experimental and numerical investigations of
high strength cold-formed lapped z purlins
under combined bending and shear
CAO HUNG PHAM
annabel f. davis
bonney r. emmett
RESEARCH REPORT R938
August 2013
ISSN 1833-2781
SCHOOL OF CIVIL
ENGINEERING
SCHOOL OF CIVIL ENGINEERING
EXPERIMENTAL AND NUMERICAL INVESTIGATIONS OF HIGH STRENGTH
COLD-FORMED LAPPED Z PURLINS UNDER COMBINED BENDING AND SHEAR
RESEARCH REPORT R938
CAO HUNG PHAM
ANNABEL F. DAVIS
BONNEY R. EMMETT
August 2013
ISSN 1833-2781
Experimental and Numerical Investigations of High Strength Cold-Formed Lapped Z Purlins
under Combined Bending and Shear
Copyright Notice
School of Civil Engineering, Research Report R938
Experimental and Numerical Investigations of High Strength Cold-Formed Lapped Z Purlins
under Combined Bending and Shear
Cao Hung Pham
Annabel F. Davis
Bonney R. Emmett
August 2013
ISSN 1833-2781
This publication may be redistributed freely in its entirety and in its original form without the consent of the
copyright owner.
Use of material contained in this publication in any other published works must be appropriately referenced,
and, if necessary, permission sought from the author.
Published by:
School of Civil Engineering
The University of Sydney
Sydney NSW 2006
Australia
This report and other Research Reports published by the School of Civil Engineering are available at
http://sydney.edu.au/civil
School of Civil Engineering
The University of Sydney
Research Report R938
Page 2
Experimental and Numerical Investigations of High Strength Cold-Formed Lapped Z Purlins
under Combined Bending and Shear
ABSTRACT
Plain C or Z- sections are two of the most common cold-formed steel purlins in use for roof systems
throughout the world. Especially for Z- sections, their lapping ability provides continuity and double thickness
material at the support regions results in greater performance and more economical designs. At the region just
outside the end of the lap, the purlin may fail under a combination of high bending and shear. Design methods
for these sections are normally specified in the Australian/New Zealand Standard for Cold-Formed Steel
Structures (AS/NZS 4600:2005) or the North American Specification for Cold-Formed Steel Structural
Members (2007). Both the Effective Width Method (EWM) and the newly developed Direct Strength Method
(DSM) can be used for the design. The DSM presented [Chapter 7 of AS/NZS 4600:2005, Appendix 1 of (AISI
2007)] is developed for columns and beams and is limited to pure compression and pure bending. Recently,
shear, and combined bending and shear have been added to the 2012 Edition of the North American
Specification. The situation of combined bending and shear as occurs in a continuous purlin system is not
considered in detail. Hence, this report presents a testing program performed at the University of Sydney to
determine the ultimate strength of high strength cold-formed lapped Z purlins with two different lap lengths.
Tests were also conducted both with and without straps screwed on the top flanges. These straps provide
torsion/distortion restraints which may enhance the capacity of the purlins. Numerical simulations using the
Finite Element Method (FEM) were also performed. The simulations are compared with and calibrated against
tests. The accurate results from FEM allowed extension of the test data by varying the lap lengths and section
thicknesses. The results of both the experimental tests and FEM were used and plotted on the recently
proposed DSM design interaction curves. Proposals for an extension to the DSM in combined bending and
shear are given in the report.
KEYWORDS
Cold-formed Z-purlins; Bolted lapped connetions; High strength steel; Direct strength method; Combined
bending and shear; Numerical simulations.
School of Civil Engineering
The University of Sydney
Research Report R938
Page 3
Experimental and Numerical Investigations of High Strength Cold-Formed Lapped Z Purlins
under Combined Bending and Shear
TABLE OF CONTENTS
ABSTRACT .......................................................................................................................................................... 3 KEYWORDS ........................................................................................................................................................ 3 TABLE OF CONTENTS....................................................................................................................................... 4 INTRODUCTION ................................................................................................................................................. 5 EXPERIMENTS ON LAPPED Z-SECTIONS UNDER COMBINED BENDING AND SHEAR ............................ 6 Test Rig Design and Tests Specimens ............................................................................................................ 6 Specimen Nomenclature, Dimensions and Coupon Test Results ................................................................... 8 Lapped Bolt Configuration ............................................................................................................................... 8 Tests With Straps and Without Straps ............................................................................................................. 9 NUMERICAL SIMULATION AND VALIDATION OF TEST RESULTS ............................................................... 9 Finite Element Modelling of Test Rig and Z-Purlin .......................................................................................... 9 Initial Geometrical Imperfection ..................................................................................................................... 10 FE Model Validation of Test Results .............................................................................................................. 10 DIRECT STRENGTH METHOD (DSM) OF DESIGN FOR COLD-FORMED SECTIONS ............................... 13 DSM Design Rules for Flexure ...................................................................................................................... 13 Local Buckling Strength .............................................................................................................................. 13 Distortional Buckling Strength .................................................................................................................... 13 DSM Design Rules for Pure Shear ................................................................................................................ 14 Proposed DSM Design Rules in Shear ...................................................................................................... 14 DSM Design Rules for Combined Bending and Shear .................................................................................. 14 COMPARISON OF DSM DESIGN RULES FOR COMBINED BENDING AND SHEAR WITH TESTS AND
NUMERICAL SIMULATION RESULTS AND PROPOSALS ............................................................................. 15 CONCLUSION ................................................................................................................................................... 19 ACKNOWLEDGEMENT .................................................................................................................................... 19 REFERENCES .................................................................................................................................................. 20 School of Civil Engineering
The University of Sydney
Research Report R938
Page 4
Experimental and Numerical Investigations of High Strength Cold-Formed Lapped Z Purlins
under Combined Bending and Shear
INTRODUCTION
The development of high strength cold-formed steel members has led to an increase of steel purlin usage in
both commercial and industrial structures e.g. roof systems, wall studs, girts, steel framed housing, etc. Most
commonly utilized purlins are both C and Z-sections with attractive attributes such as high strength to selfweight ratio, ease of prefabrication and installation, versatility and high structural efficiency. With section
thicknesses typically ranging from 1.0 mm to 3.0 mm, cold-formed members have been fabricated with a
common yield stress of 350 MPa for normal steel and recently up to 550 MPa for high strength steel. In roof
systems, the various possible arrangements of purlins include single span, double spans, sleeved multi-span
or overlapped multi-span. Fig. 1 illustrates the general member arrangement for a multi-span purlin system
with overlaps. In practice, multi-span purlin systems with overlaps are widely used due to their high structural
efficiency based on a high level of continuity between purlin members. The other advantages of using this
arrangement are the ease of transportation with effective stacking and the simple installation procedure of the
connections. Fig. 2 shows the lapped Z purlin configuration which is attached and bolted to the rafter via a
cleat. This configuration is widely used in Australia.
