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Homework 4 in MEE 322: Structural Mechanics | Two-Dimensional Stress Analysis

Problem 1

The state of stress at a point is given in the figure. Find τ and τxy directly using force equilibrium. Do NOT use the stress transformation equations.

Problem 2

The state of stress at a point is given in the figure. Find σ and τxy using stress transformation equations.

Problem 3

The state of stress at a point is given in the figure. Find the principal stresses, principal directions and the maximum shear stress using

(a) Eigenvalue problem approach

(b) Stress transformation equations

Problem 4

The shaft shown in the figure has a gear at B with a force of 2098 N in -y and 6456 N in +z applied at its tip. The force along z produces a torque that drives the component attached at C, which produces an equal and opposite torque to that produced at the gear as well as forces on the shaft of 6000 N along +y and +z. The bearing at A can be considered a spherical hinge, whereas the bearing at D can be considered a planar hinge in the y-z plane.

(b) Draw bending moment and torsion diagrams for the shaft and show diagrams of the cross-section of the shaft where the critical points occur, i.e., the locations where the maximum normal stresses due to bending and maximum shear stress due to torsion coincide. Indicate the internal reactions (bending and torsion moments) in this diagram, as well as the locations of the critical points.

(b) If the diameter of the shaft is 33 mm, find the stresses at the critical point and use them to find the maximum shear stress at that location as well as the maximum and minimum principal stresses. Note: the bending normal stress can be taken as σx, while the torsion shear stress can be assumed to be τxy for the effects of this calculation, all other stresses can be assumed to be zero.

(c) It is known that the material of the shaft is such that it will fail if the maximum shear stress reaches 300 MPa. Is the shaft safe? If so, calculate the factor of safety.

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Homework 4 in MEE 322: Structural Mechanics

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Homework 3 in MEE 322 Structural Mechanics | Normal and Shear Stresses Under Combined Loading Part 2

Problem 1

Three forces act on the tip of a L-shaped rod with a cross-sectional radius of 0.5 in.

(a) Determine the normal and shear stress at points A and B and draw the stress cube at those points based on the given coordinate system.

(b) Determine the maximum normal stress on the cross-section and locate the point at which it occurs.

Problem 2

The simply supported solid shaft has a radius of 15 mm and is under static equilibrium. Pulley C has a diameter of 100 mm. The pulleys B and D have the same diameter as each other. The forces on pulley B are at an angle of 45 to the negative z-axis. The forces on pulley C and pulley D are in the z and -y direction. The shaft dimensions are in mm.

(a) Determine the maximum bending and torsional stresses in the shaft.

(b) Locate the point(s) on the cross-section where the bending stress is maximum.

Problem 3

The structural part of a setup to measure net belt tensions in pulleys is shown in the figure. The belt tensions at both sides of the pulley at B (radius 10 cm) are P and F=0.1*P along z and a reaction force is measured from the pulley at C (radius 2 cm), which is connected to a load cell at E with an axial member parallel to x. Pulleys are rigidly attached to rod AD, which is made with a ductile steel rod 60 cm long and 1.27 cm in diameter. Length AB=0.20 m, and length DC=0.15 m. There is a spherical hinge at A and a plane hinge at D. The latter constrains motion in the x-z plane only.

(a) Draw bending moment and torsion diagrams for this structure as functions of the unknown tension P and use them to draw a diagram of the critical section showing internal loads (bending and torsion moments) and the critical points.

(b) Use your results from part a to determine the maximum normal stress due to bending and the maximum shear stress due to torsion in terms of the unknown tension P. Calculate the maximum value that P can have if only bending stresses are considered (with σallow = 350 MPa) and then if only torsion stresses are considered (with τallow = 175 MPa).

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Homework #2 in MEE 322 Structural Mechanics

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Homework 2 in MEE 322: Structural Mechanics | Normal and Shear Stress Under Combined Loading

Problem 1

The shaft with a circular cross-section is supported by two bearings at O and C. The bearings do not exert any moments or axial force on the shaft, and they act to constrain motion along the x and y axes. Find the minimum diameter required for the shaft if the maximum normal stress in the shaft cannot exceed 250~ \text{MPa}. All dimensions are in \text{mm}.

