Additional equations come from Introduction to Introduction to Statically Statically c ompatibility relationships , Indeterminate Analysis Indeterminate Analysis which ensure continuity of displacements throughout the structure . The remaining Support reactions and internal equations are constructed from forces of statically determinate member constitutive equations , structures can be determined i.e., relationships between using only the equations of stresses and strains and the equilibrium . However, the integration of these equations analysis of statically indeter- y y o er the cross section over the cross section . minate structures requires additional equations based on the geometry of deformation of the structure . 1 2 Another consequence of Design of an indeterminate statically indeterminate structure is carried out in an structures is that the relative iterative manner , whereby the variation of member sizes (relative) sizes of the structural influences the magnitudes of members are initially assumed the forces that the member the forces that the member and used to analyze the structure. d d l h will experience . Stated in Based on the computed results another way, stiffness ( large (displacements and internal member size and/or high member forces), the member modulus materials ) attracts sizes are adjusted to meet force . governing design criteria. This iteration process continues until iteration process continues until Despite these difficulties with Despite these difficulties with the member sizes based on the statically indeterminate results of an analysis are close to structures, an overwhelming those assumed for that analysis. majority of structures being built today are statically 3 indeterminate . 4 Also see pages 78 - 100 in your class notes. 1
Advantages Statically Indeterminate Structures Statically indeterminate structures typically result in smaller stresses and greater stiffness (smaller deflections) 5 as illustrated for this beam. 6 Statically indeterminate structures introduce redundancy, which may insure that failure in Determinate beam is unstable one part of the structure will not if middle support is removed result in catastrophic or collapse or knocked off! failure of the structure. 7 8 2
Disadvantages of Statically Indeterminate Structures Statically indeterminate structure is self-strained due to support settlement, which produces stresses, as illustrated above. 9 10 Indeterminate Structures: Influence Lines Influence lines for statically indeterminate structures provide the same information provide the same information as influence lines for statically determinate structures, i.e. it represents the magnitude of a response function at a particular location on the structure as a unit load moves structure as a unit load moves across the structure. Statically indeterminate struc- tures are also self-strained due to temperature changes and fabrication errors. 11 12 3
Qualitative Influence Our goals in this chapter are: Lines for Statically Inde- 1.To become familiar with the terminate Structures: shape of influence lines for the Muller-Breslau’s Principle support reactions and internal forces in continuous beams forces in continuous beams In many practical applications, it I i l li i i and frames. is usually sufficient to draw only 2.To develop an ability to sketch the qualitative influence lines to the appropriate shape of decide where to place the live influence functions for loads to maximize the response indeterminate beams and functions of interest. The frames. a es Muller Breslau Principle pro Muller-Breslau Principle pro- 3.To establish how to position vides a convenient mechanism distributed live loads on to construct the qualitative continuous structures to influence lines, which is stated maximize response function as: values. 13 14 Procedure for constructing qualitative influence lines for The influence line for a force (or indeterminate structures is: (1) moment) response function is remove from the structure the given by the deflected shape of restraint corresponding to the p g the released structure by the released structure by response function of interest, (2) removing the displacement apply a unit displacement or constraint corresponding to the rotation to the released structure response function of interest at the release in the desired from the original structure and response function direction, and giving a unit displacement (or (3) draw the qualitative deflected rotation) at the location and in rotation) at the location and in shape of the released structure the direction of the response consistent with all remaining function. support and continuity conditions. 15 16 4
Notice that this procedure is Uniformly distributed live identical to the one discussed for loads are placed over the statically determinate structures. positive areas of the ILD to maximize the drawn response However , unlike statically function values . Because the function values . Because the d t determinate structures, the i t t t th influence line ordinates tend to influence lines for statically diminish rapidly with distance indeterminate structures are from the response function typically curved. location, live loads placed more than three span lengths away can be ignored. Once the live Placement of the live loads to Placement of the live loads to load pattern is known, an maximize the desired response indeterminate analysis of the function is obtained from the structure can be performed to qualitative ILD. determine the maximum value of the response function. 17 18 QILD for R A QILD’s for R C and V B 19 20 5
Live Load Pattern to Maximize Forces in Multistory Buildings Building codes specify that members of multistory b f lti t buildings be designed to support a uniformly distributed live load as well as the dead load of the structure. Dead and live loads are normally considered separately since considered separately since the dead load is fixed in position whereas the live load QILD’s for (M C ) - , must be varied to maximize a (M D ) + and R F particular force at each section of the structure. Such 21 22 maximum forces are typically produced by 3. Axial column force (do not patterned loading . consider axial force in beams): Qualitative Influence Lines: (a) Sketch the beam line qualitative displacement lit ti di l t 1. Introduce appropriate unit diagrams. displacement at the desired (b) Sketch the column line response function location. qualitative displacement 2. Sketch the displacement diagrams maintaining equality diagram along the beam or of the connection geometry column line (axial force in column line (axial force in before and after deformation. column) appropriate for the unit displacement and assume zero axial deformation. 23 24 6
4. Beam force: (c) Sketch remaining beam line qualitative displacement (a) Sketch the beam line diagrams maintaining con- qualitative displacement nection geometry before and diagram for which the release after deformation. after deformation. has been introduced has been introduced. (b) Sketch all column line qualitative displacement diagrams maintaining connection geometry before and after deformation. Start the column line qualitative the column line qualitative displacement diagrams from the beam line diagram of (a). 25 26 Load Pattern to Vertical Maximize F Reaction F Load Pattern to Column Moment Maximize M M 27 28 7
M QILD and Load Pattern for QILD and Load Pattern for End Beam Moment M Center Beam Moment M Expanded Detail for Beam End Moment 29 30 Envelope Curves Qualitative influence lines for positive moments are given, Design engineers often use shear influence lines are influence lines to construct shear presented later. Based on the and moment envelope curves for p qualitative influence lines, critical qualitative influence lines, critical continuous beams in buildings or live load placement can be for bridge girders. An envelope determined and a structural curve defines the extreme analysis computer program can boundary values of shear or be used to calculate the member bending moment along the beam end shear and moment values due to critical placements of for the dead load case and the design live loads. For example, critical live load cases. consider a three-span continuous beam. 31 32 8
a b c d e a b c d e 1 2 3 4 1 2 3 4 QILD for (M c ) + Three-Span Continuous Beam a b c d e a b c d e 1 2 3 4 1 2 3 4 QILD for (M a ) + QILD for (M d ) + a b c d e a b c d e 1 2 3 4 1 2 3 4 QILD for (M b ) + QILD for (M e ) + 33 34 a b c d e a b c d e 1 2 3 4 1 2 3 4 Critical Live Load Placement Critical Live Load Placement Critical Live Load Placement Critical Live Load Placement for (M a ) + for (M b ) + a b c d e a b c d e 1 2 3 4 1 2 3 4 Critical Live Load Placement Critical Live Load Placement for (M a ) - for (M b ) - 35 36 9
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