Stability Of Structures: Additional Topics

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Stability Of Structures: Additional Topics

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Lecture 19: STABILITY OF STRUCTURES: ADDITIONAL TOPICS

TABLE OF CONTENTS Page

§19.1. Unified Buckling Formula §19.1.1. Effective Length . . . . . . . . . . . . §19.1.2. Critical Load in Terms of Slenderness Ratio . . . . §19.2. Failure Mode: Buckling Versus Yield §19.2.1. Failure Envelopes . . . . . . . . . . . . §19.2.2. Long Versus Short Columns . . . . . . . . . §19.3. The Southwell Plot §19.3.1. Effect of Imperfections . . . . . . . . . . §19.3.2. “Virtual” Southwell Plot Simulation with Mathematica §19.4. The ITLL Buckling Demo §19.4.1. Module Description . . . . . . . . . . . §19.4.2. Experimental Procedure . . . . . . . . . . §19.4.3. Specimen Dimensions . . . . . . . . . . §19.4.4. Restraint Beam Attachment Details . . . . . . §19.4.5. Miscellaneous Reminders . . . . . . . . . §19.5. Labwork On Column Buckling - Friday Nov 5 at ITLL §19.5.1. Description . . . . . . . . . . . . . . §19.5.2. Structural Stability Homeworks . . . . . . . §19.5.3. Experiment Scheduling . . . . . . . . . .

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19–3 19–3 19–3 19–4 19–4 19–5 19–5 19–6 19–7 19–7 19–8 19–9 19–10 19–10 19–10 19–12 19–12 19–12 19–12

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§19.1

UNIFIED BUCKLING FORMULA

This Lecture covers additional topics that are useful for Homework 8 and Midterm Exam 3, as well as the “Labwork” demo on Friday 11/5, which is to be reported as an extended homework problem. §19.1. Unified Buckling Formula §19.1.1. Effective Length The effect of different end conditions on the critical load of a column can be unified by expressing it in the form π2E I Pcr = 2 (19.1) Lef f Here L e f f denotes the effective length of the column. This length turns out to have a simple physical interpretation: distance between the inflexion points of the buckling curve associated with Pcr .

P P

P

P

L L eff = L

L eff =2L

L

L eff =0.7 L

L

L eff =L/2

fictitious continuation about fixed end

pinned-pinned (Euler column)

free-fixed (cantilever)

pinned-fixed

fixed-fixed

Figure 19.1. Effective lengths of columns with different end conditions.

Values of L e f f for three common support conditions are given in Figure 19.1. (The value of 0.7 L listed for the pinned-fixed case is correct to 3 places; a more accurate value is 0.6992 L but 0.7 is easier to remember.) The unified definition (19.1) allows us to extend results derived for the Euler column to other cases, simply by replacing the appropriate effective length in the formula. §19.1.2. Critical Load in Terms of Slenderness Ratio In formula (19.1), I (the√ minimum second moment of inertia of the cross section) may be replaced 2 by A r , in which r = + I /A is the radius of gyration of the column cross section. Examples of r for three column cross sections: •

If the section√is a b × h rectangle with h ≤ b, I = b h 3 /12, A = b h, then r 2 = I /A = h 2 /12 and r = 12 h/ 3 ≈ 0.2887 h.

•

If a solid circle of radius R, I = π R 4 /4, A = π R 2 , then r 2 = I /A = R 2 /4 and r = 12 R. 19–3

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Failure by yield (short columns)

1.2

Failure by buckling (long columns)

σcr = σ cr /σ Y

1.0 0.8 0.6

Steel

Fir Wood

Aluminum

0.4 0.2 0.0

0

50

100

150

200

Slenderness ratio s = Leff /r Figure 19.2. Column failure modes: buckling versus yield plotted in terms of the slenderness ratio. Solid lines (“Euler hyperbolas”) represent failure by buckling; dashed line failure by yield; black dots mark the critical slenderness ratios.

