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Strength of Materials I EGCE201 กำลังวัสดุ 1. Instructor: ดร.วรรณสิริ พันธ์อุไร ( อ . ปู ) ห้องทำงาน : 6391 ภาควิชาวิศวกรรมโยธา E-mail: โทรศัพท์ : 66(0) 2889-2138 ต่อ 6391. Symmetric Bending of Beams.

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strength of materials i egce201 1

Strength of Materials I EGCE201กำลังวัสดุ 1

Instructor: ดร.วรรณสิริ พันธ์อุไร (อ.ปู)

ห้องทำงาน:6391 ภาควิชาวิศวกรรมโยธา


โทรศัพท์: 66(0) 2889-2138ต่อ6391

symmetric bending of beams
Symmetric Bending of Beams
  • A beam is any long structural member on which loads act perpendicular to the longitudinal axis.

Learning objectives

  • Understand the theory, its limitation and its applications for strength based design and analysis of symmetric bending of beams.
  • Develop the discipline to visualize the normal and shear stresses in symmetric bending of beams.
pure bending
Pure Bending

Independent of

material model

deformation of symmetric member
Deformation of symmetric member

Under action of M and M’, the member will bend but will remain symmetric with respect to the plane containing the couples.







There exists a surface // to the upper and lower faces of the member

(see as a line on the cross-section) where no elongation and the bending

normal stress is zero. This surface is called the neutral axis.


The length of arc DE for both deformed and undeformed

L = rq

Consider an arc some distance y above the neutral surface

The length of arc JK can be expressed as

L’ = (r- y)q


The deformation d of JK

d = L’-L = (r- y)q - rq = - yq

The normal strain is max when y is the largest


Derive flexure formula


From earlier,


Derive flexure formula (continued)

This equation can be satisfied only if

The first area moment of the cross section about its NA = 0


Derive flexure formula (continued)

Taking the moment about the z axis = 0

is the 2nd area moment of the cross section w.r.t. z axis


Discuss flexure stress


Top Surface (+y)


Bottom Surface (-y)


Not only bending about z axis produces a normal

stress in x direction but also bending about y axis.

If the bending moment is about the y axis, a similar

relationship exists.

member of several materials
Member of several materials
  • Assume a bar is made of two different materials bonded together. The bar will deform as previously shown.
  • The normal strain in x direction will vary linearly with distance from the N.A.

Method of transformed section

The resistance to bending would be

the same if each section were made

of the same material,where the 2nd

material was multiplied by n


Season’s Greeting!

Try it for yourself at home

Transform section to all steel

beams bending analysis
Beams bending analysis
  • Beams carry loads perpendicular to their longitudinal axis.
  • Internal shear forces and
  • bending moments develop
  • along the span of a beam.

In designing a beam, it is critical to determine the internal

shear force (V) and bending moment distribution (M). This

is accomplished by constructing shear and bending moment



Steps in constructing a V and M diagram

In general, the load distribution across the width of the

beam is assumed to be applied uniformly. Therefore,

a beam can be analyzed in 2 dimensions rather than 3.

1. Determine the reactions at each support.


Shear forces and bending moments in beams

Neither half is in equilibrium


The force imbalance that exists must be counteracted

so that static equilibrium is maintained. This is done

Through internal forces and moments.


Calculating internal V and M as a function of x by isolating

a segment of beam a distance x from the left end whose

width is dx.



eliminate last term = 0

diagram by inspection
Diagram by inspection

V = constant

M = f(x)

V = f(x)

M = f(x2)

V = no effect

M = spike