Force Force and Displacement Matrix Method

The analysis of a structure by the matrix method may be described by the following steps:
1. Problem statement
2. Selection of released structure
3. Analysis of released structure under loads
4. Analysis of released structure for other causes
5. Analysis of released structure for unit values of redundant
6. Determination of redundants through the superposition equations.
7. Determine the other displacements and actions. The following are the four flexibility matrix equations for calculating redundants member end actions, reactions and joint displacements
where for the released structure
8.All matrices used in the matrix method are summarized in the following tables

1. Noida Institute of Engineering and Technology, Greater Noida Force and D isplacement Matrix Method Aayushi Assistant Professor Civil Engg . Department 5/17/2020 1 Unit: 4 Aayushi RCE-502, DOS 1 Unit 4 Subject Name : Design of Structure I Course Details : B Tech 5 th Sem

2. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 2 Content Course Objective Course O utcome CO-PO & PSO Mapping Prerequisite & Recap Stiffness Matrix Topic outcome Objective of topic 1 Topic outcome & mapping with PO Prerequisite & Recap Introduction Numerical on spring Syllabus of unit 4 Topic objective

3. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 3 Content Summary of stiffness matrix method The flexibility matrix method Topic Objective Topic Outcome Topic outcome and mapping with PO Procedure of flexibility matrix Numerical of flexibility matrix Matrix in table form Summary of flexibility matrix Youtube links Daily Quiz Objective of topic 2 Introduction

4. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 4 Content Old Question Papers Expected Questions in University Examination Summary References MCQs Weekly Assignment

5.   Objective 1 To impart the principles of elastic structural analysis and behavior of indeterminate structures. 2 To impart knowledge about various methods involved in the analysis of indeterminate structures. .. 3 To apply these methods for analyzing the indeterminate structures to evaluate the response of structures . 4 To enable the student get a feeling of how real-life structures behave 5 To make the student familiar with latest computational techniques and software used for structural analysis. . 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 5 Course Objective

6.   Students will be able CO1 To Identify and analyze the moment distribution in beams and frames by Slope Deflection Method, Moment Distribution Method and Strain Energy Method. CO2 To provide adequate learning of indeterminate structures with Muller’s Principle; Apply & Analyze the concept of influence lines for deciding the critical forces and sections while designing.. CO3 To learn about suspension bridge, two and three hinged stiffening girders and their influence line diagram external loading and analyze the same. CO4 To Identify and analyze forces and displacement matrix for various structural. CO5 To understand the collapse load in the building and plastic moment formed. CO6 Apply the concepts of forces and displacements to solve indeterminate structure. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 6 Course Outcome

7. CO PO1 PO2 PO3 PO4 PO5 PO6 PO7 PO8 PO9 PO10 PO11 PO12 RCE502.1 3 3 1 2 1 - - - 2 - - 3 RCE502.2 3 2 2 2 3 - - 1 2 - 2 2 RCE502.3 3 - 2 2 1 - - - 2 - 2 2 RCE502.4 3 1 2 2 1 - - - 2 - 2 2 RCE502.5 3 2 2 2 1 - - - 2 - 2 2 RCE502.6 3 2 3 2 1 - - - 2 - 2 3 AVG. 3 2 2 2 1 - - - 2 - 2 2 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 7 CO-PO and PSO Mapping COs PSO1 PSO2 PSO3 PSO4 RCE502.1 2 1 2 3 RCE502.2 2 1 2 2 RCE502.3 2 2 2 2 RCE502.4 2 1 2 3 RCE502.5 2 1 2 3 RCE502.6 2 1 2 3 Avg. 2 1 2 3

8. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 8 Prerequisite and Recap Basics of strength of material Basics of engineering mechanics Basic of Matrix Mathematics

9. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 9 Syllabus of Unit 4 Basic Force and Displacement Matrix method for analysis of beams, frames and trusses

10. 5/17/2020 10 Objective of Unit 4 Differentiate between the direct stiffness method and the displacement method. Analysis simple structures by the direct stiffness matrix and Flexibility matrix. Aayushi RCE-502, DOS 1 Unit 4

