Cases of eccentric loading in bolted joints

Cases of eccentric loading in bolted joints

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  Cases of Eccentricloading in bolted joints
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  • SUBJECT :-ADVANCED ENGINEERING MATHEMATICS • CLASS :- 2nd year , 4rd semester B.E Mechanical • TOPIC :- Cases of Eccentric loading in bolted joints. • NAME :- Tandel Aniket N. - 160090119115 Desai Udit K. - 160090119015 Shukla Kartikay D. – 160090119108 • GUIDED BY :- PROF. Sumit N. Patel
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  Instructional Objectives In manyapplications, a machine member is subjected to load such that a bending moment is developed in addition to direct normal or shear loading. Such type of loading is commonly known as eccentric loading. In this lesson design methodology will be discussed for three different types of joints subjected to eccentric loading (i) Screw joint (ii) Riveted joint (iii) Welded joint
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  1. Eccentrically loadedscrewed joint: Consider a bracket fixed to the wall by means of three rows of screws having two in each row as shown in figure 1. An eccentric load F is applied to the extreme end of the bracket. The horizontal component, , causes direct tension in the screws but the vertical component, , is responsible for turning the bracket about the lowermost point in left (say point O), which in an indirect way introduces tension in the screws.
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  Figure1: Eccentrically loadedbolted joint
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  It is easyto note that the tension in the screws cannot be obtained by equations of statics alone. Hence, additional equations must be formed to solve for the unknowns for this statically indeterminate problem. Since there is a tendency for the bracket to rotate about point O then, assuming the bracket to be rigid, the following equations are easily obtained. θ ≈ tan θ = 𝑦1 𝑙1 = 𝑦2 𝑙2 = 𝑦3 𝑙3 where 𝑦1= elongation of the i-th bolt 𝑙1 = distance of the axis of the i-th bolt from point O. If the bolts are made of same material and have same dimension, then 𝑓1 = 𝑘𝑦1 where 𝑓1= force in the i-th bolt k = stiffness of the bolts Thus, 𝑓1∞ 𝑦1or 𝑓1 = α𝑙1 (α = proportionality constant)
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  Figure 2: Determinationof forces in bolts
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  Using the momentbalance equations about O, the lowermost point in the left side, the following equation is obtained. 2Σ 𝑓𝑖 𝑙𝑖= 𝐹ℎ 𝐿1+ 𝐹𝑣 𝐿2 i.e., α = 𝐹ℎ 𝐿1+ 𝐹𝑣 𝐿2 2Σ 𝑙 𝑖 2 . The factor 2 appears because there are two bolts in a row. Thus the force in the i-th screw is 𝑓𝑖= [ 𝐹ℎ 𝐿1+ 𝐹𝑣 𝐿2 2Σ 𝑙 𝑖 2 ]𝑙𝑖+ 𝐹ℎ 𝑛 , where n= total number of bolts. For safe design of the joint it is therefore required that σ = max{ 𝑓1 𝐴 } ≤ 𝑠1 Where, 𝑠1= allowable tensile stress of the bolt. Note that 𝐹𝑣 causes also direct shear in the bolt. Its effect may be ignored for a preliminary design calculation.
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  2.Eccentrically loaded rivetedjoint: Consider, now, a bracket, which carries a vertical load . The bracket, in this case, is connected to the wall by four rivets as shown in figure 3. The force, Figure 3: Eccentrically loaded rivet joint
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  in addition toinducing direct shear of magnitude 𝐹 4 in each rivet, causes the whole assembly to rotate. Hence additional shear forces appear in the rivets. Once again, the problem is a statically indeterminate one and additional assumptions are required. These are as following: (i) magnitude of additional shear force is proportional to the distance between the rivet center and the centroid of the rivet assembly, whose co-ordinates are defined as x = Σ 𝐴 𝑖 𝑥 𝑖 Σ 𝐴 𝑖 , ȳ = Σ 𝐴 𝑖 𝑥 𝑖 Σ 𝐴 𝑖 (ii) directions of the force is perpendicular to the line joining centroid of the rivet group and the rivet center and the sense is governed by the rotation of the bracket. Noting that for identical rivets the centroid is the geometric center of the rectangle, the force in the i-th rivet is 𝑓1 = α𝑙1
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  where α =proportional constant 𝑙1= distance of the i-th rivet from centroid. Taking moment about the centroid Σ 𝑓𝑖 𝑙𝑖 = FL or α = 𝐹𝐿 Σ 𝑙 𝑖 2 Thus, the additional force is 𝑓𝑖 = 𝐹𝐿 Σ 𝑙 𝑖 2 𝑙𝑖. Figure 4: Forces on rivets due to
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  The net forcein the i-th rivet is obtained by parallelogram law of vector addition as 𝑓𝑖 ′ = 𝑓𝑖 2 + ( 𝐹 4 )2 + 2. 𝐹 4 . 𝑓𝑖 cos 𝜃𝑖 where 𝜃𝑖 =angle between the lines of action ofthe forces shown in the figure. For safe designing we must have 𝜏 = max{ 𝑓𝑖 ′ 𝐴 } ≤ 𝑠𝑠 Where, 𝑠𝑠 = allowable shear stress of the rivet.
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