Many engineering structural components and machine elements take
the form of a **beam** which bends under load. Analysis
of bending and the resulting stresses and strains are the subject of
several tutorials in this series.

This brief tutorial introduces the study of beams by explaining basic terms and, particularly important, the sign conventions used in subsequent tutorials.

The diagrams below illustrate different types of support and loading.

We will use the following conventions for +ve and -ve directions.

This diagram shows the positive directions for x and y axes. Thus reaction forces R1 and R2 are positive. Loads F1 and F2 are negative.

In subsequent tutorials we will use extensively the conditions for static equilibrium ("sum of forces = 0" and "sum of moments = 0"). Be aware of the following error when writing equations for the sum of forces in the example above!

It is **not correct** to state R1 +
R2 = - F1 - F2 (for sum of forces on the y
axis = 0)

It **is correct **to state
R1 + R2 - F1 - F2 = 0 or
R1 + R2 = F1 + F2

The incorrect equation makes the mistake of a "double negative". If you are prone to this error (as I am) it helps to put all terms and their signs initially on the LHS.

This diagram shows definitions of +ve and -ve bending.

We interpret bending as the consequence of **bending moments**
generated by loading.

To viualise this consider an element of a beam and the bending moments that produce positive or negative bending on the element.

In subsequent tutorials we always consider sections
of beams originating at the left hand end (x = 0) with the cut face at the right hand end of the section
Hence the sign of the bending moment M is determined by the
direction of the moment shown on the **right hand face**
of the elements above.

**Note that the rotational direction of M on the right hand
face of the elements above is the reverse of the normal sign convention for moment
loads where clockwise moments are +ve and anti-clockwise moments are
-ve**.

This difficulty regarding the sign of bending moments can be avoided by taking sections from the right hand end of the beam and reversing the +ve direction of the x axis. I find this counter intuitive and prefer to reverse the sign of M.

Vertical loads and resultant reaction forces generate vertical
**shearing forces** in a beam. This diagram shows
definitions of +ve and -ve shear.

To viualise this consider an element of a beam and the shearing forces that produce positive or negative shear on the element.

In subsequent tutorials we always consider sections
of beams originating at the left hand end (x = 0) with the cut face at the right hand end of the section
Hence the sign of the shearing force V is determined by the
direction of the force shown on the **right hand face**
of the elements above.

**Note that the direction of V on the right hand
face of the elements above is the reverse of the sign convention for
forces where upward forces are +ve **.

This difficulty regarding the sign of shearing forces can be avoided by reversing the sign convention for vertical forces. I find this counter intuitive and prefer to reverse the sign of V.

**Note that the sign conventions for bending moment M and
shearing force V for positive and negative bending and shear as defined
above are universally adopted and must not be changed.**

I welcome feedback at:

alistair@alistairstutorials.co.uk