8.0 ANALYSIS OF STRUCTURAL FRAMES/TRUSSES
8.1 Introduction
A structure having several bars/members riveted or welded together is known as a frame or a truss. If the
frame has sufficient members to keep it in equilibrium when the frame is supporting an external load,
then the frame is known as a perfect frame. Though in actual practice the members are welded or riveted
together at their joints, for calculation purposes, the joints are assumed to be hinged or pin-jointed. See
Figure 8.1 for practical examples of frames and Figure 8.2 for the common joints found in frames.
Figure 8.1: Practical examples of frames/trusses
Figure 8.2: Joints on members of a frame/truss
,8.2 Types of Frames
The different types of frames are:
(i) Perfect Frames
(ii) Imperfect Frames
8.2.1 Perfect Frames
A frame which is composed of such members which are just sufficient to keep the frame in equilibrium
when it is supporting an external load is known as a perfect frame. The simplest perfect frame is a triangle
as shown in Figure 8.3. A triangle consists of 3 members and 3 joints.
Figure 8.3
The three members are AB, BC and AC whereas the three joints are A, B and C. this frame can easily be
analysed by the conditions of equilibrium. Supposing we add two members CD and BD and a joint D to
the triangle ABC, we get frame ABCD as shown in Figure 8.4 (a). This frame can also be analysed the
condition of equilibrium. This frame is also known as a perfect frame. Suppose we add a set of two
members and a joint again, we get a perfect frame as shown in Figure 8.4 (b). Hence, for a perfect frame,
the number of joints and number of members is given by:
𝑛 = 2𝑗 − 3
𝑤ℎ𝑒𝑟𝑒: 𝑛 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑚𝑒𝑚𝑏𝑒𝑟𝑠
𝑗 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑗𝑜𝑖𝑛𝑡𝑠
Figure 8.4
8.2.2 Imperfect Frames
For an imperfect frame, the number of members will either be more or less than (2𝑗 − 3). Hence for an
imperfect frame, the following equation will apply:
, 𝑛 ≠ 2𝑗 − 3
(𝑖) 𝐼𝑓 𝑛 < 2𝑗 − 3, 𝑡ℎ𝑒 𝑓𝑟𝑎𝑚𝑒 𝑖𝑠 𝑠𝑎𝑖𝑑 𝑡𝑜 𝑏𝑒 𝒅𝒆𝒇𝒊𝒄𝒊𝒆𝒏𝒕
(𝑖𝑖) 𝐼𝑓 𝑛 > 2𝑗 − 3, 𝑡ℎ𝑒 𝑓𝑟𝑎𝑚𝑒 𝑖𝑠 𝑠𝑎𝑖𝑑 𝑡𝑜 𝑏𝑒 𝒓𝒆𝒅𝒖𝒏𝒅𝒂𝒏𝒕
8.3 Assumptions made in determining forces in Frames
(i) The frame is assumed to be a perfect frame.
(ii) The frame is assumed to carry loads only at its joints and not on its members.
(iii) All the members of the frame are assumed to be pin-jointed.
8.4 Reactions of supports of a Frame
Frames are generally simply supported, i.e. having a pinned support on one end and a roller support on
the other end. The line of action of the reaction forces in the frame are illustrated in Figure 8.5 below:
Figure 8.5
The reactions at the supports of a frame are determined by the conditions of equilibrium. The external
load on the frame and the reactions at the supports must form a system of equilibrium.
8.5 Analysis of a Frame
The analysis of a frame consists of the following two steps:
(i) Determination of the reactions at the supports.
(ii) Determination of the forces in the members of the frame.
In order to determine the reactions at the supports, we will make use of equilibrium conditions.
Determination of the forces in the members of the frame will be on condition that every joint should be
in equilibrium and therefore, the forces acting on every joint should form a system of equilibrium. The
following are the methods of analysing frames:
(i) Method of joints.
(ii) Method of sections.
(iii) Method of tension coefficients.
(iv) Graphical method.
