ArticlePDF Available

Design of beam-column splice connections according to Eurocode 3

Authors:
  • Pell Frischmann

Abstract and Figures

Ricardo Pimentel of the SCI discusses the design of beam-column splice connections considering second-order effects due to combined flexural and lateral torsional buckling according to Eurocode 3. https://www.newsteelconstruction.com/wp/wp-content/uploads/2020/10/NSC_Oct20-tech.pdf
Content may be subject to copyright.
37
NSC
October 20
Ricardo Pimentel of the SCI discusses the design of beam-column splice connections
considering second-order effects due to combined flexural and lateral torsional buckling
according to Eurocode 3.
Introduction
Buckling phenomena cause additional internal forces within members due to
local second order eects (P-δ). Recent NSC articles [1], [2], [3] introduced these
eects, giving theoretical background and practical applications. Reference
[3] provides a detailed worked example of the assessment of the second order
bending moment on columns due to strut action for column splices designed
under pure compression. Members subjected to major axis bending that are
susceptible to lateral torsional buckling are also subjected to second order
eects, because the major axis bending induces a horizontal deection (minor
axis - δh ), vertical deection (major axis - δv ) and a cross-sectional rotation (θ) as
illustrated in Figure 1. Such deformations will increase as the applied bending
moment increases. When the bending moment is close to the so-called elastic
critical moment, the deformation increases rapidly and failure occurs.
Addressing second order eects
Whilst for a strut an equivalent initial bow imperfection can be back-calculated
relatively easily and amplied to account for the second order eect, the
problem for lateral torsional buckling phenomena oers a much more complex
challenge. Although the eects of the vertical displacement and rotation
have an impact on the lateral torsional buckling resistance of the member,
the consideration of an equivalent horizontal out of plane bow imperfection
oers a good approximation to establish the initial member imperfection. EN
1993-1-1 clause 5.3.4 (3) supports this approach. A precise analysis including
the amplication of the initial member imperfection is complex and usually
undertaken by numerical analysis with advanced nite element model
tools. A reasonable approximation can be achieved by manual methods, as
demonstrated in this article. The process described is useful when designing
splice connections in unrestrained beams.
Lateral torsional buckling failure criteria
The design buckling resistances for buckling phenomena according to
Eurocode philosophy are calibrated based on an elastic cross section failure,
where all imperfections (such as residual stresses, lack of straightness, etc.)
are accounted for by an equivalent imperfection factor α. Second order local
eects are implicitly considered by the Eurocode design method (section 6.3).
Reference [1] explains this concept for a strut. Using the same principles for
an element subjected to lateral torsional buckling, the buckling failure can be
understood as a critical stress, for which two components can be identied: (i)
component due to major axis bending (σMy ); (ii) component due to the second
order bending moment under minor axis bending (σMz,Pδ,LTB ).
Out of plane bending moment due to lateral torsional buckling
If the buckling failure is considered as an elastic cross section failure (with a
material yield strength of ƒy ), the following condition can be established:
ƒy = σMy + σMz,Pδ,LTB
According to Eurocode nomenclature, the buckling resistance can be
established as the product of the reduction factor for buckling phenomenon
χ multiplied by the design characteristic resistance. As the characteristic
resistance is directly proportional to the material resistance, the stress at lateral
torsional buckling failure can be established as χLT  ƒy (described as the critical
buckling stress). The stress σMz,Pδ,LTB can be dened based on cross section
properties and the second order bending moment Mz,Pδ,LTB , which leads to:
ƒy = χLT · ƒy +
M
z,Pδ,LTB
Wel,z
Dividing the previous equation by the critical buckling stress, it can be
demonstrated that:
Mz,Pδ,LTB =
ƒ
y
χLT · ƒy
=
χ
LT
χLT · ƒy
· ƒy +
M
z,Pδ,LTB
χLT · ƒy · Wel,z
1
χLT - 1 · χLT · Mz,el,Rk
()
Where Mz,el,Rk is the out of place elastic bending resistance of the cross section.
