Application of Maxima and Minima

As an example, the area of a rectangular lot, expressed in terms of its length and width, may also be expressed in terms of the cost of fencing. Thus the area can be expressed as $A = f(x)$. The common task here is to find the value of $x$ that will give a maximum value of $A$. To find this value, we set $dA/dx = 0$.
 

Maxima and Minima | Applications

Graph of the Function y = f(x)
The graph of a function y = f(x) may be plotted using Differential Calculus. Consider the graph shown below.
 

000-graph-of-y-fx.jpg

 

As x increases, the curve rises if the slope is positive, as of arc AB; it falls if the slope is negative, as of arc BC.
 

Derivation of Sum and Difference of Two Angles

Triangle used in sum and difference of two anglesThe sum and difference of two angles can be derived from the figure shown below.
 

Solution to Problem 569 | Horizontal Shearing Stress

Problem 569
Show that the maximum shearing stress in a beam having a thin-walled tubular section of net area A is τ = 2V / A.
 

Solution to Problem 568 | Horizontal Shearing Stress

Problem 568
Show that the shearing stress developed at the neutral axis of a beam with circular cross section is τ = (4/3)(V / π r2). Assume that the shearing stress is uniformly distributed across the neutral axis.
 

Solution to Problem 567 | Horizontal Shearing Stress

Problem 567
A timber beam 80 mm wide by 160 mm high is subjected to a vertical shear V = 40 kN. Determine the shearing stress developed at layers 20 mm apart from the top to bottom of the section.
 

Solution to Problem 563 | Unsymmetrical Beams

Problem 563
A box beam is made from 2-in. by 6-in. pieces screwed together as shown in Fig. P-563. If the maximum flexure stress is 1200 psi, compute the force acting on the shaded portion and the moment of this force about the NA. Hint: Use the results of Prob. 562.
 

Solution to Problem 562 | Unsymmetrical Beams

Problem 562
In any beam section having a maximum stress fb, show that the force on any partial area A' in Fig. P-562 is given by F = (fb/c)A'(barred y') , where (barred y') is the centroidal coordinate of A'. Also show that the moment of this force about the NA is M' = fb I'/c, where I' is the moment of inertia of the shaded area about the NA.
 

Solution to Problem 561 | Built-up Beams

Problem 561
A T section has the dimensions given in Fig. P-561. Show that the neutral axis is 3 inches below the top and that INA = 166.7 in4. If the tensile stress at the bottom of the flange is 1000 psi, calculate (a) the total tensile force in the flange and (b) the total compressive force in the cross section. Also compute (c) the moment of the compressive force about the NA, and (d) the moment of the total tensile force about the NA. (e) How does the sum of (c) and (d) compare with the total applied bending moment as computed from the flexure formula?
 

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