Soru
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.
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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.### Problem 1: Stress Function for a Circular Section BarGiven stress function:
where
is the radial distance from the center of the bar,
is the radius of the bar, and
is a constant.#### Stress Distribution in Terms of Applied Force, Rate of Twist, and Warping1. **Stress Distribution:** - The stress distribution can be found by differentiating the stress function with respect to
:
- The maximum stress occurs at the outer surface (
):
2. **Rate of Twist:** - The rate of twist (
) is related to the applied torque (
) and the polar moment of inertia (
) of the cross-section:
- For a circular section,
.3. **Warping:** - The warping function (
) is related to the applied force and the moment of inertia of the cross-section:
- For a circular section,
.#### Applicability for Hollow Section- 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.### Problem 2: Stress Function for a Triangular SectionGiven stress function:
where
is the shear modulus,
is the angle of twist,
and
are the coordinates, and
is the base of the triangle.#### Shear Stress Distribution, Rate of Twist, and Warping1. **Shear Stress Distribution:** - The shear stress distribution can be found by differentiating the stress function with respect to
:
- The maximum shear stress occurs where the term inside the brackets is maximized.2. **Rate of Twist:** - The rate of twist is given directly by the stress function:
3. **Warping:** - The warping function (
) can be found by integrating the shear stress distribution over the cross-section:
- This integral will give the warping function.#### Maximum Shear Stress Position and Magnitude- 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.- 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.### Summary1. For the circular section bar, the stress function \(\emptyset = k(r^2 - a^2)\) is not applicable for a hollow section.2. For the triangular section, the stress function \(\emptyset = -G \frac{d\Theta}{dz} \left[ \frac{1}{2}(x^2 + y^2) - \frac