Knowing that for the extruded beam shown the allowable stress is 120 MPa in tension and 150 MPa

Knowing that for the extruded beam shown the allowable...

Key Concepts

Bending Stress Distribution

This concept explains that when a beam is subjected to a bending moment, the internal stresses vary linearly from the neutral axis. The fibers farthest from the neutral axis experience the maximum tensile and compressive stresses, which can be calculated using the bending formula, sigma = M*y/I, where y is the distance from the neutral axis and I is the moment of inertia of the cross?section. This distribution is foundational in assessing how applied moments affect the structural integrity of beams.

Allowable Stress

Allowable stress refers to the maximum stress that a material is permitted to carry safely under design conditions in different modes, such as tension and compression. In design, these limits ensure that the material will not experience failure before the ultimate load is reached. When the allowable stress values are different in tension and compression, the design must consider the more limiting condition to ensure structural safety.

Moment Capacity

Moment capacity is the maximum bending moment a beam can withstand without exceeding the material's allowable stress limits. It is determined by combining the material properties with the geometric properties of the section (such as the moment of inertia and the distance to the extreme fibers). In situations where the allowable tensile and compressive stresses differ, the maximum sustainable moment is controlled by the more restrictive of the two limits, ensuring that neither the tension nor the compression exceeds its allowable value.

Transcript

00:01 Excuse my diagram, but if you're getting this out of a text, you've got access to the better diagram.

00:06 And we are asked to determine the largest permissible load, w, for a beam.

00:10 Knowing the normal allowable stress is positive 80 megapascals in tension and negative 130 megapascals in compression.

00:19 Okay, so the relationships at the supports will be, and that will equal the sum of my y divided by 2.

00:33 So my rb and that will be my work times 0 .9 divided by 2 that'll be 0 .45 work okay my sheer forces so a 0 .0 v of b equal a minus 0 .2 w so my v of b equals a minus 0 .2 w so my v of b equals 0 .2 w negative and my d positive will equal.

01:50 So that will equal 0, negative 0 .2 plus 0 .45w.

02:04 That'll be w equals 0 .25w.

02:10 And then i've got c plus and c minus and d plus and d minus.

02:20 The c plus is do vc minus first.

02:25 That will equal vb plus and a 0 .5 w that'll equal 0 .25w minus 0 .5 w equals 0 .25w negative and my c plus will equal and that will equal negative negative 0 .252 .5 w plus 0 .45w equal 0 .2w and d will equal 0 .0.

03:22 Okay, got those.

03:24 Now let's do our bending moments.

03:26 I'm sick of that.

03:34 And for a, it'll be 0.

03:39 For b, it'll be a plus the derivative of b to a, a b dx.

03:46 So this will equal 0 plus 1 1⁄2 times negative 0 .2 w, and that will equal negative 0 .02w.

04:13 Then let's do e, that'll be mb plus e to b, and that will be equal to negative 0 .022 .2.

04:36 Minus 1 1 1⁄2 times 0 .25 times 0 .25 w and that will equal 0 .01125 w.

04:55 The centroid of section will equal and that will equal 1 ,200 plus 70 plus times 70 plus 1200 times 30 divided by 2 ,400 and that will equal 50 millimeters.

05:48 The moment of inertia of section will equal so that will be 40 40 times 10 to the third plus 1200 times 10 to the 2 to the 2nd plus 360 times 10 to the third plus 1 ,200 times 20 to the second...

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