Crane at 45-degree angle supports load -- Find principal boom stresses

In summary, the task involves analyzing the stresses experienced by a crane's principal boom when it is positioned at a 45-degree angle while supporting a load. This requires applying principles of physics and mechanics to determine the forces acting on the boom and calculating the resulting stress to ensure structural integrity and safety during operation.
  • #1
SoylentBlue
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Homework Statement
The crane is used to support the 350-lb load.
Determine the principal stresses acting in the boom at
points A and B. The cross section is rectangular and has a
width of 6 in. and a thickness of 3 in.
Relevant Equations
axial stress = P/A; bending stress = Mc/I
transverse shear stress=VQ/IT; torsional stress = Tc/J
The book's answer key shows a normal stress P/A as a result of adding the 350LB force from the cable to the cos(45 degrees)(350LB). It shows both of these as compressive forces. I can see how the cos(45 degrees)(350LB) is compressive; the force of the 350LB load is bending the boom clockwise, causing compression at point A. But the cable's pull from the 350LB load, intuitively, seems like it should be tension and therefore positive.

Thank you in advance for any help. I am trying to learn this on my own, so I have no professor or other students to talk with; the Internet is my professor.
 

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  • #2
I can’t clearly see in the diagram where points A and B are located in the cross section of the boom.
Could you clarify that, as well as your main question, please?
 
  • #3
The book's answer key shows, for point A, sigma = P/A = 597.49/18 in compression. The 597 is the sum of [cos(45 degrees)(350LB)] plus 350LB. cos(45 degrees)(350LB) is 247LB, which is bending the boom clockwise, therefore putting point A in compression. The additional 350 must come from the cable, which, by the arrow they have drawn, is in compression. I don't see how a cable can be in compression.
I agree that it is difficult, the way they drew this, to determine where points A & B are; I needed a magnifying glass to decipher their schematic.
The side of the boom, which we see in the illustration, is 6 inches; it is 3 inches deep.
If we look straight-on at the cross-section of the boom, where it is 6in tall and 3in wide, point A is on the bottom-right corner of the boom. Point B is on the right side edge, 3in up from the bottom; it is on the neutral axis.
Thank you in advance :^)
 
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  • #4
In order to understand the load on the boom, we need to do a free body diagram of it.
The boom "knows" nothing about the cable or the load.
All the forces reach it through the shaft of the pulley and its solid anchorage to the ground.

Crane 1.jpg


Crane 2.jpg



Crane 3.jpg
 
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  • #5
Oh, I see. It was just a matter of geometry. Thank you, thank you! :^)
 
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  • #6
@SoylentBlue ,
Now that we have identified direction and value of the actual external force that the load imposes onto the crane, shall we continue our discussion regarding internal forces deep into the boom (at cross-section A-B)?

Crane 4.jpg
 
  • #7
OK, here's a summary of the forces at points A & B:

I = 1/12(base)(height)3 = 1/12(3)(6)3 = 54

Q = y-bar*area = 1.5 * 9 = 13.5 where area is 3 high and 3 deep

For point A:

Normal force = P/A = -597/18 this is negative since it is in compression

Bending moment= Mc/I = [(7.07ft)(12inches/1ft)(3in)]/54 = -1164 psi

Moment is also negative since point A is in compression

Sigma total = -1200 psi



Tau = Tc/J = 0 there is no torsion

Tau= VQ/IT= 0 because q = 0 ; A is on outer edge, and there is no material above it



For point B:

Normal force = P/A = -597/18 this is negative since it is in compression

Bending moment= Mc/I = 0 since B is on the neutral axis

Sigma total = -33.19 psi



Tau = TC/J = 0 there is no torsion

Tau= VQ/IT= (247*13.5)/(54*3) = 20.62 this is positive since our shearing force goes up


I am still a little unsure about determining the signs of torsion and transverse shear. Do we always apply the right-hand rule? For instance, in problem 9-91 (where we have a pipe wrench on a pipe; screenshot attached), if we apply the right-hand rule, the wrench spins the pipe into the paper about the x axis, and the thumb points up; wouldn't this be positive torsion?

Thank you again. I wish I was young enough to go back to school and finish the engineering degree that I started (but never finished) back when Jimmy Carter was still President.
 

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  • #8
Addendum to last post: I did a quick review of statics, and I see that M, N and V are internal forces. The torsion and shear due to the pipe wrench are external forces. So of course the internal and external reactions would have to oppose each other, just as they show in the answer key's illustration. I am embarrassed to admit that I missed that.
 
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FAQ: Crane at 45-degree angle supports load -- Find principal boom stresses

What is the significance of analyzing boom stresses in cranes?

Analyzing boom stresses in cranes is essential for ensuring the safety and stability of the crane during operation. Understanding these stresses helps in designing the crane to withstand various loads and environmental conditions, preventing structural failures and accidents.

How do you calculate the principal stresses in a crane boom?

Principal stresses in a crane boom can be calculated using the principles of mechanics of materials. Typically, this involves applying formulas derived from bending and axial load equations, along with considering factors such as the angle of the boom, the load being lifted, and the material properties of the boom itself.

What factors affect the stresses in a crane boom at a 45-degree angle?

Several factors affect the stresses in a crane boom at a 45-degree angle, including the weight of the load, the length and cross-sectional area of the boom, the material properties (such as yield strength), and external conditions like wind or seismic activity. The angle itself also influences the distribution of forces within the boom.

What are the common materials used for crane booms, and how do they influence stress calculations?

Common materials for crane booms include steel and aluminum, each with distinct mechanical properties. Steel typically offers higher strength and stiffness, which can result in lower stresses for the same load compared to aluminum. The choice of material affects the calculations of stress, as different materials have varying yield strengths and elastic moduli.

How can the results of stress analysis improve crane design and safety?

The results of stress analysis can provide valuable insights into potential failure points and areas of high stress concentration in crane designs. By identifying these factors, engineers can modify the design to enhance safety, such as reinforcing critical areas, choosing stronger materials, or optimizing the boom geometry, thereby reducing the risk of accidents during operation.

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