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HSEHealth & Safety
Executive
Pulse pressure testing of 1/4 scaleblast wall panels with connections
Phase II
Prepared by the University of Liverpoolfor the Health and Safety Executive 2006
RESEARCH REPORT 404
HSEHealth & Safety
Executive
Pulse pressure testing of 1/4 scaleblast wall panels with connections
Phase II
G K Schleyer & G S Langdon University of Liverpool
Department of Engineering Brownlow Hill
Liverpool L69 3GH
A detailed experimental and numerical study of 1⁄4 scale stainless steel blast wall panels and their connections under pulse pressure loading has been conducted at the University of Liverpool. The panel design was based on the deep trough trapezoidal profile with welded angle connections top and bottom and free sides. Three types of panel/connection system were studied, namely a short, medium and long welded angle connection to compare the influence of the angle length. Large permanent plastic deformations were produced in the panel/connection system without rupture. Phase I established that modelling the support correctly is fundamental to the response of the blast wall and can significantly affect its ultimate capacity.
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.
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First published 2006
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ii
CONTENTS
NOTATION..................................................................................................................... V
1 INTRODUCTION ......................................................................................................... 1
2 BLAST WALL DESIGN............................................................................................... 22.1 CORRUGATED PANELS AND CONNECTIONS USED OFFSHORE ..................... 22.2 TEST PANEL DESIGN............................................................................................. 3
3 EXPERIMENTAL DATA.............................................................................................. 73.1 STATIC PRESSURE TEST DATA ........................................................................... 73.2 SHOCK PRESSURE TEST DATA ........................................................................... 83.3 HIGH RATE CONNECTION TEST DATA ................................................................ 9
4 NUMERICAL TEST DATA........................................................................................ 114.1 STATIC PRESSURE TEST SIMULATIONS........................................................... 11
4.1.1 Simulation Results ........................................................................................... 114.2 SCALING TEST DATA ........................................................................................... 13
4.2.1 Scaling Simulation Results............................................................................... 134.2.2 Scaling Effects ................................................................................................. 21
4.3 RESISTANCE CURVES......................................................................................... 22
5 SIMPLIFIED METHODS OF ANALYSIS .................................................................. 295.1 PREDICTING RESPONSE USING ISO-DAMAGE CURVES ................................ 29
5.1.1 Using resistance curves in a single-degree-of-freedom (SDOF) analysis ....... 295.1.2 Generation and use of iso-damage curves ...................................................... 30
5.2 ANALYTICAL BEAM MODELS .............................................................................. 355.2.1 Dynamic, elastic-plastic mode approximation .................................................. 355.2.2 Quasi-static, elastic-plastic model.................................................................... 38
REFERENCES ............................................................................................................. 46
APPENDIX A – WORKED EXAMPLE ......................................................................... 47
APPENDIX B – SPRING CONSTANTS....................................................................... 57
iii
iv
NOTATION
A cross-sectional area of corrugated section
Aflex cross-sectional area of angle connection
Į angle of rotation of connection
E� � scale factor
Ci generalised displacement
D Cowper-Symonds constant
ǻ axial stretching of beam x
H� strain rate
Gx extension of horizontal spring
Gy extension of vertical spring
E Young’s modulus
Fx, Fy spring forces
I second moment of area
i mode shape
Kx stiffness of horizontal spring
Ky stiffness of vertical spring
.T� � strain hardening spring constant
K elastic translation spring stiffness in x direction xe
K plastic translation spring stiffness in x directionxp
K elastic rotation spring stiffness phie
K plastic rotation spring stiffnessphip
M
L
L half span of beam
flex length of flexible angle connection
c bending moment
v
Mo fully plastic moment capacity of corrugated section
phiM Fully plastic moment capacity of rotation spring
PE potential energy loss
p(x) uniformly distributed pressure loading
q Cowper-Symonds constant
T angle of rotation of beam
Rmx, Rmx maximum resistance of linear springs
o 6c dynamic flow stress in the full-scale model
o ıc dynamic flow stress in the ¼ scale model
teq thickness of equivalent rectangular section
U strain energy
Ub flexural strain energy
UT� strain energy of angular spring
Um membrane strain energy
u difference between the projected length of beam and true length of beam
V total potential energy
w(x) transverse displacement
x distance along axis of beam from origin (origin at mid-span)
vi
1 INTRODUCTION
A study was carried out at the University of Liverpool on the response of a ¼ scale stainless steel blast
wall panel under pulse pressure loading (1999-2002). The project was funded by EPSRC, Mobil
North Sea Ltd and HSE (Offshore Division) with technical support from Mech-Tool Engineering Ltd
who designed and manufactured the test specimens. The aim was to investigate the influence of the
connection detail on the overall performance of the panel/connection system under pulse pressure
loading and to develop appropriate analytical and numerical models of the blast wall/connection
system for correlation with the test results. This was the first time that a detailed experimental study
of the behaviour of blast walls made from profiled stainless steel sheet had considered the modes of
failure and end effects of the support construction. The experimental and modelling work at Liverpool
has enabled a more appropriate assessment of the ultimate capacity of the blast wall panel based on the
influence of the connection detail [1].
The HSE funded a second phase (2003-2005), to extend the work through further experiments,
numerical simulations and simplified modelling to provide more comprehensive response data in the
quasi-static, dynamic and impulsive loading range [2]. One of the objectives of this project was to
provide design guidance on the influence of connection behaviour on the overall response of the blast
wall system. This has largely been achieved through the experimental data and numerical simulations
that engineers can use at an early stage for assessing the adequacy of numerical and analytical models
for use in design and assessment of profiled blast walls. Furthermore, most of the tests were
performed at loads in excess of the design conditions and therefore represent an upper bound extreme
load case. In none of the cases where buckling occurred (high strain regions) was there signs of
tearing in the profile or at the connection even though some high strains were recorded. FE studies [3]
have shown that strains can exceed 15%, which under current design guidance would imply tearing.
The methodology developed in this work may be used to predict large plastic deformations in blast
walls under explosion loading. Adequacy of the connection may need to be checked against standards
to ensure integrity. Three different angle connections were studied, which are characterised by a
flexible leg length. Results for both the design direction where the leg is placed in tension and the
opposite direction placing the leg in compression are given. The static resistance curves given in this
note show that as much as a 25% (approx.) increase in capacity before buckling occurs can be
achieved by using a shorter leg length.
The methods are based on an idealised SDOF structural system. The main advantages of these
methods as proposed are that they take account of connection behaviour. Some of the methods rely on
additional input and may be reserved for a few critical cases, while the approach that is recommended
as a simplified design or assessment tool can be used independently of other analyses. It is
particularly suited to large plastic deformation in the quasi-static loading domain. It can be used to
demonstrate that a blast wall system has adequate capacity in the large plastic deformation range up to
a ductility value of 30 or more provided there is adequate restraint at the supports. Normally, this type
of design calculation would be beyond the scope of a standard Biggs type calculation.
A simple welded angle connection was adopted in this study as it represented the most common type
of connection. The results and conclusions of this study have recognised the importance of the
flexibility of the angle connection. For very large deformations, the angle connection could transmit
high reaction forces into the primary structure and may experience comparatively large rotations.
These need to be assessed. The methodology proposed can provide estimates of the reaction forces
and rotations at large deformations.
This Technical Note contains experimental and numerical data from phase 2 of the project that can be
used at an early stage towards the design or assessment of profiled blast walls. This data was used to
validate analytical methods for predicting large deformations of ¼ scale blast wall panels under pulse
1
pressure loading well into the plastic range and in excess of the original design conditions. A
proposed simplified approach to assessing the influence of the blast wall connection is given in detail
together with a worked example to demonstrate the use of the method in practice to assess a full-scale
blast wall. This is also compared with the standard Biggs method. The purpose of this note is to
provide a guide to using the new method and to highlight the benefits of including the connection
behaviour in the design of the blast wall system.
2 BLAST WALL DESIGN
2.1 CORRUGATED PANELS AND CONNECTIONS USED OFFSHORE
Typical blast walls consist of stainless steel walls, about 12 m wide and 4 m high with corrugations
running top to bottom; a schematic drawing is shown in Fig. 2-1. Blast walls are connected top and
bottom to the primary steelwork by angles and are considered to behave as one-way spanning
plates/beams for analysis purposes. The connections usually consist of two angles welded together
and to the structural steelwork of the platform. A schematic of the connection detail (¼ scale) is
shown in Fig. 2-2. This appears to be the most common form of blast wall system, although others
exist.
Blast walls are ‘rated’ to a certain pressure, that is they are designed by the manufacturer to resist a
certain dynamic pressure (usually in the range of 1 to 4 bar) with some permanent deformation. There
is no agreed, common basis for determining this rating or an acceptable deformation. In general,
industry standard guidance notes [4] as issued by FABIG and the Steel Construction Institute (SCI) are
used to design blast walls for fire and blast loading. Technical Note 5 [5] provides further detailed
guidance on the design and construction of stainless steel blast walls to resist explosion loading. This
guidance is currently being updated.
