Barber J.R. Intermediate Mechanics of Materials

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Problems 275

all dimensions in mm

Figure P5.3

5.4. Find the applied mom ent for ﬁrst yield M

, the fully plastic mome nt M

, and

the shape factor for a hollow circular cylindrical beam of o uter radius 2a and inner

radius a.

5.5. A rectang ular beam of w idth b and height h is loaded by a bending moment

M a bout a horizontal axis. Find the relation between bending moment and radius of

curvature R, if the material is elastic-pe rfec tly plastic with Yo ung’s modulus E and

uniaxial yield stress S

5.6. Th e channel section of Figure P5.6 is bent by a moment M which is just sufﬁcient

to cause the ﬂanges to y ie ld. Find the value of M an d the corresponding rad ius of

curvature R, if E =210 GPa and S

=350 MPa.

all dimensions in mm

Figure P5.6

276 5 Non-linear and Elastic-Plastic Bending

5.7. Find the fully plastic moment M

for the symmetric beam section of Figure P5.7,

if the uniaxial yield stress is 300 MPa.

all dimensions in mm

Figure P5.7

5.8*. Figure P5.8 shows a composite rectangular beam, manufactured from alu-

minium alloy (E = 80 GPa, S

= 60 MPa) and steel (E = 210 GPa, S

= 300 MPa).

The beam is bent about the horizontal axis to a radius R = 7 m. Find the required

bending moment M and sketch the resulting stress distribution.

aluminium

steel

all dimensions in mm

Figure P5.8

Problems 277

5.9**. It can be shown that the curvature of an elastic-plastic beam of square cross

section a×a is





; M <

3(1 −M/M

)

;

< M < M

where M

/4 is the fully plastic moment

The beam is built in at z = 0 and loaded by a force F at z = L as shown in Figure

P5.9. If the force is just sufﬁcient to cause the beam to become fully plastic at z = 0,

show that the downward end deﬂection is then

u(L) =

40S

27Ea

Figure P5.9

Section 5.5

5.10. Find the fully plastic moment M

and the location of the corresponding neutral

axis for the beam section of Figure P5.10, if the uniaxial yield stress is 200 MPa.

all dimensions in mm

Figure P5.10

The proof of these results is a special case (b =h=a) of Problem 5.5.

278 5 Non-linear and Elastic-Plastic Bending

5,11. Find the fully plastic moment and the shape factor for the beam section of

Figure P5.11, if the uniaxial yield stress is 30 ksi.

all dimensions in inches

Figure P5.11

5.12. Find the fully plastic moment M

for the thin-walled semi-circular section of

Figure P5.12, if the yield stress of the material is S

. Assume t ≪a.

Figure P5.12

5.13. The equilateral triangular column of Figure P5.13 is loaded by an axial force

F, whose line of action is a distance h/3 from one vertex. If the force is large enough

to cause plastic collapse, ﬁnd the location of the neutral axis.

Figure P5.13

Problems 279

5.14*. Find the fully plastic moment M

for the T-section shown in Figure 5.13, if

the uniaxial yield stress is S

= 200 MPa. Find also the moment at which a second

plastic zone starts to be developed at the upper edge.

5.15*. The channel section of Figure P5.6 is bent to a radius of 16 m about the

vertical axis. Find the location of the neutral axis, the extent of the plastic zone and

the bending moment, if the relevant material properties are E = 80 GPa, S

= 100

MPa.

5.16. The beam of Problem 5.10 is bent to a radius of 1 m. Find the bending moment,

the extent of the plastic zones and the location of the neutral axis (E = 210 GPa,

=200 MPa).

5.17*. A rectangular beam of width b and height h is made from a material with

different elastic moduli in tension and compression — i.e.

= E

; e

> 0

= E

; e

< 0 .

Find the location of the neutral axis and the relation between the applied moment

M and the radius of curvature R, if E

=E and E

=2E.

Section 5.6

5.18. The I-beam of Figure P5.18 is to be bent to plastic collapse about an axis

inclined at 45

as shown. Find the magnitude M

of the required resultant moment

and its inclination

to the horizontal axis, if the material has a uniaxial yield stress

of 300 MPa. Comment on your results.

neutral

axis

all dimensions in mm

Figure P5.18

280 5 Non-linear and Elastic-Plastic Bending

5.19. The thin-walled unsymmetrical angle section shown in Figure P5.19 is bent

in such a way that the neutral axis in the fully plastic state cuts the lower leg at a

distance d from the corner, where d <a. Show that the condition of zero axial force

F =0 then demands that the corresponding intercept on the vertical leg is at a distance

(1.5a−d) from the corner.

