The 2960 should support limited routing if you have 12.2(58)SE or later. You also need the LANBASE image and you need to enable the lanbase-routing SDM template with the "sdm prefer ...." command. I don't know whether packet tracer supports all this but if it does yes you could turn the 2960 into a L3 switch.
The issue you have at the moment is the 2960 is only L2 as Alain mentioned. The IP address and default-gateway on the switch are only used so you can login to the switch to manage it. That IP cannot be used for end devices such as PCs as their default-gateway. For that you need a L3 switch (or router).
So if you want to change it to L3 there is a fair bit of reconfig to. Firstly you would need a new network between the switch and router because the 10.111.233.0/24 will be used for the PC so it cannot also be used to connect to the router if the switch is acting as L3 switch ie. you cannot route from one subnet to the same subnet, you can only L2 switch. So the full config changes would be -
Note - i'm assuming you are using vlan 1 as the vlan with the IP address 10.113.233.10 and the PC is in vlan 1. If not it doesn't matter it should still work.
new network used to connect switch to router - 10.111.234.0/30
new vlan used for switch to router connection - vlan 2
Router1
======
router(config)# int gi0/0 <-- this connects to the 2960
router(config-if)# ip address 10.111.234.1 255.255.255.252 <--- this should update the IP from it's original 10.111.233.24/24
router(config)# ip route 10.111.233.0 255.255.255.0 10.111.234.2 <-- this is needed because 10.111.233.0/24 is no longer directly connected to Router1 so it needs to know where to send the return packets. 10.111.234.2 is assigned to an SVI on the 2960 - see below.
2960 switch
=========
If you had to change the SDM template to "lanbase-routing" as mentioned above then you need to reload the switch before it takes effect ie.
switch(config)# sdm prefer lanbase-routing
switch(config)# exit
switch# reload <-- if you haven't already, save your config before reloading the switch
switch(config)# ip routing <--- this enables routing on the switch
switch(config)# no ip default-gateway 10.113.233.24 <-- no need for this when a switch is a L3 switch
switch(config)# ip route 0.0.0.0 0.0.0.0 10.113.234.1 <-- add a route to send all traffic to Router1
switch(config)# vlan 2
switch(config-vlan)# name
switch(config)# int gi0/1 <--- this connects to Router1
switch(config-if)# switchport mode access
switch(config-if)# switchport access vlan 2
the above lines create vlan in the vlan database and then assign the port connected to the router into vlan 2
switch(config)# int vlan 2
switch(config-if)# ip address 10.113.234.2 255.255.255.0
switch(config-if)# no shut
That should do it. The PC has it's default-gateway set to 10.111.233.10 which is the IP address of vlan 1 (or whatever you have used) on the switch.
Jon
