# Continuum Mechanics and Deformable Bodies

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A question pack on engineering mechanics.

Deformable Bodies Q34

How is the axial tensile strength of a cylindrical member related to the diameter of the member?

A

It increases linearly with diameter.

B

It increases in proportion to the cube of the diameter.

C

It is independent of diameter.

D

It increases approximately as the square of the diameter.

E

It increases approximately as the square root of the diamete

Deformable Bodies Q33

Which of the following correctly describes stress in the sense used in mechanics?

A

It is the vector sum of all forces acting on a system.

B

It is the sum of all torques acting on a system.

C

It is the square of the magnitude of the total force acting on a system.

D

E

It is a force per unit volume.

Deformable Bodies Q32

A rod is being used to carry a tensile load of 5000 N. The ultimate tensile stress of the material used in the rod is 1000 MPa. What should be the absolute minimum diameter of the rod to safely carry the load?

A

5 mm

B

10 mm

C

2.5 mm

D

50 mm

E

0.5 mm

Deformable Bodies Q31

A rod is used to carry a tensile load of 10000 lbf. The ultimate tensile stress of the material used in the rod is 1000 MPa. What should be the minimum diameter of the rod to safely carry the load?

A

20 mm

B

25 mm

C

50 mm

D

7.5 mm

E

2 mm

Deformable Bodies Q30

A cylindrical rod is hung in a gravitational acceleration of 9.8 m/s$^2$. The ultimate tensile stress of the rod material is 1800 MPa. The density of the rod material is 4000 kg/m$^3$. How long may we make the rod before it fails under its own weight?

A

1000 ft

B

52 miles

C

46 km

D

3 km

E

1.2 km

Deformable Bodies Q29

A cylindrical rod is hung in a gravitational acceleration of 9.8 m/s$^2$. The ultimate tensile stress of the rod material is 1800 MPa. The density of the rod material is 1000 kg/m$^3$. How long may we make the rod before it fails under its own weight?

A

46 km

B

12 km

C

184 km

D

1.8 km

E

0.2 km

Deformable Bodies Q28

A member of square cross section (10 mm on each side) and of 20 cm in length is subject to atensile load of 1 kN. The Young’s modulus of the material composing the member is 250 GPa.What is the extension of the member under load in mm?

A

12 mm

B

14 mm

C

80 mm

D

0.2 mm

E

8 mm

Deformable Bodies Q27

A member of square cross section (15 mm on each side) and of 20 cm in length is subject to a tensile load of 1 kN. The Young’s modulus of the material composing the member is 250 GPa.What is the extension of the member under load?

A

0.53 inches

B

0.35 inches

C

3.6 mm

D

5 cm

E

22 mm

Deformable Bodies Q26

The following curve is recorded in a tensile stress test for a rod of length 5 cm and diameter 5 mm. What is the 0.1%‑strain‑offset yield stress of the material?

A

200 MPa

B

100 MPa

C

50 MPa

D

10 MPa

E

1 GPa

Deformable Bodies Q25

A spherical pressure vessel is made of a material with an ultimate tensile strength of about 200 MPa. What pressure can a vessel that is 50 cm in diameter and 0.5 cm in wall thickness withstand?

A

8 x 10 $^9$ Pa

B

125 atm

C

78 atm

D

14.7 psi

E

8 x 10 $^4$ Pa

Deformable Bodies Q24

A spherical pressure vessel is made of a material with a yield strength of about 200 MPa. What pressure can a vessel that is 25 cm in diameter and 0.4 cm in wall thickness withstand?

A

13 x 10 $^6$ Pa

B

12 atm

C

1300 atm

D

14.7 atm

E

101325 Pa

Deformable Bodies Q23

A four inch inner diameter pipe with wall thickness 0.2 inches is pressurized to 150 psi. What is the percentage expansion in the pipe diameter upon pressurization if the Young’s modulus of the pipe material is 25 GPa?

A

7

B

0.08

C

1.5

D

0.0008

E

0.04

Deformable Bodies Q22

A 10 inch inner diameter pipe with wall thickness 0.25 inches is pressurized to 1500 psi. What is the percentage expansion in the pipe diameter upon pressurization if the Young’s modulus of the pipe material is 250 GPa?

A

0.08

B

0.8

C

0.04

D

0.004

E

0.4

Deformable Bodies Q21

A solid shaft of diameter 2 inches, shear modulus of 75 GPa, and length 1 m is subject to a torque of 5 kNm. What is the angular strain of the shaft?

A

32 degrees

B

C

5.8 degrees

D

1.6 degrees

E

Deformable Bodies Q20

Asolid shaft of diameter 2 inches, shear modulus of 50 GPa, and length 0.5 m is subject to a torque of 5 kNm. What is the angular strain of the shaft?

