Indeterminate Structure Session 23-26

1. Indeterminate StructureSession 23-26 Subject : S1014 / MECHANICS of MATERIALS Year : 2008

2. Indeterminate Structure

3. What is Indeterminate ? a structure is statically indeterminate when the static equilibrium equations are not sufficient for determining the internal forces and reactions on that structure…..

4. What is Indeterminate ? “Statically Indeterminate” means the # of unknowns exceeds the number of available equations of equilibrium.

5. What is Indeterminate ? Statics (equilibrium analysis)alone cannot solve the problem nR = # of reactions (or unknowns) nE = # of equilibrium equations

6. What is Indeterminate ? • If nR > nE: statically indeterminate - too many unknowns, must invoke a constraint such as a deformation relation.

7. What is Indeterminate ? • If nR = nE: statically determinate - forces in each member only depend on equilibrium.

8. Statically Indeterminate Examples Free body diagram

9. Statically Indeterminate Examples Free body diagram

10. STATISTICALLY INDETERMINATE BEAMS AND SHAFTS (CONT.) • Strategy: • The additional support reactions on the beam or shaft that are not needed to keep it in stable equilibrium are called redundants. It is first necessary to specify those redundant from conditions of geometry known as compatibility conditions. • Once determined, the redundants are then applied to the beam, and the remaining reactions are determined from the equations of equilibrium.

11. METHOD OF SUPERPOSITION • Necessary conditions to be satisfied: • The load w(x) is linearly related to the deflection v(x), • The load is assumed not to change significantly the original geometry of the beam of shaft. • Then, it is possible to find the slope and displacement at a point on a beam subjected to several different loadings by algebraically adding the effects of its various component parts.

12. STATISTICALLY INDETERMINATE BEAMS AND SHAFTS • Definition: A member of any type is classified statically indeterminate if the number of unknown reactions exceeds the available number of equilibrium equations. • e.g. a continuous beam having 4 supports

13. USE OF THE METHOD OF SUPERPOSITION • Elastic Curve • Specify the unknown redundant forces or moments that must be removed from the beam in order to make it statistically determinate and stable. • Using the principle of superposition, draw the statistically indeterminate beam and show it equal to a sequence of corresponding statistically determinate beams.

14. USE OF THE METHOD OF SUPERPOSITION • Elastic Curve • The first of these beams, the primary beam, supports the same external loads as the statistically indeterminate beam, and each of the other beams “added” to the primary beam shows the beam loaded with a separate redundant force or moment. • Sketch the deflection curve for each beam and indicate the symbolically the displacement or slope at the point of each redundant force or moment.

15. USE OF THE METHOD OF SUPERPOSITION • Compatibility Equations • Write a compatibility equation for the displacement or slope at each point where there is a redundant force or moment. • Determine all the displacements or slopes using an appropriate method

16. USE OF THE METHOD OF SUPERPOSITION • Compatibility Equations • Substitute the results into the compatibility equations and solve for the unknown redundants. • If the numerical value for a redundant is positive, it has the same sense of direction as originally assumed. Similarly, a negative numerical value indicates the redundant acts opposite to its assumed sense of direction.

17. USE OF THE METHOD OF SUPERPOSITION Equilibrium Equations Once the redundant forces and/or moments have been determined, the remaining unknown reactions can be found from the equations of equilibrium applied to the loadings shown on the beam’s free body diagram.

18. Buckling • Buckling is a mode of failure that does not depend on stress or strength, but rather on structural stiffness • Examples:

19. More buckling examples…

20. Buckling • The most common problem involving buckling is the design of columns • Compression members • The analysis of an element in buckling involves establishing a differential equation(s) for beam deformation and finding the solution to the ODE, then determining which solutions are stable • Euler solved this problem for columns

21. Euler Column Formula

22. Euler Column Formula • Where C is as follows: C = ¼ ;Le=2L Fixed-free

23. Euler Column Formula • Where C is as follows: C = 2; Le=0.7071L Fixed-pinned

24. Euler Column Formula • Where C is as follows: C = 1: Le=L Rounded-rounded Pinned-pinned

25. Euler Column Formula • Where C is as follows: C = 4; Le=L/2 Fixed-fixed

26. Buckling • Geometry is crucial to correct analysis • Euler – “long” columns • Johnson – “intermediate” length columns • Determine difference by slenderness ratio • The point is that a designer must be alert to the possibility of buckling • A structure must not only be strong enough, but must also be sufficiently rigid

27. Buckling Stress vs. Slenderness Ratio

28. Johnson Equation for Buckling

29. Solving buckling problems Find Euler-Johnson tangent point with

30. Solving buckling problems For Le/r < tangent point (“intermediate”), use Johnson’s Equation

31. Solving buckling problems For Le/r > tangent point (“long”), use Euler’s equation:

32. Solving buckling problems For Le/r < 10 (“short”) Scr =Sy

33. Solving buckling problems If length is unknown, predict whether it is “long” or “intermediate”, use the appropriate equation, then check using the Euler-Johnson tangent point once you have a numerical solution for the critical strength

34. Special Buckling Cases • Buckling in very long Pipe • Note Pcrit is inversely related to length squared • A tiny load will cause buckling • L = 10 feet vs. L = 1000 feet: • Pcrit1000/Pcrit10 = 0.0001 • Buckling under hydrostatic Pressure

35. Pipe in Horizontal Pipe Buckling Diagram

36. Far End vs. Input Load with Buckling

37. Buckling Length: Fiberglass vs. Steel