# Why does prestressing in trees make them stronger?

In "Structures: or Why Things Don't Fall Down" James Gordon talks about trees being prestressed with the outside in tension and the inside in compression. Why does it make a tree stronger to have it in tension on the surface and compression at the core and how do I read the graph in the first picture below?

Also, in the third pic it talks about doing the reverse with a concrete beam. How do you create tension using rebar?

# How To Read the Graphs

The graphs are plots of stress versus position. Stress is the force per unit area exerted on a material. Positive values are tension and negative values are compression.

The first (leftmost) plot shows the normal stress vs. position that would be expected from bending if there was no pre-stress. The middle plot shows the pre-stress on the tree. The rightmost plot shows the superposition, or sum, of the bending stress and the pre-stress.

# Why Trees are Prestressed

In mechanical engineering, strength is defined as the maximum stress that a material can tolerate before failing. Every material has unique strength limits, called their ultimate strength. In some materials this is the same in tension and compression but in others the ultimate tensile and compressive strengths are different. Most metals have approximately the same tensile and compressive strength, but materials like wood and concrete fall into the latter category, where the tensile and compressive strength are significantly different.

The article you provided states that the ultimate tensile strength of wood is greater than its ultimate compressive strength. If the tree were not prestressed, the side away from the wind would always fail first, because as shown by the first plot, the maximum tensile and compressive strengths are the same in bending without pre-stress. By having tensile pre-stress on the outside of the tree, the tree can reduce the maximum compressive stress and so stand up to stronger winds. Note that as a consequence of Newton's Third Law, there must be both tensile and compressive pre-stress to balance the forces.

## In Concrete

In concrete, the situation is different because concrete has essentially zero tensile strength. Prestressed concrete avoids failure by having the steel rebar in a state of tensile pre-stress and the concrete in a state of compressive pre-stress. This can be accomplished either by stretching the rebar while the concrete is being poured, or by pouring concrete around cables coated to prevent the concrete from sticking to the cable. In the latter case, the cables are tightened to prestress the concrete after it has cured.

• Yes, Newton's Third Law is very important because it shows that $\int \sigma\, dA = 0$, where $\sigma$ is stress, which means that any plot of stress must have an equal area above and below the x-axis (although for 3D objects, the plot of stress is properly 3D as well, so that may not be true for the 2D representation of the 3D stress) – regdoug Apr 18 '15 at 16:08
• Not all metals have equal tensile and compressive strength - cast iron, for example, has a higher safe compressive capacity than tensile, basically for exactly the same reason as concrete does - it's a brittle material with micro-defects that propagate under tensile stress. – achrn Apr 18 '15 at 21:36
• @achrn Good point, I was only using metals as an example and have revised my answer to be more clear. – regdoug Apr 18 '15 at 21:57