Plastic Analysis of Beams and Frames 1

Plastic Analysis (Structural & Stress Analysis 3)

Dr P. Mandal School of MACE

Hambly’s Paradox (1985) The figures on the right show two milking stools, one with three legs and one with four legs. Imagine that each must support a milkmaid who weighs 60 kg, and who always sits with her centre of gravity directly over the middle of the stool. The problem is to determine how many kg of her weight each leg in the both stools must carry. Hambly, E. C., Oil Rigs Dance to Newton’s Tune, Proceedings of the Royal Institution of Great Britain, 57, 79-104

Kazinczy (1914) z

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Tested 2 steel beams (about 6 m long), embedded at their ends in substantial abutments. After unloading permanent kinks formed at the two ends and at the centre; he called them hinges. Fixed end beam cannot collapse until three hinges have formed. Degree of clamping is irrelevant provided the embedment is strong enough to allow the hinges to develop.

Maier-Leibnitz

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Three tests on two-span beams (4.8 m). – – –

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Three supports were level. The central support was lowered to a point when the bending stress at the support reached the yield value. The central support was raised by the same amount, causing yield at the support.

Actual collapse loads were 13.1, 13.0 & 13.45 tonnes respectively. Collapse load is unaffected by initial imperfections, such as sinking of supports.

J. F. Baker & Michael Horne z z

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Technical officer to the Steel Structures Research Committee (SSRC) set up in 1929. Experimental work (1930) on a nine-storey hotel block, an office building & a block of residential flats. For the first time the stresses were measured in real structures. Real stresses did not match with the stresses calculated by the elastic methods. Initial imperfections, lack of fit were identified as the discrepancy factors. It raised a paradox whether a trivial defect can really affect the strength. Calculation of elastic stress is not relevant to the strength (1936). In 1948, a clause permitting plastic design was incorporated in BS 449 (The use of structural steel in building).

Example Compare the load carrying capacity of the two trusses. (a) Failure is deemed to take place when the yield stress is first reached. (b) Yielding is allowed in the members.

Hambly’s Paradox (1985) The figures on the right show two milking stools, one with three legs and one with four legs. Imagine that each must support a milkmaid who weighs 60 kg, and who always sits with her centre of gravity directly over the middle of the stool. The problem is to determine how many kg of her weight each leg in the both stools must carry. Hambly, E. C., Oil Rigs Dance to Newton’s Tune, Proceedings of the Royal Institution of Great Britain, 57, 79-104

Simulated Tensile Test

Ductility z

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The increase in strain from A to B before strain hardening is a measure of the Ductility of the material. For cold bending of curved beams, good ductility is required. Note the values of strain at first yield and at failure