M4: Basic Mechanics II (Inclined Sections & Varying Cross-Sections)

Inclined Section Area A1

The area of an inclined section expressed in terms of original area A and angle theta: A1 = A / cos(theta)

Normal Force N

The normal component of force acting on an inclined section: N = F cos(theta)

Shear Force V

The shear component of force acting on an inclined section: V = F sin(theta)

Normal Stress sigma theta

Normal stress on an inclined section: sigma_theta = (F/A) cos^2(theta) = sigma_x cos^2(theta)

Shear Stress tau theta

Shear stress on an inclined section: tau_theta =

Inclined Section Sign Convention Normal

Normal stresses are positive in tension and negative in compression

Inclined Section Sign Convention Shear

Shear stresses are positive for counter

Maximum Normal Stress Angle

Occurs at theta = 0 degrees, where sigma_max = sigma_x

Maximum Shear Stress Angle

Occurs at theta = plus or minus 45 degrees, where tau_max = sigma_x / 2

Brittle vs Ductile Failure

Brittle materials fail from maximum normal stresses, while ductile materials fail from maximum shear stresses

Varying Cross

Section Stress

Suspended Bar Normal Force

The normal force at any height x due to self