CHAPTER
1
Design of Curved Curved Beams
NOTATIONS AND SYMBOLS USED
σ
= =
Stress, MPa. Direct Dir ect str stress ess,, tens tensile ile or com compre pressi ssive, ve, MPa MPa..
=
Stre St ress ss at in inne nerr fi fibr bre, e, MP MPa. a.
=
Stre St ress ss at ou oute terr fib fibre re,, MPa MPa..
=
Normal Nor mal str stress ess due to ben bendin ding g at at inner inner fib fibre, re, MPa MPa..
=
Normal Nor mal str stress ess due to ben bendin ding g at at outer outer fib fibre, re, MPa MPa..
M b
=
Bendin Ben ding g mome moment nt for cri critic tical al sec sectio tion, n, N-m N-mm. m.
ci
=
Distan Dis tance ce of neu neutra trall axis axis fro from m inne innerr fibr fibre, e, mm.
co
=
Distan Dis tance ce of neu neutra trall axis axis from from out outer er fib fibre, re, mm mm..
e ri ro rc
= = = =
Eccentricity, mm. Distan Dis tance ce of of inner inner fibr fibre e from from cent centre re of of curva curvatur ture, e, mm. mm. Distan Dis tance ce of of outer outer fib fibre re from from cen centre tre of curv curvatu ature, re, mm mm.. Distan Dis tance ce of of centr centroid oidal al axis axis from from cent centre re of of curva curvatur ture, e, mm. mm.
rn d h
= = =
Distance Distan ce of of neutr neutral al axis axis from from cent centre re of of curva curvatur ture, e, mm. mm. Diamet Dia meter er of of circ circula ularr rod rod used used in curv curved ed beam beam,, mm. mm. Depth Dept h of curved curved beam [squ [square, are, rect rectangu angular, lar, trap trapezoi ezoidal dal or or I-sectio I-section], n], mm. mm.
σd σi σo σbi σbo
τ max = A P
= =
Maxi Ma ximu mum m sh shea earr str stres ess, s, MP MPa. a. Area of cros Area cross-s s-sect ection ion of memb member, er, (cu (curv rved ed beam beam), ), mm mm 2. Load on me member, N.
INTRODUCTION Machine frames having curved portions are frequently subjected to bending or axial loads or to a combination of bending and axial loads. With the reduction in the radius of curved portion, the stress due to curvature become greater and the results of the equations of straight beams when used becomes less satisfactory. For relatively small radii of curvature, the actual stresses
2
Design of Machine Elements - II
may be several times greater than the value obtained for straight beams. It has been found from the results of Photoelastic experiments that in case of curved beams, the neutral surface does not coincide with centroidal axis but instead shifted towards the centre of curvature. It has also been found that the stresses in the fibres of a curved beam are not proportional to the distances of the fibres from the neutral surfaces, as is assumed for a straight beam.
Differences between Straight and Curved Beams Straight beam
Curved beam
1.
The neutral axis of beam coincides with centroidal axis.
The neutral axis is shifted towards the centre of curvature by a distance called eccentricity i.e. the neutral axis lies between centroidal axis and centre of curvature.
2.
The variation of normal stress due to bending is linear, tensile at the inner fibre and compressive at the outer fibre with zero value at the centroidal axis.
The variation of normal stress due to bending across section is non-linear and is hyperbolic.
Derivation of Expression to Determine Stress at any Point on the Fibres of a Curved Beam Consider a curved beam with rc, as the radius of centroidal axis, rn, the radius of neutral surface, ri, the radius of inner fibre, ro, the radius of outer fibre having thickness ‘h’ subjected to bending moment M b. Let AB and CD be the two adjacent cross-sections separated from each other by a small angle dφ. Because of M b the section CD rotates through a small angle dα. The unit deformation of any fibre at a distance y from neutral surface is
∈=
Deformation
δ l
=
ydα (rn − y )dφ
… (1.1)
The unit stress on this fibre is, Stress = Strain × Young’s modulus of material of beam
σ = ∈ E =
Eydα (rn − y )dφ
… (1.2)
For equilibrium, the summation of the forces acting on the cross sectional area must be zero.
