Cantilever Beam Subjected to a Concentrated Load

Objective: Analysis for bending in the force plane under a concentrated force without taking into account the transverse shear deformations. The values of the maximum transverse displacement, rotation angle and bending moment are checked.

Initial data file:  Example_4_1.spr

Problem formulation: The cantilever beam is loaded by a concentrated force Р applied to its free end. Determine the maximum values of the transverse displacement w, rotation angle θ and bending moment М.

References: G.S. Pisarenko, A.P. Yakovlev, V.V. Matveev, Handbook on Strength of Materials. — Kiev: Naukova Dumka, 1988, p. 263.

Initial data:

E = 2.0·1011 Pa - elastic modulus,
ν = 0.3 - Poisson’s ratio,
L = 3 м - beam length;
I = 2.44·10-6 м4 - cross-sectional moment of inertia;
Р = 5 kN - value of the concentrated force.

 

Finite element model: Design model – general type system, 10 bar elements of type 5, 11 nodes.

 

Results in SCAD:


Bending moment diagram М (kN·m)


Values of transverse displacements w(mm)



Values of rotation angles θ (rad)

Comparison of solutions:

Parameter

Theory

SCAD

Deviations, %

Transverse displacement w, mm

-92.21

-92.21

0.00

Rotation angle Ө, rad

0.04611

0.04611

0.00

Bending moment М, kN·m

-15.0

-15.0

0.00

 

Notes: In the analytical solution, the maximum values of the transverse displacement w, rotation angle θ  and bending moment М are determined according to the following formulas:

\[ w=-\frac{P\cdot L^{3}}{3\cdot E\cdot I}; \quad \theta =\frac{P\cdot L^{2}}{2\cdot E\cdot I}; \quad M=-P\cdot L. \]