Figure 2-1: Typical blast wall construction made from corrugatedstainless steel sheet with end connections top and bottom
2
2.2 TEST PANEL DESIGN
The test panels for this project were designed and manufactured by Mech-Tool Engineering Ltd during
the first phase. The same batch of 316 stainless steel materials was used to make all the blast wall
panels, the connection test specimens and the material test specimens. The blast wall design was
based on a non-symmetric trapezoidal deep trough profile, angle connections top and bottom and free
sides. The profile dimensions are shown in Fig. 2-3 and given in Table 2-1. The dimensions of the
end connections are shown in Fig. 4 and given in Table 2-1. The mechanical properties of the blast
wall material (at 2, 3 and 4 mm thick) are summarised in Table 2-2.
The blast wall was rated using the time-domain equivalent SDOF method and assumed to behave as a
one-way span beam with cross-sectional properties of the deep trough profile. The properties are
given in Table 2-3. End fixity was assumed to be pinned at both ends. The resistance of the beam to
dynamic loading was due to inertia, elastic bending stiffness and plastic hinge formation at the mid-
span of the beam. A strain-rate enhancement factor of 1.175 was used to specify the dynamic yield
stress of the material. A pulse load of 950 mbar overpressure and load duration of 50 msec as shown
in Fig. 2-5 was applied to the SDOF model and produced a maximum permanent displacement of 14
mm, considered acceptable. The blast panel was therefore ‘rated’ at 950 mbar for triangular pulse load
duration of 50 msec.
Figure 2-2: Cross-section of the ¼ scale blast wall connection detail (side elevation view)
3
220 220 220 220
85 13535
2.0 thk 40.5
15
PressurePressure 880
Figure 2-3: Deep trough profile dimensions (all bend radii 10 mm) of test panel
4
25 10
3 thk
4 thk
2
3
3
65
35 20
85, 145, 205
195, 255, 315
100 x 75 x 12 RSA
Figure 2-4: Connection details of test panel (side elevation view)
950
Pressure (mbar)
Time (msec)
25 50
Figure 2-5: ‘Design’ pulse pressure load
Table 2-1: Description of non-symmetric trapezoidal deep trough profile of test panel
Profile width 220.0 mm
Profile depth from top surface to inside bottom surface 40.5 mm
Corner internal bend radius 10.0 mm
Base thickness of profile 2.0 mm
Cross-section area of profile 512.0 mm 2
Profile material density 7970.0 kg/m3
Profile weight per unit plan area 19.3 kg/m2
5
Table 2-2: Deep trough test panel properties
Mass of beam section / unit length 4.2 kg/m
Second moment of area 13.5 cm 4
Young’s modulus 200.0 MN/mm2
Moment of resistance 2.3 kNm
Base yield stress 265.0 N/mm 2
Design dynamic yield stress 311.4 N/mm 2
Elastic stiffness 2.1 kN/mm
Elastic-plastic stiffness 2.1 kN/mm
Plastic resistance 18.6 kN
Displacement to first yield 6.61 mm
Displacement to full plasticity 8.81 mm
Plastic displacement 14.07 mm
Maximum displacement (occurs after 32 msec) 22.88 mm
Ductility ratio 2.6
Natural elastic period 7.5 msec
Table 2-3: Mechanical Properties of 316L Stainless Steel (2, 3 and 4 mm thick)
Static Yield (0.2% proof) 276-312
Variation in dynamic Yield (compared to static Yield) +18 to -7
Cowper-Symonds constant D (at Yield) 429-2720
Cowper-Symonds constant q (at Yield) 4.08-5.78
Static UTS 615-645
Variation in dynamic UTS (compared to static UTS) +17.76 to -7.2
Further details are available in ref. [6]
2N/mm
%
2N/mm
%
6
3 EXPERIMENTAL DATA
3.1 STATIC PRESSURE TEST DATA Static pressure tests were carried out on the three types of blast wall panels as shown in Fig. 3-1. A
summary of the test results is given in Table 3-1.
195
915910
255
915910 AB
315
915910
Figure 3-1: Panel types (from left to right) short, intermediate and long angle connection (side elevation view)
7
Table 3-1: Summary of static pressure loading tests on ¼ scale blast wall panels
Connect Test #
Type
C60_1 Short
C60_2 Short
C120_1 Medium
C120_2 Medium
C180_1 Long
C180_2 Long
Peak Pressure Peak Transient
Max. Perm.
(bar) Centre Disp
(mm) Disp (mm)
Comments
0.38 3.5 0 Elastic test
1.52 63 56 Buckling of
corrugations
0.37 4.0 0 Elastic test
1.17 71.4 70 Buckling of
corrugations
0.37 4.0 0 Elastic test
1.14 80.4 84 Buckling of
corrugations
3.2 SHOCK PRESSURE TEST DATA
Shock pressure tests were performed on three blast wall panels, Fig. 3-2, each with a different
‘flexible’ angle length as shown in Fig. 3-1. A summary of the test data is given in Table 3-2.
Table 3-2: Summary of shocked pressure loading tests on ¼ scale blast wall panels
Test # Connect
Type
Mean Average Permanent Central
Displacement
(mm) Peak Pressure
(bar)
Positive Load Duration
(msec)
Impulse
(bar-msec)
1 Short (1) 0.52 83.5 18.6 1.6
2 Short (1) 0.27 70.2 9.0 0
3 Short (1) 1.23 259 102.7 22.2
4 Medium (2) 0.37 76 10.9 1.6
5 Medium (2) 1.58 136 66.6 145.4
6 Long (3) 0.57 61 12.4 1.6
7 Long (3) 2.57 88.7 40.1 185.4
8
Figure 3-2: Test panels after high-pressure testing from right to left #1-3
Figure 3-3: Typical pressure-time histories for shock tests (test #5, panel #2, medium size connection)
3.3 HIGH RATE CONNECTION TEST DATA
High rate tension/bending tests were performed on connection test specimens manufactured from the
same material used for the panels and to the same specification. The geometry of the connections was
of the same type as the panels with a short, medium and long angle connection detail.
The tests were performed at speeds (loading rates) ranging from 1 mm /sec to 300 mm/sec, which -1corresponded to initial strain rates in the range 2.1 x 10-4 to 0.19 sec .
Tests were performed in two directions (shown in Fig. 3-4):
(1) Direction 1 corresponded to a tensile mode of deformation, causing the angles to open out
and the connections to straighten.
(2) Direction 2 induced bending about the rigid angle weld-line, simulating the connection
response to the axial forces that would be induced during span shortening of a blast wall
under a pressure load.
9
The aim of these tests was to determine the parameters used to define the boundary conditions of the
panels for use in the analytical model of the blast wall/connection system, in particular the influence of
test speed on connection stiffness and moment capacity.
Pressure load
2
1
Deep trough profile section
Fig. 3-4: Schematic of connection specimen, showing load directions
Tables 3-3 and 3-4 contain summaries of the test results in directions 1 and 2 respectively.
Table 3-3: Direction 1 Test Results Summary
Connection
Type
Nominal Cross-
head Speed
(mm/sec)
Peak bending
strain
(%)
Peak axial
strain
(%)
Peak
F
(kN)
Peak
disp.
(mm)
Max
angle
(deg.)
Short
1
10
50
100
300
0.54
0.63
0.65
0.63
0.64
0.12
0.06
0.13
0.13
0.16
42.1
33.8
40.6
35.8
48.7
29.1
30.2
30.5
30.8
32.8
48
40
46
41
47
Medium
1
5
10
50
100
300
0.14
0.15
0.16
0.17
0.23
0.23
0.21
0.08
0.07
0.07
0.07
0.07
42.5
10.1
40.1
37.9
38.0
44.0
29.7
5.4
30.2
30.4
30.6
32.7
48
11
48
44
50
55
Long
1
10
50
100
300
0.09
0.09
0.12
0.13
0.10
0.08
0.08
0.08
0.07
0.08
47.5
50.2
47.1
47.1
55.9
30.4
30.3
31.4
32.8
34.0
55
49
60
56
53
10
Table 3-4: Direction 2 Tests Results Summary
Connection
Type
Nominal Cross-
head Speed
(mm/sec)
Peak bending
strain
(%)
Peak axial
strain
(%)
Peak F
(kN)
Peak
disp.
(mm)
Max
angle
(deg.)
Short
5
10
50
100
300
-
0.53
0.49
-
0.54
-
-0.04
-0.01
-
0.005
23.6
14.3
14.2
14.3
16.6
71.5
59.4
60.6
60.9
62.4
-
44
48
41
48
Medium
5
10
50
100
300
0.25
0.26
0.27
0.26
0.29
0.005
0.004
0.01
-0.02
0.01
2.13
2.15
2.38
2.48
2.36
90.5
91.3
90.7
90.3
91.4
18
35
30
35
36
Long
5
10
50
100
300
0.22
0.22
0.20
0.21
0.20
0.01
0.01
0.01
0.003
0.01
0.86
0.80
0.83
0.75
0.81
88.0
87.4
87.0
89.9
92.1
21
19
21
-
-
4 NUMERICAL TEST DATA
This section presents numerical simulations from models employed to predict the failure of short,
medium and long angle ¼ scale blast wall panels, subjected to static pressure loads. The models
incorporated the end connection detail as part of the structure analysed. The commercially available
finite element (FE) software, ABAQUS/Standard version 6.2 [10], was used to perform the
simulations. The FE models are similar to those used to model the pulse loaded panel results in phase
I of the project [1]. Full scale models of the static and pulse loaded panels were constructed to study
scaling effects and the results of these are described in section 4.2. Section 4.3 contains the resistance
curves generated from the FE output (in the form of force per corrugation versus central displacement
plots). These curves were used to define the resistance functions used in the simplified modelling.
4.1 STATIC PRESSURE TEST SIMULATIONS
Fig. 4-1 is an illustration of the short angle connection blast wall geometry, fully meshed and
restrained, used by ABAQUS/Standard to simulate its response to static pressure loading. One half of
a corrugation is modelled, due to symmetry of the blast wall panel profile, with connections at each
end. The profiled panel is modelled using S4R shell elements and the connections are modelled using
C3D8R solid elements. Static stress analyses were performed, and due to the anticipated unstable
nature of the buckling process, a small damping factor (equivalent to 2 x 10-4 of the change in strain
energy during the step) was applied to the model using the ‘stabilize’ parameter [10]. The connections
were fully clamped along the bottom face (as indicated in Fig. 4-1) and the two edges of the strip were
restrained from translating in the ‘z-direction’ (as shown in Fig. 4-1). The static pressure loads were
applied as uniformly distributed loads to the corrugated panel.
4.1.1 Simulation Results
Table 4-1 contains predictions of displacement and surface strain at the centre of the 85 mm wide
flange, and compares the values to the experimental measurements. Fig. 4-2 shows an example of a
permanently deformed profile predicted by ABAQUS.
11
1
Table 4-1: FE predictions versus experimental measurements
Peak pPanel # Source
(bar)
FE 3.24 0 0.051 0 Short 0.38
Exp 3.50 0 0.045 0
FE 69.4 64.7 2.04 1.68 Short 1.52
Exp 63.0 56.0 1.92 1.83
FE 3.6 0 0.052 0 Medium 0.37
Exp 4.0 0 0.064 0
FE 91.6 85.3 1.69 1.41 Medium 1.17
Exp 71.4 70.0 1.60 1.475
FE 4.09 0 0.056 0 Large 0.37
Exp 4.0 0 0.058 0
FE 111.0 103.2 1.50 1.34 Large 1.15
Exp 80.4 84.0 1.56 1.50
1Central Disp (mm) Strain at tension flange centre (%)
Peak Perm. Peak Permanent
The experimental surface strains were measured at the centre of the LHS corrugation.
Rotation about 1-axis andtranslation along 3-axis also
restrained to zerozero rotation about the 1-axis
(I1 = 0)
edge prevented from
translating in this direction
(U3 = 0)
I1
Bottom edges fully fixed
Fig. 4-1: FE geometry of one length of corrugation with connections employing ½ symmetry (shortoutstand panel shown)
12
Figure 4-2: Permanently deformed FE model (intermediate connection size, p = 1.17 bar)
4.2 SCALING TEST DATA
Half of one corrugation with connections was modelled in full length and appropriate boundary
conditions were employed to ensure symmetry was maintained. The corrugated panel was constructed
from S4R shell elements with a side length of approximately 16 mm (four times that of the ¼ scale
model) to eliminate any mesh dependency influence on the FE predictions. All of the linear
dimensions of the panels were scaled up by four to full-size. The damping factor applied was
equivalent to 0.5 x 10-4 of the change in strain energy during the step. This allowed the same amount
of damping to be applied to both models (as buckling is affected by the damping factor).
The (time, pressure) co-ordinate pairs from the experimental data were directly input in tabular form to
allow more accurate modelling of the pulse in the ¼ scale models. In the full-scale model, the time
data were scaled up by four. The pressure values were not changed. The material properties in the full
and ¼ scale models were identical. These are given in detail in references [6, 8]. The FE model
calculates the strain rates at each element, and selects the appropriate flow stresses from the input
material properties.
4.2.1 Scaling Simulation Results
The FE simulation results for the full-scale panels with connections are shown in Figs. 4-3 to 4-22.
For comparison with the ¼ scale predictions, the ¼ scale displacement-time histories were scaled up
(both the displacements and the times were multiplied by 4) and are also shown. Table 4-2 compares
the displacement and failure mode predictions of the full-scale and ¼ scale FE models for pulse loaded
panels.
13
1
Table 4-2: Comparison of full-scale FE predictions and ¼ scale predictions (scaled up by 4) for pulse loaded panels
Maximum mid- Permanent mid-
Peak
pressure Connection type
(test direction)
point displacement
(mm)
point displacement
(mm)
Failure mode
(bar) Full scale ¼ scale Full scale ¼ scale Full scale ¼ scale
FE x 4 FE x 4 FE
0.51 14.4 15.3 0 0 Elastic Elastic
0.91 29.9 31.0 2.0 2.2 S.P.D.1 S.P.D.
1.21 Short (A)
87.0 65.8 53.5 31.9 S.P.D. S.P.D.
1.92 351.7 337.3 328.7 312.1 Buckling Buckling
0.47 16.5 16.0 0 0 Elastic Elastic
0.94 Short (B) 35.7 35.0 4.3 3.5 S.P.D. S.P.D.
1.18 941.2 890.0 Buckling Buckling
0.43 16.8 15.1 0 0 Elastic Elastic
1.02 Medium (A) 63.2 51.9 26.9 15.4 S.P.D. S.P.D.
1.48 446.7 494.7 417.7 465.6 Buckling Buckling
0.52 20.8 19.6 0 0 Elastic Elastic
0.97 Medium (B) 230.3 54.3 194.0 13.9 Buckling S.P.D.
1.25 1153 942.9 908.5 Buckling Buckling
0.54 23.4 23.0 0 0 Elastic Elastic
0.99 75.1 59.4 36.2 20.2 S.P.D. S.P.D.
1.37 Long (A)
525.7 522.9 488.6 481.0 Buckling Buckling
1.81 577.1 616.4 547.4 588.7 Buckling Buckling
0.47 20.8 22.0 0 0 Elastic Elastic
1.00 Long (B) 1267 1123 1193 1047 Buckling Buckling
1.22 1500 - 1410 - Buckling Buckling
S.P.D.: small permanent displacement
14
Figure 4-3: Predictions of scaled-up ¼ scale response and full-scale displacement-time histories(peak pressure of 0.49 bar in direction A, short outstand connections)
Figure 4-4: Predictions of scaled-up ¼ scale response and full-scale displacement-time histories(peak pressure of 0.91 bar in direction A, short outstand connections)
Figure 4-5: Predictions of scaled-up ¼ scale response and full-scale displacement-time histories
(peak pressure of 1.21 bar in direction A, short outstand connections)
15
Figure 4-6: Predictions of scaled-up ¼ scale response and full-scale displacement-time histories(peak pressure of 1.92 bar in direction A, short outstand connections)
Figure 4-7: Predictions of scaled-up ¼ scale response and full-scale displacement-time histories(peak pressure of 0.43 bar in direction A, medium outstand connections)
Figure 4-8: Predictions of scaled-up ¼ scale response and full-scale displacement-time histories
(peak pressure of 1.02 bar in direction A, medium outstand connections)
16
quarter scale x 4
quarter scale x 4
Ti
quarter scale x 4
Figure 4-9: Predictions of scaled-up ¼ scale response and full-scale displacement-time histories(peak pressure of 1.48 bar in direction A, medium outstand connections)
Figure 4-10: Predictions of scaled-up ¼ scale response and full-scale displacement-timehistories (peak pressure of 0.53 bar in direction A, long outstand connections)
Figure 4-11: Predictions of scaled-up ¼ scale response and full-scale displacement-timehistories (peak pressure of 0.99 bar in direction A, long outstand connections)
17
quarter scale x 4
quarter scale x 4
Figure 4-12: Predictions of scaled-up ¼ scale response and full-scale displacement-timehistories (peak pressure of 1.37 bar in direction A, long outstand connections)
Figure 4-13: Predictions of scaled-up ¼ scale response and full-scale displacement-timehistories (peak pressure of 1.80 bar in direction A, long outstand connections)
Figure 4-14: Predictions of scaled-up ¼ scale response and full-scale displacement-timehistories (peak pressure of 0.48 bar in direction B, short outstand connections)
18
quarter scale x 4
quarter scale x 4
quarter scale x 4
Figure 4-15: Predictions of scaled-up ¼ scale response and full-scale displacement-timehistories (peak pressure of 0.93 bar in direction B, short outstand connections)
Figure 4-16: Predictions of scaled-up ¼ scale response and full-scale displacement-timehistories (peak pressure of 1.18 bar in direction B, short outstand connections)
Figure 4-17: Predictions of scaled-up ¼ scale response and full-scale displacement-timehistories (peak pressure of 0.51 bar in direction B, medium outstand connections)
19
Sensitivity near
buckling point
quarter scale x 4
quarter scale x 4
Figure 4-18: Predictions of scaled-up ¼ scale response and full-scale displacement-timehistories (peak pressure of 0.97 bar in direction B, medium outstand connections)
Figure 4-19: Predictions of scaled-up ¼ scale response and full-scale displacement-timehistories (peak pressure of 1.25 bar in direction B, medium outstand connections)
Figure 4-20: Predictions of scaled-up ¼ scale response and full-scale displacement-timehistories (peak pressure of 0.48 bar in direction B, long outstand connections)
20
quarter scale x 4
analysis not
completed
Figure 4-21: Predictions of scaled-up ¼ scale response and full-scale displacement-timehistories (peak pressure of 1.01 bar in direction B, long outstand connections)
Figure 4-22: Predictions of scaled-up ¼ scale response and full-scale displacement-time
histories (peak pressure of 1.22 bar in direction B, long outstand connections)
4.2.2 Scaling Effects
The Cowper-Symonds coefficients at yield, averaged in the two material directions, for the 2 -1mm thick 316L stainless steel were D = 1704.5 s and q = 5.2 [6].
