nework 1) Show that the stress function emptyset =k(r^2-a^2) is applicable to the soll lition of a solid circular section bar of radius a the stress distribution in the bar in terms of the applied forque,the rat e of twist , and the warping of the cross -section.Is it possible to I lise this stress function in the solution for a circular bar of a holl low section? 2)show that the stress function ( )=-G(dTheta )/(dz)[(1)/(2)(x^2+y^2)-(1)/(2a)(x^3- (2)-(2)/(27)a^2 is the correct solution for a bar having a cross- tion in the form of the triangle shown in the figure. termine the shear stress distribution the rate of twist, and warping of the cross-section Find the position and magnitude the maximum shear stress.

To solve the given problems, we need to apply the stress functions provided and analyze the stress distribution, rate of twist, and warping of the cross-sections for each case.<br /><br />### Problem 1: Stress Function for a Circular Section Bar<br /><br />Given stress function:<br />\[ \emptyset = k(r^2 - a^2) \]<br /><br />where \( r \) is the radial distance from the center of the bar, \( a \) is the radius of the bar, and \( k \) is a constant.<br /><br />#### Stress Distribution in Terms of Applied Force, Rate of Twist, and Warping<br /><br />1. **Stress Distribution:**<br /> - The stress distribution can be found by differentiating the stress function with respect to \( r \):<br /> \[ \sigma_r = \frac{d\emptyset}{dr} = 2kr \]<br /> - The maximum stress occurs at the outer surface (\( r = a \)):<br /> \[ \sigma_{max} = 2ka \]<br /><br />2. **Rate of Twist:**<br /> - The rate of twist (\(\tau_{twist}\)) is related to the applied torque (\(T\)) and the polar moment of inertia (\(J_p\)) of the cross-section:<br /> \[ \tau_{twist} = \frac{T \cdot L}{J_p} \]<br /> - For a circular section, \( J_p = \frac{\pi a^4}{4} \).<br /><br />3. **Warping:**<br /> - The warping function (\( \phi \)) is related to the applied force and the moment of inertia of the cross-section:<br /> \[ \phi = \frac{F \cdot L^2}{J_w} \]<br /> - For a circular section, \( J_w = \frac{\pi a^4}{32} \).<br /><br />#### Applicability for Hollow Section<br /><br />- The stress function \(\emptyset = k(r^2 - a^2)\) is specific to a solid circular section. For a hollow section, the stress function would need to account for the varying thickness and radius. Therefore, this stress function is not directly applicable to a hollow section.<br /><br />### Problem 2: Stress Function for a Triangular Section<br /><br />Given stress function:<br />\[ \emptyset = -G \frac{d\Theta}{dz} \left[ \frac{1}{2}(x^2 + y^2) - \frac{1}{2a}(x^3 - y^3) - \frac{2}{27}a^2 \right] \]<br /><br />where \( G \) is the shear modulus, \( \Theta \) is the angle of twist, \( x \) and \( y \) are the coordinates, and \( a \) is the base of the triangle.<br /><br />#### Shear Stress Distribution, Rate of Twist, and Warping<br /><br />1. **Shear Stress Distribution:**<br /> - The shear stress distribution can be found by differentiating the stress function with respect to \( z \):<br /> \[ \tau_z = -G \frac{d\Theta}{dz} \left[ \frac{1}{2}(x^2 + y^2) - \frac{1}{2a}(x^3 - y^3) - \frac{2}{27}a^2 \right] \]<br /> - The maximum shear stress occurs where the term inside the brackets is maximized.<br /><br />2. **Rate of Twist:**<br /> - The rate of twist is given directly by the stress function:<br /> \[ \tau_{twist} = -G \frac{d\Theta}{dz} \left[ \frac{1}{2}(x^2 + y^2) - \frac{1}{2a}(x^3 - y^3) - \frac{2}{27}a^2 \right] \]<br /><br />3. **Warping:**<br /> - The warping function (\( \phi \)) can be found by integrating the shear stress distribution over the cross-section:<br /> \[ \phi = \int \tau_z \, dA \]<br /> - This integral will give the warping function.<br /><br />#### Maximum Shear Stress Position and Magnitude<br /><br />- The maximum shear stress occurs where the term inside the brackets is maximized. For a triangular section, this typically occurs at the centroid or along the base.<br />- The magnitude of the maximum shear stress can be found by substituting the coordinates of the point where the term inside the brackets is maximized into the stress function.<br /><br />### Summary<br /><br />1. For the circular section bar, the stress function \(\emptyset = k(r^2 - a^2)\) is not applicable for a hollow section.<br />2. For the triangular section, the stress function \(\emptyset = -G \frac{d\Theta}{dz} \left[ \frac{1}{2}(x^2 + y^2) - \frac