1. CHAPTER FOUR STATICALLY INDETERMINATE STRUCTURES 4.1 Introduction In this chapter we will analyze beams in which the number of reactions exceeds the number of independent equations of equilibrium. Since the reactions of such beams cannot be determined by statics alone, the beams are said to be statically indeterminate. When a beam is statically determinate, we can obtain all reactions, shear forces, and bending moments from free-body diagrams and equations of equilibrium. Then, knowing the shear forces and bending moments, we can obtain the stresses and deflections.

2. However, when a beam is statically indeterminate, the equilibrium equations are not sufficient and additional equations are needed. Therefore, we also discuss The most fundamental method for analyzing a statically indeterminate beam are: To solve the differential equations of the deflection curve, The method of superposition and Beams with temperature Changes Throughout this chapter, we assume that the beams are made of linearly elastic materials.

3. 4.2 Types Of Statically Indeterminate Beams

5. B. a fixed end beam, (clamped beam and built-in beam.) Fig 4.2 (a) This beam has fixed supports at both ends, six unknown reactions (two forces and a moment at each support). only three equations of equilibrium, Hence, the beam is statically indeterminate to the third degree. If we select the three reactions at end B of the beam as the redundant, and if we remove the corresponding restraints, we are left with a cantilever beam as the released structure (Fig. 4-2b). Redundunts for fixed end beam If we release the two fixed-end moments and one horizontal reaction, the released structure is a simple beam (Fig. 4-2c).

6. Only four nonzero reactions (one force and one moment at each support). Two number of equilibrium equations Therefore the beam is statically Indeterminate to the second degree. If the two reactions at end B are selected as the redundants, the released structure is a cantilever beam; If the two moment reactions are selected, the released structure is a simple beam. C. Again considering the special case of vertical loads only (Fig. 4-3)

7. A.Analysis By The Differential Equations Of The Deflection Curve, B.The Method Of Superposition C.Beams With Temperature Changes 4.3.The most fundamental method for analyzing a statically indeterminate beam

8. Statically indeterminate beams may be analyzed by solving any one of the three differential equations of the deflection curve: (1)The second-order equation in terms of the bending moment (2) The third-order equation in terms of the shear force (3) The fourth-order equation in terms of the intensity of distributed load The procedure is essentially the same as that for a statically determinate beam and consists of writing the differential equation, integrating to obtain its general solution, and then applying boundary and other conditions to evaluate the unknown quantities. A. Analysis By The Differential Equations Of The Deflection Curve

9. Example 4-1 A propped cantilever beam AB of length L supports a uniform load of intensity q (Fig. 4-4). Analyze this beam by solving the second-order differential equation of the deflection curve (the bending-moment equation). Determine the reactions, shear forces, bending moments, slopes, and deflections of the beam.

10. Bending moment. The bending moment M at distance x from the fixed support can be expressed in terms of the reactions as follows: This equation can be obtained by the customary technique of constructing a free-body diagram of part of the beam and solving an equation of equilibrium. Substituting into Eq. (c) from Eqs. (a) and (b), we obtain the bending moment in terms of the load and the redundant reaction:

11. Differential equation. The second-order differential equation of the deflection curve (Eq. 9- 12a) now becomes After two successive integrations, we obtain the following equations for the slopes and deflections of the beam:

12. Boundary conditions. Three boundary conditions pertaining to the deflections and slopes of the beam are apparent from an inspection of Fig. 4-6. These conditions are as follows: (1) the deflection at the fixed support is zero, (2) the slope at the fixed support is zero, and (3) the deflection at the simple support is zero. Thus, Thus, the redundant reaction RB is now known. Eq.(4-1)

13. Reactions. With the value of the redundant established, we can find the remaining reactions from Eqs. (a) and (b). The results are Knowing these reactions, we can find the shear forces and bending moments in the beam. Shear forces and bending moments. These quantities can be obtained by the usual techniques involving free-body diagrams and equations of equilibrium. The results are Eq.(4-2a,b) Eq.(4-4) Eq.(4-3)

14. Shear-force and bending-moment diagrams for the beam can be drawn with the aid of these equations (see Fig. 4-7). From the diagrams, we see that the maximum shear force occurs at the fixed support and is equal to Also, the maximum positive and negative bending moments are Finally, we note that the bending moment is equal to zero at distance x = L /4 from the fixed support. Eq.(4-5) Eq.(4-6a,b) Fig. 4-7).