•

If a thin-wall circular tube of√ mean radius R and thickness t << R, I ≈ π R 3 t, A = 2π Rt, 2 / r = I /A = R 2 and r = R/ 2 = 0.707 R.

Substituting I = A r 2 in (19.1) gives Pcr =

π2 E I π2 E A r2 π2 E A = = , s2 L 2e f f L 2e f f

(19.2)

in which s = L e f f /r is called the slenderness ratio of the column. §19.2. Failure Mode: Buckling Versus Yield A key question in column design is: will be column fail first by yield of elastic buckling? In other words, which is the failure mode? The average axial stress at the critical load Pcr expressed as (19.2) is σcr =

Pcr π2 E = 2 , A s

(19.3)

This is called the critical stress. A key design question is: how does the critical stress compare with the yield stress σY of the column material? Evidently if σcr > σY the critical load formula is not valid, since the derivation is based on the assumption that the column is linearly elastic at buckling. To visualize the limitation of the buckling formula, we will use a graphical interpretations of (19.3). §19.2.1. Failure Envelopes Introduce two dimensionless ratios for the column material: def

σ¯ cr =

σcr , σY 19–4

def E E¯ = , σY

(19.4)

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§19.3

THE SOUTHWELL PLOT

Dividing both sides of (19.3) by σY and introducing the ratios (19.4) gives a dimensionless version σ¯ cr =

π 2 E¯ , s2

(19.5)

This is graphed in Figure 19.2 for the following materials: structural steel (E = 210 GPa, σY = 210 MPa, E¯ = E/σY = 1000), aluminum alloy (E = 70 GPa, σY = 280 MPa, E¯ = E/σY = 250), and fir wood (E = 12.6 GPa, σY = 35 MPa in compression, E¯ = E/σ y = 360). The curves delimit what are called failure envelopes. §19.2.2. Long Versus Short Columns The following three failure cases may be distinguished according to the slenderness of the column. •

Long column. If the slenderness exceeds a critical value scr defined below, the column will fail by elastic buckling, in which case σcr < σY or σ¯ cr < 1. Failure occurs at one of the solid curves, which are known as Euler hyperbolas in the literature.

•

Short column. If the slenderness is less than than the critical value scr defined below, it fail by yield at |sigma = P/A = σY , or σ¯ cr = 1 and the failure occurs at the dashed line. 2 Goldilocks column. Not too short, not too long. If s = scr , where scr = π 2 E¯ = π 2 E/σY the column will fail simultaneously by buckling and yield. That transition is marked by the black dot in Figure 19.2. It represents the most efficient use of the material, so it is an optimal design in that particular sense.

•

Remark 19.1. Most Mechanics of Materials books show failure diagrams by plotting (19.3) directly, that is,

without dividing through bt σY . As a result the horizontal scale is dimensionless (the slenderness ratio) but the vertical scale (in stress) is not. The resulting graphs depend on column material as well as physical units chosen. Example 19.1. A pinned-pinned steel column with E = 210 GPa and σY = 210 MPa has a pin-to-pin length of

L = 5 m = 5000 mm and a b × h solid rectangular cross section with b = 0.12 m = 120 mm, and h = 0.08 m = 80 mm. Will the column fail first by yield or elastic buckling?

Solution. The critical Euler buckling load is Pcr = π 2 E I /L 2 since L e f f = L for the pinned-pinned case. The minimum second moment of inertia is I = bh 3 /12 because h < b. Replace and divide by A = bh to get σcr = Pcr /A = π 2 E h 2 /(12L 2 ) = 44.8 N/mm2 = 44.8 MPa. Compare to yield: σcr < σY = 210 MPa. Thus the column will fail first by buckling . 2 2 Alternatively one can check the√slenderness ratio: √ s = L e f f /r , where L e f f = L and r = I /A = h /12. A quick computation gives s = L 12/ h = 5000 12/80 ≈ 216, which is way into the “long column” range as can be quickly checked from Figure 19.2.

Example 19.2. A fixed-fixed steel column (same E and σY as in the previous example) of length L = 6 m has a

solid circular cross section of unknown radius R. Find (1) R in mm so the column fails simultaneosuly by yield and by elastic buckling, and (2) which maximum load Pmax the so-designed column can support if the safety factor against both buckling and yield is s F = 4. 2 , r2 = Solution. For (1), equate σcr = σY and solve for R. Details: L e f f = L/2, I = (π/4)R 4 , A = π R√ √ ¯ = I /A = R 2 /4. σcr = π 2 E (R 2 /4)/(L 2 /4) = π 2 E R 2 /L 2 = σY , whence R = σY /E L/π = L/(π E) 2 L/99.34 = 6000/99.34, hence R = 60.4 mm . The area of this optimal design is A = π R = 11461

mm2 . The failure load is Pcr = σY A = 2.407 × 1012 N. Divide this by 4 to get

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Pmax = 6.02 × 1011 N

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§19.3. The Southwell Plot The Southwell plot‡ provides a clever graphical method for nondestructive critical-load testing of columns (as well as other structural components that may fail by buckling). This method is particularly useful for field tests of structures or structural components that are likely to be damaged by being taken up to and beyond the critical load, such as reinforced concrete columns or advanced composite materials. §19.3.1. Effect of Imperfections For the method to work well, the column should be subjected to eccentric axial loads, or to tiny lateral loads that “trigger” measurable lateral deflections prior to buckling. Let P be the applied axial load P < Pcr . Let vm (P) denote a measured lateral deflection at load level P. Southwell observed that the following approximation holds as P approaches Pcr : vm ≈

a

(19.6)

Pcr − 1 P

where a is a constant. Equation (19.6) may be transformed to vm = Pcr

vm −a P

(19.7)

vm experimental data points

~ Pcr slope =

vm P Figure 19.3. The Southwell plot.

‡ R. V. Southwell, On the Analysis of Experimental Observations in Problems of Elastic Stability, Proc. Roy. Soc. London, Series A, 135, pp. 601–616, April 1932. Bio note: Sir Richard Southwell was honored by a nobiliary title in 1948 because of his contributions to the British WWII effort. He was a developer of “relaxation” methods, which were used to solve systems of hundreds of equations on desk calculators prior to computers. (Before digital computers appeared ca. 1951, the term “computers” meant humans working on numerical calculations. To solve large systems of equations by relaxation, dozens of “computers” — mostly women during WWII — were gathered in a large room; each did part of the computations on desk calculators, receiving and passing results to neighbors.) One important application was the prediction of the Normandy weather for D-day: June 6, 1944. It worked.

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§19.4

THE ITLL BUCKLING DEMO

If vm /P and vm are plotted along the x and y axis, respectively, (19.7) becomes the equation of a straight line: y = mx − a, in which the slope m = Pcr . This observation furnishes a simple but effective experimental method: from measurements at several axial loads P < Pcr one obtains vm and vm /P as data points. These are plotted along the vertical and horizontal axes, respectively, as illustrated in Figure 19.3. This is the Southwell plot. A straight line is fitted to these points. Its slope estimates Pcr . The value of the constant a is of little importance. This technique has the important advantage of being non-destructive because P need not exceed the critical load Pcr . For this reason it is often used in aerospace structures fabricated of expensive materials such as composites. §19.3.2. “Virtual” Southwell Plot Simulation with Mathematica Figure 19.4 shows a Mathematica-scripted simulation of the Lab 3 beam-column specimen as a pinned-pinned column (the Euler column) with an assumed load eccentricity of e = 6 mm. The results of running the script are shown in Figure 19.5. The leftmost plot is the lateral-deflection versus load (vm vs. P) curve. The Southwell plot, constructed for load increments of 5 N (from 5 through 45 N), is the rightmost one.† Em=190000; L=600; t=(0.068)*25.4; w=25.4; Izz=w*t^3/12; e=6; lambda=Sqrt[P/(Em*Izz)]; vm=e*(Sec[lambda*L/2]-1); Pcr=N[Pi]^2*Em*Izz/L^2; Print["Pcr=",Pcr]; Plot[vm,{P,0,45}]; Splot=Table[{(vm/.P->Pv)/Pv,vm/.P->Pv},{Pv,5,45,5}]; ListPlot[Splot,Axes->True];

