Previous|Next|ContentsESDEP WG 11CONNECTION DESIGN: STATIC
LOADINGLecture 11.4.4: Analysis of Connections:Resistance to Moment
by Combined Tension and CompressionOBJECTIVE/SCOPETo review the
behaviour and the basis for design of local elements in
connections.PREREQUISITESLecture 1B.5: Introduction to Design of
Industrial BuildingsLecture 1B.7: Introduction to Design of
Multi-Storey BuildingsLecture 2.3: Engineering Properties of
MetalsLecture 2.4: Steel Grades and QualitiesLecture 11.1.2:
Introduction to Connection DesignLectures 11.2: Welded
ConnectionsLectures 11.3: Bolted ConnectionsLecture 11.4.1:
Analysis of Connections: Basic Determination of ForcesLecture
11.4.2: Analysis of Connections: Distribution of Forces in Groups
of Bolts and WeldsLecture 11.4.3: Analysis of Connections: Transfer
of Direct Tension or Compression and ShearRELATED LECTURESLecture
11.5: Simple Connections for BuildingsLecture 11.6: Moment
Connections for Continuous FramingLecture 11.7: Partial Strength
Connections for Semi-Continuous FramingSUMMARYThis group of 4
lectures (11.4.1 - 11.4.4) explains how the behaviour of local
elements in connections may be analysed so that each component may
safely be proportioned to resist the loads it is required to
transfer. It therefore develops the basic concepts of force
transfer that were presented in general terms inLecture 11.1.2.This
final lecture in concerned with the transfer of moments -
particularly at beam to column connections. The moment is broken
down into a localised tensile and a localised compressive force
acting at a suitable lever arm so as to produce the couple that
equates to the design moment. The need to consider also shear on
the column zone immediately adjacent to the connection in the case
of loading by unbalanced moments is also addressed.NOTATIONThe
notation of Eurocode 3 [1] has been adopted.1. INTRODUCTIONThe
transfer of moment through a connection may best be appreciated by
breaking the moment down into a pair of localised tensile and
compressive forces acting at a suitable lever arm so as to produce
a couple, see for example Figs. 2 and 3 ofLecture 11.4.1. The
simplest form for this in a beam to column connection would be one
in which only the beam's flanges are attached to the column so that
one flange transmits tension, the other transmits compression and
the lever arm is clearly the distance between flange centroids.
(Any co-existing shear could, of course, be transferred through a
web cleat or finplate arrangement of the sort illustrated in Fig.
17 ofLecture 11.4.3). The introduction of these localised forces
into the column requires a careful consideration of the possible
forms of failure and this topic is addressed in the first part of
this lecture.When the column is subject to unbalanced moment e.g.
because a beam is present only on one side, then the moment(s) also
produce a shearing effect on the panel of the column corresponding
to the depth of the beam(s), see for example Fig. 3 ofLecture
11.4.1. This panel zone effect is considered in the second part of
the lecture.2. TRANSFER OF TENSILE FORCES2.1 CriteriaFigure 1 shows
various forms of unstiffened beam-to-column connections intended to
transfer moments. In the tension zone, marked with a circle, the
tensile force must be transferred from the flange of the beam to
the web of the column.
The size of the tensile force which can be transferred without
stiffening the column depends on the resistance at this zone of the
connection to a series of possible forms at failure. These
possibilities are indicated in Figure 2 for welded connections and
for bolted connections.
Design consists of recognising that 'the strength of the chain
is determined by the weakest link'. For each of the potential
failure possibilities, the design resistance T is calculated. Rules
for these calculations are given in Annex J of Eurocode 3 [1]. The
smallest of the calculated values for T controls.The presentation
in Annex J of Eurocode 3 is based on beam-to-column connections.
However, several calculation rules are also applicable for other
types of connections.The calculation of the design resistance of
the individual fasteners, welds and bolts, discussed inLectures
11.2and 11.3. Individual fasteners are covered by criteria 3 and 4
in Figure 2. The other criteria are discussed below. Possibilities
for strengthening the connection and the design of such
strengthened connections are also described.2.2 Plastic Failure of
the Column FlangeA. Welded connectionFigure 3 shows a design model
for a welded connection to a column flange.
