For an example, the truss bridge is formed by the use of triangles. These bridges are used for the passage of trains over water. Each joint of the bridge must be in equilibrium which means that the two forces acting on it must be equal to zero. The directional forces are horizontal and vertical and their summations equal zero or ΣH=0 and ΣV=0. These expressions will yield two equations that can be solved for only two unknown forces. ΣH=0:
F
1
cosθ
- F
2
cosθ = 0 and ΣV=0: -
F
1
sinθ - F
2
sinθ = 0 where θ is the angle of inclination for example 45
0
or 60
0
. More complicated truss bridges may have more than two unknown forces and would need other analytical techniques such as matrices.
In addition to gravitational forces torque forces must also be balanced to zero or the bridge will rotate. Torque, is the cross product between the applied force relative to the position of an object's center of mass,
τ=F x d.
Commonly, mathematical applications in bridge design include differential calculus, integral calculus and Newton's Second Law: F=mg for which F is the force, m is the mass, and g is the gravitational acceleration.