interim phd report

Estimating deflection and stress in a telescopic cantilever beam using the tip reaction model

Interim PhD ReportJ.G. Abraham

School of Engineering and DesignBrunel University

UxbridgeMiddlesex UB8 3PH

1.0 Deflection Analysis in the Telescopic Beam: Modelling PrincipleRamamurtham and Narayan [1] explain how the material resists the external loads in a continuous beam subjected to transverse loads. To illustrate they consider the simply supported beam with negligible weight subjected to four transverse loads as shown in Figure 1.

Figure 1: Moment of Resistance [1]

The applied loads bend the beam creating a compressive resistance in the top part and tensile resistance in the bottom part. In Figure 1 MN represents the neutral plane representing the parts (fibres) which do not associate with any stresses. The parts above MN are under compression and the parts below MN are under tension The equivalent compressive force acting on the area MEFN is given by ‘C’. Similarly the equivalent tensile force acting on the area MHGN is given by ‘T’. The external loads applied and the effective shear force ‘S’ acting on the plane EFGH are assumed to be concentrated on the plane of symmetry as shown in Figure 1. The forces that act over the length AX of the beam are therefore

i. Vertical reaction at Aii. External loads and

iii. Shear force offered by section EFGHiv. Compressive resistance andv. Tensile resistance

Magnitudes of and are equal and since they act in the opposite directions their separation h forms a ‘moment of resistance’ .

Taking moments about O gives

But is the bending moment of the external forces. Hence the modelling principle for the effects of externally applied loads is that ‘Bending Moment at a section = Moment of resistance at that section’. The flexural equation

is based on this principle.

This principle applies well to model continuous beams. However this method cannot be applied to telescopic beams where there is discontinuity between sections and the discrepancy is more pronounced when there is a gap between the overlapping sections. Section 3 describes the modifications required when modelling telescopic beams.

1.1 Telescopic BeamA telescopic cantilever beam is one in which one or more pieces of beam are stacked inside an outer beam portion which is fixed at one end supporting the entire beam assembly and the inner pieces move out when application needs the full span. Such a beam assembly will have three types of beam sections (a) a section with one end fixed (the start or fixed-end section) (b) a section with one end free (end section) and (c) the middle section which connect two other sections, one at each end. A three section cantilever beam consists of all these three types and therefore will be considered in this paper. Figure 2a shows such a model where section 1 is the fixed end section with an overlap by an amount a1 with section 2 (middle section) in one end of it. On the other end section 2 has an overlap of a2 with section 3.

Figure 2: Three Section Cantilever

1.2 Modelling the Cantilever Beam Assembly – The Tip Reaction ModelThe tip reaction model proposed here models the load transfer between the beam sections through reactions at the tips of different beam sections. Consider the three section beam assembly shown in Figure 2 (b). Since a part of beam CD lies inside beam AB it will produce an upward reaction at C in beam AB and a downward reaction at B in beam AB. Thus the forces and moments acting in beam AB, the fixed-end section, can be modelled as shown in Figure 3.

Figure 3: Fixed-End Section of a Telescopic Cantilever

In a similar fashion if beam CD is considered, at C there will be a downward reaction and at B there will be an upward reaction, due to its contacts with beam AB. Beam EF also will impose reactions on CD. There will be an upward reaction at E and a downward reaction at D. Thus the forces at CD will be as those shown in Figure 4.

Figure 4: Loads Acting on the Middle Section

A similar analysis can be used to define the loads in beam EF. The loads acting on EF are shown in Figure 5.

Figure 5: Loads Acting on the Free-End Section

Thus, in the proposed ‘Tip Reaction Model’ the internal reactions are used to transmit the forces. The effects of the external loads applied on the telescopic cantilever beam can then be calculated using these reactions instead of the bending moment or the moment of resistance used in the continuous beam. Each beam section can now be considered separately and the equilibrium and compatibility principles of classic mechanics can be applied to them.

1.2.1 Reactions at the Tips

Figure 6: Loads Acting on the Free-End Section

Consider the beam section EF as shown in Figure A.1. Taking moments about E gives

But

From the above equations

In a similar fashion taking moments about D gives

Hence

Figure 7: Loads Acting on the Middle Section

Consider the beam section CD as shown in Figure A.2.Taking moment about C gives

Once the reactions are calculated in the above fashion deflection can be calculated in the following way.

1.2.2 Deflection Analysis

Deflection of the assembly is considered as the combination of deflection in the three beams AB, CD and EF in a three section telescopic cantilever beam. The deflected shapes of the different beams however are assumed to be the same in the overlapped regions. Consider the beams shown in Figure 8. The beam AB has two deflected portions AC’ and C’B’. The beam CD has three deflected portions C’B’, B’E’ and E’D’. In a similar fashion the beam EF has two deflected portions E’D’ and D’F’. The equations of the deflected shapes of the beams can be derived by integrating the flexural equation twice. There are seven different sections with different bending moments in this assembly. They are

i. Section AC in beam ABii. Section CB in beam ABiii. Section CB in beam CDiv. Section BE in beam CDv. Section ED in beam CDvi. Section ED in beam EFvii. Section DF in beam EF

The following relationships can also be said about the deflections.Deflection at C estimated from AC in AB = Deflection at C estimated from CB in AB

= Deflection at C estimated from CB in CD Deflection at B estimated from CB in AB = Deflection at B estimated from CB in CD

= Deflection at B estimated from BE in CDDeflection at E estimated from BE in CD = Deflection at E estimated from ED in CD

= Deflection at E estimated from CD in EFDeflection at D estimated from ED in CD = Deflection at D estimated from ED in EF

= Deflection at D estimated from DF in EF

Figure 8: Deflected Shapes of the Beams

The reactions at points C, B, E, D and F can be calculated using static equilibrium conditions. The bent shape equations of the seven segments are derived by integrating the

flexural equation twice where M is the sagging bending moment. The

process is started at AC and the integration constants are found by substituting the boundary condition at A. Using the equation thus derived the slope and deflection at C are calculated. The calculated values are used as the boundary conditions for the curve CB in beam AB. This process of fitting the curve using the boundary conditions calculated in the previous section is continued until all the equations are fitted.

