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TCW 3205 STRUCTURAL ANALYSIS II1. Plastic Analysis of StructuresElastic-plastic stress-strain relationshipPlastic bending without axial forceEffect of axial load on plastic momentCollapse loads and collapse mechanisms of beams and framesApplication of the Virtual Work and Static Method

2. Dynamic AnalysisSingle and 2 DOF systems (damped and undamped)Energy MethodRayleigh MethodCourse Synopsis3. Analysis of PlatesAssumptions in thin plate theoryMoment-curvature relationsEquilibrium of an elementDifferential equation of plate bendingBoundary conditionsNavier Solution for a plate simply supported on 4 sidesLevys Method for plate with various support conditionsTransformation of moments and curvature

Course Synopsis4. Yield line analysisYield line patternsGuidelines for choosing yield line patternsAssumptions used in yield line analysisVirtual Work methodEquilibrium Method

5. Laboratory and project workCourse SynopsisTypical tension test for mild steel

5Permanent deformation after straining beyond yield point

when specimen is loaded and unloaded in elastic range deformations are recoverable ( OC)

however beyond C deformations are irreversible (ST)- permanent set observed

Permanent deformation after straining beyond yield point

Plastic Analysis: the what and the why.Design based on ELASTIC behaviour conservative- stress not allowed to exceed yield stress divided by a factor of safetyResults in large cross-sections

But design based on PLASTIC behaviour is optimal & more economicalMakes full use of strength of sectionPlastic analysis is valid for steel structuresEquivalent for concrete is ultimate limit state design

In PLASTIC analysis load required to cause collapse is calculatedCollapse load often far greater than that causing yieldingDesign ensures that applied load less than this collapse load

8Simple plastic theory based on idealised stress-strain relationship of steel

Elasto-plastic/ elastic-perfectly plastic material model upper & lower yield point neglected- single yield point

same yield stress in both tension & compression

strain hardening neglected

9Plastic Analysis- Theorems1. Uniqueness Theorem

2. Lower Bound (Safe)Theorem

3. Upper Bound (Unsafe) Theorem

UNIQUENESS THEOREMFor a structure in its collapsed state the following 3 conditions must be satisfied simultaneously

(i) Equilibrium condition bending moments must be in equilibrium with applied loads

(ii) Yield condition bending moment at any point must not exceed plastic moment at that point

(iii) mechanism condition - enough plastic hinges must be formed so that all, or part of, the structure is a mechanism

2. LOWER BOUND (SAFE) THEOREM

3. UPPER BOUND (UNSAFE) THEOREM

Upper Bound theorem generally usedFormulate possible collapse mechanism and then calculate collapse loadMany possible collapse mechanisms but the critical one is that giving the lowest value of collapse loadCheck critical mechanism using Lower Bound Theorem

e.g. Several different collapse mechanisms

Assumptions: plane sections remain plane after bending E and y are the same in both tension and compression

Bending theory equation

Plastic Analysis of Beams

Beams having one line of symmetry

Consider the beam subjected to an applied moment

N.A. passes through centroid while behaviour is elastic the stress distribution is linear maximum stress at greatest distance y1 from the N.A.When =y then M= My, yield moment

As applied moment is further increased

strain at greatest distance y1 increases beyond yield strain y but stress remains constant at yRegion of plasticity develops at outermost portion of beam cross-sectionAs applied moment further increased, stress at distance y2 also reaches yPlastic regions now at both ends of the cross-section with central elastic core

Further increase of moment causes plastic regions to extend towards centre of the cross-section until cross-section is completely plasticAt this stage M= Mp, plastic momentWe are concerned with finding the value of Mp

But where is the N.A. At this stage?

F = 0;

if total cross-sectional area is A

i.e. Plastic N.A. Cuts beam cross-section into 2 equal areasfor doubly symmetric sections and singly symmetric sections where bending plane is perpendicular to axis of symmetry (e.g. Channel section), elastic and plastic N.A. Coincide

Now to calculate Mp take moments of stress resultants about N.A.

Zp is the plastic modulus of the cross-section

Ratio of plastic moment to yield moment Shape Factor

e.g. Prove that for a rectangular beam of breadth b and depth d: yield moment, My =

plastic moment, Mp =

shape factor, f =

Note that at collapse, a rectangular beam can carry a moment that is 50% greater than that at initial yielding.

e.g. Determine the shape factor for the I-section shown.(Ans: f= 1.14)