Plastic Analysis in framed structures

1. Dr.-Ing. GirmaZerayohannes Dr.-Ing. AdilZekaria Plastic Analysis in framed structures Dr.-Ing. Girma Z. and Adil Z.

2. Chapter 5- Plastic Hinge Theory in Framed Structures • 5.1 Introduction • All codes for concrete, steel and steel-composite structures (EBCS-2, EBCS-3, EBCS-4) allow the plastic method of analysis for framed structures • The requirement is that, sufficient rotation capacity is available at the plastic hinges Dr.-Ing. Girma Z. and Adil Z.

3. Chapter 5- Plastic Hinge Theory in Framed Structures • In this chapter we will introduce the plastic method of analysis for line elements. It is called the “plastic hinge theory” • The method is known as the “yield line theory” for 2D elements (e.g. slabs) • Both are based on the upper bound theorem of the theory of plasticity • Recall that the strip method is also a plastic method of analysis based on the lower bound theorem Dr.-Ing. Girma Z. and Adil Z.

4. Chapter 5- Plastic Hinge Theory in Framed Structures • Therefore the capacity of the line elements are greater or at best equal to the actual capacity of the member.  a concern for the designer, Dr.-Ing. Girma Z. and Adil Z.

5. Chapter 5- Plastic Hinge Theory in Framed Structures • 5.2 Design Plastic Moment Resistances of Cross-Sections • 5.2.1 RC Sections • Such plastic section capacities are essential in the plastic hinge theory, because they exist at plastic hinges Dr.-Ing. Girma Z. and Adil Z.

6. Chapter 5- Plastic Hinge Theory in Framed Structures • Determine using the Design Aid (EBCS-2: Part 2), the plastic moment resistance (the design moment resistance) of the RC section shown in following slide, if the concrete class and steel grade are C-25 and S-400 respectively. Dr.-Ing. Girma Z. and Adil Z.

7. Chapter 5- Plastic Hinge Theory in Framed Structures Fig. Reinforced Concrete Section Dr.-Ing. Girma Z. and Adil Z.

8. Chapter 5- Plastic Hinge Theory in Framed Structures • Steps: • Assume that the reinforcement has yielded • Determine Cc c • Determine MR,ds • Check assumption of steel yielding Dr.-Ing. Girma Z. and Adil Z.

9. Chapter 5- Plastic Hinge Theory in Framed Structures • Assume Reinforcement has yielded •  Ts = Asfyd = 2 314 (400/1.15) = 218435 N •  •  from General design chart No.1  Sd.s= 0.195  • Check the assumption that the reinforcement has yielded Dr.-Ing. Girma Z. and Adil Z.

10. Chapter 5- Plastic Hinge Theory in Framed Structures • yd = fyd/Es= 347.8/200000 = 1.739(0/00) • s = 9.4(0/00)  1.739(0/00)  reinforcement has yielded • Exercise for section with compression reinforcement Dr.-Ing. Girma Z. and Adil Z.

11. Chapter 5- Plastic Hinge Theory in Framed Structures Dr.-Ing. Girma Z. and Adil Z.

12. Chapter 5- Plastic Hinge Theory in Framed Structures • 5.2.2 Structural Steel Sections •  Consider the solid rectangular section shown in the next slide • The plastic section capacity, Mpl is: • Mpl = y(bd2/4); (bd2/4) is called the plastic section modulus and designated as Wpl • The elastic section modulus Wel = bd2/6 Dr.-Ing. Girma Z. and Adil Z.

13. Chapter 5- Plastic Hinge Theory in Framed Structures Fig. Rectangular section :– Stress Distribution ranging from elastic, partially plastic, to fully plastic Dr.-Ing. Girma Z. and Adil Z.

14. Chapter 5- Plastic Hinge Theory in Framed Structures Dr.-Ing. Girma Z. and Adil Z.

15. Chapter 5- Plastic Hinge Theory in Framed Structures y y Fig. Elasto-plastic behavior Dr.-Ing. Girma Z. and Adil Z.

16. Chapter 5- Plastic Hinge Theory in Framed Structures • From the stress distribution in the previous figure • Total bending moment M about the neutral axis Dr.-Ing. Girma Z. and Adil Z.

17. Chapter 5- Plastic Hinge Theory in Framed Structures • The ratio between Mpl and Mel which is equal to the ratio between Wpl and Wel is called shape factor pl. • For the solid rectangular section, • It is different for different sections • For I-sections pl  1.14 Dr.-Ing. Girma Z. and Adil Z.

