquake loading on fracture at welded joints
The results confirmed the general increase in severity of conditions towards the centre of the
beam flange width. The stress distribution also confirmed that the position of the maximum
stress coincided with the location of the fracture initiation. Connections strengthened with a
haunch or cover plates were found to show a very good performance and much better than the
dog bone and "pre-Northridge" connection. The material tensile properties also affect the crack
tip severity at the crack tip.
The absolute values of applied CTOD obtained from the FE analyses are entirely consistent
with the occurrence of fracture in materials with fracture toughness levels of the order of 0.1
mm to 0.2 mm CTOD.
A reasonable estimate of the applied CTOD value for particular earthquake loading can be made
by using the equivalent code static loading (such as the UBC method) in a standard finite
element analysis of the structure to determine basic stresses for input to a BS7910 type analysis.
The equivalent static loading is based on the magnitude of the accelerations due to the
earthquake, the mass of the structure and an allowance for the effect of ground conditions. This
agreement may be to some extent fortuitous as the equivalent static loading does not necessarily
give rise to the same mode shape of deformations in the structure as the true dynamic loading.
It is also necessary to make some allowance for non uniform stress distributions at beam to
column flange connections. With the typical frequencies involved in earthquake loading the
effects of rate of loading on material properties are relatively modest.
Material response to dynamic loading
Material tensile properties of ferritic steels under dynamic loading without significant effects of
stress wave, are governed by the rate parameter R which unifies material tensile strength at
different temperatures and strain rates. A minimum number of tests, e.g. five data points, are
required in order to generate material strength data for a wide range of temperature and strain
rate. In the absence of minimum test data, The following equation may be used to estimate
material tensile strength.
ì 1 ln( A / e& static ) 1
ü
s
ys (T
,e& )
=
s 0 +
S
í
-
ý
293 þ
î
T ln( A / e& )
Where σ0 is yield strength at room temperature under quasi-static loading. e&static is a quasi-static
loading rate, typically ~5x10-5 1/s. A is a material constant with typical value of 108. S is a
parameter to be fitted using test data, but if test data are not available, a value of 60,000 ± 10000
x
MPaK may be used. This equation is applicable for temperatures lower than the ambient and for
strain rate up to about 10001/s.
It is preferred to obtain dynamic fracture toughness by experimental tests. Loading rate in the
test specimens should be set to achieve the stress intensity factor rate equal to the rate of applied
stress intensity in the structure. It is possible to test a minimum number of specimens at one
temperature and one rate of stress intensity factor in order to derive a dynamic fracture
toughness temperature transition curve. In the absence of test data under dynamic loading, the
following equation may be used to estimate dynamic fracture toughness together with the static
fracture toughness transition temperature.
dy
K C = 20 + 80 exp{0.00066( R - 28.3T0 )}
where T0 is the fracture toughness transition temperature under quasi-static loading.
R=Tln(108/ e& ). KdyC has a standard deviation of 28% of the mean.
Crack arrest
It has been established that the peak value of the applied CTOD after a momentary crack arrest
may be up to twice the static value for the same stress and crack length. The resisting fracture
toughness to be used in crack arrest assessment should be that for the high strain rate conditions
determined by crack arrest tests.
RECOMMENDATIONS
From the work carried out in this project it is clear that general methods of assessment of the
significance of defects such as those given in BS7910 can be applied to assessments under
dynamic loading subjected two main differences.
Firstly a dynamic amplification factor must be applied to applied stresses which depends upon
the relationship between the natural period or frequency of the structure and the rate or
frequency of applied loading. The highest values of the DAF will occur when the applied
frequency is close to a natural frequency of the structure, i.e. close to a resonance condition. It is
necessary to check for interactions between applied loading frequencies and the natural
frequencies of the structure and individual members for several modes, perhaps up to the lowest
five natural frequencies. The magnitude of the dynamic amplification factor will also be
affected by the degree of damping in the structure and this must be taken into account.
Secondly the yield strength and fracture toughness must be adjusted to take account of the
effects of loading rate in accordance with the findings of the work on this project.
xi
xii
1
INTRODUCTION
Although there are well-established procedures for assessing the significance of defects in
welded structures in a number of countries, there are no clear guidelines for such assessments
under dynamic loading.
In principle standard procedures can be applied for any rate of loading, but there is little or no
experience as to how to allow for the effects of dynamic loading on load magnitude or rate of
loading in such assessments. In this project, work has been carried out jointly by UMIST and
TWI to investigate effects of dynamic loading on structural integrity assessments for fracture.
The UMIST work has concentrated on the effects of dynamic loading on the crack tip severity
or driving force aspect of fracture mechanics analyses in structural integrity assessments. This
has included finite element analyses of large scale structures under dynamic loading due to
waves or earthquakes, and the determination of local crack tip driving forces for assumed
defects under such conditions. The UMIST work has also included a brief investigation of
crack arrest behaviour using a finite element cohesive zone model applied to the behaviour of
local brittle zones in weldments. The TWI work has included a review of effects of dynamic
loading on tensile and fracture toughness properties of steels and weldments and some finite
element analyses of fracture and plastic collapse behaviour of fracture toughness specimens
under dynamic loading. All of the finite element analysis work was carried out using the
ABAQUS code (ABAQUS 1998).
1
2
AIMS AND OBJECTIVES
The aims and objectives of the project were as follows:(i)
To provide guidance on how and when to treat dynamic loading in structural
integrity assessments with respect to severity of loading and the effect of strain
rate on the response of the material, with particular reference to high strength
steels and welded joints.
(ii)
To provide guidance on use of dynamic toughness measurements for
assessment of crack arrest at short crack lengths by tough weld deposits,
including performance of high strength steels.
2
3 UMIST WORK ON OFFSHORE STRUCTURES UNDER
WAVE LOADING AND BRACE LOADING
3.1
GENERAL
This part of research project presents the numerical investigation on dynamic integrity
assessment of an offshore structure under normal and wave loads. To understand the principles,
initial work was carried out on a cracked plate under sinusoidal loading at different frequencies.
This was then extended to the behaviour of cracked tubes under sinusoidal loading at different
frequencies. Wave loading was then applied to a full jacket structure by either gridded wave or
Stokes 5th order wave methods with gravity loads also present. Global structural dynamic
analyses of a whole offshore jacket have been carried out in the first step, then, a set of sub­
models of joints selected from the worst locations. These were chosen as where the maximum
stresses occurred or to place a crack at the weldment between chord and brace. Through
thickness cracks were analysed under the loading extracted from the global analysis. Some
analyses were carried out under force control for investigations of local dynamic responses. A
data analysing process of the sub-model dynamic results was created and carried out to obtain
the fracture response in terms of crack tip opening displacement (CTOD) variation. Related
values of K, J and their rates of application were also calculated to investigate load rate effect.
The analysis methodology was proved to be applicable to this kind of study. Some analysis
results were compared to the available analytical solution and reasonably good agreement was
found between them. These results are used to give guidance on methods for design and
assessment of structural integrity under dynamic loading. In a final stage the behaviour of
cracked tubular members under sinusoidal loading was investigated. A more detailed report of
this work is given elsewhere (Zhao 2000 to 2002).
3.1.1 Design codes of practice
Design methods for structures, members or components under static loads to avoid failure,
collapse, buckling etc are well defined in codes and standards, such as BS5950 (British
Standard 2000), BS5400 (British Standard 2000), EuroCode3 and equivalent codes in other
countries, whilst for offshore structures the design code used almost invariably is API RP2A
(API 1993). Assessment methods of structures, members or components with defects under
static loads are established in BS7910 (British Standard 1999). However, there is little or no
guidance on fracture assessment for dynamic loaded structures, members, or components.
Some codes warn that the frequency of dynamic loading should avoid coincidence with
structural natural frequencies but without giving any detailed guidance
3.1.2 Wave types: Stokes 5th order and gridded wave data
Two types of wave theory have been adopted in the present study, firstly Stokes 5th order wave
theory, and secondly the gridded wave supplied by EQE International Ltd. Stokes 5th order wave
is defined by providing wave height and period in the input data with the wave type specified as
Stokes in the ABAQUS option. The gridded wave provides wave surface elevations, particle
velocities, accelerations and wave kinematics at user-defined times at specified matrix points
representing a structure. Both gridded and Stokes waves were applied as distributed loads to the
submerged members of the offshore structure using normal offshore design procedures.
3
3.1.3 Method for study fracture response of dynamic loaded structures
As the J-integral function is not valid for elastic plastic material when unloading and reloading
occurs as with wave loading, to gain valid results for dynamic loaded members, a method based
on extracting crack tip opening displacement CTOD from the nodes around crack tip was
adopted in the present work. Results were compared to the analytical solutions to validate the
methodology and accuracy. A program was written to process the ABAQUS results in terms of
mode I, II and III CTOD values. It was necessary to take account of a new reference plane
(after loading, the orientation of the crack rotated and crack moved to a new position) at each
crack tip for each layer of element and each time increment. The CTOD values are equivalent
values calculated by combining the separate mode components where mixed mode loading
occurs. This was done using the following relationship from the work of Yang (1996).
δ
eq
= δ + 0.246(δ + δ )
I
II III
(3.1)
3.1.4 FE model and sub-models
Finite element analysis (FEA) for this project was carried out using the software ABAQUS
(ABAQUS 1998). The package used to generate the mesh of the joints and tubes was
PRETUBE in SESAM (DNV 1993) with its own interface code PREABA for converting the
generated model into an ABAQUS input file for running the analysis.
In general, shell or plane elements were used for the coarse mesh, C3D20 solid or CPE8
elements for the fine mesh around the crack, collapsed 3D brick elements around the crack-tip
and INTER4 contact elements to prevent overlapping of crack faces in models. The mid-side
nodes were moved to ¼ points (or 1/3 in some cases to avoid element distortion) next to the
collapsed edge in order to model the 1/ r singularity for linear elastic cases. The mid-side
nodes were not moved in the case of elastic-plastic analysis in order to model the 1/ r
singularity at the crack tip. There were three layers of element across the thickness of the
weldment or tube.
According to the recommendations of the API code (API 1993), 2% material damping was
applied in the global (jacket) dynamic analysis carried out as the first stage of the analysis of the
offshore structure. A damping level of 5% of critical damping was included in the analyses of
plate and tube members.
Both material and geometry non-linearity were taken into account simultaneously in the non­
linear analysis. The stress-strain properties for the steel tube and weld material were defined
according to typical results for a Grade 355 offshore structural steel (ex Grade 50D) with a static
yield strength of 350 N/mm2. Where rate sensitive material properties were used, the yield
strength was assumed to vary with strain rate according to equation 3.2, with values of D and n
taken as 40 and 5 respectively to represent behaviour typical of structural steels.
1/ n
s YD
æ e& ö
=
1
+ ç ÷
s YS
è
D ø
3.2
(3.2)
CRACKED PLATE UNDER SINUSOIDAL LOADING
A dynamic analysis of the response for a plate containing edge cracks at both sides subjected to
sinusoidal loading with maximum stress of 70 N/mm2 (one fifth of yield) was undertaken first to
4
investigate the effects of different frequencies for a simple case and confirm general
expectations. The geometry of the plate was 1520 mm in length, 1020 mm in width and having
edge cracks of 254 mm length at each side. A quarter of the plate was modelled. A rigid surface
technique was adopted to avoid overlap of crack surfaces.
Elastic and elastic-plastic analysis analyses were carried out at various loading rates. From the
displacement profile, the crack tip opening displacements (CTOD) were calculated according to
the principles of fracture mechanics. Comparing the dynamic values to the static result, a
dynamic amplification factor (DAF) was obtained. A significant increase of DAF is observed
when the excitation frequency is close to the fundamental natural frequency.
