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Mechanics of bending
Course: Calculus-based Physics 1: (it6100)
56 Documents
Students shared 56 documents in this course
University: AMA University and Colleges
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Mechanics of bending when 𝑴𝑴>𝑴𝑴𝒚𝒚
Some materials, like steel, tends to exhibit elastic-plastic behavior when the material reaches 𝜎𝜎𝑦𝑦.
If the applied 𝑀𝑀=𝑀𝑀𝑦𝑦 is just enough to cause yielding in the top and bottom fibers then 𝑀𝑀𝑦𝑦=
1
6𝑏𝑏ℎ2𝜎𝜎𝑦𝑦
Mechanics of bending when 𝑴𝑴=𝑴𝑴𝒑𝒑
When an elasto-plastic material is subjected to 𝑀𝑀>𝑀𝑀𝑦𝑦, its stress distribution diagram reaches a
constant value – 𝜎𝜎𝑦𝑦. When all stresses in a section are equal to 𝜎𝜎𝑦𝑦, the material is said to have
reached its plastic state. The load that allows the material to reach this state is the plastic moment.
Generally,
and the moment equation is,
𝑀𝑀=𝑇𝑇1�2
3𝑦𝑦𝑌𝑌�+𝐶𝐶1�2
3𝑦𝑦𝑌𝑌�+𝑇𝑇1�𝑦𝑦𝑌𝑌+1
2�ℎ
2− 𝑦𝑦𝑌𝑌��+𝐶𝐶1�𝑦𝑦𝑌𝑌+1
2�ℎ
2− 𝑦𝑦𝑌𝑌��
𝑀𝑀=1
4𝑏𝑏ℎ2𝜎𝜎𝑦𝑦�1−4
3𝑦𝑦𝑌𝑌
2
ℎ2�
𝑀𝑀=3
2𝑀𝑀𝑌𝑌�1−4
3𝑦𝑦𝑌𝑌
2
ℎ2�
Notice that,
𝑀𝑀=3
2𝑀𝑀𝑌𝑌�1−4
3𝑦𝑦𝑌𝑌
2
ℎ2�
as M increases, 𝑦𝑦𝑌𝑌 approaches zero. This would render the material perfectly plastic.
𝑀𝑀𝑝𝑝=1
4𝑏𝑏ℎ2𝜎𝜎𝑦𝑦
𝑀𝑀𝑝𝑝=3
2𝑀𝑀𝑌𝑌 Plastic moment
Beams used in steel buildings are designed to resist plastic moments. Code (e.g. AISC) lists a
design property of beam called the shape factor (k)
𝑘𝑘=𝑀𝑀𝑝𝑝
𝑀𝑀𝑦𝑦
