Carabiners like this,
are marked with the force rating. The one in this picture is rated,
- 20kn the long way
- 8kn the short way
- 5kn with the gate open
Why are carabiners so much weaker when the gate is open?
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Being metal carabiners are ductile and deform under load. When the biner is under load and gate closed, the gate of the biner is captured in the nose, and forms a closed loop. When the gate is open, this loop is not closed and the nose is free open beyond its designed limits. If you load a closed biner to its gate open rating, and try to open the gate, it will probably be jammed (although due designed over engineering it may not be).
This does two things - the obvious, but less important part is the gate shares some of the load with the spine. However, more importantly, the gate stops carabiner opening further and deforming beyond its design limits. The shape of the biner means a majority of the force is transmitted in line with the spine, but the gate must be closed and locked to the nose to achieve full design strength.
Let's go to the source and watch some carabiners getting broken!
While these videos show wire-gate carabiners, the idea is the same for a solid gate carabiner like the one pictured in your question. When watching the video, note the distinctly different failure modes: an open carabiner "bends" until breaking, while a closed carabiner "stretches" until snapping. This partially explains why, all else being equal in terms of construction and materials, a D-shaped carabiner is stronger than an oval: the D-shape better aligns the forces along the spine of the carabiner, leading to "stretching" instead of "bending."
These distinctly different failure modes can even be used to autopsy a broken carabiner and determine how it was loaded when it failed. For example, see this Black Diamond QC Lab warning about the even more problematic scenario of loading a "nose-hooked" carabiner.