MOS

centroid for any triangle

h/3

hookes law with stress and strain

stress = E*strain

macaulays with UDL

• extend UDL to end of beam (If doesnt already)

• then add opposite UDL

  • for extension only to cancel that part

  • for normal across whole thing (a should be where UDL ends)

• in macaulays do q₀/2 [x-a]²

• a should be right where UDL starts

simply supported beam

• beam supported at both ends by a pin and a roller

centroid if one line of symmetry

e.g if symmetrical across y-axis, Xc=0

centroid formula (composite)

y_c = y_c1A₁ + y_c2A₂ + y_c3A₃ / A₁ + A₂ + A₃

• same for x

two axes of symmetry

• centroid on intersection

second moment of area (I_z and I_y) - with parallel axis for composites

I_z = I_z + A(Δy)

I_y = I_y + A(Δz)

• remember Δy/z is relative to the axis of centroid so difference between z(x)/y_c and z(x)/y_c1

centroid with beam and distributed load

• use force as height for area

• then do moment resolution to find support reactions

centroid formula for cut out shapes (think like composite)

y_c = y_c1A₁ - y_c2A₂ / A₁ - A₂

• same for x

page 32 of revisions notes

b and d in second moment formula

Iz for hollow circular ring

Iz = Ip/2

Ip for hollow circular ring

Ip = pi/2 (Ro⁴ - Ri⁴)

finding deflection steps

1. equilibrium

2. then make cut at x and choose side with less reactions

3. make equation for M(x) based on what would happen if load put there

4. then = to moment-curvature

5. then integrate and dont forget add c1x + c2

at pin or roller

v=0

v’ not 0

v’’ not 0

at fixed support

v=0

v’=0

v’’ not zero

at free end

v not 0

v’ not 0

v’’ = 0

max deflection for simply supported beam (2 supports both ends)

max is at dv/dx = 0

max deflection for cantilever (fixed one end free at other)

max at x=L

max deflection

when dv/dx = 0

macaulays function

only need one section cut furthest from left support (includes all cuts in one beam)

notes for integrating moment equation

• dont forget to add constants

• dont forget to add c1x +c2 after integrating for v(x)

moment equilibrium

if concentrated moment on beam don’t multiply by distance

macaulays brackets

if x<b then [x-b]ⁿ = 0

• eg. if x = 0 [0-b]² = 0

macaulays for M₀ concentrated load

M₀[x-a]⁰

critical buckling stress

σ_cr = P_cr / cross sec A

yield load

P_y = σ_yA

torsion of cyclinder shear stress max

shear stress max at r = R (surface of cylinder)

internal torque of cylinder

T = GI_p (dθ/dx)

at fixed end of torsion of cylinder

at x = 0 then θ (angle of twist) = 0

angle of twist formula

always gives in radians

right hand rule

when u cut section thumb points to other one then fingers curl direction of torque

• but internal torque points into section

pins

moment = 0

macaulays sign convention

• depends on beam bending effect (adds sag or adds hog) not clockwise rotation

• load down = negative

• load up = pos

max bending moment location and mag

max when dv/dx = 0 , find x then sub into v(x)

upper and lower bound buckling loads

• smaller L_e = larger buckling load

• bigger L_e = smaller buckling load