EQUILIBRIUM OF A BEAM

Lab 1 : Material lab

PROGRAM KEJURUTERAAN MEKANIKAL FAKULTI INDUSTRI KEJURUTERAAN LAB SHEET

EXPERIMENT 6: Equilibrium of a Beam EQUILIBRIUM OF A BEAM

INTRODUCTION One very common example of parallel forces in equilibrium is that of a beam, because in most cases the forces are vertical weights due to gravity. Hence the beam supports will develop vertical reactions to carry the weights on the beam, and the self weights of the beam of itself. For a beam on two supports there will be the two unknown reactions, so two equations of equilibrium must be set up. It is necessary to start by taking moments about a convenient point; if this point is at a reaction then there is only one unknown force (the other reaction) in the equation. The second reaction can then be found from vertical equilibrium. An alternative type of beam which projects from a support into mid-air is called a cantilever. Here the two unknown reactions are a “fixing” moment and a force which can be calculated independently of each other.

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LIST OF PARTS See Packing List at back of Instruction Manual. APPARATUS The apparatus basically provides a channel section aluminium alloy beam 1.04 m long which rest of knife edge supports that replace the scale pans of two weighing machines. The baseplates of the weighing machines can be clamped in any position along the bottom member of the HST1 test frame to permit the span of the beam to be varied. Three load hangers are available for loading the beam at different points along its length. An alternative arrangement can be constructed by using only one weighing machine near the middle of the beam. To the right of this support loading is applied using the load hangers, while to the left a downward load is applied at the end of the beam using the tie rod on a spring balance anchored to the bottom member of the frame. EXPERIMENT OBJECT The purpose of the experiment is to verify the use of the conditions of equilibrium in calculating the reactions of a simply supported beam or a cantilever. PROCEDURE Part 1, Beam Reactions In the first place fix the reaction balances in the test frame with the knife edges 1m apart. Rest the channel section beam over the knife edge supports with the zero of the scale lined up with the left hand support. Add a stirrup and load hanger at mid span. Use the zero adjustment on the balances to bring the pointer to zero. This is an artificial way of nullifying the self weight of the beam, stirrup and load hanger so that the balances will read only the reaction for any added load. Add the secession of the weights up to 60 N to the mid span load hanger, and record the two reaction values for each case. Because of the symmetry the reactions should be equal, and therefore will each be half of the load to satisfy vertical equilibrium. In these simple cases the experiment is used to check the obvious. Record the results in Table 1. Table 1 Reactions for a simply supported beam Beam span = 1 m Load and position from left end

Left end reaction

2

Right end reaction

(N)

(mm)

(kg)

(N)

(kg)

(N)

Now move the stirrup and load hanger to the quarter to the quarter span position and, using a 40 N load, record the reactions. Repeat this for two or three more positions measured from the left hand reaction, tabulating the results. Finally use the three stirrups and load hangers at pre-selected positions. Add a set of three loads, one at a time, to these hangers, recording the reactions as each load is applied. Part 2, Cantilever beam reactions Attach the spring balance assembly mid way between the reaction balances and move the channel section beam to the right so that the threaded tie rod of the spring balance passes through the hole in the top of the beam by the zero on the beam scale. The beam will then extend through the right hand side of the test frame and it should be levelled by adjusting the tie-rod. The beam cantilevers to the right of the upward reaction balance, while the spring balance provides a downward reaction. Any initial readings will be those due to the self weight of the cantilever. Position a stirrup and load hanger on the end of the cantilever 500 mm from the reaction balance and adjust the zero of the reaction balance. Add a succession of 5 N loads on the hanger. For each loading adjust the length of the spring balance tie rod to re-level the cantilever. (Note whether the spring balance reading changes while this is being done). Record the readings in Table 2. Table 2 Reactions for a 500 mm cantilever End Load (P) (N)

Reaction Balance Reading (kg) (N)

Spring Balance Reading (N)

Distance Between Balances (mm)

Change the position of the spring balance by moving it closer (say by 200 mm) to the reaction balance. Reposition the stirrup and load hanger so that it is the same distance of 500 mm from the reaction balance as above. Zero the balance. Add a succession of 10 N loads on the hanger. Re-level the cantilever and record the balance readings for each load. RESULTS Tabulate the readings for Part 1 and add the theoretical reactions calculated as shown in the appendix. A suitable table is given below for single loads. It can be modified to record the cumulative reactions as the set of three loads is added. Table 1a Reactions for a simply supported beam Beam span = 1 m Load and Position

Left end Reaction 3

Right end Reaction

from left end (N) (mm)

Experimental (N)

Theory (N)

Experimental (N)

Theory (N)

For Part 2 theoretical values are calculated as shown in the appendix, by using conditions of equilibrium. The “wall” fixing moment is the product of the spring balance reading and the distance between the balances. Table 2a Reactions for a 500 mm cantilever End Load (P) (N)

Reaction Balance Reading Exptl. Theory (N) (N)

Spring Balance Reading Exptl. Theory (N) (N)

Distance Between Balances (mm)

Fixing Moment

Reaction Balance Minus Load (N)

(N.m)

The cantilever should now be re-analysed by considering the diagram below, which shows a mathematical model.

L

MA B A VA

For vertical equilibrium The fixing moment

P

VA = P MA = P.L

It will be seen that VA is less than the reading of the reaction balance. Try subtracting P from the experimental reaction balance value, and then compare this force with the spring balance reading. Multiply the force by the distance between the balances and check if the resulting moment compares with the mathematical fixing moment. Repeat this analysis for equal values of P and the two distances between the balances. OBSERVATIONS How well did the experimental and theoretical results compare (try stating the differences as a percentage of the true values)? In the case of the three loads on the simply supported beam, would the order of applying the loads have affected the final reactions? This experiment is a very simple test of the principle of super-position. Explain what this means.

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Use the cantilever results to predict the forces acting as the built-in (fixed) end gets shorter. If the cantilever was a length of timber projecting from a brick wall what form of failure would take place as the part in the wall was reduced in length? Would the mathematical model have suggested this?

Department of Mechanical Engineering, University Industri Selangor Revised AUG:2005

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