Task 3. Construction shear force and bending moment diagrams for beams

Construct shear force and bending moment diagrams for four beams loaded as shown in fig. 5. Assume: F = qa; l = a; M = qa ².

Take the data necessary for computation from table 5.

table 5

variant /group No./
q, kN/m
a, m	1.1	1.2	1.3	1.4	1.5	1.6	0.7	0.8	0.9	1.5	1.2	0.6

Order of work

1. Draw the design schime for the beam; indicate all dimensions and loads using letters and digits.

2. Determine values of support reactions; show them on the computation diagram using letters and digits.

3. Determine number of the beam sections and numerate them. Mark each section beginning and end.

4. Using the method of sections write the equations for determination shear force Q and bending moment M _b for each section. Use value of boundary conditions in the equations to calculate Q and M _b.

5. Construct the diagrams Q and M _b.

Fig.5. (see also p.18)

Fig.5. Continued (see also p.19)

Fig.5. Continued (see also p.20)

Fig.5. Continued (see also p.21)

Fig.5. Continued (see also p.22)

Fig.5. Continued (see also p.23)

Fig.5. Ending

Full check of strength of the beam by principal stresses

For one of the beams of task 4 (by the teacher's choice) select I-beam section under condition of strength by normal stresses. Check up strength of the beam by shear stresses and make full check of the beam in a critical section by principal stresses. assume the allowable value of normal stress [ s ] = 160 MPа to computate.

Order of work

1. On М _b diagram determine the greatest value of bending moment M_max for the selected beam. Under condition of bending strength determine the moment of résistance W_y .

2. Select I-beam number using the "I-beams" guide and calculated value W _y. Write down the necessary dimensions of the I-beam section and draw it to scale.

3. check the strength of the beam by the greatest value of normal and shear stresses.

4. Determine the critical section of the beam. This is the section in which both bending moment and shear force are the big values.

5. Write down the values of M _b and Q in critical section - M_cr and Q_cr.

6. For each of five (or seven) points of the section (due to symmetry of the section upper half of which is only considered) calculate normal and shear stresses.

7. Using the normal and shear stresses calculate principal stresses for each point of the section.

8. Basing on conducted calculations draw the diagrams for normal, shear and principal stresses (fig. 6).

9. Determine the position of principal area (by the teather’s suggestion) and draw the directions of principal stresses (fig. 7).

10. Check up strength of the beam by the third theory of strength in dangerous points of the section.

11. Make the conclusions.

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