STEEL AND TIMBER STRUCTURE
CENG 5503
CHAPTER THREE
FLEXURAL MEMBER DESIGN
Prepared By Solomon B.
December, 2019
Objectives
i. Introduction
ii. Typical cross section of flexural members
iii. Classification of cross section
iv. Bending stress and moment capacity
v. Design of beam
Introduction
Flexural members (beams),:- are structural members that support loads which
are applied transverse to their longitudinal axes.
Beams:- transfer the transverse loads they carry to the supports by bending
and shear actions.
A beam is a combination of a tension element and a compression element.
The main uses of beams are to support floors and columns, carry roof sheeting
as purlins, side cladding as sheeting rails, and to support bridge decks.
Introduction
Beams on structure may also be referred to as
Main Beam/Girder:- beams that span between adjacent columns at wide spacing
secondary beams/joists:- beams which transfer the floor loading to the main beams
Spandrels:-Exterior beams at the floor level of buildings, which carry part of the
floor load and the exterior wall.
purlins:- a roof beams spanning between trusses
stringers:-a longitudinal bridge beams spanning between floor beams
lintels:-members supporting a wall over window or door openings
Typical cross-sections
Section should be proportioned such that to have the largest moment of inertia.
A universal beam is one of the most effective cross section for resting flexural
stress.
Rolled I sections with or without cover plates are usually used for floor beams.
Channel, T and L sections are normally used for beams in rood trusses as
purlins and common rafters.
The web area of the beam has to be adequate for resisting the shear.
Typical cross-sections
Typical cross-sections
Bending about strong axis (x-axis):
w
X X
Bending about weak axis w
Y Y
Beam types and its applications
Types of beam Optimum span Application
range (m)
Angles 3-6 For lightly loaded beam such as roof purlin and
sheeting rail
Rolled I sections 1-30 Most frequently used as a beam
Castellated beams 6-60 Long spans and light loads
Plate girders 10-100 Long spans with heavy loads such as bridges
Box girders 15-200 Long spans and heavy loads such as bridge girders
classification of cross section
The EN ES 3 2015 classifies sections into four categories. Accordingly, the
design strength of a cross-section subject to compression depends on its
classification as:
a) Class 1 (Plastic), fail after the formation of plastic mechanism
b) Class 2 (Compact), fail after reaching Mp at one section
c) Class 3 (Semi-compact), used for elastic design, where the section fails
after reaching My at the extreme fibers.
d) Class 4 (thin-walled) not preferred in hot-rolled structural steelwork
but they are extensively used in cold-formed members.
BEAMS WITH FULL LATERAL RESTRAINT
Many beams in a steel framework will be restrained laterally by
floor slab
wall or roof cladding,
by bracing members at specific points along the beams
secondary beam
end joints
Bending Stress & Moment Capacity
In a beam subjected to an increasing moment, the bending stress diagram ranges
from a linearly elastic condition with extreme fiber stress less than the design
strength fy to one in which all of the fibers can be considered to have reached the
design strength.
Elastic Theory
The bending stress are distributed linearly across any section of the beam as
shown in figure, the bending moment M is directly proportional to the
curvature.
For Class 3 sections, the extreme fiber strain attains value εy and stress
distribution is shown in figure above.
𝑀𝑦𝑦 = 𝑓𝑦 𝑍𝑥𝑥 Where: 𝑍𝑥𝑥 is elastic section modulus
Plastic Theory
Once yield strain of steel beam is exceeded the stress distribution is no
longer linear.
When the yield moment is exceeds curvature increase rapidly as the plastic
region proceeds inward until the full plastic moment is reached and a
plastic hinge is formed as shown in figure above.
𝑀𝑦𝑦 = 𝑓𝑦 𝑆𝑥𝑥 Where: Sxx is plastic section modulus
The moment resistance for the four classes defined above are
1. classes one and two:-plastic moment
𝑀𝑝𝑙 = 𝑊𝑝𝑙 × 𝑓𝑦
2. class three:-Elastic moment
𝑀𝑒𝑙 = 𝑊𝑒𝑙 × 𝑓𝑦
2. class four:-local buckling moment
𝑀𝑜 < 𝑀𝑒𝑙
Where 𝑊𝑝𝑙 is plastic section modulus
𝑊𝑒𝑙 is elastic section modulus
Design of Beams
Three aspects are considerate in the design of a beam:
1. Strength consideration
the beam has adequate strength to resist the applied bending moments and
accompanying shear forces.
