In this article we will discuss about:- 1. Types of Beams 2. Design Considerations of a Beam 3. Classification 4. Compound Elements 5. Failure Modes 6. Lateral Buckling.
Types of Beams:
A beam is the most commonly encountered structural member whose function is to support loads which are resisted by its resistance to bending and shear. Beams are mostly used to support floors, roof sheeting as in purlins, side cladding etc. In the most familiar rectangular buildings, the beams form the horizontal members spanning between columns. There may also be secondary beams meant to transfer the floor loads to main beams.
Depending on their functions beams are given alternate names as follows:
1. Floor Beam – This is a major beam usually supporting the joints. This may also be a transverse beam supporting a bridge floor.
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2. Girder – This is a floor beam used in buildings. This in general is any major beam in a structure.
3. Lintel – This is a beam spanning door opening, window opening, supporting the wall up to the floor above.
4. Purlin – This is a roof beam supported by roof trusses.
5. Rafter – This is a roof beam supported by the purlins.
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6. Spandrel Beam – This is a beam at the outer most wall of a building supporting the floor and the wall up to the floor.
7. Stringer – This is a longitudinal beam used in bridge floors and supported by floor beams. This term is also applied to a beam supporting stair steps.
Beams may be supported in various ways. They may be cantilevered, simply supported, fixed ended or continuous. Beam sections used are of various types.
The commonly used types are the following:
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i. Universal Beams (Rolled Sections):
In this, material is concentrated in the flanges are very efficient in resisting uniaxial bending.
ii. Universal Columns:
It may be used where depth limitations exist. If used as beams, they are less efficient. They are generally used for columns.
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iii. Compound Beam:
It consists of a universal beam strengthened by flange plates. A crane beam is a combination of a universal beam with a channel. This beam can resist bending in vertical as well as horizontal directions.
Floor beams supporting roof slabs and floor slabs are continuously laterally supported. The concrete floor over the beams provide the necessary lateral support to the compression flange and to prevent lateral buckling.
Beams are provided in portal frames and frames of multi-storeyed buildings. In most cases the compression flanges of these beams are also continuously laterally supported and thus prevented from lateral buckling.
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Beams are also provided in bridges. The beams may support the deck slab over the top flange or the bottom flange and the bridge is accordingly called a deck bridge or a through bridge.
In a deck bridge the deck slab is supported on the top flanges of the main beams. The compression flanges of the beams are therefore continuously laterally supported.
In a through bridge the deck slab is supported by the beam at a level close to the bottom flanges. In this case the compression flanges are not laterally supported.
Castellated Beams:
These beams are made by applying a special technique to wide flange I-beams. This technique consists of making a cut in the web of a wide flange beam in a corrugated pattern. The cut parts are separated and the upper and the lower parts are shifted and then welded.
This results in the formation of a beam with hexagonal openings, with a greater depth, greater strength and greater stiffeners, than the original beam and also having the same weight per unit length. The hexagonal openings can be used to accommodate various duct work. Beams of spans 10 m to 20 m can be made using this technique.
Design Considerations of a Beam:
The following points should be considered in the design of a beam:
(i) Bending Moment Consideration:
The section of the beam must be able to resist the maximum bending moment to which it is subjected.
(ii) Shear Force Consideration:
The section of the beam must be able to resist the maximum shear force to which it is subjected.
(iii) Deflection Consideration:
The maximum deflection of a loaded beam should be within a certain limit so that the strength and efficiency of the beam should not be affected. Limiting the deflection to a safe limit will also prevent any possible damage to finishing. Generally the maximum deflection should not exceed span/320.
(iv) Bearing Stress Consideration:
The beam should have enough bearing area at the supports to avoid excessive bearing stress which may lead to crushing of the beam or the support itself.
(v) Buckling Consideration:
The compression flange should be prevented from buckling. Similarly the web of the beam should also be prevented from crippling. Usually these failures do not take place under normal loadings due to proportioning of thickness of flange and web. But, under considerably heavy loads, such failures are possible and hence in such cases the member must be designed to remain safe against such failures.
