Example 7.2.1

Given: Consider the road sign of Example 2.2.5. The forces, moments, and torque acting at a cross-section near the base of the mast are shown and right (Cut A-A).

Req'd: The stress states at the surface of the mast, at Points A, B, C and D. Point A is on the windward side (facing the wind), Point C on the "leeward side" (back side), and Pt. B and D on the left and right sides of the mast. Draw 2-d stress elements as you look from the outside of the mast inward.

Forces, Moments and Torque acting at Cut A-A.

Road Sign

Sol'n:
Step 1: Consider the five different types of loads acting at the cross-section to act separately. Then, determine what stresses each load causes, and at what points they act.

Each load causes the mast to act as a different type of member, for example:

• Under the torque, the mast acts as a torsion member.
• Because of the shear force, the mast acts as a beam.
• The weight makes it act like an axial member.

Step 2: Consider how each of the loads affects Point A.

Loads Causing Stresses at Point A.   Coordinates:   x=0, y= -R
Load	Mast Acts as	Stress Equation for Stress at Surface.	Does Pt. A "feel" this stress?	Stress that acts at Point A
Torque, T	Shaft about z-axis		Yes – entire cross-section supports torque
Shear Force, V	Beam about x-axis		No – shear stress is zero at the "top and bottom" of a beam (Points A&C) - the mast acts as a beam bending against the shear force caused by the wind	0
Moment, Mx	Beam about x-axis		Yes (tension) – bending stress is maximum at "top and bottom" of beam
Moment, My	Beam about y-axis		No – bending stress is zero at Centroidal Axis, about which beam is bending (the y-axis for My)	0
Axial Force, W=Ws+Wm(z)	Axial Member		Yes (compression) – entire cross-section supports axial load

Point A thus feels three stresses:

1. the shear stress due to the torque;
2. normal stress (tensile) due to the bending moment about the x-axis;
3. normal stress (compressive) due to the weight.

What are the stresses at the other points?

Stress State at Pt. A

Step 3: By knowing the loads (forces, torques and moments) that occur at the cross-section, the stresses that act on the other elements can also be found:

Loads Causing Stresses:
Load	Mast Acts as	Stress Equation	Pt. A	Pt. B	Pt. C	Pt. D
Torque, T	Torsion Member about z-axis		Yes	Yes	Yes	Yes
Shear Force, V	Beam about x-axis		No	Yes	No	Yes; opposite stress of torsion
Moment, Mx	Beam about x-axis		Yes (+)	No	Yes (-)	No
Moment, My	Beam about y-axis		No	Yes (-)	No	Yes (+)
Axial Force, W=Ws+Wm(z)	Axial Member		Yes (-)	Yes (-)	Yes (-)	Yes (-)

(+) means normal stress is TENSILE.
(-) means normal stress is COMPRESSIVE.

Notes on Shear:

• At Point B, the Shear Stresses ADD as they are in the same direction (+y-direction);
• At Point D, the Shear Stresses SUBTRACT - they are in opposite directions. At Point B, the shear stress due to Shear Force, V, acts in the +y-direction; the shear stress due to torque acts in the -y-direction.
• Below, C_s = 4/3 [(R_o² + R_oR_i + R_i²) / (R_o² + R_i²)]
• I_x and I_y are the Moments of Inertia about the x- and y- axes at the centroid of the cross-section, respectively.