Figure 1. Overlapped multi-span purlin system
Figure 2. Continuous lapped Z purlin configuration
School of Civil Engineering
The University of Sydney
Research Report R938
Page 5
Experimental and Numerical Investigations of High Strength Cold-Formed Lapped Z Purlins
under Combined Bending and Shear
The structural behaviour of overlapped Z-purlins with bolted lapped connections has been investigated
extensively by the work of Ho and Chung (2004, 2006a, 2006b). They have proved experimentally that the
semi-continuity of lapped purlin system depends on the stress level, the bolted connection configuration and
lap length to section depth ratios. They have also shown that shear buckling of the web of a single section at
the end of the lap has mostly influenced the failure of the purlin at internal supports and, consequently, the
design procedure under combined bending and shear must be included. The tests by Zhang and Tong (2008)
also confirmed that the end of lap section of connection is the most critical and needs to be checked for
strength. In Chung and Ho’s paper (2005), they presented an analysis and design method to predict the
deformation characteristics of lapped connections between cold-formed steel Z sections due to global bending
and shear actions as well as local bearing in the web of sections around the bolt holes. Prior to these studies,
it was found in the tests by Ghosn and Sinno (1995) that the most common failure of the lapped connections
over the internal supports of multi-span purlin system was mainly caused by local buckling of the compressive
flange. The load-carrying capacity of lapped connections is, therefore, governed by the bending moment of
these sections. Recently, based on experimental tests and numerical analyses, Dubina and Ungureanu (2010)
have found that the critical section is also at the end of the lap and the interaction of bending with web
crippling and lateral-torsional buckling might become the relevant design criteria.
Currently, two basic design methods for cold-formed steel members are formally available in the
Australian/New Zealand Standard for Cold-Formed Steel Structures (AS/NZS 4600:2005) (Standards
Australia, 2005) or the North American Specification for Cold-Formed Steel Structural Members (NAS, S1002007). They are the traditional Effective Width Method (EWM) and the newly developed Direct Strength
Method of design (DSM) (Chapter 7 of AS/NZS 4600:2005, Appendix 1 of NAS S100-2007). The development
of the DSM for columns and beams, including the reliability of the method is well researched.
Recently, Pham and Hancock (2012a) have presented proposals for the design of cold-formed steel sections
in shear and combined bending and shear by the DSM for use in the NAS and AS/NZS. The proposals were
compared with tests in predominantly shear of both plain lipped C-section tests at the University of MissouriRolla of the 1970s and recent tests on high strength plain lipped C- and SupaCee sections at the University of
Sydney. Proposed DSM design rules for shear and combined bending and shear have recently been
approved for inclusion in Appendix 1 of the 2012 Edition of NAS S100.
The main purpose of this report is to provide test data on high strength cold-formed lapped Z-sections with
various overlap lengths to further refine the DSM proposals for combined bending and shear. Tests with and
without straps screwed on the top compression flanges were performed at the University of Sydney. The
presence of the straps is to provide torsion/distortion restraints and to ensure the continuity of the continuous
lapped Z-section purlin member. The report also presents the modelling and analysis of the experimental
specimens by using the Finite Element Method (FEM) program ABAQUS/Standard (2008) version 6.8-2. The
experimental data was utilized to evaluate the performance of the FE model. The accurate results of the
numerical simulation show that the FE analysis can be utilized to predict the ultimate loads which include the
post-buckling behaviour of cold-formed lapped Z-section purlin subjected to combined bending and shear.
Based on the reliable FE models, the extension of the test data is performed by simply varying section
thicknesses and the lap lengths. The results of both the experimental tests and FEM are plotted on the design
interaction curves. The recommendations and confirmation of proposals for an extension to the DSM in
combined bending and shear are included in the report.
EXPERIMENTS ON LAPPED Z-SECTIONS UNDER COMBINED BENDING AND SHEAR
TEST RIG DESIGN AND TESTS SPECIMENS
The experimental program comprised a total of four tests conducted in the J. W. Roderick Laboratory for
Materials and Structures at the University of Sydney. All tests were performed in the 2000 kN capacity
DARTEC testing machine, using a servo-controlled hydraulic ram. A diagram of the test set-up and overview
test photo for a continuous lapped Z purlin connection based on the simplified analysis in Fig. 1(c) is shown in
Fig. 3. The commercially available plain Z-lipped channel sections (Z20015) of 200 mm depth with a thickness
of 1.5 mm was chosen and the geometry of the Z-section is shown in Fig. 4.
School of Civil Engineering
The University of Sydney
Research Report R938
Page 6
Experimental and Numerical Investigations of High Strength Cold-Formed Lapped Z Purlins
under Combined Bending and Shear
Figure 3. Test set-up configuration and actual experiment
Figure 4. . Z-Section Geometry
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The University of Sydney
Research Report R938
Page 7
Experimental and Numerical Investigations of High Strength Cold-Formed Lapped Z Purlins
under Combined Bending and Shear
At the loading point at mid-span, the DARTEC loading ram has a spherical head to ensure that the load is
applied uniformly, and moved downwards at a constant stroke rate of 2 mm/min during testing. The load was
then transferred to two built-up channels of 10 mm thickness which were connected to the test beam
specimens by two vertical rows of three M12 high tensile bolts each. Four LVDTs (Linear Variable
Displacement Transducers) were utilized as shown in Fig. 3. All LVDTs were mounted directly to the base of
the DARTEC testing machine. This set-up allowed for the vertical displacement of the specimen to be
determined without being affected by bending of the test specimen.
At the two supports, the two beam specimens were bolted through the webs by two vertical rows of three M12
high tensile bolts each. These two rows of bolts were connected to two built-up channels of 10 mm thickness
which were subsequently connected to steel plates of 20 mm thickness as greased load transfer plates. These
greased load bearing plates rested on the half rounds of the DARTEC supports to simulate a set of simple
supports as shown in Fig. 3.
SPECIMEN NOMENCLATURE, DIMENSIONS AND COUPON TEST RESULTS
The Z-sections feature one broad and one narrow flange, sized so that two sections of the same size can fit
together snugly, making them suitable for lapping. The average measured dimensions and yield stress are
given in Table 1. In this table, a (lap length) is the distance from the row of bolts at the loading point to the
adjacent end of the lap. D is the overall depth. F and E are the average overall widths of the narrow and broad
flanges respectively. L is the overall lip size and fy is the average measured yield stress. The Z-section purlins
were tested in pairs with top flanges facing inwards and with a gap between them to ensure that the inside
assembly was possible.
Test
Section
MVw
MVw
Z20015
Z20015
Lap
Length
a (mm)
100
300
MVs
Z20015
100
MVs
Z20015
300
Internal Radius r = 5mm
Thickness
t (mm)
D
(mm)
F
(mm)
E
(mm)
L
(mm)
fy
(MPa)
1.5
1.5
202.2
203.1
72.44
71.98
79.18
79.53
17.69
17.68
542.56
542.56
1.5
1.5
203.3
202.5
71.51
72.29
79.89
79.54
17.63
17.71
542.56
542.56
Table 1. Measured Specimen Dimensions and Properties of Z20015 Sections
The test specimens were labelled in order to express the series, test type, channel section, depth and
thickness. Typical test labels for plain Z-sections “MVs-Z20015” and “MVw-Z20015” are defined as follows: (i)
- MV indicates the combined bending and shear series, (ii) - w expresses the test “without” straps screwed on
top flanges of Z purlins (alternatively “s” indicates the test “with” straps), (iii) - Z200 indicates plain Z- section
with the web width of 200 mm, (iv) – the final “15” is the actual thickness times 10 in mm.
Three coupons were taken longitudinally from the compression flange flat, the tension flange flat and the
centre of the web flat of each channel section member. The tensile coupon dimensions conformed to the
Australian Standard AS 1391 (Standards Australia 1991) for the tensile testing of metals using 12.5 mm wide
coupons with gauge length 50 mm. The tests were performed using the 300 kN capacity Sintech/MTS 65/G
testing machine operated in a displacement control mode. The mean yield stress fy was obtained by using the
0.2 % nominal proof stress and is also included in Table 1. The average Young’s modulus of elasticity was
calculated according to the tensile coupon stress-strain curves to be 205,835 MPa.
LAPPED BOLT CONFIGURATION
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The University of Sydney
Research Report R938
Page 8
Experimental and Numerical Investigations of High Strength Cold-Formed Lapped Z Purlins
under Combined Bending and Shear
The lapped configuration is detailed in Fig. 5 where each end of the lap must have one bolt through the
bottom flanges (the flanges furthest from the cladding), and one bolt through the webs near the top flanges
(the flanges connected to the cladding). This lapped configuration is widely used in Australia. By comparison,
both bolts are commonly connected in the webs near the top and bottom of the flanges in United States
practice.
Figure 5. Lapped Configuration and Test Without and With Straps
TESTS WITH STRAPS AND WITHOUT STRAPS
Two tests were conducted with six 25x25x5EA straps which were uniformly and symmetrically connected by
self-tapping screws on the top flanges as shown in Fig. 5(b). Two of these straps were attached adjacent to
each end of the lap and right at the position of the bolt through the webs. The purpose of these two straps is to
prevent distortion of the top flanges under compression caused by bending moment. The distortion may be a
consequence of unbalanced shear flow or distortional buckling. Two remaining tests were conducted without
the above six 25x25x5EA straps as shown in Fig. 5(a).
NUMERICAL SIMULATION AND VALIDATION OF TEST RESULTS
FINITE ELEMENT MODELLING OF TEST RIG AND Z-PURLIN
The Finite Element Method (FEM) can be used to undertake a geometrically and materially nonlinear inelastic
analysis (GMNIA) of cold-formed thin-walled structures. Pham and Hancock (2010a, b) presented the
modelling and analysis of the experimental specimens of a shear, and combined bending and shear test
series on cold-formed C-section using the FEM program ABAQUS (Abaqus/Standard Version 6.8-2, 2008).
Experimental data from Pham and Hancock (2010c, 2012a) was utilized to evaluate the performance of the
FE model. The ABAQUS results were generally in good agreement with experimental values especially the
ultimate loads and modes of failure.
A detailed finite element model based on that of Pham and Hancock (2010a, b) was developed to study the
structural behaviour and validate the test results of high strength, cold-formed Z-purlins under combined
bending and shear. The test rig was generated in ABAQUS using 3D-deformable solid elements and was
assigned as normal steel properties. The Z-purlins were modelled by using the 4-node shell elements with
reduced integration, type S4R. Quadrilateral element mesh was used for both test rig and Z-purlin. While the
coarse mesh size of 20 mm was used for the test rig, the finer 10 mm element mesh was selected for Z-purlin.
These mesh sizes were chosen and proved accurately in Pham and Hancock (2010a, b). For modelling of
boundary conditions, Fig. 6 shows the test rig configuration and FEM test model at one support. To simulate a
set of simple supports as shown in Fig. 6(a), the “CONN3D2” connector elements were used to connect the
bearing plates to the centre of the half round. Both ends of connector elements are hinges and the length of
the shortest connector member is the radius of half round as shown in Fig. 6(b). For bolt simulation, the “tie”
constraints were used to model contacts between the specimens and rigs where the channels were the slave
surfaces and the rigs were the master surfaces.
School of Civil Engineering
The University of Sydney
Research Report R938
Page 9
Experimental and Numerical Investigations of High Strength Cold-Formed Lapped Z Purlins
under Combined Bending and Shear
Figure 6. Test Rig and ABAQUS Test Model at Support
INITIAL GEOMETRICAL IMPERFECTION
In a nonlinear analysis, imperfections are usually introduced by perturbations in the geometry. Initial
geometrical imperfections are added onto the “perfect” model to create out-of-plane deformations of the plate
elements. In the ABAQUS model, there are three methods to define the geometric imperfections. Firstly, the
geometric imperfections can be defined by the linear superposition of buckling eigenmodes. Secondly,
specifying the node number and imperfection values directly on the data lines gives a method of direct entry.
The final method is defined by the displacements from an initial *STATIC analysis, which may consist of the
application of a “dead” load.
In this study, the first method as used in Pham and Hancock (2010a, b) is also chosen where scaled buckling
modes are separately superimposed on the initial geometry. An initial analysis is carried out on a perfect mesh
using the elastic buckling analysis to generate the possible buckling modes and nodal displacements of these
modes. The imperfections are introduced to the perfect mesh by means of linearly superimposing the elastic
buckling modes onto the mesh. The lowest buckling modes are usually the critical modes and these are used
to generate the imperfections. The imperfection magnitudes were based on two scaling factors of 0.15t and
0.64t where t is the thickness of section. These two factors were proposed by Camotim and Silvestre (2004)
and Schafer and Peköz (1998) respectively.
FE MODEL VALIDATION OF TEST RESULTS
Table 2 shows the results of four tests with two lap lengths of 100 mm and 300 mm, two of which were
conducted with the straps and the remaining half were tested without the straps attached on the top flanges as
shown in Fig. 5. Also included in Table 2 are the FE predictions of the maximum loads for no imperfection,
0.15t imperfection scaling and 0.64t imperfection scaling where t is the thickness of Z-sections scaling. The
FE models were carried out by varying the lap lengths from 100 mm to 500 mm.
By comparison between the ABAQUS ultimate loads with different imperfection scaling magnitudes, the
differences in the ultimate load results are not significant and less than 1.5% for the cases without straps and
up to 5% for those with straps. This fact has been proven in the studies of Pham and Hancock (2010a) where
the FE models are not particularly sensitive to the magnitudes of the initial geometric imperfections under
combined bending and shear loading.
School of Civil Engineering
The University of Sydney
Research Report R938
Page 10
Experimental and Numerical Investigations of High Strength Cold-Formed Lapped Z Purlins
under Combined Bending and Shear
Lapped
Length
a (mm)
Failure
Load (kN)
FE No
Imperfection
(kN)
MVw-Z20015
100
34.306
36.140
36.065
35.693
MVw-Z20015
200
-
42.194
42.059
41.717
MVw-Z20015
300
48.041
48.814
48.640
48.222
MVw-Z20015
400
-
56.867
56.695
56.114
MVw-Z20015
500
-
67.162
67.040
66.609
MVs-Z20015
100
51.886
56.301
55.665
53.871
MVs-Z20015
200
-
62.192
61.681
59.271
MVs-Z20015
300
68.001
70.390
69.566
67.318
MVs-Z20015
400
-
81.156
80.154
78.670
MVs-Z20015
500
-
94.685
92.823
91.153
Section
MV-Series
FE 0.15t
Imperfection
(kN)
FE 0.64t
Imperfection
(kN)
Table 2. Test and ABAQUS Finite Element Results
60
50
Load (kN)
40
30
MVw-300mm
FEM-Imp=0
FEM-Imp=0.15t
FEM-Imp=0.64t
20
10
0
0
3
6
9
12
15
18
Vertical Displacement (mm)
Figure 7. Load and Vertical Displacement Relations of MVw-Z20015-300mm
80
70
Load (kN)
60
50
40
Test MVs-300mm
FEM-Imp=0
FEM-Imp=0.15t
FEM-Imp=0.64t
30
20
10
0
0
3
6
9
12
Vertical Displacement (mm)
15
18
Figure 8. Load and Vertical Displacement Relations of MVs-Z20015-300mm
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Research Report R938
Page 11
Experimental and Numerical Investigations of High Strength Cold-Formed Lapped Z Purlins
under Combined Bending and Shear
As can be seen in Table 2, the ABAQUS results were generally in good agreement with the experimental
values. Figs. 7 and 8 show the load-displacement curves of test and FE results for the MVw-Z20015 and MVsZ20015 with 300 mm overlap length respectively. The load increases linearly and matches with the ABAQUS
model up to 15 kN. Above 15 kN, the test specimens slip on the bolts and are not accurately simulated by the
FEM. The explanation is due to the connection modelling. In the ABAQUS model, “tie” constraints were used
to model the contacts between the Z-purlin and the rig at the locations of the bolts, whereas the Z-purlins were
bolted through the webs by vertical rows of M12 high tensile in the test. These rows of bolts were then
connected to the test rig. All M12 high strength bolts of 830 MPa were pretensioned to 90 kNm torque to
prevent slip under initial loading. As the load increases further from 15 kN to the peak, the test curve deflects
more significantly than those of ABAQUS models. When the load is greater than the friction caused by
pretension of the high tensile bolts, further displacement is recorded due to the slip between Z-purlins and test
rigs where the diameter of the holes is normally 2 mm larger than that of the bolts due to the clearance for
easy installation. Apart from the slip, the additional deflection is also caused by localized bearing failure of Zpurlin plate at the holes especially with the case of very short overlap lengths. It is interesting to note from
Figs. 7 and 8 that, after the peak load, the load-displacement curves of the test are similar to those of
ABAQUS models.
For test observations, section failure of all tests and ABAQUS models occurred just outside the end of the laps
where combined bending and shear is found to be critical. By comparison of tests with or without straps, at
large deformation, significant cross-section distortion of both lower and upper Z-sections was observed in the
tests without straps. Fig. 9 shows the corresponding failure mode shapes of both test and ABAQUS model of
MVw-20015 test with 300 mm overlap length and without straps. Shear buckling occurred in the web of the
lower Z-section at the lap adjacent to the bolt through the web. The top flange of the lower Z-section then
buckled and was pulled in and down. Simultaneously, the top flange of the upper Z-section was twisted and
lifted due to the discontinuity of the connection in bending. Especially with very short lap lengths, the crosssection distortion was severe and large bearing deformation was found in the connected webs around the bolt
holes in the lapped connections.
Figure 9. Failure Mode Shapes of the Test and the ABAQUS Model of
MVw-Z20015 with 300mm Overlap Length and without Straps
For the tests with straps screwed on the top flanges of the Z-sections, a high stress distribution initially
developed in the top flange. The Z-section subsequently failed in the local buckling mode in the top flange due
to the compression stress caused by the bending moment. As can be seen in Fig. 5, the load dropped more
suddenly than that without the straps shown in Fig. 4 due to the local buckling mode in the flange. It can be
observed that shear buckling of the Z-section web at the end of the lap was also developed under the
combined bending and shear stresses. It is interesting to note that local buckling occurred in the top flange of
the upper Z-section which is broader and apparently more slender. No distortion at the cross-section was
observed due to the straps which may significantly increase the capacity and enhance the continuity of the
lapped connection. Fig. 10 shows the corresponding failure mode shapes of both test and ABAQUS model of
MVs-20015 test with 100 mm overlap length and straps.
School of Civil Engineering
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Research Report R938
Page 12
Experimental and Numerical Investigations of High Strength Cold-Formed Lapped Z Purlins
under Combined Bending and Shear
Figure 10. Failure Mode Shapes of the Test and the ABAQUS Model of
MVs-Z20015 with 100mm Overlap Length and with Straps
DIRECT STRENGTH METHOD (DSM) OF DESIGN FOR COLD-FORMED SECTIONS
DSM DESIGN RULES FOR FLEXURE
Local Buckling Strength
The nominal section moment capacity at local buckling (Msl) is determined from Section 7.2.2.3 of AS/NZS
4600:2005 [Appendix 1, Section 1.2.2.2 of NAS (2007)] as follows:
For
l  0.776 : M sl  M y
For l  0.776 : M sl

M
 1  0.15 ol
 My







(1)
0.4 

 M ol
 M y





0.4
My
(2)
where  l = non-dimensional slenderness used to determine M sl (  l = M y / M ol ; M y = Z f f y );
M ol = elastic local buckling moment of the section ( M ol = Z f f ol );
Z f = section modulus about a horizontal axis of the full section;
f ol = elastic local buckling stress of the section in bending.
Distortional Buckling Strength
The nominal section moment capacity at distortional buckling (Msd) is determined from Section 7.2.2.4 of
AS/NZS 4600:2005 [Appendix 1, Section 1.2.2.3 of NAS (2007)] as follows:
For d  0.673 : Msd  M y
For d  0.673 : M sd

M
 1  0.22 od
 My







(3)
0 .5 

  M od
 M y





0 .5
My
(4)
where  d = non-dimensional slenderness used to determine Msd (  d = M y / M od ; M y = Z f f y );
M od = elastic distortional buckling moment of the section ( M od = Z f fod );
Z f = section modulus about a horizontal axis of the full section;
f od = elastic distortional buckling stress of the section in bending.
School of Civil Engineering
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Research Report R938
Page 13
Experimental and Numerical Investigations of High Strength Cold-Formed Lapped Z Purlins
under Combined Bending and Shear
DSM DESIGN RULES FOR PURE SHEAR
Proposed DSM design rules in shear
The design proposal in Pham and Hancock (2012a) for the nominal shear capacity based on the AISI in DSM
format excluding tension field action is used in this paper. The equations in Section C3.2.1 of the North
American Specification (AISI, 2007) which are expressed in terms of a nominal shear stress Fv have been
changed to DSM format by replacing stresses by loads as follows:
For v  0.815 : Vv  V y
(5)
For 0.815  v  1.227 : V v  0.815 V cr V y
(6)
For v  1.227 : Vv  Vcr
(7)
V y  0.6 Aw f y
(8)
Vcr 
kv 2 EAw
d 
121   1 
 tw 
(9)
2
2
where Vy = yield load of web based on an average shear yield stress of 0.6fy;
Vcr = elastic shear buckling force of the whole section derived by integration of the shear stress
distribution at buckling over the whole section; v  Vy / Vcr ;
kv = shear buckling coefficient of the whole section based on the Spline Finite Strip Method (SFSM)
(Pham and Hancock, 2009, 2012b).
DSM DESIGN RULES FOR COMBINED BENDING AND SHEAR
In limit states design standards, the interaction is expressed in terms of bending moment and shear force so
that the upper limit interaction formula for combined bending and shear of a section with a vertically
unstiffened web is given in Clause 3.3.5 of AS/NZS 4600:2005 [Section C 3.3.2 of AISI (2007)]:
2
2
 M  V* 
  

 M   V  1
 s  v
(10)
where M* is bending action, Ms is the bending section capacity in pure bending, V* is the shear action, and Vv
is the shear capacity in pure shear.
The upper limit equation for combined bending and shear of vertically stiffened webs is also given in Clause
3.3.5 of AS/NZS 4600:2005 [Section C 3.3.2 of AISI (2007)]:
 M  V*

0.6
 V  1.3
M
s
v


(11)
In this report, the nominal shear capacity, Vv in Eqs. 10 and 11, is based on the AISI in DSM format as given
in Eqs 5-7. The nominal section moment capacity, Ms, is based on either Msl or Msd of the DSM as given in
Eqs. 1-4.
School of Civil Engineering
The University of Sydney
Research Report R938
Page 14
Experimental and Numerical Investigations of High Strength Cold-Formed Lapped Z Purlins
under Combined Bending and Shear
COMPARISON OF DSM DESIGN RULES FOR COMBINED BENDING AND SHEAR
WITH TESTS AND NUMERICAL SIMULATION RESULTS AND PROPOSALS
As shown in the previous section, the numerical simulation results were calibrated and compared with the
tests. The accurate FEM results show that the FE analysis can be utilized to predict the ultimate loads. In
order to extend the test data, the FE models of the same test rig used for the experiments in Fig. 3 are utilised
with lap length variation from 100 mm to 500 mm. In addition, four thicknesses of commercially available Z
sections of 1.2 mm, 1.5 mm, 1.9 mm and 2.4 mm are chosen for numerical simulation models to vary the
section thicknesses. The detailed results of FEM models are given in Tables 3 and 4 for lapped bolt
configurations with and without straps respectively.
Fig. 11 shows the interaction between (MFEM/Ms) and (VFEM/Vv) for ABAQUS models of lapped Z purlin under
combined bending and shear. The nominal section moment capacity (Ms) is based on the local buckling
moment (Msl) in Eqs. 1-2 and the nominal shear capacity (Vv) is based on Eqs. 5-7. For calculation of Vcr in
Eq. 9, the shear buckling coefficient (kv) of 6.7 for the whole section is used. This kv value is taken from the
interaction chart in Pham and Hancock (2012b) where they provided design guidelines for designers to predict
the elastic buckling shear coefficient (kv) of the complete section without using the Spline Finite Strip Method
software. Similarly, Fig. 12 shows the interaction between (MFEM/Msd) and (VFEM/Vv) for ABAQUS models
where the nominal section moment capacity at distortional buckling (Msd) in Eqs. 3-4 is replaced by the one at
local buckling (Msl). In each figure, whilst the solid markers represent the FE results of the lapped bolt
configuration without straps, the hollow markers show the ones with straps. For the same marker, there are
five of them from left to right in each figure which represent lap lengths from 100 mm to 500 mm respectively.
In all FEM models, the FE results were taken from Tables 3 and 4 based on the 0.15t imperfection.
As can be seen in Fig. 11, for the FEM models of the lapped Z purlins without straps, almost all results lie
below the circular domain limit (Eq. 10) especially those of thicker sections. As discussed and shown in Fig. 9,
severe cross-section distortion at the end of the lap leads to the discontinuity of the connection which reduces
significantly the capacity of the lapped Z-purlin. Although the results are more scattered especially with the
thicker sections, the failure mode seems to be bending only problem. A simple approach plotted in Fig. 11 is
proposed based on factored local buckling moment (Msl) as follows:
M   0.6M sl
(12)
For the FEM models of the lapped Z purlins with straps, all results are shifted above the circular domain limit
(Eq. 10) due to the straps screwed on the top flanges of lapped Z purlins adjacent to the lap ends. However,
the results of several thicker sections (1.9 mm and 2.4 mm) still lie below Eq. 11 linear. Also in cases with
straps, it is interesting to note that all results are likely to lie in the linear trend where the thicker sections seem
to be governed by bending and have less effect by shear. As the sections are thinner, the results are gradually
dropping lower due to the more effect of shear as seen in Fig. 11. This fact can be explained by the failure
mode shapes shown in Fig.10. As observed in the tests, due to the presence of the straps, the local buckling
initially occurred in the flange prior to the shear buckling in the web. The cross-section of Z-purlin adjacent to
the end of the lap has no or little distortion. Hence, a newly proposed linear interaction which fit all FEM
results is shown in Fig. 11 and is given as:
 M  V*
 4.4
4

 M s  Vv
(13)
In Fig. 12, when the Ms=Msd is utilized instead of Msl, the results are shifted higher with respect to the bending
axis due to the fact that Msd is normally lower than Msl. For the FEM models of the lapped Z purlins without
straps, the results lie closer but still lower the circular curve (Eq. 10) especially those of thicker sections. It is
still a bending problem. A simple linear interaction with higher factor is shown in Fig. 12 and is proposed as:
M   0.8M sd
(14)
For cases of the lapped Z purlins with straps, when the Ms=Msd is utilized instead of Msl, all results are higher
and lie significantly above Eq. 11 linear. The interaction between bending and shear is therefore conservative
and not significant. Therefore, there is no need to check for combined bending and shear in these cases.
School of Civil Engineering
The University of Sydney
Research Report R938
Page 15
Experimental and Numerical Investigations of High Strength Cold-Formed Lapped Z Purlins
under Combined Bending and Shear
1.2
1.2
1.2
1.2
1.2
Lapped
Length
a (mm)
100
200
300
400
500
FE No
Imperfection
(kN)
37.030
40.777
45.659
51.947
58.685
FE 0.15t
Imperfection
(kN)
36.872
40.659
45.354
51.353
57.961
FE 0.64t
Imperfection
(kN)
36.361
39.904
44.588
50.932
57.352
MVs-Z20015
MVs-Z20015
MVs-Z20015
MVs-Z20015
MVs-Z20015
1.5
1.5
1.5
1.5
1.5
100
200
300
400
500
56.301
62.192
70.390
81.156
94.685
55.665
61.681
69.566
80.154
92.823
53.871
59.271
67.318
78.670
91.153
MVs-Z20019
MVs-Z20019
MVs-Z20019
MVs-Z20019
MVs-Z20019
1.9
1.9
1.9
1.9
1.9
100
200
300
400
500
90.264
98.678
112.131
129.947
151.418
89.427
96.282
107.567
125.766
148.323
85.080
91.534
98.491
117.414
144.589
MVs-Z20024
MVs-Z20024
MVs-Z20024
MVs-Z20024
MVs-Z20024
2.4
2.4
2.4
2.4
2.4
100
200
300
400
500
131.133
145.651
164.820
188.597
211.989
129.198
143.002
158.430
183.669
206.955
123.669
131.577
143.848
171.853
199.269
Section
MV-Series
Thickness
t (mm)
MVs-Z20012
MVs-Z20012
MVs-Z20012
MVs-Z20012
MVs-Z20012
Table 3. Numerical Simulation Results of Lapped Z Purlins with Straps
1.2
1.2
1.2
1.2
1.2
Lapped
Length
a (mm)
100
200
300
400
500
FE No
Imperfection
(kN)
23.950
27.722
31.660
36.971
44.317
FE 0.15t
Imperfection
(kN)
23.839
27.606
31.594
36.788
44.236
FE 0.64t
Imperfection
(kN)
23.552
27.331
31.303
36.342
43.969
MVw-Z20015
MVw-Z20015
MVw-Z20015
MVw-Z20015
MVw-Z20015
1.5
1.5
1.5
1.5
1.5
100
200
300
400
500
36.140
42.194
48.814
56.867
67.162
36.065
42.059
48.640
56.695
67.040
35.693
41.717
48.222
56.114
66.609
MVw-Z20019
MVw-Z20019
MVw-Z20019
MVw-Z20019
MVw-Z20019
1.9
1.9
1.9
1.9
1.9
100
200
300
400
500
55.031
65.212
75.988
89.013
105.033
54.652
64.752
75.519
88.503
104.487
53.422
63.341
73.986
86.854
102.741
MVw-Z20024
MVw-Z20024
MVw-Z20024
MVw-Z20024
MVw-Z20024
2.4
2.4
2.4
2.4
2.4
100
200
300
400
500
82.638
99.498
116.278
136.704
161.288
81.473
98.112
114.919
135.193
159.766
79.024
94.785
111.329
131.247
155.412
Section
MV-Series
Thickness
t (mm)
MVw-Z20012
MVw-Z20012
MVw-Z20012
MVw-Z20012
MVw-Z20012
Table 4. Numerical Simulation Results of Lapped Z Purlins without Straps
School of Civil Engineering
The University of Sydney
Research Report R938
Page 16
Experimental and Numerical Investigations of High Strength Cold-Formed Lapped Z Purlins
under Combined Bending and Shear
1.4
Eq.(11)
1.2
Eq.(13)
1.0
MFEM
Msl
Eq.(10)
0.8
0.6
M*
 0.6
M sl
Eq.(12)
0.4
0.2
Z20012w
Z20012s
Z20015w
Z20015s
Z20019w
Z20019s
Z20024w
Z20024s
0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
VFEM / Vv ( AISI )
Figure 11. Interaction between (MFEM/Msl) and (VFEM/Vv), Msl based on DSM, Vv based on AISI and kv = 6.7
1.4
1.2
1.0
MFEM
Msd
Eq.(11)
0.8
0.6
M*
 0.8
M sd
Eq.(14)
Eq.(10)
0.4
0.2
Z20012w
Z20012s
Z20015w
Z20015s
Z20019w
Z20019s
Z20024w
Z20024s
0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
VFEM / Vv ( AISI )
Figure 12. Interaction between (MFEM/Msd) and (VFEM/Vv), Msd based on DSM, Vv based on AISI and kv = 6.7
School of Civil Engineering
The University of Sydney
Research Report R938
Page 17
Experimental and Numerical Investigations of High Strength Cold-Formed Lapped Z Purlins
under Combined Bending and Shear
1.4
Eq.(11)
1.2
1.0
MFEM 0.8
Msl
Eq.(10)
Eq.(15)
0.6
M*
 0.6 Eq.(12)
M sl
0.4
0.2
Z20012w
Z20012s
Z20015w
Z20015s
Z20019w
Z20019s
Z20024w
Z20024s
0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
VFEM / Vv ( AISI )
Figure 13. Interaction between (MFEM/Msl) and (VFEM/Vv), Msl based on DSM, Vv based on AISI and kv = 5.34
1.4
1.2
Eq.(11)
1.0
MFEM 0.8
Msd
M*
 0.8 Eq.(14)
M sd
0.6
Eq.(10)
0.4
0.2
Z20012w
Z20012s
Z20015w
Z20015s
Z20019w
Z20019s
Z20024w
Z20024s
0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
VFEM / Vv ( AISI )
Figure 14. Interaction between (MFEM/Msd) and (VFEM/Vv), Msd based on DSM, Vv based on AISI and kv = 5.34
School of Civil Engineering
The University of Sydney
Research Report R938
Page 18
Experimental and Numerical Investigations of High Strength Cold-Formed Lapped Z Purlins
under Combined Bending and Shear
The other alternative is to use different value of the shear buckling coefficient (kv). Figs. 13 and 14 show
interaction between (MFEM/Ms) and (VFEM/Vv) for ABAQUS models of lapped Z purlin under combined bending
and shear where the nominal section moment capacity (Ms) is based on local buckling moment (Msl) and
distortional buckling moment (Msd) respectively. In both Figs. 13 and 14, the nominal shear capacity (Vv) is
based on calculation of Vcr where the kv value of 5.34 is used. This shear buckling coefficient (kv=5.34) is
derived from the classical solution when considering a long web plate in shear as given in the existing North
American Specification (2007) and Australian/New Zealand Standard (2005). As the value of kv=5.34 is lower
than that of kv=6.7, the nominal shear capacity (Vv) is more conservatively predicted. It can be seen in both
Figs. 13 and 14 that all results are shifted horizontally to the right in comparisons with those of Figs. 11 and
12. For cases of the lapped Z purlins without straps, as the failure mode is most likely bending only, the simple
approach using Eqs. 12 and 14 is also utilised for Ms=Msl in Fig. 13 and Ms=Msd in Fig. 14 respectively.
In cases of the lapped Z purlins with straps, all results are shifted horizontally. In Fig. 13, when Ms=Msl is used,
a new linear interaction which fit all FEM results is proposed as:
 M  V*
 5.5
5

M
V
v
 s
(15)
In Fig. 14, when Ms=Msl is replaced by Ms=Msd, all results are shifted vertically along bending axis (as Msl is
generally greater than Msd) whilst the ratios of (VFEM/Vv) are unchanged as compared with Fig. 13. For cases
of lapped Z purlins with straps, all results lie well above Eq. 11 linear. The interaction is conservative and not
significant. Therefore, there is also no need to check for combined bending and shear in these cases.
CONCLUSION
An experimental program was carried out to determine the ultimate strength of high strength, cold-formed
lapped Z- sections subjected to combined bending and shear. A total of four tests of Z20015 purlin have been
performed. While two of tests were conducted with straps attached on the top flanges of Z-sections, the two
remaining tests were tested without straps. A series of ABAQUS simulations was also carried out on high
strength lapped Z-section steel purlins under combined bending and shear. The simulations were compared
and calibrated against tests. The use of FE program ABAQUS for simulating was generally in good agreement
with the experimental values and allowed extension of data. The FE models were then used to extend test
data by varying lap lengths and also section thicknesses. Four commercially available thicknesses of Z purlin
sections were selected for numerical simulation. The results of ABAQUS models were plotted as interaction
diagrams where Ms and Vv are determined by different methods. The nominal section moment capacity, Ms, is
based on either Msl or Msd of the Direct Strength Method (DSM). The shear capacity, Vv, is based on the new
DSM proposal for shear with two different values of shear buckling coefficient kv.
The DSM rules for combined bending and shear are shown not to be applicable and unconservative for cases
of lapped Z purlin without straps. The severe cross-section distortion at the end of the lap as shown in tests
and ABAQUS models has led to significant strength reduction due to the discontinuity of the lapped purlins
over the support. The failure mode is mainly due to bending problem. Therefore, a simple approach based on
factored buckling moment (Msl or Msd) is proposed. For cases of lapped Z purlin with straps and Ms=Msl, with
the use of shear buckling coefficient kv=6.7 for the whole Z-section to determine Vv, the newly proposed Eq.
13 linear interaction can be used. Alternatively, when the shear buckling coefficient kv=5.34 for the web only
from classical solution is utilized without using SFSM software, the proposed Eq. 15 linear interaction can be
used. For cases of lapped Z purlin with straps and Ms=Msd, The interaction is conservative and not significant.
Therefore, there is no need to check for combined bending and shear in both cases of kv=6.7 and kv=5.34.
ACKNOWLEDGEMENT
The authors are grateful to Professor Gregory J. Hancock for his important guidance of this project. His
encouragement, discussion and invaluable advice are much appreciated. Funding provided by the Australian
Research Council Discovery Project Grant DP110103948 has been used to perform this project.
School of Civil Engineering
The University of Sydney
Research Report R938
Page 19
Experimental and Numerical Investigations of High Strength Cold-Formed Lapped Z Purlins
under Combined Bending and Shear
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School of Civil Engineering
The University of Sydney
Research Report R938
Page 20