Answer:
Explanation:
From the given question:
Using the distortion energy theory to determine the factors of safety FOS can be expressed by the relation:
![\dfrac{Syt}{FOS}= \sqrt{ \sigma x^2+\sigma y^2-\sigma x \sigma y+3 \tau_{xy^2}}](https://tex.z-dn.net/?f=%5Cdfrac%7BSyt%7D%7BFOS%7D%3D%20%5Csqrt%7B%20%5Csigma%20x%5E2%2B%5Csigma%20%20y%5E2-%5Csigma%20x%20%5Csigma%20y%2B3%20%5Ctau_%7Bxy%5E2%7D%7D)
where; syt = strength in tension and compression = 350 MPa
The maximum shear stress theory can be expressed as:
![\tau_{max} = \dfrac{Syt}{2FOS}](https://tex.z-dn.net/?f=%5Ctau_%7Bmax%7D%20%3D%20%5Cdfrac%7BSyt%7D%7B2FOS%7D)
where;
![\tau_{max} =\sqrt{ (\dfrac{\sigma x-\sigma y}{2})^2+ \tau _{xy^2](https://tex.z-dn.net/?f=%5Ctau_%7Bmax%7D%20%3D%5Csqrt%7B%20%28%5Cdfrac%7B%5Csigma%20x-%5Csigma%20%20y%7D%7B2%7D%29%5E2%2B%20%5Ctau%20_%7Bxy%5E2)
a. Using distortion - energy theory formula:
![\dfrac{350}{FOS}= \sqrt{94^2+0^2-94*0+3 (-75)^2}}](https://tex.z-dn.net/?f=%5Cdfrac%7B350%7D%7BFOS%7D%3D%20%5Csqrt%7B94%5E2%2B0%5E2-94%2A0%2B3%20%28-75%29%5E2%7D%7D)
![\dfrac{350}{FOS}=160.35](https://tex.z-dn.net/?f=%5Cdfrac%7B350%7D%7BFOS%7D%3D160.35)
![{FOS}=\dfrac{350}{160.35}](https://tex.z-dn.net/?f=%7BFOS%7D%3D%5Cdfrac%7B350%7D%7B160.35%7D)
FOS = 2.183
USing the maximum-shear stress theory;
![\dfrac{350}{2 FOS} =\sqrt{ (\dfrac{94-0}{2})^2+ (-75)^2](https://tex.z-dn.net/?f=%5Cdfrac%7B350%7D%7B2%20FOS%7D%20%20%3D%5Csqrt%7B%20%28%5Cdfrac%7B94-0%7D%7B2%7D%29%5E2%2B%20%28-75%29%5E2)
![\dfrac{350}{2 FOS} =88.51](https://tex.z-dn.net/?f=%5Cdfrac%7B350%7D%7B2%20FOS%7D%20%20%3D88.51)
![\dfrac{350}{ FOS} =2 \times 88.51](https://tex.z-dn.net/?f=%5Cdfrac%7B350%7D%7B%20FOS%7D%20%20%3D2%20%5Ctimes%2088.51)
![{ FOS} =\dfrac{350}{2 \times 88.51}](https://tex.z-dn.net/?f=%7B%20FOS%7D%20%20%3D%5Cdfrac%7B350%7D%7B2%20%5Ctimes%2088.51%7D)
FOS = 1.977
b. σx = 110 MPa, σy = 100 MPa
Using distortion - energy theory formula:
![\dfrac{350}{FOS}= \sqrt{ 110^2+100^2-110*100+3(0)^2}](https://tex.z-dn.net/?f=%5Cdfrac%7B350%7D%7BFOS%7D%3D%20%5Csqrt%7B%20110%5E2%2B100%5E2-110%2A100%2B3%280%29%5E2%7D)
