beam columns

University of SheffieldDepartment of Civil & Structural Engineering

BS5950:Part1:2000

Design of members with Axial Load and Moment


BS5950:Part1:2000

Aim

• consider the approach to members subject to axial load and bending adopted in BS 5950 Part 1

• consider the background theory where it is relevant to the understanding of the approach adopted

• recognise that the design of such members will be influenced by

• method of frame analysis• shape of the cross section used • type of restraint provided.


BS5950:Part1:2000

Must not fail due to….

Local buckling Inadequate local capacity (tension or compression and or bending)) Overall buckling

1. major or minor axis buckling due to axial load2. major axis buckling due to major axis bending + axial load3. minor axis buckling due to minor axis bending + axial load4. minor axis buckling due to major axis bending + axial load


BS5950:Part1:2000

Section classificationb

t

T

d

b/t < 15

d/t < 120

Semi - compact section

1+ 2.0 r

b/t < 10

when r > 0 d/t < 100

Compact section

b/t < 9

d/t < 80

Plastic section

1 + r

Where

1 2 1 + 1.5 r 1

when r < 0 d/t < 100

1 + r 1

2

2

r 1

F

d t p y

c

For universal beam and column sections with equal flanges

=

275

y p

=

2

F

A p y

c = r

g

These factors vary for different shapes of section

but > or equal to 40


BS5950:Part1:2000

Local capacity

• Failure due to inadequate local capacity can occur in either tension or compression

• Where buckling is not a possibility the formulae are almost identical

1cy

y

cx

x

t

t

M

M

M

M

P

F

1cy

y

cx

x

yg

c

M

M

M

M

pA

F

• Tension

• Compression


BS5950:Part1:2000

Assumes failure @ yield

• addition of stresses due to axial load and bending should not exceed yield

• Class 1 and 2 x/sections use a plastic distribution of stress

• Class3 and 4 x/sections sections use an elastic distribution of stress


BS5950:Part1:2000

Alternatively

• for both tension and compression plus moment• class 1 or 2 UB or UC

121

z

ry

y

z

rx

x

M

M

M

M

Mrx and Mry are the plastic moment capacity in the presence of axial load.

z1 and z2 are empirical values varying for the type of section.

z1 = 2.0 for UB, UC, CHS

z1 = 5/3 for RHS

z2 = 1.0 for UB,UC

z2 = 2.0 for CHS

z2 = 5/3 for RHS


BS5950:Part1:2000

Reduction in plastic modulus

• Accurate values given in section tables

• Approximate formulae given in Appendix J

a

a

B

d

T

t

a = P

2.t.py

2/2a2

data

2

d

2

T

2

dBTS

rx


BS5950:Part1:2000

Compare the expressions

F

M

t

x

P t

M cx

1cy

y

cx

x

t

t

M

M

M

M

P

F

121

z

ry

y

z

rx

x

M

M

M

M


BS5950:Part1:2000

Tension members

• Generally not susceptible to buckling as axial tension prevents failure due to buckling caused by bending

• Theoretically possible to take account of this beneficial effect - expressions are complex

• Check bending effects independently - member treated as a laterally unrestrained beam

• Do this even when the tension and bending effects cannot occur independently


BS5950:Part1:2000

Compression – simplified method

• Combined gives:

P

Mx

P

Mx

X

X

P

My

P

My

y

y

1yy

yy

xy

xx

c

c

Zp

Mm

Zp

Mm

P

F

M

p x

cx

F

A +

y

1 x

g

c p Z

m x M

p y

cy

F

A +

y

1 y

g

c p Z

m y

Pc min of Agpcx and Agpcy

mx and my equivalent

uniform moment factors


BS5950:Part1:2000

Compression members- simplified method

• analogous to lateral torsional buckling in beams

• column buckles in a mode involving twisting and minor axis bending.

P

Mx

P

Mx

1yy

yy

b

xLT

cy

c

Zp

Mm

M

Mm

P

F

Pcy because considering buckling about the minor axis.

• significant for I and H sections buckling at low axial loads.

• not relevant for tubular sections which are not liable to suffer from lateral torsional buckling.


BS5950:Part1:2000

More exact approach- moments about major axis

F

F

d M F d

Secondary Moment

Primary Moment

M

M Moment diagram

Deflected shape

15.01

cx

c

cx

xx

cx

c

P

F

M

Mm

P

F

amplification factor allows for additional moment created by axial load at an eccentricity


BS5950:Part1:2000

LTB due to axial load and Mx

• In this case the amplification effect is less significant as the axial load is likely to be much less than the major axis strength.

