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The beam AC is simply supported with a pin at A and roller at B, and with a distributed load applied across its overhanging length BC. The beam has the same U-shaped cross section along its entire length (see the "end view" below) and is made of steel with modulus of elasticity E = 200 GPa. Find expression for the bending moment M(x) and deflection v(x) of the entire beam, and use these to find: a) The maximum normal stress in the beam, either in tension or compression. Explain how you determined the location of the largest amplitude of normal stress. b) The maximum upward and downward deflections of the beam. Hint: just like in calculus, maxima may occur when v'(x) = 0 and/or at the end(s) of the beam

The beam AC is simply supported with a pin at A and roller at B, and with a distributed load applied across its overhanging length BC. The beam has the same U-shaped cross section along its entire length (see the "end view" below) and is made of steel with modulus of elasticity E = 200 GPa. Find expression for the bending moment M(x) and deflection v(x) of the entire beam, and use these to find: a) The maximum normal stress in the beam, either in tension or compression. Explain how you determined the location of the largest amplitude of normal stress. b) The maximum upward and downward deflections of the beam. Hint: just like in calculus, maxima may occur when v'(x) = 0 and/or at the end(s) of the beam

Structural Analysis
Structural Analysis
6th Edition
ISBN:9781337630931
Author:KASSIMALI, Aslam.
Publisher:KASSIMALI, Aslam.
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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The beam AC is simply supported with a pin at A and roller at B, and with a distributed load applied across its overhanging length BC. The beam has the same U-shaped cross section along its entire length (see the “end view” below) and is made of steel with modulus of elasticity E = 200 GPa. Find expression for the bending moment M(x) and deflection v(x) of the entire beam, and use these to find: a) The maximum normal stress in the beam, either in tension or compression. Explain how you determined the location of the largest amplitude of normal stress. b) The maximum upward and downward deflections of the beam. Hint: just like in calculus, maxima may occur when v'(x) = 0 and/or at the end(s) of the beam End View (zoomed in) 14cm 2m Side View •B 12 kN/m 2m с 2cm 2cm 12cm 10cm
Transcribed Image Text:The beam AC is simply supported with a pin at A and roller at B, and with a distributed load applied across its overhanging length BC. The beam has the same U-shaped cross section along its entire length (see the “end view” below) and is made of steel with modulus of elasticity E = 200 GPa. Find expression for the bending moment M(x) and deflection v(x) of the entire beam, and use these to find: a) The maximum normal stress in the beam, either in tension or compression. Explain how you determined the location of the largest amplitude of normal stress. b) The maximum upward and downward deflections of the beam. Hint: just like in calculus, maxima may occur when v'(x) = 0 and/or at the end(s) of the beam End View (zoomed in) 14cm 2m Side View •B 12 kN/m 2m с 2cm 2cm 12cm 10cm
Problem 3. Consider an analysis of a human femur, similar to that on HW8, but this time during an ugly tackle (e.g., in soccer). This time, model the knee joint as a cantilever support, and force at the hip as 30 lbf in the negative z-axis direction as pictured below. Determine and sketch the full state of stress at points A, B, C, and D, this time at a location 15 inches below the hip joint. Consider which loading cases this force creates (e.g., axial, torsional, bending). Assume the bone behaves like a hollow cylinder with an inner radius 0.6 inches and outer radius 1.0 inches (the soft tissue inside the bone, drawn as a cross-hatch is sufficiently flexible that only the dense tissue of the bone needs to be factored into calculation of the bones moments of inertia). The area moment of inertia for a hollow cylinder is I = 7 (r+ — riª), the polar moment 4 π 4R of inertia is J = (rr), and the centroid of a semicircle is yc = 3πT Bin ZA Cross Section а-а хур B с > y 15in A 3015f с -B,D ·a 30154 0 M D ∙A,C π(7 9 a
Transcribed Image Text:Problem 3. Consider an analysis of a human femur, similar to that on HW8, but this time during an ugly tackle (e.g., in soccer). This time, model the knee joint as a cantilever support, and force at the hip as 30 lbf in the negative z-axis direction as pictured below. Determine and sketch the full state of stress at points A, B, C, and D, this time at a location 15 inches below the hip joint. Consider which loading cases this force creates (e.g., axial, torsional, bending). Assume the bone behaves like a hollow cylinder with an inner radius 0.6 inches and outer radius 1.0 inches (the soft tissue inside the bone, drawn as a cross-hatch is sufficiently flexible that only the dense tissue of the bone needs to be factored into calculation of the bones moments of inertia). The area moment of inertia for a hollow cylinder is I = 7 (r+ — riª), the polar moment 4 π 4R of inertia is J = (rr), and the centroid of a semicircle is yc = 3πT Bin ZA Cross Section а-а хур B с > y 15in A 3015f с -B,D ·a 30154 0 M D ∙A,C π(7 9 a
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