From [5], the influence of strain rate can be evaluated using equation (4.1):
1
� q§ H · 1 � ¨ ¸
' ¨ ¸V o © E D ¹ (4.1) ' 16 o
� § q�H · 1 ¨
© D ¹¸
21
-1
' ' where V is the dynamic flow stress in the scaled model, 6 o is the dynamic flow stress in the o
full-scale model, H� is the characteristic strain rate and E is the scale factor (0.25 in this case).
For H� = 0.7 s-1, eq (4.1) gives a 5.6% increase in flow stress in the ¼ scale model (this is a
typical strain rate for a pulse pressure tested panel [8]).
For H� = 0.0004 s , eq (4.1) gives a 1.5% increase in flow stress in the ¼ scale model (this is a
typical strain rate for a statically loaded panel [11]).
The strict requirement of equality of dynamic flow stresses at all scales is violated due to rate
effects – this is known as the size effect. It can be seen from these two simple calculations that
the strain rate used in eq (1) significantly influences the magnitude of the size effect. The size
effect means that full-scale models will be softer than their ¼ scale counterparts. This was also
observed in this study, where in over 90% of simulations the full-scale model predicted larger
displacements than the ¼ scale version. Whilst the change in flow stress may be small (circa
6%), buckling is sensitive to imperfections and small changes in flow stress and in one case
(C1211-7, direction B) this effect led to a change in the predicted failure mode (buckling
initiated in full scale model and not in the ¼ scale). The full-scale model predicted a permanent
displacement almost 14 times that of its ¼ scale counterpart (194 mm compared to 13.9 mm) –
this is clearly a non-conservative result and caution must be exercised when scaling up ¼ scale
results where the response is close to the point of buckling.
Similar failure modes were predicted by the FE models of the ¼ scale and full-scale blast wall
panels. Favourable agreement was found between the FE displacement predictions at ¼ and
full-scale. However, care must be taken where the ¼ scale results indicate a response that is
near to the initiation of buckling, particularly in direction B loading, as the FE results show that
the full-scale response will be softer and buckling may occur. This agrees with experimental
work on dynamically loaded structures where the damage suffered by full-scale structures will
be somewhat more severe than that predicted by a scaled test.
The influence of strain rate was also estimated and shown to have a relatively small influence on
displacement, less than 6%. This was attributed to the relatively low strain rates (0.7 s-1)
involved in the response, although a ten-fold increase in strain rate would lead to an error of
approximately 7.9%, which is still within acceptable limits given the other uncertainties
involved in determining blast wall panel response (e.g. determination of the magnitude and
temporal characteristics of the loading).
4.3 RESISTANCE CURVES
The numerical simulations of the pulse loaded (phase I) and statically pressure loaded blast wall
panels were used to generate pressure-displacement curves for each type of panel at various
strain rates. This data was used to construct a family of resistance curves, in the form of force
per corrugation versus central displacement, for each panel type and loading scenario. These
curves are shown in Figs. 4-23 to 4-25 and are used in section 5.1 to facilitate the calculation of
iso-damage curves for the three panel types.
22
PLASTIC SDOF1
PLASTIC SDOF2
Fo
rce p
er
co
rru
gati
on
(kN
)
0
5
10
15
20
25
30
35
40
0 10 20 30 40 50 60 70 80 90
Displacement (mm)
(a) Pulse loaded panels, direction A
0
5
10
15
20
25
30
35
Fo
rce p
er
co
rru
gati
on
(kN
)
0 10 20 30 40 50 60 70 80
Displacement (mm)
(b) Statically loaded panel, direction A
Figure 4-23: Resistance curves for panels with short outstand connections (curves generated using FE analysis)
23
0
5
10
15
20
25
Fo
rce
per
co
rru
gati
on
(k
N)
48 kPa pulse
93 kPa pulse
118 kPa pulse
ELASTIC SDOF
PLASTIC SDOF1
PLASTIC SDOF2
-200 -175 -150 -125 -100 -75 -50 -25 0
Displacement (mm)
Pulse loaded panels, direction B
Figure 4-23(cont): Resistance curves for panels with short outstand connections (curves
generated using FE analysis)
0
5
10
15
20
25
30
Fo
rce p
er
co
rru
gati
on
(kN
)
43 kPa pulse
102 kPa pulse
148 kPa pulse
ELASTIC SDOF
PLASTIC SDOF1
PLASTIC SDOF2
0 20 40 60 80 100 120 140
Displacement (mm)
(a) Pulse loaded panels, direction A
Figure 4-24: Resistance curves for panels with medium outstand connections (curves
generated using FE analysis)
24
0
5
10
15
20
25 F
orc
e p
er
co
rru
gati
on
(kN
)
ELASTIC SDOF
PLASTIC SDOF1
PLASTIC SDOF2
117 kPa Static
0 10 20 30 40 50 60 70
Displacement (mm)
(b) Statically loaded panel, direction A
0
5
10
15
20
25
30
Fo
rce p
er
co
rru
gati
on
(kN
)
51 kPa pulse 97 kPa pulse 125 kPa pulse ELASTIC SDOF PLASTIC SDOF1 PLASTIC SDOF2
-250 -200 -150 -100 -50 0
Displacement (mm)
(c) Pulse loaded panels, direction B
Figure 4-24(cont): Resistance curves for panels with medium outstand connections (curves generated using FE analysis)
25
Pa pu
P
0
5
10
15
20
25
Fo
rce p
er
co
rru
gati
on
(kN
)
30
53 k lse 99 kPa pulse 137 kPa pulse ELASTIC SDOF PLASTIC SDOF1 PLASTIC SDOF2
0 25 50 75 100 125 150
Displacement (mm)
(a) Pulse loaded panel, direction A, peak pressure = 1.37 bar
0
5
10
15
20
25
30
35
Fo
rce p
er
co
rru
gati
on
(kN
)
53 k
ELASTIC SDOF PLASTIC SDOF1 PLASTIC SDOF2
40
a pulse 99 kPa pulse 180 kPa pulse
Membrane response
0 25 50 75 100 125 150 175
Displacement (mm)
(b) Pulse loaded panel, direction A, peak pressure = 1.80 bar
Figure 4-25: Resistance curves for panels with long outstand connections (curves generated
using FE analysis)
26
0
0
5
10
15
20
25
30
35
40
45
Fo
rce p
er
co
rru
gati
on
(kN
) ELASTIC SDOF PLASTIC SDOF1 PLASTIC SDOF2
ponse 50
53 kPa pulse 99 kPa pulse 237 kPa pulse
Membrane res
25 50 75 100 125 150 175
Displacement (mm)
(c) Pulse loaded panel, direction A, peak pressure = 2.37 bar
0
5
10
15
20
Fo
rce p
er
co
rru
gati
on
(kN
)
25 117 kPa Static
ELASTIC SDOF
PLASTIC SDOF
PLASTIC SDOF2
0 25 50 75 100 125
Displacement (mm)
(d) Statically loaded panel, direction A
Figure 4-25(cont): Resistance curves for panels with long outstand connections (curves generated using FE analysis)
27
0
5
10
15
20
25
To
tal F
orc
e (
kN
)
30 48 kPa pulse 101 kPa pulse 122 kPa pulse ELASTIC SDOF PLASTIC SDOF1 PLASTIC SDOF2
-300 -250 -200 -150 -100 -50 0
Displacement (mm)
(e) Pulse loaded panel, direction B
Figure 4-25(cont): Resistance curves for panels with long angle connections (curves
generated using FE analysis)
28
5 SIMPLIFIED METHODS OF ANALYSIS
5.1 PREDICTING RESPONSE USING ISO-DAMAGE CURVES The primary objective of response analysis in structural design is to assess the maximum
transient (or maximum permanent) displacement caused by a particular load case (in this case, a
gas explosion load). This displacement can then be compared with previously determined
acceptance criteria for the structure. A pressure-impulse chart (or a family of P-I charts) can be
used to facilitate this process. These charts are often known as iso-damage curves, as a family
of curves plotted on one P-I chart can represent different levels of damage in a structure.