18. Chapter 5- Plastic Hinge Theory in Framed Structures • Shape factors for common cross sections (check as a home work) Dr.-Ing. Girma Z. and Adil Z.

19. Chapter 5- Plastic Hinge Theory in Framed Structures • For simply and doubly symmetric sections, the plastic neutral axis (PNA) coincides with the horizontal axis that divides the section in to 2 equal areas Dr.-Ing. Girma Z. and Adil Z.

20. Chapter 5- Plastic Hinge Theory in Framed Structures • 5.3 Plastic Hinge Theory • It is based on the hypothesis of a localized (concentrated) plastic hinge. Dr.-Ing. Girma Z. and Adil Z.

21. Chapter 5- Plastic Hinge Theory in Framed Structures • The load carrying capacity of a structure is reached when sufficient numbers of plastic hinges have formed to turn the structure into a mechanism. • The load under which the mechanism forms is called the ultimate load. • As an example, let us consider a typical interior span of a continuous beam (see next slide) Dr.-Ing. Girma Z. and Adil Z.

22. Chapter 5- Plastic Hinge Theory in Framed Structures Dr.-Ing. Girma Z. and Adil Z.

23. Chapter 5- Plastic Hinge Theory in Framed Structures • The ultimate state is reached when 3 plastic hinges form (2 over the supports plus 1 in the span) • The ultimate load Ppl corresponding to the ultimate state is: Dr.-Ing. Girma Z. and Adil Z.

24. Chapter 5- Plastic Hinge Theory in Framed Structures • Compare with the elastic strength of the continuous beam, Pel • Here section capacities are determined on the basis of linear elastic stress distribution where only the extreme fibers have plasticized • From structural analysis, • From Dr.-Ing. Girma Z. and Adil Z.

25. Chapter 5- Plastic Hinge Theory in Framed Structures • So that where pl = (16/12) = 1.3333 • Summary- in continuous beams or frames (statically indeterminate) there exist: • plastic cross-section reserve pl • plastic system reserve pl Dr.-Ing. Girma Z. and Adil Z.

26. Chapter 5- Plastic Hinge Theory in Framed Structures • In the above example with an I-section (pl = 1.14)  • plpl = 1.52  52% increase Dr.-Ing. Girma Z. and Adil Z.

27. Chapter 5- Plastic Hinge Theory in Framed Structures • 5.4 Method of Analysis • As in the linearly elastic method, • either the equilibrium method or • the principle of virtual work is applicable for the plastic method of analysis. • Examples for different types of framed structures follow Dr.-Ing. Girma Z. and Adil Z.

28. Chapter 5- Plastic Hinge Theory in Framed Structures • 5.4.1 Single span and continuous beams • (a) single span-fixed end beam • System and loading see next slide • Goal is to determine Fpl • First we solve using the equilibrium method and then repeat with the virtual method Dr.-Ing. Girma Z. and Adil Z.

29. Chapter 5- Plastic Hinge Theory in Framed Structures • (i) Equilibrium method • From FBD of element 1 • From FBD of element 2 •  Dr.-Ing. Girma Z. and Adil Z.

30. Chapter 5- Plastic Hinge Theory in Framed Structures Dr.-Ing. Girma Z. and Adil Z.

31. Chapter 5- Plastic Hinge Theory in Framed Structures • (ii) Principle of virtual work • External virtual work = internal virtual work •  Dr.-Ing. Girma Z. and Adil Z.

32. Chapter 5- Plastic Hinge Theory in Framed Structures • (b) Propped cantilevers under UDL • System and loading see next slide • NB- position of the plastic hinge in the span is not known. Must be determined from the condition of zero shear at location of Mmax • (i) Equilibrium method Dr.-Ing. Girma Z. and Adil Z.

33. Chapter 5- Plastic Hinge Theory in Framed Structures Dr.-Ing. Girma Z. and Adil Z.

34. Chapter 5- Plastic Hinge Theory in Framed Structures • Substituting xo in the expression for M(x) and equating the maximum moment to Mpl (MplM) results, after simplification in a quadratic equation in P. •  •  • (ii) Principle of virtual work • Knowledge of the location of the plastic hinge in the span is a requirement for VWM Substituting for x Dr.-Ing. Girma Z. and Adil Z.

35. Chapter 5- Plastic Hinge Theory in Framed Structures • Of course, the correct location of the plastic hinge can be determined by trial and error, i.e., keep trying new locations until the minimum Ppl is found • For the present example, check the result using the PVW •  Dr.-Ing. Girma Z. and Adil Z.

