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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 eects (P-δ). Recent NSC articles [1], [2], [3] introduced these

eects, 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

eects, because the major axis bending induces a horizontal deection (minor

axis - δh ), vertical deection (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 eects

Whilst for a strut an equivalent initial bow imperfection can be back-calculated

relatively easily and amplied to account for the second order eect, the

problem for lateral torsional buckling phenomena oers a much more complex

challenge. Although the eects 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

oers 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 amplication 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

eects 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 identied: (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 dened 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 denition, 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 – reected 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 = NEd ePδ,i = NEd e0,i kamp,i γM1 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

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ePδ,i is the bow imperfection accounting for the second order eects;

kamp,i is the amplication 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”:

N

cr,1

= π2EI

i

L

crit,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 amplication 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 amplication 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 amplication 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

eects 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 eects 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 prole along the segment. To assess other bending moment

proles, 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 eects 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 inexion (points between which the eective length is measured) of:

Mi,Pδ (x) = Mi,Pδ,max sin(πx ⁄

l

), where “x” is the position from a point of inexion

and “

l

” is the length between points of inexion (for a pinned column, this is the

column length).

Comparison with BS 5950 approach

Previous UK practice design addressed second order eects 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 dened 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

simplied 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 eects for members subjected to combined axial load and

bending are dened 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 dened according to BS 5950-1 4.8.3.3.4) are similar to the values

dened 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

prole, which must then be amplied;

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 eects 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 eects Critical design eects 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