### Stresses and Deformations in Beams

Reactions are the forces and/or couples acting at the supports and holding the beam and holding the beam in place. In some cases the user should enter a distributed load to account for the weight of the beam.

The shear V effective on a section is the algebraic sum of all forces acting parallel to and on one side of the section,

The bending moment is the algebraic sum of the moments due to applied loads and other applied moments to one side of the section of interest. Using value V bending moment can be calculated

Moment and shear diagram constructed by plotting to scale the particular entity as the ordinate for each section of the beam. Such diagrams show in continuous form the variation among the length of the beam.

Magnitude of the direct stresses (tension and compressive) can be calculated from the direct stress formula

where C0 = constant of integration evaluated from the boundary conditions.

The shortening/prolongation due to a direct axial tensile/compressive load is computed from

where

= total deformation of the member carrying the axial load

F = direct axial load

L = original total length of the member

E = modulus of elasticity of the material

A = cross-sectional area of the member

The following formula computes the relative elongation due to a direct axial tensile (compressive) load

When a shaft is subjected to a torque, it undergoes a twisting in which one cross-section is rotated relative to the other cross section in the shaft.

The relative twisting angle is defined

The angle of twist is computed from

where

= angle of twist (radians)

L = length of the shaft over which the angle of twist is being computed

G = modulus of elasticity of the shaft material in shear

The mass of the beam is not considered. If the consideration of the beam's mass is necessary, then represent it with a distributed load.

There are two very common beam arrangements. A beam that is supported at one fixed end is commonly referred to as a cantilever. The other is simply supported at two points along the length of the beam. Both of these arrangements are statically defined and are referred to as “determinate.”

The beam that is supported at one end can experience shearing force, longitudinal reaction, twisting, and bending moments.

The beam supported at two points can result in shear forces and reactions due to axial and shear forces.

Distance to the desired cross-section, where the calculation of the force factors and deformations is needed, is defined with “Distance to considered cross-section” variable.

For standard structural shapes, click on the table icon and a menu of standard shapes will appear and the software will automatically input the geometric values into the input page. According to the selected cross-section type the following geometrical data must be entered:

D diameter mm

h height mm

b flange width mm

t web thickness mm

t1 flange thickness mm

A area mm^2

IX-X moment of inertia mm^4

IY-Y moment of inertia mm^4

SX-X section modulus mm^3

SY-Y section modulus mm^3

D outer diameter mm

For the part of the stress-strain diagram that is straight, stress is proportional to strain, and the value of E is the constant of proportionality. That is,

This is the slope of the straight-line portion of the diagram. The modulus of elasticity indicates the stiffness of the material, or the resistance to deformation.

The modulus of elasticity in shear (G) is the ratio of shearing stress to shearing strain. This property indicates a material’s stiffness under shear loading, that is, the resistance to shear deformation. Generally this material property is available from published data. It is commonly provided in this software by clicking on the table icon and selecting the appropriate material.

There is a simple relationship between E, G and Poisson’s ratio:

where

E = modulus of elasticity in tension

= Poisson’s ratio

This equation is valid within the elastic range of the material.

The shearing forces input list “Forces and distributed loads” has the following parameters: distance from the beam’s beginning to the application point “Distance”; length of a distributed load “Length”, (should be 0 for a concentrated force); force/load value; angle of projection’s inclination of the force’s action line on the OYZ plane to Z-axis “a ”. The “Value” sign defines the direction of the force’s action. Force projections on coordinate axis are found using the method of superposition.

The list of bending moments has the following parameters: distance from the beam’s beginning to the application point “Distance”; value of the moment “Value” and angle of inclination of the moment action plane to Z-axis. The “Value” sign defines the direction of the moment’s action. Moment projections on OXY and OXZ planes are found using the method of superposition.

“Twisting moments” list is used for twisting moments input. The list has the following parameters:

distance from the beam’s beginning to the application point, “Distance”, in,

value of the moment, “Value”, lb.in. or N.mm The sign defines the direction of application. The default is shown in the corresponding graphic help.

The shear V effective on a section is the algebraic sum of all forces acting parallel to and on one side of the section,

The bending moment is the algebraic sum of the moments due to applied loads and other applied moments to one side of the section of interest. Using value V bending moment can be calculated

where

x = position on the beam measured along its length

M0 = constant of integration evaluated from the boundary conditions.

M0 = constant of integration evaluated from the boundary conditions.

A bending moment that bends a beam convex downward (tensile stress on bottom fiber) is considered positive, while convex upward (compressive on bottom fiber) is negative.

Moment and shear diagram constructed by plotting to scale the particular entity as the ordinate for each section of the beam. Such diagrams show in continuous form the variation among the length of the beam.

Magnitude of the direct stresses (tension and compressive) can be calculated from the direct stress formula

where

F = tensile/compressive force

A = cross sectional shape area

At the point of maximum bending stress, the flexure formula gives the stress

A = cross sectional shape area

At the point of maximum bending stress, the flexure formula gives the stress

M = magnitude of the bending moment in the section

I = moment of inertia of the cross section with respect to its neutral axis

c = distance from the neutral axis to the outermost fiber of the beam cross section

I = moment of inertia of the cross section with respect to its neutral axis

c = distance from the neutral axis to the outermost fiber of the beam cross section

A beam carrying loads transverse to its axis will experience shearing force. The resulting shearing stress can be computed from

where

I = rectangular moment of inertia of the cross section of the beam

t = thickness of the section at the place where the shearing stress is to be computed