1.2.3. Derivation of the Deflection Curve for the Section AC in Beam AB

Figure 9: A Section in AC

Consider section AC shown in Figure 9. The bending moment at the section at a distance x from A is

for

assuming the sign convention ‘Sagging is positive’.

But for the beam portion taking to be the second moment of area of

beam ACB. Integrating this twice will give

To find and substitute the boundary conditions at A i.e. when and slope

because when the slope . Integrating again

because when Thus, if the deflection equation for the section AC in the beam AB is given by

then it follows that the coefficients t are

Let where means the slope of section AC at C and

Also let where means the deflection of section AC at C.

1.2.4. Derivation of the Deflection Curve for the Section CB in Beam 1

Figure 10: A Section in CB

Consider the section CB shown in Figure 10.The methodology is similar to the one adopted for the portion AC but the boundary conditions applied correspond to point C ( and ) calculated earlier in section 1.2.3.Consider the section at a distance x from A as shown in Figure 10.

Bending moment for

But

Integrating this twice will give

When slope

Integrating again

When

Thus if the equation of the section CB in the beam AB is given by

Then the coefficients t become

1.2.5. Derivation of the Deflection Curve for the Section CB in Beam 2

Figure 11: Deflection of Beams AB and CD

Again, the methodology is similar to the one adopted earlier and the boundary conditions applied correspond to point C ( and ) calculated earlier in section 1.2.4., when, instead of beam AB, the beam CD is considered, the bending moment is from Figure 11(b):

for

Substituting this limit becomes

But where is the second moment of area of beam CBD.

Integrating once gives the slope

Integrating again gives the equation

When ,

When and when See Figure 11(a).

Thus if the equation of the section CB in the beam CD

is given by

Where the coefficients t are

The deflection of section CB at B is .

Also at B, when , the slope is

1.2.6. Derivation of the Deflection Curve for the Section BE in Beam 2

Figure 12: Deflection of Beam CD

Again the methodology is similar to the one adopted earlier and the boundary conditions applied correspond to point B ( and ) calculated earlier in section 1.2.5. Let From Figure A.6(b) bending moment

for ,which is .




When , and when

Thus if the equation of the section BE in the beam CD

is given

by

Here the coefficients t are

Deflection at E is found by substituting

1.2.7. Derivation of the Deflection Curve for the Section ED in Beam 2

Figure 13: Deflection of Beam CD

Again the methodology is similar to the one adopted earlier and the boundary conditions applied correspond to point E ( and ) calculated earlier in section 1.2.7.

From Figure 13(b) bending moment

For ,let , then the limit becomes



When which is when ,

Integrating once again gives

When which is when ,

Thus if the equation of the section ED in the beam CD

is given by


When

1.2.8. Derivation of the Deflection Curve for the Section ED in Beam 3Again the methodology is similar to the one adopted earlier and the boundary conditions applied correspond to point E ( and ) calculated earlier. Beam EF is considered here.Let

From Figure 14(b) bending moment

for

But where is the second moment of area of beam EDF.



When , and

Thus if the equation of the section CB in the beam CD is given by

Now the coefficients t are

When

Figure 14: Deflection of Beam EF

1.2.9. Derivation of the Deflection Curve for the Section DF in Beam 3Again the methodology is similar to the one adopted earlier and the boundary conditions applied correspond to point D ( and ) calculated earlier. From Figure 14 bending moment isWhere so that becomes .

But where is the second moment of area of beam EDF.



When , and

Thus if the equation of the section DF in the beam EF

is given by


is the full length of the beam assembly.

The tip deflection therefore is

2.0 Stress Analyses in the Telescopic Beam: Modelling Principle

Figure 15: Tip Reaction Model – Beam Assembly and Reactions on Individual Beams

In the ‘Tip Reaction Model’ the internal reactions transmit the effects of applied loads. The effects of the external loads applied can then be calculated using these reactions instead of the bending moment or the moment of resistance used in the continuous beam. Each beam section can now be considered separately and the principles of classic mechanics can be applied on them. This section studies the validity of the tip reaction model by comparing the bending stress values calculated using this model with those obtained from a finite element analysis using ABAQUS. It also compares the stress values at corresponding points on a continuous beam of similar dimensions subjected to similar loading. The analyses were carried out on a telescopic cantilever assembly and a continuous beam, described as follows.

2.1 Theoretical ConsiderationsBending stresses induced in a beam is a standard work by books on strength of materials. For

instance Rees [2] establishes that the bending stress induced is given by the equation

and explains its derivation.Consider the beam assembly shown in Figure 15 and assume that it is fixed at end A and carries a tip load at F. Due to self load and the applied tip load there will be tensile stresses in all three beam sections above the horizontal plane of symmetry and compressive stresses in portions below the plane of symmetry. If the depths of the sections can be assumed as , and and the beam is represented by the vertical plane of symmetry the maximum bending

stress at the top of the beams are given by , and . The bending

moment varies from point to point and these values can be calculated once the tip reactions are known.

2.2 Shear Stress

Figure 16 Shear Stress Calculation

Consider a cantilever as shown in Figure 16. Let the shear force at section x-x at a distance x from the fixed end be S.Let it be that the requirement is to find the shear stress on section x-x at a distance y1 (at EF) from the neutral axis. This again is shown in Figure 16.Let the area above EF is a and the distance of the centroid of this area from the neutral axis is

.Also let the breadth of the beam is b. The shear stress at section x-x at level from the Neutral Axis is:

2.3 Case Study FormulationA telescopic cantilever beam assembly consisting of three hollow sections of 1 mm thickness and dimensions of , and

respectively is taken as the case study example. A load of 30 N is applied at the tip of the beam assembly. Beam CD and AB have an overlap of 400 mm and beams CD and EF have an overlap of 300 mm. The second moment of area about the neutral axis for the beams AB, CD and EF are 9232 mm4 6188 mm4 and 3900 mm4 respectively. The linear densities of the beams AB, CD and EF are 0.007536 N/mm 0.006594 N/mm and 0.005652 N/mm respectively.

Three investigations have been planned and carried out. They arei. Use of tip reaction model to analytically calculate bending and shear stresses and

compare the results with those from Finite Element Analysis using ABAQUS.ii. Use a comparable stepped single beam to estimate bending and shear stresses and

compare the results with those from Finite Element Analysis using ABAQUS.iii. Compare the results from the tip reaction model with those of the single stepped

beam.

2.4 Single Beam for Analysis

Figure 17: Single Beam for Analysis (First Angle Projection)

The telescopic beam sections have a 1mm gap between sections to facilitate easy sliding. This is the situation in real applications as well. However when considering an equivalent single beam this creates a problem. ‘End view 2’ shows sections built from the outer section resulting in the inner section having a larger dimension i.e. external with a thickness of 1 mm. On the other hand if sections are built from the inner section resulting outer section will have a smaller dimension as shown in ‘End View 1’. In this analysis section with ‘End View 2’ is considered.

Figure 18: Hollow Rectangular Section

Consider the hollow rectangular section shown in Figure 18. Cross sectional area

Volume of the section of length 1 mm = Self weight of 1mm3 of steel (assuming g = 10N/m2)Therefore self weight of 1 mm long beam section

Second moments of area of a hollow rectangular section

SectionWidth

(b)Thickness (t)

Self Weight N/mm

Section 1 21 mm 1 mm 5347 0.00628Section 2 23 mm 2 mm 12460 0.01319Section 3 23 mm 1 mm 7113 0.006908Section 4 25 mm 2 mm 16345 0.014444Section 5 25 mm 1 mm 9232 0.007536

Consider the sectional view of the continuous stepped beam shown in Figure 19. The bending stress will be maximum in any section along the line A1C1B1E1D1F1. The shear stress will be

maximum along the line A2C2B2E2D2F2. A tip load of 30N will be applied and comparison of bending stresses will be carried out along A1C1B1E1D1F1 and the comparison of shear stresses will be carried out along A2C2B2E2D2F2.

Figure 19: Sectional View of the Continuous Stepped Beam

2.5 Beam Assembly for Finite Element Analysis

Figure 20: Telescope Beam Assembly for FEA

Four wear pads are introduced to make the ‘Tip Reaction Model’ analysis comparable to the FEA. Wear pad 1 of 0.5 mm thickness and 5 mm wide is glued the inner side of the free end of beam 1 as shown in Figure 7. Similarly wear pad 2 of thickness 0.5 mm and 5 mm wide is glued to the outside of beam 2. Wear pad 3 is glued to the inner end of beam 2 and wear pad 4 is glued to outside of beam 3 as shown in Figure 7. The beam assembly can slide on these wear pads. The bending stress in the assembly will be maximum along the line A1C1B1E1D1F1. The shear stress will be maximum along the line A2C2B2E2D2F2. A tip load of

10N will be applied and comparison of bending stresses will be carried out along A1C1B1E1D1F1 and the comparison of shear stresses will be carried out along A2C2B2E2D2F2.

2.6 Analysis of the Telescopic Assembly Using the Tip Reaction Model2.6.1 Calculation of Tip ReactionsConsider Figure 15(d) and the equations derived in section 1.2.1 for these calculations.

Similarly taking moments about D gives

Similarly taking moments about C gives

From earlier calculations and

Balancing forces give

Thus when the reactions are

Reaction R at A

Moment at A

2.6.2 Shear Force and Bending Moment Diagrams

2.6.2.1 Consider the beam ABIn the section ACShear force is where is the distance from A.

Bending moment

Therefore Shear force at

Bending moment at

In Section CBShear force is where is the distance from A.

Bending moment


Bending moment at

2.6.2.2 Consider the beam CDIn the section CBShear force is where is the distance from A.

Bending moment


Bending moment at

In Section BEShear force is where is the distance from A.Bending moment


Bending moment at

In Section EDShear force is Bending moment

where is the distance from A.


Bending moment at

2.6.2.3 Consider the beam EFIn the section EDShear force is where is the distance from A.

Bending moment


Bending moment at

In Section DFShear force is where is the distance from A.

Bending moment


Bending moment at

Figure 21 shows the bending moment and shear force diagram obtained using the values derived from above.

Figure 21: Shear Force and Bending Moment Diagrams for the Individual Sections

2.6.3 Calculation of Bending and Shear StressesBending stress will be maximum along the vertical plane of symmetry at the top of the beams marked ACBEDF shown in Figure 22.

Figure 22: A Telescopic Beam Assembly with Three Sections and the Vertical and Horizontal planes of symmetry shown

2.6.3.1 Beam ABConsider section AC in beam 1Bending moment at the section at a distance x from A

and the maximum bending stress


because the top of the beam is 12.5 mm away from the neutral plane .Using these equations in Microsoft Excel, gives the Values shown in Table 1.

Figure 23: Section of the Beam above the Neutral Plane

Shear force in AC is given by and the maximum shear stress is at the neutral plane.The y coordinate of the centroid of the portion of the beam above the neutral plane

Area of cross section =

Maximum shear stress is therefore

Note: The b denominator in in this case is equal to 2t.

Similarly for section CB in beam 1

Bending moment

And Shear force

2.6.3.2 Beam CDThis beam has three sections CB, BE and ED with differing loads. The beam width also has changed and thus For the section CB

and

Shear force is Similar equations are established for sections BE and ED and fed to Excel to calculate maximum bending stresses and maximum shear stresses in beam CD.

2.6.3.3 Beam EFThis beam has three sections ED and DF with differing loads. The beam width also has changed and thus . Similar equations are established for sections ED and DF and fed to Excel to calculate maximum bending stresses and maximum shear stresses in beam EF.

Figures 24 and 25 show the graphical comparison between the analytical and ABAQUS derived bending stress, and the analytical and ABAQUS shear stress, respectively.

Table 1: Analytical Bending Stress at the Top of the Beam Bending Stress Calculations

Beam AB Beam CD Beam EF

Dist. From A BM Bend. St Dist From A BM Bend. St Dist. From A BM Bend. St0 -108506 146.9162 600 0 0 1500 0 050 -105904 143.3929 650 -7513.64 13.35651 1550 -4846.19 11.80481100 -103321 139.8951 700 -15043.8 26.74232 1600 -9706.5 23.64404150 -100757 136.4229 750 -22590.4 40.15743 1650 -14580.9 35.51769200 -98210.9 132.9762 800 -30153.5 53.60186 1700 -19469.5 47.42575250 -95684.1 129.5549 850 -37733.1 67.07558 1750 -24372.2 59.36824300 -93176.2 126.1592 900 -45329.1 80.57861 1800 -29289.1 71.34515350 -90687.1 122.789 950 -52941.7 94.11094 1850 -27541.8 67.08896400 -88216.8 119.4443 1000 -60570.7 107.6726 1900 -25808.6 62.8672450 -85765.4 116.1251 1050 -58476.1 103.9491 1950 -24089.6 58.67986500 -83332.8 112.8315 1100 -56397.9 100.2549 2000 -22384.7 54.52693550 -80919.1 109.5633 1150 -54336.3 96.58999 2050 -20694 50.40843600 -78524.2 106.3206 1200 -52291.1 92.95441 2100 -19017.4 46.32434650 -68642.7 92.94126 1250 -50262.4 89.34813 2150 -17354.9 42.27467700 -58780.1 79.58741 1300 -48250.2 85.77115 2200 -15706.5 38.25942750 -48936.3 66.25907 1350 -46254.4 82.22348 2250 -14072.3 34.27859800 -39111.4 52.95624 1400 -44275.2 78.70511 2300 -12452.2 30.33218850 -29305.3 39.67892 1450 -42312.4 75.21604 2350 -10846.2 26.42019900 -19518 26.4271 1500 -40366.2 71.75628 2400 -9254.34 22.54262950 -9749.58 13.2008 1550 -33597.3 59.72364 2450 -7676.63 18.699471000 0 0 1600 -26844.8 47.7203 2500 -6113.04 14.89074

1650 -20108.9 35.74627 2550 -4563.58 11.116431700 -13389.5 23.80154 2600 -3028.26 7.3765311750 -6686.48 11.88612 2650 -1507.07 3.6710561800 0 0 2700 0 0

Table 2: Analytical Shear Stress at the Neutral Planes of the Beam SectionsBeam AB Beam CD Beam EF

Dist. from A S. Force S. Stress Dist. from A S. Force S. Stress Dist. from A S. Force S. Stress0 52.2312 1.222754 600 -150.108 -4.01469 1500 -96.7824 -3.0182550 51.8544 1.213933 650 -150.438 -4.0235 1550 -97.065 -3.02706100 51.4776 1.205112 700 -150.767 -4.03232 1600 -97.3476 -3.03587150 51.1008 1.196291 750 -151.097 -4.04114 1650 -97.6302 -3.04469200 50.724 1.18747 800 -151.427 -4.04996 1700 -97.9128 -3.0535250 50.3472 1.178649 850 -151.757 -4.05877 1750 -98.1954 -3.06231300 49.9704 1.169828 900 -152.086 -4.06759 1800 -98.478 -3.07112350 49.5936 1.161007 950 -152.416 -4.07641 1800 35.0868 1.094213400 49.2168 1.152186 1000 -152.746 -4.08523 1850 34.8042 1.0854450 48.84 1.143365 1000 42.0576 1.124844 1900 34.5216 1.076587500 48.4632 1.134544 1050 41.7279 1.116026 1950 34.239 1.067774550 48.0864 1.125723 1100 41.3982 1.107208 2000 33.9564 1.058961600 47.7096 1.116902 1150 35.7933 0.957303 2050 33.6738 1.050148600 197.8176 4.630993 1200 40.7388 1.089572 2100 33.3912 1.041334650 197.4408 4.622172 1250 40.4091 1.080754 2150 33.1086 1.032521700 197.064 4.613351 1300 40.0794 1.071936 2200 32.826 1.023708750 196.6872 4.60453 1350 39.7497 1.063118 2250 32.5434 1.014895800 196.3104 4.595709 1400 39.42 1.0543 2300 32.2608 1.006082850 195.9336 4.586888 1450 39.0903 1.045482 2350 31.9782 0.997269900 195.5568 4.578067 1500 38.7606 1.036664 2400 31.6956 0.988456950 195.18 4.569246 1500 135.543 3.62514 2450 31.413 0.9796431000 194.8032 4.560425 1550 135.2133 3.616322 2500 31.1304 0.970829

1600 134.8836 3.607504 2550 30.8478 0.9620161650 134.5539 3.598686 2600 30.5652 0.9532031700 134.2242 3.589868 2650 30.2826 0.944391750 133.8945 3.58105 2700 30 0.9355771800 133.5648 3.572232

Table 3: ABAQUS generated Bending Stress Values at the Top of the Beam SectionsBeam AB Beam CD Beam EF

Dist. from A ABAQUS Dist. from A ABAQUS Dist. from A ABAQUS0 125.988 600 0 1500 050 137.283 650 14.8504 1550 12.6175100 133.901 700 27.1929 1600 23.3846150 130.556 750 39.5999 1650 34.2009200 127.236 800 52.0375 1700 45.0521250 123.941 850 64.5046 1750 55.9611300 120.671 900 77.0008 1800 65.5431350 117.425 950 89.5608 1850 62.7675400 114.205 1000 100.783 1900 58.7733450 111.01 1050 98.9077 1950 54.8278500 107.839 1100 95.3401 2000 50.9157550 104.671 1150 91.8291 2050 47.0371600 73.4191 1200 88.3467 2100 43.192650 87.031 1250 84.8928 2150 39.3803700 74.5971 1300 81.4676 2200 35.6021750 62.132 1350 78.0709 2250 31.8574800 49.6943 1400 74.7029 2300 28.1461850 37.2856 1450 71.3447 2350 24.4684900 24.9103 1500 52.046 2400 20.8241950 12.4908 1550 55.4295 2450 17.21341000 1.09763 1600 44.2785 2500 13.6362