P-I charts are usually developed using a single-degree-of-freedom (SDOF) model of the
structure in question, which can be developed either from simple analytical approximations or
can be based on a nonlinear finite element analysis. The iso-damage curves developed herein
were developed using a SDOF model based on a tri-linear resistance curve. The resistance
curve was developed from the finite element simulations reported in section 4.
5.1.1 Using resistance curves in a single-degree-of-freedom (SDOF) analysis
The procedure for developing P-I charts is as follows [12]:
��Establish a realistic FE model and perform non-linear static analyses for a uniformly
distributed pressure. Derive a pressure-displacement curve at the locations of interest on
the panel. In this case, the panel centre. A force per corrugation versus displacement is
easily calculated from the FE output. A range of resistance curves may be developed if
the material is strain rate sensitive (which is the case for 316L stainless steel).
��Approximate the resistance curves to a tri-linear response. This approximation could be
performed either through engineering judgement (a subjective evaluation) or by equating
the energies under the resistance curves with that of the tri-linear approximation.
��Either develop a particular SDOF model or use available software to generate a SDOF
model of the structure. Transformation factors must be applied to the parameters obtained
from the tri-linear curve. The factors are used to equate the blast wall panel response with
that of a 1 DOF spring-mass model with a lumped mass and concentrated load. Factors
for transforming the mass and load will depend on the distribution of the loading, the
deformation domain (elastic, elastic-plastic or plastic) and the boundary conditions [14].
��Calculate the dynamic response of the SDOF system for all necessary pressure-impulse
combinations and develop iso-damage curves for acceptable levels of response.
In this case, the force-displacement curves developed from the FE analysis were approximated
to a tri-linear resistance curve using engineering judgement. The resistance curves are given in
Figs. 4-23 to 4-25. SDOF models were developed using BakerRisk in-house software [13]
based on a Biggs’ analysis [14]. The load-mass factor was taken to be 0.78 in the elastic range
and 0.66 in the plastic range, from table 5.1 in reference [14]. These values assume a uniformly
distributed pressure load and simply supported boundaries. These are reasonable
approximations, given that the additional stiffness of the connections is accounted for in the
resistance curves generated using FE analyses. In some cases, the panel response begins to
stiffen as the displacements increase and the tri-linear approximation would over predict the
displacements of the blast wall panels (due to membrane action in the panels causing them to
stiffen). In all cases, the resistance curves were assigned an acceptable displacement limit to
prevent the iso-damage curves extrapolating beyond the available data.
29
5.1.2 Generation and use of iso-damage curves
t
From the phase I analytical beam model, it was determined that the corrugated panel response
could be approximated by that of a rectangular cross-section beam with an equivalent thickness
eq. This approximation was based on equating the second moments of area of the two sections
[1, 8]. The iso-damage curves were developed for various levels of permanent displacement
(damage), ranging from elastic response to at least 3 times the equivalent thickness (depending
on the displacement limit of the resistance curves in Figs. 4-23 to 4-25). This approach included
elastic response, small levels of permanent displacement and buckling response. The load-time
histories are assumed to be triangular, with the total load duration being twice the rise time of
the pulse.
The iso-damage curves are shown in Figs. 5-1 to 5-4. They allow rapid assessment of a
structure, by enabling the load-impulse combinations that will induce specific levels of damage
to be easily ascertained. For example, combinations of pressure and impulse that fall to the left
and below a particular curve will not induce that level of failure whilst those to the right and
above the curve will produce damage in excess of the allowable limit.
The impulse values in Figs. 5-1 to 5-4 are based on the impulse required for one corrugation
(the pressure values are independent of the number of corrugations). To obtain impulse values
for the blast wall panels studied herein, the ‘chart impulses’ must be multiplied by four (as there
are four corrugations in the panel).
Iso-damage curves based on SDOF analyses are simple to use, but can be limited in their
applicability, particularly when:
��The response of a structure cannot be adequately described by a single deformation mode.
��Drag loading and deck accelerations are considered (i.e. two separate load quantities
acting in different directions with temporal characteristics).
��Ductile plastic deformation is considered, most particularly when variable membrane
forces have to be considered (direction A loading). In this case, however, the iso-damage
curves in Figs. 5-1 to 5-3 will be conservative in their predictions.
It is recommended that once a design passes this modelling stage, that a more sophisticated
analysis be completed, such as finite element simulation under various loading conditions. The
SDOF method is a useful screening tool in the early design stages.
30
(a) P-I Chart based on FE simulation of a pulse-loaded panel (from Fig. 4-23a)
(b) P-I Chart based on FE simulation of a statically loaded panel (from Fig. 4-23b)
Figure 5-1: Iso-damage curves for panels with short angle connections, direction A
31
(a) P-I Chart based on FE simulation of a pulse-loaded panel (from Fig. 4-24a)
(b) P-I Chart based on FE simulation of a statically loaded panel (from Fig. 4-24b)
Figure 5-2: Iso-damage curves for panels with intermediate angle connections, direction A
32
(a) P-I Chart based on FE simulation of a pulse-loaded panel (from Fig. 4-25b)
(b) P-I Chart based on FE simulation of a statically loaded panel (from Fig. 4-25d)
Figure 5-3: Iso-damage curves for panels with long angle connections, direction A
33
(a) P-I Chart based on FE simulation of a short angle panel (from Fig. 4-23c)
(b) P-I Chart based on FE simulation of an intermediate angle panel (from Fig. 4-24c)
Figure 5-4: Iso-damage curves for pulse-loaded panels with various connections, direction B
34
(c) P-I Chart based on FE simulation of a long angle panel (from Fig. 4-25e)
Figure 5-4(cont): Iso-damage curves for pulse-loaded panels with various connections,
direction B
5.2 ANALYTICAL BEAM MODELS The simplified modelling approach adopted in section 5.1 has generated some empirical iso-
damage curves that can be used to rapidly assess the response of a blast wall structure to a pulse
pressure load. However, the SDOF based model suffers from the limitations described above,
and requires the construction of a numerical model to generate the resistance curves.
Consequently, two other approaches were considered as a means of predicting response of blast
wall structures to pulse pressure loading, namely (1) a dynamic, elastic-plastic mode
approximation as developed in phase I of the project and (2) a quasi-static, elastic-plastic
simplified model.
5.2.1 Dynamic, elastic-plastic mode approximation
A general mathematical model for predicting the dynamic, elastic-plastic response of beams
with different structural geometries and end conditions, subjected to pulse loading was
developed in phase I of the project. The analysis includes membrane action, strain hardening
effects and flexible boundary conditions. The modelling scheme, which extends previous
analytical work [15], can be used as a general tool for predicting the response of any beam of
known cross-section and support conditions, to a uniformly distributed pulse pressure load. The
structural model consists of beams and springs connected to form a continuous system, as
shown in Fig. 5-5. The resistance functions of the springs, in the form of moment-rotation and
force-displacement curves, are given in Fig. 5-6. The model was used to predict the response of
the blast wall panels. Comparisons with experimental data are given in Figs. 5-7 and 5-8. In
this approach, a numerical solver (MATLAB) was used to solve the equilibrium equations. The
spring constants used in the analysis were pre-determined from the scaled connection test data.
Further details of the modelling scheme are contained in ref. [16].
