# Continuum Mechanics and Deformable Bodies

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A question pack on engineering mechanics.

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

It increases linearly with diameter.

It increases in proportion to the cube of the diameter.

It is independent of diameter.

It increases approximately as the square of the diameter.

It increases approximately as the square root of the diamete

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

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

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

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

It is a loading force per unit area.

It is a force per unit volume.

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?

5 mm

10 mm

2.5 mm

50 mm

0.5 mm

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?

20 mm

25 mm

50 mm

7.5 mm

2 mm

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?

1000 ft

52 miles

46 km

3 km

1.2 km

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?

46 km

12 km

184 km

1.8 km

0.2 km

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?

12 mm

14 mm

80 mm

0.2 mm

8 mm

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?

0.53 inches

0.35 inches

3.6 mm

5 cm

22 mm

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?

200 MPa

100 MPa

50 MPa

10 MPa

1 GPa

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?

8 x 10 $^9$ Pa

125 atm

78 atm

14.7 psi

8 x 10 $^4$ Pa

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?

13 x 10 $^6$ Pa

12 atm

1300 atm

14.7 atm

101325 Pa

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?

7

0.08

1.5

0.0008

0.04

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?

0.08

0.8

0.04

0.004

0.4

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?

32 degrees

1.6 radians

5.8 degrees

1.6 degrees

0.004 radians

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?

0.5 radians

0.44 degrees

0.88 degrees

0.09 degrees

4.4 degrees

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?

0.27 degrees

0.36 radians

0.32 degrees

4 radians

0.18 degrees

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?

0.64 degrees

0.34 radians

2.1 degrees

8 radians

0.006 degrees

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

In the center of a large beam under transverse load

At the middle of a column under a parallel load

Near the corners of a triangular window in a pressure vessel

On the edge of a spherical pressure vessel

At the apex of a free standing column

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.

0.0052

0.0035

0.001

0.01

0.0047

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.

0.077

0.097

0.0219

0.0438

0.015

How does a normal stress differ from a traction vector?

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

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.

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

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

Normal stresses and traction vectors act in opposite directions.

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

Pa m

Pa s

N m

psi

N

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

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.

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.

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.

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.

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.

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

It is a measure of hardness of materials.

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

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

It is a conventient accounting tool for tallying applied torques.

It is a method of computing resultant forces.

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

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

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

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

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.

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.

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 $_{x’y’}$

1.2 psi

2.3 psi

6.0 psi

2.5 psi

0.3 psi

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$_{x’y'}$?

2.8 psi

4.3 psi

6.2 psi

2.0 psi

1.0 psi

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

I

II

III

IV

V

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

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

Display of diffraction gratings that change with strain

Display of optical birefringence patterns that change with strain

Display of laser induced grid displacement upon strain

Display of grid rotation and alignment upon strain

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

Cube

Rhombus

Sphere

Circle

Ellipse

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?

Just above and below the hole

At both lateral edges of the hole

Far from the hole in the bulk of the material

At the top edge of the hole

Near the entire edge of the hole

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

Refractive index

Birefringence

Absorbance

Temperature‑dependent refraction

Reflection

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

Changes in inductance upon strain

Changes in radio‑frequency penetration upon strain

Changes in electrical resistance upon strain

Creation of short circuits upon strain

Changes in dielectric constant upon strain

Which of the following best describes the finite element method?

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

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

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

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

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

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?

24%

12%

36%

48%

56%

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?

30%

20%

10%

5%

15%

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?

3 MPa

1 MPa

6 MPa

9 MPa

12 MPa

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?

1 MPa

5 MPa

10 MPa

15 MPa

25 MPa

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?

30%

18%

10%

5%

15%

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?

1 MPa

5 MPa

9 MPa

15 MPa

25 MPa

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