8.5.1 Method of Joints
In this method, after determining the reactions at the supports, the equilibrium of every joint is
considered. This means that the sum of all vertical forces as well as horizontal forces actin on a join will
be equated to 0. The joint should be selected such that at any one time, there will be only two members
8.1 Introduction
A structure having several bars/members riveted or welded together is known as a frame or a truss. If the
frame has sufficient members to keep it in equilibrium when the frame is supporting an external load,
then the frame is known as a perfect frame. Though in actual practice the members are welded or riveted
together at their joints, for calculation purposes, the joints are assumed to be hinged or pin-jointed. See
Figure 8.1 for practical examples of frames and Figure 8.2 for the common joints found in frames.
Figure 8.1: Practical examples of frames/trusses
Figure 8.2: Joints on members of a frame/truss
,8.2 Types of Frames
The different types of frames are:
(i) Perfect Frames
(ii) Imperfect Frames
8.2.1 Perfect Frames
A frame which is composed of such members which are just sufficient to keep the frame in equilibrium
when it is supporting an external load is known as a perfect frame. The simplest perfect frame is a triangle
as shown in Figure 8.3. A triangle consists of 3 members and 3 joints.
Figure 8.3
The three members are AB, BC and AC whereas the three joints are A, B and C. this frame can easily be
analysed by the conditions of equilibrium. Supposing we add two members CD and BD and a joint D to
the triangle ABC, we get frame ABCD as shown in Figure 8.4 (a). This frame can also be analysed the
condition of equilibrium. This frame is also known as a perfect frame. Suppose we add a set of two
members and a joint again, we get a perfect frame as shown in Figure 8.4 (b). Hence, for a perfect frame,
the number of joints and number of members is given by:
𝑛 = 2𝑗 − 3
𝑤ℎ𝑒𝑟𝑒: 𝑛 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑚𝑒𝑚𝑏𝑒𝑟𝑠
𝑗 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑗𝑜𝑖𝑛𝑡𝑠
Figure 8.4
8.2.2 Imperfect Frames
For an imperfect frame, the number of members will either be more or less than (2𝑗 − 3). Hence for an
imperfect frame, the following equation will apply:
, 𝑛 ≠ 2𝑗 − 3
(𝑖) 𝐼𝑓 𝑛 < 2𝑗 − 3, 𝑡ℎ𝑒 𝑓𝑟𝑎𝑚𝑒 𝑖𝑠 𝑠𝑎𝑖𝑑 𝑡𝑜 𝑏𝑒 𝒅𝒆𝒇𝒊𝒄𝒊𝒆𝒏𝒕
(𝑖𝑖) 𝐼𝑓 𝑛 > 2𝑗 − 3, 𝑡ℎ𝑒 𝑓𝑟𝑎𝑚𝑒 𝑖𝑠 𝑠𝑎𝑖𝑑 𝑡𝑜 𝑏𝑒 𝒓𝒆𝒅𝒖𝒏𝒅𝒂𝒏𝒕
8.3 Assumptions made in determining forces in Frames
(i) The frame is assumed to be a perfect frame.
(ii) The frame is assumed to carry loads only at its joints and not on its members.
(iii) All the members of the frame are assumed to be pin-jointed.
8.4 Reactions of supports of a Frame
Frames are generally simply supported, i.e. having a pinned support on one end and a roller support on
the other end. The line of action of the reaction forces in the frame are illustrated in Figure 8.5 below:
Figure 8.5
The reactions at the supports of a frame are determined by the conditions of equilibrium. The external
load on the frame and the reactions at the supports must form a system of equilibrium.
8.5 Analysis of a Frame
The analysis of a frame consists of the following two steps:
(i) Determination of the reactions at the supports.
(ii) Determination of the forces in the members of the frame.
In order to determine the reactions at the supports, we will make use of equilibrium conditions.
Determination of the forces in the members of the frame will be on condition that every joint should be
in equilibrium and therefore, the forces acting on every joint should form a system of equilibrium. The
following are the methods of analysing frames:
(i) Method of joints.
(ii) Method of sections.
(iii) Method of tension coefficients.
(iv) Graphical method.
8.5.1 Method of Joints
In this method, after determining the reactions at the supports, the equilibrium of every joint is
considered. This means that the sum of all vertical forces as well as horizontal forces actin on a join will
be equated to 0. The joint should be selected such that at any one time, there will be only two members