11. Differentiate between the direct stiffness method and the displacement method. Formulate flexibility matrix of member . Define stiffness matrix. Construct stiffness matrix of a member . Analysis simple structures by the direct stiffness matrix. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 1 11 Topic Objective

12. To illustrate the concept of direct stiffness method to obtain the global stiffness matrix and solve a spring assemblage problem. 5/17/2020 12 Topic Outcomes Once the student has successfully completed this unit, he/she will be able to: To show how the potential energy approach can be used to both derive the stiffness matrix for a spring and solve a spring assemblage problem. To describe and apply the different kinds of boundary conditions relevant for spring assemblages. Aayushi RCE-502, DOS 1 Unit 4

13. 5/17/2020 13 Objective of Topic Topic-1 Name The Stiffness Matrix Method Objective of Topic-1: Differentiate between the direct stiffness method and the displacement method. Analysis simple structures by the direct stiffness matrix method. Aayushi RCE-502, DOS 1 Unit 4

14. 5/17/2020 14 Topic Outcome and mapping with PO Programme Outcomes (POs) PO1 PO2 PO3 PO4 PO5 PO6 PO7 PO8 PO9 PO10 PO11 PO12 TO1 2 1 1 1 1 1 Outcome of Topic-1: After the successfully competition of this topic students will be able to Analysis simple structures by the direct stiffness matrix method. Aayushi RCE-502, DOS 1 Unit 4

15. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 15 Prerequisite and Recap Degree of static and kinematic indeterminacy Basic concepts degree of freedom and restraints Generate Matrix Mathematics Matrix Solving

16. This section introduces some of the basic concepts on which the direct stiffness method is based. The linear spring is simple and an instructive tool to illustrate the basic concepts . The steps to develop a finite element model for a linear spring follow our general 8 step procedure . 1. Discretize and Select Element Types - Linear spring elements 2. Select a Displacement Function - Assume a variation of the displacements over each element. 3. Define the Strain/Displacement and Stress/Strain Relationships - use elementary concepts of equilibrium and compatibility . 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 16 Topic : Introduction to the Stiffness (Displacement) Method

17. 4 . Derive the Element Stiffness Matrix and Equations - Define the stiffness matrix for an element and then consider the derivation of the stiffness matrix for a linear elastic s pring element. 5. Assemble the Element Equations to Obtain the Global or Total Equations and Introduce Boundary Conditions - We then show how the total stiffness matrix for the problem can be obtained by superimposing the stiffness matrices of the individual elements in a direct manner . The term direct stiffness method evolved in reference to this method . 6. Solve for the Unknown Degrees of Freedom ( or Generalized Displacements ) - Solve for the nodal displacements . 7. Solve for the Element Strains and Stresses – The reactions and internal forces association with the bar element . 8. Interpret the Results 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 17 Topic: Procedure of Stiffness Matrix

18. Select Element Type - Consider the linear spring shown below . The spring is of length L and is subjected to a nodal tensile force, T directed along the x -axis . Note: Assumed sign conventions 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 18 Continuous…

19. 2. Select a Displacement Function - A displacement function u ( x ) is assumed . u=a1+a2x In general, the number of coefficients in the displacement function is equal to the total number of degrees of freedom associated with the element. We can write the displacement function in matrix forms as : We can express u as a function of the nodal displacements u i by evaluating u at each node and solving for a1 and a2 . 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 19 Continuous…

20. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 20 Continuous…. Solving for a2 : Substituting a1 and a2 into u gives : In matrix form : Or in another form :

21. Where N1 and N2 are defined as : The functions Ni are called interpolation functions because they describe how the assumed displacement function varies over the domain of the element. In this case the interpolation functions are linear . 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 21 Continuous…

22. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 22 Continuous…

23. 3. Define the Strain/Displacement and Stress/Strain Relationships - Tensile forces produce a total elongation (deformation ) delta of the spring. For linear springs, the force T and the displacement u are related by Hooke’s law : where deformation of the spring delta is given as: 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 23 Continuous…

