- 5.1 Introduction
- 5.2 Pure Bending of Beams of Symmetrical Cross Section
- 5.3 Pure Bending of Beams of Asymmetrical Cross Section
- 5.4 Bending of A Cantilever of Narrow Section
- 5.5 Bending of a Simply Supported Narrow Beam
- 5.6 Elementary Theory of Bending
- 5.7 Normal and Shear Stresses
- 5.8 Effect of Transverse Normal Stress
- 5.9 Composite Beams
- 5.10 Shear Center
- 5.11 Statically Indeterminate Systems
- 5.12 Energy Method for Deflections
- 5.13 Elasticity Theory
- 5.14 Curved Beam Formula
- 5.15 Comparison of the Results of Various Theories
- 5.16 Combined Tangential and Normal Stresses
- References
- Problems

## 5.5 Bending of a Simply Supported Narrow Beam

In this section, we consider the stress distribution in a narrow beam of thickness *t* and depth 2*h* subjected to a uniformly distributed loading (Fig. 5.6). The situation described here is one of plane stress, subject to the following boundary conditions, consistent with the origin of an *x, y* coordinate system located at midspan and midheight of the beam, as shown:

**Figure 5.6**. *Bending of a simply supported beam with a uniform load*.

Since no longitudinal load is applied at the ends, it would appear reasonable to state that σ_{x} = 0 at *x* = ± *L*. However, this boundary condition leads to a complicated solution, and a less severe statement is used instead:

The corresponding condition for bending couples at *x* = ±*L* is

For *y* equilibrium, it is required that

### 5.5.1 Use of Stress Functions

The problem is treated by superimposing the solutions Φ_{2},Φ_{3}, and Φ_{5} (Section 3.6) with

We then have

The stresses are

The conditions (a) are

and the solution is

The constant *d*_{3} is obtained from condition (c) as follows:

or

Expressions (e), together with the values obtained for the constants, also fulfill conditions (b) and (d).

The state of stress is thus represented by

where is the area moment of inertia taken about a line through the centroid, parallel to the *z* axis. Although the solutions given by Eqs. (5.25) satisfy the equations of elasticity and the boundary conditions, they are not exact. We can see this by substituting *x* = ±*L* into Eq. (5.25a) to obtain the following expression for the normal distributed forces per unit area at the ends:

This state cannot exist, as no forces act at the ends. From Saint-Venant′s principle, however, we may conclude that the solutions do predict the correct stresses throughout the beam, except near the supports.

### 5.5.2 Comparison of the Results with the Elementary Theory Results

Recall that the longitudinal normal stress derived from elementary beam theory is σ_{x} = -*My*/*I*; this is equivalent to the first term of Eq. (5.25a). The second term is then the difference between the longitudinal stress results given by the two approaches. To gauge the magnitude of the deviation, consider the ratio of the second term of Eq. (5.25a) to the result of elementary theory at *x* = 0. At this point, the bending moment is a maximum. Substituting *y* = *h* for the condition of maximum stress, we obtain

For a beam of length 10 times its depth, this ratio is small, . For beams of ordinary proportions, we can conclude that elementary theory provides a result of sufficient accuracy for σ_{x}. On the one hand, for σ_{y}, this stress is not found in the elementary theory. On the other hand, the result for τ_{xy} is the same as that yielded by elementary beam theory.

The displacement of the beam may be determined in a manner similar to that described for a cantilever beam (Section 5.4).

**Part B: Approximate Solutions**