The plot shown on the left of Figure 3.1 indicates variations of CTOD values for three cases,
static, elastic dynamic and elastic~plastic dynamic analyses. It can be seen that the CTOD
values under dynamic loading are significantly higher than those under static loading over the
range of frequency ratios 0.5< v / vn < 1.5 and that this dynamic amplification is over twice as
great for elastic plastic material compared to elastic material. Load versus CTOD for
elastic~plastic analysis is also shown in the rosette diagram in the same plot, where the numbers
around the outside show the ratio of applied to maximum load and the radial distance shows the
CTOD level for that loading condition. The results show a delay in the response, that is the
local CTOD response is almost 180o out of phase with the applied loading.
Figure 3.1 CTOD response for cracked plate at different frequencies
The CTOD, J, K, rates and strain variation histories were determined to reveal the loading rate
effects. For an elastic analysis, the maximum CTOD was found to be 0.13 mm at a frequency
close to the natural frequency of the plate. The range of CTOD rate varied from 67 to 172 mm/s
over the range of frequencies applied. The maximum values for K and J were 4552 N/mm3/2 and
91.5 N/mm. The J rate varied from 57346 to 146526 N/mm/s and the K rate varied from 1E7 to
4.65E6 N/mm3/2 s-1 for these analyses with second order fitted trend lines.
For the elastic~plastic case, the maximum CTOD was found to be 0.32 mm and the CTOD rate
was also obtained from the first part of the curve by the same method. The CTOD rate values
were found to be between 69 and 359 mm/s, which showed a slight increase at the beginning
and doubled later compared to the elastic case.
A strain rate calculation was also carried out for the elastic~plastic analysis. The strain rates
were found to be up to 1 to 8.7 per second. This is a significant increase compared to the static
case which is often taken to be of the order of 10-5 but it should be noted that these results are at
the high frequencies close to the natural frequency of the cracked plate itself.
5
3.3
CRACKED TUBES UNDER SINUSOIDAL LOADING
The study then focused on dynamic responses of individual cracked tubular members under
axial or flexural sinusoidal loading. The characteristics of individual cracked members, such as
the natural frequencies of axial and flexural vibration of the member, were modelled and
analyzed first, then fracture dynamic response modelling work was carried out.
Tubular members were modelled with a circumferential through thickness crack in the middle
of member. The length of tubes was 4192 mm, and the thickness and diameter were 30 mm and
800 mm respectively. Five models were created with different crack lengths of 50, 75, 100, 125
and 209mm. The choice of elements, mesh, loading, damping rate and material properties used
were similar to other models. Due to difficulties of convergence, no contact elements were
inserted between crack surfaces for this set of models. Table 3.1 lists the models, crack lengths,
and CTOD results for an axial stress level of 15 MPa from ABAQUS static analyses compared
to predictions for cracked plates and tubes using the analytical solutions by Zahoor (1985). The
analysis results indicate a reasonable match between the CTODs.
Table 3.1 Tube cases and comparison of ABAQUS CTOD values with theory (mm) for
static axial loading at 15 MPa stress
Case
1
2
3
4
5
C length (cm)
5
7.5
10
12.5
20.9
ABAQUS
1.65E-4/1.8E-4
2.10E-4
2.50E-4
3.05E-4/3.1E-4
4.71E-4/4.76E-4
Cracked plate
1.10E-4
1.62E-4
2.16E-4
2.70E-4
4.51E-4
Cracked tube
1.30E-4
2.024
2.52E-4
2.96E-4
5.19E-4
A time increment parametric study was performed and the final selection made for axial
analyses was 0.00001 seconds. This was to ensure that sufficient data points were obtained to
give accurate results and ensure a solution was achieved at high frequency levels spanning
either side of the natural frequency for axial vibrations. In flexural analysis, an increment of
0.0001 second was tested and found satisfactory for the lower natural frequency level response
in bending to get an accurate result. For dynamic loading, the Rayleigh structural damping
method was used with 5 per cent of the critical damping.
3.3.1 Axial dynamic response
The fundamental axial frequencies of uncracked and cracked members needed for analysis were
determined and are given in Table 3.2.
Table 3.2 Frequencies extracted for different cracked tube cases (Hz)
Case
Theory(UC)
Numerical(UC)
1
2
3
4
5
*extracted at 3/4T
Crack length (cm)
0
0
5
7.5
10
12.5
20.9
Freq.(T/4)
3831.12
3828.41
3838.2
3828.5
3827.01
3855.19
3847.35
6
Freq.(T)
3798.78
na
3786.19
3786.76
3800.16*
3775.95
It can be seen that the frequency for the uncracked member found by FE numerical analysis has
an excellent match with the theoretical solution (compare data in rows 1 and 2 in column 3).
This comparison provides confidence for the frequency results for cracked members. Although
the presence of the crack causes a reduction of the stiffness of the member the effect of crack
size on natural frequency is very small as the vibration of a member depends on the cross
section over its entire length of the member. It should be noted that the natural frequencies of
the tubes under axial loading are high, of the order of 3800 Hz.
The maximum value of the CTOD dynamic amplification factor (DAF) for the largest crack
model under a constant amplitude positive sinusoidal load was found to be less than 3. Analysis
results on rate-dependent material properties were also performed for the worst case model
using the dependency given by equation 3.2. Parameters defining the effect of loading rate on
material properties were selected corresponding to steel from the HSE Guidance Notes (Zhao
2002b) with D=40 and q=5 as representative for offshore structures. The DAF was reduced by
about 30 % for rate dependent properties when compared to the results without this effect.
The fracture parameters K, J and their rates, and CTOD rate were calculated for some cases to
give an insight on loading rate effects. The maximum CTOD rate for the worst case was found
to be 4.5 mm/s. The maximum rates of K and J were 7.71E5 N/mm3/2/s and 1600 N/mm/s,
respectively. The frequency at which the maximum values and rates occurred was slightly
bigger than the fundamental axial frequency with a ratio of 1.04.
3.3.2 Flexural dynamic response
Analyses of flexural responses of the member were also performed. This time, the load was
repeated for a few cycles to allow the response to build up and reach the maximum level. The
analyses included effects of material rate-dependence and damping. The natural frequency of
the tubes under flexural loading was about 600 Hz.
Static and dynamic analyses were performed to obtain DAF values using rate dependent elastic
plastic material properties. A plot of CTOD DAF for the largest crack model is shown in Figure
3.2. As mentioned before, no contact elements were inserted between the crack surfaces, but the
results were obtained after a few sinusoidal load cycles to get the maximum response. It is,
considered that in practice some energy would be dissipated by closing contact of the crack
surfaces, and hence the real DAF value would be less than the value shown in the plot. The
DAF value is larger than the values in axial responses as more cycles have been allowed to
occur to build up to a steady state response.
7
DAF(max)
6
top tip
bot. tip
5
4
3
2
1
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6
freq-ratio
Figure 3.2 Flexural DAF for CTOD in individual tube members
7
An equation of DAF was obtained for tubular members with elastic plastic material properties
and the 5% critical damping level applied as follows:
DAF = -34.375( v/v n ) 2 + 68.75(v/v n ) - 27.875
(3.3)
where v / vn is frequency ratio between that for applied loading and for the structure.
It is of interest to compare the values of DAF found in this work for dynamic analysis of
tubular members with the general structural recommendation of Bolton (2002) represented by
equation 3.4.
DAF = 1/( 1 + (v / vn )4 - (v / vn
)2 (2 - 1/ Q2 )
(3.4)
where v /
vn is the frequency ratio between applied loading and structure response
frequencies, and Q is a constant suggested by Bolton to have values of 40, 60 or 100 for
concrete, bolted steelwork or welded steelwork on solid foundations in air, respectively.
Over the range 0.9< v / vn <1.1 equation 3.4 gives significantly higher DAF values than
equation 3.3, the peak value for equation 3.4 being essentially equal to the value assumed for Q.
The reason for the lower DAF values in the present work, represented by equation 3.3, is the
fact that the analyses of the tubular members were carried out with the damping effects included
in the analysis. The use of equation 3.4 for this case would clearly give excessively high DAF
values. Outside the range 0.9< v / vn <1.1 the DAF values from the two equations are
relatively low but the values from equation 3.3 are still considered more reliable for application
to the immersed tubular member case with damping.
The parameters K, J and their rates, and CTOD rate have been determined for the worst cases in
flexural excitation. For this response, the maximum CTOD rate was found to be 2.18 mm/s, and
rates of K and J were 1.72E5 N/mm3/2/s and 774 N/mm/s, respectively. The applied loading for
the maximum response had a frequency that was slightly smaller than the fundamental flexural
frequency with a ratio of 0.973. It should be noted that these rates of loading correspond to the
high frequency conditions close to resonance of the individual tube members.
3.4
OFFSHORE JACKET STRUCTURE UNDER WAVE LOADING
The jacket structure analysed is shown in Figure 3.3. It is one currently installed in the
Southern North Sea. An initial global finite element analysis was carried out under different
types of wave loading. The period of the first mode of natural vibration of the full structure was
found to be 2.1 seconds. Members and joints were selected from the jacket at places where a
member is affected by wave action directly and the maximum stresses appear. A 100-years
return gridded wave was selected for the type of analysis that is normally used for safety checks
for most of North Sea jackets required in the UK by HSE. Sub models containing cracks were
created for two individual brace members, one multiplanar corner joint and one K-joint for
which the boundary and loading conditions used the output results from the global analysis as
the input of the sub-model.
3.4.1 Submodels of cracked brace members and multiplanar joint
The models analysed were two tubular members chosen from the jacket at the second bracing
level within the wave splash zone and a joint from one of the legs as shown in Figure 3.3. The
two tubular members were chosen as one parallel and one perpendicular to the predominant
wave direction. The first of these tubes was of 900 mm diameter and 30 mm thickness, parallel
to the wave direction and the second was of 800 mm diameter and 25 mm thickness,
8
perpendicular to the wave direction. The cracks were through thickness cracks of length 50 mm
in the circumferential direction. The lengths of tubes were 6131 and 4192 mm respectively.
Model 1
Model 2
Joint location
Wave
travelling
direction
Figure 3.3 Locations of members and joint
For the multiplanar joint model, the chord length was taken as 7715 mm with a diameter of
1400 mm and 75 mm as chord thickness. The brace lengths were 2943 and 3109 mm with
diameters of 800 and 900 mm and thicknesses of 25 and 30 mm. A through thickness
circumferential crack with a length of 100mm was located at the middle length of the chord.
The mesh size for the elements was 0.375 mm around the crack tips and 0.75 mm for the joint.
There are no contact elements inserted between crack surfaces for the members due to
numerical difficulties encountered during this particular analysis.
3.4.2 Results for cracked brace members
A simplified case was examined to verify the model and analysis methodology, which was
model 1 under tension forces to give a stress of 245 MPa applied at the ends. Non-linear static
analysis was performed to obtain J-integral values by using the ABAQUS option to get J=20.5
N/mm. A calculated value of J=21.4 N/mm was obtained from the CTODs by extracting these
displacements around crack tips from the ABAQUS result file. An analytical solution for a tube
under axial tension by Zahoor (1985) gave results for these conditions with J to be 20.2 N/mm.
Comparing the J values obtained by ABAQUS integration, calculation from extracted CTOD
and by Zahoor (1985), the three answers are very close and good matches were found
confirming the modelling and accuracy of results.
A typical dynamic fracture response of CTOD variation of the first model subjected to gridded
wave loading is shown in Figure 3.4. Somewhat different results were obtained at each end of
the crack with maximum CTOD values of 0.001 mm at the top tip and 0.0005 mm at the bottom
tip. Maximum dynamic amplification factors for this model under the 100-year return gridded
wave loading were found to be 1.33 for the top crack tip and 2.02 for the bottom crack tip.
CTOD rates were calculated and the maximum values under the gridded wave were found to be
0.1 and 0.2 mm/s for the top and the bottom crack tips respectively. K and J values and the rates
of K and J were also calculated for this model and the maximum values for K and J were found
to be 380 N/mm3/2 and 0.64 N/mm respectively at the top crack tip. Lower values were obtained
at the bottom crack tip. The maximum rates for of K and J were found to be 1.31E5 N/mm3/2/s
and 128 N/mm/s respectively at the bottom crack tip.