35
(a) Horizontal Translation Spring (b) Vertical Translation Spring
(c) Rotational Support Spring (d) Centre Rotational Spring
Figure 5-5: Physical representation of the structural model
Figure 5-6: Spring Resistance Functions
36
10
9
8
7
6
5
4
3
2
1
0
0 0.5 1 1.5 2 2.5 3
Peak Pressure (barg)
Short (P) Long (P)
Short (E) Long (E)
Intermediate (P)
Intermediate (E)
Figure 5-7: Normalised permanent displacement versus peak pressure: predictions (P)
and experiments (E)
0
2
4
6
8
10
12
0 0.5 1 1.5 2 2.5
Peak Pressure (barg)
Short (P) Long (P)
Short (E) Long (E)
Intermediate (P)
Intermediate (E)
Figure 5-8: Normalised maximum displacement versus peak pressure: predictions (P)
and experiments (E)
37
5.2.2 Quasi-static, elastic-plastic model
A quasi-static, elastic-plastic model was used to analyse the response of the blast wall
panel to pulse pressure loading. The panel behaviour was represented by a horizontal
one-way spanning beam with end connections as in the previous approach. The end
connections were modelled as a vertical pinned member of finite length and cross-
sectional area with one end pinned to the end of the beam and the other pinned to a
fixed point, as shown in Fig. 5-9. The beam member consisted of two flexural elements
connected through an angular spring element having rigid-plastic characteristics as in
the previous approaches. This spring element was used to model the plastic behaviour
of the beam at mid-span. The vertical pinned member was used to represent the
behaviour of the flexible angle and could be replaced by a vertical and horizontal non-
linear spring, Fig. 5-10. The characteristics of these equivalent springs, Fig. 5-11, were
determined by a consideration of the large displacements of the beam and the geometry
of the pinned member as it changed with the displacement of the beam. Thus, in this
approach the stiffness of the spring elements did not have to be pre-determined.
p
tr x
L
2L
Rigid/plastic angular spring
Beam
elemen
Pinned
membeflex
Figure 5-9: Beam/connection model
38
șK
Ky
Kx
p
x
L
Figure 5-10: Beam/connection model (one-half symmetry) with pinned member
replaced by linear springs
R
Fx Fy Mc
mx Rmy
K
Kș
x Ky
Mo
șį į x y
Figure 5-11: Spring characteristics
x
w3 C3
w2
w1
2L
C2
C1
Elastic small displacement mode:
ʌx cosw1 C1
2L
Plastic large displacement mode:
½¾¿
®¯1
Elastic small displacement mode:
w3 C3
x C �w 2 2
L
Figure 5-12: Displacement profiles of beam model
39
' x į
Ky
Kx
u
p
x
L
With axial restraint
Without axial restraint
Figure 5-13: Large displacement geometry under transverse loading with and without axial
restraint
Fx
Fy
ș�
x G
p
Fa
F
D�
Figure 5-14: Connection forces
Uncoupled mode shapes with generalised displacements (C1, C2, C3), as shown in Fig. 5-12,
were used to approximate the overall elastic-plastic behaviour of the beam model under
shock/pulse pressure loading. The mode shapes were uncoupled as this allowed the elastic and
plastic effects to be determined separately in the quasi-static analysis. The mode shapes
represent the displaced form of the beam as a whole and were used to formulate the total strain
energy (flexural and membrane) of the structural elements in terms of the generalised
displacements. In the quasi-static analysis, equilibrium equations were formulated and solved
in a stepwise manner within a spreadsheet program.
The mode shapes given in Fig. 5-12 were considered appropriate for this analysis as they
represent the fundamental mode shapes of a beam with simple supports undergoing initially
small displacements followed by large displacements, Fig. 5-13, due to the formation of a
plastic hinge at the mid-span of the beam. The end restraint is provided by the pinned link
representing the flexible angle. The ends of the beam are restrained both axially and vertically
due to the pinned link. The link is assumed to stretch and rotate at the pinned ends, Fig. 5-13.
40
The movement of the pinned link is assumed non-recoverable due to the formation of plastic
hinges at the pinned ends. These plastic hinges are assumed to contribute little resistance in the
overall large deformation of the beam. This is clearly a simplification of a complex mechanism
but reflects reasonably well the actual behaviour of the connection in the tests.
Energy methods were used to model flexural and membrane behaviour. Non-linear geometry
effects were incorporated into the model by considering the elongation of the beam member
during large displacements. Plastic behaviour was incorporated into the model by the use of the
angular spring at the mid-span and the movement of the pinned member as described above.
The angular spring, which connects the two elastic beam elements at the mid-span, has rigid,
plastic characteristics (the beam model is symmetrical about the mid-span of the beam). The
moment at the mid-span, Mc, was therefore idealised as elastic-plastic (bi-linear) through the
elastic beam element and the rigid, plastic spring element. The procedures adopted in the quasi-
static analysis of the beam model are explained in more detail in the following sections.
Procedures for small displacement analysis
Elastic flexural deformations up to the point of a plastic hinge forming at the mid-span of the
beam are approximated by the shape function, (x) w , equation (5.1).
ʌx w C1 cos C (5.1) � 3
2L
This shape function combines the fundamental flexural mode shape of a simply
supported beam and vertical support movement. The approach described in section
5.2.1 combined the fundamental flexural mode shape of a fixed and simply supported
beam together with an angular spring at the ends to enable variable support conditions
to be modelled. However, in this case the axial restraint at the ends was considered the
controlling factor in determining the large displacements of the beam model.
The flexural (bending) strain energy of the beam element is given by Ub , equation (5.2).
2§¨̈©
2 2 ·L M EI dL w³ ³U dx dx (5.2) ¸
¹¸b 200 2EI 2 dx
The potential energy loss due to the loading p(x) is given by PE, equation (5.3). L
PE ³� � � � �x p x w dx (5.3) 0
The total potential energy V of the system (strain energy of all elements and potential energy of
loading) is given by V ¦U � PE here¦U Ub .w
The quasi-static equilibrium equations are determined from a consideration of the minimumww
ww
V
C
V 64pLi for 0 3 1, gives and CC1 5C
potential energy of the system, which gives 0 .
4 pL
K .Thus, evaluating 3
EIʌi y
d2w The moment at the mid-span is found from the curvature of the beam, , at x = 0. Thus the
dx2
limiting elastic displacement of the beam is found from equating the moment at the centre Mc to
L 4M 2 0the plastic moment capacity of the beam cross-section, M0 to give C1
0
2 at a quasi-static
EIʌ 5 0
elastic limiting load p0 EIʌ C1 .1 64L4
41
pL EAflexThe support response, C3, is found from where K y . K y Lflex
Procedures for large displacement analysis
Plastic deformations after plastic hinge formation at the mid-span of the beam are approximated
by the shape function (x) w , equation (5.4).
x ½ w C2®1 � ¾ (5.4)
¯ L ¿ This shape function combines the fundamental plastic mode shape of a simply supported beam
with a central plastic hinge.
From a consideration of the large displacement geometry in Fig. 5-13 and the connection forces
u · 2in Fig. 5-14, u L � L2 �C , F pL and F Fy tan Į , where Į sin �1§¨ ¸ . y x ¨ ¸© Lflex ¹
The elastic membrane strain energy of the beam element is given by U , equation (5.5), inm
which ǻ is the axial stretching of the beam during bending, evaluated from the horizontal x
component of the connection force F , given in equation (5.6).x2ǻ AE
U x (5.5) m2L
ǻ L Fx (5.6) x
AE
The spring constants are evaluated as follows:
F K x (5.7) x į x
where į u �ǻ and x x
EAflexK (5.8) L
yflex
The strain energy of the angular spring at the mid-span is given by 2Kșș Uș M0ș � (5.9)
2
where Kș is the strain hardening spring constant.
The potential energy loss due to the loading p(x) is given by PE, equation (5.3). The total
potential energy V of the system (strain energy of all elements and potential energy of loading)
is given by V ¦U � PE where ¦U U � Uș . m
The quasi-static equilibrium equation is determined from a consideration of the minimum
wV potential energy of the system, which gives 0 .
wCwV
Thus, evaluating 0 gives wC2
C 2K 2 p 2M0 � >c1 � c2@C
3
32 � ș (5.10)
2 3L L L
42
AE K where c1 and c2 x .
2 2ª º ª ºL K AE x1�«¬ 1 L �«¬
»¼
»¼AEL K x
Thus, the transverse displacement w of the beam can be plotted against the transverse load p.
Dynamic effects
The response of a single-degree-of-freedom (SDOF) elastic spring-mass system to a pulse load
having a triangular shaped pressure pulse with a duration td has been studied by Biggs [14]. The
fundamental elastic period of vibration of the mass is T. It can be shown that dynamic effects
exercise a very important influence on the response of the mass when td/T < 1. On the other
hand, when td/T is very large, the response of the mass is essentially static. The deflections
calculated for the same peak load applied statically agree with the corresponding maximum
dynamic deflections within approximately 17% when td/T > 1.75.
