ME2113-1 DEFLECTION AND BENDING STRESSES IN BEAMS
Formal Report
INTRODUCTION Beams are one of the commonest components encountered in structures. In most cases, they are transversely loaded and thus undergo bending. Bending induces stresses in a beam and causes it to deflect. In the design and analysis of beam-type structures, the stresses and deflections generated must be examined to ascertain that they are within acceptable limits. Simple beam theory provides expressions for the calculation of these quantities. This experiment provides the means to investigate how beam theory can be applied in practice. OBJECTIVE AND SCOPE 1) To study the application of beam theory by subjecting a cantilever to various point loads and examining the resulting stresses and deflection. 2) Use experimental values to determine the Young’s Modulus and Poisson’s ratio of the beam material. 3) Investigating the magnitudes and signs of the strains and stresses at the two locations along the beam in terms of their relation to each other and in accordance with beam theory. EXPERIMENTAL PROCEDURE Part I 1.1
Identify the strain quantity (i.e. xx1 , zz1 ,xx2 or zz2 ) measured by each gauge and the channel to which it is connected on the switching box.
1.2
With no loads on the hanger, set the dial gauge reading to zero.
1.3
Select a channel on the switching box for the first set of strain gauge readings and zero the strain meter.
1.4
Load the beam in six steps of 250g weight increments and record the deflection of the load point and the reading of the strain gauge selected [note whether the strain readings are positive (tensile) or negative (compressive)]. Record the deflection and strain readings upon unloading as a check.
1.5
Select the other channels in turn, zero the dial gauge if necessary and repeat the loading/unloading process together with the recording of deflection and strain readings.
1.6
Tabulate all the readings and plot the following: Graph 1 -
Load P against vertical deflection v . From the slope of this graph and Eqn. (7), calculate Young’s modulus E of the beam material.
Graph 2 -
zz1 against xx1 and zz2 against xx2. From the slope of the lines, calculate Poisson’s ratio using Eqn. (6).
Graph3 -
Load P against xx1. Calculate the slope of this graph.
Graph 4 -
Determine, using Eqn. (1), the bending moment magnitudes at the two strain gauge locations, x = d1 and x = d2 , for each of the six applied loads. Then, use Eqn. (2) or (3) to calculate the theoretical magnitude of the longitudinal stresses (xx1 and xx2) at the beam surface (y = h/2) for the six loads. Use these six pairs of stress values to plot six lines showing the theoretical variation of maximum longitudinal stress with location along the beam (i.e. xx against x ). From the magnitudes of the longitudinal strain readings (xx1 and xx2) corresponding to the four values of applied load, calculate the corresponding experimental stress magnitudes (xx1 and xx2) using the value of E derived from Graph 1 and Eqn. (5). Plot these experimental points on the same graph with the six lines showing the theoretical variation of maximum stress with beam location.
Part II
Attach the gripper to the end of the beam (Fig. 2) and grip the gripper as tight as possible.
Record the highest reading xx1 of strainmeter, and evaluate your handgrip force.
