Beam with fixed ends Calculator | Concentrated load

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Beam with Fixed Ends Under a Concentrated Load

In this calculation, a beam with both ends fixed, of length L with a moment of inertia of cross section Iy 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 is determined. The bending moment takes extreme, but opposite in sign values at the ends of the beam. Also, the inclination angle β and deflection Y are determined, being are equal to zero at the points of beam restraint.

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

Beam with fixed ends under concentrated load
Beam with fixed ends under concentrated load calculation

INITIAL DATA

L - Beam length;


a - Load point coordinate (distance from 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;


Ix - 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.

Beam length (L)

Distance (a)

Point coordinate (X)

Load (F)

Moment of inertia (Ix)

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:

β1 = 0 - Slope at the left end of the beam;
Y1 = 0 - Deflection 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 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;


Ix - 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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