Statics Equilibrium

8 Any structural element, or part of it, must satisfy equilibrium.

1 So far we have restricted ourselves to a continuum, in this chapter we will consider a structural element.

Summation of forces and moments, in a static system must be equal to zero2. ) In a 3D cartesian coordinate system there are a total of 6 independent equations of equilibrium:

11 In a 2D cartesian coordinate system there are a total of 3 independent equations of equilibrium:

12 All the externally applied forces on a structure must be in equilibrium. Reactions are accordingly determined.

13 For reaction calculations, the externally applied load may be reduced to an equivalent force3.

14 Summation of the moments can be taken with respect to any arbitrary point.

15 Whereas forces are represented by a vector, moments are also vectorial quantities and are represented by a curved arrow or a double arrow vector.

16 Not all equations are applicable to all structures, Table 12.1

Structure Type

Equations

Beam, no axial forces

SFy SM2

2D Truss, Frame, Beam Grid

SFx SFy SM; SF2 SMx SMy

3D Truss, Frame

SFx SFy SF2 SMx SMy SM2

Alternate Set

Beams, no axial Force

SM? SM?

2 D Truss, Frame, Beam

SFX SM? SM?

SMA SM? SM?

Table 12.1: Equations of Equilibrium

Table 12.1: Equations of Equilibrium

17 The three conventional equations of equilibrium in 2D: YFx, YFy and SMZ can be replaced by the independent moment equations EMYM? , YM? provided that A, B, and C are not colinear.

18 It is always preferable to check calculations by another equation of equilibrium.

19 Before you write an equation of equilibrium,

1. Arbitrarily decide which is the +ve direction

2. Assume a direction for the unknown quantities

3. The right hand side of the equation should be zero

If your reaction is negative, then it will be in a direction opposite from the one assumed.

20 Summation of external forces is equal and opposite to the internal ones (more about this below). Thus the net force/moment is equal to zero.

21 The external forces give rise to the (non-zero) shear and moment diagram.

2 In a dynamic system £F = ma where m is the mass and a is the acceleration.

3 However for internal forces (shear and moment) we must use the actual load distribution.

22 In the analysis of structures, it is often easier to start by determining the reactions.

23 Once the reactions are determined, internal forces (shear and moment) are determined next; finally, internal stresses and/or deformations (deflections and rotations) are determined last.

24 Depending on the type of structures, there can be different types of support conditions, Fig. 12.1.

Symbols

Unknown Forces

"Ry

Figure 12.1: Types of Supports

Roller: provides a restraint in only one direction in a 2D structure, in 3D structures a roller may provide restraint in one or two directions. A roller will allow rotation.

Hinge: allows rotation but no displacements.

Fixed Support: will prevent rotation and displacements in all directions.

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