Problem 2

The beam ABCD shown in the figure is simply supported at A and D, and has a circular cross-section with a diameter of 80~\text{mm}. The distributed force acts at an angle of 60^\circ to the Z axis and has components only along the Z and X axis. The 1.2~\text{kN} force acts parallel to the Z axis and the 1.5~\text{kN} force acts parallel the X axis. AB = 0.6~ \text{m}, BC = 0.4~ \text{m} and CD = 1~ \text{m}.

(a) Draw shear force and bending moment diagrams for the XY and YZ planes.

(b) Determine the maximum tensile and compressive bending stress in the beam and show their locations on the cross-section of the beam.

Problem 3

The following steel structure will be made with a round bar 35~\text{mm} in diameter, such that section A-C is parallel to the y-axis and section B-E is parallel to x-axis.

An unknown moment T parallel to y is applied at point A, where there is also a spherical (ball) hinge, which constraints translations along x, y, and z axes. The plane hinge at C is contained in the x-z plane, which means that it constrains translations along the x and z axes. The force of 1000~\text{N} at D goes in -z, the force of 500~\text{N} at D goes in -y and the force of 500~\text{N} at E goes in -x. Given these conditions and the coordinate system provided, find:

(a) Diagrams of axial force, bending moment, and torsion moments for sections A-C and B-E. Use the given coordinate system to label the planes where you are making your internal reaction diagrams (x-y, x-z or z-y) and draw the corresponding axes.

(b) Calculate the maximum normal stress due to bending in this structure. Show in a diagram the point(s) in the cross-section of the structure where this stress occurs, and include the bending moments in each axis, the resultant moment, and the neutral axis. Use the given coordinate system to show the orientation of your diagram.

(c) Calculate the maximum shear stress due to torsion in this structure. Show in a diagram the point(s) in the cross-section of the structure where this stress occurs, including the torsion moment. Use the given coordinate system to show the orientation of your diagram.

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Homework 2 in MEE 322: Structural Mechanics

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Homework 1 in MEE 322: Structural Mechanics | Internal Reactions in 3D and 2D Bending

Problem 1

The structure shown in Fig. 1.1 is made with a steel bar that has a rectangular cross-section 5 cm tall and 10 cm wide.

Fig 1.1. Frame supporting point and distributed load

Given the geometry, loads and supports in the structure draw the bending moment diagrams for segments A-C and D-E of the structure. Then, use the diagrams to find the critical section and calculate the maximum bending stress in segment AC.

Problem 2

The torsion rod with variable cross-section shown in Fig. 2.1 is clamped at A and carries point torques at B (4 kN.m), C (8 kN.m) and D (unknown value T) with the senses indicated in the figure. It is known that the diameter of AB is 25 mm, the diameter of BC is 50 mm and the diameter of CD is 20 mm. Furthermore, 6LAB = 6LBC = 5LCD = 1200 mm. If the shear modulus of the material is 80 GPa, find:

a) The value of T that would make the angle of twist at point C with respect to A equal to zero.

b) The maximum value of T that can applied with the sense shown such that failure does not occur for an allowable shear stress of 1.1 GPa. Can the condition from part a be achieved without failure?

Fig. 2.1: Torsion rod with variable cross-section.

Problem 3

The cantilever beam shown in Fig. 3.1 has a rectangular cross-section with h = 120 mm and b = 80 mm. The 12 kN and 10 kN forces act parallel to the x and z axis, respectively, and pass through the centroid of the beam cross-section at the locations they act. The 10 kN force acts at the free end of the cantilever, whereas the 12 kN force acts 250 mm away from the free end. The cross-section ABCD is 750 mm away from the free end.

a) Determine the magnitude and location of the maximum tensile and compressive bending stress at the cross-section ABCD and indicate the neutral axis on it.

b) If an additional force, F, is applied on the beam parallel to the z axis at a point 500 mm away from the free end, what should be its magnitude and direction to make the bending stress at point B zero?

Fig. 3.1: Cantilever beam with loads along two axes.

