# Beam calculations. Part I

In this section, you can perform online calculations of beams under concentrated load. Calculations determine the deflection, the angle of rotation, and the bending moment at an arbitrary given point of the beam under various boundary conditions.

### Initial data:

L – beam length, millimeter;

a – coordinate of the point of application of the concentrated load, millimeters;

X – the coordinate of the solution point, millimeters;

F – load, newtons;

I_{x} – moment of inertia of the section, meters ^{4};

Е – elastic modulus of the material of the beam, pascals

## Beam calculation # 1.1

Calculation of a cantilever beam under concentrated load.

#### Boundary conditions:

R_{L} = 0 – support reaction at the extreme left point;

M_{L} = 0 – bending moment at the extreme left point;

θ_{R} = 0 – angle of rotation at the extreme right point;

Y_{R} = 0 – deflection of the beam at the extreme right point.

BEAM UNDER CONCENTRATED LOAD #.1

Beam length L, mm

Distance A, mm

X coordinate, mm

Load F, N

Section moment of inertia I_{y}, m^{4}

Elastic modulus Е, Pa

Monent at point Х, N*m

Angle of rotation at point X, deg

Deflection at point X, mm

©Copyright Caetec 2020

*Ref 8 Table 8.1*

## Beam calculation # 2.1

Calculation of the beam with a clamped end and a sliding support under concentrated load.

#### Boundary conditions:

R_{L} = 0 – support reaction at extremely left point;

θ_{L} = 0 – angle of rotation at extremely left point;

θ_{R} = 0 – angle of rotation at extremely right point;

Y_{R} = 0 – deflection at extremely right point.

BEAM UNDER CONCENTRATED LOAD #.2

Beam length L, mm

Distance A, mm

X coordinate, mm

Load F, N

Section moment of inertia I_{y}, m^{4}

Elastic modulus Е, Pa

Monent at point Х, N*m

Angle of rotation at point X, deg

Deflection at point X, mm

©Copyright Caetec 2020

*Ref 8 Table 8.1*

## Beam calculation # 3.1

Calculation of the beam with a clamped end and a hinged support under concentrated load.

#### Boundary conditions:

М_{L} = 0 – bending moment at extremely left point;

Y_{L} = 0 – deflection at extremely left point;

θ_{R} = 0 – angle of rotation at extremely right point;

Y_{R} = 0 – deflection at extremely right point.

BEAM UNDER CONCENTRATED LOAD #.3

Beam length L, mm

Distance A, mm

X coordinate, mm

Load F, N

Section moment of inertia I_{y}, m^{4}

Elastic modulus Е, Pa

Monent at point Х, N*m

Angle of rotation at point X, deg

Deflection at point Х, mm

©Copyright Caetec 2020

*Ref 8 Table 8.1*

## Beam calculation # 4.1

Calculation of the beam with pinched ends under concentrated load.

#### Boundary conditions:

θ_{L} = 0 – angle of rotation at extremely left point;

Y_{L} = 0 – deflection at extremely left point;

θ_{R} = 0 – angle of rotation at extremely right point;

Y_{R} = 0 – deflection at extremely right point.

BEAM UNDER CONCENTRATED LOAD #.4

Beam length L, mm

Distance A, mm

X coordinate, mm

Load F, N

Section moment of inertia I_{y}, m^{4}

Elastic modulus Е, Pa

Moment at point X, N*m

Angle of rotation at point X, град

Deflection at point X, mm

©Copyright Caetec 2020

*Ref 8 Table 8.1*

## Beam calculation # 5.1

Calculation of the beam with articulated supports under concentrated load.

#### Boundary conditions:

М_{L} = 0 – bending moment at extremely left point;

Y_{L} = 0 – deflection at extremely left point;

М_{R} = 0 – bending moment at extremely right point;

Y_{R} = 0 – deflection at extremely right point.

BEAM UNDER CONCENTRATED LOAD #.5

Beam length L, mm

Distance A, mm

X coordinate, mm

Load F, N

Section moment of inertia I_{y}, m^{4}

Elastic modulus Е, Pa

Moment at point X, N*m

Angle of rotation at point X, deg

Deflection at point X, mm

©Copyright Caetec 2020

*Ref 8 Table 8.1*

## Beam calculation # 6.1

Calculation of the beam with articulated and sliding supports under concentrated load.

#### Boundary conditions:

R_{L} = 0 – support reaction at extremely left point;

θ_{L} = 0 – angle of rotation at extremely left point;

М_{R} = 0 – bending moment at extremely right point;

Y_{R} = 0 – deflection at extremely right point.

BEAM UNDER CONCENTRATED LOAD #.6

Beam length L, mm

Distance A, mm

X coordinate, mm

Load F, N

Section moment of inertia I_{y}, m^{4}

Elastic modulus Е, Pa

Moment at point X, N*m

Angle of rotation at point X, deg

Deflection at point X, мм

©Copyright Caetec 2020

*Ref 8 Table 8.1*

## Other calculators

– round plates with linearly distributed load

– round plates with uniformly distributed pressure

– circular plates with variable distributed pressure

©Llc “Caetec”, 2020. Any use or copying of materials or a selection of materials from the site may be carried out only with the permission of the author (copyright holder) and only if there is a link to the site www.caetec.ru