SAMPLE CALCULATIONS Details given: b=25.6mm, h=6.06mm, L=250mm
1 I z bh3 = (25.6 x 10-3) (6.06 x 10-3)3 / 12 = 4.7476 x 10-10 m4 12 Using P=250g, At x= 50mm Bending Moment, Mxz
= -P(L - x) = - 2.45 (250 – 50) 10-3 = -0.490 Nm
Longitudinal stress, xx (theoretical)
M xz ( y ) , (y = h/2: top surface) Iz
= -(-0.490/(4.7476 10-10)) (6.06/2 10-3) = 3.127 MPa Longitudinal stress, xx (experimental) E xx
(From Graph 1, E= 60.691GPa)
= (60.691×109) (48×10-6) = 2.913 MPa At x= 150mm Bending Moment, Mxz
= -P(L - x) = - 2.45 (250 – 150) 10-3 = -0.245 Nm
Longitudinal stress, xx (theoretical)
M xz ( y ) , (y = -h/2: bottom surface) Iz
= -(-0.245/(4.7476 10-10)) (-6.06/2 10-3) = -1.564 MPa Longitudinal stress, xx (experimental) E xx
(From Graph 1, E= 60.691GPa)
= (60.691×109) (-22 ×10-6) = -1.335MPa
RESULTS Part I Please refer to attached Tables and Graphs From Graph 1, Gradient of the graph = 563.94 gmm-1 = 563.94 kgm-1 = 5532.25 Nm-1 (g=9.81ms-1) Using,
1 I z bh3 12 = (25.6 x 10-3) (6.06 x 10-3)3 / 12 = 4.7476 x 10-10 m4
From Equation (7), v L
PL3 , (assuming upwards deflection is positive) 3EI z
E = - (-5532.25) (250 x 10-3)3 / (3 x 4.7476 x 10-10) = 60.691 GPa From Graph 2, Gradient of the graph (x=50mm) = -0.3120 Gradient of the graph (x= 150mm) = -0.3228 Average gradient = [((-0.3120) + (-0.3228))/2] = -0.3174 From Equation (6), zz
Poisson’s Ratio, ν =
xx E
xx ,
zz = 0.3174 xx
From Graph 3, Gradient of the graph = 0.0542MN From Graph 4, x = L = 250mm, Mxz = 0
∴ σxx = 0
Part II Highest reading εxx1 of strain meter = 3010µ = 3010 × 10-6 From Graph 3, P = xx1 (0.0542 × 106) When xx1 = 3010 × 10-6, P
= 0.0542 × 106 × 3010 × 10-6 = 163.1N
Handgrip Force = 163.1N DISCUSSION 1. The signs of the strains are as follows: xx1 +ve
xx2 -ve
zz1 -ve
zz2 +ve
εxx1 is the strain measured from an element on the top surface in the x-direction while εxx2 is the strain measured from an element at the bottom surface in the x-direction. And εzz1 is the strain measured from an element on the top surface in the z-direction while εzz2 is the strain measured from an element at the bottom surface in the zdirection. In the axial direction, since the load P acts in the negative y-direction thus resulting in a negative bending moment Mxz along the x-axis, the element in the upper surface (positive y surface) experiences tension and the element in the lower surface (negative y surface) experiences compression. On the other hand in the transverse direction, by Poisson ratio zz
xx E
xx , the element in the upper surface
experiences compression and the element in the lower surface experiences tension. By sign convention, compression is assigned to the negative sign and tension is assigned to the positive sign. 2. In general, normal stress decreases linearly as x increases and reaches zero when x = 250mm at the edge of the beam. This can be observed from the graph having all the lines converge to a point on the axis. This shows that the normal stress is the highest
at the built-in-end and reaches zero at the free end, anywhere after x = 250mm experiences no stresses. Linearity of the normal stresses against distance from the built-in-end can also be concluded from the 12 straight lines shown in the graph which is consistent to the fact that the material is a homogeneous linear-elastic material. The slopes of the 6 different lines also suggested that the higher the load the greater the rate of drop in stress value as x increases. Other than that, the gradient of the theoretical lines are also observed to be consistently higher than the experimental lines. This is probably due to fact that the previously loaded beam has yet to recover to its original elevation before the next load is added. Therefore, slight curvature was present initially resulting in less stress required to perform the same strain. 3. Handgrip force measurements are not exactly accurate. This is due to the unequal distribution of strength when you grip the gripper. The amount of strength exerted by hand varies with the amount exerted by each finger throughout the process and may lead to inconsistent readings in the strain meter. The fluctuating strain readings also cause the measurement to be based on estimation rather than precise value. Also, the way one grips the gripper also affects the actual strength exerted by one on the beam. Furthermore, different apparatus may have the gripper set up differently which may affect the actual reading. Hence, experiment will only give rough estimate of the strength exerted. CONCLUSION From this experiment, I have understood how to apply beam theory on a cantilever that is subjected to various point loads and the relationship of their resulting stresses and deflection. I have also managed to obtain Young’s Modulus and Poisson’s ratio of a beam material through experimental values and graphs. Furthermore, by investigating the magnitudes and signs of the strains and stresses at the two locations along the beam, I understand the relation between them and how the beam is loaded. The Orientation and surface by which strain
values are taken also affects the sign and magnitudes of the measured values. All this can be explain using beam theory. Finally, the experiment also gives us an example by which strain values obtained might be inaccurate (eg. Handgrip force) and help us understand the slight deviation of experimental results from the theoretical.
