Structural Analysis 7 th Edition in SI Units Russell C. Hibbeler Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Statically Indeterminate Structures Advantages & DisadvantagesAdvantages & Disadvantages For a given loading, the max stress and deflection of an indeterminate structure are generally smaller than those of its statically determinate counterpartFor a given loading, the max stress and deflection of an indeterminate structure are generally smaller than those of its statically determinate counterpart Statically indeterminate structure has a tendency to redistribute its load to its redundant supports in cases of faulty designs or overloadingStatically indeterminate structure has a tendency to redistribute its load to its redundant supports in cases of faulty designs or overloading © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Statically Indeterminate Structures Advantages & DisadvantagesAdvantages & Disadvantages Although statically indeterminate structure can support loading with thinner members & with increased stability compared to their statically determinate counterpart, the cost savings in material must be compared with the added cost to fabricate the structure since often it becomes more costly to construct the supports & joints of an indeterminate structureAlthough statically indeterminate structure can support loading with thinner members & with increased stability compared to their statically determinate counterpart, the cost savings in material must be compared with the added cost to fabricate the structure since often it becomes more costly to construct the supports & joints of an indeterminate structure Careful of differential disp of the supports as wellCareful of differential disp of the supports as well © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Statically Indeterminate Structures Method of AnalysisMethod of Analysis To satisfy equilibrium, compatibility & force-disp requirements for the structureTo satisfy equilibrium, compatibility & force-disp requirements for the structure Force MethodForce Method Displacement MethodDisplacement Method © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Force Method of Analysis: General Procedure From free-body diagram, there would be 4 unknown support reactionsFrom free-body diagram, there would be 4 unknown support reactions 3 equilibrium eqn3 equilibrium eqn Beam is indeterminate to first degreeBeam is indeterminate to first degree Use principle of superposition & consider the compatibility of disp at one of the supportsUse principle of superposition & consider the compatibility of disp at one of the supports Choose one of the support reactions as redundant & temporarily removing its effect on the beamChoose one of the support reactions as redundant & temporarily removing its effect on the beam © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Force Method of Analysis: General Procedure This will allow the beam to be statically determinate & stableThis will allow the beam to be statically determinate & stable Here, we will remove the rocker at BHere, we will remove the rocker at B As a result, the load P will cause B to be displaced downwardAs a result, the load P will cause B to be displaced downward By superposition, the unknown reaction at B causes the beam at B to be displaced upwardBy superposition, the unknown reaction at B causes the beam at B to be displaced upward © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Force Method of Analysis: General Procedure Assuming +ve disp act upward, we write the necessary compatibility eqn at the rocker as:Assuming +ve disp act upward, we write the necessary compatibility eqn at the rocker as: © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Force Method of Analysis: General Procedure Using methods in Chapter 8 or 9 to solve for  B and f BB, B y can be foundUsing methods in Chapter 8 or 9 to solve for  B and f BB, B y can be found Reactions at wall A can then be determined from eqn of equilibriumReactions at wall A can then be determined from eqn of equilibrium The choice of redundant is arbitraryThe choice of redundant is arbitrary © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Force Method of Analysis: General Procedure The moment at A can be determined directly by removing the capacity of the beam to support moment at A, replacing fixed support by pin supportThe moment at A can be determined directly by removing the capacity of the beam to support moment at A, replacing fixed support by pin support The rotation at A caused by P is  AThe rotation at A caused by P is  A The rotation at A caused by the redundant M A at A is  ’ AAThe rotation at A caused by the redundant M A at A is  ’ AA © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Force Method of Analysis: General Procedure © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Maxwell ’ s Theorem of Reciprocal Disp: Betti ’ s Law The disp of a point B on a structure due to a unit load acting at point A is equal to the disp of point A when the load is acting at point BThe disp of a point B on a structure due to a unit load acting at point A is equal to the disp of point A when the load is acting at point B Proof of this theorem is easily demonstrated using the principle of virtual workProof of this theorem is easily demonstrated using the principle of virtual work © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Maxwell ’ s Theorem of Reciprocal Disp: Betti ’ s Law The theorem also applies for reciprocal rotationsThe theorem also applies for reciprocal rotations The rotation at point B on a structure due to a unit couple moment acting at point A is equal to the rotation at point A when the unit couple is acting at point BThe rotation at point B on a structure due to a unit couple moment acting at point A is equal to the rotation at point A when the unit couple is acting at point B © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Determine the reaction at the roller support B of the beam. EI is constant. Example 10.1 © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Principle of superposition By inspection, the beam is statically indeterminate to the first degree. The redundant will be taken as B y. We assume B y acts upward on the beam. Solution © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Compatibility equation Solution © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Draw the shear and moment diagrams for the beam. EI is constant. Neglect the effects of axial load. Example 10.4 © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Principle of Superposition Since axial load is neglected, the beam is indeterminate to the second degree. The 2 end moments at A & B will be considered as the redundant. The beam ’ s capacity to resist these moments is removed by placing a pin at A and a rocker at B. Solution © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Compatibility eqn Reference to points A & B requires The required slopes and angular flexibility coefficients can be determined using standard tables. Solution © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Compatibility eqn Solution © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Composite Structures Composite structures are composed of some members subjected only to axial force while other members are subjected to bendingComposite structures are composed of some members subjected only to axial force while other members are subjected to bending If the structure is statically indeterminate, the force method can conveniently be used for its analysisIf the structure is statically indeterminate, the force method can conveniently be used for its analysis © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

The beam is supported by a pin at A & two pin-connected bars at B. Determine the force in member BD. Take E = 200GPa & I = 300(10 6 )mm 4 for the beam and A = 1800mm 2 for each bar. Example © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Principle of superposition The beam is indeterminate to the first degree. Force in member BD is chosen as the redundant. This member is therefore sectioned to eliminate its capacity to sustain a force. Solution © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Compatibility eqn With reference to the relative disp of the cut ends of member BD, we require The method of virtual work will be used to compute  BD and f BDBD Solution © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Compatibility eqn For  BD we require application of the real loads and a virtual unit load applied to the cut ends of the member BD. We will consider only bending strain energy in the beam & axial strain energy in the bar. Solution © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Compatibility eqn For f BDBD we require application of a real unit load & a virtual unit load at the cut ends of member BD. Solution © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Compatibility eqn Sub into eqn (1) yields Solution © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Additional remarks on the force method of analysis Flexibility coefficients depend on the material and geometrical properties of the members and not on the loading of the primary structureFlexibility coefficients depend on the material and geometrical properties of the members and not on the loading of the primary structure For a structure having n redundant reactions, we can write n compatibility eqnFor a structure having n redundant reactions, we can write n compatibility eqn © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Additional remarks on the force method of analysis  BD are caused by both the real loads on the primary structure and by support settlement & dimensional changes due to temperature differences or fabrication errors in the members  BD are caused by both the real loads on the primary structure and by support settlement & dimensional changes due to temperature differences or fabrication errors in the members The above eqn can be re-cast into a matrix form or simply:The above eqn can be re-cast into a matrix form or simply: Note that f ij =f jiNote that f ij =f ji Hence, the flexibility matrix will be symmetricHence, the flexibility matrix will be symmetric © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Symmetric Structures A structural analysis of any highly indeterminate structure or statically determinate structure can be simplified provided the designer can recognise those structures that are symmetric & support either symmetric or antisymmetric loadingsA structural analysis of any highly indeterminate structure or statically determinate structure can be simplified provided the designer can recognise those structures that are symmetric & support either symmetric or antisymmetric loadings For horizontal stability, a pin is required to support the beam & truss.For horizontal stability, a pin is required to support the beam & truss. © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Symmetric Structures Here the horizontal reaction at the pin is zero, so both these structures will deflect & produce the same internal loading as their reflected counterpartHere the horizontal reaction at the pin is zero, so both these structures will deflect & produce the same internal loading as their reflected counterpart As a result, they can be classified as being symmetricAs a result, they can be classified as being symmetric Not the case if the fixed support at A was replaced by a pin since the deflected shape & internal loadings would not be the same on its left & right sideNot the case if the fixed support at A was replaced by a pin since the deflected shape & internal loadings would not be the same on its left & right side © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Symmetric Structures A symmetric structure supports an