Beam Calculators

Cantilever Beam Under a Concentrated Load

The calculation of beams in conditions of symmetrical transverse bending under the action of a concentrated and distributed loads are represented in this Section. At that, the following assumptions are used in order to simplify the calculations without introducing significant errors in the results:

Plane sections perpendicular to the beam axis before loading remain plane and perpendicular to the beam axis after loading.
Under bending, the longitudinal layers of the beam do not exert pressure on each other.

When carrying out calculations, the beam axis equation is solved:

y'' = M / EI_x

y - beam deflection;
y' - inclination angle.

In this calculation, a cantilever beam of length L with a moment of inertia of cross section I_y is considered. The beam is subjected to a concentrated load F located at a distance a from the free end.

As a result of calculations, the bending moment M at point X is determined. The bending moment varies from zero at the free end of the beam to the maximum values at the point of its fixation. The angles of inclination β and deflection Y are also determined, varying according to power-law dependence from zero at the fixed end to the extreme values at the free end of the beam.

For the calculation, the elastic modulus E of the beam should be specified.

Cantilever Beam Under Concentrated Load

Calculation of the Cantilever Beam Under Concentrated Load

INITIAL DATA

L - Beam length;

a - Load point coordinate (distance from the left end of the beam to the load point);

X - Coordinate of the point at which the bending moment, slope and deflection of the beam are calculated (distance from the left end of the beam to the calculated point);

F - Concentrated load;

I_x - Moment of inertia of the cross-section;

Е - Young's modulus of the beam material.

RESULTS DATA

M - Bending moment at the point X;

β - Beam slope angle at the point X;

Y - Beam deflection at the point X.

Metric Imperial

Beam length (L)

Distance (a)

Point coordinate (X)

Load (F)

Moment of inertia (I_x)

Young's modulus (Е)

Bending moment (М)

Angle of slope (β)

Deflection (Y)

BASIC FORMULAS

Bending moment, slope and deflection are determined by parametric equations depending on boundary conditions:

R₁ = 0 - Support reaction at the left end of the beam;
M₁ = 0 - Bending moment at the left end of the beam;
β₂ = 0 - Slope at the right end of the beam;
Y₂ = 0 - Deflection at the right end of the beam.

INITIAL DATA

L - Beam length;

a - Load point coordinate (distance from the left end of the beam to the load point);

X - Coordinate of the point at which the bending moment, slope and deflection of the beam are calculated (distance from the left end of the beam to the calculated point);

F - Concentrated load;

I_x - Moment of inertia of the cross-section;

Е - Young's modulus of the beam material.

RESULTS DATA

M - Bending moment at the point X;

β - Beam slope angle at the point X;

Y - Beam deflection at the point X.

MATERIALS PROPERTIES

Material	Young’s modulus Pa (psi)	Poisson’s ratio
Steel	1.86÷2.1×10¹¹ (2.7÷3.05×10⁷)	0.25÷0.33
Cast iron	0.78÷1.47×10¹¹ (1.1÷2.1×10⁷)	0.23÷0.27
Copper	1.0÷1.3×10¹¹ (1.45÷1.9×10⁷)	0.34
Tin bronze	0.74÷1.22×10¹¹ (1.1÷1.8×10⁷)	0.32÷0.35
Brass	0.98÷1.08×10¹¹ (1.4÷1.6×10⁷)	0.32÷0.34
Aluminum alloy	0.7×10¹¹ (1.0×10⁷)	0.33
Magnesium alloy	0.4÷0.44×10¹¹ (5.8÷6.4×10⁶)	0.34
Nickel	2.5×10¹¹ (3.6×10⁷)	0.33
Titanium	1.16×10¹¹ (1.7×10⁷)	0.32
Lead	0.15÷0.2×10¹¹ (2.2÷2.9×10⁶)	0.42
Zinc	0.78×10¹¹ (1.1×10⁷)	0.27
Glass	4.9÷5.9×10¹⁰ (7.1÷8.5×10⁶)	0.24÷0.27
Concrete	1.48÷2.25×10¹⁰ (2.1÷3.3×10⁶)	0.16÷0.18
Wood (along the grain)	8.8÷15.7×10¹⁰ (12.8÷22.8×10⁶)	-
Wood (across the grain)	3.9÷9.8×10¹⁰ (5.7÷14.2×10⁶)	-
Nylon	1.03×10¹⁰ (1.5×10⁶)	-

OTHER CALCULATORS

AREA MOMENTS OF INERTIA

BEAM CALCULATORS

TORSION OF BARS

CIRCULAR FLAT PLATES

BUCKLING

ELASTIC CONTACT

IMPACT LOADS

NATURAL FREQUENCIES

PRESSURED SHELLS

FLUID DYNAMIC

COMPOSITES

SPRINGS

THREAD CONNECTIONS

SHAFT CONNECTIONS

BEARINGS

DRIVES

FATIGUE

HEAT TRANSFER