Inertia effects are always significant when the loading is applied almost instantaneously as in
the case of shock loading. Biggs also studied the effect of a rectangular pressure pulse having a
finite load duration td. It can be shown that the maximum deflection of the mass is double the
quasi-static calculation (assuming the peak load is applied statically) when td/T > 0.5,
approximately.
Comparison of Results
The graphs of permanent displacement, Fig. 5-15, are shown together with the experimental
data points for comparison. The graphs of maximum displacement, Fig. 5-16, are shown
together with the numerical FEA simulation results for comparison. The parameters used for
this analysis are given in Table 14. Direct comparison with the experimental data and the
numerical simulations are also given in Table 15. The intermediate and long angle connection
analytical curves provide an acceptable fit through the test data for this geometry. However, the
analytical results for the short angle appear excessively over conservative compared with the
test data. In this case, the iso-damage curve (Fig. 61b) provides a more acceptable correlation.
In general, the quasi-static, elastic-plastic approach performs better in the large deflection range
and when L/Lflex < 5.
43
Table 5-1: Parameters for quasi-static analysis
Parameter Value Units
L 0.5 m
A 512 mm 2
flexA 660 mm 2
E 207 GN/m2
I 13.5×104 mm 2
Kș 6000 Nm/rad
0M 2300 Nm
0ı 311.4 N/mm 2
flexL 0.06, 0.12, 0.18 m
Table 5-2: Comparison of experimental, numerical and analytical data
Test
reference
Peak
pressure
(bar)
Connection
type
Experimental FEA Analytical
wmax wperm wmax wperm wmax wperm
C621-1 0.76 7.5 0 5.6 0 7.8 0
60CORR6 0.91 7.5 0 7.7 0.5 28 20
60CORR5 1.21 Lflex = 60mm
- 4.0 14 7.5 78 70
WBE #3 1.23 (Short)
- 22 38 30.5 78 70
C622 1.92 - 69 84.3 78 148 140
C1200 0.43 4.5 0 3.8 0 4.4 0
C1211-4 1.02 13 2.0 12.9 3.4 38 30
C1212 1.48 Lflex = 120mm
- 100 123.7 116 113 105
C1215 2 (Inter)
- 133 - - 158 150
WBE #5 1.57 - 145 161 152 122 114
C1801-1 0.54 4.7 0 5.8 0 5.5 0
C1801-3 0.99 9.4 0 14.8 5 38 30
C1803 1.37 Lflex =
- 131 131 120 93 85 180mm
C1804 1.81 - 153 154 147 138 130
C1805 2.39 (Long)
- 155 - - 178 170
WBE #7 2.56 - 185 189 183 188 180
44
0
1
2
3
4
0.5
1.5
2.5
3.5 P
ressu
re (
bar)
Short
C60CORR5
WBE#3
C622
Inter
C1211-4
C1212
WBE#5
C12XX
Long
C1801-3
C1803
C1804
C18XX
WBE#7
0 0.05 0.1 0.15 0.2 0.25
Permanent displacement (m)
Figure 5-15: Permanent displacement versus pressure for ¼ scale blast wall panels. (Analysis
data are shown as solid lines for each connection type. Experimental data are shown as
individual points.)
0
1
2
3
4
0.5
1.5
2.5
3.5
Pre
ssu
re (
bar)
Short
C60CORR5
WBE#3
C622
Inter
C1211-4
C1212
WBE#5
C12XX
Long
C1801-3
C1803
C1804
WBE#7
0 0.05 0.1 0.15 0.2 0.25
Maximum displacement (m)
Figure 5-16: Maximum displacement versus pressure for ¼ scale blast wall panels. (Analysis
data are shown as solid lines for each connection type. Numerical FEA data are shown
as individual points.)
45
REFERENCES
1. Schleyer, G.K. and Langdon, G.S., ‘Pulse pressure testing of ¼ scale blast walls and blast
wall connections’, HSE Final Report, Project D3920, 2002.
2. Schleyer, G.K., Langdon, G.S. and Jones, N., ‘Research Proposal – pulse pressure testing
of ¼ scale blast walls with connections – phase II’, HSE Proposal, 2003.
3. Louca, LA and Boh, JW. ‘Analysis and Design of Profiles Blast walls’, HSE Research
Report 146, HSE Books, 2004.
4. ‘Interim Guidance Notes for the Design and Protection of Topside Structures Against
Explosion and Fire’, SCI-P-122, Steel Construction Institute, 1991.
5. ‘Design Guide for Stainless Steel Blast Walls’, Fire and Blast Information Group
Technical Note 5, Steel Construction Institute, 1999.
6. Langdon, G.S. and Schleyer, G.K. ‘Unusual strain rate sensitive behaviour of AISI 316L
austenitic stainless steel’, IMechE J. of Strain Analysis, 39(1), 71-86, 2004.
7. Schleyer, G.K. Development and applications of a new pulse pressure loading test rig,
Proc. Conf. Structures under Shock and Impact, Thessaloniki, CMP, 251-262, 1998.
8. Langdon, G.S. ‘Failure of corrugated panels and supports under blast loading:
experimental, analytical and numerical studies’, PhD Thesis, University of Liverpool,
2003.
9. Langdon, G.S. and Schleyer, G.K, ‘Modelling the response of semi-rigid supports under
combined loading’, Engineering Structures, 26(4), 511-517, 2004.
10. ABAQUS Standard User Manual Volume 1, version 6.2, Hibbit, Karlsson and Sorenson,
Inc.
11. Langdon, G.S. and Schleyer, G.K. HSE Technical Progress Report, September 2003.
12. Czujko, J (Ed.). ‘Design of offshore facilities to resist gas explosion hazard: engineering
handbook’, CorrOcean ASA, 2001.
13. WBIGGS for Windows v.4.4, Baker Engineering and Risk Consultants, Inc., 2002.
14. Biggs, J.M. ‘Introduction to structural dynamics’, McGraw-Hill Publishing Company,
1964.
15. Schleyer GK and Hsu SS, ‘A modelling scheme for predicting the response of elastic-
plastic structures to pulse pressure loading’, Int J Impact Engng, 24, 759-777, 2000.
16. Langdon GS and Schleyer GK, ‘Inelastic deformation and failure of ¼ scale profiled
stainless steel blast wall panels. Part II: analytical modelling considerations’, Int J Impact
Engng, 31(4), 371-399, 2005.
46
Appendix A – Worked Example
This worked example demonstrates the use of the simplified methodology to determine the
deflection of a blast wall for a given level of dynamic loading beyond the elastic limit. The
calculations will be compared with the standard Biggs method which does not account for end
effects. This example highlights the benefits of assessing a blast wall into the large plastic
deformation range and shows that the performance of the blast wall is significantly enhanced by
analysing the combined support/blast wall system. It is important to recognise that the integrity
of the welded angle connection and the restraint of the primary support are fundamental to these
calculations. If in doubt about these factors, further calculations/checks should be performed.
Geometry
The description of the blast wall system given here is for the scaled-up (× 4) test panel with
intermediate size angle connection.
340 540140
8.0 thk 162
880 Dimensions in mm
Pressure Pressure
Figure A-1: Non-symmetrical trapezoidal deep trough profile dimensions (all bend radii 40
mm) of blast wall
Profile width 880.0 mm
Profile depth from top surface to inside bottom surface 162.0 mm
Corner internal bend radius 40.0 mm
Base thickness of profile 8.0 mm
Cross-section area of profile 8598.0 mm 2
Profile material density 7970.0 kg/m3
47
9153640 mm
100 40
12 thk
16 thk
8
12
12
580260
140
1020
80
Dimensions in mm
Primary support
Figure A-2: Connection details of blast wall (side elevation view)
4 m
1020 mm
Figure A-3: Side elevation of blast wall/connection
48
The following properties of the blast wall can be calculated using the deep profile dimensions 2given above and a yield stress of 311.4 N/mm for a profile width of 880 mm:
Mass of beam section / unit length 68.5 kg/m
Second moment of area 3638.0 cm 4
Young’s modulus 200.0 MN/mm2
Moment of resistance 151.5 kNm
Base yield stress 265.0 N/mm2
Dynamic yield stress 311.4 N/mm2
Other properties will be calculated at appropriate points in the example.
A spreadsheet analysis has been generated using the methodology described in section 5.2.2.
This can be used to generate non-dimensional curves of p against w/w o for different dynamic
load factors as shown in Fig. A-4.
Figure A-4: Non-dimensional design curves for quasi-static analysis
The parameters that were used to generate the spreadsheet analysis used in this example are as
follows:
Span (2L) 4 m
Profile width 0.88 m
Flexible angle length (Lflex) 0.48 m
Area of cross-section, A 8598 mm 2
Area of angle connection (Aflex) 10560 mm 2
Young’s modulus, E 200.0 MN/mm2
Second moment of area, I 3638.0 cm 4
Yield stress (dynamic) 311.4 N/mm2
Plastic moment resistance 151.5 kNm
Aflex = profile width × thickness of angle support = 880 × 12 mm = 10,560 mm2.