![\dfrac{350}{FOS}= \sqrt{ 12100+10000-11000](https://tex.z-dn.net/?f=%5Cdfrac%7B350%7D%7BFOS%7D%3D%20%5Csqrt%7B%2012100%2B10000-11000)
![\dfrac{350}{FOS}=105.3565](https://tex.z-dn.net/?f=%5Cdfrac%7B350%7D%7BFOS%7D%3D105.3565)
![FOS=\dfrac{350}{105.3565}](https://tex.z-dn.net/?f=FOS%3D%5Cdfrac%7B350%7D%7B105.3565%7D)
FOS =3.322
USing the maximum-shear stress theory;
![\dfrac{350}{2 FOS} =\sqrt{ (\dfrac{110-100}{2})^2+ (0)^2](https://tex.z-dn.net/?f=%5Cdfrac%7B350%7D%7B2%20FOS%7D%20%20%3D%5Csqrt%7B%20%28%5Cdfrac%7B110-100%7D%7B2%7D%29%5E2%2B%20%280%29%5E2)
![\dfrac{350}{2 FOS} ={ (\dfrac{110-100}{2})^2](https://tex.z-dn.net/?f=%5Cdfrac%7B350%7D%7B2%20FOS%7D%20%20%3D%7B%20%28%5Cdfrac%7B110-100%7D%7B2%7D%29%5E2)
![\dfrac{350}{2 FOS} =25](https://tex.z-dn.net/?f=%5Cdfrac%7B350%7D%7B2%20FOS%7D%20%20%3D25)
FOS = 350/2×25
FOS = 350/50
FOS = 70
c. σx = 90 MPa, σy = 20 MPa, τxy =−20 MPa
Using distortion- energy theory formula:
![\dfrac{350}{FOS}= \sqrt{ 90^2+20^2-90*20+3(-20)^2}](https://tex.z-dn.net/?f=%5Cdfrac%7B350%7D%7BFOS%7D%3D%20%5Csqrt%7B%2090%5E2%2B20%5E2-90%2A20%2B3%28-20%29%5E2%7D)
![\dfrac{350}{FOS}= \sqrt{ 8100+400-1800+1200}](https://tex.z-dn.net/?f=%5Cdfrac%7B350%7D%7BFOS%7D%3D%20%5Csqrt%7B%208100%2B400-1800%2B1200%7D)
![\dfrac{350}{FOS}= 88.88](https://tex.z-dn.net/?f=%5Cdfrac%7B350%7D%7BFOS%7D%3D%2088.88)
FOS = 350/88.88
FOS = 3.939
USing the maximum-shear stress theory;
![\dfrac{350}{2 FOS} =\sqrt{ (\dfrac{90-20}{2})^2+ (-20)^2](https://tex.z-dn.net/?f=%5Cdfrac%7B350%7D%7B2%20FOS%7D%20%20%3D%5Csqrt%7B%20%28%5Cdfrac%7B90-20%7D%7B2%7D%29%5E2%2B%20%28-20%29%5E2)
![\dfrac{350}{2 FOS} =\sqrt{ (35)^2+ (-20)^2](https://tex.z-dn.net/?f=%5Cdfrac%7B350%7D%7B2%20FOS%7D%20%20%3D%5Csqrt%7B%20%2835%29%5E2%2B%20%28-20%29%5E2)
![\dfrac{350}{2 FOS} =\sqrt{ 1225+ 400](https://tex.z-dn.net/?f=%5Cdfrac%7B350%7D%7B2%20FOS%7D%20%20%3D%5Csqrt%7B%201225%2B%20400)
![\dfrac{350}{2 FOS} =40.31](https://tex.z-dn.net/?f=%5Cdfrac%7B350%7D%7B2%20FOS%7D%20%20%3D40.31)
![FOS} =\dfrac{350}{2*40.31}](https://tex.z-dn.net/?f=FOS%7D%20%20%3D%5Cdfrac%7B350%7D%7B2%2A40.31%7D)
FOS = 4.341