Distributed Load Problem

Problem: A simply supported beam is subjected to a distributed load that varies linearly along its span. The expression for the distributed load is:

Determine the reactions at each of the supports, the deflection along the span, the slope at the endpoints, the shear and the bending moment.

Solution:

L = beam length

RA = the reaction at support A

RB = the reaction at support B

wm = slope of spanwise load function

wb = intercept of spanwise load function.

y(x) = deflection as a function of x.

V(x) = shear as a function of x.

M(x) = bending moment as a function of x.

We will solve the problem in two parts. First we sum forces and moments to deduce the values of the reactions at supports A and B. Then we will produce expressions to describe the deflection, shear and bending moments.

Now we need to plug in the boundary condition to find the value of C1:

Which gives us a complete result for shear. Now we proceed with bending moment:

Which gives us a complete
result for bending.

To compute spanwise deflection and slope we must refer to linear bending theory
which (under the assumption of constant stiffness) gives us that:

Having completed this integration we can now evaluate both remaining constants by inserting the boundary conditions:

With this behind us we can generate complete expressions for the deflection and slope:

Our only remaining errand is to evaluate the slope expression at the endpoints:

and to remember that end slopes are in radians.

Example Problem 1:

We first run a BeamCALCulator that only handles a uniform load:

Load Parameters

Load wm: 0; Slope of spanwise
load function

Load wb: 10 lbs/in; Uniform Load

Length: 36 in;

Material Parameters

Material: Aluminum 6061
T-6

Elastic Modulus: 10,100,000 psi

Yield Strength: 35,000 psi (Tensile)

Density: 0.098 pci

Cross Sectional Parameters

Outer Width W: 1.00 inches

Inner Width w: 0.85 inches

Outer Height H: 1.00 inches

Inner Height h: 0.85 inches

Example Problem 2:

We then run a BeamCALCulator that handles a linearly varying load with the uniform load case and compare the results:

Load Parameters

Load wm: 0; Slope of spanwise
load function

Load wb: 10 lbs/in; Uniform Load

Length: 36 in;

Material Parameters

Material: Aluminum 6061
T-6

Elastic Modulus: 10,100,000 psi

Yield Strength: 35,000 psi (Tensile)

Density: 0.098 pci

Cross Sectional Parameters

Outer Width W: 1.00 inches

Inner Width w: 0.85 inches

Outer Height H: 1.00 inches

Inner Height h: 0.85 inches

Example Problem 3:

We then run a BeamCALCulator that handles a linearly varying load with a tapered load case whose end support reactions are the same and compare the results:

Load Parameters

Load wm: -720/L2=.56 lbs/in2
Slope of spanwise load function

Load wb: 20 lbs/in; Uniform Load

Length: 36 in;

Material Parameters

Material: Aluminum 6061
T-6

Elastic Modulus: 10,100,000 psi

Yield Strength: 35,000 psi (Tensile)

Density: 0.098 pci

Cross Sectional Parameters

Outer Width W: 1.00 inches

Inner Width w: 0.85 inches

Outer Height H: 1.00 inches

Inner Height h: 0.85 inches