Bending Forces Or Beam Me Up Scotty (Credit for many illustrations is given to McGraw Hill publishers and an array of internet search results)

Parallel Reading Chapter 6 Section 6.1 Introduction Section 6.2 Strain Displacement Analysis Section 6.3 Flexural Stress in Linear Elastic Beams (Do Reading Assignment Problem Set 6A)

Consider the Case of a Beam With a Load in the Middle The members on top are in a Squeeze Play, while the members on the bottom are in Tension.

Lets Take a Look at That If we are in tension on one side of the Bend and compression on the other, Somewhere there must be a neutral plain

As We Move Away from the Neutral Axis the Strain Varies Linearly

Hooke’s Law Now Tells Us About Stress in Beam Bending Where E is Young’s Modules

Now We Get A Tip from Statics The forces (stress * area) above and below the neutral plane have to be equal.

Only One Way that is True That neutral plain has to go right through the centroid of the beam

So What is a Centroid? (we hope the heck this is a review) The physical center The center of mass for the beam So someone tell me where the centroid is for the 4 X 6 beam? Where is it for the 3 X 8 beam?

So How do We Find the Centroid When its’ not Stupid Obvious 90 mm 20 mm 40 mm 30 mm

Someone's bound to come up with Integrating Over the Area 90 mm 90 mm 20 mm 40 mm A less painful option is usually Available. 30 mm Most of the objects we work with break-down into Simple parts where the centroid is obvious

Lets Peg the Obvious Centroids 90 mm 20 mm 10 mm 45 mm 15 mm 40 mm 20 mm 30 mm

Next We’ll Weight Each Obvious Centroid by the Area of It’s Object

Now We’ll Divide Through by Total Area We just nailed ourselves the Centroid of a T beam (Of course finding centroids is not a key topic of this course, but if we can’t do it, it will make our lives miserable for this course).

Back to Bent Beams When our beam deflects it bends along the arc of a circle of radius ρ through an angle of θ. The radius extends from the center point of the arc of the bend to the neutral plane.

If We Take a Closer Look at Deformation in a Cross-Section The plain of our cross-section remains A nice flat cross-section – But On the compression side our nice former Rectangle puffs out increasingly toward The top (prob not a surprise if we Remember Poisson’s ratio) And get increasingly skinny on the bottom As we into higher tension areas away From the neutral plain.

The Amount of Deflection is Related To a bunch of terms including the Bending moment on the beam, Young’s Modulus (a material Property), and something called I That comes from the geometry of The beam.

Similarly, the amount of thinning or thickening is proportional to material properties Any given cross section stays a plain but The Compression size fattens up The Tension size skinnies down Not surprisingly the amount of plumping out or Skinnieing down is proportional to Poisson’s Ratio

Lets Review Our Materials Properties Young’s Modulus is the slope of the line in a stress Strain plot. It relates change in length from a tension Or compression load to the stress

When Things Stretch in One Direction – They Skinny Up in the other

The Proportion is Called Poisson’s Ratio

So Let Make Sure We Understand the Terms for the amount of Deflection There is Young’s Modulus (We know what that is) It makes sense that we might not want Excessive deflection

Checking Out More Terms M is that bending moment couple that Is deflecting the beam.

And Then There is I Right now I’m not seeing what that is

Let’s Review that Moment Term Note its just the Force * lever arm y

Handily a Look at that Last Equation gives us I Obviously We call this term the Moment of Inertia (Yes we do hope this is a review for you from Statics)

Section Modulus is a Closely Related Term

That Inertia Term is a Measure of the Ability of a Beam to Resist Bending There are Precalculated Tables of these Values available For most Structural steel shapes

Lets Try Doing Something With This Stuff The allowable Tensile Stress is 12 Ksi The allowable Compressive Stress is 16 Ksi What is the largest Moment Couple I can put on this thing? 6 in 4 in

Our Basic Equation Limiting stress 3 in Neutral Axis or Plane Since our beam is symmetric our most limiting stress will be tension

Obviously We Need S 6 in = For a rectangle 4 in

Going for the Answer

Assignment 13 Do Problems 6.3-6 and 6.3-10

What if We Tried a Different Shape 6 4 For our rectangle A 24 section modulus allowed us to put A 288,000 in*lb moment on our beam while Staying in allowable tensile stress

Use a Table of Standard Wide Flange Beams First term after W (in this case 12 inches) Second term is the weight in lbs per foot If I pick a weight of 22 lbs/ft I will get A Section Modulus of 25.4 > 24 A wide W12X22 wide flange beam will carry slightly more load than our 6X4 beam.

Here’s the Kicker The area of my wide flange beam is 6.48 in^2 Instead of 24 in^2 for my rectangular beam I get more from only 25% of the material by Using a wide flange beam!

Is there a Down Side? 4 in 6 in S = 16 Maximum Moment = 12,000 * 16 = 192,000 in*lbs

So Could Anything Go Wrong with Our Wide Flange Beam? NA S for a W12X22 beam about The weak axis S= 2.31 < < 16 for our 4X6 Things really go to crap around The weak axis.

Assignment 14 Do problems 6.4-15 and 6.4-16

Controlling Cost With Beams We might want to consider a less expensive material

The Problem with Concrete Beams Like most brittle rock materials – they Have little tensile strength We already saw in our last problem that Tensile strength can form our design Limit Lk

The Practical Trick Put steel reinforcing Rebar near the tensile Edge of the beam

Theory of Reinforced Concrete Compression load area Neutral Plane or axis (Which is not at the centroid) Concrete holds The rods out at A distance to Maximize their Inertial value Tensile load rides entirely on the steal reinforcing rods

Convert the Steel Cross Sectional Area to an Equivalent Concrete Area Here is our concrete Compression area Hear is the equivalent concrete area To replace the rebar

Lets Walk Through This We have a concrete beam E for Concrete is 25 GPa It has steel rebar Reinforcement. E for Steel is 200 GPa This is the area of The steel

Conversion to an Equivalent Concrete area is Proportional to Young’s Modulus So an equivalent concrete area is *

We Now Need to Find the Neutral Axis (Which is not at the Centroid this time) b We exploit the fact that the moment Of the top part must be equal to the Moment of the steel equivilent

Set Up Our Quadratic Equation Moment of our Equivalent concrete (steel) section about axis Moment of Concrete About neutral axis

Solving Our Quadratic 177.87mm 302.13mm Let us now assume the bending moment on this beam is 175 KN*m lets check out the resulting stresses

We Know the Forces Above and Below the Neutral Plain are Equal and Opposite Looks like we need the value of I

Get the Inertia of the Upper Compression Block around Neutral Axis Contribution to Inertia of Beam From Compression Concrete

Now the Inertia of Our Equivalent Concrete Area

Solve I for Our Equivalent Beam System

Now Let Get the Stress in Our Concrete Compression Area Our given Moment Load (from our previous calculation)

Going for the Stress on the Steel Stress in equivalent concrete Is the same. But we remember the ratio of Our Young’s Modulus * 8 =