ADVANCED MECHANICS OF MATERIALS

1. ADVANCED MECHANICS OF MATERIALS

2. INELASTIC DEFORMATION

3. INELASTIC BEHAVIOUR OF MATERIALS • SO FAR WE HAVE NOT YET DISCUSSED INELASTIC BEHAVIOUR OF STRUCTURAL MATERIALS. • THOUGH WE THAT ALL TYPES OF MATERIALS BEHAVE ELASTICALLY AND INELASTICALLY OR IN OTHER WORDS ELASTICALLY AND PLASTICALLY. • WE HAVE ALREADY DISCUSSED THAT UNDER THE APPLICATION OF LOAD, MATERIALS CAN DEFORM ELASTICALLY AND PLASTICALLY DEPENDING UPON THE VALUE OF YIELD STRENGTH OF INDIVIDUAL MATERIAL. • IN ELASTIC DEFORMATION MATERIALS CAN ATTAIN THE ORIGINAL SHAPE UPON THE REMOVAL OF LOAD AND IN PLASTIC DEFORMATION IT IS NOT SO.

4. STRESS-STRAIN DIAGRAM • STRESS-STRAIN DIAGRAMS OF MATERIALS CAN BE CONSTRUCTED AFTER PERFORMING TENSION AND COMPRESSION TESTS AT DIFFERENT VALUES OF LOAD. • SUCH DIAGRAMS CONVEY VERY IMPORTANT INFORMATION ABOUT THE MECHANICAL PROPERTIES AND TYPE OF BEHAVIOUR. LET US TAKE THE STRESS-STRAIN DIAGRAM OF STRUCTURAL STEEL. • SUCH DIAGRAM IS DERIVED FROM MEASURING THE LOAD APPLIED ON THE SAMPLE, AND MEASURING THE DEFORMATION OF THE SAMPLE, I.E. ELONGATION, COMPRESSION, OR DISTORTION.

5. A TYPICAL STRESS-STRAIN DIAGRAM FIG.1 STRESS–STRAIN CURVE SHOWING TYPICAL YIELD BEHAVIOR FOR NONFERROUS ALLOYS. STRESS IS SHOWN AS A FUNCTION OF STRAIN. 1: TRUE ELASTIC LIMIT2: PROPORTIONALITY LIMIT3: ELASTIC LIMIT4: OFFSET YIELD STRENGTH

6. STRESS-STRAIN DIAGRAM • THE SLOPE OF STRESS-STRAIN CURVE AT ANY POINT IS CALLED THE TANGENT MODULUS; THE SLOPE OF THE ELASTIC (LINEAR) PORTION OF THE CURVE IS A PROPERTY USED TO CHARACTERIZE MATERIALS AND IS KNOWN AS THE YOUNG’S MODULUS. • THE NATURE OF THE CURVE VARIES FROM MATERIAL TO MATERIAL. THE FOLLOWING DIAGRAMS ILLUSTRATE THE STRESS–STRAIN BEHAVIOUR OF TYPICAL MATERIALS IN TERMS OF THE ENGINEERING STRESS AND ENGINEERING STRAIN, • THESE STRESS AND STRAIN ARE CALCULATED BASED ON THE ORIGINAL DIMENSIONS OF THE SAMPLE AND NOT THE INSTANTANEOUS VALUES.

7. STRESS-STRAIN DIAGRAM OF STRUCTURAL STEEL FIG 2. A STRESS–STRAIN CURVE TYPICAL OF STRUCTURAL STEEL. 1. ULTIMATE STRENGTH2. YIELD STRENGTH3. RUPTURE4. STRAIN HARDENING REGION5. NECKING REGION.A: APPARENT STRESS (F/A0) B: ACTUAL STRESS (F/A)

