Curved Beam - VTU e-Learning

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Since Σyδa a is first moment of area about neutral axis, Σyδa/a Σ is the distance of centroid from neutral axis. Thus neutral axis coincides with centroid of the cross section. Cross sec sectional tional area coincides with neutral axis. From (1) and (2) CURVED BEAM Curved beams are the parts of machine members found in C - clamps, crane hooks, frames of presses, riveters, punches, shears, boring machines, planers etc. In straight beams the neutral axis of the section coincides with its centroidal axis and the stress distribution in the beam is linear. But in the case of curved beams the neutral axis of the section is shifted towards the centre of curvature of the beam causing a nonlinear [hyperbolic] distribution of stress. The neutral axis lies between the centroidal axis and the centre of curvature and will always be present within the curved beams.
Stresses in Curved Beam Consider a curved beam subjected to bending moment Mb as shown in the figure. The distribution of stress in curved flexural member is determined by using the following assumptions: i) The material of the beam is perfectly homogeneous [i.e., same material throughout] and isotropic [i.e., equal elastic properties in all directions] ii) The cross section has an axis of symmetry in a plane along the length of the beam. iii) The material of the beam obeys Hooke's law. iv) The transverse sections which are plane before bending remain plane after bending also. v) Each layer of the beam is free to expand or contract, independent of the layer above or below it. vi) The Young's modulus is same both in tension and compression. Derivation for stresses in curved beam Nomenclature used in curved beam Ci =Distance from neutral axis to inner radius of curved beam Co=Distance from neutral axis to outer radius of curved beam C1=Distance from centroidal axis to inner radius of curved beam C2= Distance from centroidal axis to outer radius of curved beam F = Applied load or Force A = Area of cross section L = Distance from force to centroidal axis at critical section σd= Direct stress σbi = Bending stress at the inner fiber σbo = Bending stress at the outer fiber σri = Combined stress at the inner fiber σro = Combined stress at the outer fiber
Page 1 and 2: CURVED BEAMS CONTENT: WHAT’S A C
Page 3: G1H1 = (R+Y) Ѳ Also Stress OR dF =
Page 7 and 8: We know, stress σ = E.e We know, s
Page 9 and 10: ∴ M = b θ A.e ..... (iii) θ Fro
Page 11 and 12: 2 3 Neutral axis and centroidal axi
Page 14 and 15: Why stress concentration occur at i
Page 16 and 17: For a straight beam: Inner most fib
Page 18: e = rc - rn = 100 - 91.024 = 8.976m
Page 21 and 22: = i.e., 4674.069+83.61t +83.61t = 4
Page 23 and 24: σri = σd + σbi = 7.0736 105.00
Page 25 and 26: Bending stress at the outer fiber
Page 27 and 28: =150.34mm To find the centroidal ax
Page 29 and 30: Distance of neutral axis to inner r
Page 31 and 32: Problem no.7 Determine the maximum
Page 33 and 34: Maximum shear stress max = 0.5x σm
Page 35 and 36: ∴ Radius of centroidal axis rc =
Page 37 and 38: Combined stress =-198.189 N/mm σ r
Page 39 and 40: x = = 101.785mm Radius of
Page 41 and 42: Problem no.10 Compute the combined
Page 43 and 44: Bending stress at the inner fiber
Page 45 and 46: C2 = = 50mm c1 = 90-50= 40mm Mome
Page 47 and 48: Problem no.11 The section of a cran
Page 49 and 50: Problem no.12 Design of steel crane
Page 51 and 52: Applied force F = 10 5 N Area of cr
Page 53 and 54: Problem no.13 The figure shows a lo
Page 55 and 56:
Draw the critical section at PP-Q a
Page 57 and 58:
25 ro = 75 + 2 = 87.5 mm A = π 4 d
Page 59 and 60:
The ring is symmetrical and is load
Page 61 and 62:
As this quantity is positive the di
Page 63 and 64:
The stress at inner (σ1Ai) ) and o
Page 65 and 66:
As per Castigliano’s theorem \
Page 67 and 68:
The stress at any angle Ѳ can be f
Page 69 and 70:
Distance of neutral axis to outer r
Page 71 and 72:
Problem 16 Determine the stress ind
Page 73 and 74:
Consider the cross section B - B At
Page 75 and 76:
Distance of neutral axis to outer r
Page 77 and 78:
Using usual notations prove that th
Page 79 and 80:
Substituting the value of the σx f
Page 81 and 82:
1 ∆ = 1 {From equation (x
Page 83 and 84:
VTU,Jan/Feb.2005 Fig.1.35 5. Determ
Page 85:
REFERENCE BOOKS • “MECHANICAL E
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