Chapter 10 Analysis of Statically Indeterminate Structures by

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

Statically Indeterminate Structures • Advantages & Disadvantages • For 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 overloading Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition

Statically Indeterminate Structures • Advantages & 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 structure • Careful of differential disp of the supports as well Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition

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

Force Method of Analysis: General Procedure • From free-body diagram, there would be 4 unknown support reactions • 3 equilibrium eqn • Beam is indeterminate to first degree • Use 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 beam Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition

Force Method of Analysis: General Procedure • This will allow the beam to be statically determinate & stable • Here, we will remove the rocker at B • As 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 upward Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition

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

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

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 support • The rotation at A caused by P is A • The rotation at A caused by the redundant MA at A is ’AA Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition

Force Method of Analysis: General Procedure Chapter 10: Analysis of Statically Indeterminate Structures by

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 B • Proof of this theorem is easily demonstrated using the principle of virtual work Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition

Maxwell’s Theorem of Reciprocal Disp: Betti’s Law • The 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 B Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition

Example 10. 1 Determine the reaction at the roller support B of the beam. EI is constant. Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition

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

Solution Compatibility equation Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method

Example 10. 4 Draw the shear and moment diagrams for the beam. EI is constant. Neglect the effects of axial load. Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition

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

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

Solution Compatibility eqn 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 bending • If the structure is statically indeterminate, the force method can conveniently be used for its analysis Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition

Example 10. 10 The beam is supported by a pin at A & two pin-connected bars at B. Determine the force in member BD. Take E = 200 GPa & I = 300(106)mm 4 for the beam and A = 1800 mm 2 for each bar. Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition

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

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

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

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

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

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 structure • For a structure having n redundant reactions, we can write n compatibility eqn Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition

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 • The above eqn can be re-cast into a matrix form or simply: • Note that fij=fji • Hence, the flexibility matrix will be symmetric Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition

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 loadings • For horizontal stability, a pin is required to support the beam & truss. Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition

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

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

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

Influence lines for Statically Indeterminate Beams • For statically determinate beams, the deflected shapes will be a series of straight line segments • For statically indeterminate beams, curve will result • Reaction at A • To 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 computed Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition

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

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

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

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

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 lines • Uniform loads would be placed only on girders AB, CD & FG in order to create the largest +ve moment at I Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition

Example 10. 11 Draw the influence line for the vertical reaction at A for the beam. EI is constant. Plot numerical values every 2 m Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition

Solution The capacity of the beam to resist reaction Ay 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. Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition

Solution Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method © 2009

Solution • Since a vertical 1 k. N load acting at A on the beam will cause a vertical reaction at A of 1 k. N, 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 x A C D B Ay 1 0. 852 0. 481 0 Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method © 2009 Pearson Education South Asia Pte Ltd Structural Analysis 7 th Edition