Prof. Stephen M. Mutuli, Dept. of Mechanical & Manufacturing Engineering

Prof. Stephen M. Mutuli, Dept. of Mechanical & Manufacturing Engineering, University of Nairobi (e-mail : [email protected])

EXERCISE No. 2 Tutorials FME 202 - Solid & Structural Mechanics Tutorial No 1 ( Shear Forces And Bending Moment Diagrams) 2wN / m Q1. 2wN / m wN / m

6wa 2 N  m 3a 6a Fig Q1

Fig Q1 shows a beam of length 9a simply supported over a span length 6a and loaded as indicated. (i) calculate the reactions at the supports (ii) draw to scale the Shear Force and the Bending Moment diagrams

Q2. 3.6wN / m wN / m

Fig Q2 shows a beam of length 9a meters and simply supported at the ends. The beam supports a distributed load varying in magnitude linearly from zero at the left-hand end to 3.6wN / m at a point distance 6a meters from the left-hand support; and another uniformly distributed load of magnitude wN / m over the remaining 3a meters. A clockwise couple of magnitude 8wa 2 N  m is applied at appoint 6a meters from the left-hand support. (i) calculate the reactions at the supports Prof. Stephen M. Mutuli, Dept. of Mechanical & Manufacturing Engineering,

1 University of Nairobi (e-mail : [email protected])

(ii) Draw to scale the Shear Force and the Bending Moment diagrams Q3. 2waN wN / m

2 2 4wa N m 4wa N  m  4a 4a Fig Q3

Fig Q3 shows a cantilever beam of length 8a meters, fixed at the left hand end and loaded as indicated by a distributed load, a point load and two clockwise couples. (i) calculate the reactions at the support (ii) draw to scale the Shear Force and the Bending Moment diagrams

Q4. wN / m

Fig Q4 shows a beam of length 9a simply supported at the two ends and supporting a distributed load varying in magnitude linearly from zero at the left-hand end to wN / m and a clockwise couple of magnitude 4wa 2 N  m at a point distance 6a from the left- hand support. (i) Calculate the reactions at the supports (ii) Draw to scale the Shear Force and Bending Moment diagrams

2 Q5. 80KN 120KN / m

120KN  m 2m 4m Fig Q5 Fig Q5 shows a beam 6m long and simply supported on two supports 4m apart. The beam carries a distributed load varying in magnitude linearly from zero at the left- hand end to 120KN / m at the right –hand end. A point load of magnitude 80KN and an anti-clockwise couple of magnitude 120KN  m are applied at the left-hand end. (i) Calculate the reactions at the supports (ii) Draw to scale the Shear Force and the Bending moment diagrams. Q6. 120KN 60KN / m

240KN  m 3m 3m 3m

Fig Q6

Fig Q6 shows a beam of length 9m and simply supported over a span of 6m . The beam supports a distributed load varying in magnitude linearly from zero at the left- hand end to 60KN / m at the right-hand end, a point load of magnitude 120KN at the center of the two supports and an anti-clockwise couple of magnitude 240KN  m at the right-hand end of the beam. (i) Calculate the reactions at the supports (ii) Draw to scale the Shear Force and the Bending Moment diagrams

3 Answers: Q1. .

(i) Y1  6.5wa , Y2  4wa (ii)Values of Shear Force and Bending Moments are given for different values of the length of beam (x) , measured from the left-hand end x Shear Bending Force Moment 0 0 6wa 2 a 2 1.1667wa 5.44wa 0  x  3a 2a  2.667wa 3.556wa 2 3a  4.5wa 0 3a 2wa 0 4a 1.667wa 1.9445wa 2 5a 1.334wa 3.556wa 2 2 6a 0.5wa 4.5wa 3a  x  9a 7a  0.667wa 4.444wa 2 8a  2.1667wa 3.056wa 2 Q2.

9a  4wa 0 (i) Y1  5.61wa , Y2  8.19wa (ii) Values of Shear Force and Bending Moments for different values of lengths of the beam (x) , measured from the left-hand end

x 0 a 2a 3a 4a 5a Shear 5.61wa 5.31wa 4.41wa 2.91wa 0.81wa 1.89wa Force

Bending 0 5.51wa 2 10.42wa 2 14.13wa 2 16.04wa 2 15.55wa 2 Moment

x 6a 6a 7a 8a 9a Shear  5.19wa  5.19wa  6.19wa  7.19wa  8.19wa Force Bending 12.06wa 2 20.06wa 2 14.37wa 2 7.68wa 2 0 Moment

Q3. (i) V  4wa , M  29.334wa 2

4 (ii) Values of Shear Force and Bending Moments for different values of lengths of the beam (x) measured from the left-hand end 0  x  4a x 0 a 2a 3a 4a Shear 4wa 3.875wa 3.5wa 2.875wa 2wa Force Bending  29.334wa 2  25.376wa 2  21.667wa 2 18.459wa 2 16wa 2 Moment

4a  x  8a x 4a 5a 6a 7a 8a Shear 2wa 2wa 2wa 2wa 2wa Force Bending 12wa 2 10wa 2  8wa 2  6wa 2  4wa 2 Moment

(i) Y1  1.222wa , Y2  1.778wa (ii) Values of Shear Force and Bending Moment for different values of the lengths of the beam (x) measured from the left- hand end

x 0 a 2a 3a 4a 5a Shear 1.222wa 1.139wa 0.889wa 1.028wa  0.111wa  0.8613wa Force Bending 0 1.1942wa 2 2.222wa 2 2.196wa 2 3.11wa 2 2.638wa 2 Moment

x 6a 6a 7a 8a 9a Shear 1.778wa 1.778wa 1.778wa 1.778wa 1.778wa Force Bending 1.333wa 2 5.332wa 2 3.554wa 2 1.776wa 2 0 Moment

(i) Y1  330KN , Y2  110KN

5 (ii)Values of Shear Force and Bending Moment for different values of the lengths of the beam (x) measured from the left-hand end. x 0 1m 2m 2m Shear  80KN  90KN 120KN 210KN Force Bending 120KN  m  203.33KN  m  306.64KN  m  306.64KN  m Moment x 3m 4m 5m 6m Shear 160KN 90KN 0 110KN Force Bending 119.91KN  m 6.88KN  m 53.75KN  m 0 Moment

(i) Y1  100KN , Y2  290KN (ii) Values of Shear Force and Bending Moment for different values of the lengths of the beam (x) measured from the left-hand end. x(m) 0 1 2 3 Shear 100 96.67 86.67 70 Force(KN) Bending 0 98.89 191.11 270 Moment(KN  m)

x(m) 3 4 5 6 Shear  49.97  73.32 103.33 139.99 Force(KN) Bending 270 208.89 121.12 0 Moment(KN  m)

x(m) 6 7 8 9 Shear 150.01 106.68 56.69 0 Force(KN) Bending 0 128.93 211.17 240 Moment(KN  m)

6 7 8 9