2.27
2 Statics of Particles
Statics is a quantitative description of forces acting on a
structure in equilibrium. Hence, we need a mathematical
description of force – action of one body on another, a
vector quantity.
Mass, length, volume are scalar quantities and have only
magnitude, while force is vector quantity and has
magnitude and direction.
Example: Ball hit by a bat.
Types of vectors reminder:
Free vector is not attached to any point or a line in a body.
Sliding vector has unique line of application.
Fixed vector has unique point of application (and, thereof,
line of application).
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In statics forces are sliding vectors. Hence, they have
unique magnitude, direction, and line of action. However,
we can slide the force along its line of action.
Resultant of Forces
As vector quantities, forces are added as vectors and the
sum is called resultant.
Note: Graphical method doesn’t provide a line of action for
the resultant, unless the forces are applied at the same
point. To find line of action of resultant, draw resultant
applied at a point of intersection of lines of actions for F1
and F2 (3D?)
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Particle Equilibrium (2D)
For the particle to be in equilibrium, resultant of all
forces acting on it should be equal to zero. In
mathematical representation this condition is called an
equilibrium equation.
Equilibrium equation (EE) in vector form:
∑ 𝐹̅ = 0
For 2D this equation can be written as two scalar
equations to be satisfied simultaneously:
∑ 𝐹̅ = (∑ 𝐹𝑥 ) 𝑖̅ + (∑ 𝐹𝑦 ) 𝑗̅ = 0
∑ 𝐹𝑥 = 0, ∑ 𝐹𝑦 = 0
Force components can be found using vector algebra.
It is critical to account all forces. To do so a Free Body
Diagram is drawn.
Free Body Diagram
Diagram of a particle, with all forces applied to it, is a Free
Body Diagram (FBD). There are three steps in drawing
FBD:
1. Imagine and draw the particle isolated from its
surroundings.
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2. Indicate all forces acting on the particle. Include active
forces, that would otherwise set the particle in motion,
and reactive forces, that are caused by constraints and
supports, so the particle stays in equilibrium. Encircling
the particle might help with accounting for all forces.
3. Mark known forces with their directions and magnitudes,
use letters for unknown quantities. If line of action of a
force is known, then a direction is assumed. However, if
during solving magnitude is found to be negative, the
direction of the force will be opposite to the one
assumed.
Analysis of Equilibrium
1. Draw FBD to include all acting forces.
2. Apply equations of equilibrium.
In case of 2D there are two equations per particle.
Hence, to solve them, there should be maximum 2
unknowns per particle.
Possible unknowns:
 Force vectors (two components, or magnitude and
angle of direction)
 Magnitudes of two forces with known directions
 Orientations of two forces with known magnitudes
 Other parameters (geometric, etc.) – 2 max.
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Elimination Technique
To eliminate an unknown force from equation(s) of
equilibrium, one should consider equilibrium of forces in
the direction perpendicular to the unknown force.
Repeat it for the second unknown force.
In 2D this technique breaks down a system of equations of
equilibrium into a set of independent equations. This
greatly simplifies the computational complexity and is less
likely to produce a computational error.
Note: Sometimes it is beneficial to eliminate a known
force.
Example 1:
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Notes:
Note 1: Often, it is convenient to eliminate even known
force to reduce the amount of computations (see step ii).
Note 2: Axes in the force EE’s should not necessarily be
perpendicular. Any two non-parallel axes will do.
Note 3: The advantage of ET would be especially
spectacular if we were asked to evaluate just one of two
unknown tensions. By straightforward method we’d still
have to solve for the system of 2 simultaneous equations.
By ET we’d get it from a single equation (i) or (ii).
Example 2:
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Example 3:
Particle in Equilibrium Example 1:
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Example 2:
Example 3:
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Example 4:
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Example 5:
Example 6: