Qualitative Deflected Shapes

Introduction

In order to achieve long lasting designs, structures need to be fully understood. Early structural analysis of deflected shapes saves time, and unnecessary expensive solutions. The analysis of the qualitative behavior of a structure is important in order to prevent cracking in ceilings and Girders and vibration of the structure. [1] Although during the initial design stage there is no sufficient evidence to truly prove the future behavior of a structure, engineers are able to predict the structural behavior by approximating deflected Beams and Frames, and bending moment diagrams based on fundamental principles and experimental data of structural deflection.

It is important to note that the ability to visualize and draw a qualitative deflected shape of a beam or a frame is very useful when conducting analysis of Indeterminate Structures. In many cases Concavity, Inflection Points , and supports are the main factors that contribute to visualizing deflected shapes.[2]

Important Terms

Qualitative: A non-numerical measure of quality of a certain object or a material.

Deflected shape: The shape of bending or deflections of a beam or frame after external loads and moments were applied on it.

Point load: A single concentrated external force or load acting at a specific location on a beam.

Distributed load: A couple of uniform external forces or loads acting on a specific area or surface on a beam, they are expressed in weight per length or weight per area.

Curvature: Is the slope of the strain profile where ø=M/EI (M = moment of inertia, E = Young’s Modulus, I = Second moment of inertia).

Inflection point: The point on a curve at which the curvature or concavity changes sign from positive to negative or negative to positive, and at which the bending moment is zero.

Frames: A rigid structure containing different beams that are connected together at different angles.

Beams: A long horizontal structural member made of timber, metal or concrete spanning over an opening or a room, usually to support the roof or floor of a structure.

Elastic curve: The curve showing the deflected shape of a beam or frame due to external loads and moments.

Qualitative Deflected Shapes of Beams

No numerical values are necessary to draw a qualitative deflected shape of a beam. Being able to have an idea of what the deflected shape of a beam is supposed to look like, allows us to gain insight of what to expect when calculating the numerical deflection at a specific point on the beam by using the Moment Area Theorem or by Integrating to Find Deflections.

The shape of the deflected beam can be predicted by inspection and based on the bending moment diagram. The qualitative deflected shape of a beam can easily be predicted from its bending moment diagram [2] by following the rules below.

Rules for beam qualitative deflected shapes

1. Axial Deformations

The deflections and rotations of a deflected beam are considered to be small. For simplicity, when constructing the qualitative deflected shape diagram of a beam (or a frame) the axial deformations are ignored. [3] In simpler terms, when deflected, the beam's length will not increase.

2. Concavity

Fig.1: Concavity sign convention

The concavity of a beam's deflected shape is easily determined from the sign of the bending moment diagram. The concavity of the elastic curve of the beam is dependent on the sign on the moment diagram.[4] Below is a description of this relationship.

1. Positive bending moment --> beam in deformed concave up

2. Negative bending moment --> beam is deformed concave down

This sign convention is illustrated in Fig.1. Positive internal moments cause compression on the top side of the beam, resulting in an upward concavity. Negative internal moments cause tension on the top side of the beam, resulting in a downward concavity.[4]

3. Inflection Points

Determining the presence and location of inflection points of a deflected beam is simple. Inflection points are located anywhere in the beam where the bending moment is zero (M = 0 kNm). An inflection point occurs when the deflected shape concavity switches from upward to downward or vice-versa.[4]

4. Supports

There is no deflection or vertical movement at the supports of a beam. The beam cannot simply float in space since it is and will always be attached to its supports, regardless of what type of supports it has (roller or pin). Beams which have fixed supports at one or both ends experience no deflection as well as no rotation at the fixed supports.[4] [2]

Fig.2: Beam deflected shape and moment diagram relationship

Fig.2 illustrates all the rules described above in one single deflected beam. The beam is free at location A, it is supported by a roller at location B, and it is supported by a pin at location C. The are two external point loads applied on the beam. By inspection we can predict that the two downward point loads will cause the beam to displace downward. From the supports rule, we can also predict that there will be no deflection at the supports (B and C). From the moment diagram the number and location of inflection points can be determined. At location D the bending moment is zero and therefore the only inflection point exists at D. The concavity of the deflected shape can also be confirmed from the moment diagram. From the diagram we see that segment AB has a negative bending moment, therefore the deflected shape of segment AB is concave down. Segment BC has a negative bending moment which indicates that its deflected shape is concave up. Point A is displaced downward due to the point load.

The next two examples show how the deflection rules described above are applied to obtained deflected shape diagrams of beams.

Beam Sample Problem 1

Find the qualitative deflected shape of the beam below by inspection:

First step is to draw the Free Body Diagram of the beam:

We must calculate reactions Ax, Ay,and Dy. This can be done by using the 3 equilibrium equations as shown below:

Now draw the complete Free Body Diagram:

Next step is to draw the shear and moment diagrams of the beam:

Now in order to draw the qualitative deflected shape of the beam we look at the moment diagram. There is a positive bending moment as shown in the moment diagram above thus the qualitative deflected shape is concave upward. Note that there are no inflection points in the deflected shape since the bending moment along the beam is always positive.

