Statical Indeterminacy

Any plane-frame structure which is in a state of equilibrium under the action of an externally applied force system must satisfy the following three conditions:

• the sum of the horizontal components of all applied forces must equal zero,

• the sum of the vertical components of all applied forces must equal zero,

• the sum of the moments (about any point in the plane of the frame) of all applied forces must equal zero.

This is represented by the following 'three equations of static equilibrium'

Sum of the horizontal forces equals zero

Sum of the vertical forces equals zero

Sum of the moments about a point in the plane of the +ve forces equals zero £M=0

In statically determinate structures, all internal member forces and external reactant forces can be evaluated using the three equations of static equilibrium. When there are more unknown member forces and external reactant forces than there are available equations of equilibrium a structure is statically indeterminate and it is necessary to consider the compatibility of structural deformations to fully analyse the structure.

A structure may be indeterminate due to redundant components of reaction and/or redundant members. i.e. a redundant reaction or member is one which is not essential to satisfy the minimum requirements of stability and static equilibrium, (Note: it is not necessarily a member with zero force).

The degree-of-indeterminacy (referred to as ID in this text) is equal to the number of unknown variables (i.e. member forces/external reactions) which are in excess of the equations of equilibrium available to solve for them.

1.5.1 Indeterminacy of Two-Dimensional Pin-Jointed Frames

The external components of reaction (r) in pin-jointed frames are normally one of two types:

i) a roller support providing one degree-of-restraint, i.e. perpendicular to the roller, ii) a pinned support providing two degrees-of-restraint, e.g. in the horizontal and vertical directions.

as shown in Figure 1.14

roller supports; providing one pinned supports: providing two restraint pcrpuiriicuiar lo (hi roller. mutually ptipcndLcularitstniiiits roller supports; providing one pinned supports: providing two restraint pcrpuiriicuiar lo (hi roller. mutually ptipcndLcularitstniiiits

Figure 1.14

It is necessary to provide three non-parallel, non-concentric, components of reaction to satisfy the three equations of static equilibrium. Consider the frames indicated in Figures

Figure 1.15

Figures 1.16

In Figures 1.15 and 1.16 the applied forces and the external components of reaction represent co-planar force systems which are in static equilibrium. In Figure 1.15 there are three unknowns, (HA, VA and VC), and three equations of equilibrium which can be used to determine their values: there are no redundant components of reaction.

In Figure 1.16 there are five unknowns components of reaction, (HA, VA, VF, HE and VE), and only three equations of equilibrium; there are two redundant reactions in this case.

The internal members of pin-jointed frames transfer either tensile or compressive axial loads through the nodes to the supports and hence reactions. A simple pin-jointed frame is one in which the minimum number of members is present to ensure stability and static equilibrium.

Consider the basic three member pinned-frame indicated in Figure 1.15. There are three nodes and three members. A triangle is the basis for the development of all pin-jointed frames since it is an inherently stable system, i.e. only one configuration is possible for any given three lengths of the members.

Consider the development of the frame shown in Figure 1.17:

ina ft* «I'll fJs jj, — node nnjjbof m, - member number ina ft* «I'll fJs jj, — node nnjjbof m, - member number

Figure 1.17

Initially there are three nodes and three members. If the number of members in the frame is to be increased then for each node added, two members are required to maintain the triangulation. The minimum number of members required to create a simple frame can be determined as follows:

ih = the initial thrw members +(2 « number of.idMioiwI joims)

in lliis fiisc h = S and lîicrclwu the minimum number of members- 1(2 xS)- 3 >]

Any members which are added to the frame in addition to this number are redundant members and make the frame statically indeterminate; e.g.

It is also essential to consider the configuration of the members in a frame to ensure that it is triangulated. The simple frames indicated in Figure 1.19 are unstable.

Figure 1.19

As indicated previously, the minimum number of reactant forces to maintain static equilibrium is three and consequently when considering a simple, pin-jointed plane-frame and its support reactions the combined total of members and components of reaction is equal to:

£ (number of members+support reactions)=(m+r)=(2n-3)+3=2«

Consider the frames shown in Figure 1.20 with pinned and roller supports as indicated.

number of jkwnis *i J

niwnber ci members mi = 3

number of support tcjk-Ihhtis r — 3

(«+/-> = A « 2 AI The ("nunc is suite* I ly detc-nttînaur number off jmnîv, hi m number of members. ttt = 11

mi mber of support recelions r ■ J5 (.ut - rj - 14 - 2*i The f'nunc is 3-Ui.lEcally detertmnalc ln-0

number of joints n number of members. m - 14

number of support ivzc Icimis r • 3

{jit ~ r> = 17 > 2« TIk- frame is mjjïcjIIv ind'eicrmiiuLC with on-.- reifuihJarti internal member

number of'jranîs a - Ji number of members. hi = Ï5

number of support rc-at inxfis r - 6

tm + 21 >2m The i'rainc ii MaJtica.lly im de I cam in ale urxl has ï resluiidameics"