A part of the tensile force is transferred by direct normal
stresses without bending the flange. The width of this part is
twc+2rc. The tensile force through this part is:Ftl= fyb. tfb(twc+
2 rc) (1-1)The remaining part of the tensile force must be
transferred via bending of the column flange to the web of the
column.For this load case a design model can be adopted, as
indicated in Figure 3, based on a plate clamped at three edges with
a line load in the middle.Application of yield line theory gives a
failure load that is proportional to the plastic moment resistance
of the plate mp.Ft2 = 2 C mp (1-2)mp = fyctfc2 (1-3)Ft2 = 2
Cfyctfc2 (1-4)The coefficient C is derived from tests [2]. A safe
lower bound appears to be:C = 14.Therefore, the total design
resistance of the unstiffened column flange follows from Equations
(1-1) and (1-4):Ft = fybtfb(twc+ 2 rc) + 7 fyctfc2 (1-5)or:Ft =
fybtfbbeff (1-6)with:beff= twc+ 2 rc+ 7 (fyc/fyb)(tfc2/tfb)
(1-7)This equation for beffis also discussed inLecture
11.2.3.Because of the limitations of the tests [3] which have been
carried out to determine C, the effective width for the
determination of Ft2is limited to 7 tfc:Ftfybtfb(twc+ 2 rc+ 7 tfc)
(1-8)In order to provide sufficient deformation capacity, it is
necessary that the flange of the beam yields before rupture of the
weld or rupture of the flange of the column occurs. To obtain this
behaviour the design resistance of the unstiffened column flange
must be at least 70% of the yield force of the beam flange:Ft0,7
Fy.bf= 0,7 fybtfbbfb (1-9)If this requirement is not fulfilled,
then the connection must be strengthened by welded stiffening
plates as indicated in Figure 4.
The application of short stiffening plates offers advantages
during fabrication, because their dimensions do not need to
correspond to the actual distance between the column flanges. The
thickness and the steel grade of the stiffening plates are normally
chosen to be equal to those of the beam flange.B. Bolted
connectionIn contrast to the situation for a welded connection,
when using a bolted arrangement the total tensile force to be
transferred via the column flange causes only bending moments (no
direct load transfer), see Figure 5.
To gain insight into the various forces present in this type of
connection, it is useful to consider first a more simple case. Two
T-stubs are chosen, connected by two bolts and loaded by a tensile
force Ft, see Figure 6.
Initially, it is assumed that the force in each bolt is 0,5
Ftand that the flange is designed to transfer these bolt forces via
bending, see Figure 7. The necessary thickness tffollows from:
FB = 0,5 Ft (1-10)0,5 Ft. m = Mpl (1-11)Ft = (1-12)ft = (1-13)Ft
= (1-14)tf = (1-15)In the above Equations, the dimensions of the
bolts and the flange are such that the tensile resistance of the
bolts governs the strength of the connection. At the onset of
failure, the flanges separate from each other over the entire
area.If the bolts are chosen to be stronger, then the ultimate
tensile force increases above the value given in Equation (1-14).
With stronger bolts the flanges of the T-stubs yield, while the
bolt deformation is reduced, see Figure 8.
Now at the onset of failure the flanges do not separate over the
entire area, but contact forces develop at the edges. These contact
forces are called prying forces. These prying forces produce an
extra bending moment in the flanges. When the prying forces are
sufficiently large, this bending moment is equal to the plastic
moment mpl. In this situation four yield lines are present.The
following analysis can now be carried out:0,5 Ft. m = 2 Mpl
(1-16)Ft = (1-17)Ft = (1-18)Ft = (1-19)tf=(Ftm/lfy)(1-20)Q =
(1-21)Q = (1-22)Ft+ 2 Q =FB=Ft.u (1-23)Ft+ 0,5Ft=Ft.u (1-24)with:=
(1-25)it follows:Ft = (1-26)Between the two extremes (Figures 7 and
8), there is an intermediate case where prying forces are present,
but where the bolts rupture before the mechanism in the flanges
with four yield lines has fully developed.With the aid of the above
Equations, a diagram can be drawn showing the relation between
plate strength and bolt strength and the accompanying failure
modes, see Figure 9. In Figure 9,andhave the following meaning:
= but 1,25 (1-27)= (1-28)It is recommended the connection is
designed such that failure mode (1) just controls,= 2/(1 + 2),
because then the deformation capacity is provided in the best way
and the bolt strength is consistent with the flange strength (m).
In failure mode (3) the deformation arises mainly from the bolt
elongations. These deformations are small compared with the plastic
deformations of the flanges in failure mode (1). The deformation
capacity of failure modes (2) and (3) can be increased by selecting
bolts with threads over the entire length of the bolt.According to
Annex H of Eurocode 3 [1], the tension zone of an unstiffened
column flange should be assumed to act as a series of equivalent
T-stubs with a total length equal to the total effective
lengthleffof the bolt pattern in the tension zone of the
connection.Using yield line theory, the effective length
leffappropriate for each of those T-stubs may be calculated, see
Figures 10 and 11.
