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### Chapter 2

1. 1. BASIC CONCEPTS OF DISPLACEMENT OR STIFFNESS METHOD: 2.1 INTRODUCTION Displacement or stiffness method allows one to use the same method to analyse both statically determinate and indeterminate structures, whereas the force or the flexibility method requires a different procedure for each of these two cases. Furthermore, it is generally easier to formulate the necessary matrices for the computer operations using the displacement method. Once these matrices are formulated, the computer calculations can be performed efficiently. As discussed previously in this method nodal displacements are the basic unknown. However like slope deflection and moment distribution methods, the stiffness method does not involve the ideas of redundancy and indeterminacy. Equilibrium equations in terms of unknown nodal displacements and known stiffness coefficients (force due a unit displacement) are written. These equations are solved for nodal displacements and when the nodal displacements are known the forces in the members of the structure can be calculated from force displacement relationship CHAPTER 2
2. 2. 2.2 STIFFNESS, STIFFNESS COEFFICIENT AND STIFFNESS MATRIX: The stiffness of a member is defined as the force which is to be applied at some point to produce a unit displacement when all other displacement are restrained to be zero. If a member which behaves elastically is subjected to varying axial tensile load (W) as shown in fig. 2.1 and a graph is drawn of load (W) versus displacement (∆) the result will be a straight line as shown in fig. 2.2, the slope of this line is called stiffness.
3. 3. FIG.2.1 FIG.2.2 (Members subjected to varying axial load ( (Graph of load verses displacement) Displacement Load ∆ ∆ ∆ W 2 2W W
4. 4. Mathematically it can be expressed as K=W/∆ ------ 2.1 In other words Stiffness ‘K’ is the force required at a certain point to cause a unit displacement at that point. Equation 2.1 can be written in the following form W = K ∆ ----- 2.2 Where, W = Force at a particular point K = Stiffness ∆ = Unit displacement of the particular point. The above equation relates the force and displacement at a single point. This can be extended for the development of a relationship between load and displacement for more than one point on a structure.
5. 5. 1 2 W = K ∆ = 0 ∆ = 0 (a) (b) (c) Fig:2.3 1 2 2 1 1 1 11 2 W = K2 21 W = K1 12 W = K2 22 1 2 ∆ =1 L ∆ =1
6. 6. Let us consider a beam of fig. 2.3 and two points (nodes) 1, and 2. If a unit displacement is induced at point ‘1’ while point ‘2’ is restrained from deflecting up or down (see the definition of stiffness). then the forces “W1”and “W2” can be expressed in terms of “∆1” in equation 2.2 as: W = K ∆ ---------- (2.2) when ∆1 = 1 W1 = K11. ∆1 = K11 See fig. 2.3(b) W2 = K21. ∆1 = Kwhere, K11 = force at 1 due to unit displacement at 1 K21 = force at 2 due to unit displacement at 1 These are known as stiffness co-efficients. If a unit displacement is induced at a point “2” while point “1” is restrained from deflecting up or down,then the forces W1 and W2 can be expressed in terms of “ D2” in equation 2.2 as;
7. 7. when ∆2 = 1 W1 = K12. ∆2 = K12 See fig. 2.3(c) W2 = K22. ∆2 = K22 where, K12 = force at 1 due to unit displacement at 2 K22 = force at 2 due to unit displacement at 2 First subscript indicates the point of force and second the point of deformation. As forces W1, W2 are proportional to the deformations ∆1 and ∆2, the following equation for the beam of fig.2.3 can be written as W1 = K11 ∆1 + K12 ∆2 ---------- (2.3) W2 = K21 ∆1 + K22 ∆2 ---------- (2.4) Rewriting this in matrix form W W K K K K 1 2 11 12 21 22 1 2       =             ∆ ∆ --------- (2.5)
8. 8. where is called stiffness matrix Elements of the stiffness matrix are known as stiffness coefficients. So stiffness coefficients can be defined as the forces at points (nodes) caused by introducing various unit deformations one at a time. is called force vector and is called displacement or deformation vector. K K K K 11 12 21 22       W W 1 2       ∆ ∆ 1 2      
9. 9. The expression (2.5) expresses the equilibrium at each of the node points in terms of stiffness co-efficients and the unknown nodal deformation and can be written as: W = K ∆ ---------- (2.6) The matrix K contains the stiffness co-efficients and it relates the forces W to the deformations ∆ and is called stiffness matrix. W and ∆ are called force and deformation vectors. The term “force” and the symbol “W” refers to the moments as well as forces and the term “deformation” and symbol “∆” refer to the both rotations and deflection. 2.3 STIFFNESS OR DISPLACEMENT METHOD FOR TRUSSES 2.3.1 Element and structure stiffness matrix. Application of stiffness method requires subdividing the structure into series of elements. The load-deformation characteristics of a structure are obtained from load-deformation characteristics of elements. It means that stiffness matrix of a structure [K] is formed from the stiffness matrices of the individual elements which make up the structure. Therefore it is important first to develop element stiffness matrix. The stiffness matrix for a truss element is developed in subsequent section.
