Establish general expressions for the axial force P(θ), shear V(θ), and moment M(θ) for the curv

Establish general expressions for the axial force P(θ),...

Key Concepts

Equilibrium Analysis

Equilibrium analysis involves applying the conditions for static equilibrium (sum of forces and moments equals zero) to a free-body diagram of a structural element. In analyzing curved bars, setting up the equilibrium of an infinitesimal element leads to differential equations whose solution provides the internal force and moment distributions.

Curved Beam Theory and Coordinate Systems

Curved beam theory extends classical beam theory to members that are not straight by considering the effects of curvature on the stress distribution. This requires using a coordinate system aligned with the geometry of the beam (often defined by an angle measured from a reference axis), which alters the equilibrium equations and necessitates special considerations in determining the axial force, shear, and bending moment distributions.

Bending Moment

The bending moment is the internal moment that induces bending stresses within a beam. It arises as a result of the applied loads and the geometry of the member, and it is responsible for the curvature in the beam. For curved structures, determining the bending moment involves integrating the effects of the load distribution while accounting for the curved geometry.

Shear Force

Shear force is the component of the internal force perpendicular to the member’s axis. It is responsible for producing shear stresses in the material and arises from loading that tends to slide one part of the section relative to another. In curved beams, shear force calculations require careful application of equilibrium principles along the curved path.

Axial Force

The axial force is the component of the internal force that acts along the direction of the member’s axis. In the context of beam theory and structural analysis, it represents the stress resultant due to normal stresses that develop when the member is loaded along its length. For curved members, the axial force must be analyzed considering the variation in orientation along the curved geometry.

Transcript

00:01 Now in this problem we have a a bent a half circle a beam that describes a it's bent into a half circle.

00:11 It's can't levered at this one end and then there's applied load at the other end now they want us to calculate the the sheer the normal force and shear force and the moment in the curve drive as a function of theta so we have theta and i'm going to measure from here from the positive x -axis and so this is the angle theta the the radius of this half circle i'm going to use a capital r just to avoid some confusion so we can basically we can take a cut and it's basically it's easiest to take a cut from here because then we actually don't need to really find these and we could actually find them by just taking it to figure out what these guys are when when theta equals pi.

01:07 So we have a reaction force vector here.

01:10 So this is going to contain both the shear and the norm force, and then we have a moment.

01:17 And so balance of forces simply says that this plus this has to be zero.

01:26 Now, we can figure out what this is in terms of x and y components.

01:32 So that's p cosine fee, sine fee, cosine fee in the x, sine fee in the y.

01:42 So we know that that is the magnitude of the load there.

01:50 And let's see here.

01:51 I think i missed a minus sign in there.

01:54 Let me just check.

01:56 Yeah, this should be minus.

02:11 So we can get that and now we know the position.

02:15 Of that vector, that force is just r in the x direction.

02:22 So what we can do is we can figure out what this is in terms of x and y very easily, because it's just the negative of this.

02:31 But what we want, we want the normal and the shear.

02:37 And so that's the normal is in the theta direction and the shear is in the radial direction.

02:43 So if we take the dot product of this with e, with the radial norm with the radial unit vector which is given by this and here's the circumferential or tangential unit vector so we dot this with with this vector and then with this vector and we'll get the this is the shear and this is the normal now we can then take the take moments and i'm going to take moments about about this point here just because it's easiest.

03:24 It's probably might be easier here, but here, if we take moments about here, we know what these moment arms are easily.

03:31 If we take moments about here, then we nend them this around and figure out what that moment arm is.

03:36 So the moment caused by this load here is minus r p sine fee...

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