9
0.0012
top crack t ip
CTOD (mm)
0.001
bottom crack tip
0.0008
0.0006
0.0004
0.0002
0
0
2
4
Time (s)
6
8
Figure 3.4 CTOD dynamic fracture response of first sub-model under gridded wave
Lower DAF values were found in the second model with a similar member and crack details but
perpendicular to the wave travel direction. Other values of CTOD, K, J and rates were close to
their relative values for the first model. Comparing these two responses, much less fluctuations
were found in the second model because of the different wave content in the two directions.
3.4.3
Results for cracked multi-planar joint:
The model, meshing and crack opening for the multi-planar joint investigated are shown in
Figure 3.5.
Figure 3.5 Mesh and crack opening for multi-planar joint
A typical dynamic fracture response of CTOD variation of the multi-planar joint subjected to
gridded wave loading is shown on the left of Figure 3.6, compared to the normal design static
response for the same wave conditions. The pattern of the CTOD variation (about 3.5 cycles
occurred within 7 seconds) roughly reflects the structural fundamental frequency that has a
period of 2.1 seconds.
10
Figure 3.6 CTOD dynamic (left) and static (right) fracture response under gridded
wave
Again the absolute values of CTOD were very low with a maximum of 0.001mm, despite the
presence of a through thickness crack of 100 mm under the maximum design wave loading of a
100 year gridded wave. Comparing CTOD responses from static and dynamic analyses, the
dynamic amplification factors were found to have values of 1.37 and 1.28 for the left and right
tips, respectively.
The maximum rates of CTOD for the left and right of tips were found to be 0.052 mm/s and
0.048 mm/s, respectively. The maximum values for K and J were found to be 254 N/mm3/2 and
0.315 N/mm respectively at the right crack tip. The K and J values were smaller at the left crack
tip and the maximum rates of K and J were found to be 1.94E4 N/mm3/2/s and 18.3 N/mm/s
respectively at the left crack tip.
3.5
CRACKED WELDMENT K-JOINT
After dynamic analysis of the cracked brace members and multi-planar joint sub-models
subjected to the 100 year gridded wave loading, a study was carried out of a crack at a tubular
K-joint to investigate the fracture response under different wave loading conditions based on
Stokes 5th order theory. A K joint, shown in Figure 3.7 was modelled with a through thickness
crack of length 200 mm located at the saddle around the weld toe. The chord had a length of
4116 mm, diameter of 900 mm and thickness 50 mm. Braces had a diameter of 800 mm, 20 mm
in thickness and 2000~3000 mm length.
Figure 3.7 Mesh of K joint and crack tip opening
11
A parametric study was performed first on the global model of the full offshore structure to get
the maximum wave effect for the response to wave data input, such as wavelength, frequency,
phase angle and travel direction. A total of ten full analyses was carried out, from different
combinations of wave heights 3.05 and 13.6 metres, wave periods of 2.1, 2.5, 17.5 and 19
seconds, and incident wave angles of 0°, 540 and 900. The element types, mesh, loading,
damping rate and material properties used were similar to the previous models. INTER4 contact
elements were inserted between the crack surfaces. The global analysis was carried out for static
and dynamic analyses under Stokes 5th order wave theory loading.
A dynamic analysis result of CTOD for a wave period at 2.5 seconds, which is close to the
period of first natural frequency of the jacket at 2.1 seconds gave a maximum CTOD value of
0.0003 mm and a maximum DAF of 1.23. For the wave periods of 17.5 and 19 seconds, well
away from the first mode period, similar analysis results indicated the DAF values to be almost
equal to or very close to unity.
The rates of CTOD were also calculated and the maximum rates were 0.00125 mm/s for the
right tip. The maximum values for K and J were found to be 200 N/mm3/2 and 0.196 N/mm,
respectively. The maximum rates for K and J were found to be 1758 N/mm3/2/s and 1.59
N/mm/s.
A typical variation of CTOD with time is shown in Figure 3.8 for a wave period at 2.5s. The
curves indicate the CTOD values at different locations for different modes of crack openings as
left or right crack-tips. A maximum CTOD value was found at the left crack-tip for mode III,
which is almost double the other peak CTOD value. Compared to a static analysis result, most
of the maximum dynamic CTOD values shown in Figure 3.8 were found to be slightly higher
than the static peak values except for the highest one which was twice the static value.
Therefore, the dynamic amplification factors (DAF) of those peak values are just over 1 and the
maximum DAF of 2 was found in this analysis for dynamic response. For a wave period well
away from the first mode period (2.07 seconds), a similar analysis result indicates the DAF
value to be almost equal to or very close to one.
Figure 3.8 Dynamic response of K- joint under Stokes 5th order wave
3.6
CONCLUSIONS ON OFFSHORE STRUCTURE RESPONSE
Analysis methods and modeling techniques have been developed and verified to determine
effects of dynamic loading on cracked members and structures relevant for offshore
12
applications. Some results were compared to available solutions with satisfactory results. All
these results for local member dynamic crack tip responses under axial or flexural loading
suggest that the fracture mechanics based structural integrity assessment of dynamic loaded
structures needs to incorporate a dynamic amplification factor (DAF) when the applied loading
rate has a frequency content close to the fundamental frequencies of the structural member. The
analysis results indicate that:
·
·
·
·
·
·
3.7
Dynamic analyses of cracked plates and tubes as individual members suggest that the
fracture mechanics based structural integrity assessment of dynamic loaded members is
significantly affected by the ratio of applied frequency to natural frequency of the member
or component. The natural frequencies of these individual components are high. Dynamic
amplification factors for crack tip driving force parameters depend also on the degree of
damping applied and material properties assumed. Typical values of peak DAF at
resonance with damping of 5% were up to about 6.5 were obtained for cracked tubes in
bending.
Dynamic amplification factors for offshore applications are much lower than those
recommended for onshore structure applications because of the inherent damping effects of
immersion in water.
Rate-dependent material properties reduce the dynamic amplification by 30 percent when
compared to the result that does not have the factor for the material selected for the analysis.
Global and local dynamic analysis results for an offshore structure suggest the fracture
mechanics based structural integrity assessment of cracked structural joints is not
significantly affected by the 100 year-return gridded wave. The dynamic amplification
factors obtained were less than 1.37 for the joint and 2.02 for the members. CTOD, K and J
values and their rates were also calculated for substantial through thickness cracks subjected
to this 100 year wave and low values found which were not substantially different from
static values. This result is because the frequency of this wave was well removed from the
natural frequency of the structure.
Rates of crack tip loading are directly related to the rate (or frequency) of applied loading.
For offshore wave loading the dominant rates of loading are low and there is little effect on
material properties.
For design guidance purposes, under dynamic loading all the fundamental frequencies of the
global and local member, axial flexural and rotation etc. need to be considered, as they may
cause a large contribution on fracture response. The effect on offshore design is limited
however, provided the natural frequency of the structure is well removed from the
frequency of applied wave loading.
RECOMMENDATIONS FOR OFFSHORE APPLICATIONS
It is necessary to check the fundamental frequency of the individual members as well as the
overall structure. The modes of action include axial, flexural, torsional and other basic modes.
The first mode shape is crucial and leads to a large response. A large DAF of CTOD value may
occur, (about 3~6.5) when the frequency of applied loading is close to the natural frequency of
the member or structure. This is larger than those typically stipulated in codes of practice for
load magnified factors to accommodate dynamic load (impact factors) by multiplying the static
load with this factor. Rate-dependent material properties need to be taken into account
especially when the rate of loading is high, however with wave loading only slow to moderate
loading rates are involved. For a repeated load action, such as wave loading, the response is
more serious than the single load action, because the response is building up at each cycle of the
load till it reaches a certain level which is bigger than a single loading response. The single load
response may be defined by using the BS7910 or R6 methods with the static load stresses
multiplied by a DAF dependent on the ratio of applied to natural frequency.
13
4 UMIST WORK ON EFFECTS OF EARTHQUAKE LOADING
ON FRACTURE AT WELDED JOINTS
4.1
GENERAL
In this investigation, finite element analysis has been used to study the local behaviour of
connections with defects within a complete building frame under the dynamic loading of an
earthquake. The purpose of the FEM analyses was to identify the effects of crack length,
connection design, and material properties on the local behaviour of sub model connections
located in the full steel frame building under such loading. The stress distribution in the region
of the column and beam flange connection was also considered.
Global model analyses were carried out on two building frames, one of four storeys height and
one of seventeen storeys height, representative of those involved in the Northridge earthquake.
The results in this report are for the four-storey building. These models were two-dimensional
representing one frame of the complete building. The earthquake loading applied was the time
history record of the ground movement recorded at the Canoga station during the Northridge
earthquake. In addition loadings were applied representing 95% of the energy content of the
loading spectrum, and representing the equivalent static design load, which would be used in
design with the UBC code
Figure 4.1 Time history acceleration record from the Canoga station (Northridge)
Normalised FFT amplitude (m/s2 )
0.6
0
5
10
15
20
Frequency (Hz)
Figure 4.2 Frequency content of the Earthquake motion at the Canoga station
Figure 4.1 shows the acceleration record from Canoga station and Figure 4.2 shows the
frequency content obtained by Fourier transformation. The mass and dead load of the building
14
were applied in terms of material density. It can be seen that the dominant frequencies are in
the range 0 to 5 Hz.
Five different sub model analyses were carried out consisted of five alternative designs of beam
to column connections, each series being defined by the configuration of the connection as
follows:
·
·
·
·
·
Basic connection as used at Northridge with no additional strengthening.
Connection with continuity plates added between the flanges of the column. Connection with beam flange width reduced by a "dog bone" shape away from weld. Connection with additional haunch section added underneath the beam section. Connection with cover plates added to the beam flanges and welded to the column flange. The sub models were three dimensional to allow effects across the widths of the welds to be
captured. Each sub model series was subdivided into three different cases of analysis. Case one
consisted of the sub model connection without initial defects; cases two and three consisted of
the connection with 2.5 mm and 5 mm height crack like defects in the sub model across the
underneath of the width of the bottom flange of the beam against the column face. Each series
of sub model connections was attached to the global frame model. A sensitivity study
concerning material properties was also investigated with effects of four different stress strain
curves examined. The results of the dynamic analyses for each series are also compared with
pseudo static analyses, pushover analyses, using the Uniform Building Code (UBC)-1994
method (ICBO 1997). More detailed reports of this work are given by Burdekin and
Kuntiyawichai (2002, 2003).
4.2
DESCRIPTION OF FINITE ELEMENT MODELS
The general purpose pre and post processing code PATRAN v9.0 was used to create the input
data for the finite element models. The complete models then were analysed by the ABAQUS
finite element package with facilities for linear and non-linear analyses. The two-dimensional
beam elements with three nodes, i.e. B22 available in ABAQUS, were used for modelling all of
the two-dimensional models. These elements have three degrees of freedom per node. The
beams were rigidly attached to the columns.
For the sub models of the connections, three-dimensional solid elements incorporating initial
cracks were used. Although the sub model with 3D solid elements within the full 2D model
proved to be computationally expensive, it provided valuable understanding of the local
behaviour of connections situated in the complete building frame. The elements used for the
modelling of sub models were first-order (linear) interpolation three dimensional linear brick
elements, i.e. C3D8 available in ABAQUS. These elements have three degrees of freedom per
node. With an adequately fine mesh, these elements are capable of providing accurate solutions
even in complex structures. The cracks were modelled with a crack tip with initial root radius
which was assumed to be 0.5 mm in all cases in order to prevent the overlap between the crack
faces.
Finally, the sub models were placed into the global model by using distributing coupling
elements (DCOUP3D) introduced with ABAQUS/Standard in the area of connection between
the solid elements of the sub model and the beam elements of the global model. This special
option offers general capabilities for transmitting loads and associating motions between one
node and a collection of "coupling" nodes. The option associates the coupling nodes with a
single node in a "rigid body" sense; translations and rotations of the node (the distributing
coupling element node) are associated with the coupling node group as a whole.