According to the Eurocode denition, χLT is the ratio between the buckling
bending resistance and the characteristic bending resistance of the cross
section. As the buckling bending resistance (Mb,Rd) should be always less
than the applied bending moment (My,Ed), it can be approximately (and
conservatively) assumed that:
M
b,Rd
· γ
M1
M
y,el,Rk
χLT =
χLT =
M
y,Ed
· γM1
M
y,el,Rk
Where γM1 is that partial factor for buckling phenomenon according to the UK
NA to BS EN 1993-1-1 [4].
This leads to:
Mz,Pδ,LTB =
M
z,el,Rk
My,el,Rk
1
χLT
- 1 ·
()
· My,Ed · γM1
Eq (1)
The complexity of the procedure is related to the calculation of χLT. For cases
where section 6.3.2.3 (2) of EN 1993-1-1 is applied, Mz,Pδ,LTB should be multiplied
by “ƒ”.
Splices of elements under compression
Splices subjected to axial compression should be designed for the following
forces:
1. NEd – Applied axial force;
2. Mi,Pδ,FB – Second order bending moment due to strut action (exural
buckling) about the axis “i”.
It should be clear that a member only experiences exural buckling under
one of its axes. The design bending moments Mi,Pδ,FB should be only considered
about the weak axis for exural buckling (i.e. the axis which shows the higher
slenderness – reected in a higher value of λ - according to EN 1993-1-1 section
6.3.1.2).
The second order bending moment due to strut action can be calculated as
follows:
Mi,Pδ,FB = NEdePδ,i = NEd e0,ikamp,iγM1 Eq. (2)
Where:
NEd is the applied axial load;
e0,i is the initial bow imperfection about axis “i” equal to α (λi
- 0.20)
W
el,i
A
;
Design of beam-column splice
connections according to Eurocode 3
38
Technical
Figure 1: Lateral torsional buckling mode shape
38
NSC
October 20
Technical
Figure 3: Tee dimensions RAINHAM
STEEL
Nationwide delivery of all Structural Steel Sections
Phone: 01708 522311 Fax: 01708 559024
MULTI PRODUCTS ARRIVE ON ONE VEHICLE
GRADES S355JR/J0/J2
Head Office: 01708 522311 Fax: 01708 559024 Bury Office: 01617 962889 Fax: 01617 962921
email: sales@rainhamsteel.co.uk www.rainhamsteel.co.uk
Beams • Columns
Channel • Angle
Flats • Uni Flats
Saw Cutting
Shot Blasting
Painting • Drilling
Hot & Cold Structural
Hollow Sections
Full range of advanced steel sections available ex-stock
ePδ,i is the bow imperfection accounting for the second order eects;
kamp,i is the amplication factor equal to
N
cr,i
N
cr,i
- N
Ed
;
Wel,i is the elastic modulus of the cross section about axis “i”;
A is the cross-section area;
α is the equivalent imperfection factor according to EN 1993-1-1 section
6.3.1.2;
λi is the non-dimensional slenderness according to EN 1993-1-1 section
6.3.1.2 about axis “i”;
Ncr,i is the elastic critical buckling load for exural buckling under the
axis “i”:
, where Ii is the second moment of area about axis “i” and
Lcrit,i is the buckling length about axis “i”;
NEd is the applied axial load on the column.
Splices of elements under bending
Splices within unrestrained segments subjected to major axis bending should
be designed for the following forces:
1. MEd,y – Applied bending moment under the major axis;
2. MEd,z – Applied bending moment under the minor axis;
3 Mz,Pδ,LTB – Second order bending moment due to lateral torsional buckling.
Beam-column splices
Beam-column splices can be exposed to the following design forces:
1. NEd – Applied axial force;
2. MEd,y – Applied bending moment about the major axis;
3. MEd,z – Applied bending moment about the minor axis;
4. Mi,Pδ,FB – Second order bending moment due to strut action (exural
buckling) about the axis “i”;
5. Mz,Pδ,LTB – Second order bending moment due to lateral torsional buckling;
6. Mi,Pδ,Amp – Moments due to the amplication of the applied bending
moments due to the strut action about the axis “i”.
