Design of Bending Members in SteelDesign of Bending Members in Steel

Steel wide Steel wide flange flange

beams in beams in an office an office buildingbuilding

Composite Steel-Concrete Girders

German Historical Museum

An example of curved steel beams

Sea to sky highway, Squamish

Steel Girders for Bridge Decks

Cantilevered arms for steel pole

What can go wrong ?What can go wrong ?STEEL BEAMS:

• Bending failure

• Lateral torsional buckling

• Shear failure

• Bearing failure (web crippling)

• Excessive deflections

Bending Strength Bending Strength

Linear elastic stresses

6

sectionsr rectangulafor 2

max

max

bhS

SMS

M

I

My

M

Design Equation:

SFM br Where Fb is the characteristic bending strengthFor steel this is Fb = Fy

For timber it is Fb = fb (KDKHKSbKT)

y

Plastic moment capacity of steel beamsPlastic moment capacity of steel beams

Yield moment

ZF

aAFaAF

TaCaM

y

TyCy

p

SFM yy

My

Fy

Plastic moment

Mp

Fy

C

T

a

Ac

At

Which is the definition of the plastic section modulus ZZ can be found by halving the cross-sectional area and multiplying the distance between the centroids of the two areas with one of the areasThis is also called the first moment of area

So, when do we use the one or the

other ??

Steel beam design equationSteel beam design equation

For laterally supported beams (no lateral torsional buckling)

Mr = Fy Z for class 1 and 2 sections

Mr = Fy S for class 3 sections

where = 0.9

Steel cross-section classesSteel cross-section classesClass 4 Real thin plate sections

Will buckle before reaching Fy at extreme fibres

Mr < My

(Use Cold Formed

Section Code

S136)

Class 3 Fairly thin (slender) flanges and web

Will not buckle until reaching Fy in extreme fibres

Mr = My

Class 2 Stocky plate sections

Will not buckle until at least the plastic moment capacity is reached

Mr = Mp

Class 1 Very stockyplate sections

Can be bent beyond Mp and can

therefore be used for plastic

analysis

Mr = Mp

Load deflection curves for Class 1 Load deflection curves for Class 1 to Class 4 sectionsto Class 4 sections

Local buckling of the Local buckling of the compression flangecompression flange

Local torsional buckling of the Local torsional buckling of the compression flangecompression flange

Local web bucklingLocal web buckling

Lateral torsional bucklingLateral torsional buckling

x

xx

y

y

y

y

Δx

ΔyθLe

Elastic buckling:Mu = ωπ / Le √(GJ EIy ) + (π/L)2EIy ECw

Moment gradient factorTorsional stiffnessLateral bending stiffness

Warping stiffness

Moment resistance of laterally Moment resistance of laterally unsupported steel beamsunsupported steel beams

Le

Mr /

Mu

Mmax = My for class 3or Mp for class 1 and 2

0.67Mmax

Mmax

1.15 Mmax [1- (0.28Mmax/Mu)]

Shear stress in a beamShear stress in a beam

Ib

VQ

Ib

VAy

τmax ≈ V/Aw

=V/wd

b=wA

N.A.

y τd

N.A.

y

A

b

τd

τmax = V(0.5A)(d/4) (bd3/12)b=1.5 V/A

Shear design of a steel I-beamShear design of a steel I-beam

d wh

Aw = d.w for rolled shapes and h.w for welded girders

Vr = φ Aw 0.66 Fy

for h/w ≤ 1018/√Fy = 54.4 for 350W steel

For welded plate girders when h/w ≥ 1018/√Fy

the shear stress is reduced to account for buckling of the web(see clause 13.4.1.1)

This is the case for all rolled shapes

Bearing Bearing failures in a failures in a steel beamsteel beam

24 , ,1.45r bi y bi yB w N t F or w F E

k

N

N+4t

N+10t

w

210 , ,0.60r bi y be yB w N t F or w F E

For end reactionsFor end reactions

For interior reactionsFor interior reactions

DeflectionsDeflections• A serviceability criterion

– Avoid damage to cladding etc. (Δ ≤ L/180)– Avoid vibrations (Δ ≤ L/360)– Aesthetics (Δ ≤ L/240)

• Use unfactored loads

• Typically not part of the code

Δ