Differential Equations of the Deflection Curve
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The St. Louis Arch sways about
one inch in a 20 mph wind. It is
designed to sway up to 18 inches
in 150 mile per hour winds. (source)
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With a wind of 110 miles per hour, the 1454-ft
Empire State Building gives 1.48 inches. (source)
example
from Mechanical Engineering Design, 5th Ed., by J.E. Shigley and C.R. Michke, McGraw-Hill, 1989, p95
terms
plane of bending - deflections will occur in a plane if the beam is symmetric around this plane deflection, ν - the displacement of any point along the beam from its original position, measured in the y direction
deflection curve
coordinate axes
angle of rotation, θ - the angle between the x-axis and the tangent to the deflection curve
slope of the deflection curve: dν/dx = tan θ
tan θ = θ for small angles
sign conventions
- the x and y axes are positive to the right and upward
- the deflection ν is positive upward
- the slope dν/dx and angle of rotation θ are positive when counterclockwise with respect to the positive x axis
- the curvature κ is positive when the beam is bent concave upward
- the bending moment M is positive when it produces compression in the upper part of the beam
Differential Equations
load equation shear-force equation bending-moment equation
solution technique
- integrate the equations
- solve for the integration constants using boundary conditions for the beam
assumptions
- material obeys Hooke's law
- slopes of the deflection curve are small
- shear deformations are negligible