Beam Calculators

Beam with One End Fixed and Other Simply Supported Under a Distributed Load

In this calculation, a beam with fixed and simply supported ends, 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 given distance a from the simply supported end.

As a result of calculations, a bending moment M at point X is determined. The bending moment takes on a value of zero at the left end of the beam. Also, the inclination angle β, varying from zero at the fixed end to extreme values at the simply supported end, and the deflection Y, taking on zero values at the points of restraint are being determined.

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

Beam with fixed end and simply supported end under distributed load

Beam with fixed end and simply supported end 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:

М₁ = 0 - Bending moment at the left end of the beam;
Y₁ = 0 - Slope 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)	-

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