### ةسدنهلا ةيلك - يزاغنب ةعماج

### ةيكيناكيملا ةسدنهلا مسق

**Mechanics of Materials Laboratory**

**Beam Deflection Test**

**Abstract**

If a beam is supported at two points, and a load is applied anywhere on the beam, the resulting deformation can be mathematically estimated. Due to improper experimental setup, the actual results experienced varied substantially when compared against the theoretical values. The following procedure explains how the theoretical and actual values were determined, as well as suggestions for improving upon the experiment. The percent error remained relatively small, around 10%, for locations close to supports. As much as 30% error was experienced when analyzing positions closer to the center of the beam.

**Table of Contents**

1. Introduction & Background ... 5

1.1. General Background ... 5 1.2. Determination of Curvature ... 5 1.3. Central Loading ... 5 1.4. Overhanging Loads ... 7 1.5. Experiment Setup ... 8 1.6. Overhanging Loads ... 8

2. Data, Analysis & Calculations ... 9

2.1. Central Loading ... 9

3. Conclusions ... 11

**1. Introduction & Background**

**1.1.**

**General Background**

If a beam is supported at two points, and a load is applied anywhere on the beam, deformation will occur. When these loads are applied either longitudinally outside or inside of the supports, this elastic bending can be mathematically predicted based on material properties and geometry.

**1.2.**

**Determination of Curvature**

Curvature at any point on the beam is calculated from the moment of loading (M),
the stiffness of the material (E), and the first moment of inertia (I.) The following
expression defines the curvature in these parameters as 1/ρ, where ρ is the radius of
curvature.
*I*
*E*
*M*
⋅
=
ρ
1
**1**

Equation 1 does not account for shearing stresses.

*Curvature can also be found using calculus. Defining y as the deflection and x as *
the position along the longitudinal axis, the expression becomes

2
3
2
2
2
1
1
+
=
*dx*
*dy*
*dx*
*y*
*d*
ρ
**2**

**1.3.**

**Central Loading**

Central loading on a beam can be thought of as a simple beam with two supports as shown below.

**Figure 1**

Applying equilibrium to the free body equivalent of Figure 1, several expressions can be derived to mathematically explain central loading.

2
0
2
2
0
0
*P*
*R*
*R*
*P*
*R*
*F*
*P*
*R*
*L*
*R*
*L*
*P*
*M*
*R*
*F*
*ay*
*c*
*ay*
*y*
*C*
*C*
*A*
*ax*
*x*
=
⇒
+
−
=
=
↑
+
=
⇒
⋅
+
−
=
=
=
=
→
+

### ∑

**Equation 3, 4, and 5**

Figure 2 and 3 act as free body diagrams for the section between AB and BC respectively.

**Figure 2**

**Figure 3**

*L*
*x*
*L*
*L*
*P*
*x*
*P*
*dx*
*y*
*d*
*I*
*E*
*L*
*x*
*x*
*P*
*dx*
*y*
*d*
*I*
*E*
≤
≤
+
−
=
≤
≤
=
2
2
2
2
0
2
2
2
2
2
**6, 7**
Integrating twice, Equation 6 becomes

*L*
*x*
*L*
*C*
*x*
*C*
*x*
*L*
*P*
*x*
*P*
*y*
*I*
*E*
*L*
*x*
*C*
*x*
*C*
*x*
*P*
*y*
*I*
*E*
≤
≤
+
+
+
−
=
≤
≤
+
+
=
2
4
12
2
0
12
4
3
2
3
2
1
3
**8, 9**

To determine the constants, conditions at certain positions on the beam can be applied. Knowing the deflection at each of the supports, as well as the slope at the top of the curve is zero, the constants can be derived to

48
16
3
0
16
3
4
2
3
2
2
1
*L*
*P*
*C*
*L*
*P*
*C*
*C*
*L*
*P*
*C* =− = =− =
**Equation 10, 11, 12, and 13**

Combining Equations 8 and 9 with 10 through 13, the expressions for deflection
can be expressed as
*L*
*x*
*L*
*L*
*P*
*x*
*L*
*P*
*x*
*L*
*P*
*x*
*P*
*y*
*I*
*E*
*L*
*x*
*x*
*L*
*P*
*x*
*P*
*y*
*I*
*E*
≤
≤
+
−
+
−
=
≤
≤
−
=
2
48
16
3
4
12
2
0
16
12
3
2
2
3
2
3
**14, 15**

**1.4.**

**Overhanging Loads**

Overhanging loading on a beam is similar to that of central loading. In overhanging loading, a simple beam is supported with two supports and two loads as shown below.

**Figure 4**

Using similar methods used previously for central loading, the equation for determination of deflection as a function of position, load, length, stiffness, and geometry can be derived as

### (

*a*

*b*

### )

*Pax*

*P*

*L*

### (

*a*

*b*

### )

*x*

*x*

*L*

*L*

*x*

*P*

*y*

*I*

*E*= − − + 2 + 0≤ ≤ 6 2 6 2 3

**16**

**1.5.**

**Experiment Setup**

Set the knife supports at determined positions along the frame and mount the beam to be tested. The material, width, thickness, and length between supports should be measured and recorded for later use.

**1.6.**

**Overhanging Loads**

Dial gages were placed along lengths of the test area and set to read zero with no applied load. Adding a hook and hanger on each ends extending outside the knife supports, record the new readings on each of the gages. In discrete intervals, add weights to both ends of the beam with the hooks applied previously. Record the new deflections read by the dial gages after each new loading.

Gage 1 Gage 2 Gage 3

Lo

**2. Data, Analysis & Calculations**

**2.1.**

**Central Loading**

Table 1 and 2 catalog the dimensions of the beam, as well as the position of the gages as measured from one of the two fixed supports.

1325.000
19.000
6.500
Test Length
Width
Thickness
**Beam Dimensions (mm)**
**Table 1**

Table 2 returns the results from six different load configurations.

**load,N** **Gage ,mm** **Load,N** **Gage,mm** **load,N** **Gage,mm**

4.9 12 4.9 24.5 4.9 37 9.81 24 9.81 52 9.81 76.5 14.7 37.5 14.7 78 14.7 117 19.6 50 19.6 104 24.5 62.5 24.5 131 29.4 75 29.4 138 34.3 87

** steel brass Aluminum **

**3. Conclusions**

When an load is applied to a beam, either centrally over at another point, the deflection can be mathematically estimated. Due to the error that occurred in this exercise, it is clear that margins in safety factors, as well as thorough testing, is needed when utilizing beam design. It is also important to ensure the scope of the testing closely models real-world practicality and Samples of different metals observed the relationship between modulus of elasticity and deflection (deflection decrease with increase modulus of elasticity) .

**4. References**

Gilbert, J. A and C. L. Carmen. "Chapter 11 – Beam Deflection Test." MAE/CE 370 – Mechanics of Materials Laboratory Manual. June 2000.

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