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Reactions are the forces and/or couples acting at the supports 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:
V = Σ F
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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 the value V, the bending moment can be calculated:
M = ∫ (V · dx) + M0
Where:
• x = position on the beam measured along its length
• 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.
Figure 1: Coordinate system of a beam.
Moment and shear diagrams are 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.
Direct Stress Calculations
Magnitude of the direct stresses (tension and compressive) can be calculated from the direct stress formula:
σ = F / A
Where:
• F = tensile/compressive force
• A = cross-sectional shape area
At the point of maximum bending stress, the flexure formula gives the stress:
σmax = (M · c) / I
Where:
• 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
Shearing Stress
A beam carrying loads transverse to its axis will experience shearing force. The resulting shearing stress can be computed from:
Ï„ = (V · Q) / (I · t)
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
• V = shearing force
Figure 2: Compressive Stress and Tensile Stress in Beam.
Torsional Stress
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:
Ï„max = (T · c) / J
Where:
• T = moment due to torque
• c = distance from the neutral axis to the outermost fiber
• J = polar moment of inertia
Beam Deflection
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. The fundamental equation from which the elastic curve can be developed is:
d2y / dx2 = M / (E · I)
If loads are applied in vertical and horizontal plane, it is necessary to use the principle of superposition:
fresultant = √(fh2 + fv2)
Where fh = deflection in horizontal plane and fv = deflection in vertical plane. Using value of bending moment, slope can be calculated:
θ = ∫ [ M / (E · I) ] dx + C0
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Axial Deformation and Twisting
The shortening/prolongation due to a direct axial tensile/compressive load is computed from:
ΔL = (F · L) / (E · A)
The relative twisting angle (θ) is computed from:
θ = (T · L) / (G · J)
Where:
• L = length of the shaft over which the angle of twist is being computed
• G = modulus of elasticity of the shaft material in shear
Note: All applied loads are considered to pass through the center of gravity of the cross-section. The mass of the beam is not considered; if necessary, represent it with a distributed load.
Support and Section Types
There are two very common beam arrangements: Cantilever (supported at one fixed end) and Simply Supported (supported at two points). Both are "determinate" arrangements.
Figure 3: Determinate beam support arrangements.
For standard structural shapes, geometrical data must be entered according to the type:
- Circle: Diameter (D)
- Rectangle: Width (b), Height (h)
- Standard Shapes: Depth (w), Flange width (b), Web thickness (t), Flange thickness (t1), Area (A), Moment of Inertia (I), Section Modulus (S).
- Hollow Tube: Inner diameter (d), Outer diameter (D).
Figure 4: Defining the distance to a considered cross-section.
Modulus of Elasticity
Modulus of Elasticity in Tension (E): The stiffness of the material or resistance to deformation. It is the slope of the straight-line portion of the stress-strain diagram (E = σ / ε).
Modulus of Elasticity in Shear (G): The ratio of shearing stress to shearing strain. Relationship between E, G, and Poisson’s ratio (ν):
G = E / [ 2 · (1 + ν) ]
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