The illustration above shows two structural steel rods. The rod on the left is three inches in diameter. It will pull apart at 254,000 pounds. The rod on the left is the same, except the ends are enlarged to five inches diameter. When you pull on the ends of the rod on the right it will pull apart at measurably less than 254,000 pounds.
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Greg provided the correct answer, which is “stress concentration.” In the configuration at the left the stress is uniformly distributed, but on the right the stress is concentrated along the outer edge. Some background.
Stress is load concentration, mechanically equivalent to pressure. It is, for example, pounds per square inch. Strain is an objects response to stress. Strain is deformation induced by stress. Strain is a dimensionless quantity, such as inches per inch. It is deformation in inches per total length in inches.
It is not stress that results in failure of a mechanical component, but ultimately strain. For a material, structural steel for example, as you exert a force to pull the sample apart it stretches. The effect is more pronounced in objects made of rubber, and the resulting strain of a steel part may not be apparent. A solid steel bar stretches just like a spring when you pull on the ends, usually not enough for you to notice. Unless you pull really hard. As you pull harder, eventually the steel stops acting like a spring and ceases to offer additional resistance the more you pull. While the load stays constant the steel continues to stretch. The steel has entered into the plastic deformation range.
If you continue to pull you will pull the steel bar apart. This is called rupture, and it’s the failure mode in tension for materials like steel.
When you pull on the sample on the right what you might not notice is that the strain in the center of the bar is not as much as the strain around the outer regions. That is because the extra thickness provided by the shoulder keeps the region adjacent to the narrow section from moving in response to the load. In the shoulder region the center is pulled down farther than the outer region. Within the thin section of the rod the outer region adjacent to the shoulder now stretches more. The strain is greater. The sample on the right, in this example, will start to rupture before the load reaches 254,000 pounds. When the outer region starts to rupture the load shifts to the inner region, and the bar pulls apart before the load reaches 254,000 pounds.
What you see in aircraft structures and in other structures that operate near the limits of materials is great care to reduce stress concentration. To the extent possible, sharp concave corners are eliminated. Structural members that neck down to smaller dimensions do so in a gradual manner.