As for elements under compression, the design bending moments Mi,Pδ,FB
should be only considered about one of the cross-sectional axes for exural
buckling. Beam-columns experience an additional bending moment Mi,Pδ,Amp
which is related to the amplication of the applied bending moments due
to the presence of axial load. The second order bending moments due to the
presence of axial force can be calculated considering the amplication factor
about the axis “i” as follows:
M
i,Pδ,Amp
= M
Ed,i
· N
cr,i
N
cr,i
- N
Ed
[]
- 1
Eq. (3)
The minor axis bending moment Mz,Pδ,Amp should be always considered. The
eects of My,Pδ,Amp and Mz,Pδ,Amp should not be considered together: designers
should consider two independent combination of action where My,Pδ,Amp or
Mz,Pδ,Amp are considered. This is because the second order eects will only
develop about one of the member axes, i.e. either LTB will govern and the beam
will deform sideways, or a major axis second order bending moment will be
generated.
The procedure described above comprises segments under a uniform
bending moment prole along the segment. To assess other bending moment
proles, designers may consider the value of Cm,i from EN 1993-1-1 Table B.3. For
such cases, the values of Mi,Pδ,Amp obtained from equation 3 may be multiplied by
the values of Cm,i .
As a summary, the design forces for a beam-column splice can be established
by the following equations:
NEd,splice = NEd Eq. (4)
MEd,y,splice = MEd,y + [My,Pδ,FB] + {My,Pδ,Amp } Eq. (5)
MEd,z,splice = MEd,z + [Mz,Pδ,FB] + {Mz,Pδ,LTB } + Mz,Pδ,Amp Eq. (6)
Pairs of eects within the square and round brackets should not be
considered simultaneously. Designers should consider them individually and
assess which combination of forces gives the most onerous design condition.
Second order bending moment distribution along an unrestrained
segment
The bending moment diagrams calculated according to equations 1, 2 and
3 represent a maximum value at mid span of an unrestrained segment. The
second order bending moments follow a sinusoidal shape between points
of inexion (points between which the eective length is measured) of:
Mi,Pδ (x) = Mi,Pδ,max  sin(πx
l
), where “x” is the position from a point of inexion
and “
l
” is the length between points of inexion (for a pinned column, this is the
column length).
Comparison with BS 5950 approach
Previous UK practice design addressed second order eects for columns, beams
and beam-column splices according to BS 5950 [6]. Further guidance was given
by SCI AD notes 243 [7] and AD 244 [8].
The second order out of plane bending moment is addressed by BS 5950
Annex B.3. While BS 5950 established the second order bending moment
based on a relationship between yield strength and bending strength for
lateral torsional buckling, the Eurocode nomenclature establishes it based on
the parameter χLT . The parameter χLT can also be understood as a relationship
between the allowable buckling stress and the yield strength. Therefore,
1 / χLT represents the same relationship as proposed by BS 5950. The factor
mLT , which considers the bending moment diagram shape along the
segment, is accounted for while calculating χLT according to EN 1993-1-1 6.3.2
(within the elastic critical bending moment - Mcr ). Both BS 5950 and Eurocode
3 approach have the same background.
Strut action is dened by Annex C.3 of BS 5950. Both BS 5950 and
EN 1993-1-1 approaches to address exural buckling are based on an elastic
cross section failure due to the combined stresses of axial load and second
37
39
NSC
October 20
RAINHAM
STEEL
Nationwide delivery of all Structural Steel Sections
Phone: 01708 522311 Fax: 01708 559024
MULTI PRODUCTS ARRIVE ON ONE VEHICLE
GRADES S355JR/J0/J2
Head Office: 01708 522311 Fax: 01708 559024 Bury Office: 01617 962889 Fax: 01617 962921
email: sales@rainhamsteel.co.uk www.rainhamsteel.co.uk
Beams • Columns
Channel • Angle
Flats • Uni Flats
Saw Cutting
Shot Blasting
Painting • Drilling
Hot & Cold Structural
Hollow Sections
Full range of advanced steel sections available ex-stock
Technical
Design forces and bending moments for splice design
order bending moments due to the strut action. If the same buckling
resistances are assumed, and considering the elastic section modulus, the
simplied method from Annex C.3 of BS 5950 tends to give conservative
values in comparison with equation 2. A similar answer for the strut moment is
obtained if the applied load is close to the buckling resistance.
Second order eects for members subjected to combined axial load and
bending are dened by Annex I.5 of BS 5950. The expression 1/(pEi ⁄ ƒc -1) gives
the same answer as [(Ncr,i ⁄ (Ncr,i - NEd ) -1] if the same buckling resistances are
assumed. The values of my and mx according to BS 5950-1 Annex I.5 (which
should be dened according to BS 5950-1 4.8.3.3.4) are similar to the values
dened by EN 1993-1-1 Table B.3.
