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

2. Bina Nusantara Indeterminate Structure

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

5. Bina Nusantara What is Indeterminate ? Statics (equilibrium analysis) alone cannot solve the problem n R = # of reactions (or unknowns) n E = # of equilibrium equations

6. Bina Nusantara What is Indeterminate ? If n R > n E : statically indeterminate - too many unknowns, must invoke a constraint such as a deformation relation.

7. Bina Nusantara What is Indeterminate ? If n R = n E : statically determinate - forces in each member only depend on equilibrium.

8. Bina Nusantara Free body diagram Statically Indeterminate Examples

9. Bina Nusantara Statically Indeterminate Examples Free body diagram

10. Bina Nusantara 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. Bina Nusantara METHOD OF SUPERPOSITION Necessary conditions to be satisfied: 1. The load w(x) is linearly related to the deflection v(x), 2. 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. Bina Nusantara 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. Bina Nusantara 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. Bina Nusantara 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. Bina Nusantara 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. Bina Nusantara 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. Bina Nusantara 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. Bina Nusantara Buckling Buckling is a mode of failure that does not depend on stress or strength, but rather on structural stiffness Examples:

19. Bina Nusantara More buckling examples…

20. Bina Nusantara 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. Bina Nusantara Euler Column Formula

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

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

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

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

26. Bina Nusantara 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. Bina Nusantara Buckling Stress vs. Slenderness Ratio

28. Bina Nusantara Johnson Equation for Buckling

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

30. Bina Nusantara Solving buckling problems For L e /  < tangent point (“intermediate”), use Johnson’s Equation

31. Bina Nusantara Solving buckling problems For L e  > tangent point (“long”), use Euler’s equation:

32. Bina Nusantara Solving buckling problems For L e  < 10 (“short”) S cr = S y

33. Bina Nusantara 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. Bina Nusantara Special Buckling Cases Buckling in very long Pipe Note P crit is inversely related to length squared A tiny load will cause buckling L = 10 feet vs. L = 1000 feet: P crit 1000/P crit 10 = 0.0001 Buckling under hydrostatic Pressure

35. Bina Nusantara Pipe in Horizontal Pipe Buckling Diagram

36. Bina Nusantara Far End vs. Input Load with Buckling

37. Bina Nusantara Buckling Length: Fiberglass vs. Steel