**I present a few problems**

- Given a cantilever subject to three
**separate**loadings:

(a) a concentrated loadat the free end,**P**

(b) a momentat the free end, and**M**

(c) a uniform loadover the entire beam.**w**

The span is.**L**

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

Answer: (a)Answer: (b)

Answer: (c)

- Given a simple beam with a
load at midspan.**P**

The span is.**L**

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

Answer:See calculation of symmetric rotation

I solved this problem using the virtual work method.

- Given a simple beam with a
load at at a distance**P**from the left support.**a**

The span is; therefore, .*L*

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

Determine also the deflection for all pointslocated left of the**x**loading.**P**

Answer:See calculation of asymmetric rotation

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

Note that ifthen the third equation is equal to the second equation.

- Given a beam (or plate) with uniform load,
, subjected to a 2 point symmetric lift**w**

The center span,, is to be determined*L*

The value,, is a ratio to be determined**a**

The left overhang = the right overhang =

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

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 ofagrees with the well-known value of**a**.**a**See calculation of centerspan deflection.

I determined a few interesting values of

See calculation of cantilever deflection.

and charted the resulting deflections.**a** - 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:

**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