8. LOW CARBON STEEL GENERALLY EXHIBITS A VERY LINEAR STRESS–STRAIN RELATIONSHIP UP TO A WELL DEFINED YIELD POINT AS SHOWN IN THE FIGURE. • THE LINEAR PORTION OF THE CURVE IS THE ELASTIC REGION AND THE SLOPE IS THE MODULUS OF ELASTICITY. AFTER THE YIELD POINT, THE CURVE TYPICALLY DECREASES SLIGHTLY. • AS DEFORMATION CONTINUES, THE STRESS INCREASES ON ACCOUNT OF STRAIN-HARDENING UNTIL IT REACHES THE ULTIMATE STRENGTH. • UNTIL THIS POINT, THE CROSS-SECTIONAL AREA DECREASES UNIFORMLY. THE ACTUAL RUPTURE POINT IS IN THE SAME VERTICAL LINE AS THE VISUAL RUPTURE POINT.

9. HOWEVER, BEYOND THIS POINT A NECK FORMS WHERE THE LOCAL CROSS-SECTIONAL AREA DECREASES MORE QUICKLY THAN THE REST OF THE SAMPLE RESULTING IN AN INCREASE IN THE TRUE STRESS. • AS SHOWN IN AN ENGINEERING STRESS–STRAIN CURVE THIS IS SEEN AS A DECREASE IN THE APPARENT STRESS. • HOWEVER IF THE CURVE IS PLOTTED IN TERMS OF TRUE STRESS AND TRUE STRAIN THE STRESS WILL CONTINUE TO RISE UNTIL FAILURE. EVENTUALLY THE NECK BECOMES UNSTABLE AND THE SPECIMEN RUPTURES OR FRACTURES.

10. PLASTIC DEFORMATION OR PLASTICITY • THE STRESS-STRAIN DIAGRAM FOR STRUCTURAL STEELS HAS A LINEAR ELASTIC REGION FOLLOWED BY A REGION OF CONSIDERABLE YIELDING. • SUCH DIAGRAM WOULD PRECISELY BE CONSISTED OF TWO STRAIGHT LINES. AFTER YIELD STRESS MATERIAL YIELDS UNDER CONSTANT STRESS AND THIS PARTICULAR BEHAVIOUR IS KNOWN AS PERFECT PLASTICITY AND THE REGION IS KNOWN AS PLASTIC REGION. • THE PLASTIC REGION CONTINUES UNTIL THE STRAINS ARE MANY TIMES LARGER THAN THE YIELD STRAIN. A MATERIAL HAVING SUCH DIAGRAM IS KNOWN AS ELASTIC-PLASTIC MATERIAL.

11. RESPONSE OF MATERIALS – MORE GENERAL TYPES AS WE DISCUSSED THAT SHAPE OF A STRESS-STRAIN DIAGRAM DEPENDS UPON THE MATERIAL ITSELF AND THE TEST CONDITIONS. HOWEVER, SOME OF THE FEATURES OF SUCH CURVES ARE SIMILAR FOR ALL STRUCURAL MATERIALS. WITH REFERENCE TO THE RESPONSE OF MATERIALS AGAINST APPLIED, MATERIALS MAY BE ELASTIC, PLASTIC, VISCOELASTIC, VISCOPLASTIC, OR FRACTURE. INELASTIC BEHAVIOUR CAN OCCUR UNDER MULTIAXIAL STRESS IN A LOAD-CARRING MEMB EVEN IF NONE OF THE INDIVIDUAL STRESS COMPONENTS EXCEEDS THE UNIAXIAL YIELD STRESS. THIS IS BECAUSE UNDER MULTIAXIAL STRESS STATES, THE INITIATION OF YIELDING IS GOVERNED BY SOME QUANTITY OTHER THAN THE INDIVIDUAL STRESS COMPONENTS THEMSELVES.