Classification of Beam Cross-Sections:
It may be realized that a thin projecting flange of an I-section is liable to buckle prematurely. The web of an I-section too is liable to buckle under compressive stress from bending and from shear. In order to prevent such local buckling it is necessary to limit outstand/thickness ratios of flanges and depth/ thickness ratios of webs. The sectional dimensions of plates constituents should be such that the following conditions are satisfied.
When designs are made by plastic analysis, the members should be able to form plastic hinges with sufficient rotation capacity (ductility) without local buckling, so as to permit redistribution of bending moments needed before reaching a collapse mechanism.
When designs are made by elastic analysis, the members must be able to reach the yield stress under compression without buckling.
Based on the above, beam sections are classified as follows in accordance with their behaviour in bending, as per IS 800 code:
(a) Class 1 (Plastic):
Cross-sections which can develop plastic hinges and have the rotation capacity required for failure of the structure by formation of plastic mechanism. The width to thickness ratio of plate elements shall be less than that specified under class 1 (Plastic) in the table ahead.
(b) Class 2 (Compact):
Cross-sections which can developed plastic moment of resistance, but have inadequate plastic hinge rotation capacity for formation of plastic mechanism, due to local buckling. The width to thickness ratio of plate elements shall be less than that specified under class 2 (Compact), but greater than that specified under class 1 (Plastic) in the table ahead.
(c) Class 3 (Semi-Compact):
Cross-sections in which the extreme fibre in compression can reach yield stress but cannot develop the plastic moment of resistance, due to local buckling. The width to thickness ratio of plate elements shall be less than that specified under class 3 (Semi- compact), but greater than that specified under class 2 (compact) in the table ahead.
(d) Class 4 (Slender):
Cross-sections in which the elements buckle locally even before reaching yield stress. The width to thickness ratio of plate elements shall be greater than that specified under class 3 (semi-compact). In such cases, the effective sections shall be calculated either by the following provision of IS 801 to account for the post local buckling strength or by deducting width of the compression plate element in excess of semi-compact section limit.
When different elements of a cross-section fall under different classes, the section shall be classified as governed by the most critical element.
Compound Elements in Built-Up Sections:
In the case of compound elements consisting of two or more elements bolted or welded together, the limiting width to thickness ratios as given in the table above should be considered on the basis of the following:
(a) Outstanding width of compound element (be) to its own thickness.
(b) The internal width of each added plate between the lines of welds or fasteners connecting it to the original section to its own thickness.
(c) Any outstand of the added plates beyond the line of welds on fasteners connecting it to original section to its own thickness.
Failure Modes for Beams:
The table below shows the main failure modes for a beam subjected to simple uniaxial bending. Which of these modes will govern in any particular case is dependent on factors like proportions of the beam the form of the loading applied and the nature of the supports provided. Besides adopting these strength limits, it is important to ensure that the beam does not undergo excessive deflection under service loads in order to fulfill the serviceability limit state.
Lateral Buckling of Beams:
Even though a beam is subjected to a loading in a principal plane it has a tendency to bend out of the plane of loading. Standard I- sections are so proportioned that the section provides a large moment of inertia about the principal axis normal to the web, while the moment of inertia of the section about the other principal axis is considerably low. The sections are proportioned in this way to provide economic beams.
As a consequence of such proportioning, under the action of the loading on the beam the compression flange acts as a column and as the compressive stresses increase, is liable to deflect or buckle side ways if it is not adequately restrained. The load on the beam at which the compression flange may buckle is much less than the load at which the beam can develop its full moment capacity.
Accordingly, beams may be classified into following types:
(1) Laterally supported (restrained) beams, and
(2) Laterally unsupported beams.
(1) Laterally Supported Beams:
A laterally supported beam is a beam whose compression flange is restrained from buckling. This is a beam whose compression flange is laterally supported in various ways. The compression flange may be connected to the concrete floor either by its embedment or by shear connectors. The compression flange may also be restrained by its connection to cross beams or bracings. Fig. 9.7 shows some methods of providing lateral support to the compression flange.
Design Bending Strength of Beams:
The design bending strength of a beam is governed by yield stress or by lateral torsional buckling strength. For a laterally supported beam, since lateral buckling is prevented, the design bending strength is governed by yield stress.
Laterally supported beams of plastic, compact or semi-compact sections are classified into the following cases:
Case (a). Web of the section not susceptible to buckling under shear before yielding.
Case (b). Web of the section susceptible to buckling under shear before buckling.
Case (a). Web of the Section not Susceptible to Buckling under Shear before Yielding:
These beams are said to be at low shear state.
When the factored design shear force does not exceed 0.6 Vd, where Vd is the design shear strength of the cross-section, the design bending strength Md shall be taken as-
Effect of Holes in the Tension Zone:
Bolt or rivet holes are likely to be present in the flanges and web of beams. Tests have shown that the failure of a beam is mostly based on the strength of the compression flange. The effect of holes in the tension flange or web on the flexural strength is ignorable. The I.S. code has suggested that the effect of bolt holes on the flexural strength may be ignored if the ultimate tensile yield strength of the flange is greater than the tensile yield strength of the flange.
where, Anƒ = Net area of the tension flange
Agƒ = Gross area of the tension flange
ƒu = Ultimate stress
ƒy = Yield stress
γml = Partial safety factor against ultimate stress
γmo = Partial safety factor against yield stress.
The above condition leads to the following specification given by the IS code. The effect of holes in the tension flange on the design bending strength need not be considered if,
When Anƒ/Agƒ does not satisfy the above requirement, the reduced effective flange area Aeƒ satisfying the above equation may be taken as the effective flange area in tension, instead of Agƒ.
The code has further specified that-
The effect of holes in the tension region of the web on the design flexural strength need not be considered if the limit given above is satisfied for the complete tension zone of the cross-section, comprising the tension flange and tension region of the web.
Fastener holes in the compression zone of the cross-section need not be considered in design bending strength calculation, except for oversize and slotted holes or holes without any fastener.
Case (b). When the Factored Shear Force Exceeds 0.6 Vd (High Shear State):
When the factored shear exceeds 0.6 times the design shear strength due to the interaction between the bending moment and shear the design bending strength of the beam section is reduced. As per I.S. code the design bending strength Mdv is calculated as-
(i) Plastic or Compact Section:
Md = Plastic design moment of the whole section disregarding high shear force effect considering web buckling effect.
Mƒd = Plastic design moment of the area of the cross- section excluding the shear area considering partial safely factor.
V = Factored applied shear force as governed by web yielding or web buckling
Vd = Design shear strength as governed by web yielding or web buckling
Ze = Elastic section modulus of the whole section.
(ii) Semi-Compact Section:
(2) Laterally Unsupported Beams:
Lateral Torsional Buckling of Beams of Symmetrical Sections:
In a beam subjected to a loading, the compression flange due to its considerable length acts as a long column and is liable to lateral buckling if it is not adequately stiff or if it is not restrained by continuous lateral support to prevent lateral buckling. As the loading on the beam is increased, the beam will go on bending about its strong axis up to a certain stage beyond which the compression flange will buckle.
Such lateral Duckling of a loaded beam about the weaker axis under the action of the loading on the plane of the stronger axis is called lateral torsional buckling. The beam is liable to fail at a load which is less than the load that can cause the beam section to reach its full plastic moment capacity.
Studies of lateral torsional buckling have been made on the basis of the following assumptions:
(i) The beam is simply supported at its ends
(ii) The beam is initially absolutely straight and has no initial deformations and is free from residual stresses.
(iii) The beam behaves elastically, not yet reaching the yield state.
