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Fundamentals of Beam Design

The breakthroughs in beam design, materials accessible for usage, and understanding of their behaviour and physical properties had enabled the construction of today’s architectural wonders. To appreciate these accomplishments, the Engineer must have a basic understanding of the material selection process, including beam cross section profile and physical characteristics, the importance of beam supports, and the ability to understand and perform basic beam deflections, shear stresses, and bending moments calculations.

Choice of material

Finally, the material used determines the beam’s strength, or how much stress it can support before failing, which is related to its Young’s modulus (E). Most materials, however, behave differently under compression and stress, which must be taken into account while designing it.

Cast iron, steel, concrete, and wood are the four most common materials utilised in beam design, and they will be discussed further below. Carbon fibre and composite materials are among the others.

Cast Iron

During the Industrial Revolution, a technique of manufacture (by blast furnace) was created that was both inexpensive and practical, and cast iron was recognised as a building material in the late 1700s. Because cast iron is strong in compression but not in tension, the first uses were bridges and other constructions that required short compression members. Figure 1 shows the Coalbrookdale iron bridge, which was constructed around 1770.

Coalbrookdale Iron Bridge (Figure 1)

Cast iron has a Young’s modulus of E? 211 Gpa, which means it is relatively strong but brittle by nature. Despite the capacity to produce beams of varied shapes and sophisticated designs, this undesired property led to a number of disastrous collapses of early bridges and limited its usage as a building material. Despite these bad connotations, it was seen as a breakthrough architectural material since it allowed traditional masonry to be replaced with elegant, slender iron beams.

Steel

Henry Bessemer devised a process for mass-producing steel in the late 1880s, heralding the birth of the skyscraper. With a Young’s modulus of E? 800 Gpa, this strong material could now be formed into I-beams and steel columns. The skyscraper was born when a sequence of these I-beams and steel columns were combined to form a structural, steel core of considerable height (figure 2) to which the floors, roof, and walls of a building could be attached. This approach was employed to build the Empire State Skyscraper in New York, which would go on to become the world’s tallest building for over forty years.

Figure 2: New York, c.1930, steel core structure

Steel as a building material has some drawbacks: it is good in both compression and tension due to the ability to specify a specific cross section profile, but it softens at high temperatures, so it is usually encased in a fire resistant material to prevent buildings from collapsing in the event of a fire. Steel also has the advantage of being able to modify its composition and thus its physical qualities. The addition of manganese, which is often an alloy of iron and carbon with a carbon percentage of between 0.2 percent and 2.14 percent, will provide a large improvement in strength at a low cost. Similarly, adding chromium or nickel to steel can harden it and increase its strength.

Concrete

Both the Ancient Egyptians and the Romans used concrete in their constructions, but as the Roman Empire fell apart, its secrets were practically forgotten until recently. One of the most significant milestones in the history of concrete is the filing of a patent for the manufacture of Portland cement in 1824, and since then, great progress has been made with the development of pre-stressed concrete beams.

Concrete is made up of three components: water, aggregate, and cement. Gravels (made up of crushed rock and sand) often make up the bulk of the aggregate in concrete. Cement, most often Portland cement, binds the elements together, giving the concrete its strength and durability.

Concrete has a wide range of applications and is particularly well adapted for compressive stresses, such as integral building columns; but, with reinforcements, this range can be increased to include thin-shell constructions, as seen in figure 3.

Figure 3 – Valencia’s El Palau de les Arts Reina Sofia

Because concrete is normally only good under compression, pre-stressed concrete has tendons (usually composed of steel), which counteract the tensile stress a concrete member would otherwise face when subjected to a load. Pre-tensioned, bonded post-tensioned, and unbonded post-tensioned are the three primary forms of pre-stressing concrete beams:

Pre-tensioned concrete: During the production process, the concrete beam is cast around pre-tensioned tendons, which are subsequently released and secured.

The tendons are put into a pre-designated conduit once the concrete is cast (poured and started the curing process) on site, and then released and secured. Unbonded post-tensioned concrete is similar to bonded post-tensioned concrete except that it allows for tendons to move within the concrete and can be changed at a later period.

Wood

Because of its high availability, durability, and strength, wood has been utilised as a building material for centuries. Wood is classed according to the tree from which it came; whether it is a hardwood or a softwood, this classification does not always correspond to its engineering properties. Balsa, for example, is categorised as a hardwood, yet its properties make it softer than many commercially available softwoods.

Wood, as an organic substance, has a tendency to adapt to its surroundings, particularly climate conditions, expanding in wet climates and contracting in dry ones. Figure 4 depicts a timber frame that will serve as the foundation for a building.

Beam Characteristics

There are a number of features of a beam that an Engineer should be aware of since they determine how the beam behaves when loaded and, in turn, represent potential failure locations or mechanisms. The most important are:

The second moment of area (also known as the second moment of inertia) is a measure of the resistance of the shape of the beam to bending. It is determined by the cross section profile of the beam.

Bending moment: can be used to compute locations prone to highest bending forces and thus most likely to yield, as seen on a bending moment diagram and commonly connected to beam deflection. It also shows which parts of the beam are in compression and which are in tension.

Beam deflection: Beam deflection is unfavourable and is proportional to the bending moment. Shear diagrams are used to show stress concentrations along a beam and to indicate places of highest shear pressures where the beam is more prone to fail due to shear.

Second moment of area

The second moment of area (I) is a form feature that predicts the beam’s resistance to bending and deflection. It’s derived from the beam’s physical cross sectional area and connects the profile mass to the neutral axis (this being a region where the beam is subject to neither compression or tension, as labelled in figure 5.).

It is dependent on the loading direction; for most beams, save hollow and solid box and circular sections, the second moment of area will differ depending on whether they are loaded horizontally or vertically.Figure 5:

a) a simple supported beam of length l with no force;

b) a simple supported beam with a point load (force) F at the centre creating

The I – beam, also known as the Universal beam, has the most efficient cross sectional profile because the majority of its material is positioned away from the neutral axis, resulting in a large second moment of area, which increases stiffness and thus resistance to bending and deflection. The following formula can be used to compute it: This is only ideal for loading parallel to the web, as seen in figure 6, as loading perpendicular to the web would be inefficient.

Box Section

Both laterally and vertically, the box section has the most efficient profile. It is less stiff because the second moment of area has a lower value. It can be calculated with the following formula:Shear and bending moment diagrams

Bending moment and shear diagrams are typically drawn alongside a beam profile diagram, as illustrated below, in order to accurately represent the beam’s behaviour.a) depicts a beam with a uniformly distributed load (udl) of size w applied along its length, l. The total force on the beam is wl.Reaction forces R are used to support the beam.Any point along the beam is represented by distance x.

These calculations are specific to the beam condition depicted, which is a uniformly distributed load with simple supports. A different formula will be necessary for a cantilever beam or one with varying degrees of freedom at the supports (this refers to restraints in the horizontal direction subjecting the beam to a rotating moment at this place). Although all formulas can be determined from first principles, look-up tables such as those found in “Roark’s formulas for stress and strain” can be used for convenience.

The Young’s modulus, E, and second moment of area, I, are shown to be dependent on the maximum beam deflection,? MAX, and deflection at distance x,? x, whereas shear force and bending moment are independent of these beam characteristics.

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