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Normal stresses in a beam of linearly elastic

material (a) side view of beam showing

distribution of normal stresses, and (b) cross

section of beam showing the z axis as the neutral

axis of the cross section

To determine the normal stress in a beam

subjected to bending we need to know the moment

acting at that point the distance from the

neutral axis and the moment of inertia of the

beam.

Flexure Formula

Positive curvature results from positive applied

moments. For positive curvature compression

occurs at the top surface. The beams shown have

non-symmetric cross-sections and the neutral axis

is therefore not at the center.

Taking moments about the neutral z-axis (for each

small slice the force is stress x area and the

distance from the z-axis is y)

Remember

Doubly symmetric cross-sectional shapesIf a

beam is doubly symmetric (i.e. symmetric in the z

and y directions, the neutral axis will be at the

center of the beam.

Where Ssection modulus and is a geometric

factor. For beam design we can calculate the

required section modulus and then select.

FIG. 5-13 Example 5-2.Wire bent around a drum

O

- For the beam shown determine the maximum tensile

and compressive stresses in the beam. - Calculate the maximum ve and ve moments
- Need to draw shear and moment diagrams
- Mpos2.025 kN.m Mneg-3.6 kN.m
- 2) Find neutral axis

3) Find moment of inertia about the neutral

axis

3) Calculate the maximum ve and ve stresses

there are four combinations Mpos c1 Mpos c2

Mneg c1 Mneg c2

Tensile

Compressive

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A simple beam of span length 21 ft must support a

uniform load q2000lb/ft as shown. Neglecting

the weight of the beam select a structural beam

of wide flange shape to support the loads.1)

Determine Mmax Draw shear diagram Calculate

M2) Calculate required section

modulus Calculate or look up max stress3)

Select a beam W12x50 S64.7

X19430 ft Mmax88,920 lb-ft SminMmax/?allow59.3

in3

Design of Beams

A cantilever beam AB of length L is being

designed to support a concentrated load at the

free end . The cross-section is rectangular.

Calculate the height as a function of distance so

that the beam is fully stressed. I.e. at every

point ??allow