2. Stability consideration
Buckling resistance of member
3. Serviceability – May need deeper beam to prevent serviceability problems such as
deflection or vibration.
Design of Beams
The following criteria should be considered for establishing the moment resistance of
flexural members:
1. yielding of the cross section or its flexural strength (ULS)
2. local buckling (Class 4 sections only) (ULS)
3. lateral-torsional buckling (ULS)
4. shear strength including shear buckling (ULS)
5. local strength at points of loading or reaction; i.e., criteria for concentrated loads (ULS)
6. deflection criterion; with respect to serviceability limits states, (SLS)
In the absence of shear force, the design value of the bending moment Msd at each
cross section shall satisfy.
Classes 1 and 2: Moment resistance is the design plastic moment resistance
Class 3: Moment resistance is the design elastic moment resistance
Classes 4: Moment resistance is the design local buckling moment resistance
The plastic moment resistance of a across-section is reduced by the presence of
shear.
But when the design value of the shear force Vsd does not exceed 50% of the
design plastic shear resistance Vpl,Rd no reductions is needed.
The design value of the shear force 𝑉𝐸𝐷 at each cross section shall satisfy
𝑉𝑒𝑑
≤ 1.0
𝑉𝑐,𝑅𝑑
where 𝑉𝑐,𝑅𝑑 is the design shear resistance.
For plastic design 𝑉𝑐,𝑅𝑑 is taken as the design plastic shear resistance, 𝑉𝑝𝑙,𝑅𝑑 given by
𝑓𝑦
𝐴𝑣 ( )
3
𝑉𝑝𝑙,𝑅𝑑 =
𝛾𝑀𝑜
Fastener holes in the web do not have to be considered in the shear verification.
Av is the shear area, which for rolled I and H sections, loaded parallel to the web is
𝐴𝑣 = 𝐴 − 2𝑏𝑡𝑓 + (𝑡𝑤 + 2𝑟)𝑡𝑡 ≥ 𝛈ℎ𝑤 𝑡𝑤
Where A cross-sectional area b overall breadth
r root radius 𝑡𝑓 flange thickness
𝑡𝑤 web thickness ℎ𝑤 depth of the web
η conservatively taken as 1.0
The plastic resistance moment of the section is reduced by the presence of shear.
If the design shear force exceeds 50% of the plastic shear resistance, the design
moment resistance of the cross-section is reduced to 𝑀𝑦,𝑣,𝑅𝑑
The reduced design plastic resistance moment for I and H cross section with equal
flanges and bending about the major axis is
𝜌𝐴𝑉 2
𝑀𝑦,𝑣,𝑅𝑑 = 𝑓𝑦 (𝑊𝑝𝑙,𝑦 − )/𝛾𝑀𝑜 ≤ 𝑀𝑦,𝑐,𝑅𝑑
4𝑡𝑤
2
2𝑉𝐸𝑑
𝜌= −1
𝑉𝑝𝑙,𝑅𝑑
Shear buckling need not be consider if
ℎ𝑤 𝜀
𝑡𝑤
≤ 72
η
for unstiffened web
ℎ𝑤 𝜀
𝑡𝑤
≤ 72
η
for web with intermediate stiffener
𝑘𝑟
where 𝑘𝜏 is the buckling factor for shear and is given by
for 𝑎 ℎ𝑤 < 1 𝑘𝜏 = 4 + 5.34(ℎ𝑤 𝑎)2
for 𝑎 ℎ𝑤 ≥ 1 𝑘𝜏 = 5.34 + 4(ℎ𝑤 𝑎)2
To prevent the possibility of the compression flange buckling in the plane of the web
ℎ𝑤 𝐸 𝐴𝑤
≤𝑘
𝑡𝑤 𝑓𝑦𝑓 𝐴𝑓𝑐
Where 𝐴𝑤 is the area of the web = (ℎ − 2𝑡𝑓 )𝑡𝑤
𝐴𝑓𝑐 is the area of the compression flange = 𝑏𝑡𝑓
𝐴𝑦𝑓 is the yield strength of the compression flange
The factor k assumes the following values:
Plastic rotation utilized, i.e. class 1 flanges: 0.3
Plastic moment resistance utilized, i.e. class 2 flanges: 0.4
Elastic moment resistance utilized, i.e. class 3 or class 4 flanges: 0.55
Beam cross-sections are proportioned such that moment of inertia about the major principal
axis is larger. This is done because of economy.
