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Introduction
Beams are an important class of structural element, and are normally horizontal.
The primary function of building structures is to support the major space enclosing
elements: commonly these are floors, roofs and walls. The total behaviour of anybuilding structure can be complicated but frequently two types of sub-structure can
be identified; vertical elements (associated with walls) and horizontal elements
(associated with floors and roofs).
Vertical elements are columns, walls and lift cores etc. Horizontal elements include
slabs, trusses, space frames and most importantly beams.
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What is a beam?
Beams support mainly vertical loads, and are small in cross-section compared with
their span. Engineering diagrams adopt simple conventions to represent beams,
supports and loads.
This section deals specifically with the engineering design of beams. Although
"beam" is a word in common usage for engineering design, it has a very particular
definition. A beamis a structural member which spans horizontally between supports
and carries loads which act at right angles to the length of the beam. Furthermore,
the width and depth of the beam are "small" compared with the span. Typically, the
width and depth are less than span/10.
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Use of BS 5950 - Part 1
BS5950 specifies rules for ensuring steel structures are safe
The current code of practice called "Structural Use of Steelwork in Building" is BS
5950 Part 1. This code gives specific guidance on the strength and stiffness of steelstructures for buildings to allow numerical calculations to be made.
Beam capacity is checked by comparing the ultimate strength with factored loadings.
It checks the strength of a structure by ensuring ultimate strength is not less than
working load x load factor
The load factor varies with the type of load.
Different load factors are applied to different types of load and load combinations.
This reflects the varying degree of confidence in the values of dead, imposed andwind loads used. Values of the load factors to be used are given in Table 2 of BS
5950.
The material strength is specified in relation to steel grade.
The ultimate strength is dependent on yield stress. Stresses are given for three
grades of steel called S275, S (These were formerly referred to as Grades 43, 50 and
55). Grade S275 (formerly Grade 43) is commonly used, although S355 is popular on
larger projects where it can offer significant economies. Higher grades are rarely
used, except for bridges, and special applications
The yield stresses pycorresponding to the different grades are given in BS 5950.
Specific guidance is given for determining the maximum moment and shear capacity
of a beam cross-section.
The maximum moment capacity is the lesser of:
Mc= py.Sx
and
Mc= 1.2 py.Zx
Sx is the plastic section modulus and Zx is the elastic section modulus.
The maximum shear capacity is
Pv= 0.6 pyAv
where Avis the shear area.
Cross-sections are classified according to their proportions.
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Open sections are classified as plastic, compact, semi-compact or slender. The
classification depends on the proportions of the webs and flanges. (Note that rolled
sections are seldom classified as slender).
The moment capacity of some sections may be reduced if shear forces are high.
The actual shear force (multiplied by the factor) is referred to by the symbol Fv. If Fv
exceeds 60% of the shear capacity of the cross-section, Pv, then this is defined as a
"high" shear load and the moment capacity for plastic and compact sections is
reduced.
For beams which are not fully restrained, bending strength may be reduced due to
lateral-torsional buckling; this is related to the slenderness ratio and cross-section
details of the beam.
If the compression side of a beam is not fully restrained against lateral torsional
buckling then the design stress py is reduced. This reduction depends on two factors
called the slenderness ratio and the D/T ratio.
The slenderness ratio l is given by:
l = LE/ry
LEis the effective length (Table 9 of BS5950).
ryis the radius of gyration = the square root of (I/A). Standard tables give values for
ry. The slenderness ratio may be reduced using the slenderness correction factor n
from table 20, to allow for the shape of actual bending moment.
The beam must be stiff enough to carry the working load without exceeding the
deflection limits specified.
Limiting deflections are given in Table 8 of BS 5950 as a proportion of the beam
span. These limiting values are compared with calculated deflections taking account
of the load, length, support conditions, and cross-section of the beam. In calculating
deflections it is normal to ignore any permanent loads, and to consider conditions in
service (ie without any load factors)
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Support conditions
Beam supports are generally classified as pinned, fixed or free.
Beams span between supports carrying the external load forces to the external
reaction forces.
The type of support influences the distribution of bending moments and shear forces.
For simple span beams the supports may bepinned, fixedor free.
A pinned support provides vertical but not rotational restraint.
A fixed support provides vertical and rotational restraint.
A free support provides no restraint which might seem to be a paradox (a free
support is often called a free end).
The type of support significantly influences the bending moments and shear forces.
For the same span L and the same loading, say a uniformly distributed load of W, the
distribution of bending moments and shear forces is quite different.