A

B

0.44 degrees

C

0.88 degrees

D

0.09 degrees

E

4.4 degrees

Deformable Bodies Q19

Drive shafts are often constructed from thin walled cylinders. Consider such a cylinder of length 0.75 m, outer diameter 3 inches, and wall thickness 0.25 inches. The material has a shear modulus of 68 GPa. What is the angular strain for an applied torque of 750 Nm?

A

0.27 degrees

B

C

0.32 degrees

D

E

0.18 degrees

Deformable Bodies Q18

Drive shafts are often constructed from thin walled cylinders. Consider such a cylinder of length 0.75 m, outer diameter 3 inches, and wall thickness 0.25 inches. The material has a shear modulus of 68 GPa. What is the angular strain for an applied torque of 1750 Nm?

A

0.64 degrees

B

C

2.1 degrees

D

E

0.006 degrees

Deformable Bodies Q17

In which of the following geometries is stress concentration most pronounced?

A

In the center of a large beam under transverse load

B

At the middle of a column under a parallel load

C

Near the corners of a triangular window in a pressure vessel

D

On the edge of a spherical pressure vessel

E

At the apex of a free standing column

Deformable Bodies Q16

Calculate the volumetric or dilatational strain for a cube subject to normal strains of e$_x$= 0.001, e$_y$ = 0.003, and e$_z$ = 0.0007, and a shear strain txy of 0.0048.

A

0.0052

B

0.0035

C

0.001

D

0.01

E

0.0047

Deformable Bodies Q15

Calculate the volumetric or dilatational strain for a cube subject to normal strains of ex= 0.001, ey = 0.02, and ez = 0.0009, and a shear strain tyz of 0.0077.

A

0.077

B

0.097

C

0.0219

D

0.0438

E

0.015

Continuum Mechanics Q5

How does a normal stress differ from a traction vector?

A

A normal stress is a force per area of unit magnitude; a traction vector is not normalized.

B

A traction vector is a shearing force per unit area; a normal stress is a force orthogonal to the area on which it is acting.

C

A traction vector is a force per area of arbitrary direction; a normal stress is orthogonal to the area on which it is acting.

D

A normal stress is a force per area of arbitrary direction; a traction vector is orthogonal to the area on which it is acting.

E

Normal stresses and traction vectors act in opposite directions.

Continuum Mechanics Q4

Which of the following is an appropriate unit for a traction vector?

A

Pa m

B

Pa s

C

N m

D

psi

E

N

Continuum Mechanics Q3

Which of the following adequately describes the difference between a first order tensor and a second order tensor?

A

The order of a tensor describes the dimensionality of an array; a first order tensor is like a vector; a second order tensor is like a two‑dimensional array.

B

The order of a tensor describes the dimensionality of an array; a zeroth order tensor is like a vector; a first order tensor is like a two‑dimensional array.

C

The order of a tensor describes the dimensionality of the elements; a first order tensor contains only primary quantities; a second order tensor contains quadratic quantities.

D

The order of a tensor describes the order of elements; a first order tensor has elements in increasing order; a second order tensor has the largest elements near the center.

E

The order of a tensor describes the geometry from which it is derived; a first order tensor is applicable to two dimensional (planar) mechanics; second order tensors are required for three‑dimensional mechanics.

Deformable Bodies Q14

Which of the following best describes the utility of Mohr’s circle?

A

It is a measure of hardness of materials.

B

It provides a way of visualizing and remembering how tensors rotate.

C

It allows for the convenient computation of torque from applied forces.

D

It is a conventient accounting tool for tallying applied torques.

E

It is a method of computing resultant forces.

Continuum Mechanics Q6

Which of the following adequately describes the differences between equilibrium relations, kinematic relations, and constitutive relations?

A

Equilibrium relationships describe materials chemistry; kinematic relationships describe velocities as a function of time; constitutive relationships describe compliance with codes for material behavior.

B

Constitutive relations describe tensor rotations; equilibrium relationships describe bending forces; kinematic relationships describe trajectories.

C

Equilibrium relations describe static objects; kinematic relations describe moving objects; constitutive relations describe material composition.

D

Equilibrium relations consider the action of external forces or tractions; kinematic relations consider the geometry of deformation; constitutive relations characterize a the response of a material to deformation.

E

Constitutive relations characterize the temperature and pressure dependence of density and shear modulus; equilibrium relations are overall force balances; kinematic relations describe the velocities of individual particles at the object surfaces.

