Beam Analysis • Structural members that offer resistance to bending caused by loads applied along their length are known as beams. • The most important of all structural members. • How do beams behave: 1. Determine the internal forces in beams. 2. Understand how these forces are related to the shape of the cross-section of a beam and the material they are constructed from. Universal I-beams Universal beams in a steel framed building Steel box girders RC beams and slabs Wings and bones Boeing 787 Dreamliner Proxima femur Types of beams • Statically determinate beams Sufficient support to prevent collapse and reactions can be calculated from equations of equilibrium. Simply supported 3 reactions R1 R2 Cantilever R3 3 reactions R1 M1 R 2 Simply supported with overhang 3 reactions R1 R2 R3 Types of beams • Statically Indeterminate Beams (next year in detail) More supports than are necessary to provide equilibrium Continuous Propped Cantilever 4 reactions 4 reactions Double built-in 6 reactions Beam Loading • Loading can be point loads or distributed loads. • The most common loadings are: – Uniformly distributed line load (UDL): (e.g. self-weight, building contents, wind loading) w kN for each metre span e.g. 5 kN/m acting on 4m span 20 kN resultant acts through centre of the UDL Beam Loading – Concentrated point load: Never in reality but good approximation Beam Loading • Less common loading: – Triangularly distributed line load: e.g. hydrostatic loading, soil pressure e.g. Koffer Dam subject to hydrostatic pressure – Discrete couple: e.g. due to connection with other structural member Load effects Tension d/ 2 neut ral axis Compression d/ 2 hogging Tension y Mz z d x Compression Rectangular X-section b Effect is internal couple with a moment, M. Called a Bending Moment Load effects Shearing V V Accompanied by internal shear force V V V Take imaginary cut through beam at Y-Y BENDING MOMENT Resisting internal moment to balance the moment due to applied force Providing an internal vertical force to balance the vertical applied force What is the effect of these two? SHEAR FORCE Expedition workshed … Relationship between w, V and M Fundamental relationship between loading and internal forces – very important. dx – small distance dM – small change in bending moment dV – small change in shear force F FBD: y V w dx V dV 0 Therefore, dV w dx A åM 0: @A ( ) ( V w dx ) = 0 : - M + M + dM + w × dx × dx 2 -V × dx = 0 \ dM -Vdx = 0 Þ dM =V dx or M = ò V dx Some common loading situations 1. No distributed load, w: V w dx wx c1 V c1 M V dx Vx c2 since w 0 Constant Linear 2. Uniformly distributed load, w: V w dx wx c1 Linear wx 2 M V dx wx c1 dx c1 x c2 2 Parabolic Note: for maximum (and minimum) bending moment, M, dM V 0 dx Therefore, maximum (and min) B.M. occurs at point of zero shear force. Shear force and bending moment diagrams • Bending moment and shear force vary along length of the beam. • Therefore, required to produce longitudinal diagrams. Shear force diagrams (SFD) • Scaled, visual representation (graph) of shear force variation along beam. • Plotted with beam centreline as longitudinal axis. + ve - ve Positive plotted vertically up – just a convention! Shear force and bending moment diagrams Bending Moment Diagrams (BMD) • Scaled, visual representation (graph) of bending moment variation along beam. compression sagging + ve tension Sagging taken as positive bending and plotted on tension side – just a convention!! tension - ve compression hogging