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Steel Designers' Manual - 6th Edition (2003)
9.1.1 Equations of static equilibrium
Element analysis 287
From Newton’s law of motion, the conditions under which a body remains in static
equilibrium can be expressed as follows:
• The sum of the components of all forces acting on the body, resolved along any
arbitrary direction, is equal to zero. This condition is completely satisfied if the
components of all forces resolved along the x, y, z directions individually add up
to zero. (This can be represented by SP x = 0, SP y = 0, SP z = 0, where P x, P y and
Pz represent forces resolved in the x, y, z directions.) These three equations
represent the condition of zero translation.
• The sum of the moments of all forces resolved in any arbitrarily chosen plane
about any point in that plane is zero. This condition is completely satisfied when
all the moments resolved into xy, yz and zx planes all individually add up to zero.
(SM xy = 0, SM yz = 0 and SM zx = 0.) These three equations provide for zero
rotation about the three axes.
If a structure is planar and is subjected to a system of coplanar forces, the
conditions of equilibrium can be simplified to three equations as detailed below:
• The components of all forces resolved along the x and y directions will individually
add up to zero (SP x = 0 and SP y = 0).
• The sum of the moments of all the forces about any arbitrarily chosen point in
the plane is zero (i.e. SM = 0).
9.1.2 The principle of superposition
This principle is only applicable when the displacements are linear functions of
applied loads. For structures subjected to multiple loading, the total effect of several
loads can be computed as the sum of the individual effects calculated by applying
the loads separately. This principle is a very useful tool in computing the combined
effects of many load effects (e.g. moment, deflection, etc.). These can be calculated
separately for each load and then summed.
9.2 Element analysis
Any complex structure can be looked upon as being built up of simpler units or
components termed ‘members’ or ‘elements’. Broadly speaking, these can be
classified into three categories:
• Skeletal structures consisting of members whose one dimension (say, length) is
much larger than the other two (viz. breadth and height). Such a line element is