# Approximate Analysis of Vertical Loads

## Approximate Analysis of Vertical Loads

### Introduction to Approximate Analysis

Approximate analysis is a tool engineers use to estimate the magnitude of member forces and moments, and support reactions in indeterminate structures. To understand approximate methods for vertical loadings it is important to understand the difference between approximate analysis of statically indeterminate structures compared with other methods such as the Force Method for Multiple Degrees of Indeterminacy, Moment Distribution Method for Continuous Beams or Slope-Deflection Method for Continuous Beams. These methods involve obtaining numerical solutions to an indeterminate system through analysis using the physical properties of structural members[1]. The ability to obtain a numerical solution for indeterminate systems means these methods are ‘exact’ methods. The approximate method relies on assumptions about the behavior of an indeterminate structural system to model it as a determinate one which may be solved. The approximate analysis of structures is an ‘inexact’ method that is often used in preliminary designs process because of its simplicity, the analysis is less complex with a tradeoff of reduced accuracy.

### Assumptions for Approximate Analysis

Approximate analysis of indeterminate structures under vertical loadings requires assumptions to be made regarding the behaviour of beams within a structures framing system. These assumptions allow engineers to reduce indeterminate beams into representative models which are determinate, and can be solved numerically. 'Under vertical loads, girders fixed between two columns are typically indeterminate to the third degree'[1] due to three reactions at each column, and only three equations of equilibrium. To reduce this to a determinate beam three assumptions must be made about the behavior of the beam under vertical loads.

#### Assumptions about the Location of Points of Inflection

For approximate analysis, one of the assumptions made regards the location of the inflection point of the deflected structure under a loading. These inflection points represent locations where zero bending moment is acting on a beam, and their location along a beam is an indicator of the amount of moment resistance exerted by the supports. The greater the moment resistance, the further the inflection point will be from the supports, indicating less rotation and deflection of the beam will occur. The exact amount of moment resistance provided by the fixed end (location of IP) depends on the physical properties of the girders and columns, as well as the type of connection between the girder and column[2].

Figure 1

If the columns offer no moment resistance, zero moment and therefore the inflection points occur at the supports, as is the case in a simply supported beam (fig 1.2). The location of the inflection point in a beam which is totally fixed, and allows absolutely no rotation at the supports occurs at a distance 0.21L away from the supports[1] (fig 1.3). As approximate analysis is generally a method for preliminary design, member’s stiffness or the exact connection type between column and girder are unlikely to be known. It is possible that little information is available about the system other than the loads which will be acting upon it. The approximate method’s uses the average of these conditions for the location of inflection point, 0.1L from the supports[1]. This represents a connection where the girder-column connection is moment-resisting, but a limited amount of rotation at the supports is possible. Making this assumption at both ends of the girder reduces the indeterminacy of the structure to one.

The final assumption made to reduce a indeterminate beam into a determinate one relates to axial forces in girders. Beam Analysis has shown us that the axial forces in a beam generated by vertical loadings are negligible[3]. The axial force within a beam under vertical loading is therefore assumed to be zero.

### The Determinate Beam

The assumptions made in approximate analysis for vertical loads allow a 3 degree indeterminate beam to be modeled as a determinate beam split into 3 parts as show by girder CD in fig 2.2 and 2.3. The beam is now modeled as a simply supported beam with a span between the assumed inflection points, resting on top of two short cantilevers attached by a fixed connection to columns. The loading from the determinate beam is transferred as a point load onto the ends of the cantilevered sections. Observation of girder CD shows that it is a determinate structure. Each cantilever section has one unknown moment and vertical reaction, which we may find with the equations of equilibrium for vertical force and moment.

### Engineering Impact

The approximate method for vertical loads has several implications when it comes to design. The approximate method can be used to obtain a reasonable estimate the maximum moment acting upon a beam, as well as the moments transferred onto columns due to vertical loads. These estimates can give crucial insight to a structural engineer about appropriate beam sizing or reinforcement as well as insight into connection requirements. At the conception or feasibility stage of a project it may not be reasonable or economical to conduct an exact analysis, making approximate methods a valuable tool.

## Procedure

The following points are important when analysing frames for vertical loads using approximate analysis

1. The girders of the frame are assumed to be behaving as simply supported beams.
2. Based on the assumption, a single girder is analyzed and same results are assumed for redundant sections.
3. The point of inflexion (zero moment point) for a typical simply supported beam, is assumed to occurs at 0.1L whereas it is at 0.21L for a fixed-fixed beam from both ends.
4. The beam is then cut at the points of inflexion and is usually divided into 3 parts.
5. Each part of the beam is then analyzed separately using equilibrium equations.

### Example

The example shown below will be solved using the approximate analysis method to analyze the structure for vertical loads

Figure 2.1 Example

The assumptions for vertical approximate analysis are:

1. The inflection points are located at 0.1L for either end
2. No axial force on girders
Figure 2.2 Simplified Determinate Frame

Then, a statically determinate frame can be obtained as shown in the figure. On each girder, the end portions with 1m in length are considered as fixed ends in order to partially restrained against rotation, the middle portion is simply supported with a length of 0.8L = 0.8(10) = 8m.

Figure 2.3 Girder CD and its V-M Diagrams

The approximate shear and bending moment diagrams of each girder are the same due to uniform distributed loads and span lengths.

The first step is using the equilibrium equation in the y direction to obtain the vertical reactions at the pin and roller of the middle portion.

$C_1 to D_1$

$+\uparrow \sum Fy=0$

$V_C_1 + V_D_1-(8m)(10 kN/m)=0$

$V_C_1 = V_D_1 = 40kN$

Then by Newton's law of action and reaction, each vertical reaction of the middle portion will exert an equal but opposite force on the according end portion. The shear and moment at each end of the girder connected to the frame can now be determined by using equilibrium.

$C to C_1$

$+\uparrow \sum Fy=0$

$V_C = V_D$

$V_C - (10kN/m)(1m) - V_C_1 = 0$

$V_C = 10kN + 40kN$

$V_C =V_D = 50kN$

$C to D$

$M_m_i_d_p_o_i_n_t = WL^2/12$ [2]

$M_m_i_d_p_o_i_n_t =10(10)^2/12$

$M_m_i_d_p_o_i_n_t = 83.3 kN/M$

$\sum M = 0$

$M_C=M_D$

$M_C=M_D=WL^2/22$ [2]

$M_C=M_D=10(10)^2/22$

$M_C=M_D= -45.45kN/M$

## References

1. Kassimali, A. (2011) Structural Analysis: SI Edition (3th ed.). Stanford, CT: Cengage Learning.
2. Hibbeler, R.C. (2012). Structural Analysis: 8th Edition. Upper Saddle River, NJ: Pearson Education.
3. http://www.engineeringwiki.org/wiki/Beam_Analysis
Figure 2.4 Result Forces and Moments