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Homework 1 in MEE 322: Structural Mechanics

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Hydrology and Floodplain Analysis by Bedient et.al. Chapter 1 Problem 7

Clear Lake has a surface area of 708,000 m2 (70.8 ha.). For a given month, the lake has an inflow of 1.5 m3/s and an outflow of 1.25 m3/s. A +1.0-m storage change or increase in lake level was recorded. If a precipitation gage recorded a total of 24 cm for this month, determine the evaporation loss (in cm) for the lake. Assume that seepage loss is negligible.

Solution:

We are given the following values:

\begin{align*}
\text{Area}, \ A&=708,000 \ \text{m}^2 \\
\text{Inflow}, \ I&=1.5 \ \text{m}^3/\text{s} \\
\text{Outflow}, \ O & = 1.25 \ \text{m}^3/\text{s} \\
\text{change in storage}, \ \Delta S & = 1.0 \ \text{m} \\
\text{Precipitation}, \ P&=24 \ \text{cm} \\
\text{time}, \ t &= 1 \ \text{month} = 30 \ \text{days}
\end{align*}

The required value is the \text{Evaporation}, \ E.

We shall use the formula

\Delta S=I+P-O-E

Solving for E in terms of the other variables, we have

E=I+P-O-\Delta S

Before we can substitute all the given values, we need to convert everything to the same unit of cm.

\begin{align*}
\text{Inflow}&=\frac{1.5\:\frac{\text{m}^3}{\text{s}}\cdot \frac{100\:\text{cm}}{1\:\text{m}}\cdot \frac{3600\:\text{s}}{1\:\text{hr}}\cdot \frac{24\:\text{hr}}{1\:\text{day}}\cdot \frac{30\:\text{days}}{1\:\text{month}}\cdot 1\:\text{month}}{708,000\:\text{m}^2} \\
\text{Inflow}&=549.1525 \ \text{cm}
\end{align*}
\begin{align*}
\text{Outflow}&=\frac{1.25\:\frac{\text{m}^3}{\text{s}}\cdot \frac{100\:\text{cm}}{1\:\text{m}}\cdot \frac{3600\:\text{s}}{1\:\text{hr}}\cdot \frac{24\:\text{hr}}{1\:\text{day}}\cdot \frac{30\:\text{days}}{1\:\text{month}}\cdot 1\:\text{month}}{708,000\:\text{m}^2} \\
\text{Outflow}&=457.6271 \ \text{cm}
\end{align*}
\Delta S=1.0 \ \text{m} \times \frac{100 \ \text{cm}}{1.0 \ \text{m}}=100 \ \text{cm}

Now, we can substitute the given values in the formula

\begin{align*}
E & =I+P-O-\Delta S \\
E& =549.1525 \ \text{cm}+24 \text{cm}-457.6271 \ \text{cm}-100 \ \text{cm} \\
E& =15.5254 \ \text{cm} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}
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Hydrology and Floodplain Analysis by Bedient et.al. Chapter 1 Problem 6

A lake with a surface area of 1050 acres was monitored over a period of time. During a one-month period, the inflow was 33 cfs, the outflow was 27 cfs, and a 1.5-in. seepage loss was measured. During the same month, the total precipitation was 4.5 in. Evaporation loss was estimated as 6.0 in. Estimate the storage change for this lake during the month.

Solution:

We are given the following values:

\begin{align*}
\text{Area}, \ A&=1050 \ \text{acres} \\
\text{Time}, \ t&=1 \ \text{month} \\
\text{Inflow}, \ I&=33 \ \text{cfs} \\
\text{Outflow}, \ O&=27 \ \text{cfs} \\
\text{Ground seepage}, \ G&=1.5 \ \text{in} \\
\text{Precipitation}, \ P&=4.5 \ \text{in} \\
\text{Evaporation}, \ E&=6.0 \ \text{in}
\end{align*}

The formula that we are going to use is:

\sum \text{Inflows}-\sum \text{Outflows}=\text{Change in Storage}, \Delta S \\
\\ 
\sum I-\sum Q=\Delta S

In this case, the inflows are \text{Inflow} \ I and \text{Precipitation}, \ P, while the others are outflows. Our formula now becomes

\color{Blue} \sum \text{Inflow}-\color{Red} \sum \text{Outflow}=\color{Green} \Delta S
\\
\color{Blue}(I+P)- \color{Red}(O+G+E)=\color{Green}\Delta S

Before substituting, we need to convert all the given to inches. More specifically the outflow O and inflow I.

