theory of plasticity

1. (Under the guidance of Prof.Manjunatha L H)
Prepared by:
Syed Zabiulla
R14MMD016
A Presentation on
Bending of Single-Notched
Bars

2. CONTENTS
2
 INTRODUCTION
 BENDING FOR ψ< 32.7◦
 BENDING FOR ψ> 32.7◦
 RESULTS

3. INTRODUCTION
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 A wide rectangular bar of metal, having a wedge shaped
notch on one side, is subjected to pure bending under
conditions of plane strain.
 The bar is assumed long enough for the yield point
couple to be independent of the precise distribution at the
ends.
 It is supposed that the bar is bent to open the notch, so
that the stress across the minimum section is tensile near
the root and compressive near the opposite side.
 If the notch is sufficiently deep, the region of plastic
deformation is confined in the neck, and the state of
stress at the yield point is independent of the thickness of
the bar.
 Depending on the notch angle π−2ψ, there are two

4. Pure bending for ψ< 32.7◦
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Figure 1a; Pure bending of a sharply notched bar for ψ< 32.7

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 In these notch, the sliplines are straight, meeting the
surface at 45◦. The field is continued round the
singularity A to form the centered fan ADE on either
side of the central axis. Near the plane surface FF,
the sliplines are again straight, and the state of
stress is a uniform compression 2k parallel to the
surface.
 In the fig 1(a), the two slipline domains are
extended across the minimum section to meet at a
neutral point N, the region AENE being in a state of
uniform tension 2k(1+ψ) acting across AN.
 The position of N is determined from the condition
that the resultant horizontal tension across the
minimum section is zero.

6. 6
 Denoting the height of the triangle FNF by d, we
get
d(2 + ψ) = a(1 + ψ)
where a is the depth of the minimum section
 The yield point couple M per unit width of the bar
is given by
 The ratio of the actual yield moment of the
notched bar to that of the unnotched bar is known
as the constraint factor, denoted by f .
 In the case of a sharp notch, the constraint factor
is therefore twice the right-hand side of (Eq.1).
(Eq.1).

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 The stress changes discontinuously at N, the
pressure jump must not exceed the value πk in
order to avoid overstressing of the rigid corners at
N.
 Therefore fig (1a) is valid for ψ<(π/2)−1=32.7◦. In
other words, the semiangle of the notch must be
less than or equal to 1 rad.

8. Pure bending for ψ> 32.7◦
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Figure 1b; Pure bending of a sharply notched bar for ψ> 32.7

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 When ψ>32.7◦, the slipline domains defined by the
stress-free surfaces are connected by a pair of
curved sliplines EF. The rigid ends of the bar rotate
by sliding along the curves EF, which must
therefore be circular arcs of some radius R.
 The fan angle θ at A cannot be greater than ψ,
since the rigid material in the corner EAE must not
be overstressed.
 If the angular span of EF is denoted by λ, it follows
from geometry and Hencky’s equations that
Since θ< ψ, we have λ>π/2. The above equations can be
solved for λ and θ to give
(Eq.2).

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 For a given notch angle, the slipline field is
completely specified by the radii b and R of the
circular arcs DE and EF, and the height d of the
center C above GG. One of the three relations
necessary for finding these parameters is provided
by the fact that the sum of the vertical projections of
GC, CE, and EA must be equal to a. Thus
(Eq.3).
• The other two relations are furnished by the
conditions of zero horizontal and vertical
resultants of the tractions across the interface
AEFG.
(Eq.4.).

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 When the ratios d/a, R/a, and b/a have been
computed from (3) and (4), the yield point couple
and the constraint factor can be found from the
formula
(Eq.5.).
The constraint factor increases with increasing ψ
until b vanishes.
• The geometrical parameters and the constraint factors
calculated from above equations are given in Table 1.

12. Results for pure bending of sharply notched
bars
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Table 1. Results for pure bending of sharply notched bars

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