Solution to Problem 639 | Deflection of Cantilever Beams

Problem 639
The downward distributed load and an upward concentrated force act on the cantilever beam in Fig. P-639. Find the amount the free end deflects upward or downward if E = 1.5 × 106 psi and I = 60 in4.
 

Cantilever beam with uniform downward load and concentrated upward load

 

Solution to Problem 638 | Deflection of Cantilever Beams

Problem 638
For the cantilever beam shown in Fig. P-638, determine the value of EIδ at the left end. Is this deflection upward or downward?
 

Cantilever Beam with Clockwise Moment Load at Midspan

 

Solution to Problem 637 | Deflection of Cantilever Beams

Problem 637
For the beam loaded as shown in Fig. P-637, determine the deflection 6 ft from the wall. Use E = 1.5 × 106 psi and I = 40 in4.
 

637-cantilever-uniform-loads.gif

 

The Cyclic Quadrilateral

A quadrilateral is said to be cyclic if its vertices all lie on a circle. In cyclic quadrilateral, the sum of two opposite angles is 180° (or π radian); in other words, the two opposite angles are supplementary.

$A + C = 180^\circ$

$B + D = 180^\circ$

 

Cyclic quadrilateral

 

The Quadrilateral

Quadrilateral is a polygon of four sides and four vertices. It is also called tetragon and quadrangle. In the triangle, the sum of the interior angles is 180°; for quadrilaterals the sum of the interior angles is always equal to 360°
 

$A + B + C + D = 360^\circ$

 

Classifications of Quadrilaterals
There are two broad classifications of quadrilaterals; simple and complex. The sides of simple quadrilaterals do not cross each other while two sides of complex quadrilaterals cross each other.
 

Simple quadrilaterals are further classified into two: convex and concave. Convex if none of the sides pass through the quadrilateral when prolonged while concave if the prolongation of any one side will pass inside the quadrilateral.
 

Solution to Problem 636 | Deflection of Cantilever Beams

Problem 636
The cantilever beam shown in Fig. P-636 has a rectangular cross-section 50 mm wide by h mm high. Find the height h if the maximum deflection is not to exceed 10 mm. Use E = 10 GPa.
 

Cantilever beam with two concentrated loads

 

Deflection of Cantilever Beams | Area-Moment Method

Generally, the tangential deviation t is not equal to the beam deflection. In cantilever beams, however, the tangent drawn to the elastic curve at the wall is horizontal and coincidence therefore with the neutral axis of the beam. The tangential deviation in this case is equal to the deflection of the beam as shown below.
 

Solution to Problem 632 | Moment Diagrams by Parts

Problem 632
For the beam loaded as shown in Fig. P-632, compute the value of (AreaAB) barred(X)A. From this result, is the tangent drawn to the elastic curve at B directed up or down to the right? (Hint: Refer to the deviation equations and rules of sign.)
 

Overhang beam with point and rectangular loads

 

Solution to Problem 631 | Moment Diagrams by Parts

Problem 631
Determine the value of the couple M for the beam loaded as shown in Fig. P-631 so that the moment of area about A of the M diagram between A and B will be zero. What is the physical significance of this result?
 

Overhang beam with moment load at free end

 

Solution to Problem 630 | Moment Diagrams by Parts

Problem 630
For the beam loaded as shown in Fig. P-630, compute the value of (AreaAB)barred(X)A . From the result determine whether the tangent drawn to the elastic curve at B slopes up or down to the right. (Hint: Refer to the deviation equations and rules of sign.)
 

Overhang beam with point load at free end

 

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