Q = First moment of the area to the outside of the axis of interest with respect to the overall centroidal axis of the cross section of interest

V =shearing force

When a torque is applied to a member, it tends to deform by twisting, causing a rotation of one part of the member relative to another. The value of the maximum torsion shear stress can be computed

t = thickness of the section at the place where the shearing stress is to be computed

Q = First moment of the area to the outside of the axis of interest with respect to the overall centroidal axis of the cross section of interest

V =shearing force

When a torque is applied to a member, it tends to deform by twisting, causing a rotation of one part of the member relative to another. The value of the maximum torsion shear stress can be computed

where

T = moment due to torque

c = distance from the neutral axis to the outermost fiber of the beam cross section

J = polar moment of inertia

When the beam is subjected to bending, the fibers on one side elongate, while the fibers on the other side shorten. These changes in length cause the beam to deflect. All points on the beam expect those directly over the support fall below their original position. The fundamental equation from which the elastic curve of a bent beam can be developed and the deflection of any beam obtained is,

c = distance from the neutral axis to the outermost fiber of the beam cross section

J = polar moment of inertia

When the beam is subjected to bending, the fibers on one side elongate, while the fibers on the other side shorten. These changes in length cause the beam to deflect. All points on the beam expect those directly over the support fall below their original position. The fundamental equation from which the elastic curve of a bent beam can be developed and the deflection of any beam obtained is,

If the loads are applied in vertical and horizontal plane or the forces are applied at angle to planes, in this case it is necessary to use a principle of superposition. Therefore result deflection at specific locations is computed from

where

fh = deflection in horizontal plane

fv = deflection in vertical plane

fv = deflection in vertical plane

Using value of bending moment, slope can be calculated

where C0 = constant of integration evaluated from the boundary conditions.

The shortening/prolongation due to a direct axial tensile/compressive load is computed from

where

= total deformation of the member carrying the axial load

F = direct axial load

L = original total length of the member

E = modulus of elasticity of the material

A = cross-sectional area of the member

The following formula computes the relative elongation due to a direct axial tensile (compressive) load

When a shaft is subjected to a torque, it undergoes a twisting in which one cross-section is rotated relative to the other cross section in the shaft.

The relative twisting angle is defined

The angle of twist is computed from

where

= angle of twist (radians)

L = length of the shaft over which the angle of twist is being computed

G = modulus of elasticity of the shaft material in shear

**All applied loads, forces and moments are considered to pass through the center of gravity of the cross-section.**__Note:__

The mass of the beam is not considered. If the consideration of the beam's mass is necessary, then represent it with a distributed load.

__Type of beam support__There are two very common beam arrangements. A beam that is supported at one fixed end is commonly referred to as a cantilever. The other is simply supported at two points along the length of the beam. Both of these arrangements are statically defined and are referred to as “determinate.”

The beam that is supported at one end can experience shearing force, longitudinal reaction, twisting, and bending moments.

The beam supported at two points can result in shear forces and reactions due to axial and shear forces.

__Distance to considered cross-section__

Distance to the desired cross-section, where the calculation of the force factors and deformations is needed, is defined with “Distance to considered cross-section” variable.

__Type of cross-sectional shape__

For standard structural shapes, click on the table icon and a menu of standard shapes will appear and the software will automatically input the geometric values into the input page. According to the selected cross-section type the following geometrical data must be entered:

__Circle:__D diameter mm

__Rectangle:__

b width mmh height mm

__Shapes:__

w depth mmb flange width mm

t web thickness mm

t1 flange thickness mm

A area mm^2

IX-X moment of inertia mm^4

IY-Y moment of inertia mm^4

SX-X section modulus mm^3

SY-Y section modulus mm^3

__Hollow round tube:__

d inner diameter mmD outer diameter mm

__Modulus of elasticity in tension__

For the part of the stress-strain diagram that is straight, stress is proportional to strain, and the value of E is the constant of proportionality. That is,

This is the slope of the straight-line portion of the diagram. The modulus of elasticity indicates the stiffness of the material, or the resistance to deformation.

__Modulus of elasticity in shear__The modulus of elasticity in shear (G) is the ratio of shearing stress to shearing strain. This property indicates a material’s stiffness under shear loading, that is, the resistance to shear deformation. Generally this material property is available from published data. It is commonly provided in this software by clicking on the table icon and selecting the appropriate material.

There is a simple relationship between E, G and Poisson’s ratio:

where

E = modulus of elasticity in tension

= Poisson’s ratio

This equation is valid within the elastic range of the material.

__Forces and distributed loads__

The shearing forces input list “Forces and distributed loads” has the following parameters: distance from the beam’s beginning to the application point “Distance”; length of a distributed load “Length”, (should be 0 for a concentrated force); force/load value; angle of projection’s inclination of the force’s action line on the OYZ plane to Z-axis “a ”. The “Value” sign defines the direction of the force’s action. Force projections on coordinate axis are found using the method of superposition.

__Note:__Distributed load has unit “lb/in” or "N/mm".__Bending moments__The list of bending moments has the following parameters: distance from the beam’s beginning to the application point “Distance”; value of the moment “Value” and angle of inclination of the moment action plane to Z-axis. The “Value” sign defines the direction of the moment’s action. Moment projections on OXY and OXZ planes are found using the method of superposition.

__Twisting moments__

“Twisting moments” list is used for twisting moments input. The list has the following parameters:

distance from the beam’s beginning to the application point, “Distance”, in,

value of the moment, “Value”, lb.in. or N.mm The sign defines the direction of application. The default is shown in the corresponding graphic help.

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