In simple construction, beam supports are commonly assumed to be pinned.
Ideal pinned and fixed supports are rarely found in practice, but beams supported on
walls or simply connected to other steel beams are regarded as pinned. Where
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beams are continuous or part of complete frames (portal etc.) the distribution of
moments and shear forces is influenced by the behaviour of the complete structure.
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Summary
Beams are commonly found in many types of structure.
Beams carry load principally by bending.
The internal forces generated in beams under load are referred to as bending
moments and shear forces.
The support conditions influence the nature of the bending moments and
shear forces.
The strength of a beam is related to its section modulus.
The stiffness of a beam is related to its second moment of area.
Local buckling and shear forces can reduce the bending strength of a beam.
Lateral torsional buckling is possible in laterally unrestrained beams, in which
case the buckling strength is related principally to the slenderness of the
beam.
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Factors affecting the design
Beams must be both strong and stiff.
The objective of the engineering design of beams is to make a choice of a beam for a
particular purpose then test by established numerical procedures whether:
the beam is strongenough
the beam is stiffenough
The behaviour of a beam is principally affected by the span, loading, how it is
supported, its material and its cross-section.
For the same span and loading, three things affect the performance of a beam.
These are:
the way the beam is supported.
the properties of the material
the geometry of the beam cross-section
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Factors affecting beam strength
Bending strength may be limited by material strength, lateral-torsional buckling or
local buckling.
As was stated previously the strength of a beam depends on its maximum momentand shear capacities, and these depend on the relevant allowable stresses.
A beam may fail in one of three ways. The three types of failure are material failure
causing a plastic hinge to form, lateral-torsional buckling along the length of the
beam, and local buckling of the beam cross-section.
A plastic hinge forms when the bending stress reaches the material yield strength.
Collapse by formation of a plastic hinge. Where the stress in the beam reaches the
yield stress, the bending moment cannot be increased and the beam collapses as
though a hinge has been inserted into the beam.
Lateral-torsional buckling is associated with the compression developed in part of the
beam cross-section due to bending.
Bending moments cause a pair of internal horizontal forces, one force is a tension
force and the other a compression force. The tension force stretches one side of the
beam. This force, like pulling a string, tends to keep the tension side of the beam
straight between supports.
However, the compression force can buckle the compression side of the beam.
Because the tension force is keeping one side taut, the beam can only bucklesideways and twist the beam - hence lateral torsional buckling.
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Lateral-torsional buckling may be resisted by restraining the beam laterally.
This buckling may be prevented by adequate lateral restraint preventing sideways
movement of the beam. If this is at discrete points, the beam may buckle between
these points.
Steel floor beams are often adequately restrained by the floor slab.
In building structures the floor slab is often able to provide effective restraint to the
floor beams, preventing lateral-torsional buckling. In such cases the beams can be
designed on the basis of the full bending strength.
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Local buckling can occur if the beam cross-section is very slender.
If some part of the beam is very slender, then this may buckle locally. In practice,
standard steel sections are proportioned so that this is not a critical design
consideration.
The dominant failure mode depends on a number of factors.
Which type of failure occurs depends on four factors:
the slenderness ratio of the beam.
the shape of the bending moment diagram.
the proportions of the individual parts (webs, flanges etc.) of the beam cross-
section.
the presence of a "high" shear force.
Any of these factors may reduce the allowable stress to below the yield stress to
ensure that the beam is safe against any type of failure.
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Estimating sizes
Simplified procedures can be used to estimate the required size of beam section.
In many cases it is not necessary to perform detailed calculations to determine beam
sizes, and simpler methods can be adopted. These include rules of thumb whichenable a simple estimate of approximate section sizes, and safe load tables which
provide a rigorous alternative to design calculations for standard cases.
Rules of thumb provide an estimate of the required depth of different types of beam
in relation to span.
A typical structural frame may include both primary (main) and secondary beams.
Roof beams tend to be more lightly loaded than floor beams, and the required
section is therefore generally somewhat smaller. Some guidance on maximum beam
spans and the required provide depth (as a proportion of the beam span) is given inthe table below.
Supporting Floor Roof
max span depth max span depth
Primary beams
(conventional
composite deck orprecast floor)
15m span/20 15m span/25
Secondary beams
(conventional
composite deck
floor)
12m span/25 12m span/30
ASB beams
(conventional
composite deck
floor)
10m span/30
Safe load tables provide more rigorous guidance on required sizes.
