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