When a beam is subjected to external loads, shear forces and bending moments develop in the beam. Therefore, a beam must resist these external shear forces and bending moments. The beam itself must develop internal resistance to resist shear forces and bending moments.
The stresses caused by the bending moments are called bending stresses. For beam design purposes, it is very important to calculate the shear stresses and bending stresses at various locations of a beam.
The bending stress varies from zero at the neutral axis to a maximum at the tensile and compressive side of the beam.
Procedures for determining bending stresses
Stress at a Given Point:
1.Use the method of sections to determine the bending moment M at the cross section containing the given point.
2. Determine the location of the neutral axis.
3. Compute the moment of inertia I of the cross- sectional area about the neutral axis.
4. Determine the y-coordinate of the given point. Note that y is positive if the point lies above the neutral axis and negative if it lies below the neutral axis.
5.Compute the bending stress from σ = -My / I. If correct sign are used for M and y, the stress will also have the correct sign (tension positive compression negative).
Bending Stress in
Symmetrical Cross Sections
Flexural Stress varies directly linearly with distance from the neutral axis. Thus for a symmetrical sections the compressive and tensile stresses will be the same. This will be desirable if the material is both equally strong in tension and compression. Symmetrical bending stresses arises in beams which have the bending axis (N.A) coincide with either of symmetrical axis. Singly or doubly symmetrical cross sections, are shown below.
If the neutral axis is an axis of symmetrical of the cross section, the
maximum tensile and compression bending stresses are equal in
magnitude and occur at the section of the largest bending
moment.
The following procedure is recommended for
determining the maximum bending stress in a prismatic beam:
1.Draw the bending moment diagram and Identify the maximum bending moment Mmax (disregard the sign)
2.Compute the moment of inertia I of the cross- sectional area
about the neutral axis.
3. Calculate the maximum bending stress from
σ
max = [Mmax]ymax / I,
where ymax is the distance from the neutral axis to the top or bottom
of the cross section.
Unsymmetrical Cross Sections:
Symmetrical sections will be desirable if the material is both equally strong in tension and compression. However, some materials are strong in compression than in tension. It is therefore desirable to use a beam with unsymmetrical cross section giving more area in the compression part making the stronger fiber located at a greater distance from the neutral axis than the weaker fiber. Unsymmetrical bending stress arises in beams which have a bending axis is not symmetrical about cross sections. Some of these sections are shown below.
If the neutral axis is not an axis of symmetry of the cross section, the maximum tensile and compressive bending stresses may occur at different sections.
1. Draw the bending moment diagram. Identify the largest positive and negative bending moments.
2. Determine the location of the neutral axis and record the distances y top and y bottom from the neutral axis to the top and bottom of the cross section.
3. Compute the moment of inertia I of the cross section about the neutral axis.
4. Calculate the bending stresses at the top and bottom of the cross section where the largest positive bending moment occurs from σ = -My / I. At the top of the cross section, where y = ytop,we obtain σtop = -Mytop/ I.
At the bottom of the cross section, we have y = - ybot, so that σbot = Mybot/ I.
5.Repeat the calculations for the cross section that carries the largest negative bending moment.
6.Inspect the four stresses thus computed to determine the largest tensile (positive) and compressive (negative) bending stresses in the beam.
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