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The bending strength of a beam is related primarily to the material strength and the
section modulus, which itself depends on the size and shape of the cross-section. It
is therefore possible to calculate the maximum bending strength for any cross-
section, assuming that there is possibility of lateral-torsional buckling. Such tables
enable the section size required for a given bending moment to be read directly. In
using such tables, it is assumed that shear forces are relatively small (as is often the
case in practical design conditions), and some guidance is given in relation to
deflection control.
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Cross-section geometry
Cross-section size and shape influence beam
performance.
The performance of any beam is dependent upon thecross-sectional geometry, not only on the physical
dimensions, but also the shape. Steel beams are
available in a variety of cross-sectional shapes. These
include open sections and closed sections or tubes. In
practice I beams are most commonly used for the beams
in buildings. Rectangular hollow sections may be used
for edge beams where particular edge details are
required.
Beams may be standard sections or specially
manufactured.
Both open and closed sections are produced in a large variety of standard (serial)
sizes by a hot-rolling process. Standard sections are generally used because they
are readily available and are economic.
Variations on standard sections include castellated and cellform beams which are
efficient for very long spans, and provide for service runs.
Other sections can be fabricated from plate or by joining standard sections by boltingor welding.
Beams may be curved.
Beams may be curved to reduce overall deflections (precambering) or simply to
create more interesting shapes. This is achieved by a specialised bending process
which can be performed at a modest surcharge to fabrication costs. For small
degrees of curvature, the beam design calculations are no different from those for a
straight beam. However for significantly curved beams, the structural action becomes
that of an arch.
Bending behaviour is related to the moment of inertia and the section modulus of the
beam.
The main bending behaviour of a beam is determined by two geometric quantities.
These are the moment of inertiaand the section moduluswhich depend on the size
and shape of the cross-section. These quantities are given in standard tables for all
rolled sections.
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Beam strength
Beams must be sufficiently strong to carry the applied bending moments and shear
forces.
A primary concern of engineering design is to ensure that a chosen beam is strongenough to carry the load imposed on it safely. This simple concept is far from simple
to quantify.
The beam has to carry both the bending moment and the shear force.
The bending moment and shear force capacities are related to the physical
properties of the cross-section and material strength.
The bending moment capacity is expressed simply as:
moment capacity = allowable bending stress x section modulus
and the shear force capacity as:
shear capacity = allowable shear stress x shear area
The shear area for any section is calculated from the area of the vertical part of the
cross-section.
Note the maximum allowable stress is the stress of yield point and is called the yield
stress.
Provided
moment capacity / actual bending moment = adequate factor of safety
and
shear capacity / actual shear force = adequate factor of safety
everywhere along the beam then the beam is strong enough.
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Beam stiffness
Beams deflect when loaded and this must be limited to avoid damage and distress.
When a beam is loaded it will deflect. This deflection must be limited so that building
occupants are comfortable and that building materials are not damaged. Forinstance, large deflections in a steel beam supporting a partition could cause
unacceptable cracking in the plaster.
Beam deflections can be calculated and depend on the modulus of elasticity of the
material, and the moment of inertia of the cross-section.
In general, the calculation of deflection is not straightforward. However, algebraic
expressions are tabulated for many standard cases.
The deflection of a particular beam is inversely proportional to:
modulus of elasticity
moment of inertia
The modulus of elasticity is constant for all structural steels so the larger the moment
of inertia the smaller the deflection.
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Basic beam behaviour
Beam analysis is generally based on simplifying assumptions.
The detailed behaviour of a beam is complex and exact analysis requires
considerable mathematical sophistication. However, the vast majority of beams canbe designed using engineering beam theory. This is based on a number of
simplifying assumptions.
The structural action of a beam is represented by internal forces called bending
moments and shear forces.
A beam is subjected to two sets of external forces. These are the loads applied to the
beam and reactions to the loads from the supports. The beam transfers the external
load set to the external reaction set by a system of internal forces.
Engineering beam theory identifies two types of internal force bending moments
and shear forces. The behaviour of any beam is characterised by the magnitude and
distribution of these forces. At any point in the beam, the internal shear force and the
internal bending moment can be represented as pairs of forces.
The bending moment and shear forces vary along the beam length and are often
represented diagramatically.
These internal forces may vary along the length of the beam and are usually
represented as separate bending moment and shear force diagrams.
The calculation of bending moments and shear forces is traditionally part of structuralanalysis and is beyond the scope of this unit.