Determining the state of stress at a point

In summary, the state of stress at point A on the cross section of the cantilever beam can be determined by calculating the shear stress using the formula τ = VQ/I, where V is the shear force, Q is the first moment of the cross sectional area, and I is the moment of inertia. To calculate Q, the centroid of the cross section must be determined and the first moment of the area must be measured from the centroid of the entire beam to the centroid of the section. The bending stress can then be calculated using the formula σ = My/I, where M is the bending moment, y is the distance from the neutral axis, and I is the moment of inertia.
  • #1
Bluestribute
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Homework Statement


Determine the state of stress at point A on the cross section at section a−a of the cantilever beam. Take P = 350kN .

Homework Equations


Um . . . σ = P/A ?

The Attempt at a Solution


Yeah I'm not quite sure. I thought it was P/A, using 350000N as "P" and 300000mm^2 as A, but alas, it wasn't. Because this section of the course uses the P/A definition, I don't know if I have to use an equation with moment of Inertia and things like that. But I'm not sure, really at all, where to start for something simple like this. I need like a kick in the right direction to have a "duh" moment.
And more duh moments to follow . . . because this section is confusing the hell out of me.
 

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  • #2
Bluestribute said:

Homework Statement


Determine the state of stress at point A on the cross section at section a−a of the cantilever beam. Take P = 350kN .

Homework Equations


Um . . . σ = P/A ?

The Attempt at a Solution


Yeah I'm not quite sure. I thought it was P/A, using 350000N as "P" and 300000mm^2 as A, but alas, it wasn't. Because this section of the course uses the P/A definition, I don't know if I have to use an equation with moment of Inertia and things like that. But I'm not sure, really at all, where to start for something simple like this. I need like a kick in the right direction to have a "duh" moment.
And more duh moments to follow . . . because this section is confusing the hell out of me.

σ = P/A is good only for loads applied axially.

Here, the load P is applied laterally and sets up two kinds of stress: Bending stress and shear stress are created by the load P at point A in section a-a.

If you want to tackle this problem, I would recommend first that you determine the reactions at the fixed end of the beam and then construct the shear force and bending moment diagrams for the indicated load.
 
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  • #3
So VQ/IT?

I have my V at 350kN.
My Q is at 3000000 (100mm * 100mm * 300mm*) <- I suck at Q calcs
My I equalled 225000000 (1/12 * 100 * 300^3)
T = . . . 100mm?

Not quite sure I have the right numbers . . . But the answer these gave was wrong
 
  • #4
Bluestribute said:
So VQ/IT?

The shear stress τ = VQ / I t

I have my V at 350kN.
My Q is at 3000000 (100mm * 100mm * 300mm*) <- I suck at Q calcs
My I equalled 225000000 (1/12 * 100 * 300^3)
T = . . . 100mm?

Not quite sure I have the right numbers . . . But the answer these gave was wrong

It's better to calculate the section properties using meters rather than millimeters. I would adjust T and I so that the magnitudes are expressed in meters and m4, respectively, and calculate Q in m3.

Q is the first moment of the cross sectional area which extends from the outermost fiber to the horizontal line drawn thru point A. This moment must be calculated about the centroid of the cross section. Q is a maximum at the centroid of the cross section, and Q = 0 at the outermost fiber.

After calculating the shear stress, then the bending moment and the corresponding bending stress must be calculated. It helps to find the reactions at the fixed end and construct the shear force and bending moment diagrams.
 
  • #5
Can you calculate Q for me (showing exactly which numbers you used). No matter what numbers I use, in any type of problem (not just this one), I can NEVER get the Q value that's required. And I know it's required because they say, "Q = ". Still can't get it.
 
  • #6
Bluestribute said:
Can you calculate Q for me (showing exactly which numbers you used). No matter what numbers I use, in any type of problem (not just this one), I can NEVER get the Q value that's required. And I know it's required because they say, "Q = ". Still can't get it.

You've got an area which is 100 mm wide x 100 mm deep from the outer fiber. Call this area Ashear.

You should be able to determine the location for the centroid of this 100 mm x 100 mm area.

You want to calculate the first moment of the 100 mm x 100 mm area, measuring from the centroid of the whole section to the centroid of Ashear. Call this distance ybar.

The moment Q = Ashear * ybar

Please show your calculations for Q when you reply. :smile:
 
  • #7
A = 100x100 = 10,000 mm^2
Y, for a square, is centered, so 50mm

10,000x50 = 500,000

That's why I can't calculate Q . . .
 
