EQUILIBRIUM OF NONCONCURRENT COPLANAR FORCE SYSTEM When a
EQUILIBRIUM OF NON-CONCURRENT COPLANAR FORCE SYSTEM When a body is in equilibrium, it has neither translatory nor rotatory motion in any direction. Thus the resultant force R and the resultant couple M are both zero, and we have the equilibrium equations for two dimensional force system Fx = 0; Fy = 0 Eq(1) M = 0 These requirements are both necessary and sufficient conditions for equilibrium. www. bookspar. com | Website for Students | VTU NOTES | QUESTION PAPERS 1
Supports: A structure is subjected to external forces and transfers these forces through the supports on to the foundation. Therefore the support reactions and the external forces together keep the structure in equilibrium. Types of supports There are different types of supports. Some of them are a) Roller Support b) Hinged or pinned support c) Fixed or built in support Some supports are shown in the figure along with the reactions that can be mobilised. www. bookspar. com | Website for Students | VTU NOTES | QUESTION PAPERS 2
Types of Supports Action on body (a) Flexible cable , belt , chain, rope BODY T Force exerted by cable is always a tension away from the body in the direction of cable (b) Smooth surfaces Contact forces are normal to the surfaces www. bookspar. com | Website for Students | VTU NOTES | QUESTION PAPERS F F 3
(c) Roller support Contact force is normal to the surface on which the roller moves. The reaction will always be perpendicular to the plane of the roller. Roller support will offer only one independent reaction component. (Whose direction is known. ) www. bookspar. com | Website for Students | VTU NOTES | QUESTION PAPERS 4
( d )pinned Support / hinged support Rh θ R Rv This support does not allow any translatory movement of the rigid body. There will be two independent reaction components at the support. The resultant reaction can be resolved into two mutually perpendicular components. Or it can be shown as resultant reaction inclined at an angle with respect to a reference direction. www. bookspar. com | Website for Students | VTU NOTES | QUESTION PAPERS 5
(e) Fixed or Built-in Support M RH Rv M www. bookspar. com | Website for Students | VTU NOTES | QUESTION PAPERS 6
(contd. ) This type of support not only prevents the translatory movement of the rigid body, but also the rotation of the rigid body. Hence there will be 3 independent reaction components of forces. Hence there will be 3 unknown components of forces, two mutually perpendicular reactive force component and a reactive moment as shown in the figure. www. bookspar. com | Website for Students | VTU NOTES | QUESTION PAPERS 7
TYPES OF BEAMS A member which is subjected to predominantly transverse loads and supported in such a way that rigid body motion is prevented is known as beam. It is classified based on the support conditions. A beam generally supported by a hinge or roller at the ends having one span(distance between the support) is called as simply supported beam. A beam which is fixed at one end and free at another end is called as a cantilever beam. A HA B MA span (a) Simply supported beam VA span (b) Cantilever beam www. bookspar. com | Website for Students | VTU NOTES | QUESTION PAPERS 8
If one end or both ends of the beam project beyond the support it is known as overhanging beam. A cantilever with a simple support anywhere along its length is a propped cantilever. A HA MA VA (c) Overhanging beam (right overhang) B span (d) Propped Cantilever beam 9 www. bookspar. com | Website for Students | VTU NOTES | QUESTION PAPERS
A beam which is fixed at both ends is called a fixed beam. A beam with more than one span is called continuous beam. HA HB MB MA HA span VA VB (e) Fixed beam VA VB VC (f) Two Span continuous beam www. bookspar. com | Website for Students | VTU NOTES | QUESTION PAPERS 10
Statically determinate beam and statically indeterminate beam: Using the equations of equilibrium given in EQ(1) , if all the reaction components can be found out, then the beam is a statically determinate beam , and if all the reaction components can not be found out using equations of equilibrium only, then the beam is a statically indeterminate beam. In the above fig (a), (b)and ( c ) are statically determinate beams , where as (d), (e) and ( f) are statically Indeterminate beams. www. bookspar. com | Website for Students | VTU NOTES | QUESTION PAPERS 11
If the number of reaction components is more than the number of non-trivial equilibrium equations available then such a beam is a statically indeterminate beam. If the number of reaction components is equal to the number of non-trivial equilibrium equations available then such a beam is a statically determinate beam If the number of reaction components is less than the number of non-trivial equilibrium equations available then such a beam is an unstable beam. 12 www. bookspar. com | Website for Students | VTU NOTES | QUESTION PAPERS
Determination of Beam reactions Since three equilibrium equations are available, for a planar structure a maximum of three unknown independent reaction components can be determined using these equations. Step I: Draw the free body diagram of the structure showing the given loadings and the reactions at the supports. Step 2: Apply the equations Fx = 0, Fy = 0, M = 0. Assuming some directions and senses for unknown forces and moments. Step 3: solve for unknown reactions. If any of them is positive, it is along the sense initially assumed while drawing the FBD. If it is negative, it is opposite to the initially assumed sense 13 www. bookspar. com | Website for Students | VTU NOTES | QUESTION PAPERS
Problems for practice (1)Find the reactions at A, B, C and D for the beam loaded as shown in the figure(Ans. RA=RB =34 k. N; RC=28. 84 k. N; MC=-140 k. Nm ; θC=-33. 69 ˚ ) 12 k. N/m 20 k. N 4 k. N/m 12 k. N/m 4 k. N/m 30 k. N A 4 B 3 C 40 k. Nm 1 m 2 m 1 m www. bookspar. com | Website for Students | VTU NOTES | QUESTION PAPERS 1 m 2 m 14
(2)A uniform bar AB of weight 50 N shown in the figure supports a load of 200 N at its end. Determine the tension developed in the string and the force supported by the pin at B. (Ans. T=529. 12 N; RB=807. 15 N, θB=64. 6˚) 2. 5 m string B 60˚ A 200 N 2. 5 m www. bookspar. com | Website for Students | VTU NOTES | QUESTION PAPERS 2. 5 m 15
(3)Find the position of the hinged support (x), such that the reactions developed at the supports of the beam are equal. . (Ans. x=2 m. ) 15 k. N 18 k. N/m 10 k. N/m 2. 0 m 1. 0 m 0. 6 1. 4 m x 3. 0 m www. bookspar. com | Website for Students | VTU NOTES | QUESTION PAPERS 16
(4)A right angled bar ABC hinged at A as shown in fig carries two loads W and 2 W applied at B &C. Neglecting self weight of the bar find the angle made by AB with vertical(Ans: θ =18. 44˚) A Lm θ B W 0. 5 L C 2 W www. bookspar. com | Website for Students | VTU NOTES | QUESTION PAPERS 17
- Slides: 17