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Unformatted text preview: Module 7: Equivalence & Distributed Loads
September 13, 2010 Module Content:
1. Equivalence calculaLons allow us to simplify a loading scenario on a structure.
2. Equivalence condiLons are ensured only if we saLsfy both force equivalence AND moment equivalence. Module Reading, Problems, and Demo:
Reading: Chapter 4
Problems: Fund. Prob. 4.26, Fund. Prob. 4.30 (equivalent systems)
Demo: none
Homework PlaGorm: hHp://www.masteringengineering.com
Course Blog: hHp://pages.shanL.virginia.edu/staLcs2010 MAE 2300 Sta=cs © E. J. Berger, 2010 7
1 Theory: Couple Moments
• couple moments are created when two equal and opposite forces exist oﬀset from each other: • from an equilibrium standpoint, the result is a pure moment (the forces cancel out because they are equal and opposite) • couple moments are free vectors, which means that the moment they create is the same regardless of the point we choose around which to take the moment MAE 2300 Sta=cs © E. J. Berger, 2010 7
2 Theory: Couple Moments
• couple moments are created when two equal and opposite forces exist oﬀset from each other: • from an equilibrium standpoint, the result is a pure moment (the forces cancel out because they are equal and opposite) • couple moments are free vectors, which means that the moment they create is the same regardless of the point we choose around which to take the moment MAE 2300 Sta=cs © E. J. Berger, 2010 7
2 Theory: Couple Moments
• couple moments are created when two equal and opposite forces exist oﬀset from each other: • from an equilibrium standpoint, the result is a pure moment (the forces cancel out because they are equal and opposite) • couple moments are free vectors, which means that the moment they create is the same regardless of the point we choose around which to take the moment MAE 2300 Sta=cs © E. J. Berger, 2010 7
2 Theory: Couple Moments
• couple moments are created when two equal and opposite forces exist oﬀset from each other: M = Fd • from an equilibrium standpoint, the result is a pure moment (the forces cancel out because they are equal and opposite) • couple moments are free vectors, which means that the moment they create is the same regardless of the point we choose around which to take the moment MAE 2300 Sta=cs © E. J. Berger, 2010 7
2 Theory: Couple Moments
• couple moments are created when two equal and opposite forces exist oﬀset from each other: M = Fd
(d is the distance between the forces) • from an equilibrium standpoint, the result is a pure moment (the forces cancel out because they are equal and opposite) • couple moments are free vectors, which means that the moment they create is the same regardless of the point we choose around which to take the moment MAE 2300 Sta=cs © E. J. Berger, 2010 7
2 Deriva=on: Huh? MAE 2300 Sta=cs © E. J. Berger, 2010 7
3 Concept: Equivalence
• we can deﬁne this: if two loading scenarios produce idenLcal eﬀects on the structure, then they are equivalent • “idenLcal eﬀects” simply means that from the standpoint of equilibrium calculaLons, the eﬀects are the same MAE 2300 Sta=cs © E. J. Berger, 2010 7
4 Concept: Equivalence
• we can deﬁne this: if two loading scenarios produce idenLcal eﬀects on the structure, then they are equivalent • “idenLcal eﬀects” simply means that from the standpoint of equilibrium calculaLons, the eﬀects are the same MAE 2300 Sta=cs © E. J. Berger, 2010 7
4 Concept: Equivalence
• we can deﬁne this: if two loading scenarios produce idenLcal eﬀects on the structure, then they are equivalent • “idenLcal eﬀects” simply means that from the standpoint of equilibrium calculaLons, the eﬀects are the same MAE 2300 Sta=cs © E. J. Berger, 2010 7
4 Theory: Equivalence Calcula=ons
• remember that we wish to calculate an equivalent set of loads whose eﬀect in the equilibrium calculaLon (i.e., it has the same “external eﬀects” on the body) • original two types:
•
• • force equivalence
moment equivalence
so we need to make sure that the original and the equivalent systems appear the same in the equilibrium equaLons equivalent loading at O
MAE 2300 Sta=cs © E. J. Berger, 2010 7
5 Theory: Equivalence Calcula=ons
• remember that we wish to calculate an equivalent set of loads whose eﬀect in the equilibrium calculaLon (i.e., it has the same “external eﬀects” on the body) • original two types:
•
• • force equivalence
moment equivalence
so we need to make sure that the original and the equivalent systems appear the same in the equilibrium equaLons equivalent loading at O
MAE 2300 Sta=cs © E. J. Berger, 2010 7
5 Theory: Equivalence Calcula=ons
• remember that we wish to calculate an equivalent set of loads whose eﬀect in the equilibrium calculaLon (i.e., it has the same “external eﬀects” on the body) • original two types:
•
• • force equivalence
moment equivalence
so we need to make sure that the original and the equivalent systems appear the same in the equilibrium equaLons equivalent loading at O
MAE 2300 Sta=cs © E. J. Berger, 2010 7
5 Concept: Distributed Load
• distributed loads act over a line or area, and oXen represent mechanics of contact between two objects • we have tools available to analyze distributed loads, and oXen we like to ﬁnd a point force equivalent to a given distribute load • in this case, MAE 2300 Sta=cs © E. J. Berger, 2010 7
6 Theory: Generalized Distributed Load
• how do we use our general rules about equivalence to simplify this distributed load? MAE 2300 Sta=cs © E. J. Berger, 2010 7
7 Theory: Generalized Distributed Load
• how do we use our general rules about equivalence to simplify this distributed load? MAE 2300 Sta=cs © E. J. Berger, 2010 7
7 Theory: Generalized Distributed Load
• how do we use our general rules about equivalence to simplify this distributed load? MAE 2300 Sta=cs © E. J. Berger, 2010 7
7 Theory: Magnitude of the Resultant
• this requires FORCE equivalence MAE 2300 Sta=cs © E. J. Berger, 2010 7
8 Theory: Loca=on of the Resultant
• this required MOMENT equivalence MAE 2300 Sta=cs © E. J. Berger, 2010 7
9 ...
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