1b

xLT

yc

c

M

Mm

P

F


BS5950:Part1:2000

Moments about minor axis

• In this case the amplification is more significant than about the major axis and the full value is used.

11

cy

c

cy

yy

y

c

P

F

M

Mm

P

F


BS5950:Part1:2000

Buckling about x-x due to My

• The amplification factor is negligible and the effect of the minor axis moment is small.

15.0 cy

yxy

xc

c

M

Mm

P

F


BS5950:Part1:2000

Moments about both axes

15.05.01

cy

yxy

cx

c

cx

xx

cx

c

M

Mm

P

F

M

Mm

P

FFor major axis buckling

11

cy

c

cy

yy

b

xLT

yc

c

P

F

M

Mm

M

Mm

P

F

For lateral torsional buckling

1)/1(

)/1(

)/1(

/5.01(

cyccy

cycyy

ccx

cxcxx

PFM

PFMm

PFM

PFMm

cx

For interactive buckling


BS5950:Part1:2000

Effect of moment shape

As applied moment tends towards double curvature, the secondary effects become less directly additive

F

F

M

F Secondary

Moment

Primary Moment

M b

M

F

d

M

F

Secondary Moment

Primary Moment

M

M

F

= +1.0 = -1.0

F

F

M

F Secondary

Moment

Primary Moment

M b

M

F

d

M

F

Secondary Moment

Primary Moment

M

M

F

= +1.0 = -1.0

Single curvature bending

Double curvature bending


BS5950:Part1:2000

• As a result the interaction expression plots increasingly higher

F Ag. Py

1.0

M/My 1.0

Strength interaction - zero slenderness

Column strength P=Pcr


BS5950:Part1:2000

• This change in the relationship can be represented by using an equivalent uniform moment factor m (very similar to that used for lateral torsional buckling of beams).

• Table 26 gives values and formula which can be used for all three modes of combined bending and axial load.


BS5950:Part1:2000

Design Summary

• For axial load and bending, check 1. Cross-section capacity

1cy

y

cx

x

t

t

M

M

M

M

P

FFor tension

1cy

y

cx

x

yg

c

M

M

M

M

pA

FFor compression

1

21

z

ry

y

z

rx

x

M

M

M

MMore exactly for class 1 and 2


BS5950:Part1:2000

Design Summary

2. BucklingTension – check cross-section capacity under tension

and moments (4.8.2.2 or 3) and buckling under moments alone (4.3).

Compression – check both

1yy

yy

xy

xx

c

c

Zp

Mm

Zp

Mm

P

F Buckling about either axis due to axial load and bending

1yy

yy

b

xLT

cy

c

Zp

Mm

M

Mm

P

F Buckling about the minor axis due to major axis bending and axial load


BS5950:Part1:2000

Design Summary

• Major axis moments only check

15.01

cx

c

cx

xx

cx

c

P

F

M

Mm

P

F

1b

xLT

yc

c

M

Mm

P

F


BS5950:Part1:2000

Design Summary

• Minor axis moments only check

11

cy

c

cy

yy

y

c

P

F

M

Mm

P

F

15.0 cy

yxy

xc

c

M

Mm

P

F


BS5950:Part1:2000

Design summary

• Axial load plus bi-axial moments

15.05.01

cy

yxy

cx

c

cx

xx

cx

c

M

Mm

P

F

M

Mm

P

F

11

cy

c

cy

yy

b

xLT

yc

c

P

F

M

Mm

M

Mm

P

F

1)/1(

)/1(

)/1(

/5.01(

cyccy

cycyy

ccx

cxcxx

PFM

PFMm

PFM

PFMm

cx


BS5950:Part1:2000


BS5950:Part1:2000

Special cases

• Tubular members – see 4.8.3.3.3– LTB may not be a problem

• Columns in simple structures– see 4.7.7– special simplified rules


BS5950:Part1:2000

Columns in simple structures

• Pin jointed braced fames• Nominal moments based on 100mm eccentricity• Equivalent uniform moment factors (m) = 1

1Zp

M

M

M

P

F

yy

y

bs

x

c

c

Mb calculated using LT as 0.5 L/ry

Distance between levels at which column is laterally restrained

beam columns

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