If the distance between the bolt rows is large, then a separate
yield line pattern around every bolt is formed, see Figure 10.The
circle pattern governs if e is large with respect to m, i.e.if e
> 1,8 m (1-29)The values for leffare: For one bolt row:leff = 4
m + 1,25 e (1-30)leff= 2m (1-31) For a combined T-stub:leff = 0,5 p
+ 2 m + 0,625 e (1-32)leff= 0,5 p +m (1-33)In contrast to welded
connections, it is possible when using bolted connections to
strengthen the connection with stiffening plates such that the
force can be transferred from the beam flange into the column web
without bending of the former, Figure 12.
If stiffening plates are used, the stiffness and strength of the
column flange is increased. The increase is beneficial for the
design strength of the bolt rows near such stiffening plates.The
strength of such bolt rows can be calculated by introducing a
T-stub with an equivalent length leff.According to Annex J of
Eurocode 3, the value of leffequals:leff = m1 (1-34)Values forare
given in Figure 13. This diagram is established on the basis of
yield line theory and test results [4]. The value ofdepends on the
geometry near the stiffening plate.
In the equation for1and2(the values on the horizontal and the
vertical axes in Figure 13), m1is the distance between the bolt and
the column web, and m2is the distance between the bolt and the
stiffening plate.Alternatively the column flange may be
strengthened by using loose backing plates as indicated in Figure
14.
The length of the backing plates should be at least the length
of lefffor the bolt pattern considered.The backing plates increase
the plastic moment on the yield line through the bolts, but not the
plastic moment at the junction of the flange with the web, see
Figures 7 and 8.Clearly backing plates are only effective if
failure mode (1) in Figure 9 is decisive, see also Figure
8.Equation (1-17) can be modified to account for the extra yield
line:Ft = (1-35)Ft = (1-36)Ft = (1-37)It should be noted that,
because of the appearance of only one yield line, a factor 2 must
be used for 2 Mp.bpin Equation (1-35).From the above Equations, it
may be concluded that, if tf= tbp, the use of backing plates gives
an increase of 50% in the design resistance of the column
flange.2.3 Yield/Rupture of the Column WebThe force Ftspreads in
the column web over a length beff, see Figure 15.
It is assumed that the failure load is reached if the average
stress due to Ftover the length beffequals the yield stress:Ft=
fyctwcbeff(1-38)For a welded connection, the same expression for
beffis used in the tension zone as in the compression zone.
Although tests [5] have shown that the strength in the tension zone
is usually greater than in the compression zone, the same equation
is chosen for reasons of simplicity.According to Eurocode 3,
befffor a welded connection is given by:beff = tfb+ 2ab+ 5 (tfc+
rc) (1-39)For a bolted connection, the effective length of the
column web in the tension zone is taken equal to the total
effective length of the equivalent T-stubs (see Section 1.2).The
column web can be strengthened by stiffening plates and/or by
welded supplementary web plates, see Figure 16.
If the web plate is only single sided and is connected by butt
welds, only half of the plate thickness may be used for the
calculation of the design resistance. The reasons are the
eccentricity and the design of the welds. If fillet welds are
present, some spaces must be left between the flange and the edge
of the plate to permit a reasonable execution of the weld. For this
reason, the force must be transferred through initially the
thickness of the web. Therefore, the effective thickness of the
combined column web and supplementary web plate should not be taken
greater than 1,4 times the web thickness (1,4 twc).3. TRANSFER OF
COMPRESSION FORCESThe action of a compression force on an
unstiffened column may cause local buckling of the column web.
Using tests on special test specimens [5], as shown in Figure 17,
and on complete beam-to-column connections, research has been
carried out to establish a design model. The derived model is that
the column web starts buckling if the average stress over a certain
effective length equals the yield stress. This model, therefore, is
basically the same as the model for the transfer of a tensile force
in the web, as discussed above.
Fc = fyctwcbeff (2-1)In Figure 18 the equations for beffare
given which apply for various designs of the compression zone. The
basic assumption in these equations is that the spread in the
column flange and the column radius have a slope 2,5 : 1, and the
slope in other parts of the connection is 1 : 1.
In the model of Figure 18b and Figure 19 with the extended end
plate, it is assumed that the end plate under the beam flange
yields before the buckling load in the column web is reached. In
this case, the compression force Fcspreads over the thickness of
the end plate.
