Performance of beams
A beam is a structural member designed to support various loading applied at points along the member. When a beam is loaded it will bend and this bendin is measured in newton millimetre or multiples of the units (e.g. kNm.). The forces applied to a beam are counteracted by the supports which are called reactions (RL left support and RR right support)
Beams are usually supported on walls piers or columns which provide the necessary equilibrium. (According to Newton’s third law there is for every action an equal and opposite reaction.) In simple terms the load on the beam is the action and the supports provide the reaction. There are three different support reaction:
Type (a) accommodate three reaction, (b) one reaction and type (c) two reaction
In this subject we will consider only simply supported beams which can be determinate completely with the three static equation. Continuous beams are beyond the scope of this subject. The two beam types are that we will use are shown below.
Beams are subjected to uniformly distributed loads (UDL), point (concentrated) loads or a combination of both. The various loading conditions to which a beam may be subjected to are shown below.
With the three static equation
simple structures (statically determinate) can be completely analysed.
To calculate the reaction of beams we use the equation Σ M = 0
The Shear force at any cross-section of the beam is equal to the algebraic sum of the external forces acting on one side of the section only.
The Bending moment, at any point of the beam, is equal to the algebraic sum of the moments (taken about the point) of the external forces (loads & reactions) on one side of the section only.
The unit of bending moment is the same as for moment of a force, i.e. the newton metre (Nm) and multiples and submultiples of this unit.
Sign convention for shear force and bending moment
The sign shear force (S) and bending moment (M) are positive (+) or negative (-) as shown below.
Determination of the reactions of a simple beam with a point load
Use the sum of the moments must be zero (Σ M = 0) equation to calculate the magnitude of the reactions
Sign convention: clockwise moments positive (+ve), anti-clockwise moments negative
There are to unknowns (RL and RR) and one must be eliminated to be able to calculate the reaction. We select the rotation point at RR because reaction RR times distance is then zero.
If this formula is used, you must be certain that RL is correctly calculated. At this stage it is better to calculate RR as well.
Determination of the reactions of a simple beam with a uniformly distributed load
Determination of the bending moment
A number of specifics loading cases occur frequently and for this cases standard formulas exist. The derivation of the bending moment formulas is dealt with in Structures 2
Standard formula for reactions, bending moments and deflections
Beam with a point load
For every load on a beam, there is a critical point at which a maximum bending moment occurs. The diagrams clearly indicate that this point is at zero shear. In other words where zero shear occurs there is the maximum bending moment.
The factors that influence the deflection have been discussed previously. A constant factor that depends on the loading of the beam is introduced in the deflection equation above.
Note: that in the deflection formula for UDL w is the load per metre, (in above equation