Question

solve the problem with clearly handwriting

Solve the following problem. Write down the final answers in the space provided. Show your work for partial credit. Problem: What is the percent increase in maximum stress in the beam in Fig bas compared to the beam in Figure a. Both beams are simply-supported at the ends, have the same length and have a circular cross section of diameter 2.0-in. The beam in Figure a is transversely loaded in one plane while the beam in figure b is loaded in two planes. Round-up your answer to the nearest %. Figure a Eigure

Answer 1

Percentage increase in maximum stress = 169.2 %

I have used vector notation to find Maximum Bending Moment Vector at Point B.

However the bending in Figure b can also be determined by drawing Bending Moment Diagram in the horizontal and vertical plane. Take the resultant of moment in both the diagram at point B to find Maximum Bending Moment.

- Calculation of maximum Stress in the beam in Figure (a) The Reaction at ABB in 4 8 3 direction will be zoo because there is no external disturbance in these direction Since the Beam is in Stalic Equilibrium, therefore we will ease slalic Equilibrium Equation to find Unknown forces, Ma -o [Skalici Equilibrium Equalion] (12 x Rey) - (246) = 0 Rey= 1 kip Fyro Islalic Equilibrium Equation] Roy + Rey -2=0 RAY = 2- Ray = 2-1 Writing Bending Memand Equalion at sechón x-x by considering Left farl of Beam - cut by Section. . We will not use vector, Wolation became Momond acts only in one direction Mx = .X-2(6-2). ie in Z-direction The moment will be maximum at hoint B.ie. at X=6,

Mx = 6 kip-in. Use Simple Bending. Equalion to find maximum bending stress- I = TidA de diameter of shaft d=2in Bending stress will be maximum af y-dl lwhere y is desfance from Neutral cris] 64 (Tolmar = 32x6 = 7.64 Ksi TX2Y Colman = 7.64 KB i . Marci mum Sboss im Figa. Reg. ox → Calculation of Marimum. Stres in the beam im. Figure (b) bin Reaction in se. direction at Aand C will be gero because there is no erternal disliwabance in 2-direction 2Kip. oina a 5 kip In this solution we will use vector Notation because now moment acts in 2-direction because of loading in 2- direction The Beam is in Slatio Equilibrium therefore we will use Stalic Equilibrium Equalion to find unknown forces

Erpress Forces in vector form which will be used in writing moment Equation in Vector form AB = 6 FAC = 12 Vector of hip force, F = -2 Vector of skip force, F2 = -5R vector of Reaction force at C, Re =. Rey + Rey R. EMA -o [ Static Equilibrium Equalison] [TAB XF ] + [602B) x Fr ] + [ thone x Re] =0 (C6 27x658)]+[669) x(+23] + [ (220)x (Reyž+ byp)] = 0 Chossual desseret Cross Prsluct. 307 - 12 h + 12 Rey K - 12 Ry] =0 (30 -12 Rey) } + (12 Rey - 121 R = 0 Equating ĝ and R component of above vector to gono, 30-12 Rez=0 12 Rey 12 = 0 Efg -o latio Equilibrium Equslain] RAY + Rey-2=0 Ray = 2-Rey = 2-1 Ray=1 / <fg=0 [Static Equilibrium Equalwon] Raz + Rez - 5-0

RAz = 5-Rez a 5-2.5 Raz = 2.5 kip Vector of Reaction force at A, RA= 12 + 205R Nowe we knows that he Moment will be maximum al B, so wel would we will write Moment Equation at hoint B. by considering left frart of Beam, Mo = (he) > (R) = (-6l) * ( 19 +2:58) [cuss Product of 2 Voolor] Mo = -6 + 15 Magnitude of Marximum Moment s, Mo = 6-6)*7152 Mo = (156) Kipin Dising Bending Equalién lo Calculate Maximum Bending Bless, Ob . Manar. To will be maximum at. | 0 1 4 = 4 (Tolman = 16-156 x 32x 16-156 de xd (Obe)mox = , 32x16-156 - 20-57.K8A TIX 23 (16) max = - 20-57 kbi Percentage Increase in Manimum Stress - One) mx-Bou Dmax x 100 Tomar = (20.57% (764) x100 7.64 Percentage Increase in Marimum Shress = 169.2%