section 4 - determinate beam and frame analysis annotated oct20

Beams and Frames - 1

Determinate Beams and Frames Beams and frames are structural components that carry forces through a combination of axial force, shear force, and bending moment. Because these members span horizontal and vertical spaces using slender members, deflections are an important consideration. In addition, bearing and lateral stability can be critical. There is an infinite variety of arrangements for beams and frames. Beams may be single or multi-span, they may be simple spans or continuous. Frames may be single or multi-storey. The materials used for modern beams and frames are endless: steel, concrete, timber are the most common. In this section, we will consider only statically determinate beams and frames.


Analysis Frames and beams are more complicated than trusses because connections are not pinned but are rigid. Therefore, moments are transferred between members. If the beam or frame is statically determinate, the internal forces can be determined using the 3 equations of static equilibrium alone. As with trusses, the stability and determinacy of the beam or frame should be checked before analysis. STABILITY AND DETERMINACY OF PLANE FRAMES The determinacy of a frame structure is the relationship of unknown member forces and reactions to the number of independent equilibrium equations available to solve for the unknowns. A frame whose unknown reactions may be determined by using the overall structural equations of equilibrium is called an externally determinate structure. A frame whose member end forces may be determined for the available equations of equilibrium is an internally determinate structure. If the frame has more unknown reactions than overall equations of equilibrium, it is an externally indeterminate structure. If member end forces cannot be determined from the available equilibrium equations alone, then the frame is internally indeterminate. A determinate frame is both internally and externally determinate. An indeterminate frame can be either internally or externally indeterminate, e.g.: A frame is stable if it can immediately develop resistance to any possible load.


A frame is determinate if it is stable and has an equal number of unknowns as it has equations of equilibrium. As a general rule, a structure can be identified as statically indeterminate or statically indeterminate by drawing FBDs of all its members, and then comparing the total number of unknown reactive force and moment components with the total number of available equilibrium equations. For a co-planar structure there are at most 3 equilibrium equations for each part, so that if there is a total of n parts and r force and moment reaction components: r = 3n, statically determinate r > 3n, statically indeterminate If a structure is statically indeterminate, the number of additional equations needed to solve for the unknown reactions is known as the degree of indeterminacy. The additional equations are obtained by relating the loads or reactions to the displacement or slope at different points on the structure. These equations are called compatibility equations.


EXAMPLE


Sometimes frames form internal loops. In order to classify these structures, it is necessary to use the method of sections and cut the loop apart. EXAMPLE


Stability In general a structure will be unstable if there are fewer reactive forces than equations of equilibrium: r < 3n OR, if there are enough reactions, the lines of action of the reactive forces intersect at a common point or are all parallel to one another.


Axial Force, Shear Force, and Bending Moment Diagrams The internal forces for a flexural member are summarized using shear force diagrams (SFD), axial force diagrams (AFD), and bending moment diagrams (BMD). There are various ways to get SFD and BMD:

1 Take sections and connect the dots 2 Find equations for SF and BM in terms of x, then plot 3 Integration with discontinuity functions 4 Graphical (visual)

EXAMPLE Consider the simple cantilever with an inclined load. Determine the AFD, SFD and BMD.

10 kN

A B C 3 m 30º 2 m


Graphical Approach for Drawing SFD and BMD

Often as designers, we are usually concerned only with the value and

location of the maximum shear force and bending moment in a structure. It

is often sufficient to sketch the shape of the diagrams with critical values

noted. We can use the relationships between loading, shear force, and

bending moment to quickly sketch the diagrams.

The loading, w(x), on a beam is related to the shear force, V(x), developed

in the beam:

)()( xwdx

xdV−= (1)

In words, this equation means that the slope of the shear force diagram at x

is equal to the negative of the value of the loading function at x.

The relationship between bending moment, M(x), and shear force is:

)()( xVdx

xdM= (2)

In words. the slope of the moment diagram at x is equal to the value of the

shear function at x.

Integrating Eq (1):

∫−=Δ 2

1

)(x

xdxxwV (3).

(note what each side of the equation represents)

In words, the change in shear force between two points on a beam is equal to

the negative of the area under the loading diagram between these two points.

Integrating Eq. (2):


∫=Δ 2

1

)(x

xdxxVM (4).

(note what each side of the e

In words, the change in bending moment between two points on a beam is

equal to the area under the shear diagram between these two points. r

The following figures illustrate these relationships: )


Example Draw the shear force and bending moment diagrams for the following beam.