10. 10. w, δ1 1 1 2 w ,δ22 L 2.3.2 Stiffness Matrix of an Axially loaded Element (An Individual Truss Member) For the development of an element stiffness matrix for a truss member, let us consider an axially load member of length ‘L’, area ‘A’ and modulus of elasticity ‘E’. The ends (nodes) of the member are denoted by 1 and 2 as shown in fig. 2.4(a). fig2,.4(a) (a) Element forces, w1, w2 and deformations, δ1, δ2 (b) Deformation introduced at node ‘1’ with node ‘2’ restrained.
11. 11. (b) Deformation introduced at node ‘1’ with node ‘2’ restrained (c)Deformation introduced at node ‘2’ with node ‘1’ restrained. δ=1 δ=0 21 k k2111 1 2 δ =1 δ =0 21 k k2111 1 2
12. 12. 1 Fig:2.4 w2 δ1=1 w δ2=1 )d)Member forces and deformations of the actual members. The vectors in fig. 2.4(a) define the forces w1, w2 and the corresponding deformation δ1 and δ2 at the ends of the member. These also define their positive directions. As shown in fig. 2.4(b) a positive deformation δ1, at node ‘1’ is introduced. while node ‘2’ is assumed to be restrained by a temporary pin support. Expressing the end forces in terms of δ: As (from stress-strain relationship) -------- (2.7) δ = wL AE
13. 13. w AE L = δ --------(2.8) when δ = 1 when δ1 = 1 -------(2.9) where , k11 is the force at 1 due to unit displacement at 1 k21 is the force at 2 due to unit displacement at 1 The first subscript denotes the location of the node at which the force acts and second subscript indicates the location of displacement. As forces and deformations are positive when they act to the right, so k11 is positive while k21 is negative. L AE kw == k AE L and k AE L 11 21= = −
14. 14. Similarly if end ‘1’ is restrained while end ‘2’ is deformed in the positive direction a distance d2 = 1 from fig. 2.4(c). k AE L and k AE L 12 22= − = ----- (2.10) where, k12 is the force at 1 due to unit displacement at 2 k22 is the force at 2 due to unit displacement at 2 To evaluate the resultant forces w1 and w2 in terms of displacement δ1 and δ2 w1 = k11 δ1 + k12 δ2 ---------- (2.11) w2 = k21 δ1 + k22 δ2 ---------- (2.12)
15. 15. Expressing in matrix form w w k k k k 1 2 11 12 21 22 1 2       =             δ δ ---------- (2.13)                 +− −+ =      2 1 2 1 δ δ L AE L AE L AE L AE w w             +− −+ =      2 1 2 1 11 11 δ δ L AE w w ---------- (2.14) It can be written as w = k δ ---------- (2.15) [ ]       +− −+ = 11 11 L AE k ---------- (2.16)
16. 16. This “k” is called element stiffness matrix. It can be observed that sum of the elements in each column of element stiffness matrix “k” equals zero. It is due to the reason that co-efficients in each column represent the forces produced by a unit displacement of one end while the other end is restrained (see fig. 2.4(b)). Since the bar is in equilibrium in the x-direction the forces must be equal to zero. Similarly all co-efficients along the main diagonal must be positive because these terms are associated with the forces acting at the node at which a positive displacement is introduced into the structure and correspondingly the force is the same (positive) as the displacement. 2.3.3 Composite stiffness matrix Equation 2.16 gives the stiffness matrix for an element of a truss. A great advantage of subdividing a structure into a series of elements is that the same element stiffness matrix can be used for all the elements of a structure. Stiffness matrix comprising of all the element stiffness matrices is called composite stiffness matrix. Composite stiffness matrix is a square matrix and its size depends upon number of members. Order of the the composite stiffness matrix is 2m × 2m, where m is the number of members. Let us consider a truss shown in fig. 2.5.
