The Simplest Form
In the simplest form, post and lintel structures consist of two vertical support columns holding up each end of a horizontal lintel, or header, as a load bearing beam. Although young children playing with blocks eventually design their own post and lintel architecture, the actual structural form is a little less trivial. These trabeated systems consisting of horizontal beams supported by structural posts must be designed with the allowable deformation and compressive, tensile, shear, and buckling stress limits in mind.
Design Analysis For Structural Posts
The axial compressive stress experienced by the column in post and lintel systems is: Stress = Force / Cross-sectional Area, or s = F/A. In most architectural designs the posts are usually much longer in length than in width or depth, which classifies them as slender columns. In these cases, it is not enough to design for axial compressive strength only, as the phenomenon known as buckling must also be taken into account. Buckling in a column is similar to bending in beams with an important difference. Beams will deflect in a known direction, opposite the application of a loading force. Columns however will buckle in the direction of weakest material strength and/or least lateral support. This is due to the consideration that materials are never perfectly homogenous nor loaded with perfect symmetry, giving rise to developing bending moments under axial compressive loading. Column buckling is also progressive and can easily give rise to catastrophic failures as a result. To ensure a structural design takes into account buckling failure with an adequate safety margin, an allowable bucking load is generally calculated as:
Allowable buckling load = k x d x A, where:
- k = scaling factor, based on slenderness ratio
- d = allowable compressive stress
- A = cross sectional area
The slenderness ratio is determined by the column length, the cross sectional dimensions, and how the ends are fixed or attached. The end attachment can range from total freedom of movement in rotation and lateral movement to a fully restrained condition. Usually, however, a pinned condition is assumed which restricts rotation to one direction with no lateral movement allowed. This leaves the scaling factor, k, dependent solely on geometric factors. If the allowable buckling load is less than the anticipated actual load, a change in column geometry or materials must be made.
Design Analysis For Lintels
- duration, magnitude, and type of loading
- clear span required
- cross section of the beam
- beam composition
Typically it is the lintel that will limit the capacity and span of a load bearing header, for example, generally due to material strength limitations.
There are three load factors for a lintel: bending stress, shear stress, and allowable deflection. The bending stress develops as a couple, with compressive stresses above the neutral beam axis and tensile stresses developing below the axis. This stress is determined as:
Max. allowable bending stress = Mmax / Z where:
Mmax = maximum bending moment and Z = section modulus
The maximum bending moment occurs where the tensile (or compressive) stress is greatest in the beam, and the section modulus depends on material composition and geometry.
Horizontal shear stresses develop in response to the changing distribution of bending stresses in the beam. This can be illustrated by bending a deck of cards held at one edge. The free card edges will displace relative to one another, which represents the distribution of shear stress if the cards were totally restrained from slippage. Beam shear stress is generally calculated as:
t = Q / A
where t is the average shear stress, Q is the horizontal shear force, and A is the transverse area. The actual shear stress is a distribution with minimum shear at the top and bottom layers and the maximum shear occurring at the neutral axis, where bending stresses transition between compressive and tensile. The maximum allowable deflection can be a consideration to prevent failure of other elements supported by the lintel, such as cracked masonry, sagging floors, etc. This is typically a numerical limit proportional to the length, such as L/240, and is tested against the following calculation:
Expected maximum deflection = C1 x ( WL3 / C2)
Where C1 is a constant dependent on end support and characteristic loading, W is the load, L is the span, and C2 is a modulus and geometric constant. Clearly shorter spans (smaller L), higher modulus and stiffer geometry (greater C2) yields smaller deflections.
The Not So Simple Form
These general calculations only scratch the surface of design for post and lintel structures. Eccentric loading, end fixity, composite materials, complex geometries, multiple support columns and lintels, seismic analyses, construction workmanship, and a host of other considerations affect the final design of the trabeated system. When actual construction is contemplated a competent structural engineer should always be consulted, since the deceptively simple design of these structures is clearly not child’s play.
About The Author:
John Moehring is a practicing Engineering Technologist in civil, geological, biological, and electrical engineering fields. And one of these days he may actually get it right.