Mechanics of Materials Engr 350 - Lecture 29 Beam Deflections
Mechanics of Materials Engr 350 Lecture 29 Beam Deflections Deflection Just like axial loadings cause a member to stretch, loads on a beam cause it to deflect Beam deflections depend on the The stiffness of the material Beam cross-sectional dimensions Loading state
We will first consider the integration method for determining the shape of the deformed beam. 2 Beam deflections by integration x coordinate: measures position along the beam (usually start from left end)
y coordinate: measures vertical position within beam (almost always from neutral axis, upward) v coordinate: measures displacement of the neutral axis (measured from x-axis to neutral surface) The x and y coordinates are the same as those used in the derivation of the flexure formula Recall the moment curvature relationship developed before (L21/sect 8.4)
3 Differentiating the elastic curve From calculus substituting the expression for the moment-curvature relationship we reach Notice that M is a function of x (most of the time it is) 4
Beam relationships Moment sign conventions: Deflection: Slope: Moment:
Shear: (if EI is a constant) Load: (if EI is a constant) 5 Relationships between beam diagrams 6 Deflections by integration of the moment equation
If the beam meets all of several assumptions, the moment equation can be integrated to give the deflections of the beam These assumptions are a. Beam deflections are small b. Planes remain plane c. E & I are constant (constant material, prismatic) Boundary conditions a. Specific values of deflection or slope that are known. b. When a beam is analyzed in sections, the boundary conditions are known at the ends of the section. (Bounds of section considered, not necessarily the beam)
c. One BC can be used only once to determine a constant of integration 8 Boundary conditions continued At a pinned or roller support we know that At the fixed end of a cantilever beam, we know two things
At the free end of a cantilever beam we know two things 9 Continuity conditions In beams, different sections can have different moment and shear equations, so we must write different deflection equations in each section. Since the beam is continuous, there is not an abrupt change in the the deflection or slope of the beam and this fact can be used as a boundary condition to solve for constants for the specific section. 1 0
Symmetry conditions Sometimes symmetry exists in a beam In this case, the slope at the mid-span must be zero Remember, only one constant of integration can be determined from each
boundary, continuity, or symmetry condition. 11 Homework for Tonight All about boundary conditions MM Module 10.1 has a beam boundary condition game 14-15 beam boundary conditions Type in name and print result Tips: Try printing from the machine before you play the game. If that doesnt work, see if you can print/save a PDF, then print that file elsewhere 12
Procedure for double integration method 1. Sketch: Sketch the beam including supports, loads, and the xv coordinate system. Sketch the approximate shape of the elastic curve. 2. Support reactions: Determine the beam reactions by considering the equilibrium of the entire beam. Show these reactions in their proper direction on the beam sketch. 3. Equilibrium: For each segment to be considered, draw a FBD that cuts through the beam segment at some distance x from the origin. Show all loads acting on the FBD. From the FBD, derive the bendingmoment equation, taking care to note the interval for which it is applicable ____________________ 4. Integration: For each segment, set the bending-moment equation equal to ____________ Integrate this differential equation twice, obtaining a slope equation __________, a deflection equation ______, and two constants of integration. 5. Boundary and continuity conditions: List the boundary conditions that are applicable for the bending-moment equation. 6. Evaluate constants: Use the boundary and continuity conditions to evaluate all constants of integration. 7. Elastic curve and slope equations:
Replace the constants of integration in step 4 with the values obtained from the boundary and continuity conditions in step 6. 8. Deflections and slopes at specific points: Calculate the deflection at specific points when required. 13 Example problem For the beam and loading shown, use the doubleintegration method to determine (a) the equation of the elastic curve for the beam. (b) the maximum deflection. (c) the slope at A. Assume that EI is constant for the beam. 14
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