Question #170896

Listed below are varying combinations of stresses acting at a point and referred to axes x and y in an elastic material. Using Mohr's circle of stresses, determine the principal stresses at the point and their directions for each combination. or (N/mm or mm Try (N/mm +5 (i) +54 +30 (ii) +30 +54 +5 36 (iii) -60 (iv) +30 +30 50 om at 11.50 to x axis 29 N/mm (i) ou +55 N/mm Ans. on at 11.5 tox axis (i) on +55 N/mm OIL (iii) on 34.5 N/mm om 61 N/mm ay at 79.5° to x axis. (iv) on --40 N/mm on 60 N/mm2 on at 18.5 to x axis

Expert's answer

In the question normal stresses along x-axis,y-axis and shear stress are given

we have to find the principal stresses and its direction (stresse are in N/mm^{2})

we know that formula for principal stress as

"\\sigma_1=\\frac{\\sigma_x + \\sigma_y}{2} + \\sqrt{(\\frac{\\sigma_x-\\sigma_y}{2})^2 +(\\tau_{xy})^2}"

"\\sigma_2=\\frac{\\sigma_x + \\sigma_y}{2} - \\sqrt{(\\frac{\\sigma_x-\\sigma_y}{2})^2 +(\\tau_{xy})^2}"

"tan 2\\theta= (\\frac{-(\\sigma_x-\\sigma_y)\/2}{\\tau_{xy}})"

(i) "\\sigma_x=54 ,\\sigma_y=30, \\tau_{xy}=5"

on putting value we get

"\\sigma_1=\\frac{\\sigma_x + \\sigma_y}{2} + \\sqrt{(\\frac{\\sigma_x-\\sigma_y}{2})^2 +(\\tau_{xy})^2}"

"\\sigma_1=\\frac{54+ 30}{2} + \\sqrt{(\\frac{54-30}{2})^2 +(5)^2}"

"\\sigma_1=55"

"\\sigma_2=\\frac{\\sigma_x + \\sigma_y}{2} - \\sqrt{(\\frac{\\sigma_x-\\sigma_y}{2})^2 +(\\tau_{xy})^2}"

"\\sigma_2=\\frac{54+ 30}{2} - \\sqrt{(\\frac{54-30}{2})^2 +(5)^2}"

"\\sigma_2=29,"

"tan 2\\theta= (\\frac{-(\\sigma_x-\\sigma_y)\/2}{\\tau_{xy}})"

"\\theta=11.48^o"

(ii)

"\\sigma_x=30,\\sigma_y=54,\\tau_{xy}=-5"

"\\sigma_1=\\frac{\\sigma_x + \\sigma_y}{2} + \\sqrt{(\\frac{\\sigma_x-\\sigma_y}{2})^2 +(\\tau_{xy})^2}"

"\\sigma_1= 55"

"\\sigma_2=\\frac{\\sigma_x + \\sigma_y}{2} - \\sqrt{(\\frac{\\sigma_x-\\sigma_y}{2})^2 +(\\tau_{xy})^2}"

"\\sigma_2=29"

"\\theta= 11.48" with x-axis

(iii)

"\\sigma_x=-60,\\sigma_y=-36, \\tau_{xy}= +5"

"\\sigma_1=-34.5"

"\\sigma_2=-61"

"tan 2\\theta= (\\frac{-(\\sigma_x-\\sigma_y)\/2}{\\tau_{xy}})"

"\\theta=79.5 ^o" with x-axis

(iv)

"\\sigma_x=30, \\sigma_y= -50, \\tau_{xy}=30"

"\\sigma_1=40,\\sigma_2=-60"

"\\theta= 18.5^o" with x-axis

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