17. 17. This truss is subdivided into three elements. Forces and deformations are shown in fig. 2.5(b). Stiffness matrix of element no.1 [ ]k AE L 1 1 1 1 1 = − −       Stiffness matrix of element no.2 [ ]k AE L 2 6 5 6 5 6 5 6 5 = − −       / / / / Stiffness matrix of element no.3 [ ]k AE L 3 6 5 6 5 6 5 6 5 = − −       / / / /
18. 18. Composite stiffness matrix of all elements is given by: [ ]K AE L c = − − − − − −                     1 1 0 0 0 0 1 1 0 0 0 0 0 0 12 12 0 0 0 0 12 12 0 0 0 0 0 0 12 12 0 0 0 0 12 12 . . . . . . . . Relationship between forces and displacement from equation w = kc δ ----------- (2.17)                                                 − − − − − − =                         6 5 4 3 2 1 6 5 4 3 2 1 2.12.10000 2.12.10000 002.12.100 002.12.100 000011 000011 δ δ δ δ δ δ L AE w w w w w w
19. 19. It can be seen that the some of the elements in each column of matrix ‘kc’ is zero. It is due to the reason that the stiffness co-efficient in each column represents the force produced by unit deformation of one end while other is restrained. Since the member is in equilibrium the sum of the forces must be zero. However all co-efficients along the main diagonal must be positive because these terms are associated with the force acting at the end at which positive deformation is introduced. As deformation is positive so force produced is also positive. 2.3.4 Structure stiffness matrix: Stiffness matrix of a structure can be generated from stiffness matrices of the elements into which a structure has been subdivided. The composite stiffness matrix [kc] describes the force deformation relationship of the individual elements taken one at a time, whereas structure stiffness matrix [K] describes the load deformation characteristics of the entire structure. In order to obtain structure stiffness matrix [K] from composite stiffness matrix [kc] a deformation transformation matrix is used which is described in the subsequent section.
20. 20. 2.3.5 Deformation transformation matrix: Deformation transformation matrix relates internal element or member deformation to the external nodal structure deformation. It is simply a geometric transformation of co-ordinates representing the compatibility of the deformations of the system. Following is the relationship between element and structure deformation. δ = T ∆ ------------ (2.18) where δ = element deformation ∆ = structure deformation T = deformation transformation matrix As work done by structure forces = work done by element forces
21. 21. [ ] [ ] [ ] [ ] 1 2 1 2 ∆ T T W w= δ ------------ (2.19) as [ ] [ ][ ]∆= KW ------------ (2.6) [ ] [ ][ ]w kc= δ ------------ (2.17) substituting values of W and w from equation (2.6) and (2.17) into equation (2.19) [ ] [ ][ ] [ ] [ ][ ] 1 2 1 2 ∆ ∆ T T cK k= δ δ [ ] [ ][ ] [ ] [ ][ ]δδ c TT kK =∆∆ ------------ (2.20) [ ] [ ][ ]∆= Tδ -------------(2.18) [ ] [ ][ ] [ ] [ ] [ ][ ][ ]∆ ∆ ∆ ∆ T T T cK T k T= [ ] [ ][ ] [ ] [ ] [ ][ ][ ]∆ ∆ ∆ ∆ T T T cK T k T= so [ ] [ ] [ ][ ]K T k T T c= -------------(2.21)
22. 22. Therefore structure stiffness matrix [K] can be obtained from composite element stiffness matrix [kc] if latter is pre-multiplied by [T]T and post-multiplied by [T]. 2.3.6 Formation of deformation transformation matrix: As we know δ = T ∆ Let Tij represent the value of element deformation “δi” caused by a unit structure displacement [∆j]. The total value of each element deformation caused by all structure deformations may be written as δ δ δ 1 2 11 1 12 2 1 21 1 22 2 2 1 1 2 2     m n n n n m m mn n T T T T T T T T T             = + + + + + +             ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ where δ1 , δ2 , δ3 --- δn represent set of element deformations and ∆1 , ∆2 --- ∆n the set of structure deformations. In matrix form
23. 23.             ∆ ∆ ∆             =             nmnmm n n m TTT TTT TTT  2 1 21 22221 11211 2 1 δ δ δ so [ ]             = mnmm n n TTT TTT TTT T     21 22221 11211 ------------(2.22) is called deformation transformation matrix. Matrix [T] is usually a rectangular matrix. Its columns are obtained by applying a unit values of structure deformation ∆1 through ∆n one at a time and determining the corresponding element deformations δ1 through δm . Formation of the transformation matrix of a truss, beam and a frame element is explained in the subsequent chapters.