  • #8
Bluestribute said:
A = 100x100 = 10,000 mm^2
Y, for a square, is centered, so 50mm

10,000x50 = 500,000

That's why I can't calculate Q . . .
You're not measuring ybar correctly. ybar must be measured from the centroid of the 100 mm x 100 mm area to the centroid of the whole beam. That distance is not 50 mm.
 
  • #9
50mm + 0.5m/2, so 300mm for Ybar?
 
  • #10
Bluestribute said:
50mm + 0.5m/2, so 300mm for Ybar?

Sigh, no.

The centroid of the whole beam is 150 mm from the outer fiber.
The centroid of the 100 mm x 100 mm area is 50 mm from the outer fiber.
What's the difference in these two measurements?
 
  • #11
So 100mmx100mm under A.
A's Ybar is at 50mm
The entire cross section Ybar is at 150mm
So Y is 100mm? And A is 100x100? So Q = 100x100x100?

And if Q is 100^3, would the thickness be 100 (the base there)?
 
  • #12
Bluestribute said:
So 100mmx100mm under A.
A's Ybar is at 50mm
The entire cross section Ybar is at 150mm
So Y is 100mm? And A is 100x100? So Q = 100x100x100?

And if Q is 100^3, would the thickness be 100 (the base there)?
Yes.
 
  • #13
Ok, so that would give me τ, but what about σ?
 
  • #14
Bluestribute said:
Ok, so that would give me τ, but what about σ?
Do you know the formula for calculating σ ?
 
  • #15
My/I right? What's y though?
 
  • #16
Bluestribute said:
My/I right? What's y though?
y is the distance from the neutral axis of the beam to the point at which σ is calculated.

In this case, y is the distance from the neutral axis of the beam to point A. The neutral axis is also located at the centroid of the beam.

M is the bending moment at section a-a, which has yet to be calculated. I = b h3 / 12 for this cross section.
 
  • #17
y: 150 (ybar) - 100 (location of A) = 50

M: (1/2)P - 0.5*350kN = 175 kN

I: I'm actually decent enough at these. (100)(300)^3(1/12)

Just confirming I'm using the correct numbers/signs
 
  • #18
Bluestribute said:
y: 150 (ybar) - 100 (location of A) = 50

M: (1/2)P - 0.5*350kN = 175 kN

I: I'm actually decent enough at these. (100)(300)^3(1/12)

Just confirming I'm using the correct numbers/signs
You should always show units with the quantities you calculate.

Your calculation of M is suspect. How did you arrive at this formula for M?

(Note: when writing formulas, it's always better to show M = whatever. What you have written for M is a bit of a head-scratcher.)
 
  • #19
I just wrote M as force*distance.

ybar in mm
 
  • #20
Bluestribute said:
I just wrote M as force*distance.

ybar in mm

Bending moment has units of force * distance; in this case kN-m. In order to calculate the correct value for σ, both y and I will have to be converted to m and m4, repectively.
 
  • #21
Well conversions aside is that the right formula? I'm going to put these in N-mm for the required units of MPa, but if I'm not calculating using the right formulas, my conversions don't matter.
 
  • #22
Got it . . . but it was negative.
I'll probably have another very similar problem soon with an I beam. Unless I can transfer these correctly to an I beam
 

1. What is stress at a point?

Stress at a point refers to the force per unit area acting on a material at a specific point. It is a measure of the internal forces within the material that can cause it to deform or change shape.

2. How is stress at a point determined?

Stress at a point is determined by applying the principles of mechanics and using equations such as Hooke's Law and the stress-strain relationship. This involves analyzing the external forces acting on the material and its geometry to calculate the internal stresses at a specific point.

3. What factors affect the state of stress at a point?

The state of stress at a point is influenced by several factors, including the magnitude and direction of external forces, the shape and size of the material, and the material's ability to resist deformation.

4. Why is it important to determine the state of stress at a point?

Determining the state of stress at a point is crucial for understanding how a material will behave under different loading conditions. It allows engineers to design structures that can withstand the expected stresses and avoid failure.

5. What are the different types of stress at a point?

The three main types of stress at a point are tensile, compressive, and shear stress. Tensile stress occurs when forces pull the material apart, compressive stress occurs when forces push the material together, and shear stress occurs when forces act in opposite directions parallel to the material's surface.

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