24. 4 . Step 4 - Derive the Element Stiffness Matrix and Equations - We can now derive the spring element stiffness matrix as follows: Rewrite the forces in terms of the nodal displacements: We can write the last two force-displacement relationships in matrix form as : This formulation is valid as long as the spring deforms along the x axis. The coefficient matrix of the above equation is called the local stiffness matrix k : 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 24 Continuous …

25. 5. Step 4 - Assemble the Element Equations and Introduce Boundary Conditions The global stiffness matrix and the global force vector are assembled using the nodal force equilibrium equations, and force/deformation and compatibility equations . where k and f are the element stiffness and force matrices expressed in the global coordinates . 6. Step 6 - Solve for the Nodal Displacements Solve the displacements by imposing the boundary conditions and solving the following set of equations : 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 25 Continuous …

26. 7. Step 7 - Solve for the Element Forces Once the displacements are found, the forces in each element may be calculated from : 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 26 Continuous…

27. Consider the following two-spring system shown below where the element axis x coincides with the global axis x . For element 1: For element 2 : Both continuity and compatibility require that both elements remain connected at node 3. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 27 Topic: The Stiffness Method – Spring Example 1

28. We can write the nodal equilibrium equation at each node as: Therefore the force-displacement equations for this spring system are : In matrix form the above equations are: 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 28 Continuous…

29. where F is the global nodal force vector , d is called the global nodal displacement vector , and K is called the global stiffness matrix . Assembling the Total Stiffness Matrix by Superposition Consider the spring system defined in the last example : The elemental stiffness matrices may be written for each element. For element 1 : For element 2: 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 29 Continuous…

30. Write the stiffness matrix in global format for element 1 as follows: For element 2: Apply the force equilibrium equations at each node. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 30 Continuous…

31. The above equations give: To avoid the expansion of the each elemental stiffness matrix , we can use a more direct, shortcut form of the stiffness matrix . The global stiffness matrix may be constructed by directly adding terms associated with the degrees of freedom in k (1) and k (2) into their corresponding locations in the K as follows: 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 31 Continuous…

32. Boundary conditions are of two general types: 1. homogeneous boundary conditions (the most common) occur at locations that are completely prevented from movement ; 2. nonhomogeneous boundary conditions occur where finite non-zero values of displacement are specified, such as the settlement of a support. In order to solve the equations defined by the global stiffness matrix, we must apply some form of constraints or supports or the structure will be free to move as a rigid body . Consider the equations we developed for the two-spring system . We will consider node 1 to be fixed u 1 = 0. The equations describing the elongation of the spring system become: 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 32 Topic :Boundary Conditions

33. Expanding the matrix equations gives : The second and third equation may be written in matrix form as: Once we have solved the above equations for the unknown nodal displacements, we can use the first equation in the original matrix to find the support reaction . 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 33 Continuous…

34. For homogeneous boundary conditions, we can delete the row and column corresponding to the zero-displacement degrees-of-freedom. Let’s again look at the equations we developed for the two spring system . However, this time we will consider a nonhomogeneous boundary condition at node 1: u 1=delta. The equations describing the elongation of the spring system become : Expanding the matrix equations gives : 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 34 Continuous…

35. By considering the second and third equations because they have known nodal forces we get : In matrix form the above equations are : For nonhomogeneous boundary conditions, we must transfer the terms from the stiffness matrix to the right-hand-side force vector before solving for the unknown displacements . Once we have solved the above equations for the unknown nodal displacements, we can use the first equation in the original matrix to find the support reaction . 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 35 Continuous…

36. The stiffness coefficients are defined in this section. Construction of stiffness matrix for a simple member is explained. A few simple problems are solved by the direct stiffness method. The difference between the slope-deflection method and the direct stiffness method is clearly brought out. . 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 36 Summary of Stiffness Matrix Method

37. T o analysis statically indeterminate structure of degree one. T o solve the problem by either treating reaction or moment as redundant . T o draw shear force and bending moment diagram for statically indeterminate beams. T o state advantages and limitations of force method of analysis. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 37 Topic Objective