15. The deflected shape of the beam as obtained from Eq. (4-8) is shown in Fig. 4-8. To determine the maximum deflection of the beam, we set the slope (Eq. 4-7) equal to zero and solve for the distance x1 to the point where this deflection occurs: Eq.(4-7) Eq.(4-8) Fig. 4-8

16. Substituting this value of x into the equation for the deflection (Eq. 4-10) and also changing the sign, we get the maximum deflection: Eq.(4-9 ) Eq.(4-10)

17. Eq.(4-11) Eq.(4-12)

18. Importance in the analysis of statically indeterminate bars, trusses, beams, frames, and many other kinds of structures. Steps of the analysis: Noting the degree of static indeterminacy and selecting the redundant reactions. write equations of equilibrium that relate the other unknown reactions to the redundants and the loads. We Assume that both the original loads and the redundants act upon the released structure. Then we find the deflections in the released structure by superposing the separate deflections due to the loads and the redundants. The sum of these deflections must match the deflections in the original beam. equations of compatibility B. Method Of Superposition

19. Since the released structure is statically determinate. The relationships between the loads and the deflections of the released structure are called force-displacement relations. When these relations are substituted into the equations of compatibility, we obtain equations in which the redundants are the unknown quantities. Therefore, we can solve those equations for the redundant reactions. Then, with the redundants known, we can determine all other reactions from the equations of equilibrium. Furthermore, we can also determine the shear forces and bending moments from equilibrium.

20. Analyze the beam given on Example 4-1 by superposition method. Determine the reactions, shear forces, bending moments, slopes, and deflections of the beam. Example 4-2

21. The next step is to remove the restraint corresponding to the redundant (in this case, we remove the support at end B). The released structure that remains is a cantilever beam (b).

24. Substituting these force-displacement relations into the equation of compatibility yields Note that this equation gives the redundant in terms of the loads acting on the original beam. The remaining reactions can be found from the equilibrium equations

25. Determination of the deflections and slopes of the original beam by means of the principle of superposition. Substituting for R B and then adding the deflections v 1 and v 2, we obtain the following equation for the deflection curve of the original statically indeterminate beam

26. Analysis with M A as Redundant selecting the moment reaction M A as the redundant. The released structure is a simple beam. The equations of equilibrium for the reactions R A and R B in the original beam are

27. The equation of compatibility expresses the fact that the angle of rotation A at the fixed end of the original beam is equal to zero. Since this angle is obtained by superposing the angles of rotation ( A ) 1 and ( A ) 2 in the released structure (Figs. 10-13c and d), the compatibility equation becomes Substituting into the compatibility equation

28. Solving this equation for the redundant, we get M A = qL 2 /8, which agrees with the previous result. Also, the equations of equilibrium Is the same results as before for the reactions R A and R B Now that all reactions have been found, we can determine the shear forces, bending moments, slopes, and deflections by the techniques already described.

29. Assignment 1. A two-span continuous beam ABC supports a uniform load of intensity q(Fig 1). Each span of the beam has length L. Using the method of superposition, determine all reactions for this beam. 2. A fixed-end beam AB (Fig.2) is loaded by a force P acting at an intermediate point D. Find the reactive forces and moments at the ends of the beam using the method of superposition. Also, determine the deflection at point D where the load is applied. Fig. 1 Fig. 2

30. 3. As shown in Fig. 3, a rigid horizontal bar is supported by a hinge at A and by two steel cables BD and CE, which are of equal length, L = 0.8 m, and cross-sectional area, A = 140 mm 2. Calculate the stress in each cable due to a force of 40 kN, applied as shown in the figure. Assume that the yielding stress is 250 MPa and that E = 200 GPa. Fig. 3

31. Best all !!