antisymmetric loading as shownA symmetric structure supports an antisymmetric loading as shown Provided the structure is symmetric & its loading is either symmetric or antisymmetric then a structural analysis will only have to be performed on half the members of the structure since the same or opposite results will be produced on the other halfProvided the structure is symmetric & its loading is either symmetric or antisymmetric then a structural analysis will only have to be performed on half the members of the structure since the same or opposite results will be produced on the other half © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Symmetric Structures A separate structural analysis can be performed using the symmetrical & antisymmetrical loading components & the results superimposed to obtain the actual behaviour of the structureA separate structural analysis can be performed using the symmetrical & antisymmetrical loading components & the results superimposed to obtain the actual behaviour of the structure © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Influence lines for Statically Indeterminate Beams For statically determinate beams, the deflected shapes will be a series of straight line segmentsFor statically determinate beams, the deflected shapes will be a series of straight line segments For statically indeterminate beams, curve will resultFor statically indeterminate beams, curve will result Reaction at AReaction at A To determine the influence line for the reaction at A, a unit load is placed on the beam at successive pointsTo determine the influence line for the reaction at A, a unit load is placed on the beam at successive points At each point, the reaction at A must be computedAt each point, the reaction at A must be computed © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Influence lines for Statically Indeterminate Beams Reaction at AReaction at A A plot of these results yields the influence lineA plot of these results yields the influence line The reaction at A can be determined by the force methodThe reaction at A can be determined by the force method The principle of superposition is appliedThe principle of superposition is applied The compatibility eqn for point A is:The compatibility eqn for point A is: © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Influence lines for Statically Indeterminate Beams Reaction at AReaction at A A plot of these results yields the influence lineA plot of these results yields the influence line The reaction at A can be determined by the force methodThe reaction at A can be determined by the force method The compatibility eqn for point A is:The compatibility eqn for point A is: By Maxwell ’ s theorem of reciprocal dispBy Maxwell ’ s theorem of reciprocal disp © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Influence lines for Statically Indeterminate Beams Shear at EShear at E Using the force method & Maxwell ’ s theorem of reciprocal disp, it can be shown thatUsing the force method & Maxwell ’ s theorem of reciprocal disp, it can be shown that © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Influence lines for Statically Indeterminate Beams Moment at EMoment at E The influence line for the moment at E can be determined by placing a pin or hinge at EThe influence line for the moment at E can be determined by placing a pin or hinge at E Applying a +ve unit couple moment, the beam then deflects to the dashed positionApplying a +ve unit couple moment, the beam then deflects to the dashed position Using the force method & Maxwell ’ s theorem of reciprocal disp, it can be shown thatUsing the force method & Maxwell ’ s theorem of reciprocal disp, it can be shown that © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Influence lines for Statically Indeterminate Beams Moment at EMoment at E The influence line for the moment at E can be determined by placing a pin or hinge at EThe influence line for the moment at E can be determined by placing a pin or hinge at E Applying a +ve unit couple moment, the beam then deflects to the dashed positionApplying a +ve unit couple moment, the beam then deflects to the dashed position Using the force method & Maxwell ’ s theorem of reciprocal disp, it can be shown thatUsing the force method & Maxwell ’ s theorem of reciprocal disp, it can be shown that © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Qualitative Influence lines for Frames The shape of the influence line for the +ve moment at the center I of girder FG of the frame is shown by the dashed linesThe shape of the influence line for the +ve moment at the center I of girder FG of the frame is shown by the dashed lines Uniform loads would be placed only on girders AB, CD & FG in order to create the largest +ve moment at IUniform loads would be placed only on girders AB, CD & FG in order to create the largest +ve moment at I © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Draw the influence line for the vertical reaction at A for the beam. EI is constant. Plot numerical values every 2m Example © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

The capacity of the beam to resist reaction A y is removed. This is done using a vertical roller device. Applying a vertical unit load at A yields the shape of the influence line. Using the conjugate beam method to determine ordinates of the influence line. Solution © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Solution © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Solution © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Since a vertical 1kN load acting at A on the beam will cause a vertical reaction at A of 1kN, the disp at A,  A should correspond to a numerical value of 1 for the influence line ordinate at A. Thus dividing the other computed disp by this factor, we obtain Solution © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method xAyAy A1 C0.852 D0.481 B0