Updated: Sep 14, 2020
This is the seventh lecture of the lecture series on welded joints. In this lecture, I have explained the design of the eccentrically loaded welded joint and solved some relevant problems. The eccentric load is considered to be in-plane load w.r.t. the weld plane.
Before starting with the design of eccentrically loaded welded joints its important to understand what type of load is known as an eccentric load. An external load whose line of action is parallel but does not coincide with the centroidal axis of the component/structure on which load is applied is known as an eccentric load. Whereas, the perpendicular distance between the centroidal axis and line of action of the load is known as eccentricity. If you consider the figure on the left, 'd' is the eccentricity whereas 'P' is the eccentric load.
Center of Mass, Center of Gravity, and Centroid are the three terms which are confusing ever since their existence in our life. The centroid is the geometric center of any structure/body.
The center of mass is a point in the body where the mass of the whole body assumed to be concentrated or it is the unique point where the weighted relative position of the distributed mass sums to zero. In, order to understand it in a better way you can say that it is a point in a body where the net external force will result in the movement of the body without rotation in the direction of the fore.
The center of gravity is a point where the whole weight of the body acts vertically downward. A body is made up of a large number of particles. Earth attracts each of these particles vertically downward towards its center. The pull of the Earth acting on a particle is equal to its weight. These forces acting on a particle of a body are parallel forces and can be reduced to a single force equal to the weight of the body. A point where this resultant force acts vertically towards the center of the Earth is called the center of gravity of the body.
The Center of mass and centroid for any given structure will be the same provided the density is the same through the structure. In case of the structure where the density is not the same throughout, for instance, take an example of laminated composite material, in such cases, centroid and center of mass will be totally different.
The center of mass and center of gravity are if the gravitational forces acting on the structure is the same throughout. In case the structure is subjected to varying gravitational pull on either side, then the center of gravity will shift towards the side experiencing large gravitational pull.
In a nutshell, we can say that if the structure has uniform density then on earth the center of mass, the center of gravity, and centroid all will lie at the same point assuming gravitational pull on the earth is the same at all places. Watch the video below for more clarity.
Since now we are clear about these terms lets see how we are going to analyze welded joints subjected to eccentric load. The figure below shows a bracket subjected to an eccentric force P, attached to the support by means of two fillet welds W1 and W2. The first is to find the center of gravity of all the welds. Since the welds are made up of the same material having the same density at all the points and the gravitation pull at all the points are also the same therefore, for this case we can say that center of gravity will coincide with the centroid (as discussed earlier in this article. The calculation of the centroid for the welds can be done assuming all the welds as a line. This assumption is based on the fact that the width of all the welds is the same therefore, we can treat the weld as a line for calculating centroid. In the figure below G is the center of gravity or centroid (as they both are the same for this case) and e is the eccentricity between the center of gravity and the line of action of force P.
If you recall the principle of Applied Mechanics, the eccentric force P can be replaced by an equal and similarly directed force (P) acting through the center of gravity G, along with a couple (M = 'P' times 'e') lying in the same plane. The effects of the force P and the couple M are treated separately as shown in fig. (c) and (d) respectively.
The stresses induced in this welded joint because of eccentric load P are shown in the figure below. The force P acting through the center of gravity causes direct shear stress in the welds. It is called the primary shear stress. It is assumed that the primary shear stress is uniformly distributed over the throat area of all welds.
The couple M causes torsional shear stresses in the throat area of welds. They are called secondary shear stresses. The secondary shear stress at any point in the weld is proportional to its distance from the center of gravity. Hence, secondary shear stress is maximum at the farthest point such as A as shown in the figure above. Secondary shear stress is also inversely proportional to the polar moment of inertia of all the welds. The polar moment of inertia is the measure of the resistance to torsional load which is dependant upon the cross-sectional area and has nothing to do with the material of the structure. Check out the video below to learn more about the polar moment of inertia.
Firstly, the polar moment of inertia of individual welds about its own center of gravity is calculated and then using a parallel axis theorem polar moment of inertia of all the welds about the center of gravity G calculated for getting secondary shear stress. The resultant shear stress at any point is obtained by vectorially adding primary and secondary shear stresses.
The detailed design procedure is explained in the video below. The video is divided into two parts for maximum retention.
For a better understanding of the concept, I have also solved a few problems.
Click here to download the study material related to this topic.
Reference: Design of Machine Elements by V. B. Bhandari.