Tensile Stress at Mid-span of Beam

[smr101]

Hi,

I'm struggling with finding the tensile stress at the mid-span of this beam. I've done the previous questions but have no idea how to begin with this one as I've never dealt with this question before.

What is the general method?

Correct answer is 1.66MPa.

Thanks.

 

[SteamKing]

Hi,

I'm struggling with finding the tensile stress at the mid-span of this beam. I've done the previous questions but have no idea how to begin with this one as I've never dealt with this question before.

What is the general method?

Correct answer is 1.66MPa.

Thanks.

Don't worry about the general method. Focus on solving each of the sub-problems first.

1.) Do you know how to find the horizontal centroidal axis of the beam cross section shown in the diagram?

2.) Can you determine the second moment of area of this cross section about the horizontal centroidal axis of this cross section?

Work on these two problems first, then we'll worry about tackling the beam bending.

 

[smr101]

Don't worry about the general method. Focus on solving each of the sub-problems first.

1.) Do you know how to find the horizontal centroidal axis of the beam cross section shown in the diagram?

2.) Can you determine the second moment of area of this cross section about the horizontal centroidal axis of this cross section?

Work on these two problems first, then we'll worry about tackling the beam bending.

Are you referring to question (a) and (b)?

Yes, I have done them and got the correct answer, I have also done (c).

 

[SteamKing]

Are you referring to question (a) and (b)?

Yes, I have done them and got the correct answer, I have also done (c).

If you know the bending moment at mid-span of the beam, then calculating the tensile stress is just the application of σ = My / I. You must understand which side of the beam cross section is in tension, and which side is in compression, though.

Can you show your work on this point?

 

[smr101]

If you know the bending moment at mid-span of the beam, then calculating the tensile stress is just the application of σ = My / I. You must understand which side of the beam cross section is in tension, and which side is in compression, though.

Can you show your work on this point?

y = 118.33 mm, too much working for this to type out.

Icc = 762.3 x 10^6 mm^44, using formula Icc = bh^3/12 + A1*h1^2 + A2*h2^2 + A3...

Bending moment is 3 m * 1000kN/m = 3kN load.

3 metres + 1/2 * 3 = 4.5 metres * 3kN
M = 13,500kNm

Apply σ = My / I as you say. Using yt = 118.33 to work out the stress in tensile.

All values are correct as checked with answers.

I did find that yc = 400 - 118.33 = 211.67 and putting that into the equation didn't give the correct compressive stress value, can't understand why.

 

[SteamKing]

y = 118.33 mm, too much working for this to type out.

This is the correct location of the centroid above the bottom of the cross section.

Icc = 762.3 x 10^6 mm^44, using formula Icc = bh^3/12 + A1*h1^2 + A2*h2^2 + A3...

Should be I = 762 × 10⁶ mm⁴. Be careful with the units.

Bending moment is 3 m * 1000kN/m = 3kN load.

The load shown in the diagram is 1 kN/m or 1000 N/m, not 1000 kN/m, which is equal to 10⁶ N/m. You've got to be careful with units.

3 metres + 1/2 * 3 = 4.5 metres * 3kN
M = 13,500kNm

Again, you're writing incorrect units here.

Apply σ = My / I as you say. Using yt = 118.33 to work out the stress in tensile.

If you want to work out the maximum tensile/compressive bending stress, y must be the distance from the neutral axis (or the horizontal centroid in this case) to the outer fiber of the beam. Due to how this particular beam is loaded, the tensile stresses occur above the neutral axis and compressive stresses below. You can check this by sketching the deflected shape of the beam (which curves down at the free end).

All values are correct as checked with answers.

I did find that yc = 400 - 118.33 = 211.67 and putting that into the equation didn't give the correct compressive stress value, can't understand why.

You must use correct units in the formulas. If M is measured in N-m, then y must be measured in meters and I must be measured in m⁴ in order to obtain σ in N/m².

It's not clear what the "correct answer 1.66 MPa" refers to. Is this the max. tensile stress? Max. compressive stress? Stress at mid-span?

 

[smr101]

This is the correct location of the centroid above the bottom of the cross section.

Should be I = 762 × 10⁶ mm⁴. Be careful with the units.

The load shown in the diagram is 1 kN/m or 1000 N/m, not 1000 kN/m, which is equal to 10⁶ N/m. You've got to be careful with units.

Again, you're writing incorrect units here.

If you want to work out the maximum tensile/compressive bending stress, y must be the distance from the neutral axis (or the horizontal centroid in this case) to the outer fiber of the beam. Due to how this particular beam is loaded, the tensile stresses occur above the neutral axis and compressive stresses below. You can check this by sketching the deflected shape of the beam (which curves down at the free end).You must use correct units in the formulas. If M is measured in N-m, then y must be measured in meters and I must be measured in m⁴ in order to obtain σ in N/m².

It's not clear what the "correct answer 1.66 MPa" refers to. Is this the max. tensile stress? Max. compressive stress? Stress at mid-span?

I am using the correct units and figures. The only figure that changes from working out the tensile stress and the compressive stress is the yc/yt value - correct?

yc = 400 - 118.33 = 211.67, using those figures doesn't bring the correct answer.

To clarify the max tensile stress = 4.98 MPa and max compressive stress = 2.09 MPa.

The tensile stress at mid-span is 1.66 MPa.

 

[SteamKing]

I am using the correct units and figures. The only figure that changes from working out the tensile stress and the compressive stress is the yc/yt value - correct?

Correct.

yc = 400 - 118.33 = 211.67, using those figures doesn't bring the correct answer.

You should get in the habit of indicating units.

The value of the bending moment at mid-span is not the same as the max. bending moment.

To clarify the max tensile stress = 4.98 MPa and max compressive stress = 2.09 MPa.

The tensile stress at mid-span is 1.66 MPa.

I am able to calculate all of these stresses.

 

[smr101]

Correct.You should get in the habit of indicating units.

The value of the bending moment at mid-span is not the same as the max. bending moment.I am able to calculate all of these stresses.

So my yc value for for compressive stress must be incorrect, yes?

I still have no idea how to work out the tensile stress at mid-span.

 

[SteamKing]

So my yc value for for compressive stress must be incorrect, yes?

Since the beam cross section doesn't change along the length, yc will be the same value at every location, mid-span or otherwise. Likewise, yt.

I still have no idea how to work out the tensile stress at mid-span.

σ = M y / I

We've established what y must be and you have calculated I. If σ at mid-span is incorrect, then the problem must be in the value of M you are using at that location.

Please show your calculation.