Beams
A beam is a structural member whose primary function is to resist transverse loads by developing internal shear forces and bending moments. Analysis begins with the construction of the shear-force diagram (SFD) and bending-moment diagram (BMD). These diagrams are related by the equilibrium differentials dV/dx = –w(x), dM/dx = V so that a concentrated force produces a jump in shear and a concentrated moment produces a jump in the moment diagram. Once the internal moment M(x) is known, the longitudinal bending stress at a distance y from the neutral axis is given by the elastic flexure formula σ = M y / I where I is the second moment of area of the cross-section about the neutral axis. Stress varies linearly with y, is zero on the neutral axis, and reaches its extreme values at the outermost fibers. Common maximum-moment expressions for determinate beams are M = PL/4 (central point load on a simple span), M = wL²/8 (uniform load on a simple span) and M = PL (end load on a cantilever).
Knowledge Check
1.According to the elastic flexure formula, how does bending stress vary with distance from the neutral axis?
2.For a simply supported beam carrying a single point load at mid-span, the maximum bending moment occurs:
3.On the shear-force diagram of a simply supported beam with a mid-span point load, the concentrated force appears as:
4.In the flexure formula σ = M y / I, the symbol y represents:
5.For a rectangular cross-section of width b and depth h, the second moment of area about the neutral axis parallel to the width is:
6.A cantilever beam with a point load at its free end experiences its maximum bending moment at:
7.The slope of the bending-moment diagram at any station is equal to:
8.The longitudinal bending stress on the neutral axis of a beam is:
9.A simply supported beam 5 m long carries a uniform load of 20 kN/m over its entire span. What is the maximum bending moment?
10.In a simply supported beam carrying a uniform distributed load, the maximum shear force occurs: