As you can imagine the capacity of a wooden post is not a single number that you can write
down and memorize but it’s rather design-specific, and changes based on the loading conditions.
In this video, we will see what these limits are and what has the biggest influence on them.
We will investigate a 4 by 4 timber post whose dimensions are 89 by 89 mm.
But to understand the limits of the carrying capacity of a column, first we need to know
how columns actually fail.
In general, a column can fail in two main modes: buckling or material yielding and crushing.
Material crushing can be seen only on short and stocky columns, this failure consists
of the wood grains being compressed to the point where they start to wrinkle locally.
Buckling on the other hand is a sudden lateral failure caused by a loss of stability.
Buckling may sound like a complicated phenomenon but in fact, it is very intuitive, we all
have intuitive knowledge as to what would make an object susceptible to buckling.
This can be illustrated by compressing a ruler.
If the ruler is short, it can carry a significantly higher pressure before it breaks.
Consequently, a long ruler is much easier to bend or buckle under even a small compressive
load.
That is, the longer the column the weaker it becomes.
This same intuition also applies to the design of columns that carry dozens of floors and
thousands of tons in the real world.
This intuitive idea was turned into sound mathematics by the Swiss mathematician Leonhard
Euler and has ever since been used in the design of columns [1].
Here in this video, we will reach this same equation by simply using our intuition.
As we already figured out, the length of the column is inversely proportional to the column’s
capacity.
To reiterate, the longer the column is the less load it can carry.
Furthermore, we know that the stiffness of the material has to also play a role because
stiffer materials are harder to bend.
The same goes for the shape of the column whose effects are captured by the moment of
inertia of the column.
If you want to learn more about the effects of the shape, check out our previous video
on “the secret of the I-beam”.
Going back to the equation we can see an additional pi squared term that arises from the solution
of the differential equation and is related to the shape of the buckled curved which follows
a sinusoidal function.
The last remaining factor is related to the end conditions of the column which we will
assume to be pin-connected.
This means that the column is held in place but is allowed to rotate about its ends which
makes the column more susceptible to bending.
Fully fixing the ends would increase the carrying capacity of the post however providing a full
moment fixity is difficult to achieve when it comes to wood connections.
Semi-rigid connections are much more common but for simplicity, we will assume no rotational
stiffness in the connection which would also produce a conservative estimation of the load-carrying
capacity.
Based on Euler’s Formula the only parameter that is not fixed and can affect the load-carrying
capacity is the length L. Theoretically speaking as the length gets smaller, the load capacity
will increase indefinitely.
But of course, this is not the case because at some point the compressive capacity of
the material itself starts to govern the strength of the column rather than the buckling being
the governing factor.
For that reason, the Euler’s curve is capped at the limit strength of the material.
Failure in the material takes the form of local budging, crushing, or wrinkling of the
wood grains.
The right side of the curve corresponds to the global buckling of the entire member in
essence a loss of stability.
If we lived in a perfect world our analysis would have stopped here.
However, wood is far from a perfect material and neither is our construction practice.
Unless the compressive load is applied perfectly at the centroid of our column, the Euler formula
is not exactly correct.
Applying the load even with a slight eccentricity creates an undesired bending of the column
that lowers its capacity.
For example, applying the load only 10mm off-centre lowers the capacity by nearly 50% when it
comes to timber posts.
But that’s only imperfections due to construction, the wood itself has many irregularities such
as knots, warping, irregular grains, and so on.
Furthermore, wood suffers from creep and rotting which are significantly worsened if the member
is exposed to a humid environment.
Codes and standards around the world try to simplify these conditions with factors that
engineers could select from a design manual.
These factors and equations are mainly based on the Euler Formula however they are modified
to fit empirical data.
As an example, we used the Canadian Wood Design Manual for comparison to the theoretical equation
of a perfect wood.
The Eurocode 5 [4], and American Wood Design Standard conclude similar results.
We assumed a 4 by 4 spruce-pine-fir timber post of a standard grade.
For the viewers outside of North America that is an 89 by 89 mm square post with a compression
strength parallel to grain of 10.8 MPa.
After performing the calculation as per the provisions in the design manual it turned
out that the code is conservative for longer columns however it allows higher loads for
shorter columns that could not buckle.
This is because when buckling is not an issue, the material can be loaded past its yielding
point but still away from its ultimate capacity which is associated with a complete loss of
strength.
Any accidental eccentricity is not directly accounted for in the code however its effects
are captured by sufficient safety factors built into the codified equations.
As a result, the maximum compressive load for a timber post of a length of 1m is around
8.7 tonnes, around 5 tonnes for a 2m long post, and only 2.25 tonnes for a 3m long column.
These numbers clearly illustrate the effects of buckling for longer members subjected to
compressive loads.
As always, this video is a short and simplified version of an actual column design practice
with its sole focus on the compressive load.
The design of end connections is ignored in this video.
Further factors such as duration of the load, treatment of the wood, combined loading conditions
and so on add additional complexity and considerations that need to be accounted for.
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