<2 ïnicrrul members * J cxicrml reaction* >

The degree of indeterminacy ID=(m+r)-2n

Compound trusses which are fabricated from two or more simple trusses by a structural system involving no more than three, non-parallel, non-concurrent, unknown forces can also be stable and determinate. Consider the truss shown in Figure 1.21(a) which is a simple truss and satisfies the relationships m=(2n-3) and ID=0.

Figure 1.21

This truss can be connected to a similar one by a pin and an additional member as shown in Figure 1.21(b) to create a compound truss comprising two statically determinate trusses. Since only an additional three unknown forces have been generated the three equations of equilibrium can be used to solve these by considering a section A-A as shown (see Chapter 3—Section 3.2.—Method of Sections for Pin-Jointed Frames: Problem 3.4).

1.5.2 Indeterminacy of Two-Dimensional Rigid-Jointed Frames

The external components of reaction (r) in rigid-jointed frames are normally one of three types:

i) a roller support providing one degree-of-restraint, i.e. perpendicular to the roller, ii) a pinned support providing two degrees-of-restraint, e.g. in the horizontal and vertical directions, iii) a fixed (encastre) support providing three degrees-of-restraint, i.e. in the horizontal and vertical directions and a moment restraint, as shown in Figure 1.22

mlltr *U|i|)Or!s: providing one iL'stiJiii; perpendicular to [lie follur.

pinned lujijiorlt: pro\ iriim; two [niiujiilly jwrpiiidieutir rcstnints iUctt supporis: providing two mulualfy pcrpiJuJietiLir rcslnjinLR and one mfiijR'iit restraint, mlltr *U|i|)Or!s: providing one iL'stiJiii; perpendicular to [lie follur.

pinned lujijiorlt: pro\ iriim; two [niiujiilly jwrpiiidieutir rcstnints iUctt supporis: providing two mulualfy pcrpiJuJietiLir rcslnjinLR and one mfiijR'iit restraint,

Figure 1.22

In rigid-jointed frames, the applied load system is transferred to the supports by inducing axial loads, shear forces and bending moments in the members. Since three components of reaction are required for static equilibrium the total number of unknowns is equal to: [(3xm)+rj. At each node there are three equations of equilibrium, i.e.

£ the vertical forces Fy=0;

£ the horizontal Fx=0; forces

£ the moments M=0, providing (3xn) equations.

The degree of indeterminacy ID=[(3m)+r]-3n

Consider the frames shown in Figure 1.23

Consider the frames shown in Figure 1.23

Figure 1.23

The existence of an internal pin in a member in a rigid-frame results in only shear and axial loads being transferred through the frame at its location. This reduces the number of unknowns and hence redundancies, since an additional equation is available for solution, i.e. Sum of the moments about the pin equals zero, i.e. E Mpin=0

Consider the effect of introducing pins in the frames shown in Figure 1.24

Figure 1.24

The existence of an internal pin at a node with two members in a rigid-frame results in the release of the moment capacity and hence one additional equation as shown in Figure 1.25(a). When there are three members meeting at the node then there are effectively two values of moment, i.e. M1 and M2 and in the third member M3=(M1+M2) The introduction of a pin in one of the members produces a single release and in two members (effectively all three members) produces two releases as shown in Figure 1.25(b).

In general terms the introduction of 'p' pins at a joint introduces 'p' additional equations. When pins are introduced to all members at the joint the number of additional equations produced equals (number of members at the joint—1).

Figure 1.25

Consider the frame shown in Figure 1.26.

Figure 1.26

The inclusion of an internal roller within a member results in the release of the moment capacity and in addition the force parallel to the roller and hence provides two additional equations. Consider the continuous beam ABC shown in Figure 1.27. in which a roller has been inserted in member AB

Figure 1.27

lD=([(3m)+r]-3n}-2 due to the release of the moment and axial load capacity at the roller ■'■ ID=([(3x2)+6]-(3x3)-2=1

Consider the same beam AB with a pin added in addition to the roller.

Figure 1.28

ID={[(3m)+r]-3n}-3 due to the release of the moment capacity at the position of the pin and the release of the moment and axial load capacity at the roller ID={[(3x2)+6]-(3x3)-3=0 The structure is statically determinate. A similar approach can be taken for three-dimensional structures; this is not considered in this text.

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