38. Concept of force method for analysis of statically indeterminate structure. 5/17/2020 38 Topic Outcomes Once the student has successfully completed this unit, he/she will be able to: Selection of the basic determinate structure Illustration of force method by numerical examples. Analysis simple structures by the direct stiffness matrix Aayushi RCE-502, DOS 1 Unit 4

39. 5/17/2020 39 Objective of Topic Topic-2 Name The Flexibility Matrix Method Objective of Topic-2: To analysis statically indeterminate structure by Flexibility matrix Method. Aayushi RCE-502, DOS 1 Unit 4

40. 5/17/2020 40 Topic Outcome and mapping with PO Programme Outcomes (POs) PO1 PO2 PO3 PO4 PO5 PO6 PO7 PO8 PO9 PO10 PO11 PO12 TO1 1 1 1 1 1 1 1 Outcome of Topic-2: After the successfully competition of this topic students will be able to analysis of Different beam and Frames by Flexibility matrix method Aayushi RCE-502, DOS 1 Unit 4

41. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 41 Prerequisite and Recap Concept of force method for analysis of statically indeterminate structure. Selection of the basic determinate structure. Illustration of force method by numerical examples .

42. In 1864 James Clerk Maxwell published the first consistent treatment of the flexibility method for indeterminate structures . With the flexibility method equations of compatibility involving displacements at each of the redundant forces in the structure are introduced to provide the additional equations needed for solution. This method is somewhat useful in analyzing beams, frames and trusses that are statically indeterminate to the first or second degree. For structures with a high degree of static indeterminacy such as multi-story buildings and large complex trusses stiffness methods are more appropriate. Nevertheless flexibility methods provide an understanding of the behavior of statically indeterminate structures. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 42 Topic: The Flexibility Matrix

43. The fundamental concepts that underpin the flexibility method will be illustrated by the study of a two span beam. The procedure is as follows 1.Pick a sufficient number of redundants corresponding to the degree of indeterminacy. 2.Remove the redundants . 3.Determine displacements at the redundants on released structure due to external or imposed actions. 4.Determine displacements due to unit loads at the redundants on the released structure. 5.Employ equation of compatibility, e.g., if a pin reaction is removed as a redundant the compatibility equation could be the summation of vertical displacements in the released structure must add to zero. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 43 Topic: The procedure of F lexibility Method

44. Original Beam FBD Released Beam The beam to the left is statically indeterminate to the first degree. The reaction at the middle support RB is chosen as the redundant . The released beam is also shown. Under the external loads the released beam deflects an amount D B . A second beam is considered where the released redundant is treated as an external load and the corresponding deflection at the redundant is set equal to D B . 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 44 Continuous…

45. A more general approach consists in finding the displacement at B caused by a unit load in the direction of R B . Then this displacement can be multiplied by R B to determine the total displacement Also in a more general approach a consistent sign convention for actions and displacements must be adopted. The displacements in the released structure at B are positive when they are in the direction of the action released, i.e., upwards is positive here. The displacement at B caused by the unit action is The displacement at B caused by R B is δ B R B . The displacement caused by the uniform load w acting on the released structure is 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 45 Continuous…

46. Thus by the compatibility equation 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 46 Continuous…

47. . If a structure is statically indeterminate to more than one degree, the approach used in the preceding example must be further organized and more generalized notation is introduced. Consider the beam to the left. The beam is statically indeterminate to the second degree. A statically determinate structure can be obtained by releasing two redundant reactions. Four possible released structures are shown. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 47 Topic: Numerical

48. The redundants chosen are at B and C . The redundant reactions are designated Q1 and Q2 . The released structure is shown at the left with all external and internal redundants shown. DQL1 is the displacement corresponding to Q1 and caused by only external actions on the released structure DQL2 is the displacement corresponding to Q2 caused by only external actions on the released structure. Both displacements are shown in their assumed positive direction. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 48 Continuous… The redundants chosen are at B and C. The redundant reactions are designated Q1 and Q The released structure is shown at the left with all external and internal redundants shown. DQL1 is the displacement corresponding to Q1 and caused by only external actions on the released structure DQL2 is the displacement corresponding to Q2 caused by only external actions on the released structure. Both displacements are shown in their assumed positive direction.

49. We can now write the compatibility equations for this structure. The displacements corresponding to Q1 and Q2 will be zero. These are labeled DQ1 and DQ2 respectively In some cases DQ1 and DQ2 would be nonzero then we would write 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 49 Continuous…

50. The equations from the previous page can be written in matrix format as where: {DQ } - vector of actual displacements corresponding to the redundant {DQL } - vector of displacements in the released structure corresponding to the redundant action [Q] and due to the loads [F] - flexibility matrix for the released structure corresponding to the redundant actions [Q] {Q} - vector of redundants 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 50 Continuous…

51. The vector {Q} of redundants can be found by solving for them from the matrix equation on the previous overhead. To see how this works consider the previous beam with a constant flexural rigidity EI . If we identify actions on the beam as Since there are no displacements imposed on the structure corresponding to Q1 and Q2 , then 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 51 Continuous…

52. The vector [ DQL ] represents the displacements in the released structure corresponding to the redundant loads. These displacements are The positive signs indicate that both displacements are upward. In a matrix format The flexibility matrix [ F ] is obtained by subjecting the beam to unit load corresponding to Q1 and computing the following displacements 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 52 Continuous…

53. Similarly subjecting the beam to unit load corresponding to Q2 and computing the following displacements The flexibility matrix is The inverse of the flexibility matrix is As a final step the redundants [Q] can be found as follows 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 53 Continuous…

54. All matrices used in the flexibility method are summarized in the following tables 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 54 Topic: Matrix in Table form

55. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 55 Continuous…

56. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 56 Continuous…

57. Analysis of released structure for unit values of redundant Determination of redundants through the superposition equations. Determine the other displacements and actions. The following are the four flexibility matrix equations for calculating redundants member end actions, reactions and joint displacements where for the released structure All matrices used in the flexibility method. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 57 Summary of Flexibility Matrix Method

58. Youtube /other Video Links 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 58 Youtube Video Links Topic Links Process Stiffness matrix method https://www.youtube.com/watch?v=AbvrPVT9OSM Click on the link Flexibility matrix method https://www.youtube.com/watch?v=AbvrPVT9OSM Click on the link Matrix method youtube.com/ watch?v =Mlmp9s0tlgc Click on the link

59. Flexibility coefficients depend upon loading of the primary structure. State whether the above statement is true or false a) True b) False Define Stiffness coefficient and flexibility coefficient? Define Degree of freedom? What do you understand by term Redundant? 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 59 Daily Quiz

60. Develop the flexibility matrix for the cantilever with coordinate as shown in Fig. , take uniform flexural rigidity. Analyze the following continuous beam using the flexibility or stiffness method of matrix analysis. Using the force method, determine the reactions and moments over supported of a continuous beam shown in Fig . 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 60 Daily Quiz

61. Analyze the continuous beam shown in Fig. using flexibility method and draw BMD . Using flexibility matrix method , find reaction at supports in following beam of Fig . Take EI as constant. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 61 Daily Quiz

62. Compare flexibility method and stiffness method . What are the type of structures that can be solved using stiffness matrix method ? List the properties of the stiffness matrix Why is the stiffness matrix method also called equilibrium method or displacement method ? Calculate the support reactions in the continuous beam ABC due to loading as shown in Fig.1.1 Assume EI to be constant throughout. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 62 Weekly Assignment

63. Analyse the continuous beam as shown in Fig . If the downward settlement of supports B and C are 10 mm and 5 mm respectively . Take EI = 184 x 10 " N - mm ' . Use flexibility matrix method Analyze the continuous beam shown in fig by stiffness method. Draw bending moment diagram and elastic curve . 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 63 Weekly Assignment

64. How many compatibility equations should be written if we have n no. of redundant reactions ? a) n – 1 b) n c) n + 1 d) n + 2 Flexibility matrix is always :- a) symmetric b) non-symmetric c) anti-symmetric d) depends upon loads applied 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 64 MCQ s