1650 33.139 2550 10.09261700 22.0305 2600 6.583151750 10.8997 2650 3.131421800 1.25995 2700 0

Table 4: ABAQUS generated Shear Stress Values at the Neutral Planes of the Beam SectionsBeam AB Beam CD Beam EF

Dist. from A ABAQUS Dist. from A ABAQUS Dist. from A ABAQUS0 4.4567 600 -2.2794 1500 -2.96650 1.16691 650 -3.7713 1550 -2.8528100 1.16152 700 -3.7877 1600 -2.7859150 1.15282 750 -3.7999 1650 -2.7821200 1.14409 800 -3.8093 1700 -2.7978250 1.13536 850 -3.8183 1750 -2.8141300 1.12665 900 -3.8278 1800 -2.9023350 1.11805 950 -3.836 1850 0.387400 1.10972 1000 -3.8485 1900 1.05227450 1.10171 1050 1.36154 1950 1.04279500 1.0934 1100 1.06147 2000 1.02896550 1.08887 1150 1.05393 2050 1.01907600 2.92067 1200 1.0442 2100 1.01047650 4.36334 1250 1.03506 2150 1.00185700 4.3712 1300 1.02669 2200 0.99912750 4.36361 1350 1.01867 2250 0.9931800 4.35545 1400 1.009 2300 0.98434850 4.34822 1450 0.9924 2350 0.97558900 4.34314 1500 0.9664 2400 0.96683950 4.35896 1550 3.06753 2450 0.958081000 3.62112 1600 3.29468 2500 0.94932

1650 3.30879 2550 0.931811700 3.30109 2600 0.923051750 3.29519 2650 0.91431800 3.23574 2700 0.90498

Table 5: Comparison of Bending Stress Values at the Top of the Beam SectionsBeam AB Beam CD Beam EF

Dist. from A ABAQUS Analytical Dist. from A ABAQUS Analytical Dist. from A ABAQUS Analytical0 125.988 146.9162 600 0 0 1500 0 050 137.283 143.3929 650 14.8504 13.35651 1550 12.6175 11.80481100 133.901 139.8951 700 27.1929 26.74232 1600 23.3846 23.64404150 130.556 136.4229 750 39.5999 40.15743 1650 34.2009 35.51769200 127.236 132.9762 800 52.0375 53.60186 1700 45.0521 47.42575250 123.941 129.5549 850 64.5046 67.07558 1750 55.9611 59.36824300 120.671 126.1592 900 77.0008 80.57861 1800 65.5431 71.34515350 117.425 122.789 950 89.5608 94.11094 1850 62.7675 67.08896400 114.205 119.4443 1000 100.783 107.6726 1900 58.7733 62.8672450 111.01 116.1251 1050 98.9077 103.9491 1950 54.8278 58.67986500 107.839 112.8315 1100 95.3401 100.2549 2000 50.9157 54.52693550 104.671 109.5633 1150 91.8291 96.58999 2050 47.0371 50.40843600 73.4191 106.3206 1200 88.3467 92.95441 2100 43.192 46.32434650 87.031 92.94126 1250 84.8928 89.34813 2150 39.3803 42.27467700 74.5971 79.58741 1300 81.4676 85.77115 2200 35.6021 38.25942750 62.132 66.25907 1350 78.0709 82.22348 2250 31.8574 34.27859800 49.6943 52.95624 1400 74.7029 78.70511 2300 28.1461 30.33218850 37.2856 39.67892 1450 71.3447 75.21604 2350 24.4684 26.42019900 24.9103 26.4271 1500 52.046 71.75628 2400 20.8241 22.54262950 12.4908 13.2008 1550 55.4295 59.72364 2450 17.2134 18.699471000 1.09763 0 1600 44.2785 47.7203 2500 13.6362 14.89074

1650 33.139 35.74627 2550 10.0926 11.116431700 22.0305 23.80154 2600 6.58315 7.3765311750 10.8997 11.88612 2650 3.13142 3.6710561800 1.25995 0 2700 0 0

Figure 24: Telescopic beam bending stresses from FEA and tip reaction analysis(Key: Analytical; FEA)

Table 6: Comparison of Shear Stress Values at the Neutral Planes of the Beam SectionsBeam AB Beam CD Beam EF

Dist. from A ABAQUS S. Stress Dist. from A ABAQUS S. Stress Dist. from A ABAQUS S. Stress0 4.4567 1.222754 600 -2.2794 -4.01469 1500 -2.966 -3.0182550 1.16691 1.213933 650 -3.7713 -4.0235 1550 -2.8528 -3.02706100 1.16152 1.205112 700 -3.7877 -4.03232 1600 -2.7859 -3.03587150 1.15282 1.196291 750 -3.7999 -4.04114 1650 -2.7821 -3.04469200 1.14409 1.18747 800 -3.8093 -4.04996 1700 -2.7978 -3.0535250 1.13536 1.178649 850 -3.8183 -4.05877 1750 -2.8141 -3.06231300 1.12665 1.169828 900 -3.8278 -4.06759 1800 -2.9023 -3.07112350 1.11805 1.161007 950 -3.836 -4.07641 1800 1.094213400 1.10972 1.152186 1000 -3.8485 -4.08523 1850 0.387 1.0854450 1.10171 1.143365 1000 1.124844 1900 1.05227 1.076587500 1.0934 1.134544 1050 1.06147 1.116026 1950 1.04279 1.067774550 1.08887 1.125723 1100 1.05393 1.107208 2000 1.02896 1.058961600 2.92067 1.116902 1150 1.0442 0.957303 2050 1.01907 1.050148600 4.630993 1200 1.03506 1.089572 2100 1.01047 1.041334650 4.36334 4.622172 1250 1.02669 1.080754 2150 1.00185 1.032521700 4.3712 4.613351 1300 1.01867 1.071936 2200 0.99912 1.023708750 4.36361 4.60453 1350 1.009 1.063118 2250 0.9931 1.014895800 4.35545 4.595709 1400 0.9924 1.0543 2300 0.98434 1.006082850 4.34822 4.586888 1450 0.9664 1.045482 2350 0.97558 0.997269900 4.34314 4.578067 1500 1.036664 2400 0.96683 0.988456950 4.35896 4.569246 1500 3.06753 3.62514 2450 0.95808 0.9796431000 3.62112 4.560425 1550 3.29468 3.616322 2500 0.94932 0.970829

1600 3.30879 3.607504 2550 0.93181 0.9620161650 3.30109 3.598686 2600 0.92305 0.9532031700 3.29519 3.589868 2650 0.9143 0.944391750 3.23574 3.58105 2700 0.90498 0.9355771800 5.15431 3.572232

Figure 25: Telescopic beam shear stresses from FEA and tip reaction analysis

(Key: Analytical; FEA)

2.7 Analysis of the Single beam Model

Figure 26: Single Beam Model

2.7.1 Shear Force and Bending Moment Diagrams for the Continuous Stepped Beam

2.7.1.1 In section ACEquating forces in the vertical direction assuming W = 10N

Bending Moment at A =

Shear force is where is the distance from A.

Bending moment


Bending moment at

2.7.1.2 In the section CBShear force is where is the distance from A.Bending moment


Bending moment at

2.7.1.3 In the section BEShear force is where is the distance from A.Bending moment


Bending moment at

2.7.1.4 In the section EDShear force is where is the distance from A.Bending moment


Bending moment at

2.7.1.5 In the section DFShear force is where is the distance from A.Bending moment


Bending moment at

The shear force diagram and the bending moment diagram for the continuous beam are shown in Figure 27.

Figure 27: Shear Force and Bending Moment Diagram of the Continuous Stepped Beam

2.7.2 Calculation of Bending and Shear StressesConsider the sectional view of the continuous stepped beam shown in Figure 19. The bending stress will be maximum in any section along the line A1C1B1E1D1F1. The shear stress will be maximum along the line A2C2B2E2D2F2.

2.7.2.1 Section ABConsidering Section AC within Section ABBending moment at the section at a distance x from A M

and the maximum bending stress


because the top of the beam is 12.5 mm away from the neutral plane .

Figure 28: Section of the Beam above the Neutral Plane

Shear force in AC is given by and the maximum shear stress is at the neutral plane.

The y coordinate of the centroid of the portion of the beam above the neutral plane

Area of cross section =

Maximum shear stress is therefore

Note: The b denominator in in this case is equal to 2t.

Similarly for section CB within section ABBending moment

M

And Shear force S

27.2.2 Beam CDThis beam has three sections CB, BE and ED with differing loads. The beam width also changes with each of these sections

For the section CB

M and

Shear force is SSimilar equations are established for sections BE and ED and fed to Excel to calculate maximum bending stresses and maximum shear stresses in beam CD.

2.7.2.3 Beam EFThis beam has three sections ED and DF with differing loads. The beam width also changes with each of these sections. Similar equations are established for sections ED and DF and fed to Excel to calculate maximum bending stresses and maximum shear stresses in beam EF.

Figures 29 and 30 show the graphical comparison between the analytical and ABAQUS derived bending stress, and the analytical and ABAQUS shear stress, respectively.

Table 7: Analytical Bending Stress at the Top of the Sections of the Stepped BeamSection AB Section CD Section EF

Dist. from A BM B. Stress Dist. from A BM B. Stress Dist. from A BM B. Stress0 -110541 149.670981 600 -79880.5 61.0894157 1500 -40832.3 37.686349150 -107882 146.071193 650 -77456.6 59.2356799 1550 -38868.4 35.8737285100 -105243 142.496913 700 -75068.7 57.4095595 1600 -36937.4 34.0915377150 -102622 138.948142 750 -72717 55.6110546 1650 -35039.4 32.3397767200 -100019 135.424881 800 -70401.4 53.8401652 1700 -33174.4 30.6184454250 -97436.1 131.927128 850 -68121.9 52.0968913 1750 -31342.4 28.9275439300 -94871.6 128.454885 900 -65878.5 50.3812328 1800 -29543.3 27.2670722350 -92326 125.008151 950 -63671.2 48.6931898 1800 -29543.3 58.0146699400 -89799.2 121.586926 1000 -61500 47.0327623 1850 -27768.5 54.5295614450 -87291.3 118.19121 1000 -61500 99.4306847 1900 -26009.5 51.0752833500 -84802.2 114.821003 1050 -59355.6 95.9635713 1950 -24266.1 47.6518356550 -82331.9 111.476305 1100 -57228.3 92.5243793 2000 -22538.5 44.2592183600 -79880.5 108.157117 1150 -55118.4 89.1131087 2050 -20826.5 40.8974313600 -79880.5 61.0894157 1200 -53025.7 85.7297596 2100 -19130.3 37.5664747650 -77456.6 59.2356799 1250 -50950.3 82.3743319 2150 -17449.7 34.2663484700 -75068.7 57.4095595 1300 -48892.2 79.0468255 2200 -15784.9 30.9970526750 -72717 55.6110546 1350 -46851.3 75.7472406 2250 -14135.7 27.7585871800 -70401.4 53.8401652 1400 -44827.7 72.4755771 2300 -12502.3 24.5509519850 -68121.9 52.0968913 1450 -42821.4 69.231835 2350 -10884.5 21.3741472900 -65878.5 50.3812328 1500 -40832.3 66.0160143 2400 -9282.48 18.2281728950 -63671.2 48.6931898 1500 -40832.3 37.6863491 2450 -7696.13 15.11302881000 -61500 47.0327623 1550 -38868.4 35.8737285 2500 -6125.48 12.0287152

