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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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**Beam Deflections As a beam is loaded,**

different regions are subjected to V and M the beam will deflect x x Recall the curvature equation The slope (q) & deflection (y) at any spanwise location can be derived

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**A cantilever with a point load**

The moment is M(x) = -Px The curvature equation for this case is The curvature is at A and finite at B Obviously, the curvature can be related to displacement

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**The slope & deflection equation**

The following relations are established For small angles,

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**Double Integration method**

Multiply both sides of this equation by EI and integrate Slope eqn Where C1 is a constant of integration, Integrating again Deflection eqn C1 and C2 must be determined from boundary conditions

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**Boundary conditions Cantilever beam Overhanging beam**

Simply supported beam

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**Statically indeterminate beam**

The reactions at the supports cannot be completely determined from the 3 eqns. The degree to which a beam is statically indeterminate is identified by a no. of reactions that can’t be determined. To determine all reactions, the beam deformations must be considered.

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**Statically indeterminate to 1st degree**

From the FBD shown, one can write Obviously not enough to solve for Ay, MA, and B

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**(1) Begin by drawing a FBD diagram for an arbitrary segment of beam**

For beam segment AC, summing moment about C we write The curvature equation is determined from Integrating twice in x, we have

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**(2) Using the B.C.s x=0, q=0, and y=0 the constants C1 = C2 = 0**

Using the 3rd B.C. (x=L, y=0) the above equation becomes or (2)

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The 2 equations generated from the original FBD of the beam and the equation generated by considering a BC at x=L from the curvature equation: 2 equations and 2 unknowns, they are now solved to determine the reactions at all supports Ans

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**Statically indeterminate to 2nd degree**

From the FBD shown, one can write Obviously not enough to solve for Ay, MA, By and MB

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**For beam segment AC, summing moment about C we write**

The curvature equation is determined from Integrating twice in x, we have

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**Using the B.C.s x=0, q=0, and y=0 the constants C1 = C2 = 0**

Using the 3rd B.C. (x=L, q=0, y=0) the above equation becomes or

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**2 equations and 2 unknowns, they are now solved to determine the reactions at all supports**

Ans

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Superposition The slope & deflection at any point of the beam may be determined by superposing the slopes and deflections caused by the distributed and concentrated loads.

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Superposition (cont.)

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Superposition (cont.)

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Superposition (cont.) The method of superposition can be used for many loading conditions, provided the equations for the slope and deflection associated with a specific type of load are known. For example,

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Tables Click on each beam in which u are interested to see the curvature equation, max deflection, and slope for each loading condition (You should start working on it before seeing!)

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**Singularity functions**

Instead of using the curvature equation, one can also use what so called the singularity functions. For the beam shown above, the bending moment in terms of singularity functions by

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Integrate twice to get

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**Next Week 1. Quiz 2. Continuation of Beam Deflections**

- theorems of area moment method - moment diagrams by parts

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