65. Which of the following primary structure is best for computational purposes? a) symmetric b) non-symmetric c) anti-symmetric d) depends upon loads applied For computational purposes, deflected primary structure ans actual structure should be ___________ a) as different as possible b) as similar as possible c) it doesn’t matter d) in between 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 65 MCQ s

66. Numerical accuracy of solution increases if flexibility coefficients with larger values are located:- a) near main diagonal b) near edges c) in between d) near side middles For stable structures, one of the important properties of flexibility and stiffness matrices is that the elements on the main diagonal ( i ) Of a stiffness matrix must be positive (ii) Of a stiffness matrix must be negative (iii) Of a flexibility matrix must be positive (iv) Of a flexibility matrix must be negative 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 66 MCQ s

67. The Castigliano's second theorem can be used to compute deflections (A) In statically determinate structures only (B) For any type of structure (C) At the point under the load only (D) For beams and frames only When a uniformly distributed load, longer than the span of the girder, moves from left to right, then the maximum bending moment at mid section of span occurs when the uniformly distributed load occupies (A) Less than the left half span (B) Whole of left half span (C) More than the left half span (D) Whole span 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 67 MCQ s

68. If in a pin-jointed plane frame (m + r) > 2j, then the frame is (Where ‘m’ is number of members, ‘r’ is reaction components and ‘j’ is number of joints ) (A) Stable and statically determinate (B) Stable and statically indeterminate (C) Unstable (D) None of the above Principle of superposition is applicable when (A) Deflections are linear functions of applied forces (B) Material obeys Hooke's law (C) The action of applied forces will be affected by small deformations of the structure (D) None of the above 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 68 MCQ s

69. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 69 Old Question Papers

70. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 70 Old Question Papers

71. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 71 Old Question Papers

72. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 72 Old Question Papers

73. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 73 Old Question Papers

74. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 74 Old Question Papers

75. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 75 Old Question Papers

76. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 76 Old Question Papers

77. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 77 Old Question Papers

78. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 78 Old Question Papers

79. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 79 Old Question Papers

80. A Fixed beam  AB  of constant flexural rigidity is shown . The beam is subjected to a uniform distributed loadof   w  moment  M = wL 2   kNm . Draw Shear force and bending moment diagrams by force method. Define coefficient of stiffness and flexibility matrix? 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 80 Expected Questions for University Exam

81. Analyze the beam shown in fig by stiffness matrix method. Take EI as constant. Analyse the continuous beam as shown in figvby stiffness matrix method if the support B sink by 10 mm. Take EI = 6000 KN cubic meter 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 81 Expected Questions for University Exam

82. The analysis of a structure by the matrix method may be described by the following steps: 1. Problem statement 2. Selection of released structure 3. Analysis of released structure under loads 4. Analysis of released structure for other causes 5. Analysis of released structure for unit values of redundant 6. Determination of redundants through the superposition equations. 7. Determine the other displacements and actions. The following are the four flexibility matrix equations for calculating redundants member end actions, reactions and joint displacements where for the released structure 8.All matrices used in the matrix method are summarized in the following tables 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 82 Summary

83. Jain, A. K., “Advanced Structural Analysis “, Nem Chand & Bros., Roorkee . Hibbeler , R.C., “Structural Analysis”, Pearson Prentice Hall, Sector - 62, Noida-201309 C . S. Reddy “Structural Analysis”, Tata Mc Graw Hill Publishing Company Limited,New Delhi. Timoshenko , S. P. and D. Young, “ Theory of Structures” , Tata Mc- Graw Hill BookPublishing Company Ltd., New Delhi. Dayaratnam , P. “ Analysis of Statically Indeterminate Structures”, Affiliated East- WestPress . Wang, C. K. “ Intermediate Structural Analysis”, Mc Graw -Hill Book PublishingCompany Ltd. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 83 References

84. 5/17/2020 Aayushi RCE-502, DOS 1 Unit 4 84 References Thank You