Introduction to Statically Introduction to Statically Indeterminate Indeterminate Analysis Indeterminate Indeterminate Analysis Analysis nalysis S pport reactions and internal Support reactions and internal forces of statically determinate structures can be determined structures can be determined using only the equations of equilibrium . However, the analysis of statically indeter- minate structures requires additional equations based on additional equations based on the geometry of deformation of the structure . the structure . 1
Addi i Additional equations come from l i f c ompatibility relationships , which ensure continuity of which ensure continuity of displacements throughout the structure . The remaining g equations are constructed from member constitutive equations , i i.e., relationships between l ti hi b t stresses and strains and the integration of these equations integration of these equations over the cross section . 2
Design of an indeterminate structure is carried out in an t t i i d t i iterative manner , whereby the (relative) sizes of the structural (relative) sizes of the structural members are initially assumed and used to analyze the structure. Based on the computed results (displacements and internal member forces), the member b f ) th b sizes are adjusted to meet governing design criteria governing design criteria. This This iteration process continues until the member sizes based on the results of an analysis are close to those assumed for that analysis. 3
Another consequence of statically indeterminate statically indeterminate structures is that the relative variation of member sizes influences the magnitudes of the forces that the member will experience will experience . Stated in Stated in another way, stiffness ( large member size and/or high e be s e a d/o g modulus materials ) attracts force . Despite these difficulties with statically indeterminate structures, an overwhelming majority of structures being b ilt t d built today are statically t ti ll indeterminate . 4
Advantages Statically I d t Indeterminate Structures i t St t 5
Statically indeterminate structures typically result in smaller stresses and greater stiffness (smaller deflections) tiff ( ll d fl ti ) as illustrated for this beam. 6
Determinate beam is Determinate beam is unstable nstable if middle support is removed or knocked off! or knocked off! 7
Staticall indeterminate Statically indeterminate structures introduce redundancy, which may insure that failure in which may insure that failure in one part of the structure will not result in catastrophic or collapse failure of the structure. 8
Disadvantages of St ti Statically Indeterminate ll I d t i t Structures 9
Statically indeterminate structure is self strained due to support is self-strained due to support settlement, which produces stresses, as illustrated above. st esses, as ust ated abo e 10
Statically indeterminate struc- tures are also self-strained due to temperature changes and f b i fabrication errors. ti 11
Indeterminate Structures: I fl Influence Lines Li Influence lines for statically y indeterminate structures provide the same information as influence lines for statically determinate structures, i.e. it represents the magnitude of a represents the magnitude of a response function at a particular location on the p structure as a unit load moves across the structure. 12
Our goals in this chapter are: 1.To become familiar with the shape of influence lines for the support reactions and internal t ti d i t l forces in continuous beams and frames and frames. 2.To develop an ability to sketch the appropriate shape of the appropriate shape of influence functions for indeterminate beams and frames. 3.To establish how to position distributed live loads on continuous structures to maximize response function maximize response function values. 13
Qualitative Influence Lines for Statically Inde- Li f St ti ll I d terminate Structures: Muller-Breslau’s Principle Muller Breslau’s Principle In many practical applications, it is usually sufficient to draw only the qualitative influence lines to decide where to place the live d id h t l th li loads to maximize the response functions of interest functions of interest. The The Muller-Breslau Principle pro- vides a convenient mechanism to construct the qualitative influence lines, which is stated as as: 14
The influence line for a force (or moment) response function is ) p given by the deflected shape of the released structure by removing the displacement i th di l t constraint corresponding to the response function of interest response function of interest from the original structure and giving a unit displacement (or g g p ( rotation) at the location and in the direction of the response f function. ti 15
Procedure for constructing qualitative influence lines for indeterminate structures is: (1) remove from the structure the f th t t th restraint corresponding to the response function of interest (2) response function of interest, (2) apply a unit displacement or rotation to the released structure at the release in the desired response function direction, and (3) dra (3) draw the qualitative deflected the q alitati e deflected shape of the released structure consistent with all remaining consistent with all remaining support and continuity conditions. 16
Notice that this procedure is p identical to the one discussed for statically determinate structures. However , unlike statically determinate structures, the influence lines for statically indeterminate structures are t picall c r ed typically curved. Placement of the live loads to maximize the desired response function is obtained from the qualitative ILD. 17
Uniformly distributed live y loads are placed over the positive areas of the ILD to maximize the drawn response function values . Because the influence line ordinates tend to influence line ordinates tend to diminish rapidly with distance from the response function p location, live loads placed more than three span lengths away can be ignored. Once the live b i d O h li load pattern is known, an indeterminate analysis of the indeterminate analysis of the structure can be performed to determine the maximum value of the response function. 18
QILD for R A 19
QILD’s for R C and V B 20
QILD’s for (M C ) - , (M D ) + and R F ( D ) a d F 21
Live Load Pattern to Maximize Forces in Multistory Buildings Building codes specify that members of multistory buildings be designed to support a uniformly distributed live load as well as the dead live load as well as the dead load of the structure. Dead and live loads are normally and live loads are normally considered separately since the dead load is fixed in position whereas the live load must be varied to maximize a particular force at each section particular force at each section of the structure. Such 22
maximum forces are typically produced by patterned loading . Qualitative Influence Lines: 1 Introduce appropriate unit 1. Introduce appropriate unit displacement at the desired response function location. p 2. Sketch the displacement diagram along the beam or g g column line (axial force in column) appropriate for the unit displacement and it di l t d assume zero axial deformation deformation. 23
3. Axial column force (do not consider axial force in beams): (a) Sketch the beam line qualitative displacement diagrams. (b) Sketch the column line ( ) qualitative displacement diagrams maintaining equality of the connection geometry f th ti t before and after deformation. 24
4. Beam force: (a) Sketch the beam line qualitative displacement q p diagram for which the release has been introduced. (b) Sketch all column line qualitative displacement diagrams maintaining diagrams maintaining connection geometry before and after deformation and after deformation. Start Start the column line qualitative displacement diagrams from the beam line diagram of (a). 25
(c) Sketch remaining beam ( ) g line qualitative displacement diagrams maintaining con- nection geometry before and after deformation. 26
27
Load Pattern to Vertical Maximize F Reaction F Load Pattern to Load Pattern to Column Moment Column Moment Maximize M M 28
QILD and Load Pattern for Center Beam Moment M 29
M QILD and Load Pattern for End Beam Moment M Expanded Detail p for Beam End Moment 30
Envelope Curves Design engineers often use i fl influence lines to construct shear li t t t h and moment envelope curves for continuous beams in buildings or continuous beams in buildings or for bridge girders. An envelope curve defines the extreme boundary values of shear or bending moment along the beam d due to critical placements of t iti l l t f design live loads. For example, consider a three-span consider a three span continuous beam. 31
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