12. YIELD CRITERIA – CONCEPTS AND ITS UNDERSTANDING EARLIER THE CONCEPT OF CRITERIA HAS BEEN LIMITED TO UNIAXIAL STRESS STATE. NOW THE YIELDING CRITERIA MAY BE APPLIED TO MULTIAXIAL STATES OF STRESS. WE ALREADY HAVE THE CONCEPT THAT FAILURE OF MATERIALS OCCUR AT THE INITIATION OF INELASTIC MATERIAL BEHAVIOUR THROUGH EITHER YIELDING OR FRACTURE. THE STUDY OF THE MATERIALS THAT YIELD IS KNOWN AS THEORY OF PLASTICITY. THIS THEORY OF PLASTICITY HAS FOLLOWING THREE COMPONENTS:

13. YIELD CRITERIA – CONCEPTS AND ITS UNDERSTANDING 1. YIELD CRITERIA, INITIATION OF YIELDING IN MATERIALS 2. FLOW RULE, PLASTIC STRAIN INCREMENTS TO THE STRESS INCREMENTS 3. HARDENING RULE, CHANGES IN MATERIAL RESPONSE BECAUSE OF PLASTIC STRAIN HOWEVER, AT THIS STAGE WE WOULD RESTRICT OURSELVES TO PREDICTING INITIATION OF YIELDING, THAT IS THE YIELD CRITERIA. IN GENERAL, A YIELD CRITERIA CAN BE ANY DESCRIPTIVE STATEMENT THAT DEFINES SPECIFIC CONDITIONS UNDER WHICH YIELDING MAY OCCUR SUCH AS THE STRESS STATE, STRAIN RATE, STRAIN ENERGY OR OTHERS.

14. DIFFERENT YIELDING CRITERIA FOLLOWINGS ARE THE VARIOUS YIELDING CRITERIA: MAXIMUM PRINCIPAL STRESS CRITEIRON MAXIMUM PRINCIPAL STRAIN CRITERION STRAIN-ENERGY DENSITY CRITERION

15. PLASTIC ANALYSIS OF STATICALLY INDETERMINATE STRUCTURES THERE ARE TWO TYPES OF SITUATIONS FOR STRUCTURES WITH RESPECT TO DETERMINATION OF FORCES AND THEIR REACTIONS. THESE ARE STATICALLY DETERMINATE AND INDETERMINATE STRUCTURES. IF IT IS POSSIBLE TO DETERMINE AXIAL FORCES IN THE STRUCTURES AND THE REACTIONS AT THE SUPPORTS BY DRAWING FREE-BODY DIAGRAMS AND THEN SOLVING EQUILIBRIUM EQUATIONS, SUCH STRUCTURES ARE KNOWN AS STATICALLY DETERMINATE STRUCTURES. HOWEVER, STRUCTURES WHICH CAN NOT BE ANALYZE EASILY TO DETERMINE AXIAL FORCES AND THEIR REACTIONS AT THE SUPPORTS, AND NEED EQUILIBRIUM EQUATIONS SUPPLEMENTED BY THE ADDITIONAL EQUATIONS ABOUT DISPLACEMENT OF THE STRUCTURES, ARE KNOWN AS STATICALLY INDETERMINATE STRUCTURES.

16. PLASTIC ANALYSIS OF STATICALLY INDETERMINATE STRUCTURES IN INDETERMINATE SYSTEM, THE FORCES CAN NOT BE FOUND WITHOUT FIRST FINDING THE DISPLACEMENTS, AND DISPLACEMENTS DEPEND UPON BOTH THE FORCES AND THE STRESS-STRAIN DIAGRAM OF MATERIALS. HOWEVER, FOR MATERIALS SUCH AS MILD STEEL WITH ELASTIC-PLASTIC DIAGRM, THE BEHAVIOUR OF SUCH STRUCTURE IS SIMPLE AND A PLASTIC ANALYSIS CAN BE PERFORMED WITHOUT DIFFICULTY. IN ORDER TO UNDERSTAND IT CONSIDER A SYMMETRIC THREE-BAR TRUSS AS SHOWN IN THE FIGURE. IT IS SUPPOSED THESE ARE MADE OF MILD STEEL. WHEN THE LOAD “P” IS SMALL, THE STRESSES IN THREE BARS WOULD BE LESS THAN THE YIELD STRESS (σy).

