Plastic bending - analytical calculations

[FEAnalyst]

TL;DR

What can be calculated analytically for a beam subjected to plastic bending?

Hi,

I have a question regarding plastic bending of beams (assuming bilinear elastoplastic material - with or without hardening) . In literature one can find calculations of load capacity for those beams. But what else can be calculated in such case ? Stresses ? Deflection ? Where can I find appropriate formulas for some simple sections such as rectangular one?

 

[jrmichler]

In plastic bending, part of the beam is experiencing elastic strain, part is experiencing plastic strain, and the neutral axis moves toward the inside of the bend. A quick look in Metal Forming - Mechanics and Metallurgy by Hosford and Caddell found procedures for calculating bend radii and springback. I see that this book is now in the fourth edition: https://www.amazon.com/dp/1107004527/?tag=pfamazon01-20. If you really want to analytically calculate metal deformation with plastic strain, get this book.

I took the class in the early 1990's, and don't remember much because I never used it. I do remember the need to separately integrate stress vs distance from the neutral axis for elastic and plastic strain, which implies that a single formula might be difficult to derive. Especially if you want to include strain hardening, strain rate hardening, and neutral axis shift.

 

[FEAnalyst]

Problem solved - in an old Polish book ("Elasticity and plasticity. A selection of tasks and examples" by W. Krzys and M. Zyczkowski) , I've found a formula for the deflection of a cantilever beam having rectangular section and subjected to point load at the free end (assuming elastic perfectly-plastic material): $$y_{max}=\frac{20 b^{2} h^{3} Q^{3}}{27 E P^{2}} - \frac{2 \sqrt{3} b h L Q^2}{3 E P} \cdot \left( 1- \frac{P L}{b h^2 Q} \right)^{\frac{1}{2}} - \frac{4 \sqrt{3} b^2 h^3 Q^3}{9 E P^2} \cdot \left( 1- \frac{P L}{b h^2 Q} \right)^{\frac{3}{2}}$$
where: $b$ - beam section's width, $h$ - half of beam section's height, $Q$ - yield strength, $E$ - Young's modulus, $P$ - force magnitude, $L$ - length of the beam.

Stresses in the plastic regime can't exceed yield stress in this case since the material is perfectly plastic.

More complex cases (simply-supported beams, materials with hardening, perhaps other types of cross-sections) should also be possible to solve analytically but would require more extensive derivations.