ESDEP WG 11CONNECTION DESIGN: STATIC LOADINGLectures 11.4.1:
Analysis of Connections I:Basic Determination of
ForcesOBJECTIVE/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 ConnectionsRELATED
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.In
this first lecture the general principles used in determining the
forces for which each component in a connection must be designed
are explained. These make use of the fundamental structural
concepts of equilibrium, stiffness and deformations to decide how
the externally developed loads are shared between the various
components. This then leads to the idea of load paths as being the
most effective way that applied forces can pass through the
connection.NOTATIONThe notation of Eurocode 3 [1] has been
adopted.1. INTRODUCTIONInLecture 11.1.2it was shown in general
terms how the structural adequacy of connections can be checked by
considering the resistance of the local elements of the
connection.The resistance of a local element is determined on the
basis of the resistance of the individual bolts or welds and
plates.The resistance of welds and bolts is covered inLectures
11.2and11.3. In this Lecture 11.4.1 the resistance, stiffness and
deformation capacity of a number of components are discussed.Thus
the material ofLectures 11.1,11.2and11.3is brought together to
explain how the individual components in connections can be safely
proportioned. This involves both a determination of the forces to
which each is subjected and the ways in which, acting in
combination, the parts of the connection transfer these forces from
the supported member to the supporting member.Having established
the principles,Lectures 11.4.2- 11.4.4 apply these to the
consideration of the transfer of different types of internal forces
within connections e.g. direct tension, shear, tension as part of a
moment etc., whilstLectures 11.5- 11.8 fully develop the ideas to
cover the design of particular connection types.2. DETERMINATION OF
FORCES2.1 Forces on the ConnectionFor the determination of the
forces on the connection, a static analysis is carried out. Such an
analysis includes the determination of the design loads on the
structure and the definition of the design basis for the
structure.In defining the design basis, consideration of the
structural behaviour of the connections is necessary. Are the
connections pinned, or rigid, or semi-rigid? Are they partial
strength or full strength connections? More details about the
influence of the type of connection on the distribution of forces
in the structure are given inLectures
11.1,11.5,11.6,11.7and11.8.2.2 Force Distribution in the
ConnectionAfter the determination of the normal forces, shear
forces and bending moments on the connections, an internal
distribution of forces in the connection is chosen.The distribution
of forces in the connection may be determined in whatever rational
way is best, provided that:a. The assumed internal forces are
inequilibriumwith the applied forces and moments.b. Each part of
the connection is capable ofresistingthe forces assumed in the
analysis.c. The deformations imposed by the chosen distribution are
within thedeformation capacityof the fasteners, welds and other
parts of the connection.Figure 1 gives an outline of the
determination of the load on the individual elements of the
connections, and the verification of their resistance.
It is not necessary and it is often not possible to determine
therealdistribution of forces in the connection. Arealistic
assumptionof internal forces, in equilibrium with the external
forces on the connection, is sufficient. In fact selecting this
assumption is the most difficult part of the analysis. It requires
a sound understanding of the structural behaviour of the connection
when it is loaded.The following rules apply:a. The distribution of
forces in the parts to be connected requires considerationIf, for
instance, an I-section loaded in bending and shear, has to be
connected, then the shear force is largely concentrated in the web,
whilst the flanges carry most of the bending moment. A simple and
usually acceptable assumption for the load transfer in the
connection is to connect the web for the full shear force and the
flanges for the full bending moment, see Figure 1.b. The stiffness
of the various parts in the connection requires
considerationDeformations caused by loads acting in the plane of a
plate are much smaller than those produced by loads acting
perpendicular to a plate (normal force versus bending moment). In
many cases the understanding of the influence of the stiffness
ratio on the force distribution can be improved by considering the
situation after a small deformation of the connection has occurred.
This approach is illustrated in Section 3 by examples.c. The
assumed force distribution must be consistent for all parts in the
connectionViolations against this rule may occur if a separate
calculation is carried out for the different parts in the
connection. An example is given in Figure 2. The indicated
distribution of forces for the calculation of the bolts is not
consistent with the distribution of stresses in the beam assumed to
design the welds between the beam and the end plate. Overloading of
the welds in the top flange of the beam results.
2.3 Basic Load Cases for Local ElementsThe analysis of the
structural behaviour of connections can be carried out by
considering a number of basic load cases for local elements. For a
T-connection this analysis is demonstrated in Figure 3. The load
transfer in nearly every type of connection can be modelled with
the five basic load cases. Use of these cases permits a systematic
and clear presentation of the calculation methods, despite the wide
variety of possible connection types. Eurocode 3 (Chapter 6 and
Annex J) follows this approach.
