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**Strength of Materials I EGCE201 กำลังวัสดุ 1**

Instructor: ดร.วรรณสิริ พันธ์อุไร (อ.ปู) ห้องทำงาน: 6391 ภาควิชาวิศวกรรมโยธา โทรศัพท์: 66(0) ต่อ 6391

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**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.

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Pure Bending Independent of material model

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**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. except

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**compressed (compression) elongated (tension)**

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.

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**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

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The deformation d of JK d = L’-L = (r- y)q - rq = - yq The normal strain is max when y is the largest

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**Derive flexure formula**

Because From earlier,

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**Derive flexure formula (continued)**

This equation can be satisfied only if The first area moment of the cross section about its NA = 0

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**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

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**Discuss flexure stress**

Compression Top Surface (+y) Tension Bottom Surface (-y)

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**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.

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Example

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**Compute the area moment of inertia**

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**A normal stress in the x direction due to Mz**

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**A normal stress in the x direction due to My**

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**Transverse Deformations due to bending**

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**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.

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**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

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**Example – transformed section to all aluminum**

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**Try it for yourself at home Transform section to all steel**

Season’s Greeting! Try it for yourself at home Transform section to all steel

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**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 diagrams.

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**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.

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**Review of support conditions**

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**Shear forces and bending moments in beams**

Neither half is in equilibrium

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**The force imbalance that exists must be counteracted**

so that static equilibrium is maintained. This is done Through internal forces and moments.

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**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.

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= eliminate last term = 0

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**Diagram by inspection V = constant M = f(x) V = f(x) M = f(x2)**

V = no effect M = spike

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**Faculty of Engineering Mahidol University, Thailand**

Dr. Wonsiri Punurai (Bo)

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