Calculation example
Consider a UB 533 × 165 × 66 beam-column element with an unrestrained
segment of 5 m length subjected to an axial load of 150 kN and a linear bending
moment diagram between 165 kNm and 82.5 kNm. A splice connection is
located at 1/3 (1.67m) of the unrestrained segment length, closer to the point
of maximum bending moment. The bending moment at the splice location
is therefore 137.5 kNm. The calculation of the second order design forces
to design the splice connection is summarized in the table below. Member
resistances are taken from the Blue Book.
Conclusions
1. Lateral torsional buckling failure can be considered by means of an
equivalent initial horizontal bow imperfection under the minor axis of the
prole, which must then be amplied;
2. Considering the member lateral torsional buckling capacity, it is possible to
estimate the cross-section forces at failure;
3. The failure criteria for lateral torsional buckling is assumed to be elastic failure
of the cross section considering major axis bending and the second order
bending moment due to lateral torsional buckling; strut action eects also
need to be accounted for in beam-columns;
4. EN 1993-1-1 approaches for beam and beam-column splices follow the same
principles as BS 5950.
References
1 Pimentel, R., Stability and second order of steel structures: Part 1: fundamental
behaviour; New Steel Construction; vol 27 No 3 March 2019;
2 Pimentel, R., Stability and second order of steel structures: Part 2: design
according to Eurocode 3; New Steel Construction; vol 27 No 4 April 2019;
3 Eurocode 3 - Design of steel structures - Part 1-1: General rules and rules for
buildings; BSI, 2014;
4 NA BS EN 1993-1-1+A1 UK National Annex to Eurocode 3 - Eurocode 3 - Design
of steel structures - Part 1-1: General rules and rules for buildings; BSI, 2014;
5 Henderson, R., Bearing splice in a column; New Steel Construction; vol 28 No 3
March 2020;
6 BS 5950, Structural use of steelwork in building: Part 1: Code of practice for
design - Rolled and welded sections, BSI, 2000;
7 SCI Advisory Desk Notes: AD 243: Splices within unrestrained lengths;
8 SCI Advisory Desk Notes: AD 244: Second order moments
Section properties and resistances, critical
loads (S355); UB 533 × 165 × 66 Eurocode buckling
Resistances EN 1993-1-1
P-δ eects Critical design eects for splice design
A = 83.7 cm Nb,rd,y = 2890 kN kamp,y = 1.005 NEd,splice = NEd = 150 kN
Wel,y = 1340 cm Nb,rd,z = 598 kN kamp,z = 1.267 MEd,y,splice = MEd,y = 137.5 kNm
Wel,z = 104 cm Mb,rd = 225 kNm λz= 2.04 MEd,z,splice = Mz,Pδ,FB + Mz,Pδ,LTB
MEd,z,splice = 1.3 + 16.2 = ±17.5 kNm
Wpl,y = 1560 cm Note: C1 ≈1.35 e0,z = 7.8 mm (α = 0.34) The set of design actions presented above
give the most onerous design scenario
according to equations 5 and 6.
Wpl,z = 166 cm ePδ,z = 9.9 mm
Iy = 35000 cm Mz,Pδ,FB,max = 1.5 kNm
Iy = 859 cm Mz,Pδ,FB (@ 1.67 m) = 1.3 kNm
My,pl,Rd = 554 kNm Mz,Pδ,LTB,max = 18.7 kNm Note: the value of χLT was calculated as
Mb,rd / My,rd = 225/554 = 0.41
Mz,pl,Rd = 59 kNm Mz,Pδ,LTB (@ 1.67 m) = 16.2 kNm
My,el,Rd = 474 kNm My,Pδ,Amp,max = 0.86 kNm
Mz,el,Rd = 36.9 kNm My,Pδ,Amp (@ 1.67 m) = 0.74 kNm
Ncr,y = 29017 kN (Cm,y is assumed as 1 considering the low
value of My,Pδ,Amp,max )
Ncr,z = 712 kN
ResearchGate has not been able to resolve any citations for this publication.
ResearchGate has not been able to resolve any references for this publication.