InLectures 11.4.3and11.4.4calculations for the five basic load
cases are presented for a number of connection designs. For each
part of the connection, a number of possible failure modes can be
identified. They may refer to: the fasteners (welds or bolts). the
members which are connected. extra parts in the connection, e.g.
plates and angle cleats.It has to be demonstrated that the weakest
link in the connection system (chain) is strong enough to carry the
load that acts on it.The design of the fasteners (welds and bolts)
is dealt with inLectures 11.2and11.3. The design of other parts in
the connection is dealt with in the presentLectures 11.4.3.
DISTRIBUTION OF FORCES3.1 Influence of Stiffness DifferencesWhen
distributing normal forces, shear forces and bending moments in the
connection, the stiffness differences in the connection must be
taken into account. In particular, the deformations due to normal
forces in the plane of a plate are much smaller than the
deformations due to forces acting perpendicular to it.A calculation
for the example of Figure 4 gives a good demonstration of this
principle. The plate 10010010 mm, clamped on one side, is loaded
with 1000 N perpendicular to the plate surface.
The deflection follows from:=== 0,2 mm (3-1)The same plate is
loaded with an in-plane tensile force of the same magnitude. The
displacement of the end of the plate is now:l === 0,0005 mm
(3-2)Both plates are now connected, see Figure 5, causing both
displacements at the interface to be equal. A load of 1000 N is
applied to this structure. The load is carried by both plates,
shared in proportion to the stiffness ratio. The plate loaded in
tension is 0,2/0,0005 = 400 times stiffer than the plate that is
bent. Consequently, nearly the whole load is carried by the plate
loaded in tension, see Figure 5b.
This knowledge is used to determine the distribution of forces
for the brace connection shown in Figure 6, e.g. in a floor
structure. In this connection many distributions of the forces in
the connections, each obeying equilibrium, are possible.
First it is assumed that the force is carried by both
connections, whilst the direction stays the same, see Figure 7. For
the analysis, the force 0,5 F is resolved as Fs= 0,35 F and Ft=
0,35 F.
The deformation in the shear direction (Fs) is much smaller than
the deformation in the tensile direction (Ft). The result is that
the deformation1 at point (1) is very different from the
deformation2 at point (2). The deformations1 and2 cannot be
accommodated by the gusset plate!This means that the deformation at
point (1) caused by Fs(S1) must be the same as the deformation at
point (2) caused by Ft(S2).Therefore, Fsis much larger than Ft. The
distribution of forces in Figure 7 isincorrect.The correct
distribution is indicated in Figure 8. The force F effectively
causes only shear in the bolt groups (1) and (2). The tension load
in the bolts can be ignored.
Conclusion:If large differences in the stiffness between two
possible types of load transfer exist, then ignore the load
transfer that gives the larger deformations (bending deformation of
the plate), and assume that all load is transferred in the way that
gives the smaller deformations (deformation in the plane of the
plate).This approach also applies to welded structures, e.g. see
Figure 9 which illustrates the connection of a plate to a square
hollow section. The assumed force distribution where the welds are
only loaded in shear is correct.
The stiffness ratio in the connection may influence the
assumption for the calculation of the bending moments. An example
is given in Figure 10. In the connection in Figure 10a, the
rotation of the bolted connection is larger than the rotation of
the plate which is welded in the plane of the web of the column.
Therefore, the hinge for the calculation of moments is assumed to
be the bolt row. The bolts are loaded by a shear force V. The welds
must be designed for a shear force V and a bending moment V.e.
In the connection in Figure 10b, the plate is welded to the
non-rigid wall of the square hollow section. Here the more logical
place for the hinge is this wall. The weld is now only loaded in
shear and consequently, the bolt row is loaded in shear (V) and
bending (V.e).3.2 Free Centre of Rotation and Forced Centre of
Rotationa. Free centre of rotationThe plates in Figure 11 are
connected by bolts arranged in an arbitrary pattern. The connection
is loaded by a bending moment M. The plates are assumed to be
rigid, compared with the stiffness of the fasteners. Therefore, the
rotationbetween the plates is the result of the deformation of the
fasteners. The plates rotate around the centre of rotation.
In the case of small deformations of the fasteners, a linear
relation between the bolt forces Riand the displacementsimay be
assumed, giving bolt forces Fiproportional to the distance rito the
centre of rotation and the rotation, see Figure 12.
i= ri (3-3)Ri= (ri/rmax)Rmax (3-4)Rxi= (yi/ri)Ri=
(yi/rmax)Rmax(3-5)Ryi= (xi/ri)Ri= (xi/rmax)Rmax(3-6)If the load on
the connection is a pure bending moment, equilibrium requires that
the resultant forces in the x and y directions must be zero:Rxi=
Rmax/rmaxyi= 0yi= 0 (3-7)Ryi= Rmax/ rmaxxi= 0yi= 0 (3-8)The centre
of rotation is therefore located at the centroid of the bolt
group.M =ri. Ri=Rmax/ rmax) . Rmax= (Rmax/rmax)ri2 (3-9)Rmax =
(3-10)This situation with the centre of rotation at the centroid of
the bolt group is called "free centre of rotation".If an eccentric
force acts on a bolt group with free centre of rotation, the
following analysis can be carried out, see Figure 13.