As a results the beam is weak in resistance to torsion and to bending about the minor axis if
nor held in line by the floor construction or by bracing or they may become unstable.
The instability is a sidewise bending accompanied by twisting and is referred as lateral-
torsional buckling (LTB).
LTB involves both lateral deflection (u) and twisting about a vertical axis through the web
(φ)
Members not provided with full lateral restraint must be checked for lateral torsional buckling
resistance.
lateral torsional buckling of a
cantilever
design factors which will influence the lateral stability can be summarized as:
the length of the member between adequate lateral restraints;
the shape of the cross-section;
the variation of moment along the beam;
the form of end restraint provided;
the manner in which the load is applied, i.e. to tension or compression
flange.
LTB Resistance of real beams
Significant difference exist between the assumption which for the basic of the theory and
the characteristics of real beams.
Since the theory assumes elastic behavior , it provides an upper bound on the true strength.
Comparison of test data with Mcr
Three regions can be seen form the previous slide
Stocky beams which are able to attain Mpl , with values of λ𝐿𝑇 below 0.4
Slender beams which fail at moment close Mcr , with values of λ𝐿𝑇 above 1.2
Medium slender beams which fail to reach moment close Mpl or Mcr , with
values of 0.4 < λ𝐿𝑇 < 1.2
Only stocky beams are not designed for LTB, but slender and medium slender
beams, design must be based on consideration of inelastic buckling, imperfection,
residual stress, etc.
Resistance to lateral-torsional buckling need not be checked separately (and the
buckling resistance moment Mb may be taken as equal to the relevant moment
capacity Mc) in the following cases:
CHS, square RHS or circular or square solid bars;
RHS, unless LE/ry exceeds the limiting value
I, H, channel or box sections, if λLT does not exceed λL0,
bending about the minor axis;
In order to prevent the possibility of a beam failure due to lateral torsional buckling,
the designer needs to ensure that the buckling resistance, 𝑀𝑏,𝑅𝑑 exceeds the design
moment, 𝑀𝐸𝑑 , i.e.
𝑀𝐸𝑑
≤1
𝑀𝑏,𝑅𝑑
The buckling resistance is a function of
(1) the elastic critical moment, 𝑀𝑐𝑟
(2) the buckling factor, 𝑥𝐿𝑇
Basic assumption To analysis the elastic critical moment, 𝑀𝑐𝑟
1) The beam is of uniform section with equal flanges.
2) Beam ends are simply supported in the lateral plane and prevented from
lateral movement and twisting about the longitudinal axis but are free to
rotate on plan.
3) The section is subjected to equal and opposite in plane end moments.
4) The loads are not destabilizing.
The elastic critical moment is then given by
𝜋2 𝐸𝐼𝑧 𝐼𝑤 𝐿𝑐𝑟 2 𝐺𝐼𝑇
𝑀𝑐𝑟 = +
𝐿𝑐𝑟 𝐼𝑧 𝜋2 𝐸𝐼𝑧
Where
𝐿𝑐𝑟 = length of beam between points which have lateral restraint
𝐼𝑧 = second moment of area about the minor axis (z–z)
𝐼𝑦 = second moment of area about the major axis (y–y)
𝐼𝑇 = torsional constant 𝐼𝑤 = warping constant
E = modulus of elasticity (210000 N mm−2)
𝐸
G = shear modulus = = 8100 𝑁mm-2
2(1+𝑉)
There are two methods of calculating 𝑥𝐿𝑇 (reduction factor) for lateral torsional
buckling.
1. General case
According to the general case 𝑥𝐿𝑇 is given by
Wy = Wpl,y for class 1 or 2 sections
= Wel,y for class 3 sections
= Weff,y for class 4 sections
𝛼𝐿𝑇 is an imperfection factor for LTB and is given as
2. buckling factor applicable for rolled section only
In order to take account of the bending moment curve between points of lateral
restraint the reduction factor 𝑥𝐿𝑇 may be modified as follows:
EC 3–5 distinguishes between two types of forces applied through a flange to the web:
(a) forces resisted by shear in the web (loading types (a) and (c)).
(b) forces transferred through the web directly to the other flange (loading type (b)).
For loading types (a) and (c) the web is likely to fail as a result of
1. crushing of the web close to the flange accompanied by yielding of the flange,
the combined effect sometimes referred to as web crushing
2. localized buckling and crushing of the web beneath the flange, the combined
effect sometimes referred to as web crippling.
For loading type (b) the web is likely to fail as a result of
I. web crushing
II. buckling of the web over most of the depth of the member
design resistance of webs to local buckling is given by
𝑓𝑦𝑤 𝐿𝑒𝑓𝑓 𝑡𝑤
𝐹𝑅𝑑 =
𝛾𝑚1
where: 𝑓𝑦𝑤 is the yield strength of the web
𝑡𝑤is the thickness of the web
𝛾𝑚1 is the partial safety factor = 1.0
𝐿𝑒𝑓𝑓 is effective length of web which resists transverse forces = 𝑥𝐹 𝑙𝑦
𝑥𝐹 is the reduction factor due to local buckling
𝑙𝑦 is the effective loaded length, appropriate to the length of the stiff bearing 𝑠𝑠
Reduction factor (𝑥𝐹 )
◦ 𝑘𝐹 is longitudinal stiffeners
Effective loaded length (𝑙𝑦 )
the effective loaded length for loading types (a) and (b) is given by
𝑙𝑡 = 𝑠𝑠 + 2𝑡𝑓 (1 + 𝑚1 + 𝑚2 ) ≤ 𝑎
fyf bf
m1 =
fyw t w
Length of the stiff bearing (𝑠𝑠 ) should be taken as the
distance over which the applied load is effectively distributed
at a slope of 1:1
For loading type (c) ly is taken as the smallest value obtained from equations given
as follows 𝑚 𝑙 1 𝑒
2
𝑙𝑦 = 𝑙𝑒 + 𝑡𝑓 + + 𝑚2 𝑎𝑛𝑑
2 𝑡𝑓
𝑙𝑦 = 𝑙𝑒 + 𝑡𝑓 𝑚1 + 𝑚2
𝑘𝐹 𝐸𝑡𝑤 2
𝑙𝑒 =
2𝑓𝑦𝑤 ℎ𝑤
Interaction between shear force and bending moment
when web is subjected to combined bending and shear the effect should satisfy
the following equation
𝜂2 + 0.8𝜂1 ≤ 1.4
Web stiffeners may be required due to
I. Heavy concentrated loads are applied to the flanges of section parallel to web
II. Longer Span of beam
stiffeners generally take the form of flat plates welded to the web between the
flanges of the beam
Buckling resistance of stiffeners by examining the resistance of the cruciform section
comprising the stiffeners plus length of the web equals to a maximum value of
15𝜀𝑡𝑤 on each side of the stiffeners.
To determine the buckling resistance it is necessary to calculate the
second moment of area and the radius of gyration of the area of the
cruciform section.
Effective
buckling resistance 𝑁𝑏,𝑅𝑑 =
𝑥𝐴𝑓𝑦
𝛾𝑀1
cross section
of stiffener
Where
x is the reduction factor for flexural buckling and is given by the following
expression:
• Steel beam shall be proportioned such that deflections are with in limits.
• Since deflection is a serviceability issue, it must be checked with service loads.
• Recommended vertical deflection limits
Maximum moments and deflections Simply supported beams
