Normal and Shearing Stress - Combined Loading

erobz · Mar 25, 2023

Diagram of a vertical pipe AB fixed to a base, with a horizontal arm at the top carrying a 1500 N downward force and a 1200 N force at its end, with dimensions 45 mm, 75 mm and 20 mm marked on the figure.

Here is my combined loading:

Diagram of a solid cylinder under combined loading with a downward force F = 1.5 kN, shear V = 1.2 kN, torque T = 67.5 kN·mm, and bending moments Mx = 67.5 kN·mm and Mx = 90 kN·mm shown as purple arrows.

The book solution for normal and shear stresses respectively are:

a) ##20.4~\text{MPa}, 14.34 ~\text{MPa} ## - I find both

b) ##-21.5~\text{MPa}, \boxed{19.98~\text{MPa}}## - I find the normal stress, but I'm not getting the book answer for the shear stress.

I'm getting Torsion + Shear Flow ##\approx 25.6 \text{MPa}##

I'm just brushing up for the heck of it...No professor to ask. Does anyone get solution in the book?

Lnewqban · Mar 25, 2023

@erobz
I may be wrong, but for the cross-section shared by points a and b, my values are:

Fx = 0
Mx = 90 kN-mm

Fy = 1.5 kN
My = 108 kN-mm

Fz = 1.2 kN
Mz = 67.5 kN-mm

Would you mind to verify?

erobz · Mar 25, 2023

Lnewqban said:

@erobz
I may be wrong, but for the cross-section shared by points a and b, my values are:

Fx = 0
Mx = 90 kN-mm

Fy = 1.5 kN
My = 108 kN-mm

Fz = 1.2 kN
Mz = 67.5 kN-mm

Would you mind to verify?

Diagram of a solid blue cylinder with a free-body set of loads at its top centre: axial force F = 1.5 kN (red arrow), shear force V = 1.2 kN (red arrow), torque T = 108 kN·mm (yellow curved arrow), and bending moments Mx = 90 kN·mm and Mz = 67.5 kN·mm (purple curved arrows).

Sorry, I had some bad editing there. Yeah, those match my values.

erobz · Mar 25, 2023

Never mind. I believe I found the problem. For the shear flow in a pipe (or other closed tubular geometry depending on position I suppose):

$$\tau = \frac{QV}{I (2t)} $$

Its either that or you only use a quarter of the pipe in ##Q## computation( I was using a half - with single wall thickness). To use a single pipe wall thickness ##t##, the first moment ##Q## is computed for one quarter of the pipe.Adding that to the Torsion gets me the book answer...

erobz · Mar 25, 2023

Here is my worked solution if anyone is interested.

Mathcad worksheet listing F = 1.5 kN, V = 1.2 kN, Mz = 67.5 kN·mm, Mx = 90 kN·mm, T = 108 kN·mm, Do = 42 mm, Di = 35 mm, with I = 7.908×10^-8 m^4, J = 1.582×10^-7 m^4 and t = 3.5 mm.

Mathcad worksheet for Point (a) computing combined stresses: compressive normal stress -3.543 MPa, bending stress 23.899 MPa, total normal stress 20.356 MPa, shear flow 0 MPa, torsion shear stress 14.339 MPa, total shear stress 14.339 MPa.

Mathcad worksheet for point (b) computing normal stress under compressive and bending loads, giving sigma_b = -21.467 MPa, and shear stress from shear flow and torsion, giving tau_b = 19.978 MPa, with a hollow circular cross-section diagram.

Normal and Shearing Stress - Combined Loading

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