STRENGTH OF MATERIALJJ 310 CHAPTER 4 (a)

BENDING STRESS*

Learning Outcomes:At the end of this lecture, student should be able to;

Understand the bending stress Name the symbols in the general bending formulaUnderstand the standard section tableUnderstand the centroidCalculate the second moment of area(inertia), I

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What is bending stress?Bending stress is the normal stress that is induced at a point in a body subjected to loads that cause it to bend.When a load is applied perpendicular to the length of a beam (with two supports on each end), bending moments are induced in the beam. *

Assumptions in bending theory1) The material is homogeneous/isotropic and obeys Hookes Law. (it is linearly elastic and will not deform plastically)2) The beam is initially straight with a cross section that is constant throughout the beam length.3) The material has same value of Young modulus both in compression and tensional condition.4) Cross-sections remain plane and perpendicular to the neutral surface after bending. 5) The beam is subject to pure bending. This means that the shear force is zero, and that no torsional or axial loads are present.

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General Bending Stress Formula

Where;M = Bending Moment (Nm)I = Second moment of area @ moment inertia (m4)E = Young modulus (N/m2 @ Pa)R = Radius of curvature (m) = Bending stress (N/m2 @ Pa)y = Distance from neutral axis (m)

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What is Centroid?The centroid of a body is the center of its mass.

The point at which it would be stable, or balance, under the influence of gravity.

It is also often called the center of gravity.*

Second moment of area for basic shape*

ShapeCentroidSecond moment of areaRectangular

*NA- Neutral axis

y = d/2

I = bd3/12 Circle

y = d/2

I = d4/64 @I = r4 /4

Formula to find second moment of area, ISingle shape

I = bd3 12Combination shape

I = ( INA + Ah2 )*

Example: Find I at neutral axis (Single shape)

Solution:I = bd3/12I = 20 (253) 12 I = 26.04 x 103 mm4 # @ b = 20 mm I = 26.04 x 10-9 m4 # d = 25 mm *20 mm25 mmPNmm4 m4 (x 10-12) bd

Steps to find I for combination shape*

Example: Calculate I about the x axis (combination shape)

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Step 1: Find A and y from x-x

* 10 40 b =10d = 40b = 50d = 2050 20 12

Part Area, A (mm2)y from x-x (mm)10 x 40 = 40040/2 = 2050 x 20 = 100020/2 + 40

Step 2: Find y and h *h = Distance from sentroid of each part to NA

Step 3: Find IPN and Ah2Part 1 Part 2*

Step 4: Find I= ( IPN + Ah2 )

Ixx = ( INA + Ah2 )

= ( INA + A1h12 ) + ( INA + A2h22 )

= [ (53.3 x 103) + (183.18 x 103) ] + [ (53.3 x 103) + (183.18 x 103) ] = (236.5 x 103) + (107.3 x 103)

= 343.76 x 103 mm4 @ 343.76 x 10-9 m4 #*mm4 m4 (x 10-12)

Exercise: Find y and I about the x axis Exercise 1Exercise 2*120 mm100 mmAnswer: y = 72.86 mm I xx = 32.92 x 106 mm440 mmxx30 mm30 mm120 mm120 mmAnswer: y = 52.5 mm I xx = 14.71 x 106 mm4