chapter 7a-part a

CIVL222 STRENGTH OF MATERIALS

Chapter 7a-Part A

Stresses in Beams – Pure Bending

Instructor: Assoc.Prof.Dr. Mürüde Çelikağ

Bending• Today’s Objective:

• Students will be able to:

a) Determine the stress in a beam member caused by bending

In-class Activities:• Flexural formula• Unsymmetrical bending•Composite Beams• Concept Quiz

CIVL222Dr. Mürüde Çelikağ2

Bending Deformation of a Straight Member



• When a bending moment is applied to a straight prismaticbeam, the longitudinal lines become curved and verticaltransverse lines remain straight and yet undergo a rotation



• A neutral surface is where longitudinal fibers of the materialwill not undergo a change in length.

• The bottom material is stretched and the top material iscompressed – leaving a neutral surface somewhere inbetween.



Assumptions:

a) Plane section remainsplane

b) Length of longitudinalaxis remains unchanged

c) Plane section remainsperpendicular to thelongitudinal axis

d) In-plane distortion ofsection is negligible

Before deformation

After deformation



Now lets look at a segment in isolation: x does not change inlength but x does.

xxx

x

0

lim

From the strain equation:

yx 0

lim

Substituting in terms of angle :

y



y

It should be noted that longitudinal normal strain varieslinearly with distance y from the neutral axis, ie:

yE

E

x

xx

Hooke’s Law

stress


The Flexure FormulaThe stress resultants that are related to the normal stress xacting on the cross section are

IdAy 2

Then

EIM moment-curvature equation

Bending stress variation

dAyEydAyEdAyM

ydAEdAN

AAAx

AAx

2


The Flexure Formula

Bending stress variation

But we also know that

yE

yE

1

Substitute into moment-curvature equation

yEEIM Solve for

IMy

Flexure Formula

EIM


The Flexure FormulaProcedure for analysisInternal moment

• Section member at point where bending or normal stress isto be determined and obtain internal moment M at thesection.

• Centroidal or neutral axis for cross-section must be knownsince M is computed about this axis.

• If absolute maximum bending stress is to be determined,then draw moment diagram in order to determine themaximum moment in the diagram.


The Flexure Formula


The Flexure Formula

AyydAA

• The z and y axes must pass through the centroid of the cross section.• Therefore NEUTRAL AXIS Passes through the centroid of the

cross section

From statics

ii

iii

ii

iii

A

Ayy

A

Axz

~ ,

~

Location of Neutral Axis


The Flexure Formula

Procedure for analysisSection property

• Determine moment of inertia I, of cross-sectional area aboutthe neutral axis.

• Methods used are discussed in Textbook Appendix A.

• Refer to the course book’s inside front cover for the values ofI for several common shapes.


The Flexure FormulaMoment of Inertia of composite area

A composite area is made by adding or subtracting a series of“simple shaped areas” like

Rectangles:

Triangles:

Circles:

3

centroidalx-axis

;12

bhI

3

centroidalx-axis

;36

bhI

4

cen tro id al cen tro id a lx -ax is y-ax is 4

rI I du e to sym m etry

3

centroidaly-axis 12

hbI

3

centroidaly-axis 36

hbI

2xy'y AdII 2

yx'x AdII PARALLEL-AXISTHEOREM


The Flexure Formula

Procedure for analysisNormal stress

• Specify distance y, measured perpendicular to neutral axisto point where normal stress is to be determined.

• Apply equation = -My/I, or if maximum bending stress isneeded, use max = Mc/I.

• Ensure units are consistent when substituting values into theequations


The Flexure Formula

+ve y

-ve y

x stress at anypoint y on the crosssection z

x IMy

CIVL22221

Example #1


Example #1 (Continued)



dF



M= FC×80=36000×80=2880000 N.mm

FC

FT


Example #2A beam having a tee-shaped cross section is subjected to equal 18 kN.m bendingmoments. As shown in left figure. The cross-sectional dimensions of the beamare shown in right figure. Determine:(a) The centroid location, the moment of inertia about the z axis, and the

controlling section modulus about the z axis.(b) The bending stress at point H, state whether the normal stress at H is tension

or compression.(c) The maximum bending stress produced in the cross section. State whether

the stress is tension or compression.

CIVL22227

Unsymmetric BendingMoment applied along principal axis

If y and z are the principal axes. ∫yz dA = 0(The integral is called the product of inertia)

FR = Σ Fx; 0 = ∫ σ dAA

(MR)y = Σ My; 0 = ∫ z σ dAA

(MR)z = Σ Mz; 0 = ∫ – y σ dAA


Unsymmetric Bending

Moment arbitrarily applied

= +

y

y

z

z

IzM

IyM


Unsymmetric Bending

= +

Alternatively, identify the orientation of the principal axes (ofwhich one is the neutral axis). Orientation of neutral axis:

tantany

z

IIa


Unsymmetric Bending

Note: The angle α, which is measured positive from the +zaxis toward the +y axis, will lie between the line of actionof M and the y axis; ie θ ≤ α ≤ 90°.

= +


Example #3


Composite BeamsAPPLICATIONS

Reinforced concrete beam Sandwich panel

Fig. 6.38(b) Fig. 6.38(a)


Composite Beams

Transformed homogeneous beam obtained through a transformation

factor:n = E1

E2

dF = σ dA = σ’ dA’

σ dz dy = σ’ n dz dy

σ = n σ’

and


Example #4


Example #5


1) Provided that the bending formation of a straight member is small and within elastic range. Which of the following statements is incorrect?

A) Plane section remains plane C) The length of the longitudinal axis remains unchanged

B) Cross section remains perpendicular D) In-plane distortion of cross section is to the longitudinal axis not negligible

2) Which of the following statements is incorrect for bending of a straight member?

A) Bending stress is proportional to C) bending stress is inversely the moment proportional to the moment of

inertia of the section

B) Bending stress is inversely proportional D) bending stress is not a function to the second moment of area of the section of the location

READING QUIZ


1) Which of the following statements is true?

The flexure formula for a straight member can be applied only

A)when bending occurs about axes that represent the principal axes of inertia for the section.

B) The principal axes have their origin at the centroid.C) The principal axes are orientated along an axis of symmetry, if there

is one, and perpendicular to it.D)All of the above.

CONCEPT QUIZ


chapter 7a-part a

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