Barber J.R. Intermediate Mechanics of Materials

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Problems 175

Figure P3.36

Invert the matrix to ﬁnd the corresponding compliance matrix C. Notice that C

is a full matrix, whereas K is banded. Verify that both matrices are symmetric and

that the inequalities (3.80, 3.83, 3.84) are satisﬁed for this case.

3.37. The pin-jointed bars in Figure P3.37 have a circular cross section of diameter 10

mm and are made of steel with Young’s modulus E =210 GPa. Find the 2×2 stiffness

matrix for the system relating the forces F

and the corresponding displacements

. You might ﬁnd it helpful to review the kinematics of Example 3.5 and Figure

3.12.

0.5 m

u ,

Figure P3.37

3.38*. The structure shown in Figure P3.38 consists of two circular steel bars, each

of diameter 10 mm, built in at B,C and welded together at A. Use equations (3.89,

3.90) to construct the 3×3 stiffness matrix relating the forces and moments F

and the corresponding displacements u

. For steel, E =210 GPa.

We recall from §3.3.2 that bars are much stiffer in extension than in bending.

Discuss the implications of this result in the present problem.

176 3 Energy Methods

0.5 m

u ,

0.5 m

θ , M

Figure P3.38

3.39. The strain energy in a structure can be written in terms of the stiffness matrix

and the displacements in the form

U =

∑

i=1

∑

j=1

i j

Use scenarios analogous to those used in §3.8.2 to establish inequalities that must

be satisﬁed by the coefﬁcients K

i j

of the stiffness matrix.

3.40. Use equations (3.89, 3.90) to construct the stiffness matrix for the built-in angle

of Figure P3.40 subjected to out-of-plane loading F and moments M

, M

. Notice

that the coordinate system used here differs from that used in equations (3.89, 3.90).

F,u

M ,θ

Figure P3.40

Problems 177

Section 3.10

3.41. The bars in Figure P3.41 are each made of steel of cross-sectional area A = 1

. Use the result given in Problem 3.3 to express the total strain energy stored in

the structure as a function of the displacement components, u

, i= (1,6). Hence ﬁnd

the 6 ×6 stiffness matrix relating the forces F

and the corresponding displacements

. Remember that any given product term (e.g. u

) occurs twice in the sum and

that K

i j

. For steel, E =30 ×10

psi.

u ,

10 in

Figure P3.41

3.42. Both segments of the angled beam in Figure P3.42 have the same ﬂexural

rigidity EI and length L.

Figure P3.42

178 3 Energy Methods

Find the angle

if the displacement of the point A is to be in th e same direction

as the force F.

3.43. Figure P3.4 3 shows a stepped steel shaft supported in bearings at A,C and

loaded b y a 2 kN force at B. Use Castigliano’s second the orem to ﬁnd the angu-

lar misalignment at the bearings (i.e. th e local slope of the beam) due to the shaft

deﬂection. For steel, E = 210 GPa.

200

2 kN

all dimensions in mm

Figure P3.43

3.44. Figure P3 .44 shows a tool clip which is made of 1 ×

inch spring steel plate,

for which E =30 ×10

psi,

=0.3. The separation between the grips is

inch in the

undeformed po sition and diameter of the tool handle is 1 inch. If the coefﬁcient of

friction is 0. 3, what is the maximum weight of tool that can be supported in a vertical

position.

Tool

Figure P3.44

3.45. Fig ure P3.45 shows a beam of length L and ﬂexural rigid ity EI, which is simply

supported at B and supported by a spring of stiffness k at A. Find the slope of the

beam at A when th e beam is subjected to a uniformly distributed loa d w

per unit

length.

Problems 179

3.46. A student, asked to compute the downward vertical displacement u

at C for

the beam in Figure P3.46, equates it to the partial derivative

∂

F, where U is

the strain energy of the beam. Why is this an incorrect application of Castigliano’s

second theorem? Use the theorem to calculate the correct value of u

, if the beam has

ﬂexural rigidity EI and length 2L. What is the physical interpretation of the incorrect

result

∂

Figure P3.46

3.47. A thin semi-circular beam of radius R and ﬂexural rigidity EI is subjected to a

vertical force F at the end, as shown in Figure P3.47. Find the horizontal and vertical

components (u

respectively) of the displacement at the point A.

