## Equations of Conditions

25 If a structure has an internal hinge (which may connect two or more substructures), then this will provide an additional equation (EM = 0 at the hinge) which can be exploited to determine the reactions.

26 Those equations are often exploited in trusses (where each connection is a hinge) to determine reactions.

27 In an inclined roller support with Sx and Sy horizontal and vertical projection, then the reaction R would have, Fig. 12.2.

 Rx Sy Ry Sx
 Sx Ry ■if1'0" Syl Sy Rx i\ Sx / ^—-1

Figure 12.2: Inclined Roller Support 12.2.4 Static Determinacy

28 In statically determinate structures, reactions depend only on the geometry, boundary conditions and loads.

29 If the reactions can not be determined simply from the equations of static equilibrium (and equations of conditions if present), then the reactions of the structure are said to be statically indeterminate.

30 The degree of static indeterminacy is equal to the difference between the number of reactions and the number of equations of equilibrium (plus the number of equations of conditions if applicable), Fig. 12.3.

- 3 equations 3 unknowns statically determinate

-^^ 3 equations 4 unknowns indeterminate to the 1st degree

-O-^ 4 equations 4 unknowns determinate

/r--O-~7\ 4 equations 4 unknowns determinate but unstable

7777 7777

cr--t~L 3 equations 4 unknowns indeterminate

7777 7777 7777

Figure 12.3: Examples of Static Determinate and Indeterminate Structures

31 Failure of one support in a statically determinate system results in the collapse of the structures. Thus a statically indeterminate structure is safer than a statically determinate one.

32 For statically indeterminate structures, reactions depend also on the material properties (e.g. Young's and/or shear modulus) and element cross sections (e.g. length, area, moment of inertia).

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