15
4.3
MATERIAL PROPERTIES
Both elastic and elastic-plastic material properties were considered in this study. The elastic­
plastic material models for the analyses included bi-linear (mises) and Ramberg-Osgood models
in which incremental theory of plasticity with (constant) isotropic hardening was employed with
the von Mises yield surface expressed in terms of Cauchy (true) stress. The conventional
engineering strain, e E and engineering (nominal) stress, s E values were converted to true
strain and true stress for input to the FE analysis. Table 4.1 shows the four material properties
used.
Table 4.1 Material properties
Material No.
Material
Young's Modulus
Yield stress
Work hardening
(MPa)
(MPa)
Slope (MPa)
3
M1
Elastic
210x10
-
-
M2
Elastic-plastic
210x103
350
21x103
M3
Elastic-plastic
210x103
350
210
Elastic-plastic
3
350
s u 550 MPa
M4
210x10
at 15% strain
4.4
RESULTS
Because of uncertainties about J values under dynamic loading with unloading effects
occurring, the applied crack tip severity was evaluated by the crack tip opening displacement
(CTOD) concept. The relative displacements of the nodes along the crack faces were extracted
and used to derive the crack tip opening displacement. The maximum values of these during the
Northridge earthquake loading spectrum (Canoga station) were then plotted against the position
across the width of the beam flange.
4.4.1 Applied CTOD Distributions from Dynamic Sub Model Analyses
The results of the FE analyses for full earthquake dynamic loading are shown in Figure 4.3 for
the 2.5 mm height flaw and Figure 4.4 for the 5 mm height flaw. The results for each of the
four material types M1 to M4 are shown in the parts (a) to (d) respectively of each figure. Each
part figure shows the results for the maximum applied CTOD distribution across the width of
the flange for each of the five sub models at any time during the whole occurrence of the
earthquake time history loading.
It can be seen that for both flaw sizes and each material case, the highest applied CTOD values
occur for the basic model case with no additional strengthening. Furthermore, these highest
values occur at the centre of the width of the beam flange with lower values occurring out
towards the edges of the flange. This is consistent with the findings for general stress
distributions across the flange width from previous investigations at UMIST (Burdekin and
Suman 1998).
The inclusion of continuity plates produces some reduction in the peak values of CTOD for all
material cases, as does the reduced beam width (dog bone) model. However much the greatest
reduction in applied CTOD values is obtained by the use of either a haunch section or the
16
addition of cover plates. In these cases the applied CTOD values were reduced to very low
values for all material types.
The effect of the two different flaw size assumptions can be seen by comparing the
corresponding part figure for the same material in Figures 4.3 and 4.4. It can be seen, as
expected, that the peak applied CTOD values for the 5 mm flaw are greater than those for the
2.5 mm flaw.
The effect of the different material assumptions can be seen by comparison of parts (a) to (d) of
each of Figures 4.3 and 4.4. Significant differences occur as a result of the different material
assumptions. The peak CTOD values for the basic model with the 2.5 mm high flaw vary from
0.03 mm for the elastic case (M1) to 0.11 mm for the Grade 355 steel (M4), whilst with the 5
mm height flaw the corresponding values are 0.035 mm for M1 and 0.18 mm for M4.
4.4.2 Applied CTOD Distributions from UBC Equivalent Static Loading Sub
Model Analyses
Each sub model connection was also analysed by using pseudo static analyses. The global
model was applied by the equivalent lateral loads. It was noted that the mode shape of the
deformed structure under the full dynamic earthquake loading did not necessarily correspond to
that for the UBC equivalent static loading. The results of the FE analyses for applied CTOD are
shown in Figures 4.5 and 4.6 for 2.5 mm and 5 mm height flaws respectively in the basic sub
model. It can be seen that the static analyses produce the same type of distribution of applied
CTOD across the width of the beam flange as the full dynamic loading. For the elastic material
case (M1), the static and dynamic loads give good agreement. As more plasticity is allowed in
the material cases however, differences occur between the dynamic and static results. It can be
seen from Figures 4.5(d) and 4.6(d) for the Grade 355 steel (M4) that the UBC static loading
gives increased estimates of the peak applied CTOD at 0.15 mm for the 2.5 mm height flaw and
0.22 mm for the 5 mm height flaw. For this particular material case it would be conservative to
use the UBC loading in a BS7910 type analysis as it would over estimate the applied CTOD
compared to the full dynamic load case.
4.5
CONCLUSIONS ON DYNAMIC LOADING DUE TO EARTHQUAKES
Finite element analyses have been carried out to investigate the factors that contributed to
failure of the welded connections in moment resisting steel framed building as involved in the
Northridge earthquake. The particular structure chosen for analysis was a four storey frame
typical of ones in which a significant number of brittle fractures had been reported. Analyses
have been carried out under both full dynamic earthquake and equivalent UBC static loading.
Five different geometries of sub model were incorporated into the global model, these being the
basic beam to column connection with no strengthening, the case with added continuity plates
between the column flanges, the case with the beam flange width reduced (dog bone) and the
cases with an additional haunch or additional cover plates.
The results confirmed the general increase in severity of conditions towards the centre of the
beam flange width. The stress distribution also confirmed that the position of the maximum
stress coincided with the location of the fracture initiation. Connections strengthened with a
haunch or cover plates were found to show a very good performance and much better than the
dog bone and "pre-Northridge" connection. The material tensile properties also affect the crack
tip severity at the crack tip.
17
The absolute values of applied CTOD obtained from the FE analyses are entirely consistent
with the occurrence of fracture in materials with fracture toughness levels of the order of 0.1
mm to 0.2 mm CTOD.
A reasonable estimate of the applied CTOD value for particular earthquake loading can be made
by using the equivalent code static loading (such as the UBC method) in a standard finite
element analysis of the structure to determine basic stresses for input to a BS7910 type analysis.
The equivalent static loading is based on the magnitude of the accelerations due to the
earthquake, the mass of the structure and an allowance for the effect of ground conditions. This
agreement may be to some extent fortuitous as the equivalent static loading does not necessarily
give rise to the same mode shape of deformations in the structure as the true dynamic loading.
It is also necessary to make some allowance for non uniform stress distributions at beam to
column flange connections. With the typical frequencies involved in earthquake loading the
effects of rate of loading on material properties are relatively modest.
18
d) Material model M4
c) Material model M3
19
Figure 4.3 Variation of CTOD across the bottom flange for all cases (crack height = 2.5 mm, dynamic)
b) Material model M2
a) Material model M1
d) Material model M4
c) Material model M3
20
Figure 4.4 Variation of CTOD across the bottom flange for all cases (crack height = 5 mm, dynamic)
b) Material model M2
a) Material model M1
d) Material model M4
c) Material model M3
21
Figure 4.5 Comparison of CTOD results for 2.5 mm crack height for full earthquake dynamic and UBC static loading (basic model)
b) Material model M2
a) Material model M1
d) Material model M4
c) Material model M3
22
Figure 4.6 Comparison of CTOD results for 5 mm crack height for full earthquake dynamic and UBC static loading (basic model)
b) Material model M2
a) Material model M1
5
TWI WORK ON MATERIAL RESPONSE TO DYNAMIC
LOADING
5.1
TENSILE PROPERTIES UNDER DYNAMIC LOADING
5.1.1
General
Flow stress of steels as a function of temperature is shown schematically in Figure 5.1. The flow
stress versus temperature diagram may be divided into three regions (I, II and III) with the
temperature range of T<T0, T0 <T < T1, T>T1, respectively. Significant increases in flow stress
are observed in regions I and III as temperature decreases, while flow stress in region II is
relatively less varying. Region I is the focus of the present work, because a toughness transition
occurs in this temperature regime for most ferritic steels.
o
Temperature, C
-273
1100
-173
-73
27
127
227
327
427
527
627
727
700
800
900
1000
1000
LOW CARBON STEEL
900
Yield strength, MPa
800
700
600
500
400
I
300
Thermally activated
yielding regime
200
II
III
100
T0
T1
0
0
100
200
300
400
500
600
Temperature, Kelvin
Figure 5.1 Flow stress variation with temperature (scales are indicative only)
The flow stress is the stress required to maintain plastic deformation by moving dislocations
through local and global obstacles. The magnitude of flow stress depends on temperature that is
a measure for the level of thermal energy within a material. In region I of Figure 5.1, the plastic
deformation is widely regarded as a thermally activated process such that the flow stress may be
partitioned into an effective stress s* and an internal stress sa, on the basis of temperature and
strain rate sensitivity, i.e.
s f =s a +s *
(5.1)
A schematic diagram showing the stress partition is depicted in Figure 5.2, where sp is the
effective stress at T=0 Kelvin. The flow stress varies from a maximum of sp + sa to an athermal
value equal to the internal stress sa at temperature T0. The effective stress s* is temperature and
23
strain rate sensitive. σ* can be characterised by activation enthalpy necessary for dislocation to
move
sf = sa + s*
sf
Flow stress, MPa
sp
s*
sa
I
Thermally activated
yielding regime
T0
T
0
50
100
150
200
250
300
350
400
Temperature, Kelvin
Figure 5.2 Flow stress partitions of an effective stress and an internal stress
(temperature scale is indicative)
through the local barriers and the associated short range stress fields. The internal stress sa is
also determined by obstacles which oppose dislocation motion, but in this case the associated
stress fields are of long range such that thermal fluctuations are ineffective in surmounting
them, and hence sa is not temperature or strain sensitive.
It can be shown (Ellwood et al 1984, Lei et al 1996) that the effective stress is a unique function
of the activation enthalpy H = kTln(A/ e& ) as follows:
1
æ
H ö
1- m
s * =
s p çç1
-
÷÷
è
H 0 ø
(5.2)
So that flow stress can now be described by the following Equation:
1
ì kT ln( A / e& ) ü1-m
s f = s a + s p í1 -
ý
H0
î
þ
(5.3)
where:
sa
sp
k
T
A
= the internal stress, MPa
= the Peierls stress at T=0, K
= the gas constant, 1.38E-23, JK-1
= temperature, Kelvin
= the frequency factor, a typical value is 108
24
e
&
H0
m
= strain rate, s-1
= the activation enthalpy associated with local barriers; Joule
= an integer
( )
If combinations of T and e& are such that T1ln( A / e&1 ) =T2ln( A / e&2 ) , the yield strength s ys
1
at
(T1 , e&1 ) is equal to (s ys )2 at (T2 , e&2 ) . This means that yield strength at temperature T and
strain rate e& may be obtained by pseudo-static test at temperature T * which is defined by:
T * =T
5.1.2
ln( A / e& )
ln( A / e&static )
(5.4)
Empirical yield strength-temperature-strain rate equation
Irwin (1968) developed an empirical equation for yield strength-temperature relation under
pseudo-static strain rate as follows:
(s ) = (s )
ys T
ys -100 o F
+
14500 (ksi - K)
- 40.4 ksi
T (F) + 459
(5.5)
Adapting to metric units and substituting yield strength at temperature T1 (Kelvin) for
s ys -100 o F , the above equation can be re-written as:
( )
(s ) = (s )
ys T
ys T
1
æ1 1 ö
+
S çç -
÷÷
è
T T1 ø
(5.6)
where T is absolute temperature in Kelvin and S is a constant equal to 55543 (MPa-K)
according to Eq.(5.5). Note that a material specific S may be more appropriate to achieve a
better fitting to test data.
As discussed earlier, the yield strength at T and strain rate e& is equal to the yield strength at T*
defined by Eq.(5.4) under quasi-static loading rate. Substituting T* for T in Eq.(5.6) and room
temperature 293K for T1, one obtains yield strength as a function of temperature and strain rate
as follows:
ì 1 ln( A / e& static ) 1 ü
s ys (T
, e&
)
=
s 0 +
S
í
-
ý
293 þ
î
T ln( A / e& )
(5.7)
where σ0 is quasi-static yield strength at room temperature (T=293K), S is a material constant.