∫ σdA = 0
i.e.,
or
E ydα dA n – y ) dφ
∫ (r E
dα
ydA n – y
dφ ∫ r
=0 =0
… (1.3)
3
Design of Curved Beams
Also the external moment M b applied is resisted by internal moment. From equation 1.2 we have,
∫ y (σdA) = M E y2dαdA (rn – y) dφ
∫
i.e., E
dα dφ
y2 dA
∫ (r
n
− y)
= M =
M
i.e.,
= M … (1.4)
Edα ( − y)dA + rn dφ
∫
ydA n − y)
∫ (r
… (1.5)
Note: In equation 1.5, the first integral is the moment of cross sectional area with respect to neutral surface and the second integral is zero from equation 1.3.
dα = E Ae dφ
M
Therefore,
… (1.6)
Here ‘e’ represents the distance between the centroidal axis and neutral axis. i.e.,
e
= − r
c
r
n
Rearranging terms in equation 1.6, we get
Substituting
dα dφ
=
M AeE
dα dφ
=
M in equation 1.2, AeE
σ=
Stress
... (1.7)
Ey dα (rn − y ) dφ becomes 1
C H
C d α
A
Outer fibre
h
y Centroidal axis
1
D
e B r n r i
D v
M b
d α
d φ
Neutral axis
r c M b
r o Inner fibre O Centre of curvature Figure: 1.1
4
Design of Machine Elements - II
= σ=
i.e.,
Ey M (rn − y ) AeE My (rn – y) Ae
… (1.8)
From figure 1.1, rn = v + y or y = rn – v Therefore, from equation 1.3 ydA
∫ (r
n
– y)
(rn – v ) dA (rn – y)
= ∫
= rn ∫
dA – dA = 0 v
∫
dA
= rn ∫ rn
or
=
V
−A=0
A dA v
…(1.9)
∫
Note: Since e = r c – rn , equation 1.9 can be used to determine ‘e’. Knowing the value of ‘e’, equation 1.8 is used to determine the stress σ .
Straight and Curved Beams ( a ) Straight beam
Consider a straight beam having moment of inertia ‘I’ subjected to bending moment M b as shown in (Fig. 1.2).
Centre of gravity axis coinciding with neutral axis M b
Normal stress due to bending (Compressive) −σ b M b
B −y
+y
A σ b
Normal stress due to bending (Tensile) Figure: 1.2
5
Design of Curved Beams
From fundamental equation of bending, M b I i.e., Bending stress
=
σb E = y
;
R
σb =
M b I
M b I
=
σb y
y
For a given beam, M b and ‘I ’ are constant and hence σb α y, i.e., the variation of bending stress is linear and is directly proportional to its distance from centre of gravity axis which coincides with neutral axis. The maximum bending (tensile) stress σb tensile at A and compressive at B are equal in magnitude at ‘A’ and ‘B’ as shown in the Fig. 1.2. ( b ) Curved beam
Figure 1.3 shows a curved beam subjected to bending moment M b. ri =
Let
Distance of inner fibre from centre of curvature, C
ro =
Distance of outer fibre from centre of curvature
rc =
Distance of centroidal axis (CG axis) from centre of curvature
rn =
Distance of neutral axis from centre of curvature
σbo
co
e ci
σbi
r i
M b
r c r n
r o M b
C
Centre of curvature
Figure: 1.3
The neutral axis is shifted towards the centre of curvature by a distance called eccentricity ‘e’ . The value ‘e’ should be computed very accurately since a small variation in the value of ‘e’ causes a large variation in the values of stress. i.e.,
e = rc – rn ci = Distance between neutral axis and inner fibre = rn – ri co = Distance between outer fibre and neutral axis = ro – rn
