ORGANIZATIONAL BEHAVIOR

Plastic analysis and designIntroduction the ultimate load rather than the yield stress is regarded as the structural design criterion Since the ultimate load is reached when the stress in steel is in the plastic range, this method is called plastic design method. The strength of steel beyond the yield stress is fully utilized.Basis of plastic theoryMaterial behaviourDuctility of Steel a structure only collapses when it has exhausted all means of standing de Courcy

Material behaviour

ABCDEFAB- Elastic curve, B and C Yield PointsCD- Plastic strain flowDE- Strain HardeningEF- NeckingF- FailureStress-Strain Curve (Actual Curve)Cross sectional behaviour subjected to bending The cross section does not yield simultaneously through the section, but outside the region yield first, redistributing stress and delaying failure beyond what is predicted by elastic method. The stress distribution from the neutral axis is the same as the shape of the stress-strain curve of the material. Plastic bending An applied moment causes the outside fibers of a cross section to exceed the materials yield strength The assumption made to pure bending remain valid, except that the stresses need not be proportional to the strain.

Moment-curvature characteristics of general cross section

Moment-curvature characteristics of general cross section

The corresponding stresses at five different stages of loading are examined below:Stage 1 Elastic Behaviour The applied moment causes stresses over the cross-section that are all less than the yield stress of the material. Stage 2 Yield Moment The applied moment is just sufficient that the yield stress of the material is reached at the outermost fibres of the cross-section. All other stresses in the cross section are less than the yield stress. This is limit of applicability of an elastic analysis and of elastic design. Since all fibres are elastic, the ratio of the depth of the elastic to plastic regions, = 1.0 . Moment-curvature characteristics of general cross section

The corresponding stresses at five different stages of loading are examined below: Stage 3 Elasto-Plastic Bending The moment applied to the cross sect ion has been increased beyond the yield moment. Since by the idealized stress-strain curve the material cannot sustain a stress greater than yield stress, the fibers at the yield stress have progressed inwards towards the center of the beam. Thus over the cross section there is an elastic core and a plastic region. The ratio of the depth of the elastic core to the plastic region is 1.0< < 0. Since extra moment is being applied an d no stress is bigger than the yield stress, extra rotation of the section occurs: the moment rotation curve losses its linearity and curves, giving more rotation per unit moment (i.e. looses stiffness). .Moment-curvature characteristics of general cross section

The corresponding stresses at five different stages of loading are examined below:Stage 4 Plastic Bending The applied moment to the cross section is such that all fibres in the cross section are at yield stress. This is termed the Plastic Moment Capacity of the section. Since there are no fibres at an elastic stress, = 0 . An attempt at increasing the moment at this point simply results in more rotation, once the cross-section has sufficient ductility. That is, in steel the cross section classification must be plastic and in concrete the section must be under-reinforced.Stage 5 Strain Hardening Due to strain hardening of the material, a small amount of extra moment can be sustained.Moment-curvature characteristics of general cross section

Fully plastic moment of a section

The plastic moment capacity Mp, of a section is defined as the moment of resistance of a fully plasticised or yielded cross sectionConsider the stress distribution of a fully yielded or plasticised section of a beam:AcAtytycCTfyfyN.A.(plastic neutral axisFully plastic moment of a section

From the previous figure, The equilibrium condition that the total axial force on the section be zero requires that the total force in compression and the total force in tension over that section are equal.Therefore,A = Ac +At or At = Ac = (A/2)Ac = area of cross section in compressionAt = area of cross section in tensionA = total area of the cross section

In a fully plasticised section, the neutral axis divides the section in two equal parts, and is called plastic nuetral axis. For symmetric cross sections, the plastic neutral axis is same as the elastic neutral axis which passes through the centroid of the cross sectionFully plastic moment of a section

The total compressive and tensile forces which are equal and opposite in direction form a couple that provides the plastic moment resistancePlastic moment resistance, MpMp= (fy * Ac)(yc) + (fy* At)(yt)= fy(A/2)(yc + yt) = fy*ZpWhere,Zp= the plastic modulus of the section= A (yc + yt)/2yc, yt= center of gravities of the area of the cross sectionFully plastic moment of a section

Plastic ModulusThe plastic modulus of a completely yielded section is defined as the combined statistical moment of the cross sectional areas above and below the plastic neutral axis or equal area axis. It is the resisting modulus of the plastic section.Shape factor

The ratio of the plastic modulus to the elastic modulus is called the shape factor of the cross section.This constant varies with the shape of the cross section.Significance of Shape factor

The value of the shape factor gives an indication of reserve capacity of a section from the stage of onset of yielding at extreme fibers to full plastification of the section. Thus, it is a good measure of the efficiency of the cross section in bending.A section with higher shape factor is generally more ductile and gives greater deflection at collapse.Significance of Shape factor

3. A section with higher shape factor provides a longer reaction time or warning before impending collapse.

4. Greater is the shape factor, higher will be the collapse load factor.Review questionsDefine elastic modulus and plastic modulus of a section. How are they related to each other?What is shape factor and its significance to design of steel structure? What is the reason for the difference of shape factor of various sections (e.i. rectangle, circle, I beam)?

Review questionsI. Consider the rectangular section belowbhDetermine the general expression for computing the shape factor of the given section.Review questionsII.Determine the shape factors of the following sections:Triangular section with base b and height hHollow and solid circular sections with external and internal diameters D and d, respectively