Beam Sample Problem 2

Find the qualitative deflected shape of the beam below by inspection:

Solution: The beam is supported by a fixed support on the right and a pinned support on the left.Therefore, the deflection is equal to zero at these locations. The point load causes the beam to bend downward. At the hinge location the moment is zero creating an inflection point. Note that at the hinge the slope of the beam in no longer continuous due to the ability of the beam to freely rotate about that point (the moment is released). [5]

Therefore, the deflected shape is as follows:

Qualitative Deflected Shapes of Frames

Qualitative deflected shapes of frames are trickier to visualize than deflected shapes of beams if one looks at the entire frame, however the same principles apply. The qualitative deflected shape of a frame can be approximated before (if the person doing the inspection is experienced) or after a bending moment diagram is completed. Depending on the complexity of the structure and external loads being applied the qualitative deflected shape diagram of a frame may take longer to complete. The best way to approach a frame deflected shape problem is to look at each member separately as if they were beams. Once the moment diagram of each member is completed, the deflected shape of each member can be drawn (following the rules for beam deflected shapes). [3]

The final qualitative deflected shape diagram of a frame can be achieved by putting together the individual deflected shape diagrams of each member while following the rules listed below.

Rules for frame qualitative deflected shapes

1. Corner Compatibility

Fig.3: Illustration of corner compatibility. a) and b) are acceptable, c) is not acceptable

Fig. 3 shows the three possible scenarios a corner of a frame can deform. Following the sign convention explained in section 3.1.1, a) illustrates the positive moments acting on each on the frame members. In this case the top of both members is in tension, while the bottom of the members are in compression. Case b) shows the opposite situation, the frame members are subjected to negative moments. The top of each member is in compression and the bottom in tension. Note that the corner angle remain at 90 degrees after deformation occurs because of the small angle deflection assumption, this is further explained by the corner angles rule below.[3]. Case c) is unrealistic and not acceptable when analyzing the deflection of frames. The moments at each member are of opposite signs, causing the the tension and compression of sides to be on opposite sides.

2. Corner Angles

All corners in a frame must remain at 90 degrees after deflection has occurred.[3]. In all deformations and deflections, of beams and frames, the deflection angles are assumed to be very small, so small that it would not cause the corner angles to deform significantly.[3] From a design point of view, any frames under loading which undergo considerable permanent angle deformation should raise a red flag, and its safety should be carefully considered.[3] This 90 degree rule is also illustrated in Fig.3.

The next example shows how the qualitative beam and frame deflection rules are applied to complete a qualitative deflected shape diagram of a frame.

Frame Sample Problem 1

Find the qualitative deflected shape of the following frame.

This frame is pin supported at A and is roller supported at B. It is statically determinate, therefore to solve for the reaction forces we can use our three equations of equilibrium.

Using equations of equilibrium for the Calculation of Support Reactions:

$+ M_A=0 \\ (-50kN)(2m) -(30kN)(1m) - (10kN/m)(3m)(5.5m) +By(7m)=0 \\ B_y=42.14kN \uparrow$

$+\uparrow F_y =0 \\ A_y=30kN + (10kN/m)(3m) - 42.14 kN \\ A_y=17.86 kN \uparrow \\$

$+\rightarrow F_x= 0 \\ A_x= 50kN \leftarrow \\$

Now that all the support reactions are found, we need to find the reaction at each joint so that each member may be properly analysed. See () for an in depth process of how to find the joint reactions.

In order to create the shear force, axial force, and bending moment diagrams, every load must be parallel or perpendicular to the member it is being applied to. In this example the forces acting on the inclined member need to be represented parallel and perpendicular to member BD. In other words, there needs to be a change in coordinate system.

By using the laws of SOH CAH TOA, this may be easily done. To demonstrate the concept, let us convert the vertical reaction at B to the new coordinate system.

$B_p_a_r_a_l_l_e_l= 42.14\sin(53.1) = 33.7kN \\ B_p_e_r_p = 42.14\cos(53.1) = 25.3 kN$

Now that all the loads are properly oriented, the axial, shear and bending moment diagrams may be created. It can be noted that to find the deflected shape, only the moment diagram is required, however all will be solved for this problem.

The moments of each member combined result in the following system

Knowing that a positive moment bends a member concave up, and a negative moment bends a moment concave down, and knowing that frames must follow the rule of corner compatibility, the deflected shape for the frame is as follows.

References

1. "Mechanics of Materials-Deflection",learncivilengineering,[online],(2012), available: http://www.learncivilengineering.com/wp-content/uploads/2012/12/Mechanics-of-Materials-Deflection.pdf. November 2013.
2. Kassimali, A. (2011). Structural:SI Edition (4th ed.). Stamford,CT: Cengage Learning.
3. "Gambhir, M.L. (2011). Fundamentals of Structural Mechanics and Analysis: Eastern Economy Edition. New Delhi: Asoke K. Ghosh and PHI: Learning Provate Limited.
4. Hibbeler, R.C. (2011). Mechanics of Materials (8th ed.). Upper Saddle River, NJ: Pearson Prentice Hall.
5. Logan, D. (2012). A First Course in the Finite Element Method (5th ed.). Stamford, CT: Cengage Learning.