Figure P5.19

Determine d for the case where the applied moment M acts about the horizontal

axis and hence determine the inclination of the neutral axis for this case. Assume

that the thickness of the section t ≪a throughout the calculation.

What interpretation do you place on the fact that all potential neutral axes that

cut the section in only one place (chosen to make F = 0) will correspond to the same

stress distribution?

5.20. The thin-walled I-beam of Figure P5.20 is bent by a fully plastic moment such

that the neutral axis at angle

cuts through the ﬂanges as shown. Find the inclination

of the resultant moment and sketch the variation of

with

Problems 281

neutral

axis

Figure P5.20

5.21*. The thin-walled section of Figure P5.21 is subjected to a bending moment

about the horizontal axis. Find the fully plastic moment and the inclination of the

corresponding neutral axis, if the yield stress of the material is S

Figure P5.21

282 5 Non-linear and Elastic-Plastic Bending

5.22*. The rectangular section of Figure P5.22 is loaded to plastic collapse by a

moment about an axis inclined at an angle

to the horizontal. Find the inclination of

the neutral axis

and the fully plastic moment M

as functions of

. The yield stress

for the material is 20 ksi. Sketch a graph of the resulting expressions. Hint: You will

need to consider separately the cases where the neutral axis cuts (i) the sides and (ii)

the top and bottom of the section.

all dimensions in inches

Figure P5.22

Section 5.7

5.23. The hollow rectangular section of Figure P5.23 is subjected to a bending mo-

ment just sufﬁcient to yield the upper and lower walls. The moment is then released.

Find the residual radius of curvature of the beam R

and the residual stress distribu-

tion, if E =210 GPa and S

=350 MPA.

all dimensions in mm

Figure P5.23

Problems 283

5.24. A bar of solid circular cross section of radius a is loaded by the fully plastic

moment and then unloaded. Find the residual stress distribution in the bar, if the yield

stress of the material is S

5.25*. In a manufacturing process, it is desired to form an initially straight bar of

square cross section 25 mm × 25 mm to a curved bar of mean radius 1m. This is

to be done by passing the bar between rollers, in the direction of the velocity V in

Figure P5.25, such that the central region is in a state of pure bending at a radius of

curvature R. What must be the radius R, if the beam is to spring back to the required

radius of 1m when the load is removed? Assume that the bar is of steel with E =210

GPa and S

=210 MPa.

1 m

rollers

Figure P5.25

5.26. The section of Figure P5.11 is loaded until a second plastic zone is just about

to develop at the upper edge. Find the radius of curvature at this state if E =30 ×10

ksi and S

=30 ksi.

The beam is now unloaded. What will be the residual curvature?

5.27. A 1 mm diameter steel wire (E =210 GPa, S

= 210 MPa) is close coiled into

a helix around a 25 mm diameter rigid cylindrical drum. What will be the inside

diameter of the coil when it is released from the drum.

5.28*. The T-section of Figure 5.13 is loaded by the fully plastic moment and then

unloaded. Show that the assumption of elastic unloading leads to stresses exceeding

the yield stress between the elastic and fully plastic neutral axes.

The actual unloading process will involve additional yield in a small central re-

gion, the rest of the section remaining elastic. Find the extent of this yield zone and

the ﬁnal residual stress distribution. Take care that your solution satisﬁes the condi-

tion of zero axial force (F =0). The yield stress is S

=200 MPa.

284 5 Non-linear and Elastic-Plastic Bending

5.29*. A circular ring of radius 200 mm and square cross section 4 mm×4 mm is to

be manufactured from steel of Young’s modulus E =200 GPa and tensile yield stress

=200 MPa.

400π

(a)

400

weld

(b)

(c)

2500 N/m

all dimensions in mm

Figure P5.29

(i) The ﬁrst stage in the process is to take the initially straight and stress-free bar of

cross section 4 mm×4 mm and length 400

mm of Figure P5.29 (a), apply end

moments M sufﬁcient to bend it into a circle and then weld the ends together, as

shown in Figure P5.29 (b).

Find the moment M required and the corresponding bending stress distribution.

(ii) In an attempt to reduce the residual stress in the ring, a uniform radial force

2500 N/m per unit circumference is now applied, as shown in Figure P5.29 (c),

and then released.

Find the ﬁnal stress distribution in the ring.