Preliminaries: Beam Deflections Virtual Work

1 Section 6: The Flexibility Method -BeamsWashkewicz College of Engineering1 preliminaries : beam Deflections Virtual WorkThere 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 methodsSince 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 theory that supports calculating Deflections using Virtual work will be reviewed and several examples are 6.

2 The Flexibility Method -BeamsWashkewicz College of Engineering2 Consider the following arbitrarily loaded beamIdentifyactionunit a todue on acting Stress~actionunit a todue beam in thesection any at M oment loads external todue beam in thesection any at M oment dAIymm(x)mM(x)M Section 6: The Flexibility Method -BeamsWashkewicz College of Engineering3 The force acting on the differential area dAdue to a unit action isThe stress due to external loads isThe displacement of a differential segment dAby dxalong the length of the beam isdAIymdAf ~~IyM dxIEyMdxEdx Section 6: The Flexibility Method -BeamsWashkewicz College of Engineering4 The work done by the force acting on the differential area dAdue to a unit action as the differential segment of the beam (dAby dx)displaces along the length of the beam by an amount isThe work done within a differential segment (now A by dx) due to a unit action applied to the beamis the integration of the expression above with respect to dA, ,dxdAIEymMdxIEyMdAIymfdW 22~ dxEIMmdxIEIM mdxdAyEIMmWdxdAIEymMdWTBTB ccsegmentaldifferneticcA 22222 Section 6.

3 The Flexibility Method -BeamsWashkewicz College of Engineering5 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, , The external work done along the entire length of the beam due to a unit action applied to the beam isWithor 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. dxEIxmxMWLI nternal 0 D 1 ExternalW dxEIxmxMdxEIxmxMWWLLI nternalExternal D D 001 Section 6: The Flexibility Method -BeamsWashkewicz College of EngineeringExample 6: The Flexibility Method -BeamsWashkewicz College of EngineeringExample Coefficients by Virtual workSection 6: The Flexibility Method -BeamsWashkewicz College of EngineeringPerspectives on the Flexibility MethodIn 1864 James Clerk Maxwell published the first consistent treatment of the flexibility method for indeterminate structures.

4 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 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.

5 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 -BeamsWashkewicz College of EngineeringThe fundamental concepts that underpin the flexibility method will be illustrated by the study of a two span beam . The procedure is as a sufficient number of redundants corresponding to the degree of the displacements at the redundants on released structure due to external or imposed displacements due to unit loads at the redundants on the released equation of compatibility, , 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 6: The Flexibility Method -BeamsWashkewicz College of EngineeringThe beam to the left is statically indeterminate to the first degree.

6 The reaction at the middle support RBis chosen as the redundant. The released beam is also shown. Under the external loads the released beam deflects an amount 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 8510 Example 6: The Flexibility Method -BeamsWashkewicz College of EngineeringA more general approach consists in finding the displacement at Bcaused by a unit load in the direction of RB. Then this displacement can be multiplied by RBto determine the total displacementAlso in a more general approach a consistent sign convention for actions and displacements must be adopted.

7 The displacements in the released structure at Bare positive when they are in the direction of the action released, , upwards is positive displacement at Bcaused by the unit action isThe displacement at Bcaused by RBis displacement caused by the uniform load wacting on the released structure is Thus by the compatibility equationEILB483 EILwB38454 DLwRRBBBBBB D D850 11 Section 6: The Flexibility Method -BeamsWashkewicz College of EngineeringIf 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 the beam to the left.

8 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 6: The Flexibility Method -BeamsWashkewicz College of EngineeringThe redundants chosen are at Band redundant reactions are designated Q1and released structure is shown at the left with all external and internal redundants the displacement corresponding to Q1and caused by only external actions on the released structureDQL2is the displacement corresponding to Q2caused by only external actions on the released displacements are shown in their assumed positive 6.

9 The Flexibility Method -BeamsWashkewicz College of EngineeringWe can now write the compatibility equations for this structure. The displacements corresponding to Q1and Q2will be zero. These are labeled DQ1and DQ2respectivelyIn some cases DQ1and DQ2would be nonzero then we would write021211111 QFQFDDQLQ022212122 QFQFDDQLQ21211111 QFQFDDQLQ 22212122 QFQFDDQLQ 14 Section 6: The Flexibility Method -BeamsWashkewicz College of Engineeringwhere:{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 21 QQQDDD QFDDQLQ 21 QLQLQLDDD 22211211 FFFFF 21 QQQThe equations from the previous page can be written in matrix format as15 Section 6.

10 The Flexibility Method -BeamsWashkewicz College of EngineeringThe vector {Q}of redundants can be found by solving for them from the matrix equation on the previous see how this works consider the previous beam with a constant flexural rigidity EI. If we identify actions on the beam asSince there are no displacements imposed on the structure corresponding to Q1and Q2, then QLQDDQF QLQDDFQ 1 PPPPPLMPP 3212 00QD16 Section 6: The Flexibility Method -BeamsWashkewicz College of EngineeringThe vector [DQL] represents the displacements in the released structure corresponding to the redundant loads. These displacements areThe positive signs indicate that both displacements are upward.