Lecture 14: TEMPERTURE, PRESTRIN & SUPPORT Effects Of Temperature, Pre-strain & Support Displacement In the previous sections we have only considered loads acting on the structure. We would also like to consider the effects of Temperature changes { } Prestrain of members { } These effects are taken in to account by including them in the calculation of actions in the restrained structure. If the changes are assumed to occur in the restrained structure, there will be actions associated with each in the restrained structure corresponding to the displacements {D}. s in the flexibility method the temperature actions { } in the restrained structure may be due to either uniform changes in temperature or to differential changes in temperature.
Lecture 14: TEMPERTURE, PRESTRIN & SUPPORT When the matrices [ ] and [ ] are found they can be added to the matrix { DL } of actions due to loads in the restrained structure. By superposition { } = { } { } { } [ S]{ D} D DL s before the superposition equation is solved for the vector of displacements {D}. Consider the possibility of known actions occurring at the restraints (or supports) of the structure. There are two possibilities to consider depending on whether the restraint actions corresponds to one of the displacements {D}. If the action does correspond to a displacement, its effect can be taken into account by including the displacement in the vector { D }. In a more general situation there will be actions at restraints that do not correspond to any of the selected displacements. In that event, the effects these actions must be incorporated in the analysis of the restrained structure in a manner similar to temperature displacements and prestrains. When actions occur at a restraint in the restrained structure a new matrix { DS } is introduced.
Lecture 14: TEMPERTURE, PRESTRIN & SUPPORT Thus the sum of all matrices representing displacements in the restrained structure will be denoted by { DS } and is expressed as follows { } = { } { } { } { } DS DL DR The generalized form of the superposition equation becomes { } = { } [ S]{ D} D DS When this expression is inverted to obtain the displacements we find that 1 [ D] = [ S] { [ ] [ ] } D DS
Lecture 14: TEMPERTURE, PRESTRIN & SUPPORT Summary Of Stiffness Method The analysis of a structure by the stiffness method may be described by the following steps: 1. Problem statement 2. Selection of restrained structure 3. nalysis of restrained structure under loads 4. nalysis of restrained structure for other causes 5. nalysis of restrained structure for unit values of displacements 6. Determination of displacements through the superposition equations, i.e., { } = { } [ S]{ D} D DS { } = { } { } { } { } DS DL DR 11 { D } = [ S ] { { } { } } D DS
Lecture 14: TEMPERTURE, PRESTRIN & SUPPORT 7. Determine the other displacements and actions. The following are the two matrix equations for calculating redundants member end actions and reactions { } = { } { }{ D} M MS MD { } = { } { }{ D} R RS RD ll matrices used in the stiffness method are summarized in the following tables
Lecture 14: TEMPERTURE, PRESTRIN & SUPPORT MTRIX ORDER DEFINITION D d x 1 Unknown joint displacements (d = number of displacements) D d x 1 ctions in the actual structure corresponding to the unknown displacements. DL d x 1 ctions in the restrained structure corresponding to the unknown displacements due to external loads. S d x d Member actions in the restrained structure due to unit displacements corresponding to the unknown displacements D.,, DR d x 1 ctions in the restrained structure due to temperature, prestrain, and restraint displacement DS d x 1 = DS DL DR
Lecture 14: TEMPERTURE, PRESTRIN & SUPPORT MTRIX ORDER DEFINITION M ML MD m x 1 Member end actions in the actual structure t (m = number of member end actions) m x 1 Member end actions in the restrained structure due to external loads except those that correspond to the unknown displacements m x d Member end actions in the restrained structure due to unit values of the unknown displacements,, MT MP MR m x 1 Member end actions in the restrained structure due to temperature, prestrain, and restraint displacement MS m x 1 = MS ML MT MP MR
Lecture 14: TEMPERTURE, PRESTRIN & SUPPORT MTRIX ORDER DEFINITION R r x 1 Reactions in the actual structure t (r = number of reactions) RL r x 1 Reactions in the restrained structure due to all external loads except those that correspond to the unknown displacements RD r x d Reactions in the restrained structure due to unit values of the unknown displacements,, RT RP RR r x 1 Reactions in the restrained structure due to temperature, prestrain, and restraint displacement RS r x 1 = RS RL RT RP RR