Cantilever beams

We first consider the deflection of a cantilever beam subjected to an acceleration a. The deflection 6 at the end of a cantilever beam of length L under a distributed load F L (N/m) is 

If the cantilever has a mass m that is subjected to an acceleration ng, the load would then be nmg and the deflection would be 

As an example, we consider a polysilicon cantilever beam that is 1 mm long, 20 pm wide, and 2 pm thick. The mass of the beam would be 

Under an acceleration of = 100 g, the cantilever beam would deflect by 0.55 pm. 

---

Fig. 2. Illustrations of forces to which adhesive bonds are subjected, (a) A standard lap shear specimen where the black area shows the adhesive. The adherends are usually 25 mm wide and the lap area is 312.5 mm. The arrows show the direction of the normal apphcation of load, (b) A peel test where the loading configuration, shown by the arrows, is for a 180° peel test, (c) A double cantilever beam test specimen used in the evaluation of the resistance to crack propagation of an adhesive. The normal application of load is shown by the arrows. This load is appHed by a tensile testing machine or other...

Fracture mechanics (qv) tests are typically used for stmctural adhesives. Thus, tests such as the double cantilever beam test (Fig. 2c), in which two thick adherends joined by an adhesive are broken by cleavage, provide information relating to stmctural flaws. Results can be reported in a number of ways. The most typical uses a quantity known as the strain energy release rate, given in energy per unit area. 

The force and moment ia a constrained system can be estimated by the cantilever formula. Leg MB is a cantilever subject to a displacement of and leg CB subject to a displacement Av. Taking leg CB, for example, the task has become the problem of a cantilever beam with length E and displacement of Av. This problem caimot be readily solved, because the end condition at is an unknown quantity. However, it can be conservatively solved by assuming there is no rotation at poiat B. This is equivalent to putting a guide at poiat B, and results ia higher estimate ia force, moment, and stress. The approach is called guided-cantilever method. 

FIG. 28-18 Specimens for stress-corrosion tests, (a) Bent beam, (h) C ring, (c) U Lend, (d) Tensile, (e) Tensile, (f) Tensile, (g) Notched C ring, (h) Notched tensile. ( ) Precracked, wedge open-loading type. (/) Precracked, cantilever beam. [Chem. Eng., 78, 159 (Sept. 20 1971).]... 

In the following, we shall consider a single cantilever beam, of square section, and will analyse the material requirements to minimise the weight for a given stiffness. The results are quite general in that they apply equally to any sort of beam of square section, and can easily be modified to deal with beams of other sections tubes, I-beams, box-sections and so on. 

Often it is not the weight, but the cost of a structure which is the overriding criterion. Suppose that had been the case with the cantilever beam that we have just considered - would our conclusion have been the same Would we still select wood And how much more expensive would a replacement by CFRP be ... 

These effects have been found by Creton et al. [79] who laminated sheets of incompatible polymers, PMMA and PPO, and studied the adhesion using a double cantilever beam test to evaluate fracture toughness Fc. For the original laminate Fc was only 2 J/m, but when interface reinforced with increasing amounts of a symmetrical P.M.M.A.-P.S. diblock copolymer of high degree of polymerisation (A > A e), the fracture toughness increased to around 170 J/m, and then fell to a steady value of 70 J/m (Fig. 9). 

Fig. 10. Cantilever beams used to measure the fracture energy of nail pullout from wood. Top plan of beam showing nail heads. Bottom method of loading beams with a load P after [58].

To evaluate the influx solution experimentally for an A/B cantilever beam configuration as shown in Fig. 1, we apply Griffith s theory at the critical moment of fracture, such that the incremental change in stored elastic energy U. with change in crack length a, is Just sufficient to overcome the fracture surface energy S... 

The stress in a cantilever beam of this type will be maximum at point A (Fig. 2.22) and is given by... 

Example 2.11 An aluminium cantilever beam is SO mm wide, 80 mm long and 2 mm deep. The loading is 200 N spread uniformly over the cantilever. If the beam is to be replaced by one made from acetal and the design criteria is that the end deflections should be the same in each beam after one month, calculate the dimensions (a) of a solid acetal beam and (b) an acetal beam with unidirectional ribs. The modulus of the aluminium is 70 GN/m. ... 

Fig. 8.11 Effect of beam deflection rate of cantilever beam specimens upon stress-corrosion crack velocity of carbon steel in carbonate-bicarbonate solution...

Another method of flexural testing that can be used is, for example, the cantilever beam method (Fig. 2-18), which is used to relate different beam designs. It provides an exam-... 

SIMPLY SUPPORTED BEAM CANTILEVERED BEAM (ONE END FIXED)... 

Fig. 3-13 Zigzag configured multiple-cantilever beam spring.

Fig. 4-16 Basic snap fit design for a cantilever beam with a rectangular cross-section.

Thus, if 5 = 1mm, / = 15.8 Hz. This very simple result is quite useful for approximately evaluating the gravity driven deflections of a stmeture given its natural frequency, or visa versa. Of course this was derived for a specific and very simple system, so it does not perfectly apply to more complex systems. Still it is a very useful rule of thumb. For a mass on the end of a cantilever beam, the above formula is correct. The lowest natural frequency of a massive cantilever beam is about 1.2 x the prediction of the above formula. 

Wind loading will only be important on tall columns installed in the open. Columns and chimney-stacks are usually free standing, mounted on skirt supports, and not attached to structural steel work. Under these conditions the vessel under wind loading acts as a cantilever beam, Figure 13.19. For a uniformly loaded cantilever the bending moment at any plane is given by ... 

The required thickness for the base ring is found by treating the ring as a cantilever beam. The minimum thickness is given by ... 

Equation 28 states, that co is proportional to t. The effect of the thickness of the gel film on the frequency of the first resonance mode has been investigated. When the buoyancy is taken into account, the experimental results have quantitatively followed Eq. 28. It has been found that the buoyancy plays an important role in the occurrence of the electric field-associated vibration of gel film. The vibration of the gel film in an electric field has thus roughly analyzed as a mechanical bending vibration of a uniform cantilever beam. 

Figure 2.10 SEM images of a cantilever beam tip (a) before and (b) after CXL nanoparticle deposition. Adapted from [77] IOP Publishing.

Hurst, K., Roberts, C. and Ashurst, W. (2009) A new method to determine adhesion of cantilever beams using beam height experimental data. Tribol. Lett.,... 

If the beam of Problem 4 were a cantilever beam with the load on the end, what would the deflection of the end be ... 

Two beams of the same material are clamped at one end to form a cantilever beam. One beam has a square cross section of thickness D, and the other beam has a circular cross section of diameter D. The beams are set in vibration by a tap near the free end. If the length of the beams is the same, which beam will have the higher frequency of vibration What is the ratio of their frequencies ... 

---