Beam Deflections Jun 2022

Hand calculate beam deflections

I thought I should reassess and resharpen my basic structural engineering skills.

I present a few problems

Given a cantilever subject to three separate loadings:

(a) a concentrated load P at the free end,
(b) a moment M at the free end, and
(c) a uniform load w over the entire beam.

The span is L.

For each loading, determine the beam deflection at the free end of the span.

Answer: (a)

$\delta = \frac{PL^3}{3EI}$

Answer: (b)

$\delta = \frac{ML^2}{2EI}$

Answer: (c)

$\delta = \frac{wL^4}{8EI}$

See calculation of cantilever deflections
Given a simple beam with a P load at midspan.

The span is L.

Determine the beam rotation at the right end of the span.

Answer:

$\theta_{B} = \frac{PL^2}{16EI}$

See calculation of symmetric rotation

I solved this problem using the virtual work method.
Given a simple beam with a P load at at a distance a from the left support.

The span is L; therefore, $b = L-a$ .

Determine the beam rotation at the right end of the span.

Determine also the deflection for all points x located left of the P loading.

Answer:

$\theta_{B} = \frac{Pab(L+a)}{6EIL}$

See calculation of asymmetric rotation

I solved this problem using the conjugate beam method (*).

Note that if

$a = \frac{L}{2}$

$\text{and therefore}$

$b = \frac{L}{2}$

then the third equation is equal to the second equation.
Given a beam (or plate) with uniform load, w, subjected to a 2 point symmetric lift

The center span, L, is to be determined

The value, a, is a ratio to be determined

The left overhang = the right overhang = $aL$

Determine the span and the overhang that minimizes the absolute value of the moment along the span

Answer: see calculation of lift load points

Determine the resulting deflection at the right end of the beam, at $x = L+2aL = (1+2a)L$

Answer: see hand calculation of deflection given minimized moment

Answer: see calculation of deflection given minimized moment using sympy.physics.continuum_mechanics.beam.Beam (**)

Note that my calculation of a agrees with the well-known value of a.

$a=0.3536$

$\frac{aL}{L+2aL} = \frac{a}{1+2a} = \frac{\sqrt{2}-1}{2} \approx 0.2071 \approx 0.21$

I derived two deflection equations:

$\Delta_{cantilever}=\frac{a(1-6{a}^2-3{a}^3)}{24}\frac{{w}{L}^4}{{E}{I}}$

$\Delta_{centerspan}=\frac{(6{a}^2-1)}{48}\frac{{w}{L}^4}{{E}{I}}$

See calculation of centerspan deflection.
See calculation of cantilever deflection.

I determined a few interesting values of a and charted the resulting deflections.

http://khoitsmahq.firstcloudit.com/images%2Ftable_chart.png

http://khoitsmahq.firstcloudit.com/images%2Fjust_chart.png

Given a pin-supported bent with uniform EI:

The beam span is L.
The column length is μL.

Determine the horizontal deflection of the frame at point B.

Answer:

$\Delta_{B} = \Big( \frac{\mu^2L^3(1+2\mu)}{12EI}\Big) H$

See calculation of frame deflection

Bonus Material

For the first 2 problems, I checked some algebra using Jupyter, Plotly, Python, and Sympy.

http://khoitsmahq.firstcloudit.com/images%2Fbeam_hand_calc_check.html

Additional Bonus Material

Quite a surprise — I discovered a Sympy package named sympy.physics.continuum_mechanics.beam.Beam (**) and checked some of my results using that package too. Note Sympy’s use of singularity functions in cells 4 and 5. I use singularity functions in cells 6 through 10.

http://khoitsmahq.firstcloudit.com/images%2Fshear_and_moment.html

def f_rotation(x, E, I):
    sf = SingularityFunction
    return (-10*sf(x, 0, 2)/3 + 
            5*sf(x, 48, 2) - 
            5*sf(x, 144, 2)/3 + 
            12800)/(E*I)
    
def f_deflection(x, E, I):
    sf = SingularityFunction
    return (12800*x - 
            10*sf(x, 0, 3)/9 + 
            5*sf(x, 48, 3)/3 - 
            5*sf(x, 144, 3)/9)/(E*I)

References

(*) Conjugate Beam Method — Wiki

(**) https://docs.sympy.org/… continuum_mechanics/beam.html

(***) A bonus from Dr. Myosotis