Beams and Loading

Beams

Long, straight members support transverse loads.

Beam Classifications

Classified according to the way they are supported, see lecture 2.

• Simply supported (1 pin, 1 roller)
• Overhanging (1 pin, 1 roller but not at ends)
• Cantilever (1 fixed end, 1 free end)

Statically Determinate

# of Unknowns < # of Equations

Normal Stress for Beams

• Top is in compression
• Bottom is in tension
• x is the neutral axis
• xz is the neutral plane
• Bending stress varies linearly with distance from neutral axis

Analyzing bending stress due to moments

${\sigma }_x=\frac{-M_{z}y}{I_z}$ $I={\int }_{}^{}{y}^{2}dA$
${\sigma }_{\max }=\frac{M_{z}c}{I_z}=\frac{M_z}{Z}$
Where I in this case is the second-area moment and Z = I/c is the section modulus and c is the magnitude of the greatest y (vertical distance)

For a circle, I is:
$I=\frac{\pi d^4}{64}$ where for circular sections, c is usually d/2

Normal Stress Due to Axial Loading and Bending

${\sigma }_{tot}=\pm \frac{F}{A}\pm \frac{My}{I}$
Where sign depends on sign convention and direction (above or below the neutral axis) from Shear Force and Bending Moments

Example Problems

Remember h corresponds to the vertical axis and b corresponds to the horizontal axis
SEE LECTURE 4 At a high level:

• Find the neutral axis
• Find the maximum amount of stress acting on the top and bottom relative to how a moment is applied Solution Steps:
• Calculate the location of the Centroids of each section that makes up the geometry
• Use those centroids to find the centroid of the entire shape
• Based on cross-section geometry, calculate I (moment of inertia) with respect to the neutral axis (using parallel axis theorem)
  • You sum the moments of inertia of the different shapes in the overall shape $\Sigma \left(\overline{I}+A{d}^2\right)$ where d is the distance between the centroid of the shape and the centroid of the overall shape
• Now using the sigma_max (max stress) equation, you can find the normal stress at a given point (very top or very bottom)
  • • is tensile, - is compressive

Asymmetry

When a moment is applied on an angle we get Two-Plane Bending