Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

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FIGURE 15.1 Stress trajectories for a simply supported beam subjected to a concentrated load
at midspan.
design of reinforcement to carry the tensile stresses. Curves drawn with their tangents
at each point in the directions of the principal stresses are called stress trajectories.
Since there are generally two nonzero principal stresses at each point (in plane stress),
there are two stress trajectories passing through each point. These curves will be
perpendicular, since the principal stresses are orthogonal; one set of curves will
represent the maximum stresses, whereas the other set of curves will represent the
minimum stresses. The trajectories for a simply supported beam with a rectangular
cross section where the beam is subjected to a concentrated load at midspan are shown
in Figure 15.1. The dashed lines represent the directions of the compressive stresses,
while the solid lines represent the tensile stress directions. Stress concentrations exist
in the vicinities of the load and reactions, and consequently, the stress trajectories
become much more complicated in those regions. Figure 15.1 omits the effect of stress
concentrations.
General calculation procedures
To determine the principal stresses and the maximum shear stress at a particular point in a
beam, the following procedures are useful:
1. Calculate the external beam reaction forces and moments (if any).
2. Determine the internal axial force (if applicable), shear force, and bending moment acting
at the section of interest. To determine the internal forces, it may be expedient to
construct the complete shear-force and bending-moment diagrams for the beam, or it
may suffice to consider simply a free-body diagram that cuts through the beam at the
section of interest.
3. Once the internal forces and moments are known, determine the magnitude of each
normal stress and shear stress produced at the specific point of interest.
a. Normal stresses are produced by an internal axial force F and by an internal bending
moment M. The magnitude of the axial stress is given by s = F/A, and the magnitude
of the bending stress is given by the flexure formula s = −My/I.
b. Shear stress produced by nonuniform bending is calculated from τ = VQ/It.
4. Summarize the stress calculation results on a stress element, taking care to identify the
proper direction of each stress.