36. Chapter 5- Plastic Hinge Theory in Framed Structures • (c) Continuous beams • System and loading  see next slide • The ultimate capacity of a continuous beam is reached when a mechanism forms in one of the spans. The ultimate load is determined as the minimum of the different mechanisms in all the spans Dr.-Ing. Girma Z. and Adil Z.

37. Chapter 5- Plastic Hinge Theory in Framed Structures Dr.-Ing. Girma Z. and Adil Z.

38. Chapter 5- Plastic Hinge Theory in Framed Structures • (i) Equilibrium method • Locations of plastic hinges are simple to determine. They are at 1, 2, 3, 4, and 5. • The two mechanisms I and II are to be investigated. It is not immediately obvious which one governs • Mechanism I Dr.-Ing. Girma Z. and Adil Z.

39. Chapter 5- Plastic Hinge Theory in Framed Structures • Mechanism II •  Mechanism II governs and Fpl=6M/l Dr.-Ing. Girma Z. and Adil Z.

40. Chapter 5- Plastic Hinge Theory in Framed Structures • (ii) Principle of virtual work • Mechanism I • Mechanism II • Mechanism II governs with Fpl=6M/l • PVW is much simpler in this case • Figure at the bottom shows the moment diagram at the ultimate capacity. Observe that the moment at all sections is less than or equal to the respective plastic section capacities Dr.-Ing. Girma Z. and Adil Z.

41. Chapter 5- Plastic Hinge Theory in Framed Structures • 5.5.1 Frames • One of the important application areas of the method of plastic hinge theory, which has been proved by experiments are frames • The procedure is one of trial and error as in continuous beams using the basic or combined modes Dr.-Ing. Girma Z. and Adil Z.

42. Chapter 5- Plastic Hinge Theory in Framed Structures • The combination procedure, based on selective combination of the elementary mechanisms leads to result more quickly • Three elementary(basic) mechanisms (basic modes of failure) are to be distinguished • They are the beam mechanism, frame mechanism, and joint mechanism (see next slide) Dr.-Ing. Girma Z. and Adil Z.

43. Chapter 5- Plastic Hinge Theory in Framed Structures Dr.-Ing. Girma Z. and Adil Z.

44. Chapter 5- Plastic Hinge Theory in Framed Structures • The beam and frame mechanisms represent independent failure mechanisms. • Joint mechanism can occur only in combination with another elementary failure mechanism. It does not represent a failure mechanism alone • Number of elementary (basic) mechanisms k is determined from: Dr.-Ing. Girma Z. and Adil Z.

45. Chapter 5- Plastic Hinge Theory in Framed Structures • k = m-n; where m=possible no of plastic hinges depending on system and loading, and n=degree of statical indeterminacy • The no of possible combination including the basic modes (elementary mechanisms) is given by: • q=2k-1 • The combination method will be explained by means of the portal frame Dr.-Ing. Girma Z. and Adil Z.

46. Chapter 5- Plastic Hinge Theory in Framed Structures Dr.-Ing. Girma Z. and Adil Z.

47. Chapter 5- Plastic Hinge Theory in Framed Structures • k = m-n = 5-3 = 2 • The no of possible combination q, which includes the basic mode I and II is: • q = 2k – 1 = 22 – 1 = 3 • See the three mechanisms in the next slide with the plastic moments. When a plastic hinge forms at a joint, it must be on the columns and the hinge must be shown on the column side of the joint Dr.-Ing. Girma Z. and Adil Z.

48. Chapter 5- Plastic Hinge Theory in Framed Structures Dr.-Ing. Girma Z. and Adil Z.

49. Chapter 5- Plastic Hinge Theory in Framed Structures • All member rotation angles are equal in this example. In more complicated structures, the relationships b/n the various rotations must be determined. • The virtual work equations are: • Mechanism I: • Mechanism II: F(h)=(M+M+M+M) Fh=4MF=4M/h Dr.-Ing. Girma Z. and Adil Z.

50. Chapter 5- Plastic Hinge Theory in Framed Structures • Mechanism III: 3F(l/2)+F(h) =(M+22M+M+M+M) (3/2)Fl+Fh=8M • Substituting the values for l and h •  Mechanism I: F=0.666M •  Mechanism II: F=1.000M •  Mechanism III: F=0.615M •  Therefore Mechanism III governs withFpl=0.615Mpl Dr.-Ing. Girma Z. and Adil Z.