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DESIGN OF CURVED BEAMS Introduction:-
Straight Beam: A beam is a straight structural structural member subjected to a system of external forces ac at right angles to its axis.
CURVED BEAMS: A beam in which the neutral axis in the unloaded condition is curved instead of straight. Or If the be is originally curved before applying the bending moment, are termed as “Curved Beams”.
Applications of Curved Beams: Curved beams find applications in many machine members such as c – c – clampers, clampers, crane hooks, fram presses, chains, links, rings, etc.., Differences between Straight Beams & Curved Beams: s.no
Straight Beams Curved Beams
Curved Beams
1
Neutral axis of the cross-section passes through the centroid of the section.
Neutral axis does not coincide with the cross section, but is shifted towards the centre of curvature of the beam.
2
The variation of bending stress is linear, magnitude being proportional to the distance of a fiber from the neutral axis.
The distribution of the stress in the case o curved beam is non-linear (Hyper-bolic) because of the neutral axis is initially curved.
State the assumptions & derive the stress equation in case of a curved beam:Assumptions:Assumptions: Sign up to vote on this title 1. The beam is subjected to pure bending. 2. Material of the beam is isotropic isotropic & homogeneous & obeys hook’s law. Not useful Useful 3. Plane sections perpendicular to the axis of the beam remain plane even after bending. 4. The stress induced induced do not exceed elastic limit.
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The bending stress, in a curved beam at any fibre at a distance y from neutral axis is given by the following expression.
=
( − )
Where, M - sending moment at the given section about centroids axis A - Area of cross-section. e - Distance from centroidal axis to neutral axis = R - - Radius of curvature of neutral axis y - Distance from the neutral axis to the fibre. If the section is symmetric such as rectangle circle I-beam, then maximum bending stress occurs at fibre and is given by,
=
( )
Where, - Distance from the neutral axis to the inner fiber - Radius of curvature of the inner fibre.
If the section is unsymmetric, then the maximum bending stress may occur either inside or outside fibre. The bending stress at outer fibre is given by,
=
( )
Where, −Distance from neutral axis to the oute r fibre. - R - - Radius of curvature of outer fibre. If the section has an axial load in addition to the .bending stress, then the resultant stress is the algebraic sum of axial ( ) and bending stress ( ). Sign up to vote on this title
∴ = +
Useful Not useful Expression for radius of neutral axis for rectangular, circular, trapezoidal, T-section and I- section Neutral Axis:
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Expression for radius of neutral axis for T-section:-
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Problems:-
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********************THE END*****************
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