The eccentric force F can be replaced by a bending moment M = F
. e and a force F through the centre of rotation. The loads on the
bolts are the summation of the loads caused by M (as explained
above) and the loads caused by F. For n bolts, each bolt carries
F/n. The resultant force on each bolt can be determined by
resolving the forces caused by M and F in the x-direction and in
the y-direction:Fx = FxM+ FxF (3-11)Fy = FyM+ FyF (3-12)R =
(3-13)For an arbitrary bolt pattern it is not easy to determine in
advance which bolt is the most heavily loaded. Several bolts have
therefore to be checked. In practice, however, the bolt pattern is
usually regular and the more severely loaded bolts are readily
identified.b. Forced centre of rotationIn an end plate connection
of the type shown in Figure 14, there is an important difference
between the stiffness of the tension zone and the compression
zone.
In the compression zone, the compression force is transmitted
directly from the flange of the beam to the web of the column. The
deformations in the compression zone are very small compared to the
deformations in the tension zone, where bending of the end plate
and bending of the column flange occurs.Because of this difference
in the stiffness, the centre of rotation is effectively located at
point (1) in Figure 14. Sometimes, to be more conservative, the
centre of rotation is taken as the lowest bolt row.If the end plate
is thick and therefore stiff, then the centre of rotation may also
be assumed at the lower end of the plate.The above situation, where
the centre of rotation is not in the centre of the bolt group, is
called a connection with a "forced centre of rotation".Assuming
that the stiffness at each bolt row is the same, the forces in the
bolt rows are directly proportional to their distance from the
centre of rotation. With the centre of rotation at point (1), the
following analysis can be carried out:h12T1+ h22T2+ h32T3+ h42T4+
h52T5+ h62T6= M (3-14)With equal bolt sizes:2T2 = 2T12T3 = 2T12T4 =
2T12T5 = 2T1From these equations, the bolt force T1in the most
heavily loaded bolt can be determined:(3-15)In reality, the
stiffness of the bolt rows may differ considerably, e.g. the
extended part of the end plate above the beam's top flange in
Figure 15 is less stiff than the part below the top flange where
the web of the beam has a stiffening effect. As a result, bolt row
number 2 will transmit a higher load than bolt row number 1.
For thin end plates the differences in the stiffness of
different bolt rows is more pronounced and the distribution of
forces in the bolt rows is more variable.With "normal" dimensions
of the end plate, it is reasonable to assume that the tension force
in the top flange of the beam is equally distributed between bolt
rows 1 and 2.If an end plate connection is loaded by a combination
of bending moment M and a tensile force FH, the situation with a
forced centre of rotation may occur, but also a free centre of
rotation is possible. This depends on the magnitude of FH, see
Figure 16.
If the centre of rotation is forced (FHis small), then FHis
transferred through the rigid point (1). The bending moment about
(1) is:M1= M + FH.=hi2(3-16)wherea is the distance between the
centre line of the beam and the compression point (1).From the
condition of horizontal equilibrium it follows that:D = hi- FH
(3-17)If:FH = hi (3-18)then D = 0. With D < 0, there is no
longer a forced centre of rotation. From Equations (3-16) and
(3-17) it follows that if:+ (3-19)there is a forced centre of
rotation, and if:+ (3-20)there is a free centre of rotation.
4. CONCLUDING SUMMARY All connection designs must satisfy three
fundamental requirements:i. Internal forces must be in equilibrium
with the external applied forces and moments.ii. Each part of the
connection must be capable of safely resisting the forces in it
assumed by the analysis.iii. The deformations required by the
assumed internal force distribution must be within the deformation
capabilities of the component parts. Using 5 basic load cases the
force transfers present in virtually every form of connection may
be obtained by suitable combination. Load transfer follows routes
in which the majority of load follows the stiffer paths. Moment
transfer by means of a group of fasteners may involve either a
"free centre of rotation" or a "forced centre of rotation".5.
REFERENCES[1] Eurocode 3: "Design of Steel Structures" ENV
1993-1-1: Part 1.1, General rules and rules for buildings, CEN,
1992.6. 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 Volume 8,
1987.Previous|Next|Contents
http://www.fgg.uni-lj.si/~/pmoze/ESDEP/master/wg11/l0410.htm