Beam Calculators

Cantilever Beam Under a Distributed Load

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 distributed load q varying from the value q₁ at the fixed end of the beam to the value q₂ 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 inclination angle β and deflection Y are also determined, varying according to power-law dependence from zero at the fixed end to 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 distributed load

Cantilever beam under distributed load calculation

INITIAL DATA

L - Beam length;

a - Coordinate of the left point of the load application (distance from the left end of the beam to the start 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);

q₁ - Value of distributed load at the right end of the beam;

q₂ - Value of distributed load at the point "a";

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)

Distributed load (q₁)

Distributed load (q₂)

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 - Coordinate of the left point of the load application (distance from the left end of the beam to the start 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);

q₁ - Value of distributed load at the right end of the beam;

q₂ - Value of distributed load at the point "a";

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)	-

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