Beam Calculators

Beam with Simply Supported Ends Under a Concentrated Load

In this calculation, a beam with both ends simply supported, 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 left end.

As a result of calculations, the bending moment M at point X, taking zero values at the ends of the beam is determined. Also, the inclination angle β, taking the extreme values at the points of restraint, and the deflection Y, equal to zero at these points, are determined.

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

Beam with simply supported ends under concentrated load

Beam with simply supported ends under concentrated load calculation

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:

М₁ = 0 - Bending moment at the left end of the beam;
Y₁ = 0 - Deflection at the left end of the beam;
М₂ = 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