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    Force Method for Analysis of Indeterminate Structures

     Number of unknown Reactions or Internal forces > Number of equilibrium equations Note: Most structures in the real world are statically indeterminate.

    Smaller deflections for similar members

    Redundancy in load carrying capacity(redistribution)

    Increased stability

    Advantages Disadvantages

    More material => More Cost

    Complex connections

    Initial / Residual / Settlement Stresses

    Methods of Analysis

    (i) Equilibrium of forces and moments(ii) Compatibility of deformation among members and at supports(iii) Material behavior relating stresses with strains(iv) Strain-displacement relations(v) Boundary Conditions

    Structural Analysis requires that the equations governing the following physical relationships be sat

    Primarily two types of methods of analysis:

    (Ref: Chapter

    Displacement (Stiffness) Method

    Express local (member) force-displacemerelationships in terms of unknown membdisplacements.

    Using equilibrium of assembled memberfind unknown displacements.

    Unknowns are usually displacements

    Coefficients of the unknowns are "Stiffnecoefficients.

    Convert the indeterminate structure to adeterminate one by removing some unknownforces / support reactions and replacing themwith (assumed) known / unit forces.

    Using superposition, calculate the force thatwould be required to achieve compatibilitywith the original structure.

    Unknowns to be solved for are usuallyredundant forces

    Coefficients of the unknowns in equations to be solved are "flexibility" coefficients.

    Force (Flexibility) Method

    For determinate structures, the force method allows us to find internal forces (using equilibrium based on Statics) irrespective of the material information. Material (stress-strain) relationships arneeded only to calculate deflections.

    However, for indeterminate structures , Statics (equilibrium) alone is not sufficient to conductstructural analysis. Compatibility and material information are essential.

    Indeterminate Structures

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    Maxwell's Theorem of Reciprocal displacements; Betti's law

    Betti's Theorem

    For structures with multiple degree of indeterminacy

    Example:

    The displacement (rotation) at a point P in a structure due a UNIT load (moment) at point Q is eto displacement (rotation) at a point Q in a structure due a UNIT load (moment) at point P.

    Virtual Work done by a system of forces  P B while undergoingdisplacements due to system of forces  P Ais equal to theVirtual Work done by the system of forces  P A whileundergoing displacements due to the system of forces  P B

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    Force Method of Analysis for (Indeterminate) Beams and Frames

    Example: Determine the reactions.

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    Examples

    Support B settles by 1.5 in.Find the reactions and draw the Shear Force andBending Moment Diagrams of the beam.

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    Example: Frames

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    Force Method of Analysis for (Indeterminate) Trusses

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    Force Method of Analysis for (Inderminate) Composite Structures

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    Part 1 Part 2

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    Systematic Analysis using the Force (Flexibility) Method

     Note: Maxwell's Theorem (Betti's Law)=> Flexibility matrix is symmetric!

    Example:

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    Analysis of Symmetric structures

    Symmetry: Structure, Boundary Conditions, and Loads aresymmetric.

    Anti-symmetric: Structure, Boundary Conditions are symmetric,Loads are anti-symmetric.

    Symmetry helps in reducing the number of unknowns to solve for.

    Examples:

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    Influence lines for  Determinate structures(Ref: Chapter 6

    Influence line is a diagram that shows the variation for a particular force/moment at specific locin a structure as a unit load moves across the entire structure.

    The influence of a certain force (or moment) in a structure is given by (i.e. it is equal to) thedeflected shape of the structure in the absence of that force (or moment) and when given acorresponding unit displacement (or rotation).

    Müller-Breslau Principle

    Example:

    Examples:

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    Draw the influence lines for the reaction and bending-moment at point C for the following beam.Example

    A B C D

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    Draw the influence lines for the shear-force and bending-moment at point C for the following beam

    Find the maximum bending moment at C due to a 400 lb load moving across the beam.

    Example

    A B C

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    Influence lines for  Indeterminate structures

    For statically determinate structures, influence lines are straight.For statically indeterminate structures, influence lines are usually curved.

    The Müller-Breslau principle also holds for indeterminate structures.

    Examples:

    Reaction Ay Shear at a point: Moment at a point

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    Vertical reaction at AMoment at A

    Draw the influence line forExample


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