Substituting the typical value 108 for A and 5x10-51/s for the quasi-static strain rate, yield
strength at T and strain rate e& can now be written as follows:
ì1
28.3
1
ü
s
ys (T
, e&
)
=
s 0 +
S
í
-
ý
8
î T ln(10 / e& )
293
þ
(5.8)
25
Yield strength test data from the open literature (Clausing 1969, Said 1993) were used to
validate Eq.(5.8). Results of this validation are shown in Figure 5.3 and Figure 5.4. A value of
60000 was used for S in this validation study. The temperature and strain rate ranges covered in
this validation were as follows:
-196 o C £T £ 20 o C (room temperature)
10 -5 1/ s £ e& £15001/ s (covered by the test data used in the validation)
The validation study showed that the parameter S was generally in between 50000 and 70000
for wide range of ferritic steels. In the absence of test data, a mean value of S equal to 60000
(MPaK) may be used. 2500.0
ABS-C
2000.0
A302-B
HY-80
A517-F
HY-130
18Ni (180)
Predicted yield strength, MPa
18Ni(250)
1500.0
1000.0
500.0
0.0
0
500
1000
1500
2000
2500
Measured yield strength, MPa
Figure 5.3 Comparison of measured and predicted yield strength of seven steels under
quasi-static loading rate
850.0
800.0
Predicted yield strength, MPa
750.0
700.0
650.0
600.0
dε / dt =0. 1 1/ s
dε / dt =15 1/ s
dε / dt =115 1/ s
dε / dt =1500 1/ s
one to one correlation
550.0
500.0
450.0
400.0
400
450
500
550
600
650
700
750
800
850
Measured yield strength, MPa
Figure 5.4 Comparison of measured and predicted yield strength of weld metal in
Grade 50D plate under dynamic loading
26
5.2
FRACTURE TOUGHNESS UNDER DYNAMIC LOADING
5.2.1 Definition of strain rate in fracture specimens
The mathematical definition of strain rate is simply:
e& =
e t1+ Dt -e t1
as Dt approaches to zero.
Dt
(5.9)
A distinction must be understood between the loading speed (e.g., the cross head speed of the
test machine) and the strain rate at a material point in the structural component (e.g., a point in
the fracture process zone at the crack tip).
For cracked components, an equation derived by Irwin (1964) for contained yielding at the
crack tip may be used to estimate the strain rate in the elastic region just outside the crack tip
plastic zone
2s K&
e& = y
E K
where:
sy
E
K&
K
(5.10)
= yield strength at strain rate e& , MPa
= the Young's modulus, MPa
= rate of applied stress intensity factor
= the applied stress intensity factor.
Eq.(5.10) is a good starting point in defining an appropriate strain for cracked structures and
components. To gain more accurate information on strain rate, detailed finite element analyses
may be used to investigate the spatial distribution in the fracture process zone.
5.2.2 General trends of strain rate effects on fracture toughness transition
Curve
Figure 5.5 illustrates general trends of effects of strain rate on fracture toughness transition
curve of most ferritic steels. In the lower shelf regime, the transition curves for different strain
rates approach a minimum toughness value, which is almost independent of strain rates; in the
transition regime, increasing loading rate decreases fracture toughness for a given temperature;
and in the upper shelf regime, increasing loading rate tends to increase toughness. For a given
temperature increased strain rate can cause a change in fracture mode from ductile to brittle.
Methods are required for the estimation of fracture toughness at a particular strain rate in the
transition regime, because the use of static fracture toughness values is unsafe in circumstances
where high loading rate apply. This has been recognised by the ASME pressure vessel code
guidance for lower bound fracture toughness data which provides both a curve for static loading
(KIc) and a curve for crack arrest (KIa) or dynamic (KId) fracture toughness, KIR.
Most of research work in the past on the effect of strain rate on fracture toughness was
concerned with defining a shift (DT) in fracture toughness transition temperature to higher
values using empirical methods. Fracture toughness at the strain rate of interest was estimated
from the static fracture toughness curve at a temperature DT lower than the temperature
of interest.
27
175
Fracture toughness, K c (MNm
-3/2
)
150
125
100
Increasing loading rate
75
50
DT
25
T0
T0
0
0
50
100
150
200
250
300
350
Temperature, K
Figure 5.5 General trends of strain rate effects on fracture toughness transition curve
of ferritic steels (scales are indicative)
5.2.3
Empirical methods for fracture toughness transition temperature shift
As outlined above, elevated loading rates, say e& ³ 5 ´10 -5 s -1 , tends to lead to a reduction in
fracture toughness of most ferritic steels in the transition regime. It results in an increase in
transition temperature that may be defined at an appropriate value of K1c, e.g., 100MPam0.5
(Figure 5.5).
Barsom (1976) measured the fracture toughness of a range of steels covering quasistatic( e& » 10 -5 s -1 ) to impact loading ( e& » 10 s -1 ). The data mainly apply to the lower transition
region. Only one of the steels exhibited toughness values in excess of 100MPam0.5 in the
experiment temperature range. The temperature shift DT, at a toughness of 70MPam0.5, was
related to the static room temperature yield stress sy by:
DT =119 - 0.12s y for 250MPa £s y £ 965MPa
=0
for s y > 965MPa
(5.11)
Eq.(5.11) was generalised for intermediate strain rates e& as follows:
Tshift = (83- 0.08s y )e& 0.17 for 250 MPa £s y £ 965MPa
10 - 2 s -1 £e& £10 s -1
(5.12)
Ainsworth (1992) observed that there was a tendency of decreasing shifts with increasing
transition temperature. He suggested that the shift depend on strain rate as well as the absolute
transition temperature, T, and derived the equation below:
Tshift = (83- 0.08s y ) e& 0.17 (1.78 - 0.00267 T )
28
(5.13)
for 250MPa < sy < 965MPa
and for 10-3 s-1 < e& < 10 s-1
where sy is in MPa and T is absolute temperature in Kelvin.
Wallin (1997) carried out a master curve analysis for experimental fracture toughness data
obtained in the transition regime and concluded that loading rate did not affect the shape of
transition curve. Following the master curve approach, he suggested that loading rate effects on
fracture toughness could be quantified by a shift in the transition temperature for a loading rate
of interest.
The transition temperature shift relative to a static loading rate K& 1 =1MPam 0.5 was expressed as
Tshift =
T01 G
G - ln(K& 1 )
(5.14)
where T01 refers to the master curve estimate of the transition temperature at the static loading
rate K& 1 =1MPam 0.5 . G is estimated from the following equation
ì
T 1.66 æ s
RT ö
1.09 ü
ö
ïæ
y
÷ ïý
G =
9.9 expíç 01 ÷ + ç
ç 722 ÷
ïîè 190
ø
è
ø ïþ
(5.15)
where s yRT is static yield strength at room temperature. The standard deviation of G is 19.4%.
Eurocode 3, Annex C deals with design against brittle fracture. For loading/strain rates higher
than quasi-static, the code applies the concept of a transition temperature shift in fracture
toughness to allow for effects of higher strain rates. The temperature shift is given by the
following equation:
(
1440
-
s ys )
é
æ
e& ö
ù1.5
lnç ÷
,
where e&
D
T =
550
ê ç ÷ú
êë è
e&0 ø úû
0 =
0.0001 s
-1
(5.16)
An appropriate strain rate must be used in Eq.(5.16) to calculate the temperature shift. Some
assumptions are given on the level of strain rates for different loading situations, which may be
used in the absence of more accurate information. These assumptions are summarised below:
& 10 -3 1/ s , e.g. structural members subjected to self-weight, floor
· Low strain rates: e
=
loading, wind and wave loading and lifting loads.
& 10 -1 1/ s , e.g. structural members subjected to impact of wheel
· Medium strain rates: e
=
loads.
· High strain rates: e& = 10 1 / s , e.g. structural members subjected to accidental loading
such as vehicle impact, explosion loads.
29
Transition temperature shift equations presented above were used to estimate temperature shifts
due to the effect of strain rate for three materials found in the open literature (Krabiell and Hahl
1981) and test results obtained at TWI (Wiesner and Bell 1996). A comparison between
measured transition temperature shifts and the predicted values by the empirical equations
above for three steels is shown in Figure 5.6.
Barsom eqn for
impact loading
o
Predicted temperature shifts, DT shift , C
120
100
Barsom eqn for
intermediate
loading
80
Ainsworth eqn
60
Wallin eqn
40
Eurocode3
20
One to one
correlation line
0
0
20
40
60
80
100
120
o
Measured temperature shifts, D T shift , C
Figure 5.6 Comparison of measured data of transition temperature shift and predicted
values
It is worth pointing out that when the above empirical equations are used (apart from Wallin’s
equation) to calculate a transition temperature shift, the strain rate definition in Eq.(8.10) must
be used.
The transition temperature shift method for quantifying the strain rate effects on fracture
toughness implies that the shape of the curve is not affected by strain rate. Therefore it may be
possible to describe dynamic fracture toughness in terms of a common parameter involving both
temperature and strain. This is explored next.
5.2.4
An empirical equation for fracture toughness under dynamic loading
Dynamic fracture toughness data of four structural and pressure vessel steels with yield strength
ranging from 300MPa to 600MPa have been collected from the open literature (Krabiell and
Hahl 1981) and tests results at TWI (Wiesner and Bell 1996). In analogous to Wallin’s master
curve approach, the collected data have been analysed statistically assuming that the probability
of obtaining a Kc value is a normal distribution and the standard deviation is a proportional to
the mean. It is postulated that dynamic fracture toughness be expressed as a function of the rate
parameter, designated here as R. The analysis process involved the following steps:
(1) Gather information on experimental data of dynamic fracture toughness, · Kdyc , · rate of applied K, 30
·
·
test temperature, yield strength as a function of temperature and strain rate (2) Estimate crack tip strain rate using the following equation: e&tip = 2
s y K&
E K
(5.17)
where σy is a function of strain rate at the crack tip, so determination of strain rate at the crack
tip by the above equation is an iterative process.
(3) Calculate the temperature-rate parameter R, which is
æ
A ö
R =
T lnç ÷
è
e& ø
(5.18)
where A is assumed to be a constant for all materials, A=108.
(4) Plot KdyC as a function of parameter R, and fit the data to the equation below
K dy c = A * exp{BR} or lnK dy c = BR + ln A
(5.19)
Coefficients A and B are obtained by the least squares method.
(5) Calculate R at Kc = 100MPa-m0.5, designated as R0
(6) Repeat the process described above (steps 1 to 5) for another data set
(7) When all available data sets have been processed following steps 1 to 6, plot Kdyc as a
function of R-R0, and fit the data to the equation below:
K dy C = 20 + 80 exp{C ( R - R0 )}
(5.20)
For the data collected, the coefficient C in this equation was determined to be 0.00066. The
mean values of dynamic fracture toughness can now be described by the following equation:
K C = 20 + 80 exp{0.00066( R - R0 )}
æ
10
è
e&
where R
=
T lnçç
8
(5.21)
ö
÷÷
, R0 is the value R for Kc = 100MPam0.5. The standard deviation was
ø
found to be 28% of the mean for the data analysed. The mean, mean+2SD and mean – 2SD and
the associated fracture toughness data are shown in Figure 5.7.
31
400
Fe E 460
Fe 510
20 MnMoNi 5 5
Weld metal in PV steel
Mean
Mean + 2SD
Mean - 2SD
Dynamic fracrure toughness, Kc, MPa-m 0.5
350
300
250
200
150
100
50
0
-4500
-4000
-3500
-3000
-2500
-2000
-1500
-1000
-500
0
500
1000
1500
T ln(A / ε rate ) - R0, K
Figure 5.7 Dynamic fracture toughness curve as a function of the rate parameter R
The quasi-static fracture toughness curve is a special case of Eq.(5.21). If the crack tip strain
rate is substituted by a quasi-static loading rate 5x10-51/s, one obtains an equation for quasi­
static fracture toughness
QS
K C = 20 + 80 exp{0.01869(T - T0 )}
(5.22)
The coefficient in front of (T-T0) in the above equation is virtually equal to 0.019 which is the
value derived by Wallin in the master curve for quasi-static fracture toughness. This suggests
that R0 can be related to T0 as follows:
æ 108
ö
æ 108 ö
÷ =
28.3T
÷
ç
R
0 =
T 0 ln
=
T lnç
0
ç e&static ÷ 0 ç 5
´ 10-5 ÷
ø
è
ø
è
(8.23)
The dynamic fracture toughness curve can now be written as follows:
dy
K C = 20 + 80 exp{0.00066(R - 28.3T0 )}
(8.24)
Therefore, values of dynamic fracture toughness can be estimated directly from Equation (8.24)
for a given temperature T and strain rate e& .
