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Sign Conventions Applied and Reaction Force Sign Conventions

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Sign Conventions Applied and Reaction Force Sign Conventions Sign Conventions Sign conventions are a standardized set of widely accepted rules that provide a consistent method of setting up, solving, and communicating solutions to engineering mechanics problems--statics, dynamics, and strength of materials problems. There are two types-static and deformation. Static sign conventions govern the directions of applied and reaction forces and moments, for example, upward or downward. Deformation sign conventions govern the directions of internal forces and moments that deform a body, for example, stretch, compress, twist or bend. For statically determinate structures, static sign conventions are used to determine external reactions. Both types are used to determine internal forces and moments. For statically indeterminate structures, both types are needed to find external reactions and internal forces and moments. Applied and Reaction Force Sign Conventions Are Static Sign Conventions Applied and reaction forces that are directed toward a positive axis are assigned positive signs. For example, referring to Figure 1, the signs of the components of the various forces are: 1. Force 1: F1x sign = +, F1y sign = +; 2. Force 2: F2x sign = +, F2y sign = -; 3. Force 3: F3x sign = -, F3y sign = -; 4. Force 4: F4x sign = -, F4y sign = + Figure 1. Note: In some ENGR 2140 worked examples, the + y direction is assumed to be in the direction of the applied force. To illustrate, in worked Ex1-5, the forces PA and PB are assumed to be in the + y-direction. In many beam analysis problems, the + y-direction is in the direction of the applied force. The correctness of the result is ok, but the orientation of the + y-axis is downward, not upward. 1 Applied and Reaction Moment Sign Conventions Are Static Sign Conventions Applied and reaction moment sign conventions follow the right hand rule (RHR-Figure 2). It is critical to be aware of the viewing direction (Figures 3 & 4) for assigning + and – signs. Always show reaction forces and moments in positive direction on top-level FBD! Figure 2 RHR Figure 3 RHR Figure 4 RHR The positive direction is in the positive z-axis direction (coming out of the x-y plane). Looking into the x-y plane along the z-axis in the negative z-direction, a positive moment appears to be counterclockwise following the curl of the fingers. However, looking out of the x-y plane along the positive z-axis direction, the moment appears to be clockwise following the curl of the fingers. Look into the x-y plane in the negative z-direction. A moment applied to the body is positive if it tries to turn the body in a counter-clockwise direction and negative if it tries to turn the body in a clockwise direction. It’s just the opposite looking out of the x-y plane! Referring to Figure 5, the forces attempt to turn the body as follows (as seen by an observer in the x-y plane looking in the - z-direction into the page): Force F2 attempts to turn the body in a clockwise direction and has a – sign. Forces F1 and F3 attempt to turn the body in a counter-clockwise direction and have + signs. Figure 5 Applied Moments 2 Internal Loading Sign Conventions Are Deformation Sign Conventions Figure 6 illustrates three sign conventions for torsional structures. The positive z-axis points to the left. Figure 7 illustrates a top-level FBD where TR is the reaction torque and is shown positive when normal to the face. z TR Figure 7 Top=Level FBD The equilibrium equation is: ∑T = TR + T1 – T2 = 0, TR = T2 – T1 Always show reaction torque in positive direction. Figure 6 3 Ways to Show Applied Torques Top Level FBD Analysis The first step: determine the external reaction forces and moments following the above sign conventions. Then, internal forces and moments of a structural member can be found. A cantilever beam is used as an example. The sign conventions are illustrated in Figures 8 and 9. Note: Always show the unknown forces and moments in their + directions, then, a – sign means the forces and moments are actually opposite in direction. Cut Plane a-a N1 N1 A B C NA RA Figure 8 Original Beam Figure 9 Top-Level FBD Top Level B Solving for the reaction forces and moments, using the topBD-level FBD in Figure 8, the applied (or external) forces, P and N1, both have –signs according to the statics sign conventions. RA, NB, and MA are shown in their + directions according to the static sign conventions. 3 Thus, the equations that determine the reaction forces and moment are: ∑Fx = NA – N1= 0, NA = N1 ∑Fy = RA – P = 0, RA= P ∑MA = -MA – d*P = 0, MA = -d*P where d is the distance from A to P. MA is negative, creating a “frowny face.” Internal Force and Moment Conventions. Figure 10 shows the widely accepted deformation sign conventions for normal and shear forces and bending moments. The deformation conventions are focused on the way the forces and moments deform the material, not on their directions. Positive normal forces act normal to the face and are tensile forces (pointing away from the face) -stretching the material. Their directions can be upward, downward, leftward, or rightward. Positive shear forces slice the material in a downward direction and have negative directional signs. They tend to turn the material in a clockwise direction. Positive shear forces are plotted on the positive y-axis. Positive bending moments act to bend the material in upwardly concave –creating “smiley faces”. Positive bending moments are plotted on the positive y-axis. N1 Figure 10 Positive NA A Shear and Moment RA Signs Figure 11 Right and Left Hand FBDs at Cutting Plane a-a Figure 11 shows two FBDs resulting from passing a cutting plane, a-a, at Point B. Applied forces, P1 and N1 and reaction forces RA and NA and reaction moment MA, result from the top-level FBD (Figure 6) analysis and are shown in their + directions. The shear force, VB, internal moment, MB, and normal force NB, are shown in their + directions. Note: The shear force, normal force, and internal moments assigned to the two FBDs (right of B and left of B) are shown equal in magnitude, but opposite in direction. 4 The reaction forces and moment are shown according to their static sign conventions. RA, NA, and MA have + signs. The internal forces and moments are shown according to their deformation sign conventions. The shear force, VB, is shown as a positive shear force slicing downward, and has a negative static sign in the -y-direction. The normal force, NB, is shown as a tensile force. It points away from the surface. It points in the + x-direction. Thus, it has positive sign. The moment, MA, is shown as a positive bending moment creating a smiley face (upwardly concave) shape. It has a positive deformation sign. Figure 12 shows the second-level FBD (the left side of the beam FBD). Based on the discussion above, the equilibrium equations are: ∑Fx =NA + NB = 0, NB = -NA NA A ∑Fy = RA- VB = 0, VB = RA RA ∑MB = MA + MB – x*RA= 0, MB = -MA + x*RA Figure 12 Left Hand Always show an internal normal force as tensile Side FBD Note: If the right side FBD were chosen, the exact same equilibrium equations would result. Therefore, the normal force NB, has a negative sign and is a compressive force, not a tensile force as assumed. The bending moment, MB, has a negative sign and acts in the opposite from the assumed direction. It is turning the beam downwardly concave (creating a frowny face), not upwardly concave (creating a smiley face) as assumed. The shear force VB, has a positive sign and acts in the direction assumed. 5 Figure 13 shows the shear force and bending moment diagrams. Positive shear and bending moments are plotted in the positive axis direction. VA = P y - VC = P + VC = 0 Shear Force x y x Moment -MA Slope = VB = RA = P 0 Mx = -MA + P*x Figure 13 Shear Force and Bending Moment Diagrams 6 .
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