Behaviour of Beams - Conceptual Beams cam be supported at one point, requiring a fixed support (i.e. a cantilever beam), at two points (simple beams, overhanging beams), and at several points. These various support conditions will have a profound effect on the moments and deflections that develop in the beam. Consider a simply supported beam subjected to a uniformly distributed load. The maximum moment occurs at the mid-span and is wL2/8. Perhaps the most common cantilever structure in nature is the tree. Many architects have been inspired by the cantilever action they have observed in nature. The cantilever has a single curvature, indicating that moments are negative throughout (upper beam fibers in tension, lower in compression). When subjected to a uniformly distributed load, the support moment is four times larger than the maximum moment in a simply supported beam. It can also be shown that the maximum deflection is 10 times larger.


From: Building Support Structures, Schueller, 2006 Often, cantilever beam action is developed when the ends of a simply supported beam are extended beyond the supports. The cantilever loads tend to counteract those in the anchor span, and reduce the maximum moment from wL2/8. For example, say spans of 0.35L are extended from each side. The maximum moment in the anchor span is now half the maximum moment in the simply supported span. However, we see we also develop negative moments over the supports. Keep in mind that uniform loading over all the spans may not give the critical moments. If, for example, live loads occur on the beam, they must be placed so as to give the worst negative and positive moments.


In general, with an increase in span, simply supported beams become less efficient because of the rapid increase in moment due to the dead load. The cantilever concept has been used to reduce these stresses for long spans.


Overhanging beams can be connected in a series, the basis for the Gerber suspended span system, which is a common approach for supporting the roof of single storey light steel buildings. The beams discussed so far have been statically determinate. Statically indeterminate beams require additional equations to solve for the bending moment diagrams as will be discussed in later sections. A beam with fixed ends can be analysed (using the Principle of Superposition) as a simply supported beam with two equal end moments applied. These end moments serve to reduce the maximum bending moment in the beam. A propped cantilever is similar except there is just a single end moment.


Procedure for Frame Analysis (Statically Determinate) 1 Find reactions: use overall equilibrium 2 Draw FBD of each individual member: unknown end actions 3 Determine member end actions 4 Draw diagrams: AFD, SFD, BMD 5 Sketch deflected shape

EXAMPLE Determine axial force, shear force, and bending moment diagrams for the following frame. Sketch the deflected shape.

6

A

B 5

3

C

1kND

3kN/m


EXAMPLE Draw the shear and moment diagrams for each of the three members of the frame. Sketch the deflected shape. Assume the frame is pin connected at A, C, and D and there is a fixed joint at B.


Conceptual Behaviour of Frames Frames can have many configurations: To appreciate the behaviour of a true frame, consider a rectangular frame with a pin and roller support and subjected to a uniformly distributed load:

w wL2/8


In this case, the roller can only develop a vertical reaction. This ensures that the columns carry no moments, and the horizontal beam behaves essentially as a simply supported beam with a maximum moment wL2/8. If, however, we have a second pin support, horizontal deflections are prevented, and moments will develop in the columns. Depending on the relative stiffness between the columns and beam, the maximum moments may be much smaller than wL2/8, making this frame much more efficient at carrying the load. Consider how changing the frame geometry changes the distribution of moments. We will compare it to a parabolic arch, which would carry the uniformly distributed load as a pure compression (no bending moments). In locations where the frame geometry deviates from the arch (like in the corners) we get large bending moments. As the geometry of the frame approaches that of the arch, we see the bending moments getting smaller. Note that there may be practical reasons why a rectangular configuration is needed.


The configuration of typical “portal frames” follows the above bending moment diagrams, with much deeper sections at the corners where the moments are high. It can be expensive to fabricate moment resisting connections, especially with steel, so a common configuration is the “braced frame.” The connections between beams and columns can be assumed to be pinned. Axial members are added to provide stability to the frame. We will look at these frames in more detail in CIVL 331.


As the number of bays in a frame is increased (as the size of the building increases), there are again a variety of configurations. The connections between the beams and columns may be moment-resisting or pinned. The roof beam may be continuous over the columns. Each of these configurations will impact the moments developed in the frame. For multi-bay, multi-storey buildings, again the type of connections specified between the beams and columns will have a dramatic effect on the moments developed. Below is a frame with moment resisting connections between all the beams and columns (this is typical for a concrete frame). The vertical loads will be carried by the continuous beams with moments and compressive forces transferred to the columns. Lateral loads (like wind) will create moments in the columns and beams. The moments will increase as we progress toward the base of the structure, much like the behaviour seen in a cantilever beam.