# Static equilibrium force and moment relationship

equilibrium when the resultant of all forces and all where F refers to forces and M refers to moments of forces. Static determinacy. If a body is in equilibrium under the action of the connection by pressing the two steel bars together. law; buoyancy forces are governed by laws describing hydrostatic forces in fluids. Some of these . relationship is known as Hooke's law and can be expressed as an equation k. F δ = Resultant moment exerted by a force system. We can show that there is no resultant force by adding the three forces using situation shown in Fig.3 to derive some mathematical relationships. Consequently, for true equilibrium in a static system, there must be no resultant force and no.

Drag the green dot to increase force F3 in Fig. In one investigation with the set up shown in Fig.

Which one of the following shows possible values for F1 and F2? F1 and F2 both equal 9 N F1 and F2 both equal 10 N F1 and F2 both equal 15 N For the point at the centre of the forces to be stationary there must be no unbalanced force in the horizontal direction.

### Calculating Forces in Beams - DT Online

For the point at the centre of the three forces in Fig. When this happens the system is described as being in equilibrium. Increasing the value of F3 increases the values of F1 and F2 required for equilibrium. Click through the stages in Fig. Forces in equilibrium The point at the centre of the forces is acted on by three forces.

We can show that there is no resultant force by adding the three forces using the rules of vector addition. In the animation of Fig. F2 is then added to F3 remembering to keep its size and direction correct. F1 is added to F2 again keeping its size and angle correct. The end point for the final vector is back at the beginning of the first, so there is no resultant force.

Complete the following statements to summarize the forces in Fig.

### Moments and equilibrium - Mechanics

The three forces acting in Fig. The length is equal to its magnitude and the arrow points in the direction the force is acting.

The two loads on the beam can be drawn as vertical lines pointing downwards scaled such that their lengths equal F1 and F2.

Similarly the two reactions can be drawn as vertical lines pointing upwards scaled such that their lengths equal R1 and R2. The total length of the downward pointing line is equal to the sum of all the downward forces and is known as the Resultant. The total length of the upward pointing line is equal to the sum of all the upward forces and is known as the Equilibrant - because it is the force required to balance the two loads and maintain balance or equilibrium Bow's Notation A labelling convention can be used to make it easier to work on more complex structures.

This is known as Bow's Notation and it is applied simply by working around the structure clockwise and labelling the space between each force in capital letters. This is known as a Space Diagram. We can now work round the beam clockwise to draw a Force Diagram as follows: The direction is known from the Space Diagram and magnitudes are calculated where possible using Moments - R2 was the only force not calculated The two lines can be merged together to produce a single Force Diagram representing all the forces on the beam and, if drawn to scale, the value of R2 can be measured.

As before, the Resultant 'ac' can be measured downwards and the Equilibrant 'ca' measured upwards note that small letters are now used to name the vectors in Force Diagrams.

Bending and Deflection We also know from experience that if we walk along a builder's plank, for example, it will wobble up and down, or deflect, and we know that it is easier to bend a rule when it is held flat rather than on edge.

If the beam is not strong enough it may deflect so much that it breaks! The Bending Strength of the beam and its Deflection can both be calculated.

## 14. Moments and equilibrium

The results will depend upon its length, cross section and what material it is made from. Bending Moments A Bending Moment diagram, as shown, can be drawn by projecting down from the Space Diagram and drawing Vectors to represent the applied loads i.

In the centre of the beam this can be seen to be equal to: Load x Span divided by 4 To understand why this is so, imagine the beam to be cut into two then the two halves placed back to back to create two cantilevers each supporting half of the load - and each cantilever will be half the length of the beam of course. It is clear that standing on the end of a diving board makes it bend the most and this is where to find the Maximum Bending Moment.

This can be expressed as Load x Span divided by 4 as shown Similarly, for a uniformly distributed load such as the beam's self-weight the Maximum Bending Moment BM is equal to Load x Span divided by 8 which can also be written as half the load x a quarter of the span Self-weight can be assumed to be the same as a single point load acting on the centre of a beam or the cantilever.

The Simple Bending Equation applies to simply supported beams and arches if the radius of curvature is greater than 10 times the depth.

Starting at the left side of the beam, R1 acts upwards and no other force acts on the beam shown until the centrally applied load W. The Shear Force on the left half of the beam is therefore constant and can be represented on a Shear Force Diagram starting with a Vector drawn upwards to represent R1.

Half way across the beam, the Shear Force then acts downwards because of the load W and is constant along the right half of the beam to R2.