The inflow and outflow, in cfs, will be divided by the given are to come up with units of inches.

The inflow is

\begin{align*}
\text{Inflow}&=\frac{33\:\frac{\text{ft}^3}{\text{s}}\cdot \frac{1\:\text{acre}}{43560\:\text{ft}^2}\cdot \frac{12\:\text{in}}{1\:\text{ft}}\cdot \frac{3600\:\text{s}}{1\:\text{hr}}\cdot \frac{24\:\text{hr}}{1\:\text{day}}\cdot \frac{30\:\text{days}}{1\:\text{month}}\cdot 1\:\text{month}}{1050\:\text{acres}} \\
\text{Inflow}& =22.4416 \ \text{in}
\end{align*}

The outflow is

\begin{align*}
\text{Outflow}&=\frac{27\:\frac{\text{ft}^3}{\text{s}}\cdot \frac{1\:\text{acre}}{43560\:\text{ft}^2}\cdot \frac{12\:\text{in}}{1\:\text{ft}}\cdot \frac{3600\:\text{s}}{1\:\text{hr}}\cdot \frac{24\:\text{hr}}{1\:\text{day}}\cdot \frac{30\:\text{days}}{1\:\text{month}}\cdot 1\:\text{month}}{1050\:\text{acres}} \\
\text{Outflow}& =18.3613\ \text{in}
\end{align*}

Now that everything is in inches, we can now substitute the values in the formula

\begin{align*}
\Delta S&=\left( I+P \right)-\left( O+G+E \right) \\
\Delta S&=\left( 22.4416 \ \text{in}+4.5 \ \text{in} \right)-\left( 18.3613 \ \text{in}+1.5 \ \text{in}+6 \ \text{in} \right) \\
\Delta S&=1.0803 \ \text{in} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

We can also state the change in storage in terms of volume by multiplying the given area

\begin{align*}
\Delta S \ \text{in volume} & =1.0803 \ \text{in}\times 1050 \ \text{acres}\times \frac{1 \ \text{ft}}{12 \ \text{in}}\\
\Delta S \ \text{in volume} & = 94.5263 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}
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Hydrology and Floodplain Analysis by Bedient et.al. Chapter 1 Problem 5

List seven major factors that determine a watershed’s response to a given rainfall.

Solution:

The seven major factors that determine a watershed’s response to a given rainfall are:

  1. Drainage Area
  2. Channel Slope
  3. Soil Types
  4. Land Use
  5. Land Cover
  6. Main Channel and tributary characteristics-channel morphology
  7. The shape, slope, and character of the floodplain
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Hydrology and Floodplain Analysis by Bedient et.al. Chapter 1 Problem 4

Explain how air masses are classified. Where are these types of air masses located?

Solution:

They are classified in two ways: the source from which they are generated, land (continental) or water (maritime), and the latitude of generation (polar or tropical).

These air masses are present in the United States. The Continental polar emanates from Canada and passes over the northern United States. The maritime polar air mass also comes southward from the Atlantic Coast of Canada and affects the New England states. Another maritime polar comes from the Pacific and hits the extreme northwestern states. The maritime tropical air masses come from the Pacific, the Gulf of Mexico, and the Atlantic (these affect the entire Southern United States). Continental tropical air masses form only during summer. They originate in Texas and affect the states bordering the north.

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Hydrology and Floodplain Analysis by Bedient et.al. Chapter 1 Problem 3

Explain the difference between humidity and relative humidity.

Solution:

Humidity is a measure of the amount of water vapor in the atmosphere and can be expressed in several ways. Specific humidity is a mass of water vapor in a unit mass of moist air while relative humidity is a ratio of the air’s actual water vapor content compared to the amount of water vapor at saturation for that temperature.

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