Figure 19.4. Southwell plot simulation via Mathematica for the Lab 3 specimen as pinned-pinned (Euler column) with load eccentricity e = 6 mm: script

25

25

20

20

15

15

10

10

5

5 10

20

30

0.3

40

0.4

0.5

0.6

Figure 19.5. Southwell plot simulation via Mathematica for the Lab 3 specimen as pinned-pinned (Euler column) with load eccentricity e = 6 mm: results

Note that the points fall neatly on a straight line, as Southwell discovered. This will not be generally the case in experiments because of a multitude of factors discussed in the main sections of this document. † The equation of lateral deflection v versus load P used in the script comes from eccentrically loaded column theory, m which is not covered in this course. So the Mathematica script is just a recipe for creating a plot.

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§19.4. The ITLL Buckling Demo This section describes an experimental lab on column buckling that will be carried out on Friday November 5 during lab hours. Prior to this (Fall 2010) offering, this used to be Lab 3, a formal experimental lab scheduled similarly to the first two ones. This year it is converted to a ”labwork” or “homelab”, meaning that the experiments will be done in front of 4 subsections of about 20 students each. The collected data will be collectively incorporated into Homework 8, which deals with stability. §19.4.1. Module Description The apparatus being used is an off-the-shelf module produced by a British company by the name of Hi-Tech Ltd.† As befits its European provenance, relevant dimensions and weights are provided in SI units. The Beam-Column Buckling Module is sketched in Figure 19.6. It consists of a mobile steel frame that supports a load arm and end conditions for the beam-column. A high-strength steel beam-column is provided with a nominal Young’s Modulus of 200–205 GPa. Pinned and clamped conditions can be simulated at the top of the supplied beam-column with different adapters. The boundary condition at the bottom of the beam can be varied from pinned to clamped through the use of a restraint beam that provides adjustable levels of torsional stiffness at this point. L

3L

load arm counterweight

knife edge stop

beam-column specimen of high-strength steel ruler

load tray

beam clamps frame

restraint beam

Figure 19.6. ITLL Beam-Column Buckling Module (aka Hi-Plan Module).

The end fixtures clamped to the beam-column also provide off-axis notches for applying eccentric loads. These notches are spaced 1.5 mm apart. Typically, an eccentric load is applied by offsetting the knife-edge contacts an equal number of notches off-center at the top and bottom of the beam-column. The ratio of the load arm lengths from the pivot point to the beam-column and from the pivot point to the load tray is 3:4 as shown in Figure 19.6. Remember to convert the loads applied at the load tray † Not a very accurate name, but the apparatus is rugged and serves its purpose.

