Laith Batarseh. internal forces

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1 Next Previous 1/8/2016 Chapter seven Laith Batarseh Home End Definitions When a member is subjected to external load, an and/or moment are generated inside this member. The value of the generated internal load is important to see if this member can resist it. Method of sections is used to determine these loads 1

2 Method of sections To illustrate how we can use the method of sections to find the internal loads, firs, we will start with the following cantilever beam F2 F1 A B Now section this beam from point B as shown in the next figure As you can see, the internal loads in the beam is now become external loads at the section area. Method of sections It is noted from applying this method on the beam that: There is a perpendicular force (N B ) to the section plane. This force is called normal force. this force is the main cause of tensile stress in the beam There is a tangential force (V B ) to the section plane. This force is called shear force. this force is the main cause of shear stress in the beam Finally, the couple moment (M B ) is called bending moment. this moment is the main cause of bending stress in the beam 2

3 Method of sections These forces and moment prevents the beam from the translational and rotational motions( i.e. elongation and bending deflection) The direction of these forces and moment on both segments satisfy Newton's 3 rd law and can be determined using equilibrium equations at one segment. For our case, the right segment is the preferred segment because it reduce the unknown reactions at the supporting point A. F x = 0 to find N B F y = 0 to find V B M B = 0 to find M B Three dimensional cases In three dimensional cases, a new shear and moment components as shown in the figure below z M z M x V z V y M y y V x M x x 3

4 Sign convection To assign the sign of normal force and shear force and the moment, we can assume: if the normal force tray to tensile the segment (tension), then its is positive. If the shear force tends to rotate the segment clockwise then its is positive. If the moment tends to concave the segment upward manner then its is positive. Analysis procedures To use the method of sections in finding the internal loads, you can follow the following procedures: Find the supports reactions using the equilibrium equations for F.B.D of the whole system Leave all the acting loads and moments on the graph and section it at the point that you desire to find the internal loads at it. Draw F.B.D for the segment that has the least number loads on it. This ensure that you will not face statically in-determent situation. Apply the equilibrium of moment at the section directly. This eliminate the normal and shear forces at the sections. then apply the equilibrium of forces equation. Finally, verify the direction of normal and shear forces beside the internal loads. 4

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7 Determine the internal normal force, shear force, and moment in the beam at points C and D. Point D is just to the right of the 5-kip load Solution 7

8 Solution Solution 8

9 Solution shear and moment diagrams Shear and moment As noted from the previous analysis, shear and moment vary along the loaded body which was in most of our cases the beam. beams are important element in structure because it can support both shear and moment stresses. The variation in moment and shear can be determined using the method of sections as we learned before. 9

10 shear and moment diagrams Shear and moment However, the values of shear and moment are not changed suddenly when we moves across the point of force action. Therefore, we must develop a method to study this variance with respect to the distances from the supporting points and the points of forces action. Simply, we can assume that the change in shear and moment through the body is a function in terms of body dimensions (mainly in the x-direction) shear and moment diagrams Shear and moment In general, each segment of the loaded body is studied separately and the functions of the shear and moments are determined for the under study segment. The definition of segment is given as: the part of the body that have two consecutive loadings. The type of loading could be concentrated force (single force) or distributed load( f(x)). To illustrate the process of drawing shear and moment diagrams, see the following examples. 10

11 shear and moment diagrams Draw both shear and moment diagrams for the following beam shear and moment diagrams Solution F.B.Ds Equilibrium equations P M A 0 P a P a b Dy 2a b 0 D Fy 0 A D 2P A P y y y y 11

12 [2] Solution Solution Shear and moment diagrams 12

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