7b-ex. Combined Loading Examples • Ex. 7b.1 Back | Index | Next

Example 7b.1

Given: Consider the road sign of Example 2b.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: Determine 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),
• C is on the "leeward side" (back side),
• B is on the left side, and
• D is on the right side 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.

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

Each load causes the mast to act as a different type of member, e.g.:

• 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.
 Load Mast Acts as Causes Stress at Surface Torque, T Shaft about z-axis Shear Force, V (y-direction) Beam about x-axis Moment, Mx Beam about x-axis Moment, My Beam about y-axis Axial Force, W=Ws+Wm(z) Axial Member along z-axis

Ix is the Moment of Inertia about the x-axis.
Iy is the Moment of Inertia about the y-axis.

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
feels three stresses:

1. 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 (–)

Notes on Normal Stresses:

• (+) means normal stress is tensile.
• (–) means normal stress is compressive.
• Ix and Iy are the moments of inertia about the x- and y-axes at the centroid of the cross-section, respectively. Due to symmetry they are equal.

Notes on Shear Stresses:

• 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.
• Below, Cs = (4/3)[(Ro2 + RoRi + Ri2) / (Ro2 + Ri2)]

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Updated: 05/21/09 DJD