StaticallyIndeterminateStructures

IntroductionIn the basic studies of Statics, we find that any structure will have, at the most, six equilibrium equations:Fx=0 , Fy=0 , Fz=0 , Mx=0 , My=0 , and Mz=0 For Statically Determinate structures, these six equations can be used to find all unknown forces. But Statically Indeterminate structures, which have six or more unknown forces, require more equations for the structure to be fully analyzed. More equations can be generated by taking into account deformation conditions at the points of support. These equations involve the physical properties of the body, including the strength and stiffness of the materials.

Outline.IntroductionDefinitions-Statically Indeterminate Structures vs. Statically Determinate StructuresAdvantages & DisadvantagesMethods of Analysis Force MethodSlope Deflection MethodExample of Force MethodConclusion & List of Resources

Statical DeterminancyA statically determinate structure can be fully analyzed using only equations of equilibrium. The internal and external forces can be determined independently of the physical properties and geometry of the structure.

Statical IndeterminacyA structure is statically indeterminate when it is subjected to more unknown reactions than there are available equilibrium equations. There are two types of indeterminacy:Internally Indeterminate structures are structures in which the internal forces cannot be determined by statics alone. It should be noted that it is possible for a structure to be both externally determinate and internally indeterminate.Externally Indeterminate structures are structures that have too many, or redundant, support reactions. Because external forces are required to find internal ones, externally indeterminate systems are always internally indeterminate as well.

AdvantagesLighter and more rigid structureAdded redundancy in structural system brings increase in overall safetyMaximum stresses are generally less than comparable statically determinate structuresGreater stiffness than determinate structures which leads to smaller deformationsRedundancies have ability to redistribute loads to other members that have been overstressed or collapsed due to earthquakes, tornadoes, etc.More economicalMajority of tall buildings or concrete bridges are made statically indeterminate

DisadvantagesAnalysis requires the calculation of displacements so that their cross sectional dimensions are required at the outsetTherefore design of such structures becomes a matter of trail and error, whereas the forces in the members of a statically determinate structure are independent of a member size.Support settlements may induce significant stresses compared to the determinate ones where support settlements do not cause stresses.Stresses due to temperature changes and fabrication errors are also createdAn engineer must design accordingly to account for these side affects of indeterminate structures

Force MethodMethod of Consistent DeformationsThe force method of solving statically indeterminate objects involves engineers working with the structure to identify a redundant force. The procedure works as a method for analyzingexternallyindeterminate structures with single or double degrees of indeterminacy.

Force Method(Continued)The choice of the redundants will vary since any of the unknown reactions can be utilized as a redundant. Then by calculating the deformation equations you end up with a Statically determinate structure.

Force MethodMethod of Consistent DeformationsFree Body diagram showing a total of 8 unknowns, an indeterminate frame structure with hinge h, which by using ordinary ridged body equations would not be solvable.This diagram shows the deflection of the structure under the loads p and 2p. This diagram is used in finding where your redundants occur.

Force MethodMethod of Consistent DeformationsCalculate the deformations corresponding to the redundants, i.e., the rotation at Support A,A0, and the translation,B0, at Support B.

Released RestraintsThen remove the support reactions (restraints) corresponding to the selected redundants from the indeterminate structure to obtain aprimary determinate structure, or sometimes referred to as a released structure.

Slope Deflection MethodThe basic idea of the slope deflection method is to consider the deflection as the primary unknowns, unlike the force method which considers the redundant forces to be the primary unknowns. In the slope deflection method a relationship is established between the moments at the ends of the members, and the corresponding displacements and rotations.