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Maximum Tensile Flexure Stress In A Section Of A Beam

Determine maximum flexure stress at section A-B of beam resting on simple supports as illustrated in figure 125.6(a).

**The strings**:
S_{7}P_{3}A_{31} (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, σ_{f} = (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 (in^{4}).

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)) = (zy^{3})/12

Clockwise moments are considered positive

Anti-clockwise moments are considered negative.

So, for moment about R_{1} (left simple support figure 125.6(a)):

ΣM_{1} = 0 = (P)(L/2) - (R_{2})(L)

Where L is distance between simple supports.

So, R_{2} = P/2.

Similarly, for moment about R_{2} (right simple support)

R_{1} = P/2.

Sum of vertical forces &Sima;F_{y} = 0

So, P - R_{1} - R_{2} = 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:

ΣM_{A-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 = (zy^{3})/12 (see figure 125.6(c))

c = y/2

So, flexure stress, σ_{f} = (Mc)/I = [(M)(y/2)]/(zy^{3})/12

= (3Px)/(zy^{2})

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.

Math

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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The Universe is composed of *matter* and *radiant energy*. *Matter* is any kind of *mass-energy* that moves with velocities less than the velocity of light. *Radiant energy* is any kind of *mass-energy* that moves with the velocity of light.

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