Knowing that M=250 N -m, determine the maximum stress in the beam shown when the radius r of the

Knowing that M=250 N -m, determine the maximum stress in the...

Key Concepts

Bending Moment

A bending moment is a measure of the tendency of a force to cause a beam or structural element to bend. It is calculated as the product of the applied force and the distance from the point of interest to the force’s line of action. In bending analyses, understanding the applied moment is crucial because it directly influences the bending stresses experienced by the material.

Bending Stress

Bending stress is the internal stress induced in a beam due to an external bending moment. It is typically determined using the relationship ? = M·c/I, where M is the bending moment, c is the distance from the neutral axis to the outermost fiber, and I is the moment of inertia. This concept is essential in predicting the failure or deformation of structural elements subjected to bending loads.

Section Modulus

The section modulus is a geometric property of a cross-section defined as the ratio of the moment of inertia, I, to the distance from the neutral axis to the outermost fiber, c. It provides a measure of the strength of a beam in bending by linking the applied moment to the material stress. The higher the section modulus, the lower the bending stress for a given moment.

Moment of Inertia

The moment of inertia reflects how the cross-sectional area is distributed about a given axis and significantly affects the beam’s resistance to bending. It is a critical factor in calculating bending stress, because sections with larger moments of inertia will experience lower stress under the same bending moment.

Stress Concentration

Stress concentration refers to the localized increase in stress that occurs near geometric discontinuities such as holes, notches, or filleted corners. These areas can exhibit stresses much higher than the nominal value, making the understanding of stress concentration factors vital for safe and efficient structural design.

Fillet Radius

The fillet radius is the curvature applied to the transition between surfaces in a beam’s cross-section. It impacts the stress distribution by reducing the severity of sharp corners, thereby lowering the stress concentration factors. Larger fillet radii generally lead to a more uniform stress distribution, enhancing the structural performance under bending loads.

Transcript

00:01 To solve this problem let us first draw the free body diagram so the free body diagram looks like this which i am drawing here just look at it carefully after that i will solve the problem further so this is the rough diagram which i am drawing here just look at it carefully so let this section be section 1 this section b section 2 this is also 2 and this section b section 3 now here this distance from here to here is of 3in.

01:00 This distance is of 0 .5 pi and you can write this distance from here to here is of 3i n here this axis is y axis this axis is x -axis and this point is point c going forward and solving the problem further so here you can write the value as y bar just look at it carefully from here to here this value is y bar and from here to here it is y b bar now going forward and solving the problem further so area of rectangle 1 a 1 is equal to b1 multiplication h1 that is equal to 4 multiplication 0 .5 which is equal to 2 i n square centroid distance y from x axis y1 will be equal to h1 by 2 which is equal to 0 .5 by 2 which is equal to 0 .25 pynn area of rectangle 2 a 2 will be equal to 2 multiplication b 2 multiplication h 2 that is equal to 2 multiplication 0 .5 multiplication 3 which is equal to 3 i n square and centroid distance y from x axis y 2 will be equal to h1 plus h 2 x 2 which is equal to 0 .5 plus 3 by 2 which is equal to 2 i n and area of rectangle 3 a 3 is equal to b 3 multiplication h 3 that is equal to 0 .5 multiplication 10 which is equal to 5 ian square and so in twilight distance y from x x x x x x 3 is equal to h1 plus s 3 by 2 that is 0 .5 plus 10 by 2 that is 5 .5 i .n now finally i can calculate the value of y bar distance of centroid from x -axis is y -bar which is equal to a1 y1 plus a2 y2 plus a3 by a1 plus a1 plus a3 a 1 plus a 3 so it will be equal to 2 multiplication 0 .25 plus 3 multiplication 2 plus 5 multiplication 5 5 .5 by 2 plus 3 plus 5 so you will get the value of y bar as 3 .4 ion distance of centroid from bottom y b bar is 10 .5 minus y bar that is 10 .5 minus 10 .5 minus 10.

04:24 Moment of inertia of rectangle 1...

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