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Section 6: the Flexibility Method - Beams

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Section 6: the Flexibility Method - Beams Section 6: The Flexibility Method - Beams Washkewicz College of Engineering Preliminaries: Beam Deflections – Virtual Work There are several methods available to calculate deformations (displacements and rotations) in beams. They include: • Formulating moment equations and then integrating to find rotations and displacements • Moment area theorems for either rotations and/or displacements • Virtual work methods Since structural analysis based on finite element methods is usually based on a potential energy method, we will tend to use virtual work methods to compute beam deflections. The theory that supports calculating deflections using virtual work will be reviewed and several examples are presented. 1 Section 6: The Flexibility Method - Beams Washkewicz College of Engineering Consider the following arbitrarily loaded beam Identify M M(x) Moment at any section in the beam due to external loads m m(x) Moment at any section in the beam due to a unit action m y ~ I Stress acting on dA due to a unit action 2 Section 6: The Flexibility Method - Beams Washkewicz College of Engineering The force acting on the differential area dA due to a unit action is ~ f ~ dA m y dA I The stress due to external loads is M y I The displacement of a differential segment dA by dx along the length of the beam is dx dx E M y dx E I 3 Section 6: The Flexibility Method - Beams Washkewicz College of Engineering The work done by the force acting on the differential area dA due to a unit action as the differential segment of the beam (dA by dx) displaces along the length of the beam by an amount is ~ dW f m y M y dA dx I E I M m y2 dA dx 2 E I The work done within a differential segment (now A by dx) due to a unit action applied to the beam is the integration of the expression above with respect to dA, i.e., c T M m y2 dW dA dx 2 E I A cB cT Mm 2 Wdiffernetialsegment 2 y dA dx EI cB Mm Mm 2 I dx dx EI EI 4 Section 6: The Flexibility Method - Beams Washkewicz College of Engineering The internal work done along the entire length of the beam due to a unit action applied to the beam is the integration of the last expression with respect to x, i.e., L M x mx W dx Internal 0 EI The external work done along the entire length of the beam due to a unit action applied to the beam is WExternal 1D With WExternal WInternal L M x mx 1 D dx 0 EI L M x mx D dx 0 EI or the deformation (D) of the a beam at the point of application of a unit action (force or moment) is given by the integral on the right. 5 Section 6: The Flexibility Method - Beams Washkewicz College of Engineering Example 6.1 6 Section 6: The Flexibility Method - Beams Washkewicz College of Engineering Example 6.2 Flexibility Coefficients by virtual work 7 Section 6: The Flexibility Method - Beams Washkewicz College of Engineering Perspectives on the Flexibility Method In 1864 James Clerk Maxwell published the first consistent treatment of the flexibility method for indeterminate structures. His method was based on considering deflections, but the presentation was rather brief and attracted little attention. Ten years later Otto Mohr independently extended Maxwell’s theory to the present day treatment. The flexibility method will sometimes be referred to in the literature as Maxwell-Mohr method. 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. 8 Section 6: The Flexibility Method - Beams Washkewicz College of Engineering 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. 9 Section 6: The Flexibility Method - Beams Washkewicz College of Engineering Example 6.3 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 DB. 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 DB. 5 RB w L 8 10 Section 6: The Flexibility Method - Beams Washkewicz College of Engineering A more general approach consists in finding the displacement at B caused by a unit load in the direction of RB. Then this displacement can be multiplied by RB 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 L3 B 48EI The displacement at B caused by RB is δB RB. The displacement caused by the uniform load w acting on the released structure is 5 w L4 D B 384 EI Thus by the compatibility equation D 5 D R 0 R B w L B B B B 11 B 8 Section 6: The Flexibility Method - Beams Washkewicz College of Engineering Example 6.4 If a structure is statically indeterminate to more than one degree, the approach used in the preceeding 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. 12 Section 6: The Flexibility Method - Beams Washkewicz College of Engineering 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. 13 Section 6: The Flexibility Method - Beams Washkewicz College of Engineering 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 DQ1 DQL1 F11Q1 F12Q2 0 DQ2 DQL2 F21Q1 F22Q2 0 In some cases DQ1 and DQ2 would be nonzero then we would write DQ1 DQL1 F11Q1 F12Q2 DQ2 DQL2 F21Q1 F22Q2 14 Section 6: The Flexibility Method - Beams Washkewicz College of Engineering The equations from the previous page can be written in matrix format as DQ DQL FQ 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 DQ1 DQL1 Q1 DQ DQL Q D D Q2 QL2 Q2 F11 F12 F F21 F22 15 Section 6: The Flexibility Method - Beams Washkewicz College of Engineering The vector {Q} of redundants can be found by solving for them from the matrix equation on the previous overhead. FQ DQ DQL 1 Q F DQ DQL To see how this works consider the previous beam with a constant flexural rigidity EI. If we identify actions on the beam as P1 2P M PL P2 P P3 P Since there are no displacements imposed on the structure corresponding to Q1 and Q2, then 0 DQ 0 16 Section 6: The Flexibility Method - Beams Washkewicz College of Engineering The vector [DQL] represents the displacements in the released structure corresponding to the redundant loads. These displacements are 13PL3 97PL3 D D QL1 24EI QL2 48EI The positive signs indicate that both displacements are upward. In a matrix format PL3 26 DQL 48EI 97 17 Section 6: The Flexibility Method - Beams Washkewicz College of Engineering The flexibility matrix [F ] is obtained by subjecting the beam to unit load corresponding to Q1 and computing the following displacements L3 5L3 F F 11 3EI 21 6EI Similarly subjecting the beam to unit load corresponding to Q2 and computing the following displacements 5L3 8L3 F F 12 6EI 22 3EI 18 Section 6: The Flexibility Method - Beams Washkewicz College of Engineering The flexibility matrix is L3 2 5 F 6EI 5 16 The inverse of the flexibility matrix is 1 6EI 16 5 F 3 7L 5 2 As a final step the redundants [Q] can be found as follows Q1 1 Q F DQ DQL Q2 6EI 16 5 0 PL3 26 3 7L 5 2 0 48EI 97 P 69 56 64 19 Section 6: The Flexibility Method - Beams Washkewicz College of Engineering The redundants have been obtained.
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