Continuum Mechanics Q2

The stress tensor in a plane contains the following elements sigma$_x$ = 4 psi, sigma$_y$ = 6 psi, and tau$_{xy}$ = 2.3 psi. In a new coordinate system $_{x’y’}$ rotated 15 degrees counter clockwise from the original coordinate system, what is tau $_{x’y’}$

A

1.2 psi

B

2.3 psi

C

6.0 psi

D

2.5 psi

E

0.3 psi

Continuum Mechanics Q1

The stress tensor in a plane contains the following elements sigma$_x$ = 4 psi, sigma$_y$ = 6 psi, and tau$_{xy}$= 2.3 psi. In a new coordinate system $_{x’y’}$ rotated 45 degrees counter clockwise from the original coordinate system, what is tau$_{x’y'}$?

A

2.8 psi

B

4.3 psi

C

6.2 psi

D

2.0 psi

E

1.0 psi

Deformable Bodies Q13

A loaded crane hook is shown in the schematic below. A long which plane is the hook likely to have the largest stress?

I. AB

II. CB

III. DE

IV. EF

V. FG

A

I

B

II

C

III

D

IV

E

V

Deformable Bodies Q12

Which of the following best describes the use of moiré interferometry for strain measurements?

A

Display of light fringes that change when light passes through gradients of changing orientation or spacing

B

Display of diffraction gratings that change with strain

C

Display of optical birefringence patterns that change with strain

D

Display of laser induced grid displacement upon strain

E

Display of grid rotation and alignment upon strain

Deformable Bodies Q11

Which of the following defect geometries has found special utility in the analytical or closed‑form modeling of crack propagation?

A

Cube

B

Rhombus

C

Sphere

D

Circle

E

Ellipse

Deformable Bodies Q10

Consider a circular hole drilled in a flat plate. The plate is then subjected to uniaxial tension in the plane of the plate from top to bottom. Where would the stress be concentrated the most in the plate?

A

Just above and below the hole

B

At both lateral edges of the hole

C

Far from the hole in the bulk of the material

D

At the top edge of the hole

E

Near the entire edge of the hole

Deformable Bodies Q9

Which of the following properties are exploited to use photoelasticity to monitor strain?

A

Refractive index

B

Birefringence

C

Absorbance

D

Temperature‑dependent refraction

E

Reflection

Deformable Bodies Q8

In which of the following ways do common electrical strain gauges work?

A

Changes in inductance upon strain

B

Changes in radio‑frequency penetration upon strain

C

Changes in electrical resistance upon strain

D

Creation of short circuits upon strain

E

Changes in dielectric constant upon strain

Deformable Bodies Q7

Which of the following best describes the finite element method?

A

Description of mechanical materials in terms of a finite number of chemical elements

B

Description of mechanical behavior in terms of a finite number of motions

C

Method for reducing the mathematical complexity of problems by writing them in terms of a finite number of algebraic relations

D

Method of approximating mechanical behavior by including only the most important truss elements

E

Representation of material behavior by a finite number of mechanical rods and springs

Deformable Bodies Q6

In wire drawing, the drawing stress can often be represented by σ = σY ln(A$_0$/A), where A$_0$ and A are the initial and final areas of the wire. If we constrain σ/σ$_y$ to be 0.9, then what is the percentage reduction (100*(1-d/d$_0$)) in wire diameter that can be achieved in a single drawing operation?

A

24%

B

12%

C

36%

D

48%

E

56%

Deformable Bodies Q5

In wire drawing, the drawing stress can often be represented by σ = σ$_Y$ ln(A$_0$/A), where A$_0$ and A are the initial and final areas of the wire. If we constrain σ/σ$_Y$ to be 0.7, then what is the percentage reduction (100*(1-d/d$_0$)) in wire diameter that can be achieved in a single drawing operation?

A

30%

B

20%

C

10%

D

5%

E

15%

Deformable Bodies Q3

What is the maximum shear stress occurring in a cylindrical pressure vessel of length 1m, diameter 0.3 m, and wall thickness 1 cm when pressurized to12 atm?

A

3 MPa

B

1 MPa

C

6 MPa

D

9 MPa

E

12 MPa

Deformable Bodies Q4

What is the maximum shear stress occurring in a cylindrical pressure vessel of length 1m, diameter 0.3 m, and wall thickness 1 cm when pressurized to 100 psi?

A

1 MPa

B

5 MPa

C

10 MPa

D

15 MPa

E

25 MPa

Deformable Bodies Q2

In wire drawing, the drawing stress can often be represented by σ = σ$_Y$ ln(A$_0$/A), where A$_0$ and A are the initial and final areas of the wire. If we constrain σ/σ$_Y$ to be 0.4, then what is the percentage reduction (100*(1-d/d$_0$)) in wire diameter that can be achieved in a single drawing operation?

A

30%

B

18%

C

10%

D

5%

E

15%

Deformable Bodies Q1

What is the maximum shear stress occurring in a cylindrical pressure vessel of length 1m, diameter 0.5 m, and wall thickness 2 cm when pressurized to 200 psi?

A

1 MPa

B

5 MPa

C

9 MPa

D

15 MPa

E

25 MPa

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