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§19.4


to those resulting at the beam-column appropriately. During loading, some friction between the load arm and frame may develop. The effect of this friction on the deflection response can be alleviated by gently tapping on the frame after loads are applied. §19.4.2. Experimental Procedure The experiment procedure consists of making load-displacement measurements for several combinations of boundary conditions and eccentric loads. Initially, data will be collected for a nominally pinned-pinned beam-column using two or three different eccentric load configurations. (The exact number will be decided upon during the labs.) This data will be used to explore the benefits of using the Southwell plot. Following this, the BC at the beam-column base will be adjusted to a nominally clamped state and then to an elastically restrained state. Do not forget to make careful measurements of the beam-column dimensions. BC Case 1: Pinned-Pinned Response For this case the restraining beam should be detached from the beam-column. Assemble the beamcolumn with zero eccentricity and with the ruler centered along the length of the beam-column as demonstrated in the demos. (Use a tiny 2 N weight on the load tray to keep the beam-column in place during assembly.) Check that the lower knife-edge is located directly beneath the upper knife-edge by measuring the distances from these boundaries to the right hand side of the frame. (How would you model errors in this alignment?) Align the ruler with a reference point on the beam to ”zero” (don’t forget your 2 N load) the measurement. Measure the lateral deflections induced by applied loads varying from ”zero” to the critical load. (You may need to help the beam-column buckle to the right as described in the demos.) Since the deflections of the beam-column will increase rapidly as the critical load is approached, reduce the loading increments near this load. It is difficult to resolve deflections to better than ±0.25 mm, so obtaining the larger deflection data near failure is important. You should have at least 10 load increments. Once you are satisfied with the repeatability of your data for this configuration, adjust the load eccentricity to 3 mm (2 notches) and repeat the load-deflection data acquisition procedure. You may need to shift the ruler or adjust your displacement measurements accordingly. Repeat the procedure yet again for 6 mm of load eccentricity. Use a spreadsheet to plot the loaddeflection data and to generate Southwell plots of deflection vs. compliance. Fit lines to the three data series in the Southwell plots to derive values for the critical loads for all three cases. How does the apparent critical load in the load-deflection plots and Southwell analysis compare to the theoretical prediction? (When measuring the geometry of the beam-column, consider which dimension(s) are most critical.) BC Case 2: Fixed-Pinned Response Return the knife-edges to the on-axis notches (no eccentricity). Attach the restraint beam to the bottom of the beam-column and position the restraint beam clamps as close as possible to the bottom of the beam-column. Be sure to tighten the clamps to both the beam and the frame. Attachment details are shown in Figure 19.7 on next page. 19–9


19–10

This will provide a nominally fixed boundary condition. In this test, we would again like to measure the lateral deflection where it is greatest. Consider the buckled deflection profile you would expect for these boundary conditions and adjust the vertical position of the ruler accordingly. You can proceed to buckle the beam-column to check your decision. Explain how you selected this ruler position in your lab report. Once the system is configured, repeat the pinned-pinned procedure for collecting and analyzing the load-deflection data. Be sure to do the three eccentricities: 0, 3 and 6 mm. How do the measured values for critical load, derived from Southwell’s plot, compare with the theory for a fixed-pinned beam-column? How “fixed” does the lower boundary condition appear to be? BC Case 3: Elastically Restrained-Pinned Response Loosen both restraint beam clamps and slide the left-most one completely off of the restraint beam. Position the remaining clamp 500 mm from the beam column and position the top grips of the restraining clamp to provide about 0.5 mm of clearance to allow lateral motion of the restraint beam. Approximate where the maximum beam-column lateral deflection will occur and position the ruler accordingly. Again, explain in the lab report how this was done. Repeat the data collection and analysis procedure one last time for this case. Be sure to do the three eccentricities: 0, 3 and 6 mm. Attachment details are shown Figure 19.7 below. How would you model the BC provided by the restraint beam? (The appropriate theory is provided in Lecture 18.) When this is done, how do the measured values for critical load, derived from Southwell’s plot, compare with the theory? §19.4.3. Specimen Dimensions From measurements taken on 11/11/03, the beam-column specimen (and restraint beam) in the buckling module seems to be nominally b = 1” = 25.4 mm wide and t = 1/16” = 1.58 mm thick. A caliper measurement gives a thickness of about 68/1000 in = 1.7 mm, a bit larger than 1/16”. [This is curious because all apparatus dimensions are supposed to be metric.] Note that the buckling load Pcr is very sensitive to the thickness t because Izz = bt 3 /12.) Please recheck these specimen dimensions. The span between knife-edges is 60 cm = 600 mm. The elastic modulus E of the high strength steel used in the specimen and restraint beam has a nominal range of 200 to 205 GPa. §19.4.4. Restraint Beam Attachment Details Fixed-Pinned Case. Slide the aluminum spacer (in blue plastic box) below restraining beam and under grip of clamps, as sketched in Figure 19.7. Tighten clamps enough to prevent rotation of the beam-column specimen, but do not overtighten. During this operation, please check that upper end of beam-column stays aligned with knife-edge. If misalignment occurs, loosen one side slightly as necessary to keep clamps even. Elastically Restrained-Pinned Case. Slide one clamp out of the way. (Also do not continue to use the aluminum spacer, save it in the blue plastic box.) Place the other clamp 50 cm (500 mm) away from the beam column as sketched in Figure 19.8. Attach the bottom clamp screw to set the distance. Keep the upper screws sufficiently loose to allow rotational motion as well as axial slip, but not so excessively loose as to allow significant vertical motions. This configuration justifies idealization of the clamp as a hinge BC in the theoretical analysis. 19–10

