Bending in Beams (wk 4)

what is Equilibrium

· Statics to work out bending moment -> FBDs

· No unbalanced forces

· Sum is zero, moments sum to zero (ΣF=0, ΣM=0)

When does bending occur

Bending is when force is perpendicular the plane of the member

→ mostly for beams

• one side is in tension

• the other side is in compression

• neutral axis = not in tension of compression

impacted by second moment of area + young’s modulus

stress vs strain graph for a bending moment

• Strain is max at bottom (compression), decreases to zero at neutral axis (no compression or tension), then increases again (tension)

• Same happens for stress

  • are proportional

how to calculate bending moment

M = Fr

→ or can use second moment of intertia

Note: bending moment is the internal couple at a cut section of a member → like how shear force is the internal force

what is Second Moment of Area (I)

• geometric property of the cross section of a beam

  • To do with flexibility -> how prone a beam is to bending

  • The bendiness of a beam

• Relies only on the overall dimensions

second moment of area for:

• rectangle

• circle

for composite shapes → sum up the moment of area of the individual/sub shapes

how does Bending in Concrete Beams differ to other materials

Concrete can undergo compression, but not tension

• since inflexible

• Cannot have bending without both compression and tension

  • Parts in compression balance parts in tension -> otherwise no equilibrium

  • comcrete is assumed to have zero tensile strength -> in simple design

• Concrete will break -> needs to be reinforced by other materials (mostly steel)

  • young’s modulus for steel >>> concrete

what is Buckling

sudden deformation of a structural member that is loaded in compression

• When force is parallel to the plane of the member (long and slender)

• Buckles in the direction with the lowest I value

How height impacts buckling failure

Long/slender members in compression are more likely to buckle

• Forces can cause it to bend to the side and break

• More likely to buckle before yielding

Short members will not buckle as severely/at all

• more likely to fail due to crushing/reaching yield stress

Buckling Load/Formula