Equilibrium

All structural analyses are based on satisfying one of the fundamental laws of physics, i.e.

Equation (1)

where

is the force system acting on a body m is the mass of the body a is the acceleration of the body

Structural analyses carried out on the basis of a force system inducing a dynamic response, for example structural vibration induced by wind loading, earthquake loading, moving machinery, vehicular traffic etc., have a non-zero value for 'a' the acceleration. In the case of analyses carried out on the basis of a static response, for example stresses/deflections induced by the self-weights of materials, imposed loads which do not induce vibration etc., the acceleration 'a' is equal to zero.

Static analysis can be regarded as a special case of the more general dynamic analysis in which:

Equation (2)

F can represent the applied force system in any direction; for convenience this is normally considered in either two or three mutually perpendicular directions as shown in Figure 1.1.

Figure 1.1

The application of Equation 2 to the force system indicated in Figure 1.1 is:

Sum of the forces in the direction of the X-axis £Fx=0 Equation 3 Sum of the forces in the direction of the Y-axis £Fy=0 Equation 4 Sum of the forces in the direction of the Z-axis £Fz=0 Equation 5

Since the structure is neither moving in a linear direction, nor in a rotational direction a further three equations can be written down to satisfy Equation 2:

Sum of the moments of the forces about the X-axis £Mx=0 Equation 6 Sum of the moments of the forces about the Y-axis SMy=0 Equation 7 Sum of the moments of the forces about the Z-axis £Mz=0 Equation 8

Equations 3 to 8 represent the static equilibrium of a body (structure) subject to a three-dimensional force system. Many analyses are carried out for design purposes assuming two-dimensional force systems and hence only two linear equations (e.g. equations 3 and 4 representing the x and y axes) and one rotational equation (e.g. equation 8 representing the z-axis) are required. The x, y and z axes must be mutually perpendicular and can be in any orientation, however for convenience two of the axes are usually regarded as horizontal and vertical, (e.g. gravity loads are vertical and wind loads frequently regarded as horizontal). It is usual practice, when considering equilibrium, to assume that clockwise rotation is positive and anti-clockwise rotation is negative. The following conventions have been adopted in this text:

x-^Lreclion; horizontal direction * positive is leltao right ■—►

y-directiwi: vertical direction - positive is upwards f +ve z-diwctioii: rotation about Lhe z-axis - positive is clockwise ^ +va

Figure 1.2

Structures in which all the member forces and external support reactions can be determined using only the equations of equilibrium are 'statically determinate' otherwise they are 'indeterminate structures'. The degree-of-indeterminacy 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, see Section 1.5

The availability of current computer software enables full three-dimensional analyses of structures to be carried out for a wide variety of applied loads. An alternative, more traditional, and frequently used method of analysis when designing is to consider the stability and forces on a structure separately in two mutually perpendicular planes, i.e. a series of plane frames and ensure lateral and rotational stability and equilibrium in each plane. Consider a typical industrial frame comprising a series of parallel portal frames as shown in Figure 1.3. The frame can be designed considering the X-Y and the Y-Z planes as shown.

Figure 1.3

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