SIMPLE BENDING

SIMPLE BENDING

WHAT IS SIMPLE BENDING ?

THE BENDING OF A BEAM WITH CONSTANT BENDING MOMENT AND COMPLETELY FREE FROM SHEAR FORCE IS CALLED "SIMPLE BENDING".IT IS ALSO CALLED AS "PURE BENDING".
THE STRESSES DEVELOPED IN THE BEAM ARE ONLY DUE TO THE BENDING MOMENT INDUCED ON THE BEAM.

• THE BENDING MOMENT IN THE B.M.D (BENDING MOMENT DIAGRAM) IS SHOWN BY A STRAIGHT HORIZONTAL LINE.

• THE SHEAR FORCE IN THE S.F.D (SHEAR FORCE DIAGRAM) IS ZERO WHERE THE SIMPLE BENDING OCCURS.

ASSUMPTIONS

• THE BEAM IS INITIALLY STRAIGHT BEFORE THE APPLICATION OF LOADS ON IT.
• THE BEAM MATERIAL IS HOMOGENEOUS AND ISOTROPIC (SAME PROPERTIES IN ALL DIRECTIONS).
• ELASTIC LIMIT OF THE BEAM IS NOT EXCEEDED.
• TRANSVERSE SECTIONS WILL REMAIN PLAIN BEFORE AND AFTER BENDING.
• EACH LAYER OF THE BEAM IS FREE TO EXPAND / CONTRACT WITH RESPECT TO THE ADJACENT LAYER.
• THE VALUE OF YOUNG'S MODULUS OF THE MATERIAL IS SAME IN TENSION AND COMPRESSION.
• THE BEAM IS SYMMETRICAL ABOUT THE PLANE OF BENDING,i.e., ABOUT THE PLANE PASSING THROUGH THE NEUTRAL AXIS

NEUTRAL AXIS

IT IS THE LAYER OF THE BEAM WHICH DOES NOT UNDERGO ANY CHANGES (EXPANSION / COMPRESSION) DURING BENDING AND IT HAS NO STRESSES INDUCED IN IT.
THE NEUTRAL AXIS PASSES THROUGH THE CENTROID OF THE BEAM SECTION.

EQUATION OF SIMPLE BENDING

M = MOMENT OF RESISTANCE  
I = MOMENT OF INERTIA 
f = BENDING STRESS
y = DEPTH OF NEUTRAL AXIS 
E = MODULUS OF ELASTICITY / YOUNG'S MODULUS
R = RADIUS OF CURVATURE

MOMENT OF RESISTANCE : THE INTERNAL MOMENT PRODUCED TO RESIST THE EXTERNAL LOAD MOMENT AND APPLIED MOMENTS IS KNOWN AS MOMENT OF RESISTANCE.IT IS DENOTED BY 'M'.
MOMENT OF INERTIA : THE RESISTANCE OFFERED BY A ROTATING BODY TO STAY IN EQUILIBRIUM CONDITION IS CALLED MOMENT OF INERTIA.IT IS DENOTED BY 'I'.
BENDING STRESS : THE STRESS INDUCED DUE TO THE BENDING OF THE BEAM WHICH CREATES TENSION AND COMPRESSION ON EITHER SIDES OF THE NEUTRAL LAYER.IT IS DENOTED BY 'f'.
DEPTH OF NEUTRAL AXIS : THE DISTANCE BETWEEN THE EXTREME FIBER OF THE BEAM TO THEM NEUTRAL AXIS OF THE BEAM.IT IS DENOTED BY 'y'.
MODULUS OF ELASTICITY : IT IS THE RATIO OF STRESS WITH RESPECT TO STRAIN.IT IS DENOTED BY 'E'.
RADIUS OF CURVATURE : IT IS THE RADIUS OF THE CIRCLE FORMED BY THE ARC OF THE BENDING BEAM.IT IS DENOTED BY 'R'.

FLEXURAL RIGIDITY

THE PRODUCT OF MOMENT OF RESISTANCE AND MOMENT OF INERTIA ABOUT THE NEUTRAL AXIS IS CALLED FLEXURAL RIGIDITY.
                                    FLEXURAL RIGIDITY = E x I

SECTION MODULUS

THE RATIO OF MOMENT OF INERTIA TO THE DEPTH OF NEUTRAL AXIS(MAX) IS CALLED SECTION MODULUS or MODULUS OF SECTION.IT IS DENOTED BY 'Z'.
THE MORE THE MODULUS OF SECTION THE MORE THE STRENGTH OF THE BEAM.

Z = I/y

BEAMS OF UNIFORM STRENGTH

THE BEAMS ARE SAID TO BE OF UNIFORM STRENGTH WHEN THE BENDING STRESS INDUCED IN THE BEAM IS CONSTANT AT EVERY SECTION THROUGH OUT THE LENGTH OF THE BEAM, i.e., THE STRESS AT EXTREME FIBERS IS SAME THROUGHOUT THE BEAM.
AN PRISMATIC BEAM CANNOT BE A BEAM OF UNIFORM STRENGTH BECAUSE THE VARIATION IN THE BENDING MOMENT CAUSES THE VARIATAION IN STRESSES INDUCED.

MODULUS OF RUPTURE

IT IS STRESS INDUCED IN THE BEAM WHEN IT IS SUBJECTED TO THE ULTIMATE BENDING MOMENT.IF THE STRESS IS INCREASED ABOVE THE MODULUS OF RUPTURE THE THE BEAM WILL FAIL.

MODULUS OF RUPTURE = ULTIMATE MOMENT / SECTION MODULUS = M(ULT) /Z