7-Statically Indeterminate Beams and Shafts PDF

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Lecture notes 07 - statically indeterminate beams and shafts...

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Statically Indeterminate Beams and Shafts

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A member of any type (Beam or Shaft) is classified as statically indeterminate if the number of unknown reactions exceeds the available number of equilibrium equations.

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The additional support reactions on the beam or shaft that are not needed to keep it in stable equilibrium are called redundants.

The number of these redundants is referred to as the degree of indeterminacy.

For example, consider the beam shown in Figure

There are four unknown support reactions (RAx, RAy, R by, and MA), and since three equilibrium equations are available for solution, the beam is classified as being indeterminate to the first degree. Either R Ay, RBy, or MA can be can be considered as the redundant force, since if anyone of these reactions is removed, the beam remains stable and in equilibrium RAx cannot be classified as the redundant force, since, if it were removed force equilibrium condition in x-direction is not satisfied.

Consider the continuous beam where there are four unknown reactions RA, RC, RE and RF and two equilibrium equations and .moments of all the forces about z_ axis through any point in the beam = 0 Equilibrium in x-direction is satisfied as there are no external forces in x- direction. Therefore the is indeterminate to the second degree, since there are four unknown reactions and only two available equilibrium equations,

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To determine the reactions on a beam (or shaft) that is statically indeterminate, it is first necessary to specify the redundant reactions. The beam can be treated as determinant beam by removing the two unknown reactions such that the beam is in stable static equilibrium and apply the removed unknown reactions as unknown redundant external forces. Now the beam is simply supported at A and F with applied external forces P1, P2, P3 and unknown redundant forces RC and RE. The boundary conditions v A =0 and v F = 0 and displacement compatibility conditions vC = vE = 0

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We can determine the redundant forces from conditions of geometry known compatibility conditions.

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Once the redundant forces are determined, the remaining reactions are determined the equations of equilibrium considering the free body diagram.

Consider another example Fixed- fixed beam

There are four reactions and two equilibrium equations. This problem can be solved in two ways.

(a) Treat MA and MD as redundant moments. Then problem can be treated as simply supported beam with applied forces P1 and P2 and unknown redundant moments MA and MD. Compatibility conditions are slopes at points A and D are zero.

(b) Treat RD and MD as redundant force and moments. Then the problem can be treated as cantilever beam fixed at end A and free at end B with applied forces P1 and P2 and redundant force RD and moment MD. . Compatibility conditions are displacement and slope at point D are zero...