5.3
CONCLUSIONS ON MATERIAL PROPERTY ASPECTS
The present research on material responses to dynamic loading has let to the following
conclusions:
32
·
Effects of temperature and strain rate on material yield strength can be estimated using the
following equation:
ì 1 ln( A / e& static ) 1
ü
s
ys (T
, e&
)
=
s 0 +
S
í
-
ý
293 þ
î
T ln( A / e& )
(8.25)
Where σ0 is yield strength at room temperature under quasi-static loading. e&static is a quasi­
static loading rate, typically ~5x10-5 1/s. A is a material constant with typical value of 108. S
is a parameter to be fitted using test data, but if test data are not available, a value of
60,000 ± 10000 MPaK may be used. This equation is applicable for temperatures lower than
the ambient and for strain rate up to about 10001/s.
· Effects of temperature and strain rate on fracture toughness can be estimated using a mean
fracture toughness curve in terms of the rate parameter R as follows:
dy
K C = 20 + 80 exp{0.00066(R - 28.3T0 )}
where T0 is the fracture toughness transition temperature under quasi-static loading.
R=Tln(108/ e& ). KdyC has a standard deviation of 28% of the mean.
33
(8.26)
6
TWI WORK ON LOCAL APPROACH IN FRACTURE
TOUGHNESS SPECIMENS
6.1
GENERAL
The essence of local approach in cleavage fracture is contained in the Beremin model (Beremin
1983), which describes probability of fracture with a Weibull probability function as follows:
é
mù
æs ö
P(s w ) =1- exp ê- çç w ÷÷ ú
ú
ê su
è
ø ú
û
ëê
(6.1)
and
1/ m
sW
é
1
ù
m
=ê
ò (s I ) dV p ú
ë V0 Vp
û
(6.2)
where:
Pr
σw
m
σu
=
=
=
=
sI
V0
=
=
Vp
=
the cumulative probability to fracture failure
the Weibull stress
is the shape parameter of the Weibull distribution
is the scaling parameter of the Weibull distribution, equal to the value of σw for
a failure probability of 0.63
the maximum principal stress acting in the plastic zone
a characteristic volume which is large enough to contain flaws but sufficiently
small to be considered as uniformly loaded.
volume of the plastic zone
The shape parameter m is independent of the reference volume V0; but V0 and σu are interrelated
through σmuV0= constant, so that the Beremin model is a two parameter model, defined by m the
shape parameter and σu the scaling parameters. For predominantly brittle cleavage fracture, m
and σu are considered to be independent of geometry, temperature and strain rate, i.e. they are
material constants. Once their values are determined from one set of test data for a given
material, fracture under other general conditions can be predicted using stress information at the
crack tip obtained from finite element analyses.
The purpose of this part of research was to employ the Beremin model to predict dynamic
fracture toughness values of SENB specimens under intermediate loading speed v=10mm/s for
a modern high yield strength steel, designated 450EMZ. A finite element model was created and
validated first. The Beremin model parameters m and σu were determined using quasi-static
fracture toughness K1C test results and crack tip stress information obtained static finite element
analyses using the validated FE model. Then crack tip stress information under the intermediate
loading speed was obtained using the same FE model which in this case included the effects of
loading rate on material stress-strain behaviour. Fracture toughness values were predicted for
the higher loading speed and compared with test results.
6.2
TENSILE PROPERTIES AND FRACTURE TOUGHNESS OF STEEL 450EMZ
Tests were carried out to determine tensile properties and fracture toughness values for the
modern steel (450EMZ) under quasi-static and dynamic loading rates. The composite of this
steel is given in Table 6.1.
34
Table 6.1 Chemical composition of steel grade 450EMZ 50mm plate (%)
C
Si
Mn
P
S
Cr
Mo
Ni
Cu
V
N
Ti
Al
CEV
0.1
0.28
1.20
0.011
0.002
0.02
0.14
0.47
0.01
0.05
0.0051
<0.002
0.032
0.347
Dynamic yield strength as a function of temperature and strain rate was be described by the
following equation (detailed discussion on temperature and strain rate dependency of yield
strength is given in Section 5 of this report):
ì
8 - 4
) 1 üï
dy
ï 1 ln(10 / 10
s ys = 436.0 + 67613í
ý
293 ï
ïî T ln(108 / e&)
þ
(6.2)
where T is temperature in K and e& is strain rate in 1/s.
The strain hardening behaviour for this steel was be described by the following equation:
s
ep n
f
)
= (1 +a
s0
e0
(6.3)
with α=0.035 and n=0.25. σ0 is the initial yield stress and ε0 is the initial yield strain equal to
σ0/E
Fracture toughness results of single edge notch bend specimens of 50x50x200mm are
summarised in Table 6.2 and Table 6.3 for quasi-static and intermediate loading rate
(v=10mm/s), respectively.
35
Table 6.2 Fracture toughness data for 450EMZ under quasi-static loading
Specimen ID
J
(N/mm)
CTOD
(mm)
KJc
(MPam0.5)
M01
M02
M03
M04
M05
M06
M07
M08
M09
M10
M11
M12
M13
M14
M15
M16
M17
M18
M19
M20
47.1
36.7
37.3
67.5
45.6
17.5
75.8
69.7
77.7
38.5
18.2
48.6
14.8
47.6
62.3
20.7
155.7
32.4
120.0
67.7
0.037
0.029
0.030
0.055
0.036
0.013
0.058
0.056
0.051
0.030
0.015
0.036
0.012
0.037
0.051
0.016
0.121
0.0296
0.0969
0.0547
103.5
91.4
92.1
123.9
101.8
63.1
131.3
125.9
132.9
93.6
64.3
105.1
58.0
104.1
119.0
68.6
188.2
85.8
165.2
124.1
Table 6.3 Fracture toughness data for 450EMZ under intermediate loading (v=10mm/s)
Specimen ID
Test temp.
(oC)
Load speed
(mm/s)
Fracture load
(kN)
J
(kJ/m2)
KJc
(MPam0.5)
M01-01
M01-02
M01-07
M01-03
M01-04
M01-05
M01-06
M01-08
M01-09
M01-10
M01-11
M01-12
-130
-130
-130
-100
-100
-100
-100
-100
-100
-100
-100
-100
10
10
10
10
10
10
10
10
10
10
10
10
52.64
48.71
55.42
130.0
70.6
76.3
100.08
74.91
63.64
102.91
92.23
100.24
10.4
9.4
11.6
66.5
22.3
28.6
46.2
21.8
17.1
(50)
38.2
47.8
49.0
46.5
51.8
123.9
71.8
81.3
103.2
70.8
62.9
107.4
93.9
105.0
36
6.3
FINITE ELEMENT MODEL
A finite element model was developed for a single edge notch bend (SENB) specimen through
calibration. This model was essentially a two dimensional representation of a SENB specimen
for the in-plane conditions, but the out-of-plane effects were allowed for by the use of a mixture
of plane strain and stress elements. Plane strain elements were used at the plastically deformed
crack tip region and the contact areas at the loading and reaction points.
Elastic-plastic static analyses with the classical incremental plasticity theory were carried out for
quasi-static loading using the general purpose finite element code ABAQUS (ABAQUS 1998).
With respect to the intermediate loading conditions, effects of loading rate on material yielding
behaviour were allowed for using rate dependent plasticity constitutive laws, but material
inertial effects, which were considered to be insignificant, were ignored .
This model was first calibrated using test results of SENB specimens. The test results included
the following:
·
·
·
Quasi-static J-integral versus applied load (Figure 6.1)
Crack mouth opening displacement versus applied load, intermediate loading speed,
v=10mm/s (Figure 6.2)
Load point displacement versus applied load, intermediate loading speed, v=10mm/s
(Figure 6.3)
·
It can be seen from Figure 6.1 to Figure 6.3 that the finite element results are in good agreement
with test results.
180.0
160.0
140.0
Load, kN
120.0
100.0
TWI QS tests at -130°C
FE results
80.0
60.0
40.0
20.0
0.0
0.0
50.0
100.0
150.0
J-Int, N/mm
200.0
250.0
Figure 6.1 J-integral versus applied load under quasi-static loading
37
300.0
160.0
140.0
120.0
Load, kN
100.0
80.0
Test 1: M01-03
test 2: M01-04
Test 3: M01-05
Test 4: M01-06
FE results
60.0
40.0
20.0
0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
CMOD, mm
Figure 6.2 Crack mouth opening displacement (CMOD) versus applied load under
intermediate loading rate (v=10mm/s)
160.0
140.0
120.0
Load, kN
100.0
Test 1: M01-03
Test 2: M01-04
Test 3: M01-05
Test 4: M01-06
FE results
80.0
60.0
40.0
20.0
0.0
0.0
0.5
1.0
1.5
2.0
2.5
Load point displacement, mm
Figure 6.3 Load point displacement versus applied load under intermediate loading
rate (v=10mm/s)
38
6.4
PREDICTION OF FRACTURE TOUGHNESS AT INTERMEDIATE LOADING
SPEED
The twenty fracture toughness values of SENB specimens obtained from tests at –130°C under
quasi-static loading (Table 6.2) were used for the determination of the Beremin model
parameters, m and σu. Probability of fracture failure for each of the twenty specimens was
estimated by the following estimator:
i - 0.5
Pr (i ) =
N
(6.4)
where i is the rank of the specimen after ordering the calculated Weibull stresses and N is the
number of tests. Two set of m and σu values, m=22, σu=2465MPa; m=33, σu=2269MPa, were
generated for the Beremin model using the fracture toughness data. A comparison is given in
Figure 6.4 between the calculated values of failure probability using Eq.(6.1) and those from
test results. It seems that both sets of parameters (m and σu) are reasonably satisfactory.
Critical Weibull stresses for pf=0.05, 0.5 and 0.95 were calculated using Eq.6.1 for the
two sets of Weibull model parameters (Table.6.4)
Table 6.4 Critical Weibull stresses
Probability of failure
Pf=0.05
Pf=0.50
Pf=0.95
m=22, σu=2465MPa
2154
2424
2591
m=33, σu=2269MPa
2073
2244
2346
Results of the Weibull stress versus J-integral for the intermediate loading speed at different
temperatures were obtained from finite element analyses for m=22 (Figure 6.5) and m=33
(Figure 6.6), respectively.
1.2
m=22,
σu=2451MPa,
calculated
1
m=22,
σu=2451MPa,
Test data
pf
0.8
m=33,
σu=2258MPa,
calculated
0.6
m=33,
σu=2258MPa,
test data
0.4
0.2
0
1500
1700
1900
2100
2300
2500
2700
2900
Weibull stress, MPa
Figure 6.4 Calculated values and test results of failure probability
39
3300
3100
m=22
2900
Weibull stress, MPa
2700
2500
2300
Quasi-static, T=-130°C
v=10mm/s, T=-130°C
v=10mm/s, T=-100°C
v=10mm/s, T=-70°C
v=10mm/s, T=-50°C
2100
1900
1700
1500
0.0
20.0
40.0
60.0
80.0
100.0
120.0
140.0
160.0
180.0
200.0
KJc, MPam0.5
Figure 6.5 Weibull stress (m=22) versus KJc for intermediate loading rate (v=10mm/s)
3300
3100
m=33
Weibull stress, MPa
2900
2700
2500
2300
Quasi-static, T=-130°C
v=10mm/s, T=-130°C
v=10mm/s, T=-100°C
v=10mm/s, T=-70°C
v=10mm/s, T=-50°C
2100
1900
1700
1500
0.0
20.0
40.0
60.0
80.0
100.0
120.0
140.0
160.0
180.0
200.0
KJc, MPam0.5
Figure 6.6 Weibull stress (m=33) versus KJc for intermediate loading rate (v=10mm/s)
40
With the critical Weibull stresses in Table 6.4, fracture toughness values for the intermediate
loading speed v=10mm/s at different temperatures were obtained using Figure 6.5 and Figure
6.6 for m=22 and m=33, respectively. Results of fracture toughness as a function of temperature
are shown in Figure 6.7 and Figure 6.8.
