Maximum Tensile Flexure Stress In A Section Of A Beam

Strings (SiPjAjk) = S7P3A31     Base Sequence = 12735     String Sequence = 12735 - 3 - 31

Expressions Of Pj Problems
Maximum Tensile Flexure Stress In A Section Of A Beam
Math Determine maximum flexure stress at section A-B of beam resting on simple supports as illustrated in figure 125.6(a).

The strings: S7P3A31 (Force - Pull).

The math:
Pj Problem of Interest is of type force (pull). Flexure Stress (bending stress) can be tensile or compressive.
Tensile flexure stress: convex side of a bent beam
Compressive flexure stress: concave side of a bent beam.

Assumptions: Principles of consistent deformation and superposition are applicable. Elasticity holds.
Beam is relatively weightless and has a neutral axis (line of zero stress in a member subject to bending).
Simple supports are those not capable of exerting a moment on the beam.

Equations of Interest:
flexure stress, σf = (Mc)/I
M = moment applied to the section bearing stress of interest (lb in)
c = distance from the neutral axis of cross section to the outermost edges of the section (inches).
I = moment of inertia of the section about its neutral axis (in4).
Summation of moments about a point or section, ΣM = 0.
Sum of forces in any direction, ΣF = 0.
Moment of inertia for a rectangular section (figure 125.6(c)) = (zy3)/12
Clockwise moments are considered positive
Anti-clockwise moments are considered negative.

So, for moment about R1 (left simple support figure 125.6(a)):
ΣM1 = 0 = (P)(L/2) - (R2)(L)
Where L is distance between simple supports.
So, R2 = P/2.

Similarly, for moment about R2 (right simple support)
R1 = P/2.

Sum of vertical forces &Sima;Fy = 0
So, P - R1 - R2 = 0 (down positve, up negative).

figure 125.6 illustrates free body diagram of the beam to the right of section A-B.
Vertical shear load acts on section A-B.
So, summation of moments about A-B gives:
ΣMA-b = 0 = Px/2 - M.
So, M = Px/2.
Where M = moment needed in beam in order to maintain equilibrium of the free body

For a rectangular section:
I = (zy3)/12 (see figure 125.6(c))
c = y/2
So, flexure stress, σf = (Mc)/I = [(M)(y/2)]/(zy3)/12
= (3Px)/(zy2)

The portion of beam to the right can also be used to calculate the moment M.
In this case, shear load P/2 in section acts upward
Moment M, in section is counterclockwise
Distance from section to right simple support is L-x
P is at play distance 1/2 from right support. The point . is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.
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