Figure P3.47

w per unit length

Figure P3.45

180 3 Energy Methods

3.48. Use Castigliano’s second theorem to obtain the exact solution for the change in

diameter BD in Problem 3.27.

3.49. Figure P3.49 shows a uniform beam subjected to a uniformly distributed load

over its entire length, L. The beam rests on three equally spaced simple supports. Use

Castigliano’s second theorem to determine the distance a between the supports if the

load is to be shared equally between them.

3.50. The right-angle bar of Figure P3.50 is built in at A and rests on a frictionless

support at C. Both segments of the bar have ﬂexural rigidity EI. Use Castigliano’s

second theorem to ﬁnd the horizontal displacement at C, when the bar is loaded by a

moment M

at B.

Figure P3.50

3.51. Figure P3.51 shows a semi-circular curved beam of radius R and ﬂexural rigid-

ity EI, which is pinned at A,C and subjected to a vertical load F at B. Use Cas-

tigliano’s second theorem to ﬁnd the reactions at the supports and the displacement

under the load.

w per unit length

L /2

Figure P3.49

Problems 181

Figure P3.51

3.52. Use Castigliano’s second theorem to ﬁnd the reaction at the simple support in

Problem 3.20 and hence to ﬁnd the deﬂection at the free end of the beam.

3.53. The L ×L square frame of Figure P3.53 is made from solid circular section bar

of diameter d. It is suspended from the mid-point of the upper segment and loaded

only by self-weight. Find the magnitude and location of the maximum tensile stress

if the material has Young’s modulus E and density

Figure P3.53

182 3 Energy Methods

3.54. The curved beam of Figure P3.54 is built-in at A and subjected to an out of

plane load F at B as shown. Find the bending moment M and torque T as functions

and hence determine the out of plane displacement at B. The beam has ﬂexural

rigidity EI and torsional rigidity GJ.

Figure P3.54

3.55. Figure P3.55 shows an open circular ring of mean radius 30 mm, whose cross

section is a circle of diameter 2 mm. The two ends of the ring are subjected to equal

and opposite 4 N forces out of the plane of the ring. Find the relative displacement

of the two ends, if the ring is made of steel (E =210 GPa,

=0.3).

2 mm

4 N

30 mm

Figure P3.55

3.56*. Figure P3.56 shows two views of an outboard mounted spur gear that is sub-

jected to a maximum force of 5 kN, which can be assumed to act half way along the

length of the gear. The bearings can be treated as simple supports at their mid-point.

Use Castigliano’s second theorem to estimate the misalignment of the gear teeth

due to shaft deﬂection, assuming the meshing gear is perfectly aligned and does not

deﬂect. The shaft is made of steel for which E = 210 GPa,

=0.3. Which end of the

gear will carry the most load as a result of this deformation?

Problems 183

Do you think your answer is sufﬁciently large to cause concern if the gear design

were based on the assumption of uniform contact along the length?

200

5 kN

all dimensions in mm

Figure P3.56

3.57*. A long coil spring of mean coil diameter 20 mm is fabricated from steel wire

of diameter 2 mm. The spring is close-coiled, which means that the adjacent coils

touch in the unloaded state. The ends of the coil are now subjected to a bending

moment M, as shown in Figure P3.57.

Find the effective ﬂexural rigidity of the spring — i.e. the constant K

in the

relation

M =

where R is the radius of curvature of the coil axis due to M. (E =210 GPa,

=0.3).

2 mm

20 mm

Figure P3.57

184 3 Energy Methods

3.58. The right angle crank ABC of Figure P3.58 is made of steel bar of diameter 15

mm and is loaded by an out of plane force of 300 N at the end C.

Find the slope of the bar at C. (E = 210 GPa,

=0.3).

300 N

15 mm

0.8 m

0.5 m

Figure P3.58