Determine the reactions at A and B when β=80^∘ .

Name: Problem 82, Chapter 4: Vector Mechanics for Engineers: Statics and Dynamics
Uploaded: 2020-06-26T01:28:39-08:00
Duration: 13 min
Channel: Henry York
Description: Determine the reactions at A and B when β=80^∘ .

Key Concepts

Static Equilibrium

This concept ensures that a structure or body remains at rest when the sum of all forces and moments acting on it is zero. It is crucial for solving problems involving support reactions because it provides the mathematical conditions (?F = 0 and ?M = 0) needed to determine unknown forces at points such as supports or connections.

Reaction Forces

Reaction forces are the support forces provided by the constraints or supports in a structure, which counteract the applied loads to maintain equilibrium. Understanding how these forces develop and their directions is essential, as different types of supports (like pinned or roller supports) can resist forces in different directions.

Free Body Diagram

A free body diagram is a simplified representation of a structure, isolated from its surroundings, showing all the external forces and moments acting on it. It is a fundamental tool in statics that aids in visualizing and setting up the equilibrium equations needed to solve for unknown reactions.

Resolution of Forces

When forces are applied at angles, resolution of forces involves breaking them down into orthogonal components, typically along horizontal and vertical axes. This process uses trigonometric functions and is essential for correctly applying the equilibrium equations when analyzing the forces in a structure.

Moment Equilibrium

Moment equilibrium requires that the sum of the moments (or torques) about any point in a structure is zero. This principle is often applied by selecting a strategic point (such as a support) to simplify the calculation of unknown reactions, as forces acting through that point do not create a moment.

Transcript

00:06 Problem 4 .82 is a 3 -force rigid body problem, where there is an applied force at c of 100 newtons, and the other two forces are reaction forces at points a and b.

00:21 Now, since point b is at a roller, it means that its force will be normal to the surface the roller contacts.

00:32 And since this is a three -force rigid body, where two forces lines of action intersect, the third forces line of action must also intersect there.

00:45 So we know that the reaction force at a also goes in that direction.

00:59 Now to find the exact angles of these reaction forces, we have to do some geometry and make some triangles.

01:07 So i'll make a right triangle here.

01:14 Call this point a, i mean e.

01:16 I'll call the point where the forces intersect point d.

01:25 Also make a right triangle here with the force at b.

01:33 I'll call that f and one last triangle here between a, b, and this point g i'll draw it out to make it a little clearer you know the angle here is 25 degrees that these two lengths are 0 .15 meters and 0 .25 meters given in the problem this angle is the same as the angle for the beam at point a so it's also 25 degrees and we're given this angle as 80 degrees in the problem, meaning that this angle must be 75 degrees to add up to 180.

03:20 Now using that angle, we can find points bd, and then later the sides of the reaction at b, right triangle.

03:45 So i take the tangent of the angle 75, set equal to its opposite side, bd, over its adjacent side.

04:06 The 0 .25 meters, we get that bd is equal to 0 .933 meters.

04:40 I know that this angle here at point b is 25 degrees, since the force at b is normal to the beam af, and we know that the beam af is 25 degrees to the horizontal.

05:01 And using the sign of that angle, we can find fd, which is .394 meters.

05:31 And using the law pythagorean's theorem, we can find bf, which is 0 .846 meters.

06:17 Now, we already know the direction of the reaction force of b, which is 25 degrees.

06:24 To find the reaction force angle of a, we need to find the lengths bg and ag of the small triangle at the top left...

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