Design of beams in structural steel 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 any building 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.
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 beam is 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.
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 be pinned, fixed or 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 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.
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 steel structures 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 and wind 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 p y corresponding 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: M c = p y .S x and M c = 1.2 p y .Z x Sx is the plastic section modulus and Zx is the elastic section modulus. The maximum shear capacity is P v = 0.6 p y A v where A v is the shear area. Cross-sections are classified according to their proportions. 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 F v . If F v exceeds 60% of the shear capacity of the cross-section, P v , 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 lateraltorsional 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 /r y L E is the effective length (Table 9 of BS5950). r y is 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)
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 can be 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 structural analysis and is beyond the scope of this unit.
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. For instance, 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.
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 strong enough 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
•
shear capacity / actual shear force = adequate factor of safety
and
everywhere along the beam then the beam is strong enough.
Cross-section geometry Cross-section size and shape influence beam performance. The performance of any beam is dependent upon the crosssectional geometry, not only on the physical dimensions, but also the shape. Steel beams are available in a variety of crosssectional 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 bolting or 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 inertia and the section modulus which depend on the size and shape of the crosssection. These quantities are given in standard tables for all rolled sections.
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 which enable 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 in the table below. Supporting
Floor max span
Primary beams (conventional 15m composite deck or precast floor) Secondary beams (conventional 12m composite deck floor) ASB beams (conventional 10m composite deck floor)
depth
Roof max span
depth
span/20
15m
span/25
span/25
12m
span/30
span/30
Safe load tables provide more rigorous guidance on required sizes. 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.
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 moment and 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 buckle sideways and twist the beam hence lateral torsional buckling.
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. 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.
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.