1600 -36937.4 34.0915377 2550 -4570.53 8.975231911650 -35039.4 32.3397767 2600 -3031.28 5.952579021700 -33174.4 30.6184454 2650 -1507.73 2.96075651750 -31342.4 28.9275439 2700 0.12 01800 -29543.3 27.2670722

Table 8: Analytical Shear Stress at the Neutral Planes of the Sections of the Stepped BeamSection AB Section CD Section EF

Dist. from A S. Force S. Stress Dist. from A S. Force S. Stress Dist. from A S. Force S. Stress0 53.3616 2.603921231 600 48.84 2.38327773 1500 39.6084 0.99934917750 52.9848 2.585534272 650 48.1178 2.348036059 1550 38.949 0.982712028100 52.608 2.567147314 700 47.3956 2.312794389 1600 38.2896 0.96607488150 52.2312 2.548760355 750 46.6734 1.064750323 1650 37.6302 0.949437731200 51.8544 2.530373397 800 45.9512 1.048274928 1700 36.9708 0.932800582250 51.4776 2.511986438 850 45.229 1.031799533 1750 36.3114 0.916163433300 51.1008 2.49359948 900 44.5068 1.015324139 1800 35.652 0.899526284350 50.724 2.475212522 950 43.7846 0.998848744 1800 35.652 3.573867963400 50.3472 2.456825563 1000 43.0624 0.98237335 1850 35.338 3.542391621450 49.9704 2.438438605 1000 43.0624 2.30053592 1900 35.024 3.51091528500 49.5936 2.420051646 1050 42.717 2.282083509 1950 34.71 3.479438938550 49.2168 2.401664688 1100 42.3716 2.263631098 2000 34.396 3.447962596600 48.84 2.38327773 1150 42.0262 2.245178687 2050 34.082 3.416486254600 48.84 2.38327773 1200 41.6808 2.226726276 2100 33.768 3.385009912650 48.1178 2.348036059 1250 41.3354 2.208273865 2150 33.454 3.35353357700 47.3956 2.312794389 1300 40.99 2.189821454 2200 33.14 3.322057228750 46.6734 1.064750323 1350 40.6446 2.171369043 2250 32.826 3.290580886800 45.9512 1.048274928 1400 40.2992 2.152916632 2300 32.512 3.259104545850 45.229 1.031799533 1450 39.9538 2.13446422 2350 32.198 3.227628203900 44.5068 1.015324139 1500 39.6084 2.116011809 2400 31.884 3.196151861950 43.7846 0.998848744 1500 39.6084 0.999349177 2450 31.57 3.1646755191000 43.0624 0.98237335 1550 38.949 0.982712028 2500 31.256 3.133199177

1600 38.2896 0.96607488 2550 30.942 3.1017228351650 37.6302 0.949437731 2600 30.628 3.0702464931700 36.9708 0.932800582 2650 30.314 3.0387701511750 36.3114 0.916163433 2700 30 3.007293811800 35.652 0.899526284

Table 9: ABAQUS generated Bending Stress Values at the Top of the Beam Sections of the Stepped BeamSection AB Section CD Section EF

Dist. from A B. Stress Dist. from A B. Stress Dist. from A B. Stress0 89.3542 600 51.1435 1500 31.366150 126.04 650 49.0176 1550 29.8058100 124.885 700 51.6278 1600 28.2885150 122.792 750 47.6921 1650 26.7899200 120.923 800 45.2442 1700 25.2581250 117.634 850 44.0213 1750 27.5445300 110.947 900 42.6788 1800 35.6074350 112.235 950 47.6796 1800 48.6036400 108.817 1000 47.6796 1850 44.7294450 103.227 1000 69.0299 1900 41.256500 100.078 1050 82.542 1950 39.1417550 96.7544 1100 76.8668 2000 36.2757600 77.7041 1150 74.23 2050 32.8282600 51.1435 1200 73.0918 2100 30.5938650 49.0176 1250 69.4354 2150 27.2369700 51.6278 1300 68.4102 2200 24.5266750 47.6921 1350 63.6578 2250 21.6627800 45.2442 1400 60.6837 2300 18.6692850 44.0213 1450 57.4121 2350 16.0882900 42.6788 1500 47.4053 2400 13.5938950 47.6796 1500 31.3661 2450 10.60321000 47.6796 1550 29.8058 2500 7.9004

1600 28.2885 2550 5.277091650 26.7899 2600 2.628641700 25.2581 2650 6.169131750 27.5445 2700 01800 35.6074

Table 10: ABAQUS generated Shear Stress Values at the Neutral Planes of the Stepped Beam SectionsSection AB Section CD Section EF

Dist. from A S. Stress Dist. from A S. Stress Dist. from A S. Stress0 4.5612 600 2.5874 1500 1.641250 2.7523 650 2.5678 1550 1.5974100 2.7489 700 2.4483 1600 1.3318150 2.7341 750 1.3471 1650 1.3291200 2.7245 800 1.3329 1700 1.3155250 2.7156 850 1.3217 1750 1.3007300 2.6954 900 1.3894 1800 1.2996350 2.6746 950 1.4587 1800 4.217400 2.6557 1000 1.4984 1850 4.0183450 2.6491 1000 3.9521 1900 3.9894500 2.6325 1050 2.9956 1950 3.9658550 2.6211 1100 2.7483 2000 3.9524600 2.6087 1150 2.7363 2050 3.9421600 2.5874 1200 2.7014 2100 3.9365650 2.5678 1250 2.661 2150 3.9123700 2.4483 1300 2.6023 2200 3.7743750 1.3471 1350 2.5841 2250 3.7612800 1.3329 1400 2.4997 2300 3.7549850 1.3217 1450 2.4138 2350 3.7448900 1.3894 1500 2.3746 2400 3.7369950 1.4587 1500 1.6412 2450 3.71211000 1.4984 1550 1.5974 2500 3.6822

1600 1.3318 2550 3.64711650 1.3291 2600 3.60181700 1.3155 2650 3.57361750 1.3007 2700 3.59741800 1.2996