250
SENB test data,
Quasi-static, T=130°C
SENB test data,
v=10mm/s, T=-130°C
200
KJc, MPa-m0.5
SENB test data,
v=10mm/s, T=-100°C
150
Prediction m=22,
pf=0.05
Prediction m=22,
pf=0.5
100
Prediction m=22,
pf=0.95
50
0
-140
-120
-100
-80
-60
-40
-20
0
Temperature, °C
Figure 6.7 Predicted dynamic fracture toughness values (m=22) as a function of
temperature as compared to test results
250
SENB test data,
Quasi-static, T=130°C
SENB test data,
v=10mm/s, T=-130°C
KJc, MPa-m0.5
200
SENB test data,
v=10mm/s, T=-100°C
150
Prediction m=33,
pf=0.05
100
Prediction m=33,
pf=0.5
Prediction m=33,
pf=0.95
50
0
-140
-120
-100
-80
-60
-40
-20
0
Temperature, °C
Figure 6.8 Predicted dynamic fracture toughness values (m=33) as a function of
temperature as compared to test results
41
It can be seen from Figure 6.7 and Figure 6.8 that the Beremin model underestimates fracture
toughness considerably as compared to the test data. The Beremin model with m=22,
σu=2465MPa predicts the fracture toughness better, but the predicted 50th percentile values are
virtually the lower bound of the test data.
6.5
DISCUSSION
It has been reported that experimental observations of brittle cleavage fracture are in some cases
inconsistent with the original Beremin model. One of the observations, which is also true for
this modern steel, is that fracture origins are found between the crack tip and the stress
maximum, where large plastic strains operate. Under such circumstances, Stockl et al (2000)
suggested that the original Beremin model should be modified to take into account the plasticity
effects on brittle cleavage fracture. It is out of the scope of the present work to determine the
best local approach model for fracture toughness prediction, but it may be suffice to point out
that the original Beremin model should be used with due care as to whether or not it is
applicable to a specific application.
42
7
TWI WORK ON LIMIT LOAD SOLUTIONS
7.1
GENERAL
Limit load solutions of structures and components are required for defect assessment using
BS7910 (British Standard 1999) and R6 (British Energy 2003). There are no dynamic limit
loads solutions currently available in such standards and procedures, although the principles of
calculating limit loads using finite element method should be applicable to dynamic loading.
Finite element analyses have been carried out to illustrate how limit load solutions can be
obtained using the finite element method for a SENB specimen and a wide plate under various
displacement rates up to 1m/s. Loading rate dependent material tensile properties of the modern
steel (450EMZ) have been used. Equations for estimating limit loads are suggested.
7.2
FINITE ELEMENT MODELS AND ANALYSES
Finite element models were created representing a deeply cracked (a/W=0.5) SENB CTOD
specimen of 50x50x200mm and a wide plate of 1x1x0.1m. Only one quarter of the respective
geometry of the SENB specimen and the wide plate was modelled due to a double symmetry,
with appropriate boundary condition. Finite element analyses were carried out under
displacement controlled loading. Elastic-perfectly plastic material behaviour was assumed in the
analyses. Strain rate dependency of material yield strength was described by Eq.6.2 in section
6.2. The general purpose finite element code ABAQUS (ABAQUS 1998) was used for all
analyses.
7.3
RESULTS
Results of load versus load point displacement are shown in Figure 7.1 and Figure 7.2 for the
SENB specimen and the wide plate, respectively. The dynamic limit load for the SENB
specimen can be described by the following equation:
PLDy = PLQS {0.0408Ln(v) + 1.0954}
(7.1)
The dynamic limit load for the wide plate can be described by the following equation:
PLDy = PLQS {0.0332 Ln(v) + 1.0695}
(7.2)
Under displacement controlled loading with remote loading speed v, it can be postulated that the
general equation of the dynamic limit load can be written as follows:
PLDy = PLQS {aLn(v) + b}
(7.3)
where a and b depend on dynamic yield strength, geometry and crack size.
43
200
Plane strain von Mises
solution
180
Quasi-static, T=-100°C
160
v=10mm/s, T=-100°C
140
v=100mm/s, T=-100°C
Load, kN
120
v=1000mm/s, T=-100°C
100
v=1mm/s, T=-100°C
80
60
40
20
0
0
0.5
1
1.5
2
2.5
3
3.5
Load point displacement, mm
Figure 7.1 Load point displacement versus applied load for a SENB specimen
80000
Net section solution
Quasi-static, T=-100°C
v=1mm/s, T=-100°C
v=10mm/s, T=-100°C
v=100mm/s, T=-100°C
v=1000mm/s, T=-100°C
70000
60000
Load, kN
50000
40000
30000
20000
10000
0
0
0.5
1
1.5
2
2.5
3
3.5
Load point displacement, mm
Figure 7.2 Load point displacement versus applied load for a wide plate
44
8
TWI WORK ON FAILURE ASSESSMENT DIAGRAMS
The failure assessment diagrams used in defect assessment to BS7910 (British Standard 1999)
and R6 (British Energy 2003) are derived for quasi-static loading conditions. The principles of
constructing failure assessment diagrams using the finite element method should be applicable
to dynamic loading. This part of the work is to derive failure assessment diagrams for the SENB
specimen and the wide plate mentioned earlier under displacement controlled remote loading
with loading speed up to 1m/s. Similar to construction of FADs for quasi-static loading, the two
axes of the failure assessment diagram are defined as follows:
Kr =
Je
Jp
Lr =
P
PL
Dy
(8.1)
(8.2)
For dynamic loading, the dynamic limit load is used to define the plastic collapse parameter Lr.
Failure assessment diagrams for the SENB specimen and the wide plate are shown in Figure 8.1
and Figure 8.2, respectively. It can be seen from Figure 8.1 and Figure 8.2 that when dynamic
plastic limit load is used, FADs for dynamic loading are very similar to those for quasi-static
loading.
45
1.2
Material specific FAD
BS7910 Level 2A
1
Quasi-static, T=-100°C
v=10mm/s, T=-100°C
0.8
v=100mm/s, T=-100°C
Kr
v=1000mm/s, T=-100°C
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Lr
Figure 8.1 Failure assessment diagrams for SENB specimen under dynamic loading
1.2
Material specific FAD
1
BS7910 Level 2A
v=10mm/s, T=-100°C
0.8
Kr
v=100mm/s, T=-100°C
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Lr
Figure 8.2 Failure assessment diagrams for wide plate under dynamic loading
46
9
UMIST WORK ON CRACK ARREST
9.1
OBJECTIVES
It should be noted, that despite the extra security provided by crack arrest arguments,
application of the crack arrest approach remains limited. One of the reasons of such limited
usage of the crack arrest approach is that predictive tools are much more advanced and
established for the fracture initiation methodology. Crack arrest is a highly complex
phenomenon and the analysis of crack arrest behaviour requires a detailed dynamic numerical
analysis. Increasing capabilities of modern computers and advancing modelling tools make
such analyses possible.
A discussion of assessment of crack arrest based on fracture mechanics principles was given by
Burdekin (1999) and an extensive review of crack arrest testing and methodology by Wiesner
and Hayes (1995). It was suggested (Burdekin 1999) that for arrest of cracks from local brittle
zones at welds, a simple energy balance approach could be used to estimate the toughness Gmat2
required to arrest cracks as follows:
G
mat 2
=
1
+
G
S
ìïæ a1
íçç1 ïî
è a 2
ö
é a1
ùü
÷÷
ê1 +
(1 -
2a
)ú ïý
ø
ë a 2
û ïþ
(9.1)
where fracture initiation occurs at crack length a1 and toughness Gmat1 with propagation
continuing at toughness aGmat1 until crack length a2 when the crack runs into material with
increased toughness Gmat2 and GS is the equivalent static driving force for the stress level and
crack length a2. This analysis was for elastic material under fixed load conditions. The
equation suggests that in the worst case of very low toughness in the brittle zone, (a®0), and a
short initial crack (a1®0), the toughness to arrest the crack is a factor of 2 times the static value
which would be calculated for the stress and crack length concerned.
In the current project computational modelling of the crack initiation, growth and crack arrest
behaviour under static and dynamic loading conditions has been carried out and influence of
parameters affecting crack growth and structural crack arrest behaviour has been analysed.
Further details of the work are given by Tkach and Burdekin (2002).
9.2
COMPUTATIONAL METHODOLOGY AND FINITE ELEMENT MODELS
In the present work a method for modelling crack growth and arrest is based on introducing of
cohesive surfaces into continuum. Within the cohesive surface framework, the continuum is
characterised by two constitutive relations: one describes non-linear behaviour of the bulk
material, the other relates the traction and displacement jump across the cohesive surface. The
traction-separation law is considered as a continuum model of the processes occurring in the
fracture process zone (cohesive zone) ahead of the crack tip.
The traction-separation law put forward by Xu and Needleman (1994) was used. According to
the adopted traction-separation law, interfacial properties are controlled by two parameters: the
cohesive strength s peak and the debonding energy F. It should be noted, that in the current
work further development of Xu-Needleman’s potential has been done to accommodate damage
accumulation in the fracture process zone and predict fracture behaviour in the cases when
unloading occurs either due to crack growth. The cohesive zone model has been implemented
into computational analysis through a user defined element in the finite element software
ABAQUS.
47
Two cracked geometries were considered in the current project. Firstly, crack initiation, growth
and arrest were simulated in a compact tension (CT) specimen with relative crack length
a/w=0.5 (W =50 mm ). Secondly, an attempt was made to simulate crack arrest behaviour in the
wide plate (42.2 mm thick x 850 mm long x 182 mm wide) crack arrest test carried out by TWI.
In both cases, bulk material was modelled by conventional two-dimensional elements. Rate­
sensitive plasticity theory was employed to describe the non-linear response of bulk material in
the CT specimen. Purely elastic behaviour of the bulk material was considered in the
simulation of the wide plate crack arrest test. The cohesive surface was modelled by interfacial
elements with the assumed traction-separation law. All the constitutive computations were
performed within the finite-strain framework.
The finite element models employed are shown in Figure 9.1 (a, b). In both simulations the
crack is constrained to grow along the initial crack line, where interfacial elements were placed.
9.3
ANALYSIS OF CRACK GROWTH AND ARREST
In this work the situation which can be relevant to the phenomenon of so-called local brittle
zones (LBZs) in weldments was considered. Namely, the case was modelled in which the crack
starts in a local zone with low fracture toughness and then enters the volume of material with
higher fracture toughness. To simulate this pattern of behaviour, two sets of interfacial
elements were introduced into the model. The first set of elements in the vicinity of the crack
tip was prescribed with a weaker interfacial strength (“weak interface”) than elements away
from the crack tip (“strong interface”).
The results of the modelling of the crack growth and crack arrest behaviour in the CT specimen
under dynamic loading are given in Figure 9.2(a) and Figure 9.2(b). It can be seen, that the
higher the debonding energy (F2) of the strong interface, the later the crack re-initiates in the
strong interface and the smaller the amount of subsequent crack growth which occurs. At a
certain level of debonding energy the crack reaches the strong interface and stops there with no
further re-initiation and growth. Lower debonding energy of the weak interface (F1) causes
earlier crack initiation and growth (see Figure 9.2(b)). It should be noted, that the amount of
crack extension in the strong interface is affected by the properties of weak interface. The
earlier the crack reaches the strong interface the less the amount of subsequent crack growth
which occurs in the strong interface. This is due to the fact that the amount of strain energy
accumulated by the moment of crack initiation increases with increasing crack initiation time.
