The Euler Bernoulli beam equation theory is the simple but practical tool for validating the beam deflection calculation. This beam theory is applied only for the laterally loaded beam without taking the shear deformation into account.

What is Euler Bernoulli Beam Theory?

This theory gives us an equation or relation between the deflection of the beam and the applied load intensity, as shown below:

d⁴u/dx⁴ = F/ (E*I)……………………….eqn.1.1

Where,

The axis of the beam is considered as X axis.

E is the modulus of elasticity.

u is the deflection of the beam, perpendicular to the axis of the beam.

F is the load on the beam per unit length.

Moreover, by successive derivation of “u”, the following useful relations can be obtained:

d³u/dx³ = -V/ (E*I)……………………….eqn.1.2

d²u/dx² = M/ (E*I)……………………….eqn.1.3

du/dx = θ……………………………….eqn.1.4

Where,

V is the shear force applied on the beam

M is the bending moment applied on the beam.

θ is the slope of the deflected beam
Examples of Euler-Bernoulli Beam Equation Problem statement: Create the deflection equation for a cantilever beam, which is subjected to an UDL of -F. The beam is L long, it has the modulus of elasticity E and the area moment of inertia of the beam is I.

Solution:

• By continuously integrating the eqn.1.1 with respect to x, we get:

E*I*( d³u/dx³) = F*x + K1………………………………………….…eqn.1.5

E*I*( d²u/dx²) =  (F*x² )/2+ K1*x + K2……………………………….eqn.1.6

E*I*( du/dx) =  (F*x³ )/6 + (K1*x² )/2 + K2*x + K3…………………..eqn.1.7

E*I*u =  (F*x⁴ )/24 + (K1*x³ )/6 + K2*x²/2 + K3*x+K4………….…..eqn.1.8

Where, K1, K2 and K3 are integration constants.

• The following boundary conditions (BC) will be used for finding out the different integration constants:

Boundary condition 1:  at the fixed end of the cantilever beam, the deflection will be zero.

i.e., At x=0, u = 0

So, from the eqn.1.8,

we get: K4 = 0

Boundary condition 2:  at the fixed end of the cantilever beam, the slope of the deflection will be zero.

i.e., At x=0, θ = 0

Now, by using the eqn.1.4, du/dx = 0

So, from the eqn.1.7, K3 = 0

Boundary condition 3:  at the free end of the cantilever beam, the moment will be zero, i.e., At x=L, M=0

Now, by using the eqn.1.3, d²u/dx² = 0

So, from the eqn.1.6,  (F*L ²)/2 + K1*L + K2 = 0……………………………..eqn.1.9

Boundary condition 4:  at the free end of the cantilever beam, the shear force will be zero, i.e., At x=L, V=0

Now, by using the eqn.1.2, d³u/dx³ = 0

So, from the eqn.1.5,  F*L + K1 = 0 Or, K1 =-F*L……………………..……………………..eqn.1.10

Now, from eqn.1.9 we get: K2 = 0.5*F*L²………………………………………………eqn.1.11

Now, putting all the values of the integration constants in the eqn.1.8,

we get: u = (F*x²)/(24*E*I)[x²-4*x*L + 6*x²]

This is the deflection equation for the cantilever beam.

Conclusion

The Euler Bernoulli beam theory equation is simple and widely applied beam theory useful for calculation of beam deflection and other important beam parameters. We have discussed the beam deflection formula for cantilever beam under UDL example. In similar way the theory can be customized and applied for other kinds of beams also.

Reference

• Strength of Materials: A Unified Theory – by Surya N. Patnaik, Dale A.Hopkins
• Strength of Materials – by Stephen Timoshenko