Deflections of Beams

Apply the double integration method to beams by using bending moment equations to derive slope and displacement, integrating across sections with constants of integration determined for each region.

Derive slope and displacement by double integrating the moment function Mx = 2.5x − x^2/2 and apply boundary conditions at x=0 and x=5 m to determine C1 and C2.

Apply boundary conditions for fixed and roller supports, use two moment functions for spans ab and bc, and enforce slope and displacement continuity at B in the double integration method.

Apply the double integration method to a two-span beam, deriving moment, slope, and displacement functions from boundary and continuity conditions, and determining four constants of integration.

Determine slopes and displacements at A, B, and C using the double integration method and the functions; show zero displacement at supports and a maximum clockwise slope at C.

Tackle three problems on the double integration method for beam deflections, with full solutions in resources and use the questions and answers section to ask questions.

Use the double integration method to compute the original slope and deflection of a rectangular steel beam, with E and I, converting units between inches and feet.

Apply the moment area method to beams by equating slope change to the area under the M/EI diagram between two points, a relative method using displacements from a reference point.

Use the moment area method on a simply supported beam to find the change in slope from A to B by the area under the moment diagram divided by EI.

Use the moment area method to determine slopes and displacements from the area under the moment diagram and centroid data, noting the fixed end slope is zero.

Apply the moment area method to beam deflections, calculating the distance between tangents at a point by using centroid of the moment diagram and the moment about A or B.

Apply the moment area method on a twelve-foot beam with a point load to determine slope and displacement at B, using area -144/EI and centroid at 4 ft.

Use the moment area method on a 6 m overhanging beam with a 10 kN load at 2 m from the left support to determine slope and displacement.

Solve three problems using the moment area method to master beam deflections, with solutions available in the resources section to guide your practice.

Apply the conjugate beam method by replacing the real beam with a conjugate beam loaded by the real beam’s M/e diagram; slope maps to shear, displacement to moment.

Use the conjugate beam method to determine slope and displacement of beams, solving a fixed-left-end, free-right-end case, and a simply supported beam under a uniform load to locate maximum deflection.

Apply the conjugate beam method to a fixed beam with a hinge and roller, determining reactions, bending moments, slopes, and deflection at key points.

Practice conjugate beam method problems to determine beam deflections, with full solutions provided in the resources; students are encouraged to solve all three problems themselves.

Compare three beam deflection methods: double integration with EI and boundary conditions; moment trigger method using area under the M/e diagram; and the conjugate beam method linking slope and displacement.

Compare beam deflection methods: double integration method yields exact results but is lengthy; moment method offers quick relative results for simple cases; conjugate method gives absolute results.