. Applied calculus; principles and applications . the intensity at the edge of the stripbeing taken. The sum of a finite number of such termswould give a result less than the total stress on the halfsection; but the exact value is given by hm2^ -ybAy=—l ydy Ay=0 ^0 2/1 2/1 t/0 d Sh yn^^ Sh f Shd For the center of stress, the point of apphcation of the totalstress, - D P^^ 2^ Shy^y^ Sh J Sbd\- Shd /Shd 1 , When S = St = Sc, P = Pi; hence, the total tensile andcompressive stresses form a couple with arm f d, the momentbeing ^.U^-^, o o By the mechanics of beams, if M
. Applied calculus; principles and applications . the intensity at the edge of the stripbeing taken. The sum of a finite number of such termswould give a result less than the total stress on the halfsection; but the exact value is given by hm2^ -ybAy=—l ydy Ay=0 ^0 2/1 2/1 t/0 d Sh yn^^ Sh f Shd For the center of stress, the point of apphcation of the totalstress, - D P^^ 2^ Shy^y^ Sh J Sbd\- Shd /Shd 1 , When S = St = Sc, P = Pi; hence, the total tensile andcompressive stresses form a couple with arm f d, the momentbeing ^.U^-^, o o By the mechanics of beams, if M denote the moment of the external forces acting on the beam, M = —^— Example 3. — A vertical circular section of a beam issubjected to a stress of tension and compression uniformlyvarying in intensity from zero at the horizontal diameter2 a of the section to St and Sc at the top and bottom fibers,respectively. PRESSURE OF LIQUIDS 361 Find the total stresses on the upper and lower semicirclesand the centers of stress on the Denoting by P the total stress on either semicircle, andtaking St = Sc = S; P = lim 5j -y 2x^y = — I Va^ — y^ydyA2^=o ^0 o, a Jo y>P=—-j Va^-y^yHya Jo = 2^r^(^2)V^J3^+^%in-i^TP^^^-2Exer-\ 2S TTO^ a 16 Swa Sird 8 64 ^ 64/6 32 * Hence the couple formed by the forces P has an arm,2^ = TQT^d^ and M = -^Sd^ • Y^Trd = ~o^j the section modulus. Example 4. — Find the total water pressure upon the endof a circular right cylinder immersed lengthwise, one elementof the cylinder just at the surface of the water. Find thecenter of pressure of the circular area. 362 INTEGRAL CALCULUS The intensity of pressure at a depth y being wy^approximate pressure on a strip iswy2x ^y. 0 the
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