49
Small displacement analysis
The elastic limit properties of the blast wall are calculated here. These are independent of the
angle length. The analysis assumes no moment resistance at the supports.
The limiting elastic displacement of the blast wall is calculated from:
L 4M 2 0C1
0
2 , which gives 33.8 mm.
EIʌ
The corresponding quasi-static limiting load is calculated from:
5 00 EIʌ C1 p , which gives 0.83 bar.1 464L
These values are used in the first line of the spreadsheet analysis.
0 pFor a DLF of 2, the above limiting load can be calculated from 1 , which gives 0.415 bar.
DLF
Large displacement analysis
This utilises the curves from Figure A-4, which have been generated using the spreadsheet
analysis. First, the dynamic amplification of the loading on the response of the blast wall needs
to be assessed. This can be done quite simply using the classification of loading according to
the following rules:
Quasi-static loading regime: Ȧtd !40
Dynamic loading regime: 0.4 dȦtd d40
Impulsive loading regime: Ȧtd d0.4
In this case, Ȧ dt 43.9 where Ȧ k/M . Therefore the problem may be solved as a quasi-
static loading case.
DLF calculations:
1For an elastic-plastic structure, the DLF can be modified by including the effects of ductility as
follows:
For a shock pulse load, the DLF can be calculated for the quasi-static regime from
ºª«««
»»»
1 x , !x elDLF
x el1�¼¬ 2x
For a triangular pulse load with a finite rise time, the DLF = 1 for the quasi-static regime.
TNO Green Book, CPR 16E, ‘Methods for the determination of possible damage’, 1992.
50
1
Thus for a limiting displacement (x) or ductility ratio (x / xel) and a given DLF, a design
pressure can be determined from the chart in Fig. A-4.
Two load cases will be considered, namely a shocked-up pressure pulse and a triangular
pressure pulse with a finite rise time.
Pressure (bar)
0.8
Time (msec)
200
Figure A-5: Load case 1 - shock pulse load
It is first necessary to estimate the ductility using a SDOF analysis. For this, the standard Biggs
analysis will be used. This has been determined as 4.
Therefore, using the above formula for modified DLF:
ºª«««
»»»
1 1 DLF 1.14
1xel1� 1� ¼¬ 2x 8
From Fig. A-6, the intersection of DLF = 1.14 and p = 0.8 bar gives a ductility of 7.
DLF=1.14
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
1.80
2.00
2.20
P (bar) DLF=1.14
1 4 7 10 13 16 19 22 25 28 31
w / wo
Figure A-6: Non-dimensional design curve for quasi-static analysis, DLF=1.14
51
Pressure (bar)
1.2
Time (msec)
100 200
Figure A-7: Load case 2 - triangular pulse load with finite rise time
From Fig. A-8, the intersection of DLF = 1 and p = 1.2 bar gives a ductility of 20.5.
DLF=1
2.60
2.40
2.20
2.00
1.80
1.60
1.40 P (bar)
1.20
1.00
0.80
0.60
0.40
0.20
0.00
1 4 7 10 13 16 19 22 25 28 31
DLF=1
w / wo
Figure A-8: Non-dimensional design curve for quasi-static analysis, DLF=1
Biggs analysis
To compare the influence of end effects in the analysis, the above methodology will be
compared with the standard Biggs analysis.
For a simply-supported beam with no end fixity and a uniformly distributed load,
Load transformation factor, KL = 0.64 (elastic), 0.5 (plastic)
Mass transformation factor, KM = 0.5 (elastic), 0.33 (plastic)
Use KL = 0.5 and KM = 0.33
Span, 2L = 4m
52
8M 8u 5.151 Maximum resistance, R m
o 303 kN (2L) 4
384EI 384 u 200 u106 u 3638u10�8
Elastic spring constant, k 8731.2 kN/m 5(2L)3 5u 43
Mass of beam, M m u (2L) 68.5u 4 274 kg
Equivalent mass (for SDOF analysis), Me M uK 274 u 0.33 90.42 kgM
Equivalent stiffness (for SDOF analysis), ke k uK 8731.2 u 0.5 4365.6 kN/mL
90.42 eNatural period, T 2 ʌ M2 ʌ 0.0286 sec
k 4365.6 u103
e
303 mElastic limit displacement, į R 0.0347 m el
k 8731.2
For load case 1:
Peak load, F1 = peak pressure u profile width u beam length = (0.8 u 100) u 0.88 u 4 = 281.6kN
R 303 The ratio, m 1.076
F1 281.6
The ratio, td 0.2
6.99 T 0.0286
From Fig. A-9, intersection of R m 076.1 and
td 99.6 gives a ductility ratio, F1 T
Xm į Xel elį
4 (approx.)
53
Xm
/ X
el (d
ucti
lity
rati
o)
td / T
Figure A-9: Maximum response of elastic-plastic, one-degree-of-freedom system for a
triangular shocked pulse load. Values next to curves are Rm/F1.
Therefore the maximum displacement of the blast wall is calculated from
į 4 u į 4 u 0.0347 0.1388 m el
The permanent displacement of the blast wall at the mid-span is calculated from
į į � į 0.1388 � 0.0347 0.104 m p el
For load case 2:
Peak load, F1 = peak pressure u profile width u beam length = (1.2 u 100) u 0.88 u 4 = 422.4
kN
R 303 The ratio, m 0.714
F 422.41
t 0.2 The ratio, d 6.99
T 0.0286
54
From Fig. A-10, intersection of R m 714.0 and
td 99.6 gives a ductility ratio, F1 T
Xm į Xel elį
35-38 (approx.) X
m / X
el (d
ucti
lity
rati
o)
td / T
Figure A-10: Maximum displacement of elastic-plastic, one-degree-of-freedom system for
a triangular pulse load. Values next to curves are Rm/F.
Therefore the maximum displacement of the blast wall is calculated from
į 38u į 38u 0.0347 1.319 m el
The permanent displacement of the blast wall at the mid-span is calculated from
į į � į 1.319 � 0.0347 1.284 m p el
55
Summary of results:
Table A-1: Comparison of ductility į/į results for Biggs method and proposed quasi-static el
method
Quasi-static analysis with Biggs standard
Load case modified DLF
Pressure (bar)
0.8
Time (msec)
elį/į = 7 elį/į = 4
200
Pressure (bar)
1.2
elį/į = 20.5 elį/į = 35-38
100 200
Time (msec)
For load case 1, the modified quasi-static method gives higher displacement predictions than the
standard Biggs method. At this displacement level, the effect of the supports is negligible in the
analysis.
For load case 2, the Biggs method without end effects predicts higher displacements than the
quasi-static method.
It is concluded that for relatively low levels of displacement, the standard Biggs method gives
adequate results for the purposes of preliminary design or assessment. However, for higher
levels of displacement, say į/į > 10, the influence of the connections becomes important and el
the modified quasi-static method or one of the other methods proposed in this report, which
account for axial effects, should be used.
56
Appendix B – Spring Constants
The results of the connection tests were used to derive spring constants in the x and I directions,
in the same way as the static values were derived in ref. [1, 8, and 9]. Table B-1 lists the
derived spring constants and Fig. B-1 defines the x and I directions. Further details are
available in reference [8]. It is important to note from the results that connection geometry has a
much larger effect upon the support behaviour and spring parameters than varying the test
speed.
Table B-1: Derived Spring Constants
Connection
Type
Test
speed
(mm/sec)
Kxe
(N/mm)
Kxp
(N/mm)
Rmx
(N)
Mphi
(Nm)
Max.
Angle
(deg.)
Kphi(e)
(Nm/deg)
Kphi(p)
(Nm/deg)
Short 1 250 48 21.8 3.1
10 301 123.2 900 250 40 21.2 3
50 466 139 800 215 46 10.4 0.21
100 243 128.9 850 260 41 26.1 5.1
300 192.9 116.5 775 274 47 20.36 3.8
Inter 1 260 48 20.3 2.8
5 35.7 12.8 500
10 26.3 17.4 450 260 48 23.7 2.7
50 37.4 14.9 450 260 44 21.4 3.6
100 31.9 15.1 475 265 50 21.9 1.8
300 30.6 14.8 500 255 55.2 11.3 2
Long 1 255 55 18 8.2
5 14.9 4.7 437.5
5 14.2 4.2 470
10 12.1 3 437.5 270 49 25.7 8.3
50 13.9 6.4 450 270 60 22 5
100 423 290 56 18.6 4.9
300 467 280 53 20 5.5
57
(a) I-direction (b) x-direction
Figure B-1: Definition of (a) rotation spring direction (from direction 1 tests) and (b) translation
spring direction (from direction 2 tests)
Published by the Health and Safety Executive 01/06
RR 404