# Beam calculation. Part III

In this part, the calculations of beams under action of a bending moment are performed. 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, in millimeters;

a – coordinate of the load application point, millimeters;

X – coordinate of the solution point, millimeters;

T – bending moment, newtons×meter;

Ix – moment of inertia, meters 4;

Е – elastic modulus, pascal.

## Расчет балки # 3.1

Calculation of a cantilever beam under moment load.

#### Boundary conditions:

RL = 0 – support reaction at extremely left point;

ML = 0 – bending moment at extremely left point;

θR = 0 – angle of rotation at extremely right point;

YR = 0 – deflection at extremely right point.

BENDING BEAM #1

Beam length L, mm

Distance A, mm

X coordinate, mm

Bending moment T, N*m

Section moment of inertia Iy, m4

Elastic modulus Е, Pa

Moment at point X, N*m

Angle of rotation at point X, deg

Deflection at point X, mm  www.caetec.ru

Ref 8 Table 8.1

## Beam calculation # 3.2

Calculation of the beam with a clamped end and a sliding support under the action of a bending moment.

#### Boundary conditions:

RL = 0 – support reaction at extremely left point;

θL = 0 – angle of rotation at extremely left point;

θR = 0 – angle of rotation at extremely right point;

YR = 0 – deflection at extremely right point.

BENDING BEAM #2

Beam length L, mm

Distance A, mm

X coordinate, mm

Moment T, N*m

Section moment of inertia Iy, m4

Elastic modulus Е, Pa

Moment at point X, N*m

Angle of rotation at point X, deg

Deflection at point X, mm  www.caetec.ru

Ref 8 Table 8.1

## Beam calculation # 3.3

Calculation of the beam with a clamped end and a hinged support under the action of a bending moment.

#### Boundary conditions:

МL = 0 – bending moment at extremely left point;

YL = 0 – deflection at extremely left point;

θR = 0 – angle of rotation at extremely right point;

YR = 0 – deflection at extremely right point.

BENDING BEAM #3

Beam length L, mm

Distance A, mm

X coordinate, mm

Moment T, N*m

Section moment of inertia Iy, m4

Elastic modulus Е, Pa

Moment at point X, N*m

Angle of rotation at point X, deg

Deflection at point X, mm  www.caetec.ru

Ref 8 Table 8.1

## Beam calculation # 3.4

Calculation of the beam with pinched ends under the action of a bending moment.

#### Boundary conditions:

θL = 0 – angle of rotation at extremely left point;

YL = 0 – deflction at extremely left point;

θR = 0 – angle of rotation at extremely right point;

YR = 0 – deflection at extremely right point.

BENDING BEAM #4

Beam length L, mm

Distance A, mm

X coordinate, mm

Moment T, N*m

Section moment of inertia Iy, m4

Elastic modulus Е, Pa

Moment at point X, N*m

Angle of rotation at point X, deg

Deflection at point X, mm  www.caetec.ru

Ref 8 Table 8.1

## Beam calculation # 3.5

Calculation of the beam with articulated supports under the action of bending moment.

#### Boundary conditions:

МL = 0 – bending moment at extremely left point;

YL = 0 – deflection at extremely left point;

МR = 0 – bending moment at extremely right point;

YR = 0 – deflection at extremely right point.

BENDING BEAM #5

Beam length L, mm

Distance A, mm

X coordinate, mm

Moment T, N*m

Section moment of inertia Iy, m4

Elastic modulus Е, Pa

Moment at point X, N*m

Angle of rotation at point X, deg

Deflection at point X, mm  www.caetec.ru

Ref 8 Table 8.1

## Beam calculation # 3.6

Calculation of the beam with articulated and sliding supports under the action of bending moment.

#### Boundary conditions:

RL = 0 – support reaction at extremely left point;

θL = 0 – angle of rotation at extremely left point;

МR = 0 – bending moment at extremely right point;

YR = 0 – deflection at extremely right point.

BENDING BEAM #6

Beam length L, mm

Distance A, mm

X coordinate, mm

Moment T, N*m

Section moment of inertia Iy, m4

Elastic modulus Е, Pa

Moment at point X, N*m

Angle of rotation at point X, deg

Deflection at point X, mm  www.caetec.ru 