Download - 8 beam deflection
Beam Deflection
BIOE 3200 - Fall 2015Watching stuff break
Learning ObjectivesDefine beam deflection (δ) and
identify the factors that affect itDetermine deflection and slope in
beams in bending using double integration method
BIOE 3200 - Fall 2015
Deflection in beamsDeflection is deformation from original position in y direction
BIOE 3200 - Fall 2015
Recall relationships between shear force, bending moment and normal stresses
BIOE 3200 - Fall 2015
To balance forces within beam cross section:
P = V == 𝑥𝑦
Bending moments balance normal stresses across area of cross section. This is how we relate applied loads and deformation (stress and strain).
BIOE 3200 - Fall 2015
Simplify: Recall that So:
where u0y(x) is displacement in y direction
General formula for beam deflection involves double integration of bending moment equation
Governing equation for deflection: = , where deflection δ(x) = uoy(x) Solve double integral to get equation for δ(x)
(elastic curve), or the deflected shape Shape of beam determined by change
in load over length of beamBIOE 3200 - Fall 2015
How to calculate beam deflection using double integration method
BIOE 3200 - Fall 2015
δ(x)
EI = Flexural Rigidity
Governing equation: =
Note: = tan θ ≈ θ(x) ] dx + x + - general formula for beam deflection
Sign conventions for beam deflection
X and Y axes: positive to the right and upward, respectively
Deflection δ(x): positive upwardSlope of deflection at any point and
angle of rotation θ(x): positive when CCW with respect to x-axis
Curvature (K) and bending moment (M): positive when concave up (beam is smiling)
BIOE 3200 - Fall 2015
δ(x)
What affects deflection?Bending moment
◦ Magnitude and type of loading◦ Span (length) of beam◦ Beam type (simply supported, cantilever)
Material properties of beam (E)Shape of beam (Moment of Intertia I)
BIOE 3200 - Fall 2015
How to complete double integrationx + Find C1 and C2 from boundary
conditions (supports)Example: cantilever beam with load at
free end
BIOE 3200 - Fall 2015
Using boundary conditions to calculate deflections in beamsOther examples of boundary
conditions:
BIOE 3200 - Fall 2015
Pulling it all togetherRelation of the deflection with beam
loading quantities V, M and loadDeflection = Slope = d / dxMoment M(x) = EI Shear V(x) = - dM/dx = - EI (for
constant EI)Load w(x) = dV/dx = - EI (for
constant EI)BIOE 3200 - Fall 2015
Load, moment, deformation and slope can all be sketched for a beam
BIOE 3200 - Fall 2015
Procedure for calculating deflection by integration methodSelect interval(s) of the beam to be used,
and set coordinate system with origin at one end of the interval; set range of x values for that interval
List boundary conditions at boundaries of interval (these will be integration constants)
Calculate bending moment M(x) (function of x for each interval) and set it equal to EI
Solve differential equation (double integration) and solve using known integration constants
BIOE 3200 - Fall 2015
Typical deflection equationSimply supported beam under uniform constant load:
δx = ( + (at any point x)
BIOE 3200 - Fall 2015
Load
Material Property
Shape Propert
y
L
Δmax
Span
Examples of deflection formulae
FBD for simply supported beam under constant uniform load:
δmax = (at midpoint)
δx = ( + (at any point x)
BIOE 3200 - Fall 2015
Examples of deflection formulae
Simply supported beam, point load at midspan
δmax = (at point of load)
δx = () (where x < ; symmetric about midspan)
BIOE 3200 - Fall 2015
x
Examples of deflection formulae
Cantilever beam loaded at free end
δmax = (at free end)
δx = ( - (everywhere else)
BIOE 3200 - Fall 2015
P