(iv) The beam is in simple bending i.e., it is subjected to pure end couples in the plane of the web.
On the basis of study of such simply supported beams, the elastic critical moment capacity of a beam while it undergoes lateral torsional buckling is determined to be-
where,
EIy = Flexural rigidity about the minor axis of the beam section
GIt = Torsional rigidity.
EIw = Warping rigidity.
This critical moment Mcr is due to two torsional characteristic of the section namely pure torsional resistance and warping torsional resistance. To bring this point the expression for the critical moment Mcr is rearranged as-
Girders of small length and great depth have large torsional resistance. Girders of large length and small depth have low torsional resistance against warping. For such long beams Iw is negligible so that the critical moment capacity may be taken as-
Most often I-sections are used as beams. The I-section consists of thin plate elements which can provide only low torsional rigidity. However the light gauge sections provide more resistance against warping than against torsion. These resistances are also influenced by the lengths of beams. Small lengths of deep beams provide high warping resistance. Long beams of small depths have low warping resistance.
The expression for Mcr given above is rearranged in the form given below by IS 800 code.
where, LLT = Effective length against lateral torsional buckling I.S. 800 code specifications for laterally unsupported beams.
The phenomenon of lateral buckling thus involves two types of deformations, namely lateral bending and twisting of the beam section. We will therefore consider the two types of restraints needed to prevent these deformations.
In the case of rolled and built-up beams we can provide the torsional restraints in the following ways:
(i) The beam may be supported over a seating.
(ii) At each end the web should be connected by angles.
(iii) The flanges of the beam are connected at each end by connecting angels.
(iv) At each end the beam is built into masonry walls.
The strength of the beam is reduced due to lateral buckling. To prevent or minimize lateral buckling we may adopt the following techniques.
The top compression flange of the beam can be embedded in the concrete slab. This method provides full restraint to the compression flange against lateral buckling. But this may not be possible in cantilever beam whose lower flange is in compression. If a concrete slab just rests over the beam simply supported at its ends the lateral restraint available is only due to friction. Such a restraint may be affected seriously when the floor is subjected to vibration. In the absence of such vibration, i.e. in ordinary residential structures, this method is workable.
If the entire beam is embedded in concrete, both the flanges of the beam get restrained laterally.
A main beam can be laterally supported by cross beams connected at intervals over its span.
To summarise, the factors influencing lateral torsional buckling are the following:
(i) The unrestrained length of the compression flange – The beam will be weak for long unrestrained lengths. Props provided at intermediate points will prevent lateral buckling,
(ii) End conditions of the beam – Restraints against rotation at ends assist to prevent buckling.
(iii) Shape of beam section – A section having greater lateral bending stiffness, greater torsional stiffness has greater resistance to buckling,
(iv) The way the loads are applied between restraints.
Purlins are members spanning on the roof frames running generally through top chord joints. They are meant to support the roof coverings. The span of the purlins is the spacing between adjacent trusses. Rolled sections like angles, Channels and I-sections are used as purlins. When angles and channels are used, the connections of the purlins to the rafters are made by using cleat angles. But, when I-sections are used they are bolted directly to the rafters. When I-sections and channels are used they are fixed to the rafters with their major axis parallel to the rafters. When angles are used one leg of the angle is fixed normal to the rafter.
For I-section and channel section purlins the principal axis of the purlin is parallel to the roof. But in an angle purlin this is not so. Generally in most cases of members subjected to bending due to transverse loading, the plane of bending is identical with the plane of one of the principal axes, and the neutral axis of the section is identical with a principal axis. In a purlin however, the resultant of vertical loads and wind loads does not act along a principal axis. The purlin is therefore subjected to unsymmetrical bending.
Generally the purlin section is designed as-
Let W = Vertical load on the purlin and
We = Wind load on the purlin acting normal to roof
Total load normal to roof = We + W cos θ
Total load parallel to roof = W sin θ
where θ = Inclination of roof with the horizontal
Let the above loads be factored loads.
