INELASTIC BENDING OF BEAMS

INELASTIC BENDING IS THE BENDING OF BEAM WHEN THE MATERIAL DOES NOT FOLLOW THE HOOKE’S LAW. IN MORE APPROPRIATE TERMS SUCH BENDING TAKES PLACE WHEN THE BEAM IS LOADED IN SUCH A MANNER THAT THE STRESSES EXCEED THE PROPORTIONAL LIMIT. IT IS ALWAYS POSSIBLE TO DETERMINE THE STRESSES, STRAINS, AND DEFLECTION OF BEAMS IF THE STRESS-STRAIN DIAGRAM OF THE MATERIALS IS KNOWN. IN FACT THE ANALYSIS OF AN INELASTIC BEAM IS BASED ON THE FACT THAT PLANE X-SECTIONS OF A BEAM REMAIN PLANE UNDER PURE BENDING.

HENCE, THE STRAINS IN AN INELASTIC BEAM VARY LINEARLY OVER THE HEIGHT OF THE BEAM, AND HENCE WE CAN DETERMINE THE MAGNITUDE OF STRESSES, CURVATURE AND DEFLECTIONS OF BEAM BY THE HELP OF STRESS-STRAIN DIAGRAM AND EQUATIONS OF STATICS. BY MAKING AN INELASTIC ANALYSIS OF BEAM ITS ULTIMATE LOAD-CARRYING CAPACITY CAN BE DETERMINED. OF COURSE ULTIMATE LOAD-CARRYING CAPACITY IS MUCH GREATER THANT THE LOAD AT PROPORTIONAL LIMIT. ULTIMATE LOAD-CARRYING CAPACITY IS OFTEN NEEDED FOR DESIGN PURPOSES IN ORDER TO ASCERTAIN THE FACTOR OF SAFETY AGAINST FAILURE.

IN THIS CASE FACTOR OF SAFETY WOULD BE MUCH LARGER THAN THE FACTOR OF SAFETY WITH RESPECT TO THE PROPORTIONAL LIMIT. IN ORDER TO OBTAIN THE BASIC EQUATIONS FOR INELASTIC BENDING, LET US CONSIDER A BEAM IN PURE BENDING SUBJECTED TO A POSITIVE BENDING MOMENT “M”. CONSEQUENTLY THIS BEAM WOULD DEFLECT IN THE XY PLANE WHICH IS THE PLANE OF BENDING IF BENDING IS CONSIDERED WITH REFERENCE TO Z-AXIS. Z-AXIS IS THE NEUTRAL AXIS AND ITS LOCATION IS TO BE DETERMINED. WE KNOW THAT THE STRAINS IN THE BEAM HAVE A LINEAR DISTRIBUTION IRRESPECTIVE OF THE NATURE OF THE MATERIAL.

HENCE STRAINS VARY FROM TOP TO THE BOTTOM OF THE BEAM FROM MAXIMUM TO ZERO AND THEN FROM ZERO TO MAXIMUM AS SHOWN IN THE FIG. LET US DENOTE THE STRAINS AT THE LOWER SUFRACES BY “ε1” AND AT THE UPPER SURFACES AS “ε2”, RESPECTIVELY. WE HAVE THE FOLLOWING RELATIONSHIP:

ε = -y/ρ = -кy

NOW THE STRAINS AT THE OUTERMOST SURFACES WOULD BE AS FOLLOWS:

ε1 = -кh1 AND ε2 = -кh2

STRAINS FROM THESE EQUATIONS CAN READILY BE DETERMINED IF THE VALUE OF CURVATURE AND THE POSITION OF NEUTRAL AXIS IS KNOWN. THE LOCATION OF THE NEUTRAL AXIS AND CURVATURE CAN BE FOUND BY MAKING USE OF THE STRESS-STRAIN DIAGRAM AND EQUATIONS OF STATICS. THE FIRST EQUATION EXPRESSES THAT THE RESULTANT HORIZONTAL FORCE DUE TO THE NORMAL STRESSES ACTING ON ANY X-SECTION OF THE BEAM VANISHES; THEREFORE,

∫ σ dA = 0

THE SECOND EQUATION EXPRESSES THAT THE RESULTANT OF THE STRESSES ACTING ON THE X-SECTION IS EQUAL TO THE BENDING MOMENT “M”.

∫ σ y dA = M

AFTER THE DETERMINATION OF CURVATURE, DEFLECTIONS OF THE BEAM “ν” CAN BE DETERMINED BY EQUATING THEM TO CURVATURE. HENCE

к = 1/ρ = dθ/dx = d²ν/dx²

SOME OF THE SAME TECHNIQUES DEVELOPED EARLIER TO DETERMINE DEFLECTIONS OF ELASTIC BEAMS CAN BE USED FOR INELASTIC BENDING. HOWEVER, IT IS NECESSARY TO USE MORE COMPLICATED EXPRESSIONS FOR CURVATURE IN PLACE OF THE QUANTITY “M/EI”. NOW THE SIMPLEST CASE OF INELASTIC BENDING IS PLASTIC BENDING WHICH OCCURS WHEN THE MATERIAL IS ELASTIC-PLASTIC.