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7b-ex. Combined Loading Examples• Ex. 7b.1

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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 Torqueacting at Cut A-A.

Road Sign

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

ex. Examples http://strengthandstiffness.com/7_combined/page_7b-ex.ht

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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 Actsas

Stress Equation for Stress atSurface.

Does Pt. A "feel" thisstress?

Stress that actsat Point A

Torque, TShaft 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 atCentroidal 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:

shear stress due to the torque;1.

normal stress (tensile) due to the bending moment

about the x-axis;

2.

normal stress (compressive) due to the weight.3.

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, TTorsion Member

about z-axisYes Yes Yes Yes

Shear Force, VBeam about

x-axisNo Yes No

Yes;

opposite

stress of

torsion

Moment, MxBeam about

x-axisYes (+) No Yes (–) No


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Moment, MyBeam about

y-axisNo 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) / (Ro

2+ Ri

2)]

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Updated: 05/21/09 DJD


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