Problem 6.1. Strain energy of a cantilever

Determine the strain energy of the cantilever loaded by a concentrated force, F = 5 kN at the endpoint. Length of the beam is L = 6 m, bending stiffness of the cross section is EI = 4.2 × 106 Nm2.

Solve Problem

Solve

Strain energy, U [Nm]=

Do you need help?

Steps

Step by step

Hint: integral of two functions: Lf1(x)f2(x)dx,  when at least one of the functions is linear can be performed by the so called “visual integration method”. Let f2 be linear.

Lf1(x)f1(x)dx=Ah

where A=Lf1(x)dx is the integral of function f1, i.e. the area under the function on the 0 – L interval, while h is the ordinate of funder the center of gravity of A.

[/expandsub1]

 

Step 1. Draw the moment diagram. Give the functions of the moment and the curvature along the beam’s length.

Check functions

Step 2. Determine the strain energy of the bent beam by performing the integration.

Check strain energy

The strain energy is

Eq.(6-19)

U=12LκzMzdx=12EILMz2dx=12EILF2x2dx=F2L36EI=5.02×106×6.036×4.2×106=214.3 Nm

Step 3. Determine the strain energy of the bent beam applying the visual integration method. 

Check strain energy

Since κz is linear, the result of the above integration can be obtained by multiplying the area of the moment diagram with the value of the linear curvature function at the centroid of the moment diagram:

U=12LκzMzdx=12Ah=12FL222FL3EI=F2L36EI=214.3 Nm

Results

Worked out solution

Hint: integral of two functions: Lf1(x)f2(x)dx,  when at least one of the functions is linear can be performed by the so called “visual integration method”. Let f2 be linear.

Lf1(x)f1(x)dx=Ah

where A=Lf1(x)dx is the integral of function f1, i.e. the area under the function on the 0 – L interval, while h is the ordinate of funder the center of gravity of A.

The functions of the moment and the curvature along the beam’s length are given in the Figure below.

The strain energy of the bent beam is

Eq.(6-19)

U=12LκzMzdx=12EILMz2dx=12EILF2x2dx=F2L36EI=5.02×106×6.036×4.2×106=214.3 Nm

Since κz is linear, integration can be performed by the visual integration method by multiplying the area of the moment diagram with the value of the linear curvature function at the centroid of the moment diagram:

U=12LκzMzdx=12Ah=12FL222FL3EI=F2L36EI=214.3 Nm