Consequently, the higher amount of surplus energy which has to be absorbed in the strong
interface.
It can be seen from Figure 9.3, that there is the dynamic oscillation of the applied CTOD
following the crack arrest. The finite analysis results indicate that the peak value of the applied
CTOD during the oscillation is about twice the static value which supports completely the
energy based analysis of equation 9.1. This dynamic oscillation around the equilibrium CTOD
may lead to further crack re-initiation in the strong interface. Therefore, the value of the driving
force immediately following arrest is a very important parameter, which determines whether a
sustained crack arrest will be achieved.
In the simulation of the TWI test initial (“primary”) constant load was applied to the plate.
Then, “secondary” load was applied to the crack starter tabs. Figure 9.4 represents the results of
the computational modelling. It can be seen from Figure 9.4 that although the crack arrested
when it first entered the higher toughness material, the crack then re-initiated after a short delay.
48
Sustained crack arrest in the TWI test piece was not achieved at debonding energy of the
“strong” interface up to 500kN/m for constant displacement conditions applied at the end of the
specimen. It was considered that this result was probably affected significantly by the inertia
effects of material follow up displacement and by the effect of the length of the specimen on
stored strain energy. Two further sets of analyses were then carried out. Firstly the material
density was reduced by a factor of two for the same geometry and loading conditions. Secondly
the length of the specimen was reduced by a factor of two for the original material density
corresponding to steel. In both cases the revised analyses were able to demonstrate arrest of the
fracture despite the same oscillation and enhancement of the applied CTOD. The reduced
specimen length results are included in figures 9.3 and 9.4.
Applied load
Primary load
Secondary load
Applied load
Secondary load
Primary load
(b)
(a)
Figure 9.1 Finite element models employed: (a) CT specimen and (b) the TWI test
piece
49
-3
-3
0.0
0.0
1.0x10
-3
2.0x10
-3
3.0x10
-3
4.0x10
-3
5.0x10
(a)
-6
5.0x10
F1 = 100 kN/m
speak = 890 MPa
Length of weak
interface = 0.004 m
-5
1.0x10
-5
1.5x10
F2 = 150 kN/m
F2 = 155 kN/m
F2 = 160 kN/m
F2 = 175 kN/m
speak = 890 MPa
Fixed LLD = 0.0011 m
Displacement control
-3
-3
-3
-3
-3
-3
(b)
-6
-6
5.00x10
-5
1.00x10
Time, sec
-6
-5
1.25x10
-5
1.50x10
F2 = 150000 N/m
speak = 890 MPa
Fixed LLD = 0.0011 m
Displacement control
7.50x10
speak = 890 MPa
F1 = 100 kN/m
F1 = 95 kN/m
F1 = 90 kN/m
F1 = 80 kN/m
F1 = 75 kN/m
F1 = 110 kN/m
Length of weak
interface = 0.004 m
2.50x10
0.0
1.0x10
2.0x10
3.0x10
4.0x10
5.0x10
6.0x10
50
Figure 9.2 Influence of debonding energy in strong (a) and weak (b) interfaces on the crack growth and arrest in CT specimen
Crack growth, m
6.0x10
Crack growth, m
-3
5.0x10
-3
4.0x10
-3
CTOD, m
-4
2.0x10
Fixed LLD = 0.0011 m
-4
1.0x10
0.0
0.0
CTOD, m
Crack increment, m
-5
2.0x10
-5
3.0x10
-3
F1 = 100000 N/m
speak = 890 MPa
2.0x10
-3
F2 = 150000 N/m
speak = 890 MPa
1.0x10
-3
Length of weak
interface = 0.004 m
0.0
-5
4.0x10
6.0x10
8.0x10
Crack growth, m
-4
3.0x10
6.0x10
-5
Time, sec
Figure 9.3 Crack growth and arrest in CT TWI test piece condition and “ring-up” effect.
100
Crack growth, mm
75
50
speak = 450 MPa
F 1 = 50 kN/m
25
0
0.0
F2 = 150 kN/m
F2 = 250 kN/m
F2 = 500 kN/m
Fixed secondary load
4.0x10
-4
8.0x10
-4
1.2x10
-3
1.6x10
-3
Time, Sec
Figure 9.4 Crack growth and arrest in the TWI test piece.
51
9.4
CONCLUSIONS OF CRACK ARREST
1. The ability of the developed methodology to predict crack growth and crack arrest in a
real test piece has been demonstrated.
2. Computational simulation based on the cohesive zone model has been carried out to
predict crack initiation, growth and crack arrest behaviour under static and dynamic
loading conditions. The influence of the interfacial properties on the crack growth and
subsequent crack arrest has been investigated.
3. It has been established, that the peak value of the driving force milliseconds after a
momentary crack arrest and the resisting fracture toughness are the most important
parameters affecting achievement of a sustained crack arrest.
52
10
10.1
GUIDANCE ON
ASSESSMENT
DYNAMIC
STRUCTURAL
INTEGRITY
GENERAL ISSUES
From the work carried out in this project it is clear that general methods of assessment of the
significance of defects such as those given in BS7910 can be applied to assessments under
dynamic loading subject to two main differences.
Firstly, the magnitude of the crack tip driving force at defects within structures subject to
dynamic loading is likely to be increased compared to that under static loading. This increase is
due to the effects of dynamic loading in increasing the stresses within the structure. For the
case of a stationary crack, the increase in stress intensity factor due to dynamic loading will be
directly proportional to the increase in stresses in members adjacent to a crack under
consideration for the dynamic case compared to a static case. This may be thought of as
increasing the crack tip driving force by a dynamic amplification factor (DAF).
The work carried out in this project has shown that the dynamic amplification factor depends
upon the response of the structure or component compared to the dynamic nature of the loading.
In general dynamic loading will increase the stresses and displacements within a structure, the
dynamic amplification factor depending upon the relationship between the natural period or
frequency of the structure and the rate or frequency of applied loading. The highest values of
the DAF will occur when the applied frequency is close to a natural frequency of the structure,
i.e., close to a resonance condition. It should be noted that structures will generally be able to
vibrate in a number of different modes, each having its own frequency. It is therefore necessary
to check for interactions between applied loading frequencies and the natural frequencies of the
structure for several modes, perhaps up to the lowest five natural frequencies. There is also the
possibility that the natural frequency of individual members may be close to the applied loading
frequency, although the individual members will usually have significantly higher natural
frequencies than the overall structure. The magnitude of the dynamic amplification factor will
also be affected by the degree of damping in the structure and this must be taken into account.
General expressions for the magnitude of a dynamic amplification factor are given by the
following equations:(10.1)
DAF = -15.625(w/wn)2 + 31.25(w/wn) - 12.125
or
(
)
4
2
DAF = 1/ é1+ ( w / w n ) - ( w / w n ) 2 -1/ Q 2 ù
ë
û
(10.2)
where equation 10.1 applies to an offshore structure immersed in water and equation 10.2 is a
more general expression for structures in general. The differences between these equations arise
from different damping conditions which have their greatest effect in the range of applied to
natural frequency of 0.9 to 1.1. Most of the Structural Codes which give any guidance on
dynamic loading effects advise ensuring that the natural frequency of the structure is well
removed from any applied frequency content. For example BS5400 for bridges deals with this
for foot and cycle bridges by recommending that the natural frequency for the unloaded bridge
should be not less than 5Hz. Underlying this is the assumption that the applied frequency for
human beings walking is likely to be about 1 Hz so that the ratio of applied to natural frequency
should be less than 0.2. Some guidance is given in BS 5400 Part 2 for vertical accelerations to
be designed for if the natural frequency of the bridge is less than 5 Hz with the concept of
dynamic response factors dependent on the bridge span and type of construction. In the worst
53
cases dynamic response factors of up to 15 are predicted and again it is clearly necessary to
ensure that the ratio of applied to natural frequency is kept well away from unity.
In the case of earthquake loading, the dynamic excitation is essentially a combination of
frequencies although the greatest energy content is likely to lie at frequencies less than 5 Hz.
Again dynamic excitation factors can be estimated from equations 10.1 and 10.2 for different
combinations of earthquake and structural response frequencies.
10.2
SINGLE LOAD APPLICATIONS – IMPACT OR BLAST CONDITIONS
The magnitude of the stresses in a structure under a single load application can usually be
determined by energy considerations. The potential or kinetic energy of the impact or blast
condition has to be absorbed within the structure either in the form of elastic strain energy, or by
plastic deformation. In some codes, where relatively low rates of loading have to be considered,
this is dealt with by applying an impact factor to the static load, such factors typically lying
between one and two. For impact or blast loading it is more common to carry out a finite
element analysis and specialist codes are available to do this, such as Dyna 3D or ABAQUS.
These analyses do not need to attempt to model the presence of defects but can be relatively
coarse mesh analyses to determine maximum stress levels under the rate and magnitude of
loading relevant for the structure or component concerned.
In general, it will be found that vibration of the structure commences from the moment the load
is first applied and continues after the load reaches its maximum value or is removed. This
leads to an over shoot in the maximum stresses experienced by the structure compared to those
caused by the same load applied statically, the magnitude of the over shoot depending upon the
ratio of the time to reach maximum load compared to the natural period of the structure.
The material properties may also be affected by the rate of loading under dynamic loading
conditions. The yield strength will be increased by rate of loading in rate sensitive materials and
the fracture toughness will be reduced. The magnitude of these changes depends upon the type
of material concerned but can generally be represented by equations of the following kind:
ì 1 ln( A / e& static ) 1
ü
s
ys (T
,e&
)
=
s 0 +
S
í
-
ý
293 þ
î
T ln( A / e& )
(10.3)
Where σ0 is yield strength at room temperature under quasi-static loading. e&static is a quasi-static
loading rate, typically ~5x10-5 1/s. A is a material constant with typical value of 108. S is a
parameter to be fitted using test data, but if test data are not available, a value of 60,000 ± 10000
MPaK may be used. This equation is applicable for temperatures lower than the ambient and for
strain rate up to about 10001/s.
10.3
REPEATED SINUSOIDAL LOADING
Under repeated sinusoidal loading the dynamic amplification factor depends upon the ratio of
the applied frequency to the natural frequency of the structure or component, and the degree of
damping present. It is therefore necessary to have an estimate of the natural frequency of the
structure for its lowest modes and to determine the ratio of the applied frequency to these
natural frequencies. The dynamic amplification factor can then be estimated from equation 1
above. For a loading history which is more complex than a simple sinusoidal loading, the
effects can be analysed by determining the primary frequency content from Fourier transforms.
54
The dominant frequency band for the applied loading can then be compared to the natural
frequencies of the structure to estimate the dynamic amplification factor.
The effects of the rate of loading on the material properties under repeated loading conditions
can be estimated by assuming that the strain rate is given by the following expression:Alternatively, the effects of rate of loading can be estimated by dK/dt analyses.
10.4
CRACK ARREST ANALYSIS
Successful modelling of crack propagation and arrest behaviour using finite element analysis of
a cohesive zone model has confirmed that it is essential to allow for the “ring up” or
enhancement of the crack tip driving force for dynamic conditions at arrest compared to an
equivalent static condition. Thus to carry out a crack arrest assessment for a burst of brittle
fracture such as through a local brittle zone, the static crack tip driving force for a crack of
length entering the tougher material under applied and residual stresses should be multiplied up
by a “ring up”/enhancement factor. Expressed in terms of stress intensity factor, K, a factor of
1.5 is appropriate, but in terms of CTOD or J values a factor of 2 is appropriate.
This enhanced crack tip driving force should be compared with a reduced fracture toughness
allowing for the effects of strain rate for the material concerned or using a crack arrest fracture
toughness.
55
11
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