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Unformatted text preview: Lecture 10 1 Lecture 10 1 Outline: Inelastic bending Partially Plastic Section Fully Plastic Section Shape Factor  Biaxial bending and axial load 1. Inelastic bending The kinematic assumptions imposed in Lecture 4 are still valid which only concern the strain distribution. Therefore, the linear strain distribution over the crosssection is still valid (Fig. 1) The stressbending moment relations developed in Lecture 4, however, were based on the assumption that the material is linear elastic; hence Hookes law was applicable. Under the acting bending moment, if the stress at a point causes the material point to yield, then the stress is not proportional to the strain. In this case, the flexural stress formula is not valid at every point on the crosssection (Fig.2) My I = Plastic analysis must be used to determine the stress distribution over the crosssection. y x N.A y x N.A x x E = Fig. 1. Strain distribution Fig. 2. Stress distribution From lecture 4, it can be recalled that for any given normal stress distribution the Resultant Moment acting on the crosssection can be calculated by integrating the moment of the stress acting on an infinitesimal area dA throughout the crosssection. A M ydA = 1 Mechanics of Solids Dr Emre Erkmen Lecture 10 2 Consider the response of a simplysupported beam with a rectangular crosssection loaded incrementally in four stages. The material is elasticperfectly plastic both in tension and compression as shown in the Fig. 3. P y 1 2 3 4 y Crosssection Fig. 3. Loading of a simplysupported beam in four stages At the first stage the loading P 1 is such that none of the material points reaches the yield limit. Therefore, stress is linearly distributed across the section at the midspan as shown in Fig.4. Fig. 4. The section is fully elastic under bending moment 1 A M ydA = At the second stage the loading P 2 > P 1 caused the strains at the top and bottom fibres reach the yield limit y . Therefore, the maximum stress at the midspan reached to y , however since the stresses below the top fibre and above the bottom fibre are less than y the stress distribution is still linear as shown in Fig. 5. When maximum stress reaches the elastic limit we call the acting moment yield moment of the section , i.e. 2 y M M = Fig. 5. Maximum stress reached the elastic limit under bending moment 2 A M ydA = Lecture 10 3 At the third stage the loading P 3 > P 2 causes strains at top and bottom fibres go beyond the elastic limit. For this fibres however, the stress cannot increase further. The section is partially plastic as shown in Fig. 6. > y M 3 N.A P 3 = y Fig. 6. Strains at some portion reached beyond the elastic limit under bending moment 3 A M ydA = At the fourth stage the loading P 4 > P 3 causes the stresses to reach the yield limit y at every point of the section at the midspan. The section is fully plastic as shown in Fig. 7. When section is fully as shown in Fig....
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