Answer:
Explanation:
From the given question:
Using the distortion energy theory to determine the factors of safety FOS can be expressed by the relation:
[tex]\dfrac{Syt}{FOS}= \sqrt{ \sigma x^2+\sigma y^2-\sigma x \sigma y+3 \tau_{xy^2}}[/tex]
where; syt = strength in tension and compression = 350 MPa
The maximum shear stress theory can be expressed as:
[tex]\tau_{max} = \dfrac{Syt}{2FOS}[/tex]
where;
[tex]\tau_{max} =\sqrt{ (\dfrac{\sigma x-\sigma y}{2})^2+ \tau _{xy^2[/tex]
a. Using distortion - energy theory formula:
[tex]\dfrac{350}{FOS}= \sqrt{94^2+0^2-94*0+3 (-75)^2}}[/tex]
[tex]\dfrac{350}{FOS}=160.35[/tex]
[tex]{FOS}=\dfrac{350}{160.35}[/tex]
FOS = 2.183
USing the maximum-shear stress theory;
[tex]\dfrac{350}{2 FOS} =\sqrt{ (\dfrac{94-0}{2})^2+ (-75)^2[/tex]
[tex]\dfrac{350}{2 FOS} =88.51[/tex]
[tex]\dfrac{350}{ FOS} =2 \times 88.51[/tex]
[tex]{ FOS} =\dfrac{350}{2 \times 88.51}[/tex]
FOS = 1.977
b. σx = 110 MPa, σy = 100 MPa
Using distortion - energy theory formula:
[tex]\dfrac{350}{FOS}= \sqrt{ 110^2+100^2-110*100+3(0)^2}[/tex]
[tex]\dfrac{350}{FOS}= \sqrt{ 12100+10000-11000[/tex]
[tex]\dfrac{350}{FOS}=105.3565[/tex]
[tex]FOS=\dfrac{350}{105.3565}[/tex]
FOS =3.322
USing the maximum-shear stress theory;
[tex]\dfrac{350}{2 FOS} =\sqrt{ (\dfrac{110-100}{2})^2+ (0)^2[/tex]
[tex]\dfrac{350}{2 FOS} ={ (\dfrac{110-100}{2})^2[/tex]
[tex]\dfrac{350}{2 FOS} =25[/tex]
FOS = 350/2×25
FOS = 350/50
FOS = 70
c. σx = 90 MPa, σy = 20 MPa, τxy =−20 MPa
Using distortion- energy theory formula:
[tex]\dfrac{350}{FOS}= \sqrt{ 90^2+20^2-90*20+3(-20)^2}[/tex]
[tex]\dfrac{350}{FOS}= \sqrt{ 8100+400-1800+1200}[/tex]
[tex]\dfrac{350}{FOS}= 88.88[/tex]
FOS = 350/88.88
FOS = 3.939
USing the maximum-shear stress theory;
[tex]\dfrac{350}{2 FOS} =\sqrt{ (\dfrac{90-20}{2})^2+ (-20)^2[/tex]
[tex]\dfrac{350}{2 FOS} =\sqrt{ (35)^2+ (-20)^2[/tex]
[tex]\dfrac{350}{2 FOS} =\sqrt{ 1225+ 400[/tex]
[tex]\dfrac{350}{2 FOS} =40.31[/tex]
[tex]FOS} =\dfrac{350}{2*40.31}[/tex]
FOS = 4.341