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§19.4


Tighten clamp screws to preclude rotations

Place aluminum spacer below restraint beam and under clamp grips

Figure 19.7. Attachment detail for fixed-pinned case.

Keep this clamp sufficiently loose so it acts roughly as a hinge, allowing rotation but little vertical motion Slide this clamp out of the way distance 50 cm = 500 mm

Figure 19.8. Attachment detail for elastically restrained-pinned case.

§19.4.5. Miscellaneous Reminders •

If something is wrong with the equipment, or you find that items are missing, do not proceed. Report problem to instructor or TAs.

•

At least one student in each subsection should bring a notebook to record measurements. Transcribe data later to a spreadsheet. It is recommended to bring also a pocket metric ruler.

•

For dimensions of the apparatus, beam column specimen and restraint beam, you may check Instruction Manual HST/12, “Column Buckling Failure.” This is kept with the module. All dimensions given there are in mm. But it is equally as easy to measure those yourself.

•

Once finished with all Friday demos, disssamble the apparatus and place all items and tools in the blue plastic box. Do not forget to insert the two restraint-beam attachment screws back in the beam specimen to avoid loss.

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§19.5. Labwork On Column Buckling - Friday Nov 5 at ITLL §19.5.1. Description Lab 3 on Column Buckling will be replaced this Fall (an in subsequent course offerings) by a “Labwork”, which is a combination of experimental lab and homework. A combination demo+experiment will be carried out on Friday Nov 5 at the Active Learning Classroom (ITLL 1B50). The Hi-Plan Buckling Module will be rolled into the classroom. The module need not be connected to a Labstation because no electronics is used to record data. Just old-fashioned eyeballing, and pencil-and-paper. To enhance visibility and facilitate participation, the lab will be divided into 4 subsections of about 20 students each. See below for time scheduling details. The experimental procedure is described in §19.4 of this Lecture. The use of the Southwell plot to compare experimental data with analytical predictions is described in §19.3. There are no group experiments next week, thus no sign-up sheet and no formal lab report. Comparison of collected data with analytical predictions will be the subject of an additional homework, as described below. §19.5.2. Structural Stability Homeworks There will be two homeworks assigned on Structural Stability, but no Experimental Lab report. Homework 8, which is entirely analytical, will be posted by Nov 3 and due on Thursday Nov 11. Collected data from the Labwork of Friday Nov 5 will be collectively incorporated into a new but short Homework 9, which will be due on Tuesday Nov 16. That data will be part of the homework assignment document, to be posted on CULearn by Nov 10. Homework can be done by the usual groups, formed directly by students. There is no requirement that a HW group span a lab section or subsection. §19.5.3. Experiment Scheduling Section 11. Divided into 2 subsections: 11a and 11b. 11a: 10-11AM. Students with last name starting with A through M (inclusive). Experiments: pinned-pinned, pinned-fixed and pinned restraint. Eccentricity 1.5 mm (1 notch). 11b: 11-12AM. Students with last name starting with N through Z. Experiments: pinned-pinned, pinned-fixed and pinned restraint. Eccentricity 3 mm (2 notches). Section 12. Divided into 2 subsections: 12a and 12b. 12a: 1-2PM.

Students with last name starting with A through L (inclusive). Experiments: pinnedpinned, pinned-fixed and pinned restraint. Eccentricity 4.5 mm (3 notches).

12b: 2-3PM.

Students with last name starting with M through Z. Experiments: pinned-pinned, pinned-fixed and pinned restraint. Eccentricity 6 mm (4 notches).

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Stability Of Structures: Additional Topics

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