17. IF THE LOAD IS GRADUALLY INCREASED, THE STRESSES IN THE BARS INCREASE UNTIL THE YIELD STRESS IS REACHED. IF THE BARS HAVE THE EQUAL X-SECTIONAL AREAS, THEN THE FORCES (F1 & F2) UNDER ELASTIC CONDITIONS CAN BE OBTAINED BY FOLLOWING RELATIONSHIPS: F1 = P cos²β/(1 + 2cos³β) F2 = P /(1 + 2cos³β) AS THE FORCE “F2” IS BIGGER THAN “F1”, THE AXIAL STRESS IN THE MIDDLE BAR WILL ATTAIN THE YIELD STRESS FIRST AND THE FORCE “F2” WOULD BE EQUAL TO “σyA”. THE CORRESPONDING LOAD WILL BE CALLED YIELD LOAD “Py”. THE VALUE OF “Py” CAN BE OBTAINED BY SETTING “F2” EQUAL TO “σyA” AND SOLVING FOR THE LOAD, THAT IS, Py = σyA (1 + 2cos³β)

18. NOW AS LONG AS THE LOAD IS LESS THAN YIELD LOAD, THE STRUCTURE BEHAVE ELASTICALLY AND THE FORCES IN THE BAR CAN BE CALCULATED FROM ABOVE-MENTIONED TWO EQUATIONS. NOW THE DEFLECTION AT THE YIELD POINT “δy” AT POINT “D” CAN BE CALCULATED BY THE FOLLOWING RELATIONSHIP: δy = F2L/EA = σyL/E THE FURTHER INCREASE IN LOAD CAUSES THE FORCES APPLIED AT THE INCLINED BARS “F1” TO INCREASE. HOWEVER, THE FORCE “F2” REMAINS CONSTANT AS THE MIDDLE BAR HAS BECOME PERFECTLY PLASTIC. EVENTUALLY, THE FORCES “F1” ON BOTH INCLINED BARS REACH THE VALUE “σyA”, AND THESE TWO BARS ALSO BECOME PLASTIC. THE THREE BARS CONTINUE TO ELONGATE UNDER THIS MAXIMUM VALUE OF LOAD, ULTIMATE LOAD “Pu”.

19. IF THE STRESS-STRAIN DIAGRAM IS CONSIDERED THEN THE OA REPRESENT THE BEHAVIOUR OF TRUSS UP TO YIELD STRESS. LINE AB REPRESENTS THE BEHAVIOUR ABOVE YIELD STRESS AND THE HORIZONTAL LINE BC REPRESENTS THE REGION OF CONTINUOUS PLASTIC DEFORMATION. AFTER THE ULTIMATE LOAD IS REACHED, THE STRUCTURE CONTINUES TO DEFORM. WITH STRAIN-HARDENING THE STRUCTURE IS ABLE TO SUPPORT AN ADDITIONAL LOAD. HOWEVER, WITH THE PRESENCE OF VERY LARGE DEFLECTIONS, IT IS MEANT THAT STRUCTURE HAS ULTIMATELY FAILED. THEREFORE, THE DETERMINATION OF ULTIMATE LOAD “Pu” IS OF CONSIDERABLE INTEREST TO DESIGN ENGINEERS. THIS SHOULD BE KEPT IN MIND THAT IF THE LOAD IS REMOVED BEFORE YIELD LOAD, THE STRUCTURE WOULD BEHAVE ELASTICALLY. IF YIELD STRESS IS EXCEEDED SOME PART OF THE STRUCTURE WOULD RETAIN A PERMANENT SET AFTER REMOVAL OF LOAD.

20. QUESTIONS AND QUERIES IF ANY! IF NOT THEN GOOD BYE SEE ALL OF YOU IN NEXT LECTURE ON shear STRESS AND STRAIN, allowable stresses & axially loaded members