Fundamentals of Beam Design

Choice of material

Ultimately the choice of material determines the strength of the beam, that is how much load it can support before failure occurs and generally relates to its Young’s modulus (E). However, most materials exhibit different behaviour when subject to compression and tension, which must be accounted for in its design.

The four most significant materials used in beam design, that will be examined further here are: cast iron, steel, concrete and wood. Others include carbon fibre and composite materials.

Cast Iron

Cast iron was recognised as a building material in the late 1700s when during the Industrial Revolution a method of production (by blast furnace), this being both economical and practical, was developed. Cast iron is generally strong in compression but not tension so initial applications were in the form of bridges and other structures requiring short members existing in compression. Coalbrookdale iron bridge, built c.1770 provides and excellent example of this, figure 1.

Figure 1 – Coalbrookdale Iron Bridge

Young’s modulus of cast iron: E ? 211 Gpa means it is relatively strong yet simultaneously brittle by nature. This undesirable characteristic lead to a number of catastrophic collapses of early bridges and limited its use as a building material, despite the ability to form beams of varying shapes and elaborate designs. In spite of these negative connotations it was viewed as a revolutionary building material as it enabled the replacement of traditionally masonry with sleek, slender iron beams.

Steel

In the late 1880s, Henry Bessemer developed a method for mass producing steel – a move that signified the dawn of the skyscraper. This strong material with a Young’s modulus of: E ? 800 Gpa, could now be feasibly formed into I-beams and steel columns. Combining a series of these I-beams and steel columns it was possible to construct a structural, steel core of great height (figure 2) to which the floors, roof and walls of a building could be attached, giving birth to the skyscraper. This method was used to construct the Empire State Building, New York, which was to remain the tallest building in the world for over forty years.

Figure 2 – steel core construction, New York c.1930

Using steel as a building material is not without its disadvantage, good in both compression and tension due to the ability to dictate a specific cross section profile, it softens at high temperatures so to prevent the collapse of buildings in the event of fire it tends to be encases in a fire resistant material.

Another advantage of steel is the ability to vary its composition and hence change its physical properties. Typically an alloy of iron and carbon, the carbon content commonly being between 0.2% and 2.14%, the addition of manganese will provide a significant increase in the strength at a modest cost. Similarly the addition of chromium or nickel will harden the steel and increase its ability to resist corrosion. Other alloys can be added accordingly to enhance certain physical properties or characteristics.

Concrete

Both the Ancient Egyptians and Romans used concrete in their buildings, however after the collapse of the Roman Empire its secrets were almost lost until its rediscovery in recent times. The application of a patent for the manufacture of Portland cement in 1824 signifies one of the important milestones in history of concrete and since this time significant advancements have been made with the development of pre-stressed concrete beams.

Concrete contains water, aggregate and cement. The aggregate tends to be gravels (comprising of crushed rock and sand) that form the bulk volume of the concrete. Cement, commonly Portland cement, bonds together the constituents providing the strength and durability of the concrete.

Concrete has a wide range of functions and is particularly suited for applications where it is subject to compressive forces, such as integral building columns, yet with reinforcements this range can be expanded to include thin-shell structures as show in figure 3.

Figure 3 – El Palau de les Arts Reina Sofia, Valencia

Pre-stressed concrete contains tendons (usually made of steel), as concrete is generally only good in compression these tendons offset the tensile stress a concrete member would have otherwise experience when subject to a load. There are three main types of pre-stressing concrete beams, pre-tensioned, bonded post-tensioned and unbonded post-tensioned:

1. Pre-tensioned concrete: the concrete beam is cast around already tensioned tendons in the manufacturing process; these are then released and secured.
2. Bonded post-tensioned: the tendons are inserted into a pre-designated duct after the concrete in cast (poured and begun the curing process) on site; these are the released and secured.
3. Unbonded post-tensioned concrete: these are the same as bonded post-tensioned except for the fact that they allow for movement of the tendons within the concrete and can be adjusted at a later date.

Wood

Wood has been used for centauries as a building material due to its high availability, durability and strength. Wood is classified according to the tree of its origin; it being a hardwood or softwood, this classification does not necessarily represent its engineering properties. For example, Balsa is classified as a hardwood yet its characteristics mean it is actually softer that many commercial types of softwood.

As an organic material, wood has a tendency to adapt to its surroundings specifically climate conditions where by it will expand when moisture is present and contract in dryer climates. Figure 4 shows a wooden frame, which will form the integral structure of a building.

Figure 4 – wooden framed house

Beam Characteristics

There are a number of properties of a beam that an Engineer should be aware of as they dictate beam behaviour when subject to a load and ultimately represent possible areas or mechanisms for failure. The main ones being:

• Second moment of area (also referred to as the second moment of inertia): this depends on the cross section profile of the beam and is a measure of the resistance of the shape of the beam to bending.
• Bending moment: usually illustrated on a bending moment diagram, and often related the deflection of the beam, can be used to calculate regions subject to maximum bending forces and consequently most likely to yield. It also illustrates which sections of the beam are in compression or tension.
• Beam deflection: beam deflection tends to be undesirable and correlates to the bending moment.
• Shear diagrams: these are used to illustrate stress concentrations along the beam and provide a means to identify areas of maximum shear forces where the beam is more likely to fail by shear.

Second moment of area

The second moment of area (I) is a property of the shape used to predict the resistance of the beam to bending and deflection. It is calculated from the physical cross sectional area of the beam and relates 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 dependant on the direction of loading; for most beams except both hollow and solid box and circular sections, the second moment of area will be different when loaded from a horizontal or vertical direction.

Figure 5 – a) simply supported beam of length l with no force; b) simply supported beam subject to point load (force) F at centre creating bending.

The second moment of area can be calculated from first principles for any cross section profile using the equation:

However, for common beam profiles standard formula are used:

I – beam / Universal beam

Figure 6 – I-beam cross-section profile with loading parallel to web.

The I – beam or Universal beam has the most efficient cross sectional profile as most of its material is located away from the neutral axis providing a high second moment of area, which in turn increases the stiffness, hence resistance to bending and deflection. It can be calculated using the formula:

As shown in figure 6, this is only suitable for loading parallel to the web, as loading perpendicular to the web would be less efficient.

Box section

Figure 7 – Box section cross-section profile

The box section has the most efficient profile in loading both horizontally and vertically. It has a lower value for second moment of area so is less stiff. It can be calculated by using the formula:

Bending moment and shear diagrams

Bending moment and shear diagrams are typical drawn alongside a diagram of the beam profile as shown below, this enables an accurate representation of the beams behaviour.

a) represents a beam subject to a uniformly distributed load (udl) of magnitude w, across its length, l. Total force on beam being wl.

The beam is simply supported with reaction forces R.

Distance x represents any point along the beam.

b) shear force diagram shows the regions of maximum shear, for this beam these correlate to the reaction forces.

The slope of the shear force diagram is equal to the magnitude of the distributed load.

A positive shear force will cause the beam to rotate clockwise and a negative shear force will cause the beam to rotate in an anticlockwise direction.

c) maximum bending moment occurs when no shear forces exist on the beam.

As the beam is simply supported, that is only subject to vertical reaction forces, no bending moment is experienced at these points. If the beam were restricted as in a cantilever situation then bending moments would be experienced at either end Correlating to the diagrams of beam loading, shear force and bending moments maximums and values at distance x along the beam can be calculated using the following formula:

Reaction force and maximum shear force and Shear force at distance x

Maximum bending moment and Bending moment at distance x

Maximum deflection and Deflection at distance x

These formulas are specific to this beam situation, that is a uniformly distributed load with simple supports as shown. For a cantilever beam, or one with varying degrees of freedom at the supports (this refers to restrains in the horizontal direction subjecting the beam to a turning moment at this location) then different formula will be required. All formula can be calculated from first principles but for convenience look-up-tables such as those contained in “Roark’s formulas for stress and strain” can be utilised.

The equations for maximum beam deflection, ?_MAX and deflection at distance x, ?_x are shown to be dependant on the Young’s modulus, E and second moment of area, I, where as shear force and bending moment are independent of these beam characteristics.