Table 11: Comparison of Bending Stress Values at the Top of the Beam Sections of the Stepped BeamSection AB Section CD Section EF

Dist. from A ABAQUS Analytical Dist. from A ABAQUS Analytical Dist. from A ABAQUS Analytical0 89.3542 149.670981 600 51.1435 61.0894157 1500 31.3661 37.686349150 126.04 146.071193 650 49.0176 59.2356799 1550 29.8058 35.8737285100 124.885 142.496913 700 51.6278 57.4095595 1600 28.2885 34.0915377150 122.792 138.948142 750 47.6921 55.6110546 1650 26.7899 32.3397767200 120.923 135.424881 800 45.2442 53.8401652 1700 25.2581 30.6184454250 117.634 131.927128 850 44.0213 52.0968913 1750 27.5445 28.9275439300 110.947 128.454885 900 42.6788 50.3812328 1800 35.6074 27.2670722350 112.235 125.008151 950 47.6796 48.6931898 1800 48.6036 58.0146699400 108.817 121.586926 1000 47.6796 47.0327623 1850 44.7294 54.5295614450 103.227 118.19121 1000 69.0299 99.4306847 1900 41.256 51.0752833500 100.078 114.821003 1050 82.542 95.9635713 1950 39.1417 47.6518356550 96.7544 111.476305 1100 76.8668 92.5243793 2000 36.2757 44.2592183600 77.7041 108.157117 1150 74.23 89.1131087 2050 32.8282 40.8974313600 51.1435 61.0894157 1200 73.0918 85.7297596 2100 30.5938 37.5664747650 49.0176 59.2356799 1250 69.4354 82.3743319 2150 27.2369 34.2663484700 51.6278 57.4095595 1300 68.4102 79.0468255 2200 24.5266 30.9970526750 47.6921 55.6110546 1350 63.6578 75.7472406 2250 21.6627 27.7585871800 45.2442 53.8401652 1400 60.6837 72.4755771 2300 18.6692 24.5509519850 44.0213 52.0968913 1450 57.4121 69.231835 2350 16.0882 21.3741472900 42.6788 50.3812328 1500 47.4053 66.0160143 2400 13.5938 18.2281728950 47.6796 48.6931898 1500 31.3661 37.6863491 2450 10.6032 15.11302881000 47.6796 47.0327623 1550 29.8058 35.8737285 2500 7.9004 12.0287152

1600 28.2885 34.0915377 2550 5.27709 8.975231911650 26.7899 32.3397767 2600 2.62864 5.952579021700 25.2581 30.6184454 2650 6.16913 2.96075651750 27.5445 28.9275439 2700 0 01800 35.6074 27.2670722

Figure 29: Continuous beam bending stresses from FE and Theory(Key: Analytical; FEA)

Table 12: Comparison of Shear Stress Values at the Neutral Planes of the Stepped Beam SectionsSection AB Section CD Section EF

Dist. from A ABAQUS Analytical Dist. from A ABAQUS Analytical Dist. from A ABAQUS Analytical0 4.5612 2.603921231 600 2.5874 2.38327773 1500 1.6412 0.99934917750 2.7523 2.585534272 650 2.5678 2.348036059 1550 1.5974 0.982712028100 2.7489 2.567147314 700 2.4483 2.312794389 1600 1.3318 0.96607488150 2.7341 2.548760355 750 1.3471 1.064750323 1650 1.3291 0.949437731200 2.7245 2.530373397 800 1.3329 1.048274928 1700 1.3155 0.932800582250 2.7156 2.511986438 850 1.3217 1.031799533 1750 1.3007 0.916163433300 2.6954 2.49359948 900 1.3894 1.015324139 1800 1.2996 0.899526284350 2.6746 2.475212522 950 1.4587 0.998848744 1800 4.217 3.573867963400 2.6557 2.456825563 1000 1.4984 0.98237335 1850 4.0183 3.542391621450 2.6491 2.438438605 1000 3.9521 2.30053592 1900 3.9894 3.51091528500 2.6325 2.420051646 1050 2.9956 2.282083509 1950 3.9658 3.479438938550 2.6211 2.401664688 1100 2.7483 2.263631098 2000 3.9524 3.447962596600 2.6087 2.38327773 1150 2.7363 2.245178687 2050 3.9421 3.416486254600 2.5874 2.38327773 1200 2.7014 2.226726276 2100 3.9365 3.385009912650 2.5678 2.348036059 1250 2.661 2.208273865 2150 3.9123 3.35353357700 2.4483 2.312794389 1300 2.6023 2.189821454 2200 3.7743 3.322057228750 1.3471 1.064750323 1350 2.5841 2.171369043 2250 3.7612 3.290580886800 1.3329 1.048274928 1400 2.4997 2.152916632 2300 3.7549 3.259104545850 1.3217 1.031799533 1450 2.4138 2.13446422 2350 3.7448 3.227628203900 1.3894 1.015324139 1500 2.3746 2.116011809 2400 3.7369 3.196151861950 1.4587 0.998848744 1500 1.6412 0.999349177 2450 3.7121 3.1646755191000 1.4984 0.98237335 1550 1.5974 0.982712028 2500 3.6822 3.133199177

1600 1.3318 0.96607488 2550 3.6471 3.1017228351650 1.3291 0.949437731 2600 3.6018 3.0702464931700 1.3155 0.932800582 2650 3.5736 3.0387701511750 1.3007 0.916163433 2700 3.5974 3.007293811800 1.2996 0.899526284

Figure 30: Continuous beam shear stresses from FE and Theory(Key: Analytical; FEA)

References1. Ramamrutham S. And Narayan R.; Strength of Materials, Eleventh Edition,

Dhanpatrai &Sons, Dehli 1992.2. Rees, D.W.A.; Mechanics of Solids and Structures, World Scientific 2000.3. Benham P.P and Crawford R.J .; Mechanics of Engineering Materials, English

Language Book Society/ Longman Group Limited, Essex England 1987.

interim phd report

Documents