Structures Sem 1

1 N/mm² =

1 MPa (mega pascal)

Moment of resistance formula (engineer’s theory of bending)

M/I = σ/y

M = σZ

Elastic section modulus formula

Z = I / y

2nd moment of area for a rectangle

I = bd³/12, where b is the base and d is the height.

z = I / y, how do we find I?

ΣI_cc + ΣAh² , where A is the area of the sections and h is the distance from the centroid to the neutral axis.

How to find position of the neutral axis / centroid

ȳ = Σ(Ay) / ΣA, where A is the area and y is the distance of the centroid of the section from the reference axis.

Moment of resistance equation for rectangular section unsimplified

M = (σ_max bd²) / 6

because equation is M / I = σ_max / y, rearrarange for M and we know that for a rectangle:

I = bd³/12, max y = d/2, so equation becomes:

M = (σ_max) (bd³/12 ÷ d/2)

M = (σ_max) (bd³/12 × 2/d)

so M = (σ_max bd²) / 6

Moment of resistance formula for simply supported beam (max bending moment)

M = wL² / 8, where w is the uniform load and L is the span length.

Moment of reisitance formula for cantilever beam (max bending moment)

M = wL² / 2, where w is the uniform load and L is the length of the cantilever.

Equation of max shear force for simply supported beam

V = wL / 2, where w is the uniform load and L is the span length.

Equation of max shear force for cantilever beam

V = wL, where w is the uniform load and L is the length of the cantilever.

Standard timber sizing lengths

100, 125, 150, 175, 200, 225 mm

Sagging

+ve bending: compression at top, tension at bottom

Hogging

-ve bending: tension at top, compression at bottom

Newton’s first law

An object at rest stays at rest, and an object in motion stays in motion unless acted upon by a net external force.

Newton’s second law

F = ma

Change in momentum is equal to the net force applied to an object over time.

Newton’s third law

Every action has an equal and opposite reaction

Reactions at roller

1

Reactions at hinge (external pin)

2

Reactions at encastre

3

Reactions at link

1

Young's Modulus Equation

Stress / Strain

E = σ / ε

Stress equation

σ = F / A

Strain equation

ε = ΔL / L

Shear force definition

The unbalanced force in a beam acting perpendicular to the primary axis

Poisson's Ratio definition

The ratio of lateral to longitudinal strain

What is the limit of proportionality

The point at which stress is no longer proportional to strain in a material which obeys hookes law - it has gone past its elastic limit

Modulus of elasticity definition

Ratio of direct stress to direct strain

When drawing a shear force diagram, when would we use a vertical jump vs a sloping line for change in shear?

Vertical jump = for point loads

Sloping line = for UDLs (uniformly distributed loads, the ones kN/m lots of arrows)

Point of contraflexure meaning

Point where the resultant bending moment is zero

Equation for number of degrees of freedom

2j (2 × number of joints)

Equation to know if structure is statically determinate

m + r = 2j (m = number of members, r = number of reactions, j = nodes / joints)

Centroid definition

The center of mass / gravity of a structure

Reason for calculating 2nd moment of area

Characterises the deflection of a shape under load

Bending moment definition

The internal force that occurs in a structure when an external force bends the structure

Elastic section modulus definition

A measurement of how much a beam can bend without breaking

It is a ratio of the second moment of area to the distance from the neutral axis to the outer fibers

Description of tension failure in reinforced concrete

Failure only occurs after large deflections of beam because the concrete is weak in tension but the steel will yield when subjected to excessive stress

Description of compression failure in reinforced concrete

Failure will be sudden without warning because concrete resists compressive stresses however concrete is brittle

Description of shear failure in reinforced concrete

Sudden failure and often fails at a very low load