The projection of the end plate should obey the following
condition:fyWpl.endplateFcle (2-2)fybptp2Fcle (2-3)le (2-4)If the
condition of Equation (2-4) is not fulfilled, then a hard point is
formed at the edge of the end plate. The effective length beffis
then smaller:beff = 5 (tc+ rc) (2-5)The moment arm, however,
between the tensile force in the tension zone and the compression
force is somewhat greater. This is an advantage.In the case of
large axial forces in the column, the local buckling load in the
compression zone is reduced. As long as the axial stressnis smaller
than 0,5 fy, the influence can be ignored. For greater values ofn,
the design resistance for the compression force Fcshould be
calculated with the following equation:Fc= fyctwcbeff[1,25 - 0,5n/
fyc] (2-6)In this equation the second part is the reduction
factor:R = 1,25 - 0,5 [n/ fyc] but R1,0 (2-7)The column can also be
strengthened in the compression zone with stiffening plates between
the flanges or with a supplementary web plate.4. TRANSFER OF SHEAR
FORCES (SHEAR ZONE)In non-symmetric connections, such as T- and
corner connections, the column web is also loaded by a shear force
Fv. Loading by a shear force also occurs in symmetric connections
that are loaded asymmetrically.For instance in the T-connection of
Figure 20, the tensile force in the upper flange of the beam must
be transferred through the shear panel to be in equilibrium with
the compression force in the lower flange of the beam.
Assuming that the web is sufficiently stocky for shear buckling
not to occur, the design resistance of the shear panel is:Fv =
hctwc (3-1)The column web can be strengthened with diagonal plates
or with one or two supplementary web plates, see Figure 21. When
diagonal plates are designed, care should be taken to avoid
problems with the installation of the bolts; see for example Figure
21b.
The action of stiffeners sometimes can be better understood when
thinking in terms of tension and compression; see for example the
arrangement in Figure 22.
5. CONCLUDING SUMMARY In order to transfer moment a connection
must be capable of resisting local tensile and compressive forces.
Design must address each item in the "chain" of components involved
in transferring the loads, with connection resistance being
controlled by the weakest link in the chain. For bolted connections
a T-stub model provides a satisfactory explanation of all important
aspects of behaviour; yield line theory supported by test evidence
forms the basis for detailed rules. For semi-continuous framing, an
acceptable balance between tensile resistance and adequate
ductility may be achieved by arranging for mode 1 failure to
control; this corresponds to yielding of the plates without the
development of excessive bolt prying forces. In other moment
resisting frames, it is usually more economic to use connections
that are governed by Mode 2 or Mode 3, i.e. with thicker
end-plates. For unsymmetrical beam-to-column connection
arrangements, including unbalanced loading of symmetrical
connections, the resistance of the column web panel in shear should
be checked.6. REFERENCES[1] Eurocode 3: Design of Steel Structures:
European Prestandard 1993-1-1: Part 1.1: General rules and rules
for buildings, CEN, 1992.[2] Wood, R. H., "Yield Line Theory",
Research Paper nr. 22, Building Research Station, Watford, England,
1955.[3] Zoetemeijer, P., Summary of the research on bolted
beam-to-column connections, Delft University of Technology, Faculty
of Civil Engineering, Stevin Laboratory report 6-90-02, 1990. This
report is also published as a background report for Eurocode 3,
Chapter 6.[4] WRC and ASCE, "Commentary on Plastic Design in
Steel", Progress Report 6: Connections, Journal Eng. Mech, Div.,
ASCE, 86, EM2, April 1960, pp 107-140.[5] Graham, J. D.,
Sherbourne, A. N., Khabbaz, R. N., and Jensen, C. D., Welded
Interior Beam-to-Column Connections, Welding Research Council
Bulletin nr 63, August 1960.7. ADDITIONAL READING1. Owens, G. W.
and Cheal, B. D., "Structural Steelwork Connections", Butterworths
& Co. (Publishers) Limited, 1989.2. Kulak, G. L., Fisher, J. W.
and Struik, J. H. A., "Guide to Design Criteria for Bolted and
Riveted Joints", Willey - Interscience, 2nd Edition, 1987.3.
Ballio, G. and Mazzolani, F. M., "Theory and Design of Steel
Structures", Chapman & Hall 1983.4. W. F. Chen "Joint
Flexibility in Steel Frames", Journal of Constructional Steel
Research Vol 8,
1987.Previous|Next